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ALFRED NORTH WHITEHEAD. Sc.D., F.R.S.
FELLOW OF TIUNITY COI.LEOE. CAMHRIlMJK, FROEKSSOR OF PllllAlSOl'IlY
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PREFACE
T HE mathematical treatment of the principles of mathematics, which is
the subject of the present work, has arisen from the conjunction of two
different studies, both in the main verv modern. On the one hand we have
the work of analysts and geometers, in the way of formulating and systematising
their axioms, and the work of Cantor and others on such mutters as the theory
°f aggregates. On the other hand we have symbolic logic, which, after a
necessary period of growth, has now, thanks* to Peano and his followers,
acquired the technical adaptability and the logical comprehensiveness that arc
essential to a mathematical instrument for dealing with what have hitherto
been the beginnings of mathematics. From the combination of these two
studies two results emerge, namely (1) that what were formerly taken, tacitly
or explicitly, as axioms, are either unnecessary or demonstrable; (2) that the
same methods by which supposed axioms arc demonstrated will give valuable
results in regions, such as infinite number, which had formerly been regarded
as inaccessible to human knowledge. Hence the scope of mathematics is
enlarged both by the addition of new subjects and by a backward extension
into provinces hitherto abandoned to philosophy.
The present work was originally intended by us to be comprised in a
second volume of The Principles of Mathematics. With that object in view,
the writing of it was begun in 1900. But as we advanced, it became in¬
creasingly evident that the subject is a very much larger one than we had
supposed; moreover on many fundamental questions which had been left
obscure and doubtful in the former work, we have now arrived at what we
believe to be satisfactory solutions. It therefore became necessary to make
our book independent of The Principles of Mathematics. We have, however,
avoided both controversy and general philosophy, and made our statements
dogmatic in form. The justification for this is that the chief reason in favour
of any theory on the principles of mathematics must always be inductive,
i.e. it must lie in the fact that the theory in question enables us to deduce
ordinary mathematics. In mathematics, the greatest degree of self-evidence
is usually not to be found quite at the beginning, but at some later point;
hence the early deductions, until they reach this point, give reasons rather
for believing the premisses because true consequences follow from them, than
for believing the consequences because they follow from the premisses.
In constructing a deductive system such as that contained in the present
work, there are two opposite tasks which have to be concurrently performed.
On the one hand, we have to analyse existing mathematics, with a view
to discovering what premisses are employed, whether these premisses are
mutually consistent, and whether they are capable of reduction to more
fundamental premisses. On the other hand, when we have decided upon our
S remisses, we have to build up again as much as may seem necessary of the
ata previously analysed, and as many other consequences of our premisses
as are of sufficient general interest to deserve statement. The preliminary
labour of analysis does not appear in the final presentation, which merely
sets forth the outcome of the analysis in certain undefined ideas and
VI
PREFACE
undomonst rated prop>silions. It is not claimed that the analysis could not
have hern carried farther: we have no reason to suppose that it is impossible
to find simpler ideas and axioms by means of which those with which we
start could be defined and demonstrated. All that is affirmed is that the
ideas and axioms with which we start are sufficient, not that they are
necessary.
In making deductions from our premisses, we have considered it essential
tocarry them up to the point where we have proved as much as is true in
whatever would ordinarily lie taken for granted. Hut we have not thought
it desirable to limit ourselves too strictly to this task. It is customary to
consider only particular cases, even when, with our apparatus, it is just as
easy to deal with the general case. For example, cardinal arithmetic is
usually conceived in connection with finite numbers, but its general laws hold
equally for infinite numbers, and are most easily proved without any mention
of the distinction between finite and infinite. Again, many of the properties
commonly associated with series hold of arrangements which are not strictly
serial, but have only some of the distinguishing properties of serial arrange¬
ments. In such cases, it is a defect in logical style to prove fora particular
class of arrangements what might just as well have been proved more generally.
An analogous process of generalization is involved, to a greater or less degree,
in all our work. Wo have sought always the most general reasonably simple
hypothesis from which any given conclusion could be reached. For this reason,
especially in the later parts of the l*»ok, the importance of a proposition
usually lies in its hypothesis. The conclusion will often be something which,
in a certain class of cases, is familiar, but the hypothesis will, whenever possible,
be wide enough to admit many cases besides those in which the conclusion is
familiar.
We have found it necessary to give very full proofs, because otherwise it
is scarcely possible to see what hypotheses are really required, or whether our
results follow from our explicit premisses. (It must be remembered that we
are not affirming merely that such and such propositions are true, but also
that the axioms stated by us are sufficient to prove them.) At the same time,
though full proofs are necessary for the avoidance of errors, and for convincing
those who may feel doubtful as to our correctness, yet the proofs of propo¬
sitions may usually be omitted by a reader who is not specially interested in
that part of the subject concerned, and who feels no doubt of our substantial
accuracy on the matter in hand. The reader who is specially interested in
some particular portion of the book will probably find it sufficient, as regards
earlier portions, to read the summaries of previous parts, sections, and
numbers, since these give explanations of the ideas involved and statements of
the principal propositions proved. The proofs in Part I. Section A. however,
are necessary since in the course of them the manner of stating prools is
explained. The proofs ol the earliest propositions are given without the
omission of any step, but as the work proceeds the proofs are gradually
compressed, retaining however sufficient detail to enable the reader by the
help of the references to reconstruct proofs in which no step is omitted.
The order adopted is to some extent optional. For example, we have treated
cardinal arithmetic and relation-arithmetic before series, but we might have
treated senes first. To a great extent, however, the older is determined by
logical necessities. J
PREFACE
Vll
A very largo part of the labour iuvolvotl in writing tin* pivsrnt work lias
boon expended on the contradictions and paradoxes which have inlivlvd logic
and the theory of aggregates. We have examined a great number of hypo¬
theses for dealing with these eontradictions; many such hypotheses have been
advanced by others, and about as many have been invented by ourselves.
Sometimes it has cost 11 s several months' work to convince ourselves that
a hypothesis was untenable. In the course of such a prolonged study, we
have been led, as was to be expected, to modify our views from time to time;
but it gradually became evident to us that some form of the doct rine of types
must be adopted if the contradictions were to be avoided. The particular
form of the doctrine of types advocated in the present work is not logically
indispensable, and there are various other forms equally compatible with the
truth of our deductions. We have particularized, both because the form of
the doctrine which we advocate appears to us the most probable, and because
it was necessary to give at least one perfectly definite theory which avoids
the contradictions. But hardly anything in our book would be changed by the
adoption of a different form of the doctrine of types. In fact, we may go
farther, and say that, supposing some other way of avoiding the contradictions
to exist, not very much of our book, except what explicitly deals with types,
is dependent upon the adoption of the doctrine of types in any form, so soon
ns it has been shown (as we claim that we have shown) that it is possible
to construct a mathematical logic which does not lead to contradictions. It
should be observed that the whole effect of the doctrine of types is negative:
it forbids certain inferences which would otherwise be valid, but does not
permit any which would otherwise be invalid. Hence we may reasonably
expect that the inferences which the doctrine of types permits would remain
valid even if the doctrine should be found to be invalid.
Our logical system is wholly contained in the numbered propositions, which
are independent of the Introduction and the Summaries. The Introduction
and the Summaries are wholly explanatory, and form no part of the chain of
deductions. The explanation of the hierarchy of types in the Introduction
differs slightly from that given in *12 of the body of the work. The latter
explanation is stricter and is that which is assumed throughout the rest of
the book.
The symbolic form of the work has been forced upon us by necessity:
without its help we should have been unable to perform the requisite
reasoning. It has been developed as the result of actual practice, and is not
an excrescence introduced for the mere purpose of exposition. The general
method which guides our handling of logical symbols is due to Peano. His
great merit consists not so much in his definite logical discoveries nor in the
details of his notations (excellent as both are), as in the fact that he first
showed how symbolic logic was to be freed from its undue obsession with the
forms of ordinary algebra, and thereby made it a suitable instrument for
research. Guided by our study of his methods, we have used great freedom
in constructing, or reconstructing, a symbolism which shall be adequate to
deal with all parts of the subject. No symbol has been introduced except
on the ground of its practical utility for the immediate purposes of our
reasoning.
A certain number of forward references will be found in the notes and
explanations. Although we have taken every reasonable precaution to secure
Vlll
PREFACE
the accuracy of these forward references, we cannot of course guarantee their
accuracy with the same confidence as is possible in the case of backward
references.
Detailed acknowledgments of obligations to previous writers have not very
often been possible, as we have had to transform whatever we have borrowed,
in order to adapt it to our system and our notation. Our chief obligations
will In* obvious t«> ever}* reader who is familiar with the literature of the
subject. In the matter of notation, we have as far as possible followed Pcano,
supplementing his notation, when necessary, by that of Frege or by that ot
Schroder. A great deal of the symbolism, however, has had to be new. not
so much through dissatisfaction with the symbolism of others, os through the
fact that we deal with ideas not previously symbolised. In all questions of
logical analysis, our chief debt is to Frege. Where we differ from him, it is
largely because the contradictions showed that he, in common with all other
logicians ancient and modern, had allowed some error to creep into his pre¬
misses; hut apart from the contradictions, it would have been almost impossible
to detect this error. In Arithmetic and the theory of series, our whole work
is based on that of Georg Cantor. In Geometry we have had continually
before us the writings of v. Staudt. Pasch. Peano. Pieri, and Veblen.
We have derived assistance at various stages from the criticisms ol friends,
notably Mr G. Cl. Berry of the Bodleian Library and Mr R. G. Hawtrcy.
We have to thank the Council of the Royal Society for a grant towards the
expenses of printing of £200 from the Government Publication Fund, and also
the Syndics of tin* University Press who have liberally undertaken the greater
portion of the expense incurred in the production of the work. The technical
excellence, in all departments, of the University Press, and the zeal and courtesy
of its officials, have materially lightened the task of proof-correction.
The second volume is already in the press, ami both it and the third will
appear as soon as the printing can be completed.
A. N. W.
B. R.
Camiikidoe,
Xavrml/rr, l!»10.
CONTENTS OF VOLUME I
I* AUK
PREFACE . v
ALPHABETICAL LIST OF PROPOSITIONS REFERRED TO BY
NAMES . xii
INTRODUCTION TO THE SECOND EDITION .... xiii
INTRODUCTION. 1
Chapter I. Preliminary Explanations of Ideas and Notations 1
Chapter II. The Theory of Logical Types.37
Chapter III. Incomplete Symbols .6G
PART 1. MATHEMATICAL LOGIC
Summary of Part I ........ 87
Section A. The Theory of Deduction.90
#1. Primitive Ideas and Propositions.91
•2. Immediate Consequences of the Primitive Propositions . 98
•3. The Logical Product of two Propositions . . . . 109
•4. Equivalence and Formal Rules . . . . . . 115
•5. Miscellaneous Propositions.123
Section B. Theory of Apparent Variables.127
•9. Extension of the Theory of Deduction from Lower to Higher
Types of Propositions.127
•10. Theory of Propositions containing one Apparent Variable . 138
•11. Theory of two Apparent Variables.151
•12. The Hierarchy of Types and the Axiom of Reducibility . 161
•13. Identity . . . . . . . . . . 168
•14. Descriptions.173
Section C. Classes and Relations.187
•20. General Theory of Classes . . . . . . . 187
•21. General Theory of Relations ...... 200
•22. Calculus of Classes.• . 205
•23. Calculus of Relations.213
•24. The Universal Class, the Null Class, and the. Existence of
Classes . 216
•25. The Universal Relation, the Null Relation, and the Existence
of Relations . 228
CONTENTS
Section I). Lor.ic op Relations.
*30. Descriptive Functions .......
• 31. Converses of Relations ......
•32. Referents and Relataof a given Term with respect to a givei
Relation .........
• 33. Domains, Converse Domains, and Fields of Relations .
• 34. The Relative Product of two Relations
• 35. Relations with Limited Domains and Converse Domains
• 36. Relations with Limited Fields .....
•37. Plural Descriptive Functions .....
• 38. Relations and Classes derived from a Double Descriptive
Function .........
Note to Section I) ......
Section E. Products and Sums ok Classes ....
•40. Products and Sums of Classes of Classes
•41. The Product and Sum of a Class of Relations
• 42. Miscellaneous Pro|Misitions ......
•43. The Relations of a Relative Product to its Factors
PART LI. PROLEGOMENA TO CARDINAL ARITHMETIC
Summary of Part II
Section A. Unit Classes and Couples.
• 50. Identity and Diversity as Relations ....
•51. Unit Classes.
• 52. The Cardinal Number I.
• 53. Miscellaneous Propositions involving Unit Classes
•54. Cardinal Couples .......
•55. Ordinal Couples ........
• 56. The Ordinal Number 2 , .....
Section B. Sub-Classes, Sub-Relations, and Relative Types
• 60. The Sub-Classes of a given Class ....
•61. The Sub-Relations of a given Relation .
•62. The Relation of Membership of a Class
• 63. Relutive Types of Classes.
•64. Relative Types of Relations.....
•65. On the Typical Definition of Ambiguous Symbols.
Section C. One-Many, Many-One, and Onb Onb Relations .
•70. Relations whose Classes of Referents and of Relata belong to
given Classes.
•71. One-Many, Many-One, and One-One Relations .
•72. Miscellaneous Propositions concerning One-Many, Many-One
and One-One Relations
•73. Similarity of Classes.
•74. On One-Many and Many-One Relations with Limited Fields
PACK
231
232
•238
242
247
25G
205
277
279
296
299
302
304
315
320
324
329
331
333
340
347
352
359
366
377
386
388
393
395
400
410
415
418
420
42C
441
455
468
CONTENTS
XI
I’ACK
Section D. Selections. 17 S
*80. Elementary Properties of Selections ..... 483
•81. Selections from Many-One Relations ..... 496
•82. Selections from Relative Products ..... • r *01
•83. Selections from Classes of Classes.508
•84. Classes of Mutually Exclusive Classes . . . . . •“» 17
•85. Miscellaneous Propositions ....... f>25
•88. Conditions for the Existence of Selections .... 536
Section E. Inductive Relations.343
•90. On the Ancestral Relation.f>49
•91. On Powers of a Relation.558
•92. Powers of One-Many and Many-One Relations . . 573
•93. Inductive Analysis of the Field of a Relation . . . 579
•94. On Powers of Relative Products.588
•95. On the Equi-factor Relation.596
•96. On the Posterity of a Term.607
•97. Analysis of the Field of a Relation into Families . . 623
APPENDIX A
•8. The Theory of Deduction for Propositions containing Apparent
Variables.635
APPENDIX B
•89. Mathematical Induction.650
APPENDIX C
Truth-Functions and others.659
LIST OF DEFINITIONS.667
ALPHABETICAL LIST OF PROPOSITIONS
REFERRED TO BY NAMES
Name
Number
Abs
*2 01 .
: yO ~ p . I). ^ p
Add
*13.
K : 7 . D • p v 7
Ass
*335
V : p . p D 7 . D. 7
Assoc
*15.
P : p v (7 v r ). D . 7 v (y> v r)
Comm
*204
H p . D . 7 D *•: D : 7 . D . p D »
Comp
*343
L./O'/./Or.Ds/j.D.r/.r
Exp
*33.
H p. t\ . D . r : D : p . D . 7 D r
Fact
*345.
Fs./Of.Drjt.r.D.f.r
Id
♦208.
Imp
*331.
H :.]). D . q D r: D : p. 7 . D . r
Perm
*14.
l-: y> v 7 . D . 7 v y>
Simp
* 202 .
H : 7 . D. yO 7
ii
*326.
: y>. 7 . D . y>
•i
*327.
h : />. 7 . D. 7
Sum
*16.
h:. 7 Dr.D:y)V 7 .D.y)vr
Syll
*205.
H 7 D r. D : y) D 7 . D . y> D r
it
*206.
h y>D 7 . D : 7 D r. D . p D r
•i
*333.
h : yO 7.7 D r. D . yO r
••
*334.
h : 7 D »* • yO 7 . D . yO r
Taut
* 12 .
K : y> v p . D . y>
Transp
*203.
h:y»D'v 7 .D.^D~y»
••
*215.
H ~ y> D 7 . D . 7 D y>
ft
*216.
H y» D 7 . D, <v 7 3 iv yj
M
*217.
H ~ 7 D ~ y>. D . y» 3 7
II
*337.
^ !.y». 7 , D , r: D : y>, iv j’. D ,
II
*41.
HpDf . = ,>v^fvy)
VI
*411.
Hy» = 7. = .<vyj = l v 7
INTRODUCTION TO THE SECOND EDITION*
In preparing this new edition of Prnicipiti Muthemativa. the authors have
thought it best to leave the text unchanged, except as regards misprints and
minor errors+, even where they were aware of possible improvements. The
chief reason for this decision is that any alteration of the propositions would
have entailed alteration of the references, which would have meant a very
great labour. It seemed preferable, therefore, to state in an introduction the
main improvements which appear desirable. Some of these are scarcely open
to question ; others are, as yet, a matter of opinion.
The most definite improvement resulting from work in mathematical logic
during the past fourteen years is the substitution, in Part I, Section A, of the
one indefinable “ p and q are incompatible ” (or, alternatively, •' p and q are
both false”) for the two indcfinables “not -p' and “ p or q." This is due to
Dr H. M. ShefferJ. Consequentially. M. Jean Nicod§ showed that one
primitive proposition could replace the five primitive propositions #1’2 , 3‘4’5'6.
From this there follows a great simplification in the building up of
molecular propositions and matrices; *9 is replaced by a new chapter, #8,
given in Appendix A to this Volume.
Another point about which there can be no doubt is that there is no need
of the distinction between real and apparent variables, nor of the primitive
idea “assertion of a propositional function.” On all occasions where, in
Prircipia Mathematica , we have an asserted proposition of the form “ V .fx"
or U V .fp,” this is to be taken as meaning “ 1-. (x) .fx ” or “ V . ( p) . fp Con¬
sequently the primitive proposition *111 is no longer required. All that is
necessary, in order to adapt the propositions as printed to this change, is the
convention that, when the scope of an apparent variable is the whole of the
asserted proposition in which it occurs, this fact will not be explicitly indicated
unless “ some ” is involved instead of “ all.” That is to say, “ H . <f>x " is to mean
“ I-. ( x ) . <j)X ”; but in “ I-. ( gx). <£x ” it is still necessary to indicate explicitly
the fact that “some” x (not “ all ” x’s) is involved.
” It is possible to indicate more clearly than was done formerly what are
the novelties introduced in Part I, Section B as compared with Section A.
• In this introduction, as well as in the Appendices, the authors are under great obligations
to Mr F. P. Ramsey of King’s College, Cambridge, who has read the whole in MS. and contributed
valuable criticisms and suggestions.
., t In regard to these we are indebted to many readers, but especially to Drs Behmann and
Boscovitoh, of Gflttingen.
$ Traru. Amer. Math. Soc. Vol. xrr. pp. 481—488.
§ “A reduction in the number of the primitive propositions of logic,” Prate. C'amb. Phil. Soc.
Vol. six.
XIV
INTRODUCTION
They are three in number, two being essential logical novelties, and the third
merely notational.
(1) For the “ p” of Section A. we substitute " <£-r.” so that in place of
“ h .</>). //> we havc*'l m .(4>.x)'f (fa)." Also, if we have "h •f(p,q , r . • ••)»”
we may substitute <f>>\ <f>y. 4>z,... for p, » /. r,... or <f>.r, <f>y lor />, q. and yfrz, ...
for ;•.and so on. We thus obtain a number of new general propositions
different from those of Section A.
(2) We introduce in Section B the new primitive idea " (gx). t.e.
existence- 1 impositions, which do not occur in Section A. In virtue of the
abolition of tin- real variable, general propositions of the form “(/>)•//> ’do
occur in Section A, but "(3 p )•.//>" docs not occur.
(3) By means of definitions, we introduce in Section B general propositions
which are molecular constituents of other propositions; thus " (x) . (f>x . v . p is
to mean " (x ). tf>r v p."
It. is these three novelties which distinguish Section B from Section A.
One point in regard to which improvement is obviously desirable is the
axiom of rcducibility (*12111). This axiom has a purely pragmatic justifica¬
tion : it leads to the desired results, and to no others. But clearly it is not
the sort of axiom with which we can rest content. On this subject, however,
it cannot be said that a satisfactory solution is as yet obtainable. Dr Leon
Chwistek* took the heroic course of dispensing with the axiom without
adopting any substitute; from his work, it is clear that this course compels
us to sacrifice a great deal of ordinary mathematics. There is another course,
recommended by Wittgenstein! for philosophical reasons. This is to assume
that functions of propositions are always truth-functions, and that a function
can only occur in a proposition through its values. There are difficulties in
the way of this view, but perhaps they are not insurmountable}. It involves
the consequence that all functions of functions are extensional. It requires us
to maintain that " A believes p " is not a function of p. How this is possible
is shown in Tract at us J.ugico-l’lnlusophiciis {luc. cit. and pp. 19—21). We arc
not prepared to assert that this theory is certainly right, but it has seemed
worth while to work out its consequences in the following pages. It appears
that everything in Vol. I remains true (though often new proofs are required);
the theory of inductive cardinals and ordinals survives; but it seems that the
theory of infinite Dedekindian and well-ordered series largely collapses, so
that irrationals, and real numbers generally, can no longer be adequately
dealt with. Also Cantor’s proof that 2" > n breaks down unless n is finite.
Perhaps some further axiom, less objectionable than the axiom of reducibility,
might give these results, but we have not succeeded in finding such an axiom.
• In liis “ Theory of Constructive Types.” See references at the end of this Introduction,
t Tractalus Logico-Philosophicut, «5-54 ff.
* See Appendix C.
NTHOIHH'TION
XV
It- shouM be stated that a now and very |Hmvrf..l moMiml in matlu n.atieal
lo Kl0 has been invented by Dr 11. M. She Her. This meth.Hl. however, would
demand a complete re-writing of Prineipia Mathematic,. We recommend
this task to Dr Shorter. since what has so far been published by him is
scarcely sufficient to enable others to undertake the necessary reconstruction.
We now proceed to the detailed development of the above general sketch.
I.
ATOMIC AXD MOLKCULAU
I'KOPOSITIOXS
Our system begins with “atomic propositions.- We accept these as a
datum, because the problems which arise concerning them belong to the
philosophical part of logic, and are not amenable (at any rate at present) to
mathematical treatment.
Atomic propositions may be defined negatively as propositions containing
no parts that are propositions, and not containing the notions “air or “some.”
Thus “ this is red,” “this is earlier than that," are atomic proposition®.
Atomic propositions may also be defined positively—and this is the better
course—as propositions of the following sorts:
Ri (.r), meaning “x has the predicate
!/) [° r x R«.y\ meaning "x has the relation R, (in intension) to y"\
Ri(x,y, z), meaning have the triadic relation R 3 (in intension)";
ft, (x, y, e, w), meaning “x,y.z,w have the tctradic relation It, (in intension)";
and so on ad infinitum, or at any rate as long as possible. Logic does not
know whether there are in fact n-adic relations (in intension); this isanempirical
question. We know as an empirical fact that there are at least dyadic relations
(m intension), because without them series would be impossible. But logic is
not interested in this fact; it is concerned solely with the hypothesis of there
being propositions of such-and-such a form. In certain cases, this hypothesis is
itself of the form in question, or contains a part which is of the form in question;
in these cases, the fact that the hypothesis can be framed proves that it is
true. But even when a hypothesis occurs in logic, the fact that it can be
framed does not itself belong to logic.
Given all true atomic propositions, together with the fact that they are all,
every other true proposition can theoretically be deduced by logical methods!
lhat is to say, the apparatus of crude fact required in proofs can all be con¬
densed into the true atomic propositions together with the fact that every
true atomic proposition is one of the following: (here the list should follow).
. used, this method would presumably involve an infinite enumeration,
since it seems natural to suppose that the number of true atomic propositions
18 infinite, though this should not be regarded as certain. In practice,
generality is not obtained by the method of complete enumeration, because
tni 8 method requires more knowledge than we possess.
b&w i .
XVI
INTRODUCTION'
We must now advance to molecular propositions. Let p. 7 , r, s, t denote,
to begin with, atomic propositions. We introduce the primitive idea
p q,
which may be read "p is incompatible with q," m and is to be true whenever
either or both are false. Thus it may also be read "p is false or 7 is false”;
or again, ‘ p implies 1101 - 7 .” Hut as we are going to define disjunction, impli¬
cation, and negation in terms of pi 7 . these ways of reading p 7 arc better
avoided to begin with. The symbol "pi 7 ” is pronounced: "p stroke 7 .” We
now pul
~p . = • p p Df.
P D 7 . m . P ~7 Df,
j)V7- = .~p]~7 Df,
p.q . = .~(p!r/) Df.
Thus all the usual truth-functions can be constructed by means of the stroke.
Note that by the above,
p D 7 . * -p |(71 7 ) Uf-
We find that
p . D . 7 . r . = . p | ( 7 1 r).
Thus p D 7 is a degenerate case of a function of three propositions.
We can construct new propositions indefinitely by means of the stroke;
for example, (p 1 7 ) j r, p | (71 r),(pl 7 )|(r|s),andsoon. Note that the stroke obeys
the perinutativc law (p| q) = (7 Ip) but not the associative law (pl 7 >|r = />|( 7 l>).
(These of course are results to be proved later.) Note also that, when we
construct a new proposition by means of the stroke, we cannot know its truth
or falsehood unless either (a) we know the truth or falsehood of some of its
constituents, or ( 6 ) at least one of its constituents occurs several times in a
suitable manner. The case (a) interests logic as giving rise to the rule of in¬
ference, viz.
Given p and | (7 jr), we can infer r.
This or some variant must be taken as a primitive proposition. For the
moment, we are applying it only when p, 7 , r are atomic propositions, but we
shall extend it later. Wc shall consider (/>) in a moment.
In constructing new propositions by means of the stroke, wc assume that
the stroke can have on either side of it any proposition so constructed, and
need not have an atomic proposition on either side. Thus given three atomic
propositions p, 7 , r, we can form, first, p 17 and 71 r, and thence (p j 7 ) | r and
p | ( 7 1 r). Given four, p. 7 , r, s, we can form
l(pl7)Mk (p\q)\(r\s), pltal( r l*)l
and of course others by permuting p, 7 , r, 5 . The above three are substantially
• For what follows, see Nicod, *• A reduction in the number of the primitive propositions of
logic.” Proe. Camb. Phil. Soc. Vol. xix. pp. 32—41.
INTRODUCTION
XVII
•i
different propositions. Wo have in fact
10 * l«?)l r ) |* • = v~q . r : v :<w,
(P17) i (** I • = • p • q • v . r .
P: { 71 ( r l*)l • = : v :»/ .*%»#• v-^s.
All the propositions obtained by this method follow from one rule: in
p | 9 , substitute, for/) or 7 or both, propositions already constructed by means
of the stroke. This rule generates a definite assemblage of new propositions
out of the original assemblage of atomic propositions. All the propositions so
generated (excluding the original atomic propositions) will be called “ mole¬
cular propositions.” Thus molecular propositions are all of the form p\q, but
the p and q may now themselves be molecular propositions. If p is p , p s ,
p, and p a may be molecular; suppose p,-p,|!p„. p„ may be of the form
Pm |Pm, and so on; but after a finite number of steps of this kind, we are to
arrive at atomic constituents. In a proposition p | q, the stroke between p and
q is called the “principal” stroke; if p = p,|p„ the stroke between p, and p.. is
a secondary stroke; so is the stroke between q x and q t if q = q x j q s . If p x =p u | p„ t
the stroke between p xx and p„ is a tertiary stroke, and so on.
Atomic and molecular propositions together are “ elementary propositions.”
Thus elementary propositions are atomic propositions together with all that
can be generated from them by means of the stroke applied any finite number
of times. This is a definite assemblage of propositions. We shall now, until
further notice, use the letters p, q, r, s, t to denote elementary propositions,
not necessarily atomic propositions. The rule of inference stated above is to
hold still; i.e.
If P’ r are elementary propositions, given p and p|( 9 |r), we can infer r.
This is a primitive proposition.
We can now take up the point ( 6 ) mentioned above. When a molecular
proposition contains repetitions of a constituent proposition in a suitable
manner, it can be known to be true without our having to know the truth or
falsehood of any constituent. The simplest instance is
pI(pIp),
which is always true. It means “p is incompatible with the incompatibility
of p -with itself,” which is obvious. Again, take "p.q.D . p.” This is
t(pl9)l(pk))l(plp). .
Again, take “~p.D.~pv~ q." This is
Again, “p.D.pv^” is
(PI P> I |(p I ?) I (p | ?))•
p I [{(p I p) I (? I q)} I l(p I p) I ($ I $))].
All these are true however p and q may be chosen. It is the fact that we can
build up invariable truths of this sort that makes molecular propositions
important to logic. Logic is helpless with atomic propositions, because their
6 2
INTRODUCTION'
xviii
truth or falsehood can only be known empirically. But the truth of molecular
propositions of suitable form can be known universally without empirical
evidence.
The laws of logic, so far as elementary propositions are concerned, are all
assertions to the effect that, whatever elementary propositions p, q, r, ... may
be, a certain function
F{p,q, r,...),
whose values are molecular propositions, built up by means of the Stroke, is
always true. The proposition "/'(/>) is true, whatever elementary proposition
p may be" is denoted by
ip ). F ( p).
Similarly the proposition "F(p,q. r,...) is true, whatever elementary pro¬
positions />, q, r.... may be is denoted by
(p.q. r ....) • F(p, q, r....).
When such a proposition is asserted, we shall omit the “(p.q.r, ...)” at the
beginning. Thus
" h • F{p, q. r,...)"
denotes the assertion (as opposed to the hypothesis) that F(p,q.r,...) is true
whatever elementary propositions p. q, r, ... may be.
(The distinction between real and apparent variables, which occurs in
Frege and in Principia Mathemntica. is unnecessary. Whatever appears as a
real variable in Principal Mathematica is to be taken as an apparent variable
whose scope is the whole of the asserted proposition in which it occurs.)
The rule of inference, in the form given above, is never required within
logic, but only when logic is applied. Within logic, the rule required is different.
In the logic of propositions, which is what concerns us at present, the rule
used is:
Given, whatever elementary propositions p, 7 , r may be, both
*'h . F(p, 7 ,r,...)” and . F(p, 7 , r, ...)| \G(p,q, r, ...)\H(p, q, r,
we can infer "h . //(/>, 7 , r ,...).”
Other forms of the rule of inference will meet us later. For the present,
the above is the form we shall use.
Nicod has shown that the logic of propositions (*1—*5) can be deduced,
by the help of the rule of inference, from two primitive propositions
^-p 1 (p\p)
and b :pD q .s\q5p\$.
The first of these may be interpreted as “/> is incompatible with not-/),” or
as “ p or not-/),” or as ** not (p and not-/))," or as "p implies /).” The second
may be interpreted as
p0q.D:qD'>*s,5.p'5~s,
INTRODUCTION
XIX
which is a form of the principle of the syllogism. Written wholly in terms of
the stroke, the principle becomes
l/>! (9I 9>1 I [!(•'* I 9> I U/> 1 i ( P I *))) I ! q) ! Up! *> i (p | x»JJ.
Nicod has shown further that these two principles may be replaced by
one. Written wholly in terms of the stroke, this one principle is
\p K91 r)\ | [(* | (t ! t)\ I 1(5: q) I ((/»I *) | (/>. *•)))].
It will be seen that, written in this form, the principle is less complex than
the second of the above principles written wholly in terms of the stroke.
When interpreted into the language of implication, Nicod’s one principle
becomes
p . D . q . r : D . ID t . s J q D p \ s.
In this form, it looks more complex than
/ 09 .D.«| 9 D/>|« (
but in itself it is less complex.
From the above primitive proposition, together with the rule of inference,
everything that logic can ascertain about elementary propositions can be
proved, provided we add one other primitive proposition, viz. that, given a
proposition (p, q, r, ...) . F (p, q, r, ...), we may substitute for p, q, r, ...
functions of the form
and assert
f 3 (p, q, r ,...),
A(p. q. r,...)
(P,q,r, •••) • F\f x (p,q,r, ...), /,(p, q,r, ...),f 9 (p, q, r,...), ...),
where f t f%,f ,... are functions constructed by means of the stroke. Since
the former assertion applied to all elementary propositions, while the latter
applies only to some, it is obvious that the former implies the latter.
A more general form of this principle will concern us later.
II. ELEMENTARY FUNCTIONS OF INDIVIDUALS
1. Definition of ” individual ”
We saw that atomic propositions are of one of the series of forms:
Ri (*), R »C x, y), R , (a, y, z), R< (x, y, z,w), ....
Here R Xt R 2 , RR 4 , ... are each characteristic of the special form in which
they are found: that is to say, R n cannot occur in an atomic proposition
Rm(xi, x 2 , ... x m ) unless n = m, and then can only occur as R m occurs, not as
••• occur. On the other hand, any term which can occur as the
a? a occur in R n (x lt x a> ... x n ) can also occur as one of the x's in R m (x 1 ,x t ,... x m )
even if m is not equal to n. Terms which can occur in any form of atomic
proposition are called “ individuals” or “ particulars”; terms which occur as the
R’b occur are called “ universals.”
We might state our definition compendiously as follows: An “ individual”
J s anything that can be the subject of an atomic proposition.
XX
INTRODUCTION
Given an atomic proposition R„( x,,x-, ... x„), we shall call any of the x’s
a "constituent" of the proposition.and R„ a " component ‘ of the proposition *.
We shall say the same as regards any molecular proposition in which
... x„) occurs. Given an elementary proposition p \q, where p and 7
may be.atomic or molecular, we shall call p and 7 "parts" of pi 7 ; and any
parts of p or 7 will in turn be called parts of /> 7 . and so on until we reach the
atomic parts of p 7 . Thus to say that a proposition /• " occurs in ’’ /> J 7 and to
say that r is a " part " of p 7 will be synonymous.
2. Definition of an elementary function of an individual
Given any elementary proposition which contains a part of which an
individual a is a constituent, other propositions can be obtained by replacing
a by other individuals in succession. We thus obtain a certain assemblage
of elementary propositions. We may call the original proposition 4>a, and
then the propositional function obtained by putting a variable x in the
place of a will Ik- called <£.r. Thus tf,r is a function of which the argument
is .<• and tin* values are elementary propositions. The essent ial use of ” <f>.r"
is that it collects together a certain set of propositions, namely all those that
are its values with different arguments.
We have already had various special functions of propositions. If p is a
part of some molecular proposition, we may consider the set of propositions
resulting from the substitution of other propositions for p. If we call the
original molecular pro|K>sition fp, the result of substituting 7 is called fij.
When an individual or a proposition occurs twice in a proposition, three
functions can be obtained, by varying only one. or only another, or both, of
the occurrences. For example,/) |/> is a value of any one of the three functions
P 1 7 . 7 1 p, 7 ! 7 . where 7 is the argument. Similar considerations apply when an
argument occurs more than twice. Thus p\(p\p) is a value of 7 | (/* |«). or
71( r 17>» or 7 i ( 7 1 **), or 7 |(r », or 7 K 7 I 7 ). When we assert a proposition
"I -.(p).Fp'‘ the p is to be varied whenever it occurs. We may similarly
jissert a proposition of the form "(x). meaning "all propositions of the
assemblage indicated by <f>x are true"; here also, every occurrence of x is to be
varied.
3. " Always true " and " sometimes true "
Given any function, it may happen that all its values arc true; again, it
may happen that at least one of its values is true. The proposition that all
the values of a function <f> ( x,y,z ,...) are true is expressed by the symbol
" 2 ,...).<t>(x,y,
unless we wish to assert it, in which case the assertion is written
This terminology is taken from Wittgenstein.
INTRODUCTION
XX!
We have already had assertions of this kind where the variables were ele¬
mentary propositions. We want now to consider t he case where the variables
are individuals and the function is elementary, i.e. all its values are elementary
propositions. We no longer wish to confine ourselves to the case in which it.
is asserted that all the values of <f> (.r, y ,...) are true; we desire to be able
to make the proposition
{x,y,z ,...). </>(.r,y,-.•••)
a part of a stroke function. For the present, however, wo will ignore this
desideratum, which will occupy us in Section III of this Introduct ion.
In addition to the proposition that a function <f>.v is "always true”
(i.e. (a?). <t>.v), we need also the proposition that <f>x is "sometimes true," i.e. is
true for at least one value of x. This we denote by
‘•(3.r). <*>*."
Similarly the proposition that <f>(x,y,z ,...) is "sometimes true" is denoted by
"(3*.y.*. y,
We need, in addition to (x, y, z,...). <t>(x,y,z, ...)and (gx,y, z ,...). <f>(x t y,z ,...),
various other propositions of an analogous kind. Consider first a function of
two variables. We can form
(a*) = (y) • </> (*. y), (*): (3y) • <t> (*.y). <3y) : ( x ) • </> (*, y), (y) • (a*) • <t> (*. y)-
These are substantially different propositions, of which no two arc always
equivalent. It would seem natural, in forming these propositions, to regard
the function <f>(x,y ) as formed in two stages. Given <f>(a,b), where a and b
are constants, we can first form a function <f>(a,y), containing the one variable
y\ we can then form
(y) • <t> (<*, y) and ( 3 y) . <f> (a, y).
We can now vary a, obtaining again a function of one variable, and leading
to the four propositions
(x) (x, y), (gx) : (y) . <f> (x, y), (x) : (gy) . <f> (x, y), (gx) : (gy) . «*> (x, y).
On the other hand, we might have gone from <f> (a, b) to <f> (x, 6), thence to
(x). <ft (x, 6) and (g#) . <f> (x, b ), and thence to
(y) i(x).<f> (x, y), (gy) : (x) . 0 (x, y), (y) : (gx) . <f> (x, y), (gy) : (gx) . <f> (x, y).
All of these will be called "general propositions"; thus eight general
propositions can be derived from the function (f> (x, y). We have
(x) : (y). <f> (x, y) s = s (y) : (x). <f> (x, y),
(3*) s (3y) '<f>(x,y)z = i (gy): (gx) . <f> (x, y).
But there are no other equivalences that always hold. For example, the dis¬
tinction between “ (x) : (gy). <f> (x, y) " and “ (gy) : (x) . <f> (x, y) ” is the same
as the distinction in analysis between “ For every e, however small, there is a
8 such that...” and "There is a 8 such that, for every e, however small, ....”
XXII
INTRODUCTION
Although it might seem easier, in view of the above considerations, to
regard ever}' function of several variables as obtained by successive steps, each
involving only a function of one variable, yet there are powerful considerations
on the other side. There are two grounds in favour of the step-by-step method;
first, that only functions of our variable need be taken ns a primitive idea;
secondly, that such definitions as the above seem to require either that, we
should fiist vary ./•. keeping // constant, or that we should first vary y, keeping
j' constant. The former seems to be involved when "(//) or "(3//)" appears
to the left of “( ur )’ or "('•K).’ the latter in the converse case. The grounds
agam»t the step-by-step method are that it interferes with the method of
matrices, which brings order into the successive generation of types of pro¬
positions and functions demanded by the theory of types, and that it requires
us, from the start, to deal with such propositions as (y). y), which are
not elementary. Take, for example, the proposition : 7 . D . p v7." This
will be
H :>./.v 9l
wr b !• (7) :• (p) : 7. D . p v 7,
and will thus involve all values of either
(7): 7 . D . /> v 7 considered as a function of p,
or (p) s 7. D. p v 7 considered us a function of 7.
This makes it im|>ossiblo to start our logic with elementary propositions, as
we wish to do. It is useless to enlarge the definition of elementary propositions,
since that only increases the values of 7 or p in the above functions. Hence
it seems necessary to start with an elementary function
T *' •••
before which we write, for each x r , either “(x r )" or "(gx r ), M the variables in
this process being taken in any order xve like. Here <f> (j-,, x it x,, ... x„) is
called the " matrix,” and what comes before it is called the " prefix.” Thus in
<3*) (*• y)
" y)" is the matrix and “ (gx): (y) ” is the prefix. It thus appears that
a matrix containing n variables gives rise to n!2" propositions by taking its
variables in all possible orders and distinguishing " (j* f ) ” and “ (gj* r ) ” in each
case. (Some of these, however, arc equivalent.) The process of obtaining such
propositions from a matrix will be called “generalization," whether we take
“all values” or “some value," and the propositions which result will be called
" general propositions."
We shall later have occasion to consider matrices containing variables that
are not individuals; we may therefore say:
A “ matrix ” is a function of any number of variables (which may or may
not be individuals), which has elementary propositions as its values, and is
used for the purpose of generalization.
INTRO IH’OTION
Will
A “ general proposition ” is one derived from a matrix by generalization.
Wo shall add one further definition at this stage:
A “first-order proposition** is one derived by generalization from a malrix
in which all the variables are individuals.
4. Methods of pror in tj general propositions
There are two fundamental methods of proving general propositions, one
for universal propositions, the other for such as assort existence. The method
of proving universal propositions is as follows. Oiven a proposition
F (P> 7 . r > •••>•**
where F is built up by the stroke, and p, 7 , r, ... are elementary, we may re¬
place them by elementary functions of individuals in any way we like, putting
p =f (x t , x^, •••
q a-,, ... a„).
and so on, and then assert the result for all values of a\, x a , ... <r„. What we
thus assert is less than the original assertion, since p, 7 . r,... could originally
take all values that are elementary propositions, whereas now they can only
take such as are values of f,f 3 ,f 3 . (Any two or more of f,f>, fj, ... may
be identical.)
For proving existence-theorems we have two primitive propositions, namely
* 8 * 1 . • (a*. y) • <t> a 1 ( 4> x 1 4>y ) and
*811. I-. (ax) • <t> x I (4> a I 4> b )
Applying the definitions to be given shortly, these assert respectively
<t> a.D . ( 3 *) . 4>x
and (x). <f>x . D . <j>a . <f>b.
These two primitive propositions are to be assumed, not only for one variable,
but for any number. Thus we assume
<f> (a,, a„ ... a„) . D . ( 3 *,, x,,... x n ) . <f> (x,, x,.... x n ),
( x lt x a , ... x n ) . <f> ... x n ) .0 . <f> (a,, a,, ... a n ) . <f> (&,, b 2 , ... b n ).
The proposition (x) . <f>x . D . <f>a . <f>b, in this form, does not look suitable for
proving existence-theorems. But it may be written
( 3 x) - ~ <frx . v . <f>a . tf>b
or ~ <fja v ~ <f>b . D . (a 31 ) - ~ 4* x >
in which form it is identical with *9*11, writing <f> for ~<f>. Thus our two
primitive propositions are the same as *9T and *9'II.
For purposes of inference, we still assume that from (x) . <f>x and
(x) . tfjx D \frx we can infer (x) . i/rx, and from p and pDj we can infer q, even
when the functions or propositions involved are not elementary.
XXIV
INTRODUCTION*
Existence-theorems arc very often obtained from the above primitive
propositions in the following manner. Suppose we know a proposition
b ./(*,*).
Since <f>.r . D . (gy). <£y. we can infer
i.e. I-: (x): (gy) ./(*, y).
Similarly b s (y): (g.r) .fir. y).
Again, since <t> (x. y ). D . (gr. ut). <f> (z. w), we can infer
and b . <gy,x) ./(x. y).
We may illustrate the proofs both of universal and of existence propo¬
sitions by a simple example. We have
b .</>)./> Op.
Hence, substituting <t>r for />,
K (/). D <t> r.
Hence, as in the case of/(x, x) above.
b :(x): (gy). <*u 0 4 >y,
*■ • (•/) • (:*l r ) • <t> r 3
• ( 3 *. y) • 4> x 3 «#>//•
Apart from special axioms asserting existence-theorems (such as the axiom of
reducibility, the multiplicative axiom, and the axiom of infinity), the above
two primitive propositions give the sole method of proving existence-theorems
in logic. They are, in fact, always derived from general propositions of the
form (x)./(x,x) or (x)./(x,x.x) or etc., by substituting other variables for
some of the occurrences of x.
III. GENERAL PROPOSITIONS OF LIMITED SCOPE
In virtue of a primitive proposition, given (x) . <f>x and (x) . <£x 0 \f/x, we
can infer (x). yfrx. So far, however, we have introduced no notation which
would enable us to state the corresponding implication (as opposed to inference).
Again, (gx) . <px and (x, y ). £x 0 yjry enable us to infer (y). yjry; here again,
we wish to be able to state the corresponding implication. So far, we have only
defined occurrences of general propositions as complete asserted propositions.
Theoretically, this is their only use, and there is no need to define any other.
But practically, it is highly convenient to be able to treat them as parts
of stroke-functions. This is entirely a matter of definition. By introducing
suitable definitions, first-order propositions can be shown to satisfy all the
propositions of *1—#5. Hence in using the propositions of *1—*5, it will
no longer be necessary to assume that p, q, r, ... are elementary.
The fundamental definitions are given below.
INTRODUCTION
XXV
When a general proposition occurs as part of another, il. is said !•» have
limited scope. If it- contains an apparent, variable .r, the scope of .r is said to
be limited to the general proposition in question. Thus in /> }(.r). </>.r', the
scope of .r is limited to (.r) . <f>x, whereas in (x) . p <f>.v the scope of .r extends
to the whole proposition. Scope is indicated by clots.
The new chapter *8 (given in Appendix A) should replace *1* in Priiictpio
Mathematica. Its general procedure will, however, be explained now.
The occurrence of a general proposition as part of a stroke-function is
defined by means of the following definitions:
10*0 •‘Ml7 • “ • (3 *0 . </> »•! q Df.
Ka®) • ♦**•} w.
pi l(y)• • -• (3y)• p i Df -
p\ 1(3»/)• *y) • = • (y)• p \ Df -
These define, in the first place, only what is meant by the stroke when it
occurs between two propositions of which one is elementary while the other is
of the first order. When the stroke occurs between two propositions which
are both of the first order, we shall adopt the convention that the one on the
left is to be eliminated first, treating the one on the right as if it were ele¬
mentary; then the one on the right is to be eliminated, in each case, in
accordance with the above definitions. Thus
i(®) . <f>x\ | «y) . yjry) . = : (a®) : 4 >® I {(!/) • +'j\ !
-:(3*) : (3y)-^l^y.
l(®) . <f>x\ | |(ay) . ^y) . = : (a*) : <f>x | {(ay) . ^y | :
“: (a*) : (y) • <t> x I ^y>
1(3®) • «M I l(y) • 'fc/) • * : (®) : (3y) • <t> x I ^ y-
The rule about the order of elimination is only required for the sake of
definiteness, since the two orders give equivalent results. For example, in
the last of the above instances, if we had eliminated y first wc should have
obtained
(3y) •• (®) • ^y>
which requires either (a:) .~<f>x or (ay) •~ , '/ r y» aQ d is then true.
And 0*0 : (ay) • £® i 'ft/
is true in the same circumstances. This possibility of changing the order of
the variables in the prefix is only due to the way in which they occur, i.e. to
the fact that x only occurs on one side of the stroke and y only on the other.
The order of the variables in the prefix is indifferent whenever the occurrences
of one variable are all on one side of a certain stroke, while those of the other
are all on the other side of it. We do not have in general
(a®) = (y) • x (®»y) : = : (y) : ( 3 ®) • x (®* y);
XXV
INTRODUCTION
here the right-hand side is more often true than the left-hand side. But we
do have
<3 r ) : <//> • 4>* V''/ : : <y) : (gx). 4>x y\ry.
The possibility of altering the order of the variables in the prefix when the)'
are separated by a stroke i* a primitive proposition. In general it is convenient
t<» put on the left the variables of which "all" arc* involved, and on the right
those of which " some " are involved, after the elimination has been finished,
always assuming that the variables occur in a way to which our primitive
proposition is applicable.
It is not necessary for the above primitive proposition that the stroke
separating .r and y should be the principal stroke, e.rj.
P |(y)-^l]-«■•/> [(*):(ay). ^y ]•
»:(3*):<»/)•/> (4> r I 'K'/) s
s :(y):(g*).p|(<^x|^ry).
All that is necessary is that there should be some stroke which separates x
from y. When this is not the case, the order cannot in general be changed.
Take e.y. the matrix
<f>x V >\nj >fry.
This may be written (<£.r D >\nj) j (>/ry D <f>x)
or l^l(^yl^y))||^y|(^i^)I.
Here there is no stroke which separates all the occurrences of x from all those
of y, and in fact the two propositions
(y) i (3*) - </>x V >\nj . — 4>x V ~ yfry
and ( 3 *):(y). «^rv>fry .~<£xv~>/ry
arc not equivalent except for special values of tf> and yfr.
By means of the above definitions, we arc able to derive all propositions,
of whatever order, from a matrix of elementary propositions combined by
means of the stroke. Given any such matrix, containing a part p, we may
replace p by <f>x or <f> (x, y) or etc., and proceed to add the prefix (x) or (gx)
or (x, y) or (x): (gy) or (y): (gx) or etc. If p and q both occur, we may replace
p by <f>x and q by yfry, or we may replace both by <f>x, or one by <f>x and another
by some stroke-function of <f>x.
In the case of a proposition such as
P I !(*) : ( 3 y) • * (*. y))
we must treat it as a case of p | {(x). <£xj, and first eliminate x. Thus
P I K*) : w) • * (*. y) | . = : (gx): (y). p \ + (x, y).
That is to say, the definitions of {(x).<£x)) q etc. are to be applicable un¬
changed when <f>x is not an elementary function.
I XTROIU’I'TIOX
XXVII
The definitions of ~ p, p v q. />.»/. p D ij are to be taken over iinelian«'o«l.
Thus
~ {(a ) . <f>.r] . = : |(.r) . <f>.r] f J(.r) . <f>.r\ :
= 8 (a-**) 8 <t> r • <M :
= : ia*-> ‘ lay) • <<*>•' <*>//>.
~ |(3») • <M • = 8 (•*•) 8 ('/) • (<f>r 4>>/).
p . D . (. 1 ). <f>.r : = :/>; [j(x) . <f>.r] |(.r). :
= 8 P |(a^> 8 (ay) • *y)I :
= 8 (•* ) 8 (y)-/>!(«*> ' +//).
(.r) . <^r . D . /> : = : |(x) . <£.r{ j (;> />) :
= 8 <3-»*> • <t *■. (/»! p ): - : (3 « > • ♦*«■ =5
(.r) . <#>x . v . p : - : [~|(x) . <*>x|] | ~p :
= 8 1(3*) 8 (3y) • (4> r I i (p ! p ):
- 8 (*) • l<3//> • (<*>*! 4>y )\! (/> I p) :
■ 5 <*) 8 <y> • (tf>* 1 4>y) I (p ».
/>. V .(x).<*>x: - : (.r) :(y). (/>!/;) | (<^r;</>y).
It will be seen that in the above two variables appear where only one might
have been expected. We shall find, before long, that the two variables can be
reduced to one; i.e. we shall have
(3*) 8 (3y) . 4>x | <f>y : s . (gx) . <f>x | <f>x,
(*) • (y) - <t>* 1 4>y : = • (x) - <t >*! 4> x.
These lead to
~ {(x) .<f>x | . = . (gx). ~ <*>x,
~ {(gx) • ♦*) ~ 0 x.
But we cannot prove these propositions at our present stage ; nor, if we could,
would they be of much use to us, since we do not yet know that, when two
general propositions are equivalent, either may be substituted for the other
as part of a stroke-proposition without changing the truth-value.
For the present, therefore, suppose we have a stroke-function in which p
occurs several times, say p \ (p | p), and we wish to replace p by (x). <f>x, we
shall have to write the second occurrence of p " (y). <£y,” and the third
"(*)•$£." Thus the resulting proposition will contain as many separate
variables as there are occurrences of p.
The primitive propositions required, which have been already mentioned,
are four in number. They are as follows:
(I) b - ( 3 ®, y) '4>a\ (<f>x | <f>y), i.e. b : <£a . D . (gx) . tf>x.
, ( 2 ) b . (gx) . <f>x | (<f>a | tf>b), i.e. b : (x) . <f>x . D . <f>a . <f>b.
• (3) The extended rule of inference, i.e. from (x) . <f>x and (x) . <f>x D \Jrx
we can infer (x) . yfrx, even when <ft and yjr are not elementary.
(4) If all the occurrences of x are separated from all the occurrences of
y by a certain stroke, the order of x and y can be changed in the prefix; i.e.
xxvin
INTRODUCTION
For (g.r): (y). <f>s ^y we can substitute (y ): (gr). <f>r >\ry. and vice
versa, even when this is only a part of the whole asserted proposition.
The above primitive propositions are to be assumed, not only for one
variable, but for any number.
By means of the above primitive propositions it can be proved that all
the propositions of *1—*5 apply equally when one or more of the propositions
/>. 7 r, ... involved are not elementary. For this purpose, we make use of the
work of Nicod. who proved that the primitive propositions of *1 can all be
deduced from
h • P 3 P
and h . p D q . D . 8 7 Dp*
together with the rule of inference: “ Given /> and p (7 | r), we can infer r."
Thus all we have to do is to show that the above propositions remain true
when p. 7 , .v, or some of them, are not elementary. This is done in *8 in
Appendix A.
IV. FUNCTIONS AS VARIABLES
The essential use of a variable is to pick out a certain assemblage of
elementary propositions, and enable us to assert that all members of this
assemblage arc true, or that at least one member is true. We have already
used functions of individuals, by substituting <f>x for p in the propositions of
*1—*5, and by the primitive pro|K>sitious of * 8 . But hitherto we have always
supposed that the function is kept constant while the individual is varied, and
we have not considered cases where we have "g<£." or where the scope of "<£"
is less than the whole Jisscrted proposition. It is necessary now to consider
such cases.
Suppose a is a constant. Then " tf>a” will denote, for the various values
of <f>, all the various elementary propositions of which a is a constituent. This
is a different assemblage of elementary propositions from any that can be
obtained by variation of individuals; consequently it gives rise to new general
propositions. The values of the function are still elementary propositions,
just ns when the argument is an individual; but they are a new assemblage
of elementary propositions, different from previous assemblages.
As we shall have occasion later to consider functions whose values are not
elementary propositions, wo will distinguish those that have elementary
propositions for their values by a note of exclamation between the letter
denoting the function and the letter denoting the argument. Thus "<£! x” is
a function of two variables, x and <f >! z. It is a matrix, since it contains no
apparent variable and has elementary propositions for its values. We shall
henceforth write “<£ ! x" where we have hitherto written tf>r.
If we replace x by a constant a, we can form such propositions as
(<£).<£!a, (g 4>).<t>la.
INTRODUCTION
X X 1 X
Those are not elementary propositions, and are therefore not of the form </>! a.
The assertion of such propositions is derived from matrices by the method of
*S. The primitive propositions of *8 are to apply when the variables, or some
of them, are elementary functions as well as when they are all individuals.
A function can only appear in a niatri.r throuqh its rallies *. To obtain a
matrix, proceed, as before, by writing <p ! .r, >/r ! //. * ! r _ in place of p. q, /•,...
in some molecular proposition built tip by means of tin* stroke. We can then
apply the rules of #8 to <f>, \Jr, ••• as well as to .r, y. c . The difference
between a function of an individual and a function of an elementary function
of individuals is that, in the former, the passage from one value to another
is effected by making the same statement about a different individual, while
in the latter it is effected by making a different statement about the same
individual. Thus the passage from "Socrates is mortal'’ to "Plato is mortal”
is a passage from fix to f\ y , but the passage from "Socrates is mortal" to
"Socrates is wise” is a passage from <f >! a to \/r ! a. Functional variation is
involved in such a proposition as: "Napoleon had all the characteristics of a
great general.”
Taking the collection of elementary propositions, every matrix has values
all of which belong to this collection. Every general proposition results from
some matrix by generalizationf. Every matrix intrinsically determines a
certain classification of elementary propositions, which in turn determines the
scope of the generalization of that matrix. Thus " x loves Socrates ” picks out
a certain collection of propositions, generalized in " (x).x loves Socrates ” and
"(a*) . x loves Socrates.” But" <f >! Socrates” picks out those, among elementary
propositions, which mention Socrates. The generalizations "( <f >). <f> l Socrates"
and " (a<*>) • <f> 1 Socrates ” involve a class of elementary propositions which
cannot be obtained from an individual-variable. But any value of "<£ fSocrates”
is an ordinary elementary proposition; the novelty introduced by the variable
<f> is a novelty of classification, not of material classified. On the other hand,
<*) . x loves Socrates, (</>) . <J>! Socrates, etc. are new propositions, not contained
among elementary propositions. It is the business of *8 to show that these
propositions obey the same rules as elementary propositions. The method of
proof makes it irrelevant what the variables are, so long as all the functions
concerned have values which are elementary propositions. The variables may
themselves be elementary propositions, as they are in *1—*5.
A variable function which has values that are not elementary propositions
starts a new set. But variables of this sort seem unnecessary. Every elementary
proposition is a value of <f >! Ss ; therefore
(p) .fp. = . ( <f>, x).f(<pix) •. (a p) • fp- = • (a</». *) •/(«/> ! *)•
* This assumption is fundamental in the following theory. It has its difficulties, but for the
moment we ignore them. It takes the place (not quite adequately) of the axiom of reduoibility.
It is disoussed in Appendix 0.
t In a proposition of logic, all the variables in the matrix must be generalized. In other
general propositions, suoh as “all men are mortal,” some of the variables in the matrix are re-
placed by constants.
XXX
INTRODUCTION
Hence all secon<l-order propositions in which the variable is an elementary
proposition can be derived from elementary matrices. The question of other
second-order projxwitions will be dealt with in the next section. A function
of two variables, say if> (x. y). picks out a certain class of classes of propositions.
We shall have the class y), for given a and variable y; then the class of
all classes <f> (a, y) as a varies. Whether we an* to regard our function as
giving classes <f>(ii,y) or <f>(x./>) depends upon the order of generalization
adopted. Thus " (gx): (y) ” involves <f>(o,y), but "(y):( 3 x) M involves
</>(./•. b).
Consider now the matrix <f>lx, as a function of two variables. If we first
vary x. keeping <£ fixed (which seems the more natural order), we form a class
of propositions if) ! .r, ! y, ! •, ... which dittbr solely by the substitution of
one individual for another. Having made one such class, we make another,
and so on. until we have done so in all possible ways. But now suppose we
vary <f> first, keeping x fixed and equal to »/. We then first form the class of
all propositions of the form (f >!«, i.e. all elementary propositions of which a is
a constituent: we next form the class <f>\h\ and so on. The set of propositions
which are values of <f >! u is a set not obtainable by variation of individuals,
i.e. not of the form fs [for constant / and variable x). This is what inukes <f>
a new sort of variable, different from .r. This also is why generalization ol the
form (<f>). Fl(<b\z,x) gives a function not of the form fix [for constant /].
Observe also that whereas ,» j s a constituent of / ! «./ is not; thus the matrix
<f >! x has the peculiarity that, when a value is assigned to x, this value is a
constituent of the result, but when a value is assigned to <f>, this value is
absorbed in the resulting proposition, and completely disappears. We may
define a function <£!.2 as that kind of similarity between propositions which
exists when one results from the other by the substitution of one individual
for another.
We have seen that there are matrices containing, sis variables, functions
of individuals. We may denote any such matrix by
/U4>'-2.+l2.X 1z - — !/• z > •••)•
Since a function can only occur through its values, <f >! 2 (e.y.) can only occur
in the above matrix through the occurrence of <f >! x, <f >! y, <f> ! z ,... or of <f> l a,
<f>\b,<f>\c .where a, b, c are constants. Constants do not occur in logic, that
is to say, the a. b, c which we have been supposing constant are to be regarded
as obtained by an extra-logical assignment of values to variables. They may
therefore be absorbed into the x, y, z . Now x, y, z themselves will only
occur in logic as arguments to variable functions. Hence any matrix which
contains the variables <£!2, yfrlz, x'.'z, x, y, z and no others, if it is of the sort
that can occur explicitly in logic, will result from substituting <f>lx, if>ly, <f>lz,
i' I ^ 1 y» if ! 2 > X ! x > X *• y* X ! z > or 301,16 of them, for elementary propositions
in some stroke-function.
iNTitoiurrioN
x x x i
It is necessary here to explain what is meant when wo spo.ik of a •• maui\
that can occur explicitly in logic.' or. as we may call it. a "logical matrix."
A logical matrix is one that contains no constants. Thus /» #/ is a logical
matrix ; so is <f> !.i\ where <f> anil .r are both variable. Taking any elementary
proposition, we shall obtain a logical matrix it' we replace all its components
and constituents by variables. Other matrices result I rum logical matrices by
assigning values to some of their variables. There are. however, various ways
of analysing a proposition, and therefore various logical matrices can be derived
from a given proposition. Thus a proposition which is a value of p will
also be a value of (</>!a) | (y\r\y) and of \• (•»’. .'/)• Different forms are reiptired
for different purposes; but all the forms of matrices required explicitly in
logic are logical matrices as above defined. 'Phis is merely an illustration of
the fact that logic aims always at complete generality. The test of a logical
matrix is that it can be expressed without introducing any symbols other
than those of logic, e.g. we must not require the symbol “Socrates.” Consider
the expression
/! 2, ... g, z).
When a ydue is assigned to/, this represents a matrix containing the variables
<f>- X> ••• x » !/> * . But while / remains unassigned, it is a matrix of a
new sort, containing the new variable / We call / a “ second-order function,"
because it takes functions among its arguments. When a value is assigned,
not only to / but also to <f>, yfr, ... x, y, z, .... we obtain an elementary
proposition; but when a value is assigned to / alone, we obtain a matrix
containing as variables only first-order functions and individuals. This is
analogous to what happens when we consider the matrix <f >! x. If wc give
values to both </> and x, we obtain an elementary proposition; but if we give
a value to <f> alone, we obtain a matrix containing only an individual as variable.
There is no logical , matrix of the form /!(<£! 2). The only matrices in
which <f> ! z is the only argument are those containing <f> ! a, </>! 6, (f> ! c, . .., where
a, b, c, ... are constants; but these are not logical matrices, being derived
from the logical matrix <f >! x. Since <f> can only appear through its values, it
must appear, in a logical matrix, with one or more variable arguments. The
simplest logical functions of <f> alone are (x). <f >! x and (gx) . <f >! x, but these
are not matrices. A logical matrix
/! (<f> l x lt x t , ... x n )
is always derived from a stroke-function
F(Px,Pz, p». ...p n )
by substituting <f> ! Xj, <f> l x 2 , ... <f> ! x n for p it p„ ... p n . This is the sole method
of constructing such matrices. (We may however have x r = x, for some values
of r and 8.)
• Second-order functions have two connected properties which first-order
functions do not have. The first of these is that, when a value is assigned to
r&w i r
XXX11
INTRODUCTION
/', the result may be a logical matrix: the second is that certain constant values
of / can be assigned without going outside logic.
To take the first point first:/! {tf >! 2. x), for example, is a matrix containing
three variables,/, <f>, and x. The following logical matrices (among an infinite
number) result from the above by assigning a value to/: <f> ! x, {<f> ! x) | (</>! x),
<t >! x D <£ ! x, etc. Similarly <f >! x D <f >! y, which is a logical matrix, results from
assigning a value to /in/!(</>! 2 ,x.y). In all these cases, the constant value
assigned to/is one which can be expressed in logical symbols alone (which
was the second property of/). This is not the case with <f>\x: in order to
assign a value to <f>, we must introduce what we may call “empirical constants,"
such as “Socrates" and “mortality" and “ being Greek." The functions of x
that can be formed without going outside logic must involve a function as a
generalized variable; they are (in the simplest case) such as (<f>).<f>lx and
(a$) •*!•'••
To some extent, however, the above peculiarity of functions of the second
and higher orders is arbitrary. We might have adopted in logic the symbols
R i (*). K (•**. •/). R» (•*■,!/. i) .
where R, represents a variable predicate. II, a variable dyadic relation (in
intension), and so on. Each of the symbols /?, (x). R~{.r,y), R»{r,y,x), ••• »
a logical matrix, s«* that, if we used them, we should have logical matrices not
containing variable functions. It is perhaps worth while to remind ourselves
of the meaning of "<f >! a," where a is a constant. The meaning is as follows.
Take any finite* number of propositions of the various forms R, ( x ), R,(x,y ),...
and combine them by means of the stroke in any way desired, allowing any
one of them to be repeated any finite number of times. If at least one of
them has a as a constituent, t.e. is of the form
Rn (a,6,, 6„ ... &„-,).
then the molecular proposition we have constructed is of the form <f>la,
i.e. is a value of “ </>!«" with a suitable <f>. This of course also holds of the
proposition R„ (<t,6,,6 t ,... &„_,) itself. It is clear that the logic of propositions,
and still more of general propositions concerning a given argument, would be
intolerably complicated if we abstained from the use of variable functions;
but it can hardly be said that it would be impossible. As for the question of
matrices, we could form a matrix/!(/£,, x), of which R , (.r) would be a value.
That is to say, the properties of second-order matrices which we have been
discussing would also belong to matrices containing variable universals. They
cannot belong to matrices containing only variable individuals.
By assigning <f >! z and x in /!(</»! 2, x), while leaving/variable, we obtain
an assemblage of elementary propositions not to be obtained by means of
variables representing individuals and first-order functions. This is why the
new variable / is useful.
INTRODUCTION*
XXXIII
We can proceed in like manner t<» matrices
FI (/!(<£ ! gl($l * n ... * ! * X ! 2. ... .r. //. ...]
and so on indefinitely. These merely represent new ways of grouping ele¬
mentary propositions, leading to new kinds of generality.
V. FUNCTIONS OTHER THAN MATRICES
When a matrix contains several variables, functions of some of them can
be obtained by turning the others into apparent variables. Functions obtained
in this way are not matrices, and their values are not elementary propositions.
The simplest examples are
(*,y) and (gy). <f> !(.r, y).
When we have a general proposition (<*>). F (<*>! 2 , x, y, ...|, the only values </>
can take are matrices, so that functions containing apparent variables are not
included. We can, if we like, introduce a new variable, to denote not only
functions such as <f >! 2, but also such as
(y).</>!(2,y), (y,z)-<t>l(£,y,z), ... (ay) • <t> 10*. y), •••;
in a word, all such functions of one variable as can be derived by generalization
from matrices containing only individual-variables. Let us denote any such
function by <t> x x, or ^,.r, or x,a:, or etc. Here the suffix 1 is intended to indi¬
cate that the values of the functions may be first-order propositions, resulting
from generalization in respect of individuals. In virtue of #8, no harm can
come from including such functions along with matrices as values of single
variables.
Theoretically, it is unnecessary to introduce such variables as <f>,, because
they can be replaced by an infinite conjunction or disjunction. Thus e.g.
. <f>ix . = : (</>) . <f >! x : (<f>, y ). <f >! (x, y) z (<£): (gy) .<t>l(x,y): etc.,
(a*»> • * * = : (atf*) - <t >! X : v: (g<£): (y) . ! (x, y ): v :(g<£, y). <f >! (x, y) : v: etc.,
and generally, given any matrix/! ( <f> ! 2, x), we shall have the following pro¬
cess for interpreting (<*>,) ./! (<£, 2 , x) and (g<*>.) ./! (£,2 , x). Put
(<*>>) ./! x). = z (<*>) ./! | (y) . <f> l (2, y), x) : (<*,) ./! [(gy) . <f >! (2, y), *),
where/! {(y). <f >! (2,y), x\ is constructed as follows: wherever, in/! \<f> ! 2, x),
a value of <f>, say <f ,! a, occurs, substitute (y) . </>! (a, y), and develop by the
definitions at the beginning of *8. /! ((gy) . <f> ! (2, y), a:) is similarly con¬
structed. Similarly put
(**> •/* (*’ 1 5, x) . = : (*)./! ((y, */).«*>! (2, y. w ), x\ :
(<#») •/* l(y) s (a"') - £ * (2, y, w), *| : etc.,
where *etc. w covers the prefixes (gy) : (w) (gy,«/) (w) : (gy). We define
<p> «£ 4 ,... similarly. Then
: ,‘(«r/l.(tt*).-:(«./WW s(^*)./!(^2,a:): etc.
This process depends upon the feet that /!(</>! 2 , a:), for each value of <f> and a:,
is a proposition constructed out of elementary propositions by the stroke, and
XX XIV
INTRODUCTION
that *8 enables us to replace any of these by a proposition which is not
elementary. (fl<M •/•(<£.-• •' * is defined by an exactly analogous disjunction.
It is obvious that, in practice, an infinite conjunction or disjunction such
as the above cannot Ik* manipulated without assumptions ad hoc. We can
work out results for any segment of the infinite conjunction or disjunction,
and we can ".see” that these results hold throughout. But we cannot prove
this, because mathematical induction is not applicable. We therefore adopt
certain primitive propositions, which assert only that what we can prove in
each case holds generally. By means of these it becomes possible to manipulate
such variables as <f>
In like manner we can introduce /, ( 0 , 2 , s), where any number of in¬
dividuals and functions 0,. ••• may appear ;is apparent variables.
No essential difficulty arises in this process so long as the apparent
variables involved in a function are not of higher order than the argument to
the function. For example, x «!)*/{, which is (fly) • *Ry, may be treated
without danger as if it were of the form <f >! a*. In virtue of *8. <f>,x may be
substituted for <f>ls without interfering with the truth of any logical pro¬
position which <t>'.x is a part. Similarly whatever logical proposition holds
concerning/! (£, 3 , x) will hold concerning/, (0,2. x).
But when the apparent variable is of higher order than the argument, a
new situation arises. The simplest cases are
<*)./!<*! 2. x). (g^)./!(^!?.x).
These are functions of /, but are obviously not included among the values
for <t> lx (where <f> is the argument). If we adopt a new variable fa which is
to include functions in which 0! 2 can be an apparent variable, we shall obtain
other new functions
<*,)./!(*, 2 .*). < 3 *.>./«<*.J.*).
which are again not among values for fax (where fa is the argument), because
the totality of values of faz, which is now involved, is different from the totality
of values of 0 ! 2 , which was formerly involved. However much we may en¬
large the meaning of <f>. a function of x in which 0 occurs as apparent variable
has a correspondingly enlarged meaning, so that, however 0 may be defined,
(*)./! (*2,x) and (fl0)./! (02, x)
can never be values for fac. To attempt to make them so is like attempting
to catch one's own shadow. It is impossible to obtain one variable which
embraces among its values all possible functions of individuals.
We denote by fax a function of x in which fa is an apparent variable, but
there is no variable of higher order. Similarly fax will contain fa as apparent
variable, and so on.
INTRODUCTION
X X X V
The essence of the matter is that a variable may travel through any well-
defined totality of values, provided these values are all such that any one eon
replace any other significantly in any context. In constructing fax, the only
totality involved is that of individuals, which is already presupposed. Hut
when we allow <f> to be an apparent variable in a function of x, we enlarge the
totality of functions of x, however <t> may have been defined. It- is therefore
always necessary to specify what sort of <f> is involved, whenever <f> appears as
an apparent variable.
The other condition, that of significance, is fully provided for by the
definitions of *8, together with the principle that a function can only occur
through its values. In virtue of the principle, a function of a function is a
stroke-function of values of the function. And in virtue of the definitions in
*8, a value of any function can significantly replace any proposition in a
stroke-function, because propositions containing any number of apparent
variables can always be substituted for elementary propositions and for each
other in any stroke-function. What is necessary for significance is that every
complete asserted proposition should be derived from a matrix by generaliza¬
tion, and that, in the matrix, the substitution of constant values for the
variables should always result, ultimately, in a stroke-function of atomic
propositions. We say " ultimately,” because, when such variables as fa .3 are
admitted, the substitution of a value for fa may yield a proposition still
containing apparent variables, and in this proposition the apparent variables
must be replaced by constants before we arrive at a stroke-function of atomic
propositions. We may introduce variables requiring several such stages, but
the end must always be the same: a stroke-function of atomic propositions.
It seems, however, though it might be difficult to prove formally, that the
functions <f> x , /, introduce no propositions that cannot be expressed without
them. Let us take first a very simple illustration. Consider the proposition
(3<M . <f>i& • fao>, which we will call f{x, a).
Since <£, includes all possible values of <f >! and also a great many other values
in its range,/(x, a) might seem to make a smaller assertion than would be
made by
( 30 ). 0 ! x . <f> ! a, which we will call /. (x, a).
But in fact f(x, a) . D . f 0 (x, a). This may be seen as follows: fax has one of
the various sets of forms:
(y) . 0 ! (x, y), (y,z) . <f>l (x, y, z) t
(ay) - 0 *(*. y). (a.y>*) • 0 * (*>y> *)> •••».
-(y): (a*) • 0 • (*» y» *)» (ay) • (*) - 0 * (*. y.*) .
Suppose first that fax . = . (y) . <f> l (x, y). Then
fax . faa • = :(!/) • 4>l{x,y):(y) . <f>l(a,y)i
Di<t>l(x,b).<t>l(a,b)z
, • D : (30) . <f >! x . <f >! a.
!> •
XXX VI
INTRODUCTION*
Next, suppose 0 ,/. = . (fly ). <t>l(-r, y). Then
<t> t x. tf>,a . = :(gy)-0!(*.y): (g*>- <f>Ua,z):
D : (ay. :): <f >! (x.y) v <*>! (x, j) . <£ ! (a. y) v <f >! (a, z ):
D: (g£). <£ !x. <f>! a,
because <f>! (x. y) v <f >! (x, j) is of the form <f >! x. when y and r are fixed. It is
obvious that tins method of proof applies to the other cases mentioned above.
Hence
We can satisfy oprsclves that the same result holds in the general form
(<t>X *)• = • (3</»)./! ( <t >! 2, x)
by a similar argument. We know that /!(<£! 2 , x) is derived from some
stroke-function
F (/>. 7. r ,...)
by substituting $ ! x, <f >! a. <#>! fc, ... (where d. 6. ... are constants) for some of
the propositions ji, if. r, ... and y,! x. y 7 ! .r. y,! x. ... (where <7,, *7,. are
constants) for others of /». 7, r.while replacing any remaining propositions
/>, if, r, ... by constant propositions. Take a typical case; suppose
We then have to prove
4>,«|(4>,x <*>,/>). 3 .(g$). $!« (<£ ! x | <£ ! 6),
where </>,x may have any of the forms enumerated above.
Suppose first that ^,x. = . (y). <f >! (x, y). Then
4>,«|(<f>,x 0,6>. » : (gy) :(*, w). <t >! (<*.y)l I 4 > * (x. z) | <t >! (b, «•)) :
3 : (ay) • <t> * («. y)l 1* •• (*. y) 1 4 > ! (*. y)l :
D:( 3 *).<*>!a|<*!*|«M 6 )
because, for a given y. <f >! (x, y) is of the form 4 >! x.
Suppose next that <£,x . = . (gy). *f >! (x, y). Then
<M S (<M! «M) • = *• (y) : (a*. «*) • ! («*. y>! !^> ! ( x * -)! ! (&. w)):
D :(g^). irla\(ylrlx\yfrl b).
putting \Jr ! x . = . <f >! (x, z) v if >! (x, w). Similarly the other wises can bo dealt
with. Hence the result follows.
Consider next the correlative proposition
<*.) ./!(<*>,2 .x). = .(<*>)./!«*>I 2.x).
Here it is the converse implication that needs proving, i.e.
(<*,)./!(*! 2. x). D. (*,)./! (<#>,2. x).
This follows from the previous case by transposition. It can also be seen in¬
dependently os follows. Suppose, as before, that
/! (<f>,2, x). = . (<#>,a) | (</>,x | </>,£»),
and put first «/>,x. = . (y). <t >! (x, y).
Then (<J>,a) | (</>,x | <*>,fc). = : (gy): (z, w).<f>l (a, y) | {<£! (x, z) | if >! (6, w)).
INTRODUCTION
X X X V11
Thus we require that, given
we should have (g;/) : (z, w) . <f >! («i, //) | \<f >! (*, z) | <f >! ( b . w)\.
Now
(yfr) . yfr ! o | (>/r ! x j yfr ! b) . D <f >! (ci, j) . D . <f> ! (.i\ z). <f> l (b. z) z
4 >! (u, «*). D . <f >! (.r, w) . </>! (6, w)
D :.</>! (a, j) . <f >! (a, /<•) . D . 0 ! (.r, z). <f>l (b, w)
D <£! (a, m») . D : <£! (a, z ). D . <f>l (.r, z ). <f >! (6. tv) (1)
Also (a t w) . D : 0 ! (a, «•). D . <£! (<r, j) . </>! (/>. «•) (* 2 )
(l).(2).Ds.(^).^!a|(^!o:!>fr!6):D:.(ay):^!(a^).D.<^Har^).0!(6. w)
which was to be proved.
Put next = - ( 3 !/) • <£*(*• y)-
Then (</>,a) | (<£,* | <#>,&) . - : (y) : (g*. w) . <*>! (a. y) i |</> • (*. *)!</>! (&. w)|.
In this case we merely put z = w = y and the result follows.
The method will be the same in any other case. Hence generally:
<+.) •/! (<M. *) • =’• <«*>) •/* (<t> * 2, a:).
Although the above arguments do not amount to formal proofs, they suffice
to make it clear that, in fact, any general propositions about <f>! 2 are also
true about <£, 2 . This gives us, so far as such functions arc concerned, all that
cotild have been got from the axiom of rcducibility.
Since the proof can only be conducted in each separate case, it is necessary
to introduce a primitive proposition stating that the result holds always. This
primitive proposition is
h :(*)./!<*! 2 , x). D./! x) Pp.
As an illustration: suppose we have proved some property of all classes defined
by functions of the form <f >! 2, the above primitive proposition enables us to
substitute the class D‘i 2 , where R is the relation defined by <f >! ( 2 , p), or b y
(a 2 ) - <f> l ( 2 , f), z), or etc. Wherever a class or relation is defined by a function
containing no apparent variables except individuals, the above primitive pro¬
position enables us to treat it as if it were defined by a matrix.
We have now to consider functions of the form tf> t x, where
<f> 2 x . = . (<f >)./! ( <f> l 2 , x) or <f> 2 x . = . (g<#>) ./! ( <f >! 2, x).
We want to discover whether, or under what circumstances, we have
(<t>) • 9 1 (<t> ! 2 * x) . D . g ! (<^ 2 , x). (A)
Let us begin with an important particular case. Put
gl(tf>l 2 ,x). = .<f>laD<pix.
. Then ( <f >) . g l (<f> ! 2 , x) . = . x = a, according to * 13 * 1 .
XXXVIII
INTRODUCTION
Wc want to prove
(0). 0! a D 0! x . D . <t>.u D 0.x,
U. (0). 0! a 0 0 Sx. D : (0)./! (0 ! 2.a). D . (0)./! (0! 2 . 4 r):
< 30 ) •/! (0 ! 2. «). D . (%!</>)./! (0 ! 2, *).
Now y'! (0 ! 2 ,./■) must be derived from some stroke-function
/’(/>. 7. r, ...)
by substituting for some of p, 7. r, ... the values 0 ! .r, 0 ! &, 0 ! c. ... where
b, c .... are constants. As soon as 0 is assigned, this is of the form 0 ! .r. Hence
(0). 0! n D 0! x . D :«<^>>:/! (0 ! 2,«) • 3 ./!(0 ! 2, *):
D:(0)./!(0!2.o). D .(0)./!(0! 2.a):
<a0)-/!(0!2.«).^.(a^)-/!(0i2.x).
Thus generally ( 0 ). 0 !a D 0 I/.D. ( 0 ..). 0 .ci D 0 ; x without the need of any
axiom of rcducibility.
It must not, however, be assumed that (A) is always true. The procedure
is as follows: /! (0! 2,a) results from some stroke-function
F(p. 7, /•. ...)
by substituting for some of p, 7, r, ... the values 0 ! x, 0 ! ci, 0 ! 6, ... (a, b, ...
being constants). We assume that, e.g.
0.x.« .<0)./!(0! 2. a).
Thus 0 ,x. = .( 0 ). F(<f >! x, 0 ! a, 0 ! 6, ...). (B)
What we want to discover is whether
(0) • #! (0 ! 2, a). D . <7! (0.2, a).
Now g ! ( 0 ! 2 , a) will be derived from a stroke-function
G(/>. 7. r, ...)
by substituting 0 !a, 0 ! «\ 0 ! 6 ', ... for some of p, 7. r. To obtain
*7! (0a2, a), we have to put 0 ,x, 0 ,o', 0 , 6 \ ... in G(/>, 7, r, ...), instead of
0 ! a\ 0 !0 ! b' . We shall thus obtain a new matrix.
If(0)-«7!(0! 2 ,a) is known to be true because G(p, 7, r, ...) is always
true, then 7 ! (0,2, a) is true in virtue of *8, because it is obtained from
G (p, 7, r, ...) by substituting for some of 7, r, ... the propositions 0 ,a,
0 ,«\ 0 , 1 *',... which contain apparent variables. Thus in this case an inference
is warranted.
We have thus the following important proposition:
Whenever ( 0 ). g ! (0 ! 2 , x) is known to be true because gl(<f>lz,x) is
always a value of a stroke-function
G(p, q, r, ...),
which is true for all values of p,q,r .then g ! (0,2.x) is also true, and so
(of course) is (0,) .g ! (0,2, x).
INTH01>l , l*TlUX
X X XIX
'l'his, however, does not cover the case where (</>).</! (*f >! ••') is not a
truth of logic, but a hypothesis, which may be true for some values ol .»• ami
false for others. When this is the case, the inference to g l . .r) is some¬
times legitimate and sometimes not: the various eases must be investigated
separately. We shall have an important illustration of the failure of the
inference in connection with mathematical induction.
VI. CLASSES
r l he theory of classes is at once simplified in one direction and complicated
in another by the assumption that functions only occur through their values
and by the abandonment of the axiom of reducibility.
According to our present theory, all functions of functions are extonsional.
i.e.
<f>x = x yfrx . D ./(<£3) =f(>\rz).
This is obvious, since <*> can only occur in /(<£2) by the substitution of values
of <f> for p, q, r, ... in a stroke-function, and, if 4>x = yjrx, the substitution of
<f>x for p in a stroke-function gives the same truth-value to the truth-function
as the substitution of yfrx. Consequently there is no longer any reason to
distinguish between functions and classes, for we have, in virtue of the above,
<t>x = x yfrx . D .
We shall continue to use the notation 5 (<f>x), which is often more convenient
than <f >£; but there will no longer be any difference between the meanings of
the two symbols. Thus classes, as distinct from functions, lose even that
shadowy being which they retain in *20. The same, of course, applies to
relations in extension. This, so far, is a simplification.
On the other hand, we now have to distinguish classes of different orders
composed of members of the same order. Taking classes of individuals as the
simplest case, &(<f>lx) must be distinguished from £( fax ) and so on. In
virtue of the proposition at the end of the last section, the general logical
properties of classes will be the same for classes of all orders. Thus e.g.
aC£ .0Cy. D .aCy
will hold whatever may be the orders of a, 0, y respectively. In other kinds of
cases, however, trouble arises. Take, as a first instance, p l K and s*k. We have
x ep*K . = : a e k . D„ . x e a.
Thus p‘rc is a class of higher order than any of the members of k. Hence the
hypothesis (or) ./a may not imply /(p‘<c), if a is of the order of the members
of k. There is a kind of proof invented by Zermelo, of which the simplest
example is his second proof of the Schroder-Bernstein theorem (given in *73).
This kind of proof consists in defining a certain class of classes k, and then
showing that p‘fce/c. On the face of it, “p‘/c e k ” is impossible, since p‘/c is
xl
INTRODUCTION
not of the same order as members of *. This, however, is not all that is to be
said. A class of classes * is always defined by some function of the form
r., »/ = , ...). F(.r t €a. x.ea. ... y.co, y t ea. ...),
where F is a stroke-function, and "acK means that the above function is
true. It may well happen that the above function is true when p*K is sub¬
stituted for a. and the result is interpreted by *8. Does this justify ns in
asserting />‘* e k *
Let. us take an illustration which is important in connection with
mathematical induction. Put
k = a ( R“a Cfl.ftt a).
Then C p‘x . a < p‘« (see <40-81)
mi that, in a sense, />** * k. That is to say, if we substitute for a in the
defining function of *. and apply *8. we obtain a true proposition. By the
definition of *90.
R+n ■= />**.
Thus R 0 *a is a second-order class. Conse<piently. if we have a hypothesis
(a) .fa. where a is a first-order class, we cannot assume
(a).fa.O .j\R+*a). (A)
By the proposition at the end of the previous section, if (a), fa is deduced by
logic from a universally-true stroke-function of elementary propositions.
/(/f* 4 «) will also be true. Thus we may substitute for a in any asserted
proposition “h./a' which occurs in Principia Mathematica. But when
(a) .fa is a hypothesis, not a universal truth, the implication (A) is not, prima
facie, necessarily true.
For example, if k — a ( R*‘a C a . a c a), we have
atif.D:an/3fif . = . R“(a r\ 0)C 0 . a c 0.
Hence a c « . R“(a r\0)C0.a€0.O. p*x C0 (1)
In many of the propositions of *90, as hitherto proved, we substitute p*tc for
a, whence we obtain
i.e.
or
li' “</3n p‘K) C 0.ac0.D.p*K C 0 (2)
zc 0. ali+z. D..*. W€0 m €0 . aR+x : D . j* € 0
aR+x . D z € 0 . aR+z . . w c 0 : a e 0 : 3 . x c 0.
This is a more powerful form of induction than that used in the definition of
But the proof is not valid, because we have no right to substitute p‘ic
for a in passing from (1) to (2). Therefore the proofs which use this form of
induction have to be reconstructed.
INTRODUCTION
xli
It will bo found that the form to which wo can reduce most. the fallacious
inferences that scorn plausible is the following:
Given M h . (.r) . /(.r,.r)'* wo can infer " b : (.r) : (;•!//>. t( r, //).** Thus given
" ^ ” we can infer “ b : (a): (g£) ./(a, £)." But this depends upon
the possibility ot a = 8. If. now, a is of one order and 8 of another, wo do
not know that a = 8 is possible. Thus suppose we have
a e k . . ga
and we wish to infer g8> where 8 is a class of higher order satisfying
The proposition
(8) :• a € k . D a . ga : D : e * . D . q8
becomes, when developed by *8,
(£)::(ga):.a* * . D .</a : D :/9c * . D . < 7 #
This is only valid if a = /9 is possible. Hence the inference is fallacious if 8
is of higher order than a.
Let us apply these considerations to Zermelo’s proof of the Schruder-
Bernstein theorem, given in *73 8 ff. We have a class of classes
* = 3 (a C D‘R . 8 - C l‘R C a . R“a C a)
and we prove p‘/c e k (*73'81), which is admissible in the limited sense ex¬
plained above. We then add the hypothesis
x~c (8 ~ G‘i2) vi R“p‘*
and proceed to prove p‘tc — l*x c * (in the fourth line of the proof of *73 82).
This also is admissible in the limited sense. But in the next line of the same
proof we make a use of it which is not admissible, arguing from p'tc - i‘x e k
to p l K C p*K — i*x, because
a e k . D. . p*K C o.
The inference from
a e k . D. .p*K C a to p‘ic — l‘x e k . D . p*K Cp‘/e — l‘x
is only valid if p t K—i t x is a class of the same order as the members of *.
For, when a e k . ,p*K C a is written out it becomes
(a) ::: (g/9) ( x ) ::ae*.D:./9e*.D.x€/9: D . x e a.
This is deduced from
a e k . Dz.acrc.D.xea : D . x e a
by the principle that/(a, a) implies (g>9) ./(a, £). But here the 8 must be
of the same order as the a, while in our^case a and 8 are not of the same
order, if a = p‘/c — i‘x and 8 is an ordinary member of k. At this point, there¬
fore, where we infer p'tc Cp‘/c — i*x, the proof breaks down.
It is easy, however, to remedy this defect in the proof. All we need is
x~e(J3 — Q.‘R) vi R“ p ‘k . D . x~ep*K
or, conversely.
xep‘* . D . xe(8— d‘R) vi R“p‘tc.
INTRODUCTION
xlii
Noxv
ep l K . D :.a **. D. : a — f 4 J*~e * :
D. : -<£ - (I 4 K C a - i«jp) . v . -!£“<a - i‘.r) C a - t 4 .r* :
tt 44 (a- f 4 .r)
D :..r* £- (I 4 /f : v : a f «f. D 4 • < /f 44 a.
Hence. by *72*341.
.*<//*. D .s<(0-(\‘R) v rt 4< /> 4 *
which gives llie required result.
Wi- assume that a-f 4 .» is »f no higher order than a; this can he secured
by taking a to b«- of at least the second order, since i*j\ and therefore — l*x,
is of tin* second order. We may always assume our classes raised to a given
order, but not raised indefinitely.
Thus the Schroder-Bernstein theorem survives.
Another difficulty arises in regard to sub-classes. We put
Cl 4 a «£(£Ca) Df.
Now " Cn is significant when & is of higher order than a, provided its
members are of the same type as those of a. But when we have
£ C a . D, ./$,
the fS must be of some definite type. As a rule, we shall be able to show
that a proposition of this sort holds whatever the type of tf, if we can show
that it holds when (3 is of the same type ns a. Consequently no difficulty
arises until we come to Cantor's proposition *2" > n, which results from the
proposition
^J(CI 4 a)sm a|
which is proved in *102. The proof is ns follows:
li € 1 1 . D 4 /* = a . (I 4 /* C Cl 4 a . £-3 |x*a- R‘x j. D :
~
'/«a . y t R l ;j .1.,. y • 'J~' R*y • : 3 :y««. + R'y :
D:£~*(l 4 7f.
As this proposition is crucial, we shall enter into it somewhat minutely.
Let a — jt (A ! x), and let
xR {*<* !*)). = ./!<*!*,*).
Then by our data,
A Sx.D.( a ^) ./!(<#>! 5.x),
/l(^!2.x)./!(^!2,y).D.x = y,
/l(<f>llx)./!(yJr!3,x).D.<f>ly=,yJr!y.
With these data,
xea — R*x . = : A!x:/!(<£ !z,x). D* ! x.
£ = *((<*>): A !*:/!(*! 2.*).
Thus
INTRODUCTION
xliii
Thus £ is defined by n function in which 0 appears as apparent variable. If
we enlarge the initial range of 0. we shall enlarge the range of values involve.I
m the definition off. There is therefore no way of escaping from the result,
that f is of higher order than the sub-classes of o contemplated in the
definition of CTa. Consequently the proof of 2" > n e.. I lapses when the
axiom of reducibility is not assumed. We shall find, however, that t he propo¬
sition remains true when n is finite.
With regard to relations, exactly similar questions arise as with regard to
classes. A relation is no longer to be distinguished from a function of two
variables, and wo have
0 (•«•, 9) - 0“ (^. 9) • = s 0 y) . . 0 (.r, y).
The difficulties ns regards p *\and Rl'Pare less important than those concerning
P*k and Cl 4 a, because p*\ and R1‘P are less used. But a very serious difficulty
occurs as regards similarity. We have
asrajS.B. (gP) . R e 1 1 . a - D‘P . & = CI‘P.
Here R must be confined within some type; but whatever type we choose,
there may be a correlator of higher type by which a and /9 can be correlated.
Thus we can never prove ~(a sm /9), except in such special cases as when
either a or is finite. This difficulty was illustrated by Cantor’s theorem
2” > n, which we have just examined. Almost all our propositions are con¬
cerned in proving that two classes are similar, and these can all be interpreted
so as to remain valid. But the few propositions which are concerned with
proving that two classes are not similar collapse, except where one at least of
the two is finite.
VII. MATHEMATICAL INDUCTION
All the propositions on mathematical induction in Part II, Section E and
Part III, Section C remain valid, when suitably interpreted. But the proofs
of many of them become fallacious when the axiom of reducibility is not
assumed, and in some cases new proofs can only be obtained with considerable
labour. The difficulty becomes at once apparent on observing the definition
of “ xR^y" in *90. Omitting the factor "xcC'-R,” which is irrelevant for
our purposes, the definition of “xR+y" may be written
zRw . w . 0 ! z D 0 ! w : D* . <t> l x D <£ ! y, (A)
i.e. “ y has every elementary hereditary property possessed by x.” We may,
instead of elementary properties, take any other order of properties; as we
shall see later, it is advantageous to take third-order properties when R is
one-many or many-one, and fifth-order properties in other cases. But for
preliminary purposes it makes no difference what order of properties we take,
and therefore for the sake of definiteness we take elementary properties to
begin with. The difficulty is that, if 0, is any second-order property, we
cannot deduce from (A)
zRw . . <p 2 z D 0*u/ : D . <f> s x 3 0 2 y. (B)
xliv
INTRODUCTION*
Suppose, for example, that = .(*)./!($ S3.*); then from (A) we can
deduce
zliw . D. , r ./! < <f> ! 3. :) D*/! (<*>! 3, itr): D s/S <<#>! 3.- 0 * ./'• <4> '• 2, y) :
D: <f> : .r. D. <£*//. (C)
But in general our hypothesis here is not implied by the hypothesis ot (B).
It wc put <f> : z . = . ( 3 +)./! (<f> S 3.we get exactly analogous results.
Hence in order to apply mathematical induction to a second-order property,
it is not sufficient that it should be itself hereditary, but it must be composed
of hereditary elementary properties. That is to say, if the property in question
is <f> 2 z, where <f>.z is either
(*)•/!(+Si.*) »r wt>)./U<t>'.2.z).
it is not enough to have
zliir. O.^.^.zOtft.W,
but we must have, for each elementary <fj.
zliw . 0 : .„ ./! (<*>! 3 . z) Of l ( 0 ! 3 , w).
One inconvenient consequence is that, prirnd facie, an inductive property
must not be of the form
rli+ z . 4 >! z
or StVoiiiVR.^lS
or «« NC induct. <f >!«.
This is inconvenient, because often such properties are hereditary when <f>
alone is not, i.e. we may have
.r/f* z . <t >! z . zliw . 0 t>te . -rtf* w . <f >! w
when we do not have
<f>lz .zRw.O t '*.<t>lw,
and similarly in the other cases.
These considerations make it necessary to re-examine all inductive proofs.
In some cases they are still valid, in others they are easily rectified ; in still
others, the rectification is laborious, but it is always possible. The method of
rectification is explained in Appendix B to this volume.
There is, however, so far as wc can discover, no way by which our present
primitive propositions can be made adequate to Dedekindian and well-ordered
relations. The practical uses of Dedekindian relations depend upon *211*63—
*155)2, which lead to *214*3—*34, showing that the series of segments of a series
is Dedekindian. It is upon this that the theory of real numbers rests, real
numbers being defined as segments of the series of rationals. This subject is
dealt with in *310. If we were to regard ;is doubtful the proposition that the
series of real numbers is Dedekindian, analysis would collapse.
The proofs of this proposition in Principia Mathematica depend upon the
axiom of reducibility, since they depend upon *211*64, which asserts
X.CD ‘P t .0.s‘\eV‘Pe.
INTRODUCTION
xlv
For reasons explained above, if a is of the order of members of X, (a) ./a may
not imply /($‘X), because s*\ is a class of higher order than the members of
X. Thus although we have
s*X = 7>‘ViV‘X.
yet we cannot infer except when s‘X or s‘i J , M X is, for some special
reason, of the same order as the members of X. This will be the case when X
is finite, but not necessarily otherwise. Hence the theory of irrationals will
require reconstruction.
Exactly similar difficulties arise in regard to well-ordered series. The
theory of well-ordered series rests on the definition *250 01 :
Bord - P (Cl ex*C t P C a<minj») Df,
whence P € Bord . = : a C C*P . a ! a . . 3 ! a - P“a.
In making deductions, we constantly substitute for a some constructed class
of higher order than C*P. For instance, in *250122 we substitute for a the
class C*P r\ p‘P**(a r> C‘P ), which is in general of higher order than a. If this
substitution is illegitimate, we cannot prove that a class contained in C‘P
and having successors must have an immediate successor, without which the
theory of well-ordered series becomes impossible. This particular difficulty
might be overcome, but it is obvious that many important propositions must
collapse.
It might be possible to sacrifice infinite well-ordered series to logical
rigour, but the theory of real numbers is an integral part of ordinary mathe¬
matics, and can hardly be the object of a reasonable doubt. We arc therefore
justified in supposing that some logical axiom which is true will justify it.
The axiom required may be more restricted than the axiom of reducibility,
but, if so, it remains to be discovered.
The following are among the contributions to mathematical logic since the
publication of the first edition of Principia Mathematica.
D. Hilbert. Axiomatisches Dcnken, Mathematieche Annalcn, Vol. 78. Die logischen
Grundlagen dor Mathematik, ib. VoL 88. Neue Begriindung dcr Mathematik,
Abhandlungen aus dem mathemalischen Seminar der Uamburgischen UnivcrsitiU , 1922.
P. Bernays. Ueber Hilbert’s Gedanken zur Grundlegung dcr Arithmetik, Jahresbericht
der deutschen MathenuUiker- Vereinigung, Vol. 31.
H. Behmakn. Beitrage zur Algebra dor Logik. Malhematische Annalcn, Vol. 86.
L. CHWI8TBK. Ueber die Antinomien der Prinzipien der Mathematik, Mathematiichc
Zeitachrift, Vol. 14. The Theory of Constructive Types. Annalcs de la Societe
MathAmatique de Pologne, 1923. (Dr Chwistek has kindly allowed us to read in MS.
a longer work with the same title.)
H. Weyl. Dae KorUinuum , Veit, 1918. Ueber die neue Grundlagenkrise der Mathematik,
Mathematieche Zeitachrift , Vol. 10. Randbemerkungen zu Hauptproblemen der
Mathematik, Mathematieche Zeitechrift , VoL 20.
xlvi
INTRODUCTION
L K. J. Hkoiker. Begri.ml....g .lor Mengenlchn- .m»l.hang.g ynm iogwchon bat* dos
l.lo~on*l. DritU-n. Vcrkaadtlingt* <1. A'. Ah*k»it r. " *. .,*■/.a,.pm, Amstcr-
|9|8, 1!||9. Intmti.ini'tittcbc Mongonlol.r
liltr- Ycfrinigitinj, \ " 1 . 2 *.
\ T'JIKI uai m-Tar-ki. Sur le Icnnc |iriiiiitif de la Unique, Fundonunta ifothcmoticae,
Tom. IV. Sur les " truth-functions’* au mm.-* «le MM. Resell et WlnleheAil, i .
Tom. V. Sur quelqucs thcoriiiies qui equivalent .i Paxioinc «lu clmix. ib.
V. liKHNSTKis. Die Mengenlclire tieorg Cantors u...t dor Finitismus. do-
dott.*rhrn Mallmi'itiL'-.-- Yfrci*uju*g % Vol. 2*.
.1. KuNin. Vr-H-llagr* iIt LogiL\ Arithmrtil; n„d M.H.jod'h", \ e.t, 1014.
(_« | Lewis. .1 Surrri/ of SyiHlndie Logir, University of California, 1018.
II M. SiU KFKH. Total determinations of deductive systems with S|*cinl reference to the
Al-ehraof l.ogic. Dolinin of (hr A nor ion, Mathematic/1 Sorutj/, Vol. XVI. Irons. Arner.
Moth. Soc. Vol. XIV. pp. 4*1-488. The fir nr ral thron, of notation"! re/atmti,, < am-
I .ridge, Mass. 1021.
.1. f;. |\ NICOD. A reduction in the iminUr of the primitive pm|Kwitions of logic. /Voc.
Comb. rhi/. For. Vol. XIX.
L. WlTTGKNKTKIN. Tru<-to In* loyiro- PJ> ilosopk iem, Kcgnn Paul, 1022.
M. ScHuNWiNKBI.. Uehcr die Itaustcine dcr n.athenmtischen b*gik, Math. Annaten, \ »l. 02
u.nu.:
INTRODUCTION
stcc. JS'o: ___
<
*v. SPlN N ''',J
The mathematical logic which occupies Part I of the present work has
been constructed under the guidance of three different purposes. In the first
place, it aims at effecting the greatest possible analysis of the ideas with
which it deals and of the processes by which it conducts demonstralions,
and at diminishing to the utmost the number of the undefined ideas and
undemonstrafod propositions (called respectively primitive ideas and primitive
propositions) from which it starts. In the second place, it is framed with a
.view to the perfectly precise expression, in its symbols, of mathematical
propositions: to secure such expression, and to secure it in the simplest and
most convenient notation possible, is the chief motive in the choice of topics.
In the thin! place, the system is specially framed to solve the paradoxes
which, in recent years, have troubled students of symbolic logic and the
theory of aggregates; it is believed that the theory of types, as set forth in
what follows, leads both to the avoidance of contradictions, and to the
detection of the precise fallacy which has given rise to them.
Of the above three purposes, the first and third often compel us to adopt
methods, definitions, and notations which are more complicated or more
difficult than they would be if we had the second object alone in view. This
applies especially to the theory of descriptive expressions (#14 and *30) and
to the theory of classes and relations (*20 and *21). On these two points,
and to a lesser degree on others, it has been found necessary to make some
sacrifice of lucidity to correctness. The sacrifice is, however, in the main
only temporary: in each case, the notation ultimately adopted, though its
real meaning is very complicated, has an apparently simple meaning which,
except at certain crucial points, can without danger be substituted in
thought for the real meaning. It is therefore convenient, in a preliminary
explanation of the notation, to treat these apparently simple meanings as
primitive ideas, i.e. as ideas introduced without definition. When the notation
has grown more or less familiar, it is easier to follow the more complicated
explanations which we believe to be more correct. In the body of the work,
where it is necessary to adhere rigidly to the strict logical Older, the easier
order of development could not be adopted ; it is therefore given in the
Introduction. The explanations given in Chapter I of the Introduction are
such as place lucidity before correctness; the full explanations are partly
supplied in succeeding Chapters of the Introduction, partly given in the body
of the work.
The use of a symbolism, other than that of words, in all parts of the book
which aim at embodying strictly accurate demonstrative reasoning, has been
r&w i • 1
2
XTRODUCTIOX
forei’il on us by \ Ik* consistent pursuit of the- above three purposes. The
reasons for this extension of symbolism beyond the tamiliar regions of number
and allied idea' are many :
( I ) The ideas here employed are more abstract than those familiarly con¬
sidered in language. Accordingly there are no words which are used mainly
in the exact consistent -enso*. which are required here. Any use of words
would rc«|iiire unnatural limitations to their ordinary meanings, which would
be in fact more difficult to remember consistently than are the definitions of
entirely new symbols.
<-> The grammatical structure of language is adapted to a wide variety
o! usages. Thus it po>*e>^es no unique simplicity in representing the few
simple though highly abstract, processes and ideas arising in the deductive
trains o( reasoning employed here. In fact the very abstract simplicity of the
ideas of this work defeats language. Language can represent complex ideas
moreea-ily The proposition "a whale i^ big represents language at its best,
giving terse expression to a complicated fact; while the true analysis of "one
is a number " leads, in language, to an intolerable prolixity. Accordingly
terseness is gained by using a symbolism especially designed to represent the
ideas and processes of deduction which occur in this work.
(3) The adaptation of the rules of the symbolism to the processes of
deduction aids the intuition in regions too abstract for the imagination
readily to present to the mind the true relation between the ideas employed.
For various collocations of symbols become familiar as representing im¬
portant collocations of ideas; and in turn the possible relations—according
to the rules of the symbolism—between these collocations of symbols become
familiar, and these further collocations represent still more complicated
relations between the abstract ideas. And thus the mind is finally led to
construct trains of reasoning in regions of thought in which the imagination
would be entirely unable to sustain itself without symbolic help. Ordinary
language yields no such help. Its grammatical structure docs not represent
uniquely the relations between the ideas involved. Thus, "a whale is big”
and "one is a number both look alike, so that the eye gives no help to the
imagination.
(4) The terseness of the symbolism enables a whole proposition to be
represented to the eyesight as one whole, or at most in two or three parts
divided where the natural breaks, represented in the symbolism, occur. This
is a humble property, but is in fact very important in connection with the
advantages enumerated under the heading (3).
(5) The attainment of the first-mentioned object of this work, namely
the complete enumeration of all the ideas and steps in reasoning employed
INTRODUCTION
3
in mathematics, necessitates both terseness ami the presentation of each pro¬
position with the maximum of formality in a form as characteristic of itself
as possible.
Further light on the methods ami symbolism of this hook is thrown by a
slight consideration of the limits to their useful employment:
(a) Most mathematical investigation is concerned not with the analysis
ot the complete process of reasoning, but with the presentation of such an
abstract ot the proof as is sufficient to convince a properly instructed mind.
For such investigations the detailed presentation of the steps in reasoning is
of course unnecessary, provided that the detail is carried far enough to guard
against error. In this connection it may be remembered that the investiga¬
tions of Weierstrass and others of the same school have shown that, even in
the common topics of mathematical thought, much more detail is necessary
than previous generations of mathematicians had anticipated.
(/3) In proportion as the imagination works easily in any region of
thought, symbolism (except for the express purpose of analysis) becomes only
necessary us a convenient shorthand writing to register results obtained
without its help. It is a subsidiary object of this work to show that, with
the aid of symbolism, deductive reasoning can be extended to regions of
thought not usually supposed amenable to mathematical treatment. And
until the ideas of such branches of knowledge have become more familiar,
the detailed type of reasoning, which is also required for the analysis of the
steps, is appropriate to the investigation of the general truths concerning
these subjects.
CHAPTER I
PRELIMINARY EXPLANATION'S OF IDEAS AND NOTATIONS
The notation adopted in the present work is based upon that of Peano,
and t lie following explanations are to some extent modelled on those which
he prefixes to his Formulario Mathematico. His use of dots as brackets is
adopted, and so an* many of his symbols.
Variables. The idea of a variable, as it occurs in the present work, is
more general than that which is explicitly used in ordinary mathematics.
In ordinary mathematics, a variable generally stands for an undetermined
number or (piantity. In mathematical logic, any symbol whose meaning is not
determinate is called a variable, and the various determinations of which its
meaning is susceptible are called the values of the variable. The values may
be any set of entities, propositions, functions, classes or relations, according
to circumstances. If a statement is made about " Mr A and Mr B,” “ Mr A
and " Mr B ” are variables whose values are confined to men. A variable may
either have a conventionally-assigned range of values, or may (in the absence
of any indication of the range of values) have as the range of its values all
determinations which render the state-ment in which it occurs significant.
Thus when a text-book of logic asserts that "A is A." without any indication
as to what A may be. what is meant is that any statement of the form
"A is A is true. We may call a variable restricted when its values are
Coi ifine*I to some only of those of which it is capable: otherwise, we shall call
it unrestricted. 'Phils when an unrestricted variable occurs, it represents any
object such that the statement concerned can be made significantly (i.e. either
truly or falsely) concerning that object. For the purposes of logic, the
unrestricted variable is more convenient than the restricted variable, and we
shall always employ it. Wc shall find that the unrestricted variable is still
subject to limitations imposed by the manner of its occurrence, i.e. things
which can be said significantly concerning a proposition cannot be said
significantly concerning a class or a relation, and so on. But the limitations
to which tin* unrestricted variable is subject do not need to be explicitly
indicated, since they are the limits of significance of the statement in which
the variable occurs, and are therefore intrinsically determined by this state¬
ment. This will be more fully explained later*.
To sum up, the three salient facts connected with the use of the variable
arc: (1) that a variable is ambiguous in its denotation and accordingly undefined;
(2) that a variable preserves a recognizable identity in various occurrences
throughout the same context, so that many variables can occur together in the
• Cf. Chapter II of the Introduction.
CHAP. I]
TilK VAKIMtl.K
sjuuo context each with its separate identity: and (3* that either the imigo of
possible determinations of two variables may be the same, so that, a possible
determination of one variable is also a possible determination of the other, or
the ranges of two variables may be ditVerent. so that, if a possible determina¬
tion of one variable is given to the other, the resulting complete phrase is
meaningless instead of becoming a complete unambiguous proposition (true
or talse) as would be the case if all variables in it had been given any suitable
determinations.
The uses of various letters. Variables will be denoted by single letters, and
so will certain constants; but a letter which has once been assigned to a constant
by a definition must not afterwards be used to denote a variable. The small
letters of the ordinary alphabet will all be used for variables, except /> and s
after *40, in which constant meanings are assigned to these two letters. The
following capital letters will receive cons bint meanings: B, C. D, K. F, I and f.
Among small Greek letters, we shall give constant meanings to e, i and (at a
later stage) to rj, 0 and o>. Certain Greek capitals will from time to time be
introduced for constants, but Greek capitals will not be used for variables. Of
the remaining letters, p, q, r will be called propositional letters, and will stand
for variable propositions (except that, from *40 onwards, p must not be used
for a variable); f g, <f>, 0 and (until *33) F will be called functional
letters, and will be used for variable functions.
The small Greek letters not already mentioned will be used for variables
whose values arc classes, and will be referred to simply as Greek letters. Ordinary
capital letters not already mentioned will be used for variables whose values
are relations, and will be referred to simply as capital letters. Ordinary small
letters other thanp, q, r, s.f g will be used for variables whose values are not
known to be functions, classes, or relations; these letters will be referred to
simply as sviall Latin letters.
After the early part of the work, variable propositions and variable functions
will hardly ever occur. We shall then have three main kinds of variables:
variable classes, denoted by small Greek letters; variable relations, denoted by
capitals; and variables not given as necessarily classes or relations, which will
be denoted by small Latin letters.
In addition to this usage of small Greek letters for variable classes, capital
letters for variable relations, small Latin letters for variables of type wholly
undetermined by the context (these arise from the possibility of “systematic
ambiguity,” explained later in the explanations of the theory of types), the
reader need only remember that all letters represent variables, unless they have
been defined as constants in some previous place in the book. In general the
structure of the context determines the scope of the variables contained in it;
but the special indication of the nature of the variables employed, as here
proposed, saves considerable labour of thought.
G
INTKOIHCTION
[CHAP.
Thefill" l‘i mental functions of /impositions. An aggregation of propositions,
con«*idcied as wholes not neccs'aiily unambiguously determined, into a single
propition ..complex than it-* constituents, is a function irith propositions
ns urtfoments. The general idea of .such an aggregation of propositions, or ol
variables representing propitious, "ill not he employed in this work. But
there are four special cases which an- of fundamental importance, since all the
aggregations of sulMudinate propositions into one complex proposition which
occur in the sequel are formed out of them step by step.
They me (I) the Contradictory Function, ( 2 ) the Logical Sum, or Dis¬
junctive Function, (.'{> the Logical Product, or Conjunctive Function, ( 4 ) the
Implicative Function. These functions in the sons© in which they are required
in this work are not all independent: and it two of them arc taken as primitive
undefined ideas, the other two can 1 m- defined in terms of them. It is to some
extent—though not entirely—arbitrary as to which functions are taken jus
primitive. Simplicity of primitive ideas and symmetry of treatment seem to
be gained by taking the first two functions as primitive ideas.
The Contradictory Function with argument p. whore p is any proposition,
is the proposition which is the contradictory of />. that is. the proposition
asserting that p is not true. This is denotes I by ^ p. Thus ^ p is the
contradictory function with p as argument and means the negation of the
proposition />. It will also he referred to as the propition not-p. Thus ~p
means not-yr, which means the negation of p.
The Logical Sum is a propitionnl function with two arguments p and 7.
and is the proposition asserting p or 7 disjunctively, that is, asserting that at
least one of (he two yinmDy is true. This is denoted by p v 7. Thus /ivy is
the logical sum with y> and 7 as arguments. It is also called the logical sum of
p and 7. Accordingly y» v 7 means that at least p or 7 is true, not excluding the
case in which both are true.
'fhe Logical Product is a propositional function with two arguments p ami
7, and is the prop it ion asserting y> ami 7 conjunctively, that is, asserting that
both y> and 7 are true. This is denoted by p . 7, or—in order to make the dots
act as brackets in a way to be explained immediately—by p 17, or by p 7,
or by put/. Thus y>*7 is the logical product with p and 7 as arguments. It
is also called the logical product of y» and 7. Accordingly p . 7 means that both
y> and 7 are true. It is easily seen that this function can be defined in terms
of the two preceding functions. For when p and 7 are both true it must be
false that either ^p or is true. Hence in this book y>. 7 is merely a
shortened form of symbolism for
~ ~ p v ~ 7).
If any further idea attaches to the proposition “both p and 7 are true," it is
not required here.
>]
FUNCTIONS OF l*KOPOSITION'S
The Implicative Function is a pnqiositional function with t wo .-irgumnits
p and 7. and is tlu? proposition that cither not-y> or 7 is true, that is. it is the
proposition ^ pv 7. Thus if p is true, ^ p is false, and accordingly the only
alternative left by the pro{>osition pv q is that 7 is true. In other welds
it P and v 7 are both true, then 7 is true. In this sense the proposition
^ pv q will be quoted as stating that p implies 7. The idea contained in
this propositional function is so important that it requires a symbolism which
with direct simplicity represents tlu* proposition as connecting /» and 7
without the intervention of ^ p. Hut ** implies” as used here expresses
nothing else than the connection between p and 7 also expressed by the
disjunction “not -p or 7.” The symbol employed for "y> implies 7." i.e. for
“ ~pvq,’’ is “/O7.” This symbol may also be read "if />. then 7.” The
association of implication with the use of an apparent variable produces
an extension called "formal implication.” This is explained later: it is an
idea derivative from “ implication ” as here defined. When it is necessary
explicitly to discriminate " implication ” from " formal implication.” it is called
“material implication.” Thus “ material implication” is simply “implication'
as here defined. The process of inference, which in common usage is often
confused with implication, is explained immediately.
These four functions of propositions arc the fundamental constant (i.e.
definite) propositional functions with propositions as arguments, and all other
constant propositional functions with propositions as arguments, so far ns they
are required in the present work, are formed out of them by successive steps.
No variable propositional functions of this kind occur in this work.
Equivalence. The simplest example of the formation of a more complex
function of propositions by the use of these four fundamental forms is furnished
by “equivalence." Two propositions p and 7 are said to be “equivalent”
when p implies 7 and 7 implies p. This relation between p and 7 is denoted
by "p = 7.” Thus “ p = 7 ” stands for “(pDg).(gD p).’’ It is easily seen that
two propositions are equivalent when, and only when, they are both true or
are both false. Equivalence rises in the scale of importance when we come
to “ formal implication ” and thus to “ formal equivalence.” It must not
be supposed that two propositions which are equivalent are in any sense
identical or even remotely concerned with the same topic. Thus “Newton
was a man ” and “ the sun is hot ” arc equivalent as being both true, and
“ Newton was not a man ” and “ the sun is cold " are equivalent as being both
false. But here we have anticipated deductions which follow later from our
formal reasoning. Equivalence in its origin is merely mutual implication as
stated above.
Truth-values. The “ truth-value ” of a proposition is truth if it is true,
and falsehood if it is false*. It will be observed that the truth-values of
• This phrase is due to Frege.
8
INTRODUCTION
[chap.
p v 7 V • 7 * /' ^ 7- ~ />• /> = 7 depend only upon those of /> and 7. namely the
truth-value of "p v</' is truth if the truth-value of either p or 7 is truth,
and is falsehood otherwise ; that of"/#. 7 is truth if that of both p and 7 is
truth, and is falsehood otherwix*; that of “/O7 is truth if either that ot p
is falsehood or that of 7 is truth; that of* p is the opposite of that of p\
and that of " /> = 7 is truth if p and 7 have the same truth-value, and is
falsehood otherwise. Now the only ways in which propositions will occur
in the present work are ways derived from the above by combinations and
repetitions. Hence it i> easy to x-e (though it cannot be formally proved
except in each paiticul'.r case) that if a proposition /» occurs in any propo¬
sition / (/») which we shall ever have occasion to deal with, the truth-value
Ol /( p) will de|M*nd. lint upon the particular pro|xisition p, but only upon
its truth-value: i.e. if /> = 7, we shall have /< //) 2/(7). Thus whenever two
propositions are known to In- equivalent, either may be substituted for the
other in anv formula with which we shall have occasion to deal.
We may call a function J\ p) a " truth-function “ when its argument p is
a proposition, and the truth-value of /ip) depends only upon the truth-
value o| jt. Such functions are by no means the only common functions of
propositions. For example, ".1 believes p " is a function of p which will
vary its truth-value for diflorent arguments having the same truth-value:
A may believe one true pro|tosition without believing another, and may
believe one false proposition without believing another. Such functions
are not excluded from our consideration, and arc included in the scope of
any general propositions we may make about functions; but the particular
functions of propositions which we shall have occasion to construct or to con¬
sider explicitly are all truth-functions. This fact is closely connected with a
characteristic of mathematics, namely, that mathematics is always concerned
with extensions rather than intensions. The connection, if not now obvious, will
become more so when we have considered the theory of classes and relations.
Asxertion-xiyn. The sign "b,” called the "assertion-sign,” means that
what follows is asserted. It is required for distinguishing a complete propo¬
sition, which we assert, from any subordinate propositions contained in it but
not asserted. I11 ordinary written language a sentence contained between full
stops denotes an asserted proposition, and if it is false the book is in error.
The sign " b” prefixed to a proposition serves this same purpose in our sym¬
bolism. For example, if “b(/Op)’’ occurs, it is to be taken as a complete
assertion convicting the authors of error unless the proposition “pOp" is
true (as it is). Also a proposition stated in symbols without this sign “ b ”
prefixed is not asserted, aud is merely put forward for consideration, or as a
subordinate part of an asserted proposition.
Inference. The process of inference is as follows: a proposition “/>” is
asserted, and a proposition "p implies 7 ” is asserted, and then as a sequel
ASSKRTION AXI> IXFKRKNCK
l \ ASSKRTION AXI) IXFKRKXCK \)
the proposition “7” is assorted. Tlu* tins! in inference is tin* belief t hat, if tin*
two former assertions are not in error, the Hnal assertion is not. in error.
Accordingly whenever, in symbols, where /» ami 7 have of course special
determinations,
“!-/>•* ami " I- (/> D 7) "
have occurred, then t, \ m q" will occur if it is desired to put. it on record. The
process of the inference cannot be reduced to symbols. Its sole record is the
occurrence of" h 7.’* It is of course convenient, even at the risk of repetition,
to write "b/>” and “ b (/> D 7)" in close juxtaposition before proceeding to
“ h 7 ” as the result of an inference. When this is to be done, for the sake of
drawing attention to the inference which is being made, we shall write
instead
which is to be considered as a mere abbreviation of the threefold statement
“ h p " and " h (p D 7) ’’ and M h 7.'*
Thus may be read “p. therefore 7," being in fact the same
abbreviation, essentially, as this is; for " p, therefore q" does not explicitly
state, what is part of its meaning, that p implies 7. An inference is the
dropping of a true premiss ^ it is the dissolution of an implication.
The use of dots. Dots on the line of the symbols have two uses, one to
bracket off propositions, the other to indicate the logical product of two
propositions. Dots immediately preceded or followed by “v " or ‘O” or
“ — ” or - h/* or by “<*) ” “(or. y),” "<*. y , z)'\ .. or “(a*).” « (g*. y)f “(g*r, y. z)’\..
or “[(lx)(ft>x)]” or “[i£‘y]” or analogous expressions, serve to bracket ofF a
proposition; dots occurring otherwise serve to mark a logical product. The
general principle is that a larger number of dots indicates an outside bracket,
a smaller number indicates an inside bracket. The exact rule as to the scope
ot the bracket indicated by dots is arrived at by dividing the occurrences of
dots into three groups which we will name I, II, and III. Group I consists of
dots adjoining a sign of implication (D) or of equivalence (=) or of disjunction
(v) or of equality by definition (= DO- Group II consists of dots following
brackets indicative of an apparent variable, such as (x) or (x, y) or (gx) or
(a*»y) or [(?#) or analogous expressions*. Group III consists of dots
which stand between propositions in order to indicate a logical product.
Group I is of greater force than Group II, and Group II than Group III.
The scope of the bracket indicated by any collection of dots extends backwards
or forwards beyond any smaller number of dots, or any equal number from a
group of less force, until we reach either the end of the asserted proposition
or a greater number of dots or an equal number belonging to a group of
equal or superior force. Dots indicating a logical product have a scope which
works both backwards and forwards; other dots only work away from the
• The meaning of these expressions will be explained later, and examples of the nee of dots in
oonneotion with them will be given on pp. 16, 17.
10
INTRODUCTION
[chap.
adjacent sign of di*»junction, implication. or equivalence, or forward from tlie
adjacent symbol of one of the other kinds enumerated in Group II.
.Some examples will serve to illustrate the use of dots.
' /> v y. 3 .7 v// means the proposition " * p or y implies '7 or />.’ ” When
we tissert this proposition, instead of merely considering it, we write
“ h : // v 7 . 3 . 7 v />.
where the two dots after the assertion-sign show that what is asserted is the
whole of what follows the assoition-sign. since there are not. as many as two
dots anywhere else. If we had written "p : v : 7 . 3 . 7 v//,” that would mean
tie- pro|M»iti.»n - either/> i^ true, or 7 implies 7 or//."' If we wished to assert
this, we should have to put three dots after the assertion-sign. If we had
written "p v 7 . 3 . 7 : v : />. that would mean the proposition " either '/> or 7'
implies y. or // is true.'" The forms p . v . 7 . 3 . y v />" and * // v 7 . 3 . 7 . v . />’’
have 110 ineaiiiiio
’ /O7. 3 : 7 3 r . 3 . //3 r" will mean “ if /> implies 7, then if 7 implies /*,
// implies r." If we wi*h to assert this (which is true) we write
■* h :. // 3 7.3 :7 3 /•. 3 . // 3 r ."
Again •// 3 y . 3 . y 3 /•: 3 .// 3 /•'■ will mean ** if ‘y> implies y* implies y
implies /•,' then // implies r. This is in general untrue. (Observe that
/> 3 y is sometimes mo't conveniently read as ' p implies 7," and sometimes
as
it p, then 7.") "//3 y . y 3 /•. 3 .// 3 /•' will mean "if /> implies 7, and
/ implies #•, then // implies #•." In this formula, the first dot indicates a logical
product; hence the scope of the second dot extends backwards to the begin¬
ning of the pro|K/sition. ' p 3 y : y 3 r . 3 . /> 3 r" will menu "/> implies y; and
i! y implies r, then p implies r." (This is not true in general.) Here the two
dots indicate a logical product; since two dots do not occur anywhere else, the
seopo of these two dots extends backwards to the beginning of the proposition,
and forwards to the end.
" p v 7. 3 p , v . 7 3 r : 3 ,p v r" will mean “ if either p or 7 is true, then
il either p or *y implies r’ is true, it ft/llows that either p or r is true.” If
this is to lie asserted, wc must put four dots after the assertion-sign, thus:
" h ;; y» v y . 3 ;. />. v • y 3 r i 3 .yj v r."
( I'liis proposition is proved in the body of the work; it is *2 73 .) If we wish
to assert (what is equivalent to the above) tbe proposition: "if either p or 7
is true, and either p or *y implies r' is true, then either p or r is true," we
write
" b p v 7 : p . v . 7 3 r : 3 . p v r.”
Here the first pair of dots indicates a logical product, while the second pair
docs not. Thus the scope of the second pair of dots passes over the first pair,
ami back until we reach the three dots after the assertion-sign.
Other uses of dots follow the same principles, and will be explained as
they are introduced. In reading a proposition, the dots should be noticed
>1
PKFIXITIOKS
II
tij-st-, as they show its structmv. In a proposition containing s«-\vrnl signs <<l
implication or equivalence. the one with the greatest nniuher of «l«*ts Im-I'oiv
or alter it is the principal one: everythin*; that goes he lore this one is stated
l>y the proposition to imply or he equivalent to everyth in** that cmnes alter it.
Definitions. A definition is a declaration that a certain newlv-i lit rod need
symbol or combination of symbols is to mean the same as a certain other
combination of symbols of which the meaning i> already known. Oi. if the
defining combination of symbols is one which only acquires meaning when
combined in a suitable manner with other symbols*. what is meant is that
any combination of symbols in which the newly-defined symbol or combination
of symbols occurs is to have that meaning (if any) which results from substi¬
tuting the defining combination of symbols for the newly-defined symbol or
combination of symbols wherever the latter occurs. We will give the names
of definiendum and definiens respectively to what is defined and to that which
it is defined as meaning. We express a definition by putting the definiendum
to the left and the definiens to the right, with the sign ** —” between.and the
letters “Df” to the right of the definiens. It is to be understood that the
sign "=a” and the letters “Df” are to be regarded as together forming one
symbol. The sign " — ” without the letters "Df ” will have a different meaning,
to be explained shortly.
An example of a definition is
Df.
It is to be observed that a definition is, strictly speaking, no part of the
subject in which it occurs. For a definition is concerned wholly with the
symbols, not with what they symbolise. Moreover it is not true or false,
being the expression of a volition, not of a proposition. (For this reason,
definitions are not preceded by the assertion-sign.) Theoretically, it is
unnecessary ever to give a definition: we might always use the definiens
instead, and thus wholly dispense with the definiendum. Thus although we
employ definitions and do not define “definition,'’ yet "definition” does not
appear among our primitive ideas, because the definitions are no part of our
subject, but are, strictly speaking, mere typographical conveniences. Prac¬
tically, of course, if we introduced no definitions, our formulae would very soon
become so lengthy as to be unmanageable; but theoretically, all definitions are
superfluous.
In spite of the fact that definitions are theoretically superfluous, it is
nevertheless true that they often convey more important information than is
contained in the propositions in which they are used. This arises from two
causes. First, a definition usually implies that the definiens is worthy of
careful consideration. Hence the collection of definitions embodies our choice
'•This case will be fully considered in Chapter III of the Introduction. It need not further
concern ue at present.
12
INTRODUCTION
... ..W^VV.4».> [CHAP.
«»t subjects ami our judgment as to what is most important. Secondly, when
what is defined is (sis often occurs) something already familiar, such ns cardinal
or ordinal numbers, the definition contains an analysis of a cotnniou idea, and
may therefore expre->notable advance. (’antor’s definition of the continuum
illustrate* this: his definition amounts to the statement that what he is de¬
fining is the object which has the proju-rlies commonly associated with the
word continuum, though what precisely constitutes these properties had
Hot before been known. In such cases, a definition is a " making definite it
gives definiteness m, idea which had previously been mole or less vague.
I'oi- tlu se reasons, it will be found, in what follows, that the definitions
aie what i* mo>l important, and what most deserves the reader’s prolonged
attention.
Some impel t int remarks must be made respecting the variables occurring
in the i/i'liiurn.s ami the ilcjinienihini. Hut these will la- deferred till the
icu ion of an "apparent variable' lias been introduced, when the subject can be
considered as a whole.
Sum,nun/ of U,uj statements. There are. in the above, three primi¬
tive ideas which are not ••defined’* but only descriptively explained. Their
piimitivcncss is only relative to our exposition of logical connection and is
not absolute; though of course such an exposition gains in importance ac-
cording to the simplicity of its primitive ideas. These ideas are symbolised
by "'>-/> and /»v 7." and by ,, K* prefixed to a proposition.
Three definitions have been introduced:
/>•'/. = pw ^1/) I)f.
. = ,^p V iy Df.
y# = #y . =» .yOr/.ry Dyj ])f.
Primitive propositions. Some propositions must be assumed without proof,
since all inference proceeds from propositions previously asserted. These, as
far as they concern the functions of propositions mentioned above, will be
found stated in *1. where the formal and continuous exposition of the subject
commences. Such propositions will be called “primitive propositions.” These,
like the primitive ideas, are to some extent a matter of arbitrary choice; though,
as in the previous ease, a logical system grows in importance according as the
primitive propositions are few and simple. It will be found that owing to the
weakness of the imagination in dealing with simple abstract ideas no very
great stress can be laid upon their obviousness. They are obvious to the in¬
structed mind, but then so are many propositions which cannot be quite true,
:ls being disproved by their contradictory consequences. The proof of a logical
system is its adequacy and its coherence. That is: (1) the system must embrace
among its deductions all those propositions which we believe to be true and
capable of deduction from logical premisses alone, though possibly they may
I]
PRIMITIVE PROPOSITIONS
13
require some slight, limitation in the form of an increased stringency of enun¬
ciation; and (2) the system must lead to no contradictions, namely in pursuing
our inferences we must never be led to assort both /» and not-/>. i.e. both " h . p"
mul "b . ~/>” cannot legitimately appear.
The following are the primitive propositions employed in the calculus of
propositions. The letters “Pp" stand for ‘•primitive proposition."
( 1 ) Anything implied by a true premiss is true Pp.
This is the rule which justifies inference.
(2) h:pvp.D./> Pp.
i.e. if/) or p is true, then p is true.
( 3 ) b : q . D . p v q Pp.
i.e. if q is true, then p or q is true.
( 4 ) I ~:pvq.D.qvp Pp,
i.e. if p or q is true, then q or p is true.
( 5 ) b :pv(qvr). D .q v(pvr) Pp,
t.e. if either p is true or “q or r” is true, then either 7 is true or “p or r” is
true.
(<») b:.qDr.D:pvq.D.pvr Pp,
i.e. if q implies r, then "p or 7” implies ”p or r.”
( 7 ) Besides the above primitive propositions, we require a primitive pro¬
position called "the axiom of identification of real variables.” When we have
separately asserted two different functions of x, where x is undetermined, ii
is often important to know whether we can identify the x in one assertion
with the x in the other. This will be the case—so our axiom allows us to
infer if both assertions present x as the argument to some one function, that
is to say, if (f>x is a constituent in both assertions (whatever propositional func¬
tion 0 may be), or, more generally, if <f>(x, y, z ,...) is a constituent in one
assertion,and <t> ( x , u, v ,...) is a constituent in the other. This axiom introduces
notions which have not yet been explained; fora fuller account, see the remarks
accompanying *3 03 , * 17 , *1 71 , and *1 72 (which is the statement of this
axiom) in the body of the work, as well as the explanation of propositional
functions and ambiguous assertion to be given shortly.
Some simple propositions. In addition to the primitive propositions we
have already mentioned, the following are among the most important of the
elementary properties of propositions appearing among the deductions.
The law of excluded middle:
b .p v~p.
This is * 2-11 below. We shall indicate in brackets the numbers given to the
following propositions in the body of the work.
The law of contradiction (* 3 * 24 ):
h . (p . ~/>).
*
II
INTRODUCTION'
[CHAP.
The law of double negation (* 413 ):
I" • /> = M •>-/>).
I lie principle ot transjtontion, i.e. "if /> implies 7, then not-9 implies not-/),"
•in‘1 'ice versa: this principle lias various forms, namely
( *4 I ) I* : /O. 'N «y D v
<*4-11) b :/» =#y. =
I*4' 14) I- :./>. */. D ~r. D . v«y,
as well as others which are variants of these.
The law of tautology, in the two forms:
(*4 24 ) I- :y#. 5 ./»./#,
(*4 2 .>) h :/>. = ./) v/>.
P is true is equivalent to ‘ /j is true and p is true." as well as to "p is true
or p is true." Fr.a formal point of view, it is through the law of tautology
and Its conseipiences that the algebra of logic is chielly distinguished from
ordinary algebra.
The law of absorption:
(* 471 ) h.y)D)y.s:yi. = .yj.(y,
' *• > implies 7 " i> equivalent to "p is equivalent to p . 7." This is called the
law of absorption In-cause it shows that the factor 7 in the product is absorbed
by the fact or /#, it /1 implies 7. This principle enables us to replace an impli¬
cation (/O7) by an equivalence (/;. = ./>.7) whenever it is convenient to
llo St I
An analogous and very important principle is the following:
(*4 73 ) h 7 . D :/). = ./». 7.
Logical addition and multiplication of propositions obey the associative
and commutative laws, and the distributive law in two forms, namely
<*4 4 ) h :./). 7 v r. = : p . 7 . v . />. r.
(*4 41) h y». v . 7 , /•: = ; y/ v 7 . yj v r.
The second of these distinguishes the relations of logical addition and multi¬
plication from those of arithmetical addition and multiplication.
Propositional functions. Let <f>x be a statement containing a variable x
ami such that it becomes a proposition when x is given any fixed determined
meaning. I hen 0.# is called a ••propositional function”; it is not a proposition,
since owing to the ambiguity of .r it really makes no assertion at all. Thus
“j- is hurt really makes no assertion at all. till we have settled who x is. Yet
owing to the individuality retained by the ambiguous variable x, it is an am¬
biguous example from the collection of propositions arrived at by giving all
possible determinations to .r in “x is hurt” which yield a proposition, true or
lalse. Also it \r is hurt” and "y is hurt" occur in the same context, where y is
PROPOSITIONAL FUNCTION'S
. m>rwi i iurt.ii, pi'Krrioxs i~
i •)
another variable, then ncwnliug to the .h-U-imi,unions giv,... :1 ,„| ,, ,|„. v
can be settled to be (possibly) the same ,motion or (possibly) ,lilt,
propositions. But apart from somedeterminatio., give. to.,a,id , A Ma y relain
in that context, their ambiguous .lirterentiation. Thus "... is hint" is an am¬
biguous ••yalue" of a propositional lunetion. When we wish to sin-alt ol 'ihe
propositional function corresponding to is hurt" we shall write ".?■ is hurt "
Inis a is hurt is the propos.tional lunetion and is hurt" is an nmbig,..
value ol that function. Accordingly though ".r is hurt" and -,/ is hurt" •/„„
Sa '" e co '" e,t can b '- distinguished, is hurt" and ' is hurt" convey
no distinction of meaning at all More generally. 0., is an ambiguous value ol
the propositional function 0.7, and when a definite signification „ is substituted
ror .r, <pa is nil unambiguous value of 0. 7.
Propositional functions are the fundamental kind from which the more usual
kinds of function, such as ••sin.r" or "log.," or "the father of are derive,I.
I esc derivative functions are considered later, and are called "descriptive
functions. The functions of propositions considered above me a particular
case of propositional functions. 1
The range of values and total variation. Thus corresponding to anv propo¬
rtional function 0.?, there is a range, or collection, of values, consisting of all
the propositions (true or false) which can be obtained by giving every possible
determmation to * in 0a-. A value of a- for which 0., is true will bo said to
satisfy 0.?. Now in respect to the truth or falsehood of propositions of this
range three important cases must be noted and symbolised. These cases are
given by three propositions of which one at least must be true. Either ( 1 ) all
propositions of the range are true, or (2) some propositions of the range are
( 1 ) "? P r ^P° s,tion of the ran S e is The statement (1) is symbolised
*>y (*).<K and ( 2 ) is symbolised by “(a*). 0 *” No definition is given of
hese two symbols, which accordingly embody two new primitive ideas in our
system. The symbol "(ar) . 0x” may be read “ 0 * always.” or “0* is always true,”
or 0 * ,s true for all possible values of The symbol “( 3 *). 0 *” may be
read “there exists an a: for which <f>x is true ” or “there exists an .r satisfying
<PA and thus conforms to the natural form of the expression of thought.
Proposition ( 3 ) can be expressed in terms of the fundamental ideas now on
hand. In order to do this, note that - ~ <f>x ” stands for the contradictory of 0*.
Accordingly ~ 05 is another propositional function such that each value of 05
contradicts a value of ~ 05 , and vice versa. Hence “(*). ~ 0*” symbolises the
proposition that every value of 05 is untrue. This is number ( 3 ) as stated above.
. Ifc ls aQ obvious error, though one easy to commit, to assume that cases
U) and ( 3 ) are each other’s contradictories. The symbolism exposes this fallacy
at once, for (1) is ( x).<px , and ( 3 ) is (x).~<f>x, while the contradictory of (1) is
~ K®). 0 a;(. For the sake of brevity of symbolism a definition is made, namely
~ (;r). <f>x . = . ~ {(*) . 0 x j Df.
16
INTRODUCTION
[CHAP.
Definit ions of which the object is to gain some trivial advantage in brevity
by a slight adjustment of symbols will be said to be of “merely symbolic import,"
in contradistinction to those definitions which invite consideration of an im¬
portant idea.
The proposition (x). <f>r is called the “total variation ‘ of the function tjjr.
For reasons which will lie explained in Chapter II. we do not take negation
as a primitive idea when proposition.** of the forms (x). <f>j and (3x) .«/>.» are
concerned, but we define the negation of (x).^x, i.e. of "$x is always true," as
being ' <£ / is sometimes false, i.e. "(W). ^ <f >>and similarly we define the
negation of (j|x). as being (x). Thus wc put
^ ;(x) . <Jm ■] . ss . ($|.r) . <f>.r l)f,
^ . i.r) . ^ (f,.r |)f.
In like manner wc define a disjunction in which 011c of the propositions is
of the form ' (.r). <£./ or * (gx). tf).r in terms of a disjunction of propositions
not of this form, putting
(x). <p.r . v . y »: ■.(d.f'V/) Df,
i.e. “either <f>.< is always true, or y* is true" is to mean '“<f>.r or/>’ is always true."
with similar definition^ in other cases. This subject is resumed in Chapter II,
and in *1) in the body of the work.
Apparent variables. The symbol "(x). <f>r ■" denotes one definite proposition,
and there is no distinction in meaning between “(.r). <f>.i •" and "(//)• <f>‘/ when
they occur in the same context. Thus the “x * in "(x). is not an ambiguous
constituent of any expression in which "(x). ^r" occurs; and such an ex¬
pression does not cease to convey a determinate meaning by reason of the
ambiguity of the x in the The symbol “(.r). <f>.r" has some analogy to
the symbol
•. .6
I dx
for definite integration, since in neither case is the expression a function of#.
The range of x in “(x).<£./ or " (3*r). <t*x“ extends over the complete
field of the values of x for which has meaning, and accordingly the
meaning of ”(x).<f>x" or "(H r )>0- r " involves the supposition that such n field
is determinate. The x which occurs in “(x).</>x” or “(%jx). «/»x" is called
(following Penno) an “apparent variable.” It follows from the meaning of
“ • </>•' that the x in this expression is also an apparent variable. A
pro|x>sit.ion in which x occurs as an apparent variable is not a function of x.
Thus e.g. “(x).x = x" will mean “everything is equal to itself.” This is an
absolute constant, not a function of a variable x. This is why the x is called
an apparent variable in such cases.
Besides the "range" of x in “(x).<£x" or which is the field
of the values that x may have, we shall speak of the "scope" of x, meaning
1 ]
APPARENT VAR IA HI.ES
17
the function ot which all values or some value are being affirmed. If we are
asserting all values (or some value) of "*r." " is the scope of.r; if we are
asserting all values (or some value) of "</uO p." "<f>.cDp" is the scope of .r;
if we are asserting all values (or some value) of "<f>x D yjrx,” “<f>.v D y/r.r" will be
the scope of x, and so on. The scope of x is indicated by the number of dots
after the "(x) ’ or "(gpr)”; that is to say, the scope extends forwards until
we reach an equal number ot dots not indicating a logical product, or a greater
number indicating a logical product, or the end of the asserted proposition in
which the “(x)” or “( 3 - 0 ” occurs, whichever of these happens lirst*. Thus c.g.
*‘(.r) : <f>x . D . +x”
will mean “ <f>x always implies ^x,” but
.D.*x"
will mean “if <f>x is always true, then yfrx is true for the argument x.”
Note that in the proposition
(x) . <f>x . D . yjrx
the two x’s have no connection with each other. Since only one dot follows
the x in brackets, the scope of the first x is limited to the u <f>x'’ immediately
following the x in brackets. It usually conduces to clearness to write
(x) . <f>x . D . yjry
rather than (x) . <f>x . D . yfrx,
since the use of different letters emphasises the absence of connection between
the two variables; but there is no logical necessity to use different letters,
and it is sometimes convenient to use the same letter.
Ambiguous assertion and the real variable. Any value “«/»x” of the function
<f& can be asserted. Such au assertion of an ambiguous member of the values
of <f,ai is symbolised by
“Ktf.x.”
Ambiguous assertion of this kind is ^primitive idea, which cannot be defined
in terms of the assertion of propositions. This primitive idea is the one which
embodies the use of the variable. Apart from ambiguous assertion, the con¬
sideration of “ <f>x," which is an ambiguous member of the values of <px, would
be of little consequence. When we are considering or asserting “<£x,” the
variable x is called a “ real variable.” Take, for example, the law of excluded
middle in the form which it has in traditional formal logic:
“ a is either b or not 6.”
Here a and b are real variables: as they vary, different propositions are
expressed, though all of them are true. While a and b are undetermined, as in
the above enunciation, no one definite proposition is asserted, but what is
asserted is any value of the propositional function in question. This can only
• This agrees with the roles for the occurrences of dots of the type of Group II as explained
above, pp. 9 and 10.
18
IXTRODt'CTION
[CHAP.
be legitimately asserted if. whatever value may be chosen, that value is true,
i.e. if all the value*! arc true. Thus the above form of the law of excluded
middle is equivalent to
“ (*/. h ). a is either b or not
i.e. t.» “ it is always true that a is either b or not. b.' But these two, though
equivalent, arc not identical, and we shall find it necessary to keep them
distinguished.
When we assert something containing a real variable, as in e.g.
we are asserting ##«*/value of a propositional function. When we assert some¬
thing containing an apparent variable, as in
or “K(5 J/).jbi;'
we are asserting, in the first case all values, in the second case some value
(undetermined), of the propositional function in question. It is plain that
we can only legitimately assort 11 unit value" if till values are true; for other¬
wise. since the value of the variable remains to be determined, it might he so
determined as to give a false proposition. Thus in the above instance, since
we have
h . x ss .r
we may infer b . (x ). x = x.
And generally, given an assertion containing a real variable x. we may trans¬
form the real variable into an ap|urcnt one by placing the x in brackets at
the beginning, followed by as many dots as there arc after the assertion-sign.
When we assert something containing a real variable, wc cannot strictly
be said to be asserting a proposition, for we only obtain a definite proposition
by assigning a value to the variable, and then our assertion only applies to
one definite case, so that it has not at all the same force as before. When what
we assert contains a real variable, we arc asserting n wholly undetermined one
of all the propositions that result from giving various values to the variable.
It will be convenient to speak of such assertions as asserting a propositional
function. The ordinary formulae of mathematics contain such assertions; for
example
“sin 3 .r + cos’1”
does not assert this or that particular case of the formula, nor does it assert,
that the formula holds for all possible values of x, though it is equivalent to
this latter assertion; it simply asserts that the formula holds, leaving x wholly
undetermined; and it is able to do this legitimately, because, however x may
be determined, a true proposition results.
Although an assertion containing a real variable does not, in strictuess,
us>*ert a proposition, yet it will be spoketi of as asserting a proposition except
when the nature of the ambiguous assertion involved is under discussion.
Definition and real variables. When the dejiniens contains one or more
real variables, the deriniendnin must also contain them. For in this case we
have a tunction ot the real variables, and the deti niemlu in must have the same
meaning as the definiens for all values of these variables, which requires that
the symbol which is the definiendnm should contain the letters representing
the real variables. This rule is not always observed by mathematicians, ami
its infringement has sometimes caused important- confusions of thought,
notably in geometry and the philosophy of space.
In the definitions given above of "p . ij" and " p D ij" and "p = ij ," p and 7
are real variables, and therefore appear on both sides of the definition. In
the definition ot j(x) . 4 >r\ " only the function considered, namely </>?, is a
real variable; thus so far ns concerns the rule in <piestion, .#• need not appear
on the left. But when a real variable is a function, it is necessary to indicate
how the argument is to be supplied, and therefore there are objections to
omitting an apparent variable where (as in the case before us) this is the
argument to the function which is the real variable. This appears more
plainly if, instead of a general function <£.?, we take some particular function,
say and consider the definition of ^ |(.r) . x = «J. Our definition gives
~ {(x) . x — a}. — . (gx) . ~ (x = a) Df.
But if we had adopted a notation in which the ambiguous value "x = r/,"
containing the apparent variable x, did not occur in the definiendum, we
should have had to construct a notation employing the function itself, namely
“£ = a." This does not involve an apparent variable, but would be clumsy in
practice. In fact we have found it convenient and possible—except in the
explanatory portions—to keep the explicit use of symbols of the type
either as constants [e.g. 5 = a) or as real variables, almost entirely out of this
work.
Propositions connecting real and apparent variables. The most important
propositions connecting real and apparent variables are the following:
( 1 ) “When a propositional function can be asserted, so can the proposition
that all values of the function are true.” More briefly, if less exactly, “ what
holds of any, however chosen, holds of all." This translates itself into the rule
that when a real variable occurs in an assertion, we may turn it into an apparent
variable by putting the letter representing it in brackets immediately after
the assertion-sign.
( 2 ) “ What holds of all, holds of any," i.e.
h : (x) . <f>x . D . tf>y.
This states “if <f>x is always true, then <f>y is true.”
( 3 ) “ If (^y i 8 true, then <f>x is sometimes true,” i.e.
\-z<f>y.D. (gx). <*>x.
ST
y Acc. J\'o; _ __
20
INTRODUCTION
[CHAP.
Ad asserted proposition of the form " (gx). <f>x ” expresses an "existence-
theorem,” namely " there exist* an x for which <f>.r is true.” The above pro¬
position gives what is in practice the only way of proving existence-theorems:
we always have to find some particular »/ for which 4 >y holds, and thence to
infer " (g.r). If we were to assume what is called the multiplicative
axiom, or the equivalent axiom enunciated by Zermelo, that would, in au
important class of cases, give an existence-theorem where no particular instance
of its truth can he found.
In virtue of '• Y : (x). $x. D . <f>y" and " b : 4 >y . D . (gx). $x," we have
* 1 -: (./). (f>r . D . (gx). tfyr” i.e. *' what is always true is sometimes true." This
would not he the case if nothing existed; thus our assumptions contain the
assumption that there is something. This is involved in the principle that
what holds of all, holds of any; for this would not he true if there were no
" any/*
( 4 ) "If <f)j is always true, and yjr.r is always true, then *<£x .yjrx ’ is always
true," i.e.
h !. (x) . 4 >.r : (x) . yfrx : D . (x) . <t>j .
(This requires that <f) and yfr should be functions which take arguments of the
same ty/ie. We shall explain this requirement at a later stage.) The converse
also holds; i.e. we have
h (x) . 4 >.r . yfrx.D : (x) . <f>j : (.r). yjrx.
It is to some extent optional which of the prop>sitions connecting real
and apparent variables are taken as primitive propositions. The primitive
propositions assumed, on this subject, in the body of the work (* 9 ), are the
following:
(1) b:$x.D.(g
( 2 ) V :<t>j-v<f>y. D.(g*).<£-,
i.e. if either <fu• is true, or <f>y is true, then (g*). <f>c is true. (On the necessity
for this primitive proposition, see remarks on *911 in the body of the work.)
( 3 ) If we can assert tfty, where y is a real variable, then we can assert
(x). <ftx\ i.e. what holds of any, however chosen, holds of all.
Formal implication and formal equivalence. When an implication, say
tf>x .D . \frx, is said to hold always, i.e. when (x): tf>x . D . yjr.r, we shall say that
<f>x formally implies yjr.c; and propositions of the form “ (x) : <f>x. D . >/rx” will
be said to state formal implications. In the usual instances of implication,
such as " ‘ Socrates is a man * implies * Socrates is mortal/ ” we have a propo¬
sition of the form " <f>x . 0 . yf/x " in a case in which " (x): <f>x . D . yfrx ” is true.
In such a case, we feel the implication as a particular case of a formal impli¬
cation. Thus it has come about that implications which are not particular
cases of formal implications have not been regarded as implications at all.
There is also a practical ground for the neglect of such implications, for, speaking
I]
FORMAL IMPLICATION
*21
generally, they can only be knoini when it is already known either that their
hypothesis is false or that their conclusion is true; and in neither of these
cases do they serve to make us know the conclusion,since in the first, case the
conclusion need not be true, and in the second it is known already. Thus
such implications do not serve the purpose for which implications are chiefly
useful, namely that of making us know, by deduction, conclusions of which we
were previously ignorant. Formal implications, on the contrary, do serve this
purpose, owing to the psychological fact that we often know "(.r):</>./•.D.x/r.r'*
and <t>i/, in cases where yjry (which follows from these premisses) cannot easily
be known directly.
These reasons, though they do not warrant the complete neglect of impli¬
cations that are not instances of formal implications, are reasons which make
formal implication very important. A formal implication states that, for all
possible values of .r, if the hypothesis <f >.r is true, the conclusion yfrx is true.
Since " <fxc. D . yfrx" will always be true when <f>.v is false, it is only the values
of x that make tf>x true that are important in a formal implication ; what is
effectively stated is that, for all these values, yfrx is true. Thus propositions
of the form "all a is / 9 ,” “ no a is / 9 " state formal implications, since the first
(as appears by what has just been said) states
(x) : x is an a . D . x is a / 3 ,
while the second states
(x) : x is an a . D . x is not a / 3 .
And any formal implication " (x) : <f>x . D . yjrx " may be interpreted as : “All
values of x which satisfy* <f>x satisfy yfrx while the formal implication
“ 0*0 : <f>x . D ■ yfrx ” may be interpreted as: “ No values of x which satisfy <f>x
satisfy ^x. M
We have similarly for "some a is /9 ” the formula
(gx) . x is an a . x is a / 9 ,
and for " some a is not /3 ” the formula
(gx) . x is an a . x is not a / 9 .
Two functions <f>x, yfrx are called formally equivalent when each always
implies the other, i.e. when
(x) : <f>x . = . yjrx,
and a proposition of this form is called a formal equivalence. In virtue of
what was said about truth-values, if <f>x and yjrx are formally equivalent, either
may replace the other in any truth-function. Hence for all the purposes of
mathematics or of the present work, $2 may replace y\r 2 or vice versa in any
proposition with which we shall be concerned. Now to say that <f>x and yjrx
are formally equivalent is the same thing as to say that <f >2 and yfr 2 have the
8 ame extension, i.e. that any value of x which satisfies either satisfies the other.
• ▲ value of x is said to tatiafy or <p2 when *x is true for that value of x.
22
INTRODUCTION
[CHAP.
Thus wlu-nevt-r a constant function occurs in our work, the truth-value of the
proposition in which it occurs depends only upon the extension of the function.
A proportion containing a function an«l having this property (i.e. that its
truth-value depends only ujion the extension of (f> 3 ) will be called an exten-
sional function of <f> 2 . Thus the functions of functions with which we shall be
specially concerned will all lie exlensional functions of functions.
What has just la-on said explains the connection (noted above) between
the fact that the functions of profit ions with which mathematics is specially
concerned are all truth-functions and the fact that mat hematics is concerned
with extensions rather than intensions.
(Urnlenient abbreviation. The following definitions give alternative and often
more convenient notations:
4 >.r . D, . ylfS z = : (.r): 4 >x . D . \frx J)f,
<f>* . ", . yfr.r = . yfex Df.
This notation " <f>> . D,. y^.r is due to IVano. who. however, has no notation
lor the general idea "{.•). <f>x." It may be noticed as an exercise in the use
of dots as brackets that wo might have written
4>.r yfrx .-.(d.^rD \fr.r Df.
4 >r =, yfr.r . = . (x). <f>r = yfr.r Df.
In practice however, when <f>x and \fr.r are special functions, it is not possible
to employ fewer dots than in the first form, and often more are required.
The following definitions give abbreviated notations for functions of two
or more variables :
(•*•.!/) • <f> ( r > >j) • - • (■*) : (y) • <t> (•*.!/) I)f .
and so on for any number of variables;
<t> (.r, y) . D x .„ . t (r. y ): = : (x-. y ): <f> (x*. y ). D . yjr (.r. y) Df.
and so on for any number of variables.
Identity. The propositional function "x is identical with y ” is expressed by
* = !!■
This will be defined (cf. *13 01 ). but, owing to certain difficult points involved
in the definition, we shall here omit it (cf. Chapter II). We have, of course,
I-. x- = x* (the law of identity),
Y:x = y. = .y = x,
Yzx = y.y = z.D.x = 2.
The first of these expresses the reflexive property of identity: a relation is
called reflexive when it holds between a term and itself, either universally, or
whenever it holds between that term and some term. The second of the
above propositions expresses that identity is a symmetrical relation : a relation
is called symmetrical if, whenever it holds between x and y, it also holds
’]
IPKNT1TV
23
between y and .r. Tlu* tliiixl proposition express's licit, identity is a transitin'
relation: a relation is called transitin' it', whenever it holds between .i-and //
and between y and it holds also between .*• and
^ 0 shall find that no new definition of t he sign of equality is re*|iii i*«mI in
mathematics: all mathematical equations in which the sign of equality is used
in the ordinary way express some identity, and thus use the sign of equality
in the above sense.
If x and y are ideutica), either can replace the other in any proposition
without altering the truth-value of the proposition ; thus we have
h : x « y . D . <f>.v = <f>y.
This is a fundamental property of identity, from which the remaining properties
mostly follow.
It might be thought that identity would not have much importance, since
it can only hold between x and y if x and y are different symbols for the same
object. This view, however, does not apply to what we shall call "descriptive
phrases," i.e. " the so-and-so." It is in regard to such phrases that identity is
important, as we shall shortly explain. A proposition such ns " Scott was the
author of Waverley" expresses an identity in which there is a descriptive
phrase (namely “ the author of Waverley "); this illustrates how, in such cases,
the assertion of identity may be important. It is essentially the same case
when the newspapers say "the identity of the criminal has not transpired."
In such a case, the criminal is known by a descriptive phrase, namely " the
man who did the deed," and wc wish to find an x of whom it is true that
“ ®=the man who did the deed.” When such an x has been found, the identity
of the criminal has transpired.
Classes and relations. A class (which is the same as a manifold or aggre¬
gate) is all the objects satisfying some propositional function. If a is the class
composed of the objects satisfying we shall say that a is the class determined
by i p£. Every propositional function thus determines a class, though if the
propositional function is one which is always false, the class will be null,
i.e. will have no members. The class determined by the function <f>$ will be
represented by 2(</>2)*. Thus for example if <f*x is an equation, z(<f>z) will be
the class of its roots; if <f>x is “x has two legs and no feathers,” 2(<f>z) will
be the class of men; if <f>x is “ 0 < x < 1,” 2(<f>z) will be the class of proper
fractions, and so on.
It is obvious that the same class of objects will have many determining
functions. When it is not necessary to specify a determining function of a
class, the class may be conveniently represented by a single Greek letter.
Thus Greek letters, other than those to which some constant meaning is
assigned, will be exclusively used for classes.
• Any other letter may be used instead of z.
21
INTRODUCTION*
[CHAP.
There are two kinds of difficulties which arise in formal logic; one kind
arises in connection with classes and relations and the other in connection
with descriptive functions. The point of the difficulty for classes and relations,
so far as it concerns classes, is that a class cannot In- an object suitable as an
argument to any of it> determining functions. If a represents a class and </>•'
one of its determining functions [so that a = 3 (<fc)]. it is not sufficient that
<f>a be a false proposition, it must be nonsense. Thus a certain classification
of what apjK-ar to be objects into things of essentially different types seems
to be rendered neeessaiy. This whole question is discussed in Chapter II, on
the theory of typos, and the formal treatment in the systematic exposition,
which forms the main body of this work, is guided by this discussion. The
part of the systematic ox|»osition which is specially concerned wit h the theory
of classes is * 20 , and in I his Introduction it is discusM-d in Chapter III. It is
sufficient to note here that, in the complete treatment of *20. we have avoided
the decision as to whether a class of things has in any sense an existence as
one object. A decision of this question in either way is indifferent to our logic,
though pmhaps, if we had regarded some solution which held classes and re¬
lations to lie in some real sense objects as 1h>(Ii true and likely to be universally
received, we might have simplified one or two definitions and a few preliminary
propositions. <)ur symbols, such as " " and a and others, which represent
classes and relations, an* merely defined in their use, just as standing for
5* a* a*
?x* + + di’ 1
has no meaning apart from a suitable function of x, »/. z on which to operate.
The result of our definitions is that the way in which we use classes corre¬
sponds in general to their use in ordinary thought and speech; and whatever
may be the ultimate interpretation of the one is also the interpretation of
the other. Thus in fact our classification of types in Chapter II really
performs the single, though essential, service of justifying us in refraining
from entering on trains of reasoning which lead to contradictory conclusions.
The justification is that what seem to be propositions arc really nonsense.
The definitions which occur in the theory of classes, by which the idea ot
a class (at least in use) is based on the other ideas assumed as primitive,
cannot be understood without a fuller discussion than can be given now
(cf. Chapter II of this Introduction and also * 20 ). Accordingly, in this pre¬
liminary survey, we proceed to state the more important simple propositions
which result from those definitions, leaving the reader to employ in his mind
the ordinary unanalysed idea of a class of things. Our symbols in their usage
conform to the ordinary usage of this idea in language. It is to be noticed
that in the systematic exposition our treatment of classes and relations requires
no new primitive ideas and only two new primitive propositions, namely the
two forms of the “Axiom of Reducibility " (cf. next Chapter) for one and two
variables respectively.
CLASSKS
I]
•>
— •
»
The propositional function **.r is a member of t he class a" will be oxpivsscd,
following Peano, by the notation
.i* e a.
Here € is chosen as the initial of the won I tV ti. ".rco 1
an a. Thus "x t nmn” will mean ".r is a man." ami so on.
convenience we shall put
,i'*vfa. = .<v(,rfa) l>f.
• r » y e a . = . .v € a . / e a Df.
may be read " x is
For t ypograph ic:1 1
For "class” we shall write **Cls'’;
thus “ a € CIs " means a is a class."
We have
h : .r € 3 (</>-) . = - <£*\
i.e. u, x is a member of the class determined by <f>z' is equivalent to './•
satisfies <f> 2 ,' or to ' <f>x is true.' ”
A class is wholly determinate when its membership is known, that is, there
cannot be two different classes having the same membership. Thus if <f>x, yjrx
are formally equivalent functions, they determine the same class; for in that
case, if a; is a member of the class determined by <f> 2 , and therefore satisfies <f>.v,
it also satisfies yfrx, and is therefore a member of the class determined by yjrfi.
Thus we have
h 2 {<f>z)*=z(yfrz). = : <f>x . =, . yfrx.
The following propositions are obvious and important:
h a = 3 (<f>z) . = : x € a. = x . <f>x,
i.e. a is identical with the class determined by when, and only when, "x is
an a” is formally equivalent to <px;
h:.a = / 9 . = :a:ea. =«.*€£,
i.e. two classes a and /9 are identical when, and only when, they have the same
membership;
V ,&(xc a) = a,
i.e. the class whose determining function is “ a: is an a ” is a, in other words,
a is the class of objects which are members of a ;
b . 3 ( <f>z) e CIs,
i.e. the class determined by the function <f >2 is a class.
It will be seen that, according to the above, any function of one variable
can be replaced by an equivalent function of the form " x e a.” Hence any
extensional function of functions which holds when its argument is a function
of the form “Sea,” whatever possible value a may have, will hold also when
its argument is any function <f> 2 . Thus variation of classes can replace varia¬
tion of functions of one variable in all the propositions of the sort with which
we are concerned
-2 ; -
26
INTRODUCTION
[CHAP.
In an exactly analogous manner we introduce dual or dyadic relations,
i.e. relations between two terms. Such relations will be called simply
"relations"; relations between more than two terms will he distinguished as
multifile relations, or (when the number of their terms is specified) as triple,
«|iiadruple,...relations, or as triadic, tetradic_relations. Such relations will
not concern us until we come to(!eometry. For the present, the only relations
we are concerned with are dual relations.
Relations like classes, are to be taken in e.>tension, i.c. if li and .S’ are
relations which hold between the same pairs of terms, 11 and .S' are to be
identical. We may regard a relation, in the sense in which it is required for
our purjM.ses, as a class of couples: i.e. the couple (x, y) is to be one of the
class of couples constituting the relation li if x has the relation R to y*.
This view of relations as classes of couples will not, however, be introduced
into our symbolic treatment, ami is only mentioned in order to show that it
is possible so to understand the meaning of the word relation that a relation
shall Ik* determined by its extension.
Any function <f> ( ».//) determines a relation li between x and y. If we
regard a relation as a class of couples, the relation determined by <f>(x,y) is
the class of couples y) for which <p u*. //) is true. The relation determined
by the function </>(.#-, y) will Ik- denoted by
sy4> {x, y).
We shall use a capital letter for a relation when it is not necessary to specify
the determining function. Thus whenever a capital letter occurs, it is to be
understood that it stands for a relation.
The propositional function " x has the relation li to y " will bo expressed
by the notation
xliy.
This notation is designed to keep as near as possible to common language,
which, when it has to express a relation, generally mentions it between its
terms, as in “ x loves y,“ " x equals y." “ x is greater than y," and so on. For
“ relation " we shall write ’* Rel thus "Re Kel M means "li is a relation."
Owing to our taking relations in extension, we shall have
h U )* (x,y). = :<f> (x, y) ^ (x, y),
i.e. two functions of two variables determine the same relation when, and only
when, the two functions are formally equivalent.
We have V . z \xy<f> (x, y)\w.= .<f> (z, w),
• Such n couple has a tense, i.e. Ihe couple (x, y) is different from the couple (y, x), unless
x = y. Wo shall call it a "couple with sense,” to distinguish it from the class consisting of x
and y. It may also be called an ordered couple.
J 1 OAU'l'Ll’S OF CLASSICS 27
i.t*. has to ir the ivlation determined by the function <£(.#•. //)" is equivalent
to <£ (j, «»>;
H A‘ = .iy/</> (.r, //) . = : .1 Hi/ . = x „ . if> ^.»\ #/).
I- A* = £ . = : .#• A*// . = x . y • •* «S//.
^ • *«V? (•• !*•/) = A*,
h . J.rv<#» f.r, //)) t* l\el.
These propositions are analogous to those previously given for classes. It
results from them that any function of two variables is formally equivalent. to
some function of the form a* Kg; hence, in extensional functions of two variables,
variation of relations can replace variation of functions of two variables.
Both classes and relations have properties analogous to most of those of
propositions that result from negation and the logical sum. The logical prod net
of two classes a and 0 is their common part, i.e. the class of terms which are
members of both. This is represented by a n 0. Thus we put
a r» = a . .v e 0) Df.
This gives us hs/ean^.s.. vea.xe 0.
i.e. “ x is a member of the logical product of a and 0 ” is equivalent to the
logical product of "a: is a member of a ” and “ x is a member of 0."
Similarly the logical sum of two classes a and 0 is the class of terms which
arc members of either; wc denote it by a v# 0. The definition is
a w 0 = 2 (x € a . v . x e 0) Df,
and the connection with the logical sum of propositions is given by
The negation of a class a consists of those terms x for which "xea" can
be significantly and truly denied. We shall find that there are terms of other
types for which "xea” is neither true nor false, but nonsense. These terms
are not members of the negation of a.
Thus the negation of a class a is the class of terms of suitable type which
are not members of it, i.e. the class £(x~ea). We call this class “—a ” (read
“not-a”); thus the definition is
— a = 5 (x~e a) Df,
and the connection with the negation of propositions is given by
I- : x e — a . = . x~e a.
In place of implication we have the relation of inclusion. A class o is said
to be included or contained in a class 0 if all members of a are members of 0,
i.e. if x e a . D x . x e 0. We write “ a C 0 ” for “ a is contained in 0.” Thus we
put
aC0. = zxea.D x .xe0 Df.
a*
28
INTRODUCTION
[CHAP.
Most of the formulae concerning p. q. p v 7 , ~ p . /O 7 remain true if we
substitute a r\ 0, a v 3. - a. a C /3. In place of equivalence, we substitute
identity: for " /»= 7 " was defined as “ p D 7 . 7 D y>," but " a C /3. /9 C a ’’ gives
' .tea . = x whence a = &.
The follow ing are some propositions concerning classes which arc analogues
of propositions previously given concerning propositions:
l-.o a/}*-(-qo- 3 ),
i.e. the common part of a and /3 is the negation of" not-a or not-/3
h .re(a v -a),
i.e. " .r is a member of a or not-a ;
1 - • X'W (a r\ - a).
i o. " .<• is not a member of l»oth a and not-a
Ka--(-a),
h:aC/3. = .-£C-a.
H:a-/9.s.-a--/9,
ha=oftfl,
ho*flva.
The two lust arc the two forms of the law of tautology.
The law of absorption holds in the form
h:aC/3. = .a = an/3.
Thus for example "all Cretans are liars” is equivalent to "Cretans arc
identical with lying Cretans."
Just as we have h : p D 7.7 D r . D . p D r,
so we have \-:aC0.0Cy.'2.aCy.
This expresses the ordinary syllogism in Barbara (with the premisses
interchanged); for "aC/3" means the same as "all a's are /9's,” so that the
above proposition states: "If all a's are /3's, and all /3's are 78 , then all a's
are y’s.” (It should be observed that syllogisms are traditionally expressed
with " therefore," as if they asserted both premisses and conclusion. This is,
of course, merely a slipshod way of speaking, since what is really asserted is
only the connection of premisses with conclusion.)
The syllogism in Barbara when the minor premiss has an individual
subject is
t'ixe&.fiCy.O.xey,
e.rj. " if Socrates is a man. and all men are mortals, then Socrates is a
mortal." This, as was pointed out by Peano, is not a particular case of
"aC/ 3 ./ 3 C 7 .D.aC 7 ," since " x € /9 " is not a particular case of “ a C
This point is important, since traditional logic is here mistaken. The nature
and magnitude of its mistake will become clearer at a later stage.
I]
CALCULUS OF CLASSICS
2«»
For relations, wo have precisely analogous definitions and propositions.
We put
A* rt.S’ = .rf/ (. r/\i /. jr&ijS I)f,
which leads to l- : jr(lt f\ S) //. = . xRy . xSy.
Similarly Rv S = .?// . v . .rSy) Df.
- R = xy \~(xRy)\ Df.
i? C .S'. = : .rA*y . D x> v . .cSy Df.
Generally, when we require analogous but different symbols for relations
and for classes, we shall choose for relations the symbol obtained by adding
a dot, in some convenient position, to the corresponding symbol for classes.
(The dot must not be put on the line, since that would cause confusion with
the use of dots as brackets.) But such symbols require and receive a special
definition in each case.
A class is said to exist when it has at least one member: "o exists" is
denoted by “ g ! a." Thus we put
a !a.«.(a*).*e« Df.
The class which has no members is called the “ null-class," and is denoted by
“A.” Any propositional function which is always false determines the null-
class. One such function is known to us already, namely " x is not identical
with x,” which we denote by “ x^x." Thus we may use this function for de¬
fining A, and put
A = £(***) Df.
The class determined by a function which is always true is called the
universal classy .nd is represented by V; thus
V = £(* = *) Df.
Thus A is the negation of V. We have
h . (x) .xeV,
te - « is a member of V ’ is always true and
h . ( x ) . x~c A,
ie. ‘"a; is a member of A’ is always false." Also
h : a = A . = . ~g ! a,
te - “a is the null-class” is equivalent to “a does not exist.”
For relations we use similar notations. We put
a i R - —. (a*, y ) • xR y>
l - e - “glil” means that there is at least one couple x , y between which
the relation R holds. A will be the relation which never holds, and V the
relation which always holds. V is practically never required; A will be the
relation a£) {x =f= x . y + y). We have
h .(x, y).~(iAy),
and \-:R = A. = .~nlR.
30
INTRODUCTION
[CHAP.
There are no classes which contain objects of more than one type. Ac¬
cordingly there is a universal class and a null-class proper to each type of
object. But these syinlxil* need not be distinguished, since it will be found
that there is no possibility of confusion. Similar remarks apply to relations.
Ih'scii/itions. By a **description" we mean a phrase of the form "the
so-and-so" or of some equivalent form. For the present, we confine our
attention to the in the singular. We shall use this word strictly, so as to
imply uniqueness: e.y. we should not say ".I is the son of IS" if H had other
sons besides ,|. Thus a description of the form “the so-and-so’ will only
have an application in the event of then* being one so-and-so and no more.
Hence a description requires s*»me promotional function <f>7 which is satisfied
by mu* value of .r and by no other values; then “the x which satisfies <f>r
is a description which definitely describes a certain object, though we may
not know what object it dcscrilies. For example, if y is a man, " x is the
father of y " must be true for one, and only one, value of x. Hence "the
lather of y is a description of a certain man, though we may not know what
man it describes. A phrase containing “ the " always presupposes some initial
propositional function not containing " the "; thus instead of ".r is the father
of y" we ought to take as our initial function " x begot y then “ the father
of //’’ means the one value of x which satisfies this propositional function.
If <f)7- is a propositional function, the symbol “(i.r)(<£./)’’ is used in our
symbolism in such a way that it can always be read as "the x which satisfies
</>/•.’’ But we do not define " (lx)(<t>x)" as standing for " the x which satisfies
0r," thus treating this last phrase as embodying a primitive idea. Every use
of " (ix)(<t>x),'‘ where it apparently occurs as a constituent of a proposition
in the place of an object, is defined in terms of the primitive ideas already
on hand. An example of this definition in use is given by the proposition
" E!(!.»•)which is considered immediately. The whole subject is treated
more fully in Chapter III.
The symbol should be compared and contrasted with " x(<f>x) ’’ which in
use can always be read as " the .r’s which satisfy Both symbols arc in¬
complete symbols defined only in use, and as such arc discussed in Chapter III.
The symbol " x(tftx) " always has an application, namely to the class determined
by (fjr ; but u (ix)($x)" only has an application when tf>x is only satisfied by
one value of x, neither more nor less. It should also be observed that the
meaning given to the symbol by the definition, given immediately below, of
E! (ix)(<t>x) dues not presuppose that we know the meaning of "one." This is
also characteristic of the definition of any other use of ( ix)(<f>x ).
We now proceed to define " E! (ix)(^x)” so that it can be read “ the x
satisfying <f>r exists." (It will be observed that this is a different meaning ot
existence from that which we express by " 3 .") Its definition is
E! (tx)(<f>x) . = : ( 3 c) : <f>x. = x .x = c Df,
I]
1'INSCRIPTIONS
31
t.e. “ the .v satisfying <f>r exists " is to mean
is true when .r is c but not otherwise.”
'* there is an object c such that </>./•
The following are equivalent forms:
H h! ( hr) {<$>••'). = : (ye): tfn :: </>.#•. D r . .#• = c,
H E! (hr) (<f>.r) . = : (gc) . <f>c z <f>.r . </>y . D r y . .r = #/.
b E ! t IJ-) » ). = : (yrl: </>»•: .r + c . D r . ~<£./.
The last of these states that ** the x satisfying <f>x exists" is equivalent t«>
“there is an object c satisfying <f>.r, anil every object other than c does not
satisfy <£.T» ”
The kind of existence just defined covers a great many cases. Thus fbi
example “ the most perfect Being exists " will mean :
(gc) : .r is most perfect. 5 , . .?• = c,
which, taking the last of the above equivalences, is equivalent to
(gc) : c is most perfect: x + c. D x . x is not most perfect.
A proposition such as “Apollo exists" is really of the same logical form,
although it docs not explicitly contain the word the. For “Apollo" means
really “ the object having such-and-such properties," say “ the object having
the properties enumerated in the Classical Dictionary*.” If these properties
niake up the propositional function <f>x, then “Apollo" means “(?.r) (<£a),"
and “Apollo exists" means “E l (ix) (<f>x)." To take another illustration,
" the author of Waverley" means “ the man who (or rather, the object which)
wrote Waverley." Thus “ Scott is the author of Waverley " is
Scott ■= (lx) (x wrote Waverley).
Here (as we observed before) the importance of identity in connection with
descriptions plainly appears.
The notation “ (ix) (<f>x),” which is long and inconvenient, is seldom used,
being chiefly required to lead up to another notation, namely “ R'y," meaning
" object having the relation R to y." That is, we put
R‘y = (lx)(xRy) Df.
The inverted comma may be read “of." Thus “ R‘y ” is read “the R of y.”
Thus if R is the relation of father to son, “i2‘y” means “the father of y";
if R is the relation of son to father, means “the son of y,” which will
only “ exist ” if y has one son and no more. R*y is a function of y, but not
a propositional function; we shall call it a descriptive function. All the
ordinary functions of mathematics are of this kind, as will appear more fully
in the sequel. Thus in our notation, “siny" would be written “sin *y ," and
sin would stand for the relation which sin ‘y has to y. Instead of a variable
descriptive function fy, we put R‘y, where the variable relation R takes the
The same principle applies to many uses of the proper names of existent objects, e.g. to all
UBee of proper names for objects known to the speaker only by report, and not by personal
acquaintance.
32
INTRODUCTION
[chap.
place of the variable function f A descriptive function will in general exist
while y belongs to a certain domain, but not outside that domain : thus if "'e
are dealing with positive rationals. y/y will be significant if y is a perfect
square, but not otherwise; if we are dealing with real numbers, and agree
that “ \ //' is to mean the fugitive square root (or, is to mean the negative
square root), N 'y will be significant provided y is positive, but not otherwise:
and so on. Thus every descriptive function has what we may call a “domain
of definition " «»r a “domain of existence.'’ which may be thus defined: If the
function in question is R‘y, its domain of definition or of existence will be
the class of those arguments y for which we have E! !(*•/, t.r. for which
K!(l.i )(.i7ty). i.e. for which there is one .r. and no more, having the relation
li to y.
11 R is any relation, wo will sp.ak of 11* y as the “ associated descriptive
function.” A great many of the constant relations which we shall have occasion
to introduce arc only or chiefly im|K>rtant on account of their associated descrip¬
tive functions. In such cases, it. is easier (though less correct) to begin by
assigning tin- meaning of the descriptive function, and to deduce the meaning
of the relation from that of the descriptive function. This will be done in the
following explanations of notation.
\’m ious descri/itive functions of mint ions. If li is any relation, the converse
of R is the relation which holds between y and x whenever 11 holds between
.*• and y. Thus yrenter is the converse of less, before of after, cause of effect
NO
husband of wife, etc. The converse of li is written * Cnv‘7? or li. The defi¬
nition is
R - xy (yRx) Df,
Cnv‘7? = li I)f.
The second of these is not a formally correct definition, since we ought to
define “ Cnv ” and deduce the meaning of Cnv*R. But it is not worth while
to adopt this plan in our present introductory account, which aims at simplicity
rather than formal correctness.
A relation is called symmetrical if li = li, i.e. if it holds between y and x
whenever it holds between x and y (and therefore vice versa). Identity,
diversity, agreement or disagreement in any respect, are symmetrical relations.
A relation is called asymmetrical when it is incompatible with its converse,
i.e. when Rr\ li = A, or, what is equivalent,
xi?y.D x>y .~(y/tr).
Before and after, greater and less, ancestor and descendant, are asym¬
metrical, as are all other relations of the sort that lead to series. But there are
many asymmetrical relations which do not lead to series, for instance, that of
* The second of those notations is taken from Schroder’s Algebra und Logik der Relative.
I>KSCRImvK FI* NTTIONs
33
wifo's brother*. A relation may be neither symmetrical nor asymmetrical;
or example, this holds of the relation of inclusion between classes: aC ami
f Ca will ^ oth true if a = fS, but otherwise* only one of them, at most, will
be true. The relation brother is neither symmetrical nor asymmetrical, for if
.r is the brother of y. y may be either the brother or the sister of .r.
In the propositional function xRy t we call .r the referent and y the re/atum.
1 he class X(xRy\ consisting of all the Ss which have the relation A‘ to y. is
billed the class of referents of y with respect to i*; the class 7) (xRy), consisting
of all the y’s to which .r has the relation R, is called the class of relata of .r
with respect to R, These two classes are denoted respectively by R‘y and 7 7‘.c.
rh us
7?y = (xRy) Df.
Df.
The arrow runs towards y in the first case, to show that we are concerned
with things having the relation R to y; it runs away from a- in the second
case, to show that the relation R goes from x to the members of T^r. It runs
in fact from a referent and towards a relatum.
The notations R‘y, R*x are very important, and are used constantly. If
R is the relation of parent to child, R‘y = the parents of y, Ji*x = the children
of We have
b : x « RU
xRij
and
b : y € R‘x . = . xRy.
These equivalences are often embodied in common language. For example,
we say indiscriminately u x is an inhabitant of London ” or “a: inhabits London.”
we put " R” for "inhabits,” "x inhabits London” is “xR London," while "x
is an inhabitant of London ” is London.”
Instead of R and R we sometimes use s g l R, gs‘R, where *' sg ” stands for
sagitta,” and “ gs ” is “ sg ” backwards. Thus we put
sg‘R = ~R Df,
gs *R=*R Df.
These notations are sometimes more convenient than an arrow when the
re ation concerned is represented by a combination of letters, instead of a
single letter such as R. Thus e.g. we should write sg‘(R r* S ), rather than put
an arrow over the whole length of ( R r\ S).
The class of all terms that have the relation R to something or other is
°a led the domain of R. Thus if R is the relation of parent and child, the
lh . Tl11 , 8 re ktion not strictly asymmetrical, but is so except when the wife’s brother is also
e sister’s husband. In the Greek Church the relation is strictly asymmetrical.
R&W i
3
INTRODUCTION
34
[chap.
domain of R will he the class of parents. We represent the domain of R by
"\yR." Thus we put
i>‘/f=*kay).wty: Df.
Similarly the class of all terms to which something or other has the relation
R is calleil the converse domain of R ; it is the same as the domain of the
converse of R. The converse domain of R is represented by “C VR thus
(I ‘ R = y !<g.r). r R>/[ Df.
The sum of the domain and the converse domain is called the field, and is
represented by C*R: thus
C*R= 0‘/e w (|‘/e Df.
The field is chi.-Hy important in connection with series. If R is the ordering
relation of a series, t u R will be the class of terms of the series. I)*/? will be all
tie* terms except the last (if any), and <1 'R will be all the terms except the
first (if any). The first term, if it exists, is the only member of D *R n — d*R,
since it is the only term which is a predecessor but not a follower. Similarly
the last term (if any) is the only member of (I'Kn-D 1 //. The condition
that a series should have no end is <l‘/f C 1 VR. i.e. “every follower is a pre¬
decessor"; the* condition for no beginning is D‘PC(l 4 /f. These conditions
are equivalent respectively to !)*/( * C‘R and <I 4 /( = C U R.
The relative /noilnet of two relations R and S is the relation which holds
between .r and z when there is an intermediate term y such that x has the
relation R to y and y has the relation .S' to z. The relative product of R and
S is represented by R .V; thus we put
R S = rz |(gy). xRy . ySz | Df,
whence \- :x{R S) z . = . (gy). xRy . ySz.
Thus " paternal aunt” is the relative product of sister and father; “paternal
grandmother” is the relative product of mother and father; " maternal grand¬
father" is the relative product of father and mother. The relative product is
not commutative, but it obeys the associative law, i.e.
h.{l>\Q)\R-P\(Q\R).
It also obeys the distributive law with regard to the logical addition of
relations, i.e. we have
KP|(Qv/tf)-(PiQ)«<P|P).
MQw/?)iP = (Q|P)c,(fl|P).
But with regard to the logical product, we have only
h.P!(QA«)C(/>|Q)n(P|/JX
MQA*)|PG(Q|P)A(Q|/0.
The relative product does not obey the law of tautology, i.e. we do not
have in general R R = R. We put
fr=R\R Df.
35
*] PLURAL DESCRIPTIVE FUNCTIONS
Thus paternal grandfather = (father) 3 ,
maternal grandmother = (mother ) 3
A relation is called transitive when R S GR. i.e. when, if xRy and i/R:, we
always have xRz, i.e. when
xRy . yRz . . .vRz.
Relations which generate series are always transitive; thus e.g.
•** > If • •/ >
If P is a relation which generates a series. P may conveniently be read
precedes ; thus u xPg . yPz . D ,xPz" becomes “ if .v precedes yarn! y
precedes z, then .r always precedes z." The class of relations which generate
series are partially characterized by the fact that they are transitive and
asymmetrical, and never relate a term to itself.
If P is a relation which generates a series, and if we have not merely P : Q P,
but P- = P, then P generates a series which is compact ( iiberall dicht), i.e. such
that there are terms between any two. For in this case we have
xPz . D . (ay) . xPy . yPz,
i.e. if x precedes z, there is a term y such that x precedes y and y precedes z,
i.e. there is a term between x and z. Thus among relations which generate
series, those which generate compact series are those for which P* =» P.
Many relations which do not geuerate series are transitive, for example,
identity, or the relation of inclusion between classes. Such cases arise when
the relations are not asymmetrical. Relations which are transitive and sym¬
metrical are an important class: they may be regarded as consisting in the
possession of some common property.
Plural descriptive functions. The class of terms x which have the relation
R to some member of a class a is denoted by R“a or R/a. The definition is
R“a = £{('ay).yca.xRy\ Df.
Thus for example let R be the relation of inhabiting, and a the class of towns;
then R“a = inhabitants of towns. Let R be the relation “ less than " among
rationals, and a the class of those rationals which are of the form 1 — 2~ n , for
integral values of n; then R**a will be all rationals less than some member
of a, i.e. all rationals less than 1. If P is the generating relation of a series,
and a is any class of members of the series, P“a will be predecessors of as, i.e. the
segment defined by cl If P is a relation such that P*y always exists when
yea, P“a will be the class of all terms of the form P*y for values of y which
are members of a; i.e.
P“a = £ ((ay ). y e a . x = P‘y).
Thus a member of the class “ fathers of great men ” will be the father of y,
where y is some great man. In other cases, this will not hold; for instance,
let P be the relation of a number to any number of which it is a factor; then
3—2
30
INTRODUCTION
[CHAP. I
P“ (oven numbers) = factors of even numbers, but this class is not composed
of terms of the form "the factor of .r, where .r is an even number, because
numbers do not have only one factor apiece.
Unit classes. The class whose only member is .r might be thought to he
identical with x, but Peano and Frege have shown that this is not the case.
(The reasons why this i> not the case will be explained in a preliminary way
in Chapter II of the Introduction.) We denote by " C.r" the class whose only
member is x: thus
iV = .Ky = .r> Of.
i.e. " Ux ' means "the class of objects which are identical with x."
The class consisting ot x and »/ will be i*x\j i‘u ; the class got by adding
x to a class a will be a v i*x: tin- class got by taking away x from a class a
will be a - Us. (We write a — & as an abbreviation for a n — /9.)
It will be observed that unit classes have been defined without reference
to the number 1 ; in fact, we use unit classes to define the number 1. This
number is defined as the class of unit classes, i.e.
1 m a |(gx). a = f‘x) l)f.
This leads to
b a c I . = : (ax): y « a . =„. \j - x.
From this it appears further that
l~: a < 1 . = . E !(»/)(/*«),
whence h : 2 (<f>z) « 1 . ■ . E! ( ix) (tf>x),
i.e. " 2 is a unit class " is equivalent to " the x satisfying <f>? exists.”
v
If «f 1, t‘a is the only member of a. for the only member of a is the only
term to which a has the relation i. Thus u t‘a'' takes the place of u (ix)(<l>x),"
if a stands for z(<f>z). In practice, "l‘a’’ is a more convenient notation than
"(f- r )(</>' ). and is generally used instead of "(u)(<f>x).''
The above account has explained most of the logical notation employed
in the present work. In the applications to various parts of mathematics,
other definitions arc introduced; but the objects defined by these later defi¬
nitions belong, for the most part, rather to mathematics than to logic. The
rentier who has mastered the symbols explained above will find that any
later formulae can be deciphered by the help of comparatively few additional
definitions.
CHAPTER II
THE THEORY OF LOGICAL TYPES
The theory of logical typos, to bo explained in the present Chapter, re¬
commended itself to us in the first instance by its ability to solve certain
contradictions, of which the one best known to mathematicians is Burali-Fortis
concerning the greatest ordinal. But the theory in question is not wholly
dependent upon this indirect recommendation: it has also a certain consonance
with common sense which makes it inherently credible. In what follows, we
shall therefore first set forth the theory on its own account, and then apply it
to the solution of the contradictions.
I. The Vicious-Circle Principle.
An analysis of the paradoxes to be avoided shows that they all result from
a certain kind of vicious circle*. The vicious circles in question arise from
supposing that a collection of objects may contain members which can only be
defined by means of the collection as a whole. Thus, for example, the collection
of propositions will be supposed to contain a proposition stating that " all
propositions are either true or false.” It would seem, however, that such a
statement could not be legitimate unless “all propositions” referred to some
already definite collection, which it cannot do if new propositions are created
by statements about “all propositions.” We shall, therefore, have to say that
statements about “all propositions” are meaningless. More generally, given
any set of objects such that, if we suppose the set to have a total, it will con¬
tain members which presuppose this total, then such a set cannot have a total.
By saying that a set has “no total,” we mean, primarily, that no significant
statement can be made about “all its members.” Propositions, as the above
illustration shows, must be a set having no total. The same is true, as we shall
shortly see, of propositional functions, even when these are restricted to such
as can significantly have as argument a given object a. In such cases, it is
necessary to break up our set into smaller sets, each of which is capable of a
total. This is what the theory of types aims at effecting.
The principle which enables us to avoid illegitimate totalities may be
stated as follows: “Whatever involves all of a collection must not be one of
the collection”; or, conversely: “If, provided a certain collection had a total,
it would have members only definable in terms of that total, then the said
collection has no total.” We shall call this the “vicious-circle principle,” be¬
cause it enables us to avoid the vicious circles involved in the assumption of
illegitimate totalities. Arguments which are condemned by the vicious-circle
* See the last section of the present Chapter. Cf. also H. Poincar^, “ Les math^matiques et
la iogiqne,'* Revue de M€taphysique et de Morale, Mai 1906, p. 307.
38
INTRODUCTION
[chap.
I»rinci|»l«- will hi* called ”\ i« i«.n<-cirr|.- fallacies." Such arguments, in certain
circninstaii« «-s. may had to contradictions. hut it often happens that the con¬
clusion** t«» which th#*y !• ■;»•! ar«• in fact true, though the arguments are
fallacious. Take. f.»r example, the law of excluded middle, in the form "all
propositions are it no or false.” If from this law we argue that, because the
law of excluded middle is a pro|M»ition, therefore the law of excluded middle
is true or false. wv iiicm a vicious-circle fallacy. "All propositions" must he
in some wav limited In-foie it Im-couh-s a legitimate totality, and any limita¬
tion which makes it legitimate must make anv statement about the totality
tail outside the totality. Similarly, the imaginary sceptic, who asserts that
knows nothing, and is refuted by In-ing asked if he knows that he knows
nothing, lias asserted iiohscum-, and has been fallaciously refuted by an
argument which involves a vicious-circle fallacy. In order that the sceptic's
ass, rtion may Income significant, it is necessary to place some limitation
upon the things of which he is asserting his ignorance, because the things
of which it is possible to be ignorant form an illegitimate totality. But ns
s»h.ii as a suitable limitation has been placed by him upon the collection of
propositions of which he is asserting his ignorance, the proposition that he is
ignorant of every member of this collection must not itself be one of the
collection. Hence any significant scepticism is not open to the above form of
refutation.
The paradoxes of symbolic logic concern various sorts of objects: propo-
Mtions. classes, cardinal and ordinal numbers, etc. All these sorts of objects,
as we shall show, represent illegitimate totalities, and are therefore capable of
giving rise to vicious-circle fallacies. But by means of the theory (to be
explained in Chapter III) which reduces statements that are verbally con¬
cerned with classes and relations to statements that are concerned with
propositional functions, tin* paradoxes arc reduced to such as are concerned
with propositions ami propositional functions. The paradoxes that concern
propositions are only indirectly relevant to mathematics, while those that
more nearly concern the mathematician arc all concerned with propositional
functions. We shall therefore proceed at once to the consideration of propo¬
sitional functions.
II. The Xutnre of Propositionul Functions.
By a "propositional function" we mean something which contains a
variable x, and expresses a proposition as soon as a value is assigned to .r.
'I hat is to say, it differs from a proposition solely by the fact that it is
ambiguous: it contains a variable of which the value is unnssigned. It agrees
with the ordinary functions of mathematics in the fact of containing an
unnssigned variable; where it differs is in the fact that the values of the
function are propositions. Thus e.y. "x is a man" or "sin x= 1 " is a propo¬
sitional function. We shall find that it is possible to incur a vicious-circle
PROPOSITIONAL FUNCTIONS
30
n]
fallacy at. the very outset, by admitting as |x>ssible arguments to a propositional
function terms which presuppose the function. This form of the fallacy is wrv
instructive, and its avoidance leads, as we shall see. to the hierarchy of types.
The question as to the nature of a function* is by no means an easy one.
It would seem, however, that the essential characteristic of a function is
ambiguity. JTnke, for example, the law of identity in the form "A is A," which
is the form in which it is usually enunciated. It is plain that, regarded
psychologically, we have here a single judgment. But what are we to say of
the object of the judgment i We are not judging that Socrates is Socrates,
nor that Plato is Plato, nor any other of the definite judgments that arc
instances of the law of identity. Yet each of these judgments is, in a sense,
within the scope of our judgment. We are in fact judging an ambiguous
instance of the propositional function "A is A." We appear to have a single
thought which does not have a definite object, but has as its object an
undetermined one of the values of the function "A is A." It is this kind of
ambiguity that constitutes the essence of a function. When we speak of "<f>x”
where x is not specified, we mean one value of the function, but not a definite
one. We may express this by saying that "tfrx" ambiguously denotes <f>a t <f>b, (f>c,
etc., where <f>a, <f>b, <f>c, etc., are the various values o f u <f>x."
When we say that “<£x” ambiguously denotes <f>a, 4>b, tf>c, etc., we mean
that “tfrx” means one of the objects (f>a , <f>b, <f>c, etc., though not a definite
one, but an undetermined one. It follows that only has a well-defined
meaning (well-defined, that is to say, except in so far as it is of its essence to
be ambiguous) if the objects <f>a, <f>b, <f>c, etc., are well-defined. That is to say,
a function is not a well-defined function unless all its values are already well-
defined. It follows from this that no function can have among its values
anything which presupposes the function, for if it had, we could not regard
the objects ambiguously denoted by the function as definite until the function
was definite, while conversely, as we have just seen, the function cannot be
definite until its values are definite. This is a particular case, but perhaps the
most fundamental case, of the vicious-circle principle. A function is what
ambiguously denotes some one of a certain totality, namely the values of the
function; hence this totality cannot contain any members which involve the
function, since, if it did, it would contain members involving the totality,
which, by the vicious-circle principle, no totality can do.
It will be seen that, according to the above account, the values of a
function are presupposed by the function, not vice versa. It is sufficiently
obvious; in any particular case, that a value of a function does not presuppose
the function. Thus for example the proposition “ Socrates is human ” can be
perfectly apprehended without regarding it as a value of the function "x is
human.” It is true that, conversely, a function can be apprehended without
.. * "When the word “function ” is used in the sequel, “propositional function" is always meant,
v^) ther function s will not be in question in the present Chapter.
10
INTRODUCTION
[chap:
its being necessary to apprehend its values severally and individually. If this
were not the case, n** function could be apprehended at all, since the number
of values (true and falsi-) of a function is necessarily infinite and there are
necessarily possible arguments with which we are unacquainted. What is
necessary is not that the values should lie given individually and extensionally,
but that the totality of the values should be given intcnsionally, so that, con¬
cerning any aligned object, it is at least theoretically determinate whether or
not the said object i' a value of the function.
It is necessity practically D» distinguish the function itself from an
undetermined value of the function. We may regard the function itself as
that which ambiguously denotes, while an undetermined value of the function
is that which is ambiguously denoted. If the undetermined value is written
wo will write the function itself •<£?." (Any other letter may he used
in place of j-.) Thus we should sty is a proposition,’ but "<f)7 is a propo¬
sitional function." When wo say "<f>r is a proposition ” wc mean to state
something which is true for every possible value of x. though we do not decide
what value x is to have. We are making an ambiguous statement about any
value oi the function. But when we say •* <f>x is a function," wc are not making
an ambiguous statement. It would be more correct to say that we are making
a statement about an ambiguity, taking the view that a function is an am¬
biguity. The function itself. <fy7. is the single thing which ambiguously denotes
its many values; while </»». where x is not specified, is one of the denoted
objects, with the ambiguity belonging to the manner of denoting.
We have seen that, in accordance with the vicious-circle principle, the
values of a function cannot contain terms only definable in terms of the
function. Now given a function 4>x. the values for the function* are all pro¬
positions of the form <p.r. It follows that there must be no propositions, of
the form <f>x, in which x has a value which involves <f>x. (If this were the case,
the values of the function would not all be determinate until the function
was determinate, whereas we found that the function is not determinate unless
its values are previously determinate.) Hence there must be no such thing as
the value for <f>7 with the argument <f>x, or with any argument which involves
<f>2. That is to say, the symbol "<f> ( <f>x)" must not express a proposition, as
docs if (f>a is a value for <f>7. In fact "<£ (<£.?)’’ must be a symbol which
does not express anything: we may therefore say that it is not significant. Thus
given any function 4>7, there are arguments with which the function has no
value, as well as arguments with which it has a value. We will call the
arguments with which <f»x has a value "possible values of x.” We will say
that 4 >x is "significant with the argument x" when <f>x has a value with the
argument x.
• We shall speak in this Chapter of "values for $&" and of "values of $x % '' meaning in each
case the same thin*, namely $a, $b, $c, etc. The distinction of phraseology serves to avoid
ambiguity where several variables are concerned, especially when one of them is a function.
POSSIBLE ARGUMENT* FOR FUNCTIONS
II
»]
When it is said that r.ij. " <f> is meaningless, ami fhoivfoiv n«*ith«*r
true nor false, it is necessary to avoid a misunderstanding. If•</> (</>?)" wmv
interpreted as meaning "the value for <f>: with the argument <J>: is true.'*
that would be not meaningless, but false. Il is false for the same reason for
which “the King of France is bald" is false, namely because there is no such
thing as “the value tor <f>7 with the argument Hut- when, with some
argument a, wo assert <f>a, we are not meaning to assert “the value for </>.»• wit h
the argument a is true ": we are meaning to assert the actual proposition
which is the value for d>7 with the argument a. Thus for example if <f>7 is
“.«• is a man,” «/> (Socrates) will be “Socrates is a man," not "the value lor
the function '.7 is a man,' with the argument Socrates, is true." Thus
in accordance with our principle that “<£(<£?)" is meaningless, we cannot
legitimately deny “the function ‘7 is a man’ is a man.” because this is
nonsense, but we can legitimately deny “the value for the function '7 is a
man’ with the argument '7 is a man' is true," not on the ground that the
value in question is false, but on the ground that there is no such value for
the function.
We will denote by the symbol u {x) . 4>x” the proposition “<£.»• always*,"
xe. the proposition which asserts all the values for <f>7. This proposition
involves the function <f>7, not merely an ambiguous value of the function. The
assertion of <f>x, where x is unspecified, is a different assertion from the one
which asserts all values for <f>7, for the former is an ambiguous assertion,
whereas the latter is in no sense ambiguous. It will be observed that u (x).<f>.c"
does not assert u <f>x with all values of x," because, as we have seen, there must
be values of x with which “ <f>x " is meaningless. What is asserted by “(.z*).<£a"
is all propositions which are values for <f>7; hence it is only with such values
of x as make “<f>x” significant, i.e. with all possible arguments, that <f>x is asserted
when we assert “(a;). <f>x‘' Thus a convenient way to read “(#). <f>x n is “<f>x is
true with all possible values of x.” This is, however, a less accurate reading
than "<f>x always,” because the notion of truth is not part of the content of
what is judged. When we judge “all men are mortal," we judge truly, but
the notion of truth is not necessarily in our minds, any more than it need be
when we judge “Socrates is mortal."
III. Definition and Systematic Ambiguity of Truth and Falsehood.
Since “(#) . <f>x" involves the function <p7, it must, according to our
principle, be impossible as an argument to <f>. That is to say, the symbol
“<f> {(#) . (f>x )" must be meaningless. This principle would seem, at first sight,
to have certain exceptions. Take, for example, the function “p is false,” and
consider the proposition “(p) . p is false." This should be a proposition
asserting all propositions of the form “ p is false.” Such a proposition, we
* We use “always” as meaning “in all cases,” not “at all times." Similarly “sometimes”
will mean “in some cases.”
42 INTRODUCTION [CHAP.
should bo inclined to say, must be false, because "p is false” is not always
true. Hence we should be led to the proposition
“'(/>). /> is false) is false,"
i.e. we should be led to a proposition in which "(/>)-p is false" is the argu¬
ment to the function "p is false," which we had declared to be impossible.
Now it will be seen that "(/>). P * s false." in the above, purports to be a
proposition about all prop»sitions, and that, by the general form of the vicious-
cirele principle, there must be no propositions about all propositions. Never¬
theless, it seems plain that, given any function, there is a proposition (true or
fals«*) asserting all its values. Hence we are led to the conclusion that “p is
false" and •*// is false' must not always In* the values, with the arguments p
and //, for a single function " p is false." This, however, is only possible if the
word "false" really has many different meanings, appropriate to propositions
of different kinds.
That the words "true" and "false” have many different meanings, accord¬
ing to the kind of proposition to which they arc applied, is not difficult to
see. Let us take any function tf>r. and let <f»i be one of its values. Let us call
the sort of truth which is applicable to <fm "first truth." (This is not to assume
that this would be first truth in another context: it is merely to indicate that
it is the first sort of truth in our context.) Consider now the proposition
(•>’)• If this has truth of the sort appropriate to it, that will mean that
• •very value <f>.r has "first, truth." Thus if we call the sort of truth that is
appropriate to (x ). “second truth." we may define "|(.r).<£.rj has second
truth" as meaning "every value for <f& has first truth," i.e. r has first
truth).” Similarly, if we denote by "( 3 /) . <£.» " the proposition "</>.r sometimes,
i.e. :is we may less accurately express it. "</>»• with some value of .r," we find
that (g.r). <f>.r has second truth if there is an .r with which (f>x has first truth ;
thus wo may define " |(g.r). <*>.r) has second truth" as meaning "some value
for <f>r has first truth," i.e. "(gx) . (</>.** has first truth)." Similar remarks apply
to falsehood. Thus "|(.r). has second falsehood" will mean "some value
for has first falsehood,” i.e. "(gx) . (<f>x has first falsehood)," while
" !<:-K) • «M has second falsehood" will mean "all values for <f>£ have first
falsehood," i.e. "(x).{<f>x has first falsehood)." Thus the sort of falsehood that
can belong to a general proposition is different from the sort that can belong
to a particular proposition.
Applying these considerations to the proposition “(p). p is false," we see
that the kind of falsehood in question must be specified. If, for example,
first falsehood is meant, the function "p has first falsehood" is only signi¬
ficant when p is the sort of proposition which has first falsehood or first
truth. Hence "(p). p is false” will be replaced by a statement which is
equivalent to "all propositions having either first truth or first falsehood
have first falsehood." This proposition has second falsehood, and is not
II]
TRUTH AND FALSEHOOD
•13
a possible argument to the function “p has first falsehood." Thus the
apparent exception to the principle that •*$ {(.r). <£.r)" must be meaningless
disappears.
Similar considerations will enable us to deal with - n«.t-/i" and with "p or 7 .”
It might seem as if these were functions in which any proposition m iRht
appear as argument. But this is due to a systematic ambiguity in the mean¬
ings of “not" and “or," by which they adapt themselves to propositions of any
order. To explain fully how this occurs, it will be well to begin with a
definition of the simplest kind of truth and falsehood*
The universe consists of objects having various qualities and standing
in various relations. Some of the objects which occur in the universe are
complex. When an object is complex, it consists of interrelated parts. Let
us consider a complex object composed of two parts a and b standing to each
other in the relation R. The complex object “a-in-the-relation-/2-to-6" may
be capable of being perceived ; when perceived, it is perceived as one object.
Attention may show that it is complex ; we then judge that a and b stand in
the relation Ji. Such a judgment, being derived from perception by mere
attention, may be called a “judgment of perception." This judgment of
perception, considered as an actual occurrence, is a relation of four terms,
namely a and b and Ji and the percipient. The perception, on the contrary, is
a relation of two terms, namely “a-in-the-rclation-Zi-to-^," and the percipient.
Since an object of perception cannot be nothing, we cannot perceive “a-in-the-
relation-ft-to- 6 ” unless a is in the relation R to b. Hence a judgment of
perception, according to the above definition, must be true. This does not
mean that, in a judgment which appears to us to be one of perception, we
are sure of not being in error, since we may err in thinking that our judgment
has really been derived merely by analysis of what was perceived. But if our
judgment has been so derived, it must be true. In fact, we may define truth,
where such judgments are concerned, as consisting in the fact that there is a
complex corresponding to the discursive thought which is the judgment. That is, ’
when we judge ,€ a has the relation R to b ," our judgment is said to be true
when there is a complex “a-in-the-relation-i 2 -to- 6 ,” and is said to be false
when this is not the case. This is a definition of truth and falsehood in rela¬
tion to judgments of this kind.
It will be seen that, according to the above account, a judgment does not
have a single object, namely the proposition, but has several interrelated
objects. That is to say, the relation which constitutes judgment is not a
relation of two terms, namely the judging mind and the proposition, but is a
relation of several terms, namely the mind and what are called the constituents
of the proposition. That is, when we judge (say) “this is red," what occurs
is a relation of three terms, the mind, and “this," and red. On the other hand,
when we perceive “the redness of this,” there is a relation of two terms, namely
‘It
INTRODUCTION
[CHAP.
tin,* mind ami the complex object "the redness of this." When a judgment
occurs, there is a certain complex entity, composed of the mind and the
various objects of the judgment. When the judgment is true, in the case of
the kind ot judgments we have been considering, there is a corresponding
complex of the ••hjects of the judgment alone. Falsehood, in regard to our
present class of judgments, consists in the absence of a corresponding complex
composed of the objects alone. It follows from the above theory that a
' propnxiiii.n." in the sense in which a proposition is supposed to be the object
ot a judgment. iN a fal^abstraction, because a judgment has several objects,
not one. It is the several ness of the objects in judgment (as opposed to
perception) which ln> led people to speak of thought as “discursive,” though
they do not appear to have realized clearly what was meant by this epithet.
Owing to the plurality «»t the objects of a single judgment, it follows that
what we call a “proposition' (in the sense in which this is distinguished from
the phrase expressing it) is not a single entity at all. That is to say, the phrase
which expii's^es a proposition is what we call an "incomplete" symbol*; it
dni-s not have meaning in itself, but requires some supplementation in order
to acquire a complete meaning. This fact is somewhat concealed by the
circumstance that judgment in itself supplies a sufficient supplement, and that
judgment in itself makes no rcrlml addition to the proposition. Thus “the
proposition Socrates is human"' uses "Somite* is human" in a way which
rei| ii ires a supplement of some kind Ik* fore it acquires a complete meaning;
but when I judge “Socrates is human," the meaning is completed by the act of
judging,and we no longer have an incomplete symbol. The fact that propositions
are" incomplete symbols ' is important philosophically.and is relevant at certain
points in symbolic logic.
The judgments we have been dealing with hitherto are such as arc* of the
same form as judgments of perception, i.e. their subjects are always particular
and definite. But there are many judgments which are not of this form. Such
* are “nil men are mortal," “I met a man,"“some men are Greeks.” Before
dealing with such judgments, we will introduce some technical terms.
We will give the* name of “a complex” to any such object as “a in the re¬
lation R to b " or “« having the quality 7 ," or “« and b and c standing in the
relation Broadly speaking, a complex is anything which occurs in the
universe and is not simple. We will call a judgment elementary when it
merely asserts such things as “a has the relation R to b ”" a has the quality q"
or "a and b and c stand in the relation S. ’ Then an elementary judgment is
true when there is a corresponding complex, and false when there is no corre¬
sponding complex.
But take now such a proposition as “all men are mortal.” Here the
judgment does not correspond to one complex, but to many, namely “Socrates
• Sec Chapter III.
GENERAL JUDGMENTS
II]
15
is mortal, "Plato is mortal, "Aristotle is mortal,'' etc. (For the moment, it
is unnecessary to inquire whether each of these docs not require further
treatment before we reach the ultimate complexes involved. For purposes of
illustration, "Socrates is mortal" is here treated as an elementary judgment,
though it is in fact not one. as will be explained later. Truly elementary
judgments are not very easily found.) We do not mean to deny that there
maybe some relation of the concept man to the concept mortal which maybe
equivalent to "all men are mortal,'* but in any case this relation is not. the
same thing as what we affirm when we say that all men are mortal. Our
judgment that all men are mortal collects together a number of elementary
judgments. It is not, however, composed of these, since ( e.g .) the fact that
Socrates is mortal is no part of what we assert, as may be seen by considering
the fact that our assertion can be understood by a person who has never heard
of Socrates. In order to understand the judgment "all men are mortal,” it is
not necessary to know what men there are. We must admit, therefore, as a
radically new kind of judgment, such general assertions as "all men are mortal.”
We assert that, given that x is human, a: is always mortal. That is, we assert
"x is mortal" of every x which is human. Thus we are able to judge (whether
truly or falsely) that all the objects which have some assigned property also
have some other assigned property. That is, given any propositional functions
4>x and yjr$, there is a judgment asserting \frx with every x for which we have
<f>x. Such judgments we will call general judgments.
It is evident (as explained above) that the definition of truth is different
in the case of general judgments from what it was in the ease of elementary
judgments. Let us call the meaning of truth which we gave for elementary
judgments "elementary truth." Then when we assert that it is true that all
men are mortal, we shall mean that all judgments of the form "x is mortal,”
where a: is a man, have elementary truth. We may define this as "truth of
the second order” or “second-order truth.” Then if we express the proposition
“all men are mortal” in the form
"(#) . x is mortal, where a: is a man,”
and call this judgment p, then "p is true” must be taken to mean "p has
second-order truth,” which in turn means
’ “(#) . *x is mortal’ has elementary truth, where x is a man.”
In order to avoid the necessity for stating explicitly the limitation to
which our variable is subject, it is convenient to replace the above interpre¬
tation of “all men are mortal" by a slightly different interpretation. The
proposition “all men are mortal” is equivalent to U€ x is a man’ implies 'x is
mortal,’ with all possible values of x.” Here x is not restricted to such values
are men, but may have any value with which “‘x is a man’ implies 'a; is
mortal’ ” is significant, i.e. either true or false. Such a proposition is called a
“formal implication.” The advantage of this form is that the values which the
variable may take are given by the function to which it is the argument: the
46 INTRODUCTION [CHAP.
values which the variable may take are all those with which the function is
significant.
We use the symbol “{.•■). <f>> " to express the general judgment which
asserts all judgments of the form Then the judgment “all men are
mortal is equivalent to
“(j-).'j- is a inau' implies \r is a mortal,”’
i.e. (in virtue of the definition of implication) to
•'(/) ..*• is not a man or x is mortal."
As we have just seen, the meaning of truth which is applicable to this pro-
pi icable to “x is a
.r). <f>r, the sense
that
ue w
» a single corre¬
sponding complex : tin- eoriespouding complexes are as numerous as the possible
values of x.
in which <f>x
len it points
position is not the same as the meaning of truth which is ap
man" or to “x is mortal." And generally, in any judgment (
in which this judgment i*» or may be true is not the same as
is or may he true. If 4>> is an elementary judgment, it is tr
tn a correspondin'' complex. But (.r). Ax does not imint t
It follows from the above that such a proposition as “all the judgments
made by Kpimenidos are true” will only be prima facie capable of truth if all
his judgments are of the same order. If they are of varying orders, of which
the ath is the highest, we may make a assertions of the form "all the judg¬
ments of order m made by Epimenides are true." where m has all values up
to n. But no such judgment can include itself in its own scope, since such a
judgment is always of higher order than the judgments to which it refers.
Let us consider next what is meant by the negation of a proposition of
the form We observe, to begin with, that in some cases,” or
"(fix sometimes." is a judgment which is on a par with "<fix in all cases, or
••</»./ always." The judgment "<*m sometimes" is true if one or more values of
x exist for which <fix is true. We will express the proposition "</> « sometimes
by the notation “(gx). where " 3 " stands for “there exists.” and the
whole symbol maybe read “there exists an x such that </m\” We take the
two kinds of judgment expressed by “(x).<Jm” and "(3 x). <fix as primitive
ideas. We also take as a primitive idea the negation of an elementary pro¬
position. We can then define the negations of (x). </>x and (g x). <fix. 'll * 0
negation of any proposition p will be denoted by the symbol “~p. Then the
negation of (x) . <f>x will be defined as meaning
"(3 x).~£x” m
and the negation of (gx). (f>x will be defined as meaning “(x) . ~ (fix. ^ “ ls>
in the traditional language of formal logic, the negation of a universal affir¬
mative is to be defined as the particular negative, and the negation of the
particular affirmative is to be defined as the universal negative. Hence the
meaning of negation for such propositions is different from the meaning o
negation for elementary propositions.
SYSTEMATIC AMBIGUITY
•17
n]
An analogous explanation will apply to disjunction. Consider the state¬
ment "either p, or <f>.v always.'' We will denote the disjunction of two
propositions p, q by "p v 7 ." Then our statement is "p . v . (./•). We will
suppose that p is an elementary proposition, and that- <f>.r is always an elemen¬
tary proposition. We take the disjunction of two elementary propositions as
a primitive idea, and we wish to define the disjunction
"P • v • (•*•) • </>•*•.*’
This may be defined as “(x) . p v <f>x," i.e. "either p is true, or <f>x is always true”
is to mean “ l p or <j>x' is always true." Similarly we will define
"P • v . (a-r> . <f> v"
as meaning "(g.r) .p v <f>x," i.e. we define “either p is true or there is an x
for which <f>x is true" as meaning “there is an .r for which either p or <f>.v is
true.’ Similarly we can define a disjunction of two universal propositions:
(f) • ^* r • v • (y) • will be defined as meaning “(x,y). <f>x v \J/y," i.e.
either <f>x is always true or yfry is always true” is to mean “* <f>x or \fry‘ is
always true." By this method we obtain definitions of disjunctions con¬
taining propositions of the form (x) . <£.v or (gx). <f>x in terms of disjunctions
of elementary propositions; but the meaning of "disjunction” is not the same
for propositions of the forms (#). <f>x, ( 3 -*) . <f*x, as it was for elementary pro¬
positions.
Similar explanations could be given for implication and conjunction, but
this is unnecessary, since these can be defined in terms of negation and
disjunction.
IV. Why a Given Function requires Arguments of a Certain Type.
The considerations so far adduced in favour of the view that a function
cannot significantly have as argument anything defined in terms of the
function itself have been more or less indirect. But a direct consideration
of the kinds of functions which have functions as arguments and the kinds
of functions which have arguments other than functions will show, if we are
not mistaken, that not only is it impossible for a function tf>2 to have itself
or anything derived from it as argument, but that, if yfrS is another function
such that there are arguments a with which both "<£a” and are sig¬
nificant, then yfrS and anything derived from it cannot significantly be
argument to </>2. This arises from the fact that a function is essentially
an ambiguity, and that, if it is to occur in a definite proposition, it must
occur in such a way that the ambiguity has disappeared, and a wholly
unambiguous statement has resulted. A few illustrations will make this clear.
Thus “( x ) . which we have already considered, is a function of <f>£; as soon
as «/>£ is assigned, we have a definite proposition, wholly free from ambiguity.
But it is obvious that we cannot substitute for the function something which
is not a function: means “4>x in all cases," and depends for its
significance upon the fact that there are “cases" of <f>x, i.e. upon the
INTRODUCTION
(CHAP.
ambiguity which is characteristic of a function. This instance illustrates
the fact that, when a (unction can occur significantly as argument.something
which is not a function cannot occur significantly as argument. But con¬
versely. when something which is not a function can occur significantly
as argument, a function cannot occur significantly. Take, e.y. "x is a man.
ami consider " <f>r is a man." Here there is nothing to eliminate the
ambiguity which constitutes <f>r\ there is thus nothing definite which is
said to be a man. A function, in fact, is not a definite object, which could
be or not b«- a man; ir is a mere ambiguity awaiting determination, and
in ordt*r that it may occur significantly it must receive the necessary deter¬
mination. which it obviously does not receive if it is merely substituted
for something determinate in a i»ro|>osition•. This argument does not, how¬
ever, apply directly as against such a statement as " <f>x) is a man.
Common sense would pronounce such a statement to be meaningless, blit it
cannot be condemned on the ground of ambiguity in its subject. Wc need
here a new objection, namely the following: A proposition is not a single entity,
but a relation of several; hence a statement in which a proposition appeal's
as subject will only be significant if it can 1 m- reduced to a statement about
the terms which appear in the proposition. A proposition, like such phrases
as"the so-and-so.” where grammatically it appears as subject, must be broken
up into its constituents if wc are to find the true subject or subjectst.. But
in such a statement as "p is a man." where /> is a proposition, this is not
]M)ssible. Hence 4 >r < > s 51 uian" is meaningless.
V. The Hierarchy of Functions and J’ropositions.
Wc are thus led to the conclusion, both from the vicious-circlc principle
and from direct inspection, that the functions to which a given object a can
be an argument are incapable of being arguments to each other, and that they
have no term in common with the functions to which they can be arguments.
We are thus led to construct a hierarchy. Beginning with a and the other
terms which can be arguments to the same functions to which a can be argu¬
ment, wc come next to functions to which a is a possible argument, and then
to functions to which such functions are possible arguments, and so on. But
the hierarchy which has to be constructed is not so simple as might at first
appear. The functions which can take a as argument form an illegitimate
totality, and themselves require division into a hierarchy of functions, 'lhis
is easily seen as follows. Let /(<J>2, x) be a function of the two variables tf>3
and x. Then if, keeping x fixed for the moment, we assert this with all possible
values of <f>, we obtain a proposition:
(<*>) -/(^3, x).
• Note that statements concerning the significance of a phrase containing "•p: " concern the
ujmbol ami therefore do not fall under the rulo that the elimination of tlio functional
ambiguity is necessary to significance. Significance is a property of signs. Cf. pp. 40, 41.
t Cf. Chapter III.
THE HIERARCHY OF FUNCTIONS
10
n]
Here, it .r is variable, wo have a function of .r; but as this function involves
a totality ot values ot <f>~ 9 , it cannot itself be one of the values included in
the totality, by the vicious-circle principle. It follows that the totality of values
of tf)s concerned in (<£). /(<£?, x) is not the totality of all functions in which
x can occur as argument, and that there is no such totality as that of all func¬
tions in which .v can occur as argument.
It follows from the above that a function in which </>* appears as argument
requires that should not stand for any function which is capable of a
given argument, but must be restricted in such a way that none of the
functions which are possible values of should involve any reference to
the totality of such functions. Let us take as an illustration the definition
of identity. We might attempt to define 4 *.r is identical with y" as meaning
"whatever is true of x is true of y," i.e. "</>.*• always implies </>//.” But here,
since we are concerned to assert all values of u <f>.r implies <f>y ” regarded as a
function of </>, we shall be compelled to impose upon <f> some limitation which
will prevent us from including among values of <f> values in which "all possible
values of <p” are referred to. Thus for example “x is identical with a" is a
function of x\ hence, if it is a legitimate value of <f> in “<f>x always implies
<t>y," we shall be able to infer, by means of the above definition, that if is
identical with a, and x is identical with y, then y is identical with a.
Although the conclusion is sound, the reasoning embodies a vicious-circle
fallacy, since we have taken “(<£). <f>x implies <f>a" as a possible value of <f>x,
which it cannot be. If, however, we impose any limitation upon </>, it may
happen, so far as appears at present, that with other values of </> we might
have <f>x true and <f>y false, so that our proposed definition of identity would
plainly be wrong. This difficulty is avoided by the "axiom of reducibility,”
to be explained later. For the present, it is only mentioned in order to
illustrate the necessity and the relevance of the hierarchy of functions of a
given argument.
Let us give the name “a-functions" to functions that are significant for a
given argument a. Then suppose we take any selection of a-functions, and
consider the proposition "a satisfies all the functions belonging to the selection
in question.” If we here replace a by a variable, we obtain an a-function; but
by the vicious-circle principle this a-function cannot be a member of our
selection, since it refers to the whole of the selection. Let the selection consist
of all those functions which satisfy f Then our new function is
(£)• (/(£*) implies <M-
where x is the argument. It thus appears that, whatever selection of
a-funotions we may make, there will be other a-functions that lie outside our
* When we speak of “values of <p2” it is <p, not *, that is to be assigned. This follows from
the explanation in the note on p. 40. When the function itself is the variable, it is possible and
simpler to write <p rather than <p£, except in positions where it is necessary to emphasize that an
argument must be supplied to secure significance.
B&w 1
4
50
INTRODUCTION
- [CHAP.
selection. Such '/-functions, as the al»ove instance illustrates, will always
arise through taking a function of two arguments. <f >3 and x ,and asserting all
some of the values resulting from varying <f>. What is necessary, therefore,
in order to avoid vicious-circle fallacies, is to divide our //-functions into
"types," each of which contain> no functions which refer to the whole of that
type.
When something is assorted or denied about all possible values or about
some (undetermined) possible values of a variable, that variable is called
after IV.ino. The presence of the words at/ or some in a proposition
indicates the presence of an apparent variable; but often an apparent variable
is really present where- language does not at once indicate its presence. Thus
for example "A is mortal" means "there is a time at which A will die.” Thus
a variable time occurs as apparent variable.
The clearest, instances «.f propositions not containing apparent variables
are such as express immediate judgments of perception, such as "this is red"
or "this is painful." where "this" is something immediately given. In other
judgments, even where at first sight no variable appears to be present, it
olten happens that then- really is one. Take (say) ".Socrates is human." To
Socrates himself, the won I "Socrates” no doubt stood for an object of which
ho tt’sw immediately aware, and the judgment "Socrates is human" contained
no apparent variable. But to us, who only know Socrates by description, tin-
word Socrates cannot mean what it meant to him; it means rather "the
person having such-and-such properties," (say)“ the Athenian philosopher who
drank the hemlock." Now in all propositions about "the so-and-so" there is
an apparent variable,as will 1 m- shown in Chapter III. Thus in what we have
in mind when we say "Socrates is human" there is an apparent variable,
though there was no apparent variable in the corresponding judgment as
made by Socrates, provided we assume that there is such a thing as immediate
awareness of oneself.
W hatever may be the instances of propositions not containing apparent
variables, it is obvious that propositional functions whose values do not contain
apparent variables arc the source of propositions containing apparent variables,
in the sense in which the function 0a 1 is the source of the proposition (.r). <f>x.
For the values for <f>x do not contain the apparent variable ./•, which appears
in (x).tf>x\ it they contain an apparent variable y, this can be similarly
eliminated, and so on. This process must come to an end, since no proposition
which we can apprehend can contain more than a finite number of apparent
variables, on the ground that whatever we can apprehend must be of finite
complexity. Thus we must arrive at last at a function of as many variables
as there have been stages in reaching it from our original proposition, and
this function will be such that its values contain no apparent variables. We
may call this function the matrix of our original proposition and of any other
MATRICES
51
«]
propositions and tunctions to be obtained l»y turning some ot’ the arguments
to the function into apparent variables. Thus for example, if we have a matrix-
function whose values are <f> (.r, y), we shall derive from it
(y) . <f> (.r, y\ which is a function of .r.
(.r). <f> (.r, y), which is a function of //.
$ (* 1 *. y), meaning “ <f> (.r, y) is true with all possible values of .r and y."
This last is a proposition containing no real variable, i.e. no variable except
apparent variables.
It is thus plain that all possible propositions and functions are obtainable
from matrices by the process of turning the arguments to the matrices into
apparent variables. In order to divide our propositions and functions into types,
we shall, therefore, start from matrices, and consider how they are to be divided
with a view to the avoidance of vicious-circle fallacies in the definitions of the
functions concerned. For this purpose, we will use such letters as a , b, c, .v, y, z, w,
to denote objects which are neither propositions nor functions. Such objects
we shall call individuals. Such objects will be constituents of propositions or
functions, and will be genuine constituents, in the sense that they do not
disappear on analysis, as (for example) classes do, or phrases of the form “the
so-and-so.”
The first matrices that occur are those whose values are of the forms
<f>x, yfr (x, y), *(x,y,* ...),
t.e. where the arguments, however many there may be, are all individuals.
The functions (f> , yjr, x since (by definition) they contain no apparent
variables, and have no arguments except individuals, do not presuppose any
totality of functions. From the functions \fr, x ••• we may proceed to form
other functions of *, such as (y) . yfr (x, y), (gy) . y/r (x, y), (y, z) . x (*, 'J> *)•
(y) : (a*) • X( x > V* z )> and so on. All these presuppose no totality except that
of individuals. We thus arrive at a certain collection of functions of x,
characterized by the fact that they involve no variables except individuals.
Such functions we will call "first order functions.”
We may now introduce a notation to express “any first-order function.”
We will denote any first-order function by “<£!£” and any value for such a
function by “<£ ! x” Thus "<f> ! x" stands for any value for any function which
involves no variables except individuals. It will be seen that “ <f >! x” is itself
a function of two variables, namely <f >! 2 and x. Thus <f >! x involves a variable
which is not an individual, namely <f >! 2 . Similarly “( x ) . <f> l x” is a function
of the variable <f >! 5, and thus involves a variable other than an individual.
Again, if a is a given individual,
“<f>lx implies <f >! a with all possible values of <f>"
18 a function of x, but it is not a function of the form <ft ! x, because it involves
an (apparent) variable <f> which is not an individual. Let us give the name
"predicate” to any first-order function <f> ! at. (This use of the word “predicate”
52
INTRODUCTION
[CHAP.
is only proposed for the pur|»oses of the present, discussion.) Then the state¬
ment "<f >! .r implies <f> ! a with all possible values of <f>' may be read “all the
predicates of x are predicates of a. This makes a statement about .r, but does
not attribute to ./• a prei/icate in the special sense just defined.
Owing to the introduction of the variable first-order function <f >! 2, we
now have a new set of matrices. Thus "<£!.r" is a function which contains no
apparent variables,but contains the two real variables <f>! 3 and x. (It should
be observed that when <f> is assigned, we may obtain a function whose values do
involve individuals as apparent variables, for example if <f> lx is (//). yfr(x,y).
But so long as <f> is variable, <t >! x contains no apparent variables.) Again,
if o is a definite individual. <f>la is a function of the one variable <f> l z.
If n and li are definite individuals, implies yfr l b" is a function of the
t wo variables <f >! 3, yfr ! 3, and so on. We are thus led to a whole set of new
matrices,
f (<f >! 2). <n<f>lz,yfrl 2 ). /’(</>! 2 , .#•), and so oil.
These matrices contain individuals and first-order functions as arguments, but
(like all matrices) they contain no apparent variables. Any such matrix, if it
contains more than one variable, gives rise to new functions of one variable
by turning all its arguments except one into apparent variables. Thus we
obtain the functions
(<f>). g (<f> l 3 , yfr ! 3 ). which is a function of yfr ! 3.
(•') - /’’<0 ! 3. x), which is a function of <f >! 3.
(</>). /-’(«/>! 3, x), which is a function of .r.
We will give the name of nee mil-order matrices to such matrices as have
first-order functions among their arguments, and have no arguments except
first-order functions and individuals. (It is not necessary that they should
have individuals among their arguments.) We will give the name of second-
order functions to such as either are second-order matrices or are derived from
such matrices by turning some of the arguments into apparent variables. It
will be seen that either an individual or a first-order function may appear as
argument to a second-order function. Second-order functions are such as con¬
tain variables which are first-order functions, but contain no other variables
except (possibly) individuals.
We now have various new classes of functions at our command. In the first
place, we have second-order functions which have one argument which is a
first-order function. We will denote a variable function of this kind by the
notation /*!(<£! 3), and any value of such a function by _/’!(</>! 3). Like
if) ! x, f ! (</>! 3 ) is a function of two variables, namely f\ (<£ ! 3) and if> ! 3. Among
possible values of f!(<f>l$) will be <f> ! a (where a is constant), ( x).<f>lx,
(yx) .if) lx, and so on. (These result from assigning a value to /, leaving
</> to be assigned.) We will call such functions “predicative functions of
first-order functions.”
n]
SECOND-ORDER FUNCTIONS
►3
In the second place, we have second-order functions of two arguments, one
of which is a first-order function while t he other is an individual. Let us denote
undetermined values of such functions by the notation
/! (0 ! 3 . .*•>.
As soon as x is assigned, we shall have a predicative function of <f>lc. If our
function contains no first-order function as apparent variable, we shall obtain
a predicative function of* if we assign a value to <f >! j. Thus, to take the
simplest possible case, if/! ( <f> ! z,x) is <f >! .r.the assignment of a value to </> gives
us a predicative function of x, in virtue of the definition of " <f> lx." But if
./-(</>! z, x) contains a first-order function as apparent variable, the assignment
ot a value to <f >! z gives us a second-order function of x.
In the third place, we have second-order functions of individuals. These
will all be derived from functions of the form /'! (<f >! s t x) by turning </> into an
apparent variable. We do not, therefore, need a new notation for them.
We have also second-order functions of two first-order functions, or of two
such functions and an individual, and so on.
We may now proceed in exactly the same way to third-order matrices,
which will be functions containing second-order functions as arguments, and
containing no apparent variables, and no arguments except individuals and
first-order functions and second-order functions. Thence we shall proceed, as
before, to third-order functions; and so we can proceed indefinitely. If the
highest order of variable occurring in a function, whether as argument or ns
apparent variable, is a function of the nth order, then the function in which
it occurs is of the n + 1th order. We do not arrive at functions of an infinite
order, because the number of arguments and of apparent variables in a function
must be finite, and therefore every function must be of a finite order. Since
the orders of functions are only defined step by step, there can be no process
of "proceeding to the limit/’ and functions of an infinite order cannot occur.
We will define a function of one variable as predicative when it is of the
next order above that of its argument, i.e. of the lowest order compatible with
its having that argument. If a function has several arguments, and the highest
order of function occurring among the arguments is the nth, we call the function
predicative if it is of the n -t- 1th order, i.e. again, if it is of the lowest order
compatible with its having the arguments it has. A function of several
arguments is predicative if there is one of its arguments such that, when the
other arguments have values assigned to them, we obtain a predicative function
of the one undetermined argument.
It is important to observe that all possible functions in the above hierarchy
can be obtained by means of predicative functions and apparent variables. Thus >
as we saw, second-order functions of an individual x are of the form
; ( <t >) •/! (<*> 1 2. *) or (a*) . / l(<f> l 2, x) OT (<*», +) .f\ (<f> l 2, +! 2, x) or etc.,
where f is a second-order predicative function. And speaking generally, a
54
INTRODUCTION
[CHAP.
non-predicative function ot the wth order is obtained from a predicative function
of the //til order by tinning nil the arguments of the n — 1 th order into apparent
variables. <( )t her arguments also may Im- turned into apparent variables.) Thus
we need not introduce as variables any functions except predicative functions.
Moreover, to obtain any function ot one variable ./*, we need not go beyond
predicative functions ot tiro variables. For the function (yjr ). /'!(</>! 2 , yfs lz,.r),
where j is given, is a function of 0 ! 2 ami ./, and is predicative. Thus it is of
the form !(0! and therefore (0. yfr )./! <0 ! 3, ! 2,.r) is of the form
(0). /’! (0 ! 3,./ ». Thus shaking generally, by a succession of steps we find that,
if 0! a is a predicative function of a sufficiently high order, any
predicative function of.#- will be of one of the two forms
assigned non-
( 0 ) ./’!(</»! f/..r), <'.| 0 ). /’! (0 ! «,*•),
where F is a predicative function of 0 ! n and ./-.
I'he nature of the above hierarchy of functions maybe restated as follows.
A function, as we saw at an earlier stage, prestip|x»scs as part of its meaning
the totality of its values, or, what comes to the same thing, the totality of
its possible mguments. The arguments to a function may be functions or
propositions or individuals. (It will be remembered that individuals were
defined as whatever is neither a proposition nor a function.) For the present
we neglect the case in which the argument to a function is a proposition.
Consider a function whose argument is an individual. This function pre¬
supposes the totality of individuals: but unless it contains functions as
apparent variables, it does not presuppose any totality of functions. If,
however, it does contain a function as apparent variable, then it cannot
be defined until some totality of functions has been defined. It follows that
we must first define the totality of those functions that have individuals
as arguments and contain no functions as apparent variables. These are
the predicative functions of individuals. Generally, a predicative function
of a variable argument is one which involves no totality except that of
the possible values of the argument, and those that are presupposed by any
one of the possible arguments. Thus a predicative function of a variable
argument is any function which can be specified without introducing new
kinds of variables not necessarily presupposed by the variable which is the
argument.
A closely analogous treatment can be developed for propositions. Pro¬
positions which contain no functions and no apparent variables may be called
elementary propositions. Propositions which are not elementary, which contain
no functions, and no apparent variables except individuals, may be called
first-order propositions. (It should be observed that no variables except
apparent variables can occur in a proposition, since whatever contains a j'eal
variable is a function, not a proposition.) Thus elementary and first-order
propositions will be values of first-order functions. (It should be remembered
n]
THE AXIOM OP KEDUCIHIMTY
r»r>
that a function is not a constituent in one of its values: thus for example
the function is human ” is not a constituent of the proposition “Socrates
is human. ) Elementary and first-order propositions presuppose no totality
except (at most) the totality of individuals. The} • are of one or other of the
three forms
9! .v ; (.r) . <f >! .r; (g.r) . <f >! .r.
where <f> l x is a predicative function of an individual. If follows that, if />
represents a variable elementary proposition or a variable first-order propo¬
sition, a function fp is either/(^»! .«•) or/|(.r) . <f >! .«•) or/(<g.r) . cf >! .»•). Thus
a function of an elementary or a first-order proposition may always be reduced
to a function of a first-order function. It follows that a proposition involving
the totality of first-order propositions may be reduced to one involving the
totality of first-order functions; and this obviously applies equally to higher
orders. The propositional hierarchy can, therefore, be derived from the
functional hierarchy, and we may define a proposition of the nth order as
one which involves an apparent variable of the n — 1th order iu the functional
hierarchy. The propositional hierarchy is never required in practice, and is
only relevant for the solution of paradoxes; hence it is unnecessary to go into
further detail as to the types of propositions.
VI. The Axiom of Reducibility.
It remains to consider the “axiom of reducibility.” It will be seen that,
according to the above hierarchy, no statement can be made significantly
about “all a-functions,” where a is some given object. Thus such a notion
as “all properties Of a,” meaning “all functions which are true with the
argument a,” will be illegitimate. We shall have to distinguish the order
of function concerned. We can speak of “ all predicative properties of a," “ all
second-order properties of a,” and so on. (If a is not an individual, but an
object of order n, “second-order properties of a” will mean “functions of
order n + 2 satisfied by a.”) But we cannot speak of “ all properties of a.”
In some cases, we can see that some statement will hold of “ all uth-order
properties of a,” whatever value n may have. In such cases, no practical
harm results from regarding the statement as being about “ all properties of
a,” provided we remember that it is really a number of statements, and not
a single statement which could be regarded as assigning another property to
a, over and above all properties. Such cases will always involve some syste¬
matic ambiguity, such as that involved in the meaning of the word “truth,”
as explained above. Owing to this systematic ambiguity, it will be possible,
sometimes, to combine into a single verbal statement what are really a number
of different statements, corresponding to different orders in the hierarchy.
•This is illustrated in the case of the liar, where the statement “all A’s
statements are false ” should be broken up into different statements referring
to his statements of various orders, and attributing to each the appropriate
kind of falsehood.
INTRODUCTION
[CHAP.
50
The axiom of rcducibility is introduced in order to legitimate a great
mass of reasoning, in which, primn facie, we are concerned with such notions
as “all properties of a " or “all o-functions.’ and in which, nevertheless, it
seems scarcely possible to suspect any substantial error. In order to state
tin* axiom, we must first define what is meant by “formal equivalence." Two
functions </m. 0.2 are said to be * formally equivalent" when, with every possible
argument x, <f>.r is equivalent to \J/s, i.e. and are either both true or
both false. Thus two functions are formally equivalent when they arc satisfied
by the same set arguments. The axiom of rcducibility is the assumption
that, given any function </>.“. there is a formally equivalent predicative function,
i.e. there is a predicative function which is true when (f>.r is true and false
when <f>> is false. In symbols, the axiom is:
I- : <3^r): <t>> . =,. 0 i-r.
I'or t\\M variables, wv require a similar axiom, namely: Given any function
//), there is a formally equivalent predicative function, i.e.
H : (:•!>/'): 0( r. y>. . 0 ! (.r. //).
In order to explain t he pm |>oses of the axiom of rcducibility, and the nature
of the grounds for supposing it true, we shall first illustrate it by applying it
to some particular cases.
If we call a predicate of an object a predicative function which is true of
t hat object, then the predicates of an object are only some among its properties.
Take for example such a projs.sition as " Napoleon had all the qualities that
make a great general." We may interpret this as meaning "Napoleon had all
the predicates that make a great general." Here there is a predicate which is
an apparent variable. If we put “/(<£ ! 2)" for “0! 2 is a predicate required
in a great general." our projiositioii is
( <f >>: f(<f >! 2) implies 0 ! (Napoleon).
Since this refers to a totality of predicates, it is not itself a predicate of
Napoleon. 11 by no means follows, however, that there is not some one predicate
common and peculiar to great generals. In fact, it is certain that there is such
a predicate. For the number of great generals is finite, and each of them
certainly possessed some predicate not possessed by any other human being
—for example, the exact instant of his birth. The disjunction of such predicates
will constitute a predicate common and peculiar to great generals*. If we
call this predicate 0-! 2, the statement we made about Napoleon was equi¬
valent to 0 ! (Napoleon). And this equivalence holds equally if wo substitute
any other individual for Nai>oleon. Thus we have arrived at a predicate which
is always equivalent to the property we ascribed to Napoleon, i.e. it belongs
to those objects which have this property, and to no others. The axiom of
rcducibility states that such a predicate always exists, i.e. that any property
• When u (finite) set of predicates is given by actual enumeration, their disjunction is a
predicate, because no predicate occurs as apparent variable in the disjunction.
Ix ] THE AXIOM OF RKIWCIIHLITY 57
of nn object belongs to the same collection of objects as those that possess
some predicate.
We may next illustrate our principle by its application to identity. In
this connection, it has a certain affinity with Leibni/.'s identity of indiscernibles.
It is plain that, if .r and y are identical, and <f>.v is true, then <f>y is true. Here
it cannot matter what sort of function <f>.e may be: the statement must hold
for any function. But we cannot say, conversely: “If, with all values of <f>.
<px implies <f>y, then .r and .y are identical"; because “all values of </>” is
inadmissible. If we wish to speak of “all values of <£," we must confine
ourselves to functions of one order. Wo may confine <f> to predicates, or to
second-order functions, or to functions of any order we please. But we must
necessarily leave out functions of all but one order. Thus we shall obtain, so
to speak, a hierarchy of different degrees of identity. We may say “all the
predicates of x belong to y," “all second-order properties of x belong to //,"
and so on. Each of these statements implies all its predecessors: for
example, if all second-order properties of x belong to y, then all predicates
of x belong to y, for to have all the predicates of x is a second-order property,
and this property belongs to x. But we cannot, without the help of an axiom,
argue conversely that it all the predicates of x belong to ;/, all the second-order
properties of x must also belong to y. Thus we cannot, without the help of
an axiom, be sure that x and y arc identical if they have the same predicates.
Leibniz’s identity of indiscernibles supplied this axiom. It should be observed
that by" indiscernibles” he cannot have meant two objects which agree as to
dll their properties, for one of the properties of x is to be identical with x,
and therefore this property would necessarily belong to y if x and y agreed
in all their properties. Some limitation of the common properties necessary
to make things indiscernible is therefore implied by the necessity of an axiom.
For purposes of illustration (not of interpreting Leibniz) we may suppose the
common properties required for indiscernibility to be limited to predicates.
Then the identity of indiscernibles will state that if x and y agree as to
all their predicates, they are identical. This can be proved if we assume the
axiom of reducibility. For, in that case, every property belongs to the same
collection of objects as is defined by some predicate. Hence there is some
predicate common and peculiar to the objects which are identical with x.
This predicate belongs to x, since x is identical with itself; hence it belongs
to y, since y has all the predicates of x; hence y is identical with x. It
follows that we may define x and y as identical when all the predicates of x
belong to y, i.e. when ( <f >) : tf> l x . D . <f >! y. We therefore adopt the following
definition of identity*:
= : (<p) : <f> l x . D . <f >! y Df.
* Note that in this definition the second sign of equality is to be regarded as combining with
J ‘Df" to form one symbol; what ia defined is the sign of equality not followed by the letters “Df.”
?• : . . ..
58
INTRODUCTION*
[CHAP.
But apart from the axiom «f reducibility, or some axiom equivalent in this
connection, we should be compelled to regard identity as indefinable, and to
admit (what seems impossible) that two objects may agree in all their pre¬
dicates without being identical.
Tin* axiom of reducibility is even more essential in the theory of classes.
It should be observed, in the first place, that if we assume the existence of
classes, the axiom of reducibility can be proved. For in that case, given any
function <f>: of whatever order, there is a class a consisting of just those
objects which satisfy </>3. Hence is equivalent to “x belongs to a."
But " x belongs to a ” is a statement containing no apparent variable, and is
therefore a predicative function of .»•. Hence if we assume the existence ot
classes, the axiom of reducibility becomes unnecessary. The assumption ot
the axiom of reducibility is therefore a smaller assumption than the assump¬
tion that there are classes. This latter assumption has hitherto been made
unhesitatingly. However, both on the ground of the contradictions, which
require a more complicated treatment if classes are assumed, and on the ground
that it is always well to make tin* smallest assumption required for proving
our theorems, we prefer to assume the axiom of reducibility rather than the
existence of classes. But in order to explain the use of tin* axiom in dealing
with classes, it. is necessary first to explain the theory of classes, which is a
topic belonging to Chapter III. We therefore postpone to that Chapter the
explanation of the use of our axiom in dealing with classes.
It is worth while to note that all the pur|K>ses served by the axiom of
reducibility are equally well served if we assume that t here is always a function
of the nth order (where n is fixed) which is formally equivalent to <f>?. what¬
ever may be the order of Here we shall mean by "a function of the nth
order" a function of the nth order relative to the arguments to thus if
these arguments are absolutely of the /nth order, we assume the existence of
a function formally equivalent to whose absolute order is the m + nth. The
axiom of reducibility in the form assumed above takes n = 1, but this is not
necessary to the use of the axiom. It is also unnecessary that n should be the
same for different values of ni; what is necessary is that n should be constant
so long as m is constant. What is needed is that, where extcnsional functions
of functions are concerned, wc should be able to deal with any n-function by
means of some formally equivalent function of a given type, so as to be able
to obtain results which would otherwise require the illegitimate notion of
“all ({-functions ”; but it does not matter what the given type is. It docs
not appear, however, that the axiom of reducibility is rendered appreciably
more plausible by being put in the above more general but more complicated
form.
The axiom of reducibility is equivalent to the assumption that “any
II]
THE AXIOM OF RKIWCIMLITY
combination or disjunction of predicates* is equivalent to a single pivdicale,”
i.e. to the assumption that, if we assert that .r has all the predicates that,
satisfy a function f (^>! ?), there is some one predicate which will have
whenever our assertion is true, and will not have whenever it is false, and
similarly if we assert that .r has some one of the predicates t hat satisfy a function
.t For by means of this assumption, the orderof anon-predicative function
can be lowered by one; hence, after some finite number of steps, we shall be able
to get from any non-predicative function to a formally equivalent predicative
function. It does not seem probable that the above assumption could be
substituted for the axiom of rcducibility in symbolic deductions, since its use
would require the explicit introduction of the further assumption that by a
finite number of downward steps we can pass from any function to a predicative
function, and this assumption could not well be made without developments
that are scarcely possible at an early stage. But on the above grounds it seems
plain that in fact, if the above alternative axiom is true, so is the axiom of
redueibility. The converse, which completes the proof of equivalence, is of
course evident.
VII. Reasons for Accepting the Axiom of Reducibilitg.
That the axiom of redueibility is self-evident is a proposition which can
hardly be maintained. But in fact self-evidence is never more than a part of
the reason for accepting an axiom, and is never indispensable. The reason
for accepting an axiom, as for accepting any other proposition, is always
largely inductive, namely that many propositions which are nearly indubitable
can be deduced from it, and that no equally plausible way is known by which
these propositions could be true if the axiom were false, and nothing which is
probably false can be deduced from it. If the axiom is apparently self-evident,
that only means, practically, that it is nearly indubitable; for things have
been thought to be self-evident and have yet turned out to be false. And if
the axiom itself is nearly indubitable, that merely adds to the inductive
evidence derived from the fact that its consequences are nearly indubitable :
it does not provide new evidence of a radically different kind. Infallibility is
never attainable, and therefore some element of doubt should always attach
to every axiom and to all its consequences. In formal logic, the element of
doubt is less than in most sciences, but it is not absent, as appears from the
fact that the paradoxes followed from premisses which were not previously
known to require limitations. In the case of the axiom of rcducibility, the
inductive evidence in its favour is very strong, since the reasonings which it
permits and the results to which it leads are all such as appear valid. But
although it seems very improbable that the axiom should turn out to be false,
* Here the combination or disjunction is supposed to be given intensionully. If given exten-
sionally (t.e. by enumeration), no assumption is required; but in this case the number of
predicates concerned mast be finite.
GO
INTRODUCTION'
[CHAP.
it is by no means improbable that it should be found to be deductible from
some other more fundamental and more evident axiom. It is possible that the
use of the vicious-circle principle, as embodied in the above hierarchy of types,
is more drastic than it need be, and that by a less drastic use the necessity
for the axiom might be avoided. Such changes, however, would not render
anything false which had been asserted on the basis of the principles explained
above: they would merely provide easier proofs of the same theorems. There
would seem, therefore, to Ik* but the slenderest ground for fearing that the
use of (he axiom of redueibility may load us into error.
V 111. The Contradictions.
We are now in a position to show how the theory of types affects the
.solution of the contradictions which have beset mathematical logic. For this
purpose, we shall begin by an enumeration of some of the more important and
illustrative of these contradictions, and shall then show how they all embody
vicious-circle fallacies, and are therefore all avoided by the theory of types. It
will he noticed that these paradoxes do not relate exclusively to the ideas of
numher and (plantiiy. Accordingly no solution can be adequate which seeks
to explain them merely as the result of some illegitimate use of these ideas.
The solution must he sought in sonic such scrutiny of fundamental logical
ideas as has been attempted in the foregoing pages.
(I) The oldest contradiction of the kind in question is the Rpimenides.
lCpiineiiides the ('retail said that all Cretans were liars, and all other state¬
ments made by ('retans were certainly lies. Was this a lie? The simplest form
of this contradiction is afforded by the man who says "I am lying”; if he is
lying, lie is speaking the truth, and vice versa.
(*2> Let iv he the class of all those classes which are not members of
themselves. Then, whatever class .r may be, "x is a ir" is equivalent to “.r is
not an .r.” Hence, giving to x the value tv, "w is a w ” is equivalent to
" to is not a to."
(.'*) Let T be the relation which subsists between two relations R and 6*
whenever R does not have the relation R to S. Then, whatever relations
R and S may be. " R has the relation T to S" is equivalent to "R does not
have the relation R to S." Hence, giving the value T to both R and S,
"T has the relation T to T " is equivalent to "T does not have the relation
'it . 'it »*
1 to /.
(4) Burali-Forti's contradiction* may be stated as follows: It can be
shown that every well-ordered scries has an ordinal number, that the series of
ordinals up to and including any given ordinal exceeds the given ordinal by
one, and (on certain very natural assumptions) that the scries of all ordinals
(in order of magnitude) is well-ordered. It follows that the series of all
• "Unu questiono sui mimcri trnnsfiniti,” JRendieonti del circolo maUmatico di Palermo, Vol.
xi. (1897). See -258.
ENUMERATION OF CONTRADICTION'S
<)1
n]
ordinals has an ordinal number, fl say. But in that case the series of all
ordinals including H has the ordinal number 11 + 1 . which must be greater
than 12. Hence fl is not the ordinal number of all ordinals.
(5) The number of syllables in the English names of finite integers
tends to increase as the integers grow larger, and must gradually increase
indefinitely, since only a finite number of names can be made with a given
finite number ot syllables. Hence the names of some integers must consist of
at least nineteen syllables, and among these there must be a least. Hence "the
least integer not nameable in fewer than nineteen syllables" must denote a
definite integer; in fact, it denotes 111,777. But “the least integer not
nameable in fewer than nineteen syllables" is itself a name consisting of
eighteen syllables; hence the least integer not nameable in fewer than nine¬
teen syllables can be named in eighteen syllables, which is a contradiction*.
(6) Among transfinite ordinals some can be defined, while others can not;
for the total number of possible definitions is N 0 f, while the number of trans¬
finite ordinals exceeds N 0 . Hence there must be indefinable ordinals, and
among these there must be a least. But this is defined as “ the least indefinable
ordinal," which is a contradiction*.
(7) Richard’s paradox§ is akin to that of the least indefinable ordinal. It
is as follows: Consider all decimals that can be defined by means of a finite
number of words; let E be the class of such decimals. Then E has N 0 terms;
hence its members can be ordered as the 1st, 2nd, 3rd. Let N be a number
defined as follows: If the fith figure in the nth decimal is p, let the nth
figure in N be p + 1 (or 0, if p = 9). Then N is different from all the members
of E, since, whatever finite value n may have, the nth figure in N is different
from the nth figure in the nth of the decimals composing E, and therefore N
is different from the nth decimal. Nevertheless we have defined N in a finite
number of words, and therefore N ought to be a member of E. Thus A r both
is and is not a member of E.
In all the above contradictions (which are merely selections from an
indefinite number) there is a common characteristic, which we may describe
as self-reference or reflexiveness. The remark of Epimenides must include
itself in its own scope. If all classes, provided they are not members of them¬
selves, are members of w, this must also apply to w\ and similarly for the
* This contradiction was suggested to us by Mr O. O. Berry of the Bodleian Library.
t Ro * 8 the number of finite integers. See «123.
X Cf. Kdnig, "Ueber die Ornndlagen der Mengenlehre nnd dan Kontinuumproblem,” Math.
Annalen, Vol. mi. (1905); A. C. Dixon, "On ‘well-ordered’ aggregates," Proc. London Math.
Soc. Series 2, Vol. iv. Part I. (1906); and E. W. Hobson, "On the Arithmetic Continuum," ibid.
The eolation offered in the last of these papers depends upon the variation of the " apparatus of
definition," and is thus in outline in agreement with the solution adopted here. But it does not
invalidate the statement in the text, if "definition” is given a constant meaning.
§ Of. Poincare, "Les math£matiquea et la logique,” Revue de Mitaphyeique et de Morale,
Mai 1906, especially sections vn. and ix.; also Peano, Reoista de Mathematica, Vol. vm. No. 6
(1906), p. 149 fl.
62
INTRODUCTION
[CHAP.
analogous relational contradiction. In the cases of names and definitions, the
paradoxes result from considering non-nameability and indefinability as ele¬
ments in names and definitions. In the case of Burali-Forti's paradox, the
series whose ordinal number causes the difficulty is the series of all ordinal
numbers. In each contradiction something is said about all cases of some kind,
and from what is said a new cav seems to In- generated, which both is and is not
of the same kind as the eases of which all were concerned in what was said.
But this is the characteristic of illegitimate totalities, as we defined them in
stating tin* vicious-circle principle. Hence all our contradictions are illustra¬
tions of vicioiis-cirelc fallacies. It only remains to show, therefore, that the
illegitimate totalities involved are excluded bv the hierarchy of types which
We have const meted.
( I ) When a man says "I am lying,” we may interpret his statement as:
"There is a proposition which I am affirming and which is false.” That is to
say, he is asserting the truth of some value of the function "I assert p, and p
is false." But we saw that tin- word "false" is ambiguous, and that, in order
to make it unambiguous, we must specify the order of falsehood, or. what comes
to the same thing, the order of the pro|>osition to which falsehood is ascribed.
We saw also that, if p is a proposition of the nth order, a proposition in which
p occurs as an apparent variable is not of the nth order, but of a higher order.
Hence the kind of truth or falsehood which can belong to the statement "there
is a projMjsition p which I am affirming and which Inis falsehood of the nth
order ” is truth or falsehood of a higher order t han the nth. Hence the state¬
ment of Kpimcnides does not fall within its own scope, and therefore no
contradiction emerges.
If we regan I the statement" I am lying" as a compact way of simultaneously
making all the following statements: "I am asserting a false proposition of the
first order," " I am asserting a false proposition of the second order,” and soon,
we find the following curious state of things: As no proposition of the first
order is being asserted, the statement *'I am asserting a false proposition of
the first order" is false. This statement is of the second order, hence the
statement "I am making a false statement of the second order” is true. This
is a statement of the third order, and is the only statement of the third order
which is being made. Hence the statement "I am making a false statement
of the third order” is false. Thus we see that the statement "I am making a
false statement of order 2a + 1" is false, while the statement "I am making
a false statement of order 2«" is true. But in this state of things there is no
contradiction.
(2) In order to solve the contradiction about the class of classes which are
not members of themselves, we shall assume, what will be explained in the
next Chapter, that a proposition about a class is always to be reduced to a
statement about a function which defines the class, i.e. about a function which
n l vicious-circle FALLACIES (13
is satisfied by the members of the class and by no other arguments. Tims a
class is an object derived from a function and presupposing the function, just
as, for example, (.r).<#>.r presupposes the function Hence a class cannot,
by the vicious-circle principle, significantly be the argument to its defining
function, that is to say, if we denote by the class defined by <f>z, the
symbol “ <f> [2 (<f>c)\ ” must be meaningless. Hence a class neither satisfies nor
does not satisfy its defining function, and therefore (as will appear more fully
in Chapter III) is neither a member of itself nor not a member of itself. This
is an immediate consequence of the limitation to the possible arguments to a
function which was explained at the beginning of the present Chapter. Thus
it a is a class, the statement "a is not a member of a" is always meaningless,
and there is therefore no sense in the phrase*'the class of those classes which
are not members of themselves.” Hence the contradiction which results from
supposing that there is such a class disappears.
(3) Exactly similar remarks apply to “the relation which holds between
R and & whenever R does not have the relation R to S." Suppose the
relation R is defined b}' a function <f>{x, y), i.c. R holds between x and y
whenever <f> (x, y) is true, but not otherwise. Then in order to interpret
"R has the relation R to S,” we shall have to suppose that R and jS can
significantly be the arguments to <p. But (assuming, as will appear in
Chapter III, that R presupposes its defining function) this would require
that (f> should be able to take as argument an object which is defined in
terms of <f>, and this no function can do, as we saw at the beginning of this
Chapter. Hence “R has the relation R to S ” is meaningless, and the contra¬
diction ceases.
(4) The solution of Burali-Forti’s contradiction requires some further
developments for its solution. At this stage, it must suffice to observe that
a series is a relation, and an ordinal number is a class of series. (These state¬
ments are justified in the body of the work.) Hence a series of ordinal numbers
is a relation between classes of relations, and is of higher type than any of the
senes which are members of the ordinal numbers in question. Burali-Forti's
“ordinal number of all ordinals” must be the ordinal number of all ordinals of
a given type, and must therefore be of higher type than any of these ordinals.
Hence it is not one of these ordinals, and there is no contradiction in its being
greater than any of them*.
(5) The paradox about “the least integer not nameable in fewer than
nineteen syllables” embodies, as is at once obvious, a vicious-circle fallacy.
For tbe word “ nameable” refers to the totality of names, and yet is allowed
to occur in what professes to be one among names. Hence there can be no
such thing as a totality of names, in the sense in which the paradox speaks
“ The eolation of Barali-Forti’s paradox by means of the theory of types is given in detail in
•266.
61
INTRODUCTION
[CHAP.
of "names.' It is easy to see that, in virtue of the hierarchy of functions,
the theory of types renders a totality of "names" impossible. We may, in
fact, distinguish names of different orders as follows: (a) Elementary names
will be such as are true ( * proper names." i.e. conventional appellations not
involving any description, (h) First-order names will be such as involve a
description by means of a first-order function; that is to say, if <f >! ./* is a first-
order function, "the term which satisfies <£!./•" will be a first-order name,
though there will not always be an object named by this name, (c) Second-
oider names will be such as involve a description by means of a second-order
function; among such names will be those involving a reference to the totality
of first-order names. And so we can proceed through a whole hierarchy. But
at no stage can we give a meaning to the word "nameable” unless we specify
the onler of names to be employed; and any name in which the phrase "name-
able by name' •*! older w" occurs is necessarily of a higher order than the nth.
Thus the paradox disappears.
'file solutions of the paradox about the least indefinable ordinal and
of Bichard's paradox are closely analogous to the above. The notion ol
"definable." which occurs in both, is nearly the same as "nameable,” which
occurs in our fifth paradox: "definable" is what "nameable" becomes
when elementary names are excluded, i.e. "definable" means "nameable by
a name which is not elementary." But here there is the same ambiguity
as to type as there was before, and the same need for the addition of words
which specify the type to which the definition is to belong. And however
the type may be specified, “the least ordinal not definable by definitions of
this type" is a definition of a higher type; and in Richard's paradox, when
we confine ourselves,as we must, to decimals that have a definition of a given
type, the number A r , which causes the paradox, is found to have a definition
which belongs to a higher type, and thus not to come within the scope of our
previous definitions.
An indefinite number of other contradictions, of similar nature to the
above seven, can easily he manufactured. In all of them, the solution is
of the same kind. In all of them, the appearance of contradiction is pro¬
duced by the presence of some word which has systematic ambiguity of
type, such as truth, falsehood, function, property, class, relation, cardinal,
anti nal. name, definition. Any such word, if its typical I ambiguity is over¬
looked. will apparently generate a totality containing members defined in
terms of itself, and will thus give rise to vicious-circle fallacies. In most
cases, the conclusions of arguments which involve vicious-circle fallacies
will not be self-contradictory, but wherever we have an illegitimate totality,
a little ingenuity will enable us to construct a vicious-circlc fallacy leading
to a contradiction, which disappears as soon as the typically ambiguous words
are rendered typically definite, i.e. are determined as belonging to this or that
type.
VIOIOUS-CIROLK FALI.At'I KS
n]
<;r>
Thus the appearance of contradiction is always due to the presence of words
embodying a concealed typical ambiguity, and the solution of the apparent
contradiction lies in bringing the concealed ambiguity to light.
In spite of the contradictions which result from unnoticed typical
ambiguity, it is not desirable to avoid wools and symbols which have
typical ambiguity. Such words and symbols embrace practically all the
ideas with which mathematics and mathematical logic are concerned: the
systematic ambiguity is the result of a systematic analogy. That is to say. in
almost all the reasonings which constitute mathematics and mathematical
logic, we are using ideas which may receive any one of an infinite number of
different typical determinations, any one of which leaves the reasoning valid.
Thus by employing typically ambiguous words and symbols, we are able to make
one chain of reasoning applicable to an)-one of an infinite number of different
cases, which would not be possible if we were to forego the use of typically
ambiguous words and symbols.
Among propositions wholly expressed in terms of typically ambiguous
notions practically the only ones which may differ, in respect of truth or false¬
hood, according to the typical determination which they receive, are existence-
theorems. If we assume that the total number of individuals is n, then the
total number of classes of individuals is 2", the total number of classes of classes
ol individuals is 2 2 ", and so on. Here n maybe either finite or infinite,and in
either case 2 n > n. Thus cardinals greater than n but not greater than 2 H exist
as applied to classes of classes, but not as applied to classes of individuals, so
that whatever maybe supposed to be the number of individuals, there will be
existence-theorems which hold for higher types but not for lower types. Even
here, however, so long as the number of individuals is not asserted, but is
merely assumed hypothetically, wc may replace the type of individuals by any
other type, provided we make a corresponding change in all the other types
occurring in the same context. That is, we may give the name “relative in¬
dividuals” to the members of an arbitrarily chosen type t, and the name
“relative classes of individuals” to classes of “relative individuals,” and so on.
Thus so long as only hypothetical are concerned, in which existence-theorems
for one type are shown to be implied by existence-theorems for another, only
relative types are relevant even in existence-theorems. This applies also to cases
where the hypothesis (and therefore the conclusion) is asserted , provided the
assertion holds for any type, however chosen. For example, any type has at
least one member; hence any type which consists of classes, of whatever order,
has at least two members. But the further pursuit of these topics must be left
to the body of the work.
R&w i
6
CHAPTER III
INCOMPLETE SYMBOLS
(1) Descriptions. By an “ incomplete " symbol we mean a symbol which
is not supposed to have any meaning in isolation, but is only defined in
d [°
certain contexts. In ordinary mathematics, for example, . and I are in-
complete symbols: something has to be supplied before we have anything
significant. Such symbols have what may be called a "definition in use.
Thus if wc put
V, “j£ + 5S3 + ir« ,,f '
a# 4 dip rV
we define the use of V*. but V* by itself remains without meaning. This dis¬
tinguishes such symbols from what (in a generalized sense) we may call proper
names: "Socrates,” for example, stands for a certain man, and therefore has
a meaning by itself, without the need of any context. If we supply a context,
as in "Socrates is mortal," these words express a fact of which Socrates him¬
self is a constituent: there is a certain object, namely Socrates, which does
have the property of mortality, and this object is a constituent of the complex
fact which we assert when we say "Socrates is mortal." But in other cases,
this simple analysis fails us. Suppose we say: "The round square does not
exist." It seems plain that this is a true proposition, yet we cannot regard it
as denying the existence of a certain object called " the round square." I'or
if there were such an object, it would exist: wc cannot first assume that there
is a certain object, and then proceed to deny that there is such an object.
Whenever the grammatical subject of a proposition can be supposed not to
exist without rendering the proposition meaningless, it is plain that the
grammatical subject is not a proper name. i.e. not a name directly representing
sonic object. Thus in all such cases, the proposition must be capable of being
so analysed that what was the grammatical subject shall have disappeared.
Thus when wc say " the round square does not exist," we may, as a first
attempt at such analysis.substitute *' it is false that there is an object# which
is both round and square.” Generally, when " the so-and-so " is said not to
exist, we have a proposition of the form*
•*~E !(!#)(<*>#),”
i.e. ~{<gc) # = c),
or some equivalent. Here the apparent grammatical subject (l#) (4> x ) has
completely disappeared; thus in E! (»x) (<£#),” (»#)(£#) is an incomplete
symbol.
• Cf. pp. so. 31.
CHAP. Ill]
DESCRIPTIONS
(J7
By an extension of the above argument, it can easily be shown that
(U*)(<^r) is always an incomplete symbol. Take, for example, the following
proposition: “Scott is the author of Waverley." [Here "the author of
Waver ley” is (ix) (.r wrote Waverley).] This proposition expresses an identity;
thus if “ the author of Waverley ” could be taken as a proper name, and sup¬
posed to stand for some object c. the proposition would be "Scott is c." But
if c is any one except Scott, this proposition is false; while if c is Scott, the
proposition is “Scott is Scott,” which is trivial, and plainly different from
“Scott is the author of Waverley.” Generalizing, we see that the proposition
a = (ix)(<f>.c)
is one which may be true or may be false, but is never merely trivial, like
a = a; whereas, if (l.r) (<f>x) were a proper name, a = (Mr)(<£.< ) would necessarily
be either false or the same as the trivial proposition a —a. We may express
this by saying that a = (ur)(<£.r) is not a value of the propositional function
a — y> from which it follows that (ix)(<f>x) is not a value of y. But since y
may be auything, it follows that (ix)(<f>x) is nothing. Hence, since in use it
has meaning, it must be an incomplete symbol.
It might be suggested that “ Scott is the author of Waverley ” asserts that
“Scott” and “the author of Waverley” are two names for the same object.
But a little reflection will show that this would be a mistake. For if that
were the meaning of “ Scott is the author of Waverley,” what would be required
for its truth would be that Scott should have been called the author of
Waverley: if he had been so called, the proposition would be true, even if
some one else had written Waverley; while if no one called him so, the pro¬
position would be false, even if he had written Waverley. But in fact he was
the author of Waverley at a time when no one called him so, and he would
not have been the author if every one had called him so but some one else
had written Waverley. Thus the proposition “Scott is the author of Waverley”
is not a proposition about names, like “Napoleon is Bonaparte”; and this
illustrates the sense in which “ the author of Waverley ” differs from a true
proper name.
Thus all phrases (other than propositions) containing the word the (in the
singular) are incomplete symbols: they have a meaning in use, but not in
isolation. For “the author of Waverley” cannot mean the same as “Scott,”
or “ Scott is the author of Waverley ” would mean the same as “ Scott is
Scott,” which it plainly does not; nor can “ the author of Waverley ” mean
anything other than “ Scott,” or “Scott is the author of Waverley ” would be
false. Hence “the author of Waverley” means nothing.
It follows from the above that we must not attempt to define “ (ix) (<f>x),”
but must define the uses of this symbol, i.e. the propositions in whose symbolic
expression it occurs. Now in seeking to define the uses of this symbol, it is
important to observe the import of propositions in which it occurs. Take as
68
INTRODUCTION
[CHAP.
an illustration: “The author of Waverley was a poet.” This implies (1) that
Waverley was written, (2) that it was written by one man, and not in collabora¬
tion. (3) that the one man who wrote it was a poet. If any one of these fails,
the proposition is false. Thus “ the author of ‘ Slawkenburgius on Noses’ was
a poet’ is false, because no such book was ever written; "the author of'The
Maid’s Tragedy* was a poet" is false, because this play was written by
Beaumont and Fletcher jointly. These two possibilities of falsehood do not
arise if we say “Scott was a poet.” Thus our interpretation of the uses ot
(/./)(<£./•) must be such as to allow for them. Now taking <f>x to replace
".»• wrote Waverley. it i> plain that any statement apparently about (i.r)(<f>. r)
requires (I) (g.r). <«£.*•) and (2) <£.'• .«/>//. D /iJ# . .#• = »/; here (I) states that at
least one object sati.-fies </>.»•, while (2» states that at most one object satisfies
<f>.r. The two together are Ciplivalent to
<3C) : <f>> . =,..> = c,
which we defined as K !(!.»•)< 4>x).
Thus “ K !(;./•)(</>./) mii't be part «>f what is affirmed by any proposition
about (ix)(<fr.r). If our proposition is/ •( lx)($x)\, what is furtiier affirmed is
Jc t if 4 >r • =* • * ™ c. Thus we have
/1(m) («/».»){ . = :(yc):f .=,.j-c:./c Df.
i.c. " the x satisfying </>* satisfies fc " is to mean: “There is an object c such
that <f>r is true when, and only when, .*• is c, and fc is true,” or, more exactly:
M There is a c such that *<£./•’ is always equivalent to ‘ x is c’ and fc." In this,
“( tx) (<£.*•)" has completely disappeared; thus "(U ) (<£./ )’” is merely symbolic,
and does not directly represent an object, as single small Latin letters are
assumed to do*.
The proposition “ a * (ix)(<ftx)" is easily shown to be equivalent to
“<£./•. = x . x = a" For, by the definition, it is
(3c) : 4>r . = x . x = c : a = c,
i.e. “ there is a c for which *p.r. = x ..r = c. and this c is a" which is equivalent
to '* </>.*•. = x . .»• = a.“ Thus “ Scott is the author of Waverley " is equivalent to:
‘"x wrote Waverley' is always equivalent to \r is Scott,'”
i.e. " x wrote Waverley " is true when x is Scott and false when x is not Scott.
Thus although )” has no meaning by itself, it may be substituted
for if in any propositional function fy. and wc get a significant proposition,
though not a value of fy.
When f\{tx)(<f>x)\ t as above defined, forms part of sonic other proposition,
we shall say that (ix) (<£.r) has a secondary occurrence. When (ix)(<f>x) has
a secondary occurrence, a proposition in which it occurs may be true even
when (ix)(<f>x) does not exist. This applies, e.g. to the proposition: “There
• We shall generally write "/(o') (*x)” rather than "/{(»x) in future.
09
III]
THE SCOPE OF A DESCRIPTION
is no such person os the King of France." We may interpret this as
~ |E ! (</>.» )j,
or as ~ {(go) . c = (l.r)(<f>.c )|,
if “ <f>.v” stands for “x is King of France." In either case, what is asserted is
that a proposition p in which ( i.v)(<p.r) occurs is false, and this proposition p
is thus part of a larger proposition. The same applies to such a proposition
as the following: “ If France were a monarchy, the King of France would be
of the House of Orleans."
It should be observed that such a proposition as
~/ ((w) (<*>*) 1
is ambiguous; it may deny /((i.r) (<£» )). in which case it will be true if
(?.r)(<£.r) does not exist, or it may mean
(gc): <f>.r . =, . x — c : ~fc,
in which case it can only be true if (i.v)(<f>.v) exists. In ordinary language,
the latter interpretation would usually be adopted. For example, the propo¬
sition " the Kiug of France is not bald " would usually be rejected as false,
being held to mean “ the King of France exists and is not bald,” rather than
"it is false that the King of France exists aud is bald.” When (lx) (<f>x)
exists, the two interpretations of the ambiguity give equivalent results; but
when (ix)(<f>x) does not. exist, one interpretation is true and one is false. It
is necessary to be able to distinguish these in our notation; aud generally, if
we have such propositions ns
yfr(ix)(<f>x).0 .p,
p.D.yJr (lx) (<t>x ),
yfr (7x) (<f>x) . D . x(lx) (<f>x),
and so on, we must be able by our notation to distinguish whether the whole
or only part of the proposition concerned is to be treated as the “/(?#) (<f>x)”
of our definition. For this purpose, we will put “ [(*r)(«fcr)]” followed by dots
at the beginning of the part (or whole) which is to be taken as f(ix)(<f>x), the
dots being sufficiently numerous to bracket off the /(ix)(<f>x); i.e. f(ix) (<f>x)
is to be everything following the dots until we reach an equal number of dots
not signifying a logical product, or a greater number signifying a logical pro¬
duct, or the end of the sentence, or the end of a bracket enclosing ,l [(ix) (</>x)].”
Thus
[(7*) ( <px )] . yjr (ix ) ( 4>x) . D .p
will mean (gc) : <f>x .= x .x = cz yfre : D . p,
but [(,*) (0*)] : yfr (jx) (0*) .O.p
will mean (gc) s <f>x . = x . x = c : yfre . D . p.
It is important to distinguish these two, for if (ix)(<f>x) does not exist, the
first is true and the second false. Again
[(7X) (**)] . ~ ^ 0 *) (<£*>
TO
INTRODUCTION
[CHAP.
will mean (^r): </m c: ~ >/rc,
while [( #./-><<£« )] . yfr (lx)(<f>x)\
will mean .<gc>: <£» . = T . .r = c :'/'t-j.
Hero again, when d«»cs not exist, the first is false and the second true.
In order to avoid thi** ambiguity in propositions containing (?.»•) (<£.»), we
amend our definition, or rather our notation, putting
(U'X</m )]./o* )<<*>./>. = :<gc): <t> x. = x ..r = c:/c Df.
By means of this definition, we avoid any doubt as to the portion of our
whole asserted proposition which is to he treated as the "/(tx)(<f>x)” of the
definition. This portion will be called the scope of (ix )(</>./•). Thus in
[( ix)(4> > >]./( lx) ( <f>.r) . D . p
the scope of ( lx) ( <ft.r ) isy't /./)(<£-); but in
(t ix) t <t> > >J:/( ix) ( </>j ). D . p
t he scope is j\ ix)(<frx) . D . p:
in ^ |[(l.*)(^r)]./(»j-)(^.r)|
the scope is /‘(#.»•)( tfrx); but in
[( lx) ( <f> > )] . ^ / ( j.r) ($.#•)
the scope is lx) (<f>.r).
It will he seen that when (i.«)(0.») has the whole of the proposition
concerned for its scope, tin- proposition concerned cannot be true unless
E!( ix) ((f>x): hut when (/.«)(<£.»> has only part of the proposition concerned
for its scope, it may often be true even when (ix) (<f>x) does not exist. It. will
be seen further that when K! (lx)(j>x), we may enlarge or diminish the scope
of ( ix) (<f>x) as much as we ph ase without altering the truth-value of any
proposition in which it occurs.
If a proposition contains two descriptions, say ( ix)(<f>x ) and (ix)(yfrx),
we have to distinguish which of them has the larger scope, i.e. we have to
distinguish
( 1) [( hr) (<£•'• o : [( »•'•) (>/'•» )] •/1( hr) ( <f>x), ( ix) (>/r.r)|,
(2) [(lr)(yfrx)] : [( lx)(4>x)) ./[(?*) (4>.r), (lx) (^)j.
The first of these, eliminating (j.r )(<£.»•), becomes
(3) (gc):^.r.s x .j = ct [(lx) (>fr.r)] . / |c, (ix) (yjrx )),
which, eliminating (ix)(\Jrx). becomes
W (ac) <t>.r . = x . .r = C <gc/) : yfrx . = x . x = c :/(c, d),
and the same proposition results if, in (1), we eliminate first (hr)(>Jr. r) and
then (lx)(<f>x). Similarly (2) becomes, when (ix)(<f>x) and (ix)(yfr.r) are
eliminated,
<5) (3 rf ).= x .r = d (gc) : <fsr . = x .x = c :/(c, d).
(4) and (5) are equivalent, so that the truth-value of a proposition contain¬
ing two descriptions is independent of the question which has the larger scope.
m]
CLASSES
71
It will be found that, in most cases in which descriptions occur, their
scope is, in practice, the smallest proposition enclosed in dots or other brackets
in which they are contained. Thus for example
[t , r ) f£••)] • ^ ('•*■) 1 4 >• 3 • [( » (</>•'*>] • X 1 *•' > (<#»•«•>
"ill occur much more frequently than
[^m)(«Jm)] : \Jr (hr)(<f>.r ). D . % (hr) (</>.<).
For this reason it is convenient to decide that, when the scope of an occurrence
of (hr) (*fhv) is the smallest proposition, enclosed in dots or other brackets, in
which the occurrence in question is contained, the scope need not be indicated
by “[(»*) (<f>.v)V' Thus e.g.
p . D . a — ( hr) (</>.*■)
will mean p . D . [(to*) (£.r)] . a = (».r) (<£•<•);
and p . D . (ga) . a = (u ) (</m)
will mean p . D . (ga) . [(?.r) (<^r)] . a = (?.r) (</>.«•);
and p . D . a =J= (?.r) (<£.r)
will mean p . D . [(7x) (£.r)] . ~ {a = (?.r) (<£.<•));
but p . D . ^ |a = (7a:) (<f>x)}
will mean p . D . ~ ([(7a:) (<£•*)] • « — (»•**) (♦•«•)).
This convention enables us, in the vast majority of cases that actually
occur, to dispense with the explicit indication of the scope of a descriptive
symbol; and it will be found that the convention agrees very closely with the
tacit conventions of ordinary language on this subject. Thus for example, if
“(7#) (<f>x) n is “tho so-and-so,” “a+ (?x)(<£*)” is to be read “a is not the
so-and-so,” which would ordinarily be regarded as implying that “ the so-and-
so” exists; but [a = (ix)(<f>x)}” is to be read “it is not true that a is the
so-and-so,” which would generally be allowed to hold if “ the so-and-so ” does
not exist. Ordinary language is, of course, rather loose and fluctuating in its
implications on this matter; but subject to the requirement of definiteness,
our convention seems to keep as near to ordinary language as possible.
In the case when the smallest proposition enclosed in dots or other
brackets contains two or more descriptions, we shall assume, in the absence
of any indication to the contrary, that one which typographically occurs
earlier has a larger scope than one which typographically occurs later. Ihus
(ix)(<f>x)=‘(ix)(yfrx)
will mean (gc) : <px . = x . x = c : [(7a:) (^a:)] . c = (lx) (yjrx),
while (ix) (yjrx) = (lx) (<f>x)
will mean (g d) : ypx . = x . x = d : [(7a:) (<£a:)] . (lx) (<f>x) = d.
These two propositions are easily shown to be equivalent.
(2) Classes. The symbols for classes, like those for descriptions, are, in
our system, incomplete symbols: their uses are defined, but they themselves
are not assumed to mean anything at all. That is to say, the uses of such
72
INTRODUCTION
-- [CHAP.
symbols are so defined that, when the de/iniei ms substituted for the definiendum ,
there no longer remains any symbol which could be supposed to represent
a class. 'I Ims classes, so far as we introduce them, are merely symbolic or
linguistic conveniences, not genuine objects as their members are if they are
individuals.
It is an old dispute whether formal logic should concern itself mainly with
intensions or with extensions. In general, logicians whose training was mainly
philosophical have decided for intensions, while those whose training was
mainly mathematical have decided for extensions. The facts seem to be that,
while mathematical logic requires extensions, philosophical logic refuses to
supply anything except intensions. Our theory of classes recognizes and
reconciles these two apparently opposite facts, by showing that an extension
(which is tin* same as a class) is an incomplete symbol, whose use always
acquires its meaning through a reference to intension.
In the case ot desciiptions, it was possible to prove that they are in¬
complete symbols. In the case of classes, we do not know of any equally
definite proof, though arguments of more or less cogency can be elicited from
the ancient problem of the One and the Many*. It is not necessary for our
purposes, however, to assert dogmatically that there are no such things as
classes. It is only necessary for us to show that the incomplete symbols
which we introduce as representatives of classes yield all the propositions for
the sake ot which classes might be thought essential. When this has been
shown, the mere principle of economy of primitive ideas loads to the non-
nitinduction of classes except as incomplete symbols.
lo explain the theory of classes, it is necessary first to explain the dis¬
tinction between estensional and intensional functions. This is effected by
the following definitions:
The truth-value of a proposition is truth if it is true, and falsehood if it is
false. (This expression is due to Frege.)
'I wo propositions are said to be equivalent when they have the same truth-
value, i.e. when they are both true or both false.
Two propositional functions arc said to be formally equivalent when they
are equivalent with every possible argument, i.e. when any argument which
satisfies the one satisfies the other, and vice versa. Thus “u* is a man" is
formally equivalent to •*£ is a featherless biped”; **.£• is an even prime” is
formally equivalent to “.r is identical with 2.”
A function of a function is called ej:tensional when its truth-value with any
argument is the same as with any formally equivalent argument. That is to
Briefly, these arguments reduce to the following: If there is such an object as a class, it
must be in some sense one object. Yet it is only of classes that many can bo predicated. Hence,
if we admit classes as objects, we must suppose that the same object can be both one and many,
which seems impossible.
>n]
EXTENSION A L FUNCTIONS OF FUNCTIONS
73
say , f is an extonsional function of <f>; if. provided yfr: is formally oquiva-
lent to <p 2 t is equivalent to f\\jrs). Here the apparent variables </> ami
are necessarily of the type from which arguments can significantly he
supplied to/' We find no need to use as apparent variables any functions
of non-predicative types; accordingly in the sequel all extonsional functions
considered are in fact functions of predicative functions*.
A function of a function is called intensional when it is not extonsional.
The nature and importance of the distinction between intensional and
extensional functions will be made clearer by some illustrations. The pro¬
position “\r is a man ’ always implies ‘ .r is a mortal ’” is an extensional function
of the function “a 1 is a man,” because we may substitute, for “.»• is a man,”
"x is a featherless biped,” or any other statement which applies to the same
objects to which “ x is a man ” applies, and to no others. Hut the proposition
“A believes that *x is a man’ always implies ‘a- is a mortal’” is an intensional
function of “a? is a man,” because A may never have considered the question
whether feathcrless bipeds are mortal, or may believe wrongly that there are
featherless bipeds which are not mortal. Thus even if "x is a featherless
biped” is formally equivalent to “a- is a man,” it by no means follows that a
person who believes that all men are mortal must believe that all featherless
bipeds are mortal, since he may have never thought about feathcrless bipeds,
or have supposed that feathcrless bipeds were not always men. Again the
proposition “ the number of arguments that satisfy the function <f >! $ is n ” is
an extensional function of <f >! 2, because its truth or falsehood is unchanged if
we substitute for </>!2 any other function which is true whenever <£!2 is true,
and false whenever (f >! 2 is false. But the proposition “A asserts that the
number of arguments satisfying <f >! 2 is n” is an intensional function of <£ ! 2,
since, if A asserts this concerning <£!2, he certainly cannot assert it concerning
all predicative functions that are equivalent to <f >! 2, because life is too short.
Again, consider the proposition “two white men claim to have reached the
North Polo.” This proposition states “ two arguments satisfy the function
is a white man who claims to have reached the North Pole.’” The truth or
falsehood of this proposition is unaffected if we substitute for "it? is a white
man who claims to have reached the North Pole ” any other statement which
holds of the same arguments, and of no others. Hence it is an extensional
function. But the proposition “it is a strange coincidence that two white
men should claim to have reached the North Pole,” which states “ it is a
strange coincidence that two arguments should satisfy the function ‘a? is a
white man who claims to have reached the North Pole,”’ is not equivalent to
“it is a strange coincidence that two arguments should satisfy the function
is Dr Cook or Commander Peary.’” Thus “ it is a strange coincidence that
<t> should be satisfied by two arguments” is an intensional function of <f>l£.
• Cf. p. 63.
71
INTRODUCTION
[CHAP.
The above instances illustrate the fact that the functions of functions with
which mathematics is specially concerned are extensional, and that intensions!
functions of functions only occur where non-mathematical ideas are introduced,
such as what somebody believes or affirms, or the emotions aroused by some
fact. Hence it is natural, in a mathematical logic, to lay special stress on
e.iicnsionnl functions of functions.
When two functions arc formally equivalent, we may say that they have
the same e.rteiisinn. In this definition, we are in close agreement with usage.
We do not assume that there is such a thing as an extension: we merely
define the whole phrase “having the same extension." We may now say that
an extensional function of a function is one whose truth or falsehood depends
only upon the extension of its argument. In such a case, it is convenient to
regard the statement concerned as being about the extension. Since exten-
sional functions are many and important, it is natural to regard the extension
as an object, called a class, which is supposed to be the subject of all the
equivalent .statements about various formally equivalent functions. Thus
*•'/• wo say " there were twelve Apostlo." it is natural to regard this state¬
ment as attributing the property of being twelve to a certain collection of
men, namely those who were Apostles, rather than as attributing the property
of being satisfied by twelve arguments to the function ".#* was an Apostle.”
This view is encouraged by the feeling that there is something which is
identical in the case of two functions which “have the same extension." And
if we take such simple problems as “ how many combinations can be made of
a things t" it seems at first sight necessary that each "combination " should
be a single object which can be counted as one. This, however, is certainly
not necessary technically, and we see no reason to suppose that it is true
philosophically. The technical procedure by which the apparent difficulty is
overcome is as follows.
We have seen that an extensional function of a function may be regarded
as a function of the class determined by the argument-function, but that an
intensionnl function cannot be so regarded. In order to obviate the necessity
of giving different treatment to intensionnl and extensional functions of
functions, we construct an extensional function derived from any function of
a predicative function yfr ! 2, and having the property of being equivalent to
the function from which it is derived, provided this function is extensional,
as well as the property of being significant (by the help of the systematic
ambiguity of equivalence) with any argument <f>z whose arguments are of the
same type as those of yfr ! 2. The derived function, written "f |2 * s de¬
fined as follows: Given a function f (yfr ! 2), our derived function is to be "there
is a predicative function which is formally equivalent to <f>z and satisfies/.”
If f>z is a predicative function, our derived function will be true whenever
/(02) is true. If f (<pz) is an extensional function, and <f>z is a predicative
ni)
DEFINITION OF CLASSES
■»
/•')
function, our derived function will not be true unless/’(<£-) is true; thus in
this ease, our derived function is equivalent t oj\<f>z). If r\<f>~) is not an ex-
tensional function, and if <f >2 is a predicative function, our derived function
may sometimes be true when the original function i* false. But in any case the
derived function is always exteusional.
In order that the derived function should be significant for any function
<£-, of whatever order, provided it takes arguments of the right type, it is
necessary and sufficient that /(yfr !2) should be significant, where y/r ! 2 is any
predicative function. The reason of this is that we only require, concerning
an argument <£?, the hypothesis that it is formally equivalent to some predi¬
cative function \Jrl 2, and formal equivalence has the same kind of systematic
ambiguity as to type that belongs to truth and falsehood, and can therefore
hold between functions of any two different orders, provided the functions
take arguments of the same type. Thus by means of our derived function we
have not merely provided extensional functions everywhere in place of in-
tensional functions, but we have practically removed the necessity for con¬
sidering differences of type among functions whose arguments are of the same
type. This effects the same kind of simplification in our hierarchy as would
result from never considering auy but predicative functions.
If f(\y ! 2) can be built up by means of the primitive ideas of disjunction,
negation, (x). <f>. r, and (ft*). <f>x, as is the case with all the functions of
functions that explicitly occur in the present work, it will be found that, in
virtue of the systematic ambiguity of the above primitive ideas, any function
</>2 whose arguments are of the same type as those of yfr ! 2 can significantly
be substituted for yfrl2 in f without any other symbolic change. Thus in
such a case what is symbolically, though not really, the same function /can
receive as arguments functions of various different types. If, with a given
argument </>2, the function f(<f> 2), so interpreted, is equivalent to /(>/r!2)
whenever yfr ! 2 is formally equivalent to <f>z, then {2(<£*)) is equivalent to
/(£$) provided there is any predicative function formally equivalent to </>2.
At this point, we make use of the axiom of reducibility, according to which
there always is a predicative function formally equivalent to </>2.
As was explained above, it is convenient to regard an extensional function
of a function as having for its argument not the function, but the class de¬
termined by the function. Now we have seen that our derived function is
always extensional. Hence if our original function was f(yjr l 2), we write the
derived function f {2 (<f>z)\, where “2 (<pz)” may be read “ the class of arguments
which satisfy <£2,” or more simply "the class determined by <£2.” Thus
“/{$($*)}*' will mean: “There is a predicative function -*fr ! 2 which is formally
equivalent to <£2 and is such that is true." This is in reality a function
of but we treat it symbolically as if it had an argument 2 ( tf>z ). By the
help of the axiom of reducibility, we find that the usual properties of classes
70
INTRODUCTION
[CHAP.
result. For example, two formally equivalent functions determine the same
class, and conver>ely, two functions which determine the same class are formally
equivalent. Also to say that .#• is a member of 2 (4>z), i.e. of the class determined
by <f>z, is true when <f>x is true, and false wheu <f>x is false. Thus all the
mathematical purposes for which classes might seem to be required are fulfilled
by the purely symbolic objects zi<f>z), provided we assume the axiom of
reducibility.
In virtue of the axiom of reilucibility, if d>2 is any function, there is
a formally equivalent predicative function yjr 1 2; then the class 2(</>s) is
identical with the class 2(^!r), so that every class can be defined by a
predicative function. Hence the totality of the classes to which a given term
can be significantly said to belong or not to belong is a legitimate totality,
although the totality of /mictions which a given term can be significantly
said to satisfy or not to satisfy is not a legitimate totality. The classes to
which a given term a belongs or does not belong are the classes defined by
u-fuiictiong; they are also the classes defined by predicative ci-functions. Let
us call them a-classes. Then "u-classes ” form a legitimate totality, derived
bom that of predicative a-functions. Hence many kinds of general state¬
ments become possible which would otherwise involve vicious-circle paradoxes.
These general statements are none of them such as lead to contradictions, and
many of them such as it is very hard to suppose illegitimate. The fact that
they are rendered possible by the axiom of reducibility, and that they would
otherwise be excluded by the vicious-circle principle, is to be regarded as an
argument in favour of the axiom of reducibility.
The above definition of “the class defined by the function <£2,” or rather,
of any proposition in which this phrase occurs, is, in symbols, as follows:
f\z (4>z)\ . = : (3 yjr): *f>x . . yfr ! 2| Df.
In order to recommend this definition, we shall enumerate five requisites
which a definition of classes must satisfy, aud we shall then show that the
above definition satisfies these five requisites.
We require of classes, if they are to serve the purposes for which they are
commonly employed, that they shall have certain properties, which may be
enumerated as follows. (1) Every propositional function must determine a
class, which may be regarded as the collection of all the arguments satisfying
the function in question. This principle must hold when the function is
satisfied by an infinite uumber of arguments as well as when it is satisfied by
a finite number. It must hold also when no arguments satisfy the function;
i.e. the “null-class” must be just as good a class as any other. (2) Two pro-
positional functions which are formally- equivalent, i.e. such that any argument
which satisfies either satisfies the other, must determine the same class; that
is to say, a class must be something wholly determined by its membership, so
that e.g. the class “ featherless bipeds ” is identical with the class “ men,” and
in]
CLASSES
77
the class “ even primes ” is identical with the class “ numbers identical with 2."
(3) Conversely, two propositional functions which determine the same class
must be formally equivalent: in other words, when the class is given, the
membership is determinate : two different sets of objects cannot vield the same
class. (4) In the same sense in which there are classes (whatever this sense
may be), or in some closely analogous sense, there must also be classes of
classes. Thus for example “ the combinations of n things in at a time." where
the n things form a given class, is a class of classes: each combination of
m things is a class, and each such class is a member of the specified set of
combinations, which set is therefore a class whose members are classes. Again,
the class of unit classes, or of couples, is absolutely indispensable; the former
is the number 1, the latter the number 2. Thus without classes of classes,
arithmetic becomes impossible. (S) It must under all circumstances be
meaningless to suppose a class identical with one of its own members. For if
such a supposition had any meaning “a € a'" would be a significant propositional
function*, and so would “a~ea." Hence, by (1) and (4), there would be a
class of all classes satisfying the function "a a." If we call this class k, we
shall have
a t k . =* . a~ e a.
Since, by our hypothesis, “/c e k ” is supposed significant, the above equivalence,
which holds with all possible values of a, holds with the value k, i.e.
But this is a contradictionf. Hence "at a” and "a^ea" must always be
meaningless. In general, there is nothing surprising about this conclusion,
but it has two consequences which deserve speciul notice. In the first place,
a class consisting of only one member must not be identical with that one
member, i.e. wc must not have i‘x = x. For we have x e i*x, and therefore, if
x = l*x, we have i l xel‘x t which, we saw, must be meaningless. It follows that
must be absolutely meaningless, not simply false. In the second
place, it might appear as if the class of all classes were a class, i.e. as if
(writing “Cls” for “class”) “Cls € Cls” were a true proposition. But this com¬
bination of symbols must be meaningless; unless, indeed, an ambiguity exists
in the meaning of “Cls,” so that, in “Cls e Cls.” the first “Cls” can be supposed
to have a different meaning from the second.
As regards the above requisites, it is plain, to begin with, that, in accordance
with our definition, every propositional function rf>z determines a class z (<f>z).
Assuming the axiom of reducibility, there must always be true propositions
about i.e. true propositions of the form f{2((f>z) J. For suppose <f>z is
formally equivalent to and suppose yfrl 2 satisfies some function /. Then
■ As explained in Chapter I (p. 25), “ita" means “x is a member of the class a," or,
more shortly, "x is an a.” The definition of this expression in terms of our theory of classes
will be given shortly.
t This is the second of the contradictions discussed at the end of Chapter II.
78
INTRODUCTION*
[CHAP.
2 (4>z) also satisfies/. Hence. given any function <£3, there are true propositions
of the form/[3(<£j)j, i.e. true propositions in which “the class determined by
<f>z” is grammatically the subject. This shows that our definition fulfils the
first of our five requisites.
The second and third requisites together demand that the classes z(<f>z) and
z(\Jfz) should be identical when, and only when, their defining functions are
formally equivalent, i.e. that we should have
z(<t>z)= z (\jsz) . = : 4>r . = x . yfrx.
Here the meaning «>f ' z i<pz) = z {y}r;y' is to be derived, by means of a two¬
fold application of the definition of/;3 (<£*)•, from the definition of
‘ X !3 = 0!3,"
which is *!3 = d!3. - :(/):/! *!2. D ./! $\z Df
by the general definition of identity.
In interpreting "3 (<£-) = 3 (yjfz)." we will adopt the convention which we
adopted in regard to (;.» )(</>/ ) and (tx)(\Jrx). namely that the incomplete symbol
which occurs first is to have the larger scope. Thus 3 (</>x) =• 3 (yfrz) becomes,
by our definition,
*(**>.
which, by eliminating zi^z). becomes
4 > J '• s x« *!•»*:. ( 30 ): =*• Olxzx'- z = 0\z,
which is equivalent to
< 3 *. 0)z<f>x y \ rx • - x • 0\ x z \ \ z ^ 0\ z ,
which, again, is e(|tiivalent to
< a X > : <f> r • s x • X ! x : s * • X 1 *’
which, in virtue of the axiom of reducibility, is equivalent to
4> r.= x .y/r.r.
Thus our definition of the use of z(<f>z) is such ns to satisfy the conditions (2)
and (.‘I) which we laid down for classes, i.e. we have
H 3 (<f>z) = z(\frz). = : <f>x. = x . yjrx.
Before considering classes of classes, it will be well to define membership
of a class, i.e. to define the symbol "xe 3 (<t>z)," which may be read "x is a
member of the class determined by <f> 3.” Since this is a function of the form
y\z(<f>z)\, it must be derived, by means of our general definition of such func¬
tions, from the corresponding function/{>/r! 3). We therefore put
= Df.
This definition is only needed in order to give a meaning to “xe2 the
meaning it gives is, in virtue of the definition of/|3(<£x)},
(3 1r):4>y.s,.1r!yz>lrlx.
It thus appears that " xez(<f>z )” implies <px, since it implies yfrlx, and yjrlx
is equivalent to <f>x ; also, iu virtue of the axiom of reducibility, <f>.v implies
“a* e 3 {<t>z),” since there is a predicative function formally equivalent to <f>,
in]
CLASSES
71)
and x must satisfy yfr, since x (ex hi/pot/icsi) satisfies <f>. Thus in virtue of the
axiom of reducibility we have
: x es (<f>=) . = . <f>x,
V.e. a* is a member of the class 3 (<f>=) when, and only when, x satisfies tin*
function <f> which defines the class.
We have next to consider how to interpret a class of classes. As we have
defined f{z(<f>z)\, we shall naturally regard a class of classes as consisting of
those values of 2 (<f>z) which satisfy f[z(<t>z)\. Let us write a for z (<f>:): then
we may write a (fa) for the class of values of a which satisfy fa*. We shall
apply the same definition, and put
F (5 (fa)} . = : ( 3 , 7 ) :/0 . = „ . 0 \ 0 : F {^! 5) Df,
where “/9” stands for any expression of the form z(\frl z).
Let us take "76 a (fa)" as an instance of /*{a(_/a)|. Then
h 7 c a (/a) . = : ( 3 / 7 ): f $. = * . gl 0 : 7 e g\ci.
Just as we put x € yjr ! 2 . = . yfr ! x Df,
so we put 7 e gl a . = . gl 7 Df.
Thus we find
h s. 7 « a (fa) . = : ( 3 * 7 ) : f/3 . s p . g! 0 : gl 7 .
If we now extend the axiom of reducibility so as to apply to functions of
functions, i.e. if we assume
we easily deduce
I" : (30) :/( 2 (^ ! *)\ •=*- 0 ! l 2 (^ ! *))>
i.e. h(2L!7)if0'*Zfi'gl/3-
Thus h : 7 e 3 (fa) . = .fy.
Thus every function which can take classes as arguments, i.e. every function
of functions, determines a class of classes, whose members are those classes
which satisfy the determining function. Thus the theory of classes of classes
offers no difficulty.
We have next to consider our fifth requisite, namely that "2 (<f>z) e 2 (<f>z)”
is to be meaningless. Applying our definition of f[2 (<f>z)\, we find that if this
collection of symbols had a meaning, it would mean
( 3 ^) : 4>x . =, . yjri x : +1 2 e 2,
i.e. in virtue of the definition
x e yfrl 2 . = . yjrl x Df,
it would mean ( 3 ^) : <t> x - = x • ^1 x • 'k 1 • £)•
But here the symbol “^1 (yfrl 2)” occurs, which assigns a function as argument
to itself. Such a symbol is always meaningless, for the reasons explained at
the beginning of Chapter II (pp. 38—41). Hence “2 (<f>z) e 2 (<f>z )” is meaning¬
less, and our fifth and last requisite is fulfilled.
• The use of a single letter, each as a or p, to represent a variable class, will be further
explained shortly.
80
INTRODUCTION
[CHAP.
As in the case of so in that of f\z(<f>z)) t there is an ambiguity
as to the scope of 3 (<£j) if it occurs in a proposition which itself is part of a
larger proposition. But in the case of classes, since we always have the axiom
of reducibility, namely .... , ,
' (^yfr) : <f>r . m, . yfrlx,
which takes the place of K! (i.O(<£' ). it follows that the truth-value of any
proposition in which «»ccurs is the same* whatever scope we may give to
2 (</>j). provided the proposition is an extensional function of whatever functions
it may contain. Hence we may .adopt the convention that the scope is to be
always tin- smallest proposition enclosed in dots or brackets in which 2 (<fjz)
occurs. If at any time a larger scope is required, we may indicate it by "[2 (<£j)] ’
followed by dots, in the same way as we did for [(i.r)(<£j )].
Similarly when two class symln.ls occur, e.'f. in a proposition of the form
f 3 ( <i>: >. 2 (>/'■.:>!. we need not remember rides lor the scopes of the two symbols,
since all choices give equivalent results, as it. is easy to prove. For the pre¬
liminary propositions a rule is desirable, so we can decide that the class symbol
which occurs first in the order of writing is to have the larger scope.
The representation of a class by a single letter a can now be understood.
For the denotation of a is ambiguous, in so far ns it is undecided ns to which
of the symbols 2 (</>£), 2 (>^;). 3<x-). etc. it is to stand for, where <f>~, ^ 3, *3,
etc. are the various determining functions of the class. According to the choice
made, diHerent propositions result. Blit, all the resulting propositions are equi¬
valent by virtue of the easily proved proposition:
"h : <f>.r 3 x yj/.v . D ./’{3 <<£->! = / j3 {yjrs)\."
Hence unless we wish to discuss the determining function itself, so that the
notion of a class is really not properly present, the ambiguity in t he denotation
of a is entirely immaterial, though, ns we shall see immediately, we are led to
limit ourselves to predicative determining functions. Thus where a is a
variable class, is really “f\z (<£r)S," where 4> is a variable function, that is, it is
3 ;.”
where <f> is a variable function. But here a difficulty arises which is removed
l»v a limitation to our practice and by the axiom of reducibility. For the deter¬
mining functions <f>z, yfrz, etc. will be of different types, though the axiom of
reducibility secures that some are predicative functions. Then, in interpreting
a as a variable in terms of the variation of any determining function, we shall
be led into errors unless we confine ourselves to predicative determining func¬
tions. These errors especially arise in the transition to total variation (cf.
pp. 15, 1(3). Accordingly
fa= .i'&yy) . <l>\x= x y\r\x .f\yjr'.z\ Df.
It is the peculiarity of a definition of the use of a single letter [viz. a] for a
variable incomplete symbol that it, though in a sense a real variable, occurs
only in the definiendum, while “«/»," though a real variable, occurs only in the
dejiniens.
RELATIONS
81
in]
Thus "fa" stands for
, , . “(30>-0!.*=*0'!;f./'|V r! 31. M
ft nd “(a) .fa" stands for
“(0) 2 (a^) • x ./i^r! ?}.”
Accordingly, in mathematical reasoning, we can dismiss the whole apparatus
of functions and think only of classes as “quasi-things,” capable of immediate
representation by a single name. The advantages are two-fold: (1) classes are
determined by their membership, so that to one set of members there is one
class, (2) the “type” of a class is entirely defined by the type of its members.
Also a predicative function of a class can be defined thus
f\ , (g^r) . <ftl X =, yfrl X . f\ {^r! z\ Df.
Thus a predicative function of a class is always a predicative function of an)'
predicative determining function of the class, though the converse does not hold.
(3) Relations. With regard to relations, we have a theory strictly analogous
to that which we have just explained as regards classes. Relations in extension,
like classes, are incomplete symbols. We require a division of functions of two
variables into predicative and non-predicative functions, again for reasons which
have been explained in Chapter II. We use the notation u <f>\ (.r, y)" for a
predicative function of x and y.
We use “0!(£, p)” for the function as opposed to its values; and we use
, ‘^9 < t > (®» y)" for the relation (in extension) determined by <f>(x,y). Wo put
/ 1*9 0 (*. y )) • = : (af) : * <*, y) . =*,„. *1 (x, y) : / {*! ($, 9 A *>f.
Thus even when y|\/r! (£,p)) is not an extensional function of yfr, f (ap <f> (x, y)j
ls an extensional function of <f>. Hence, just as in the case of classes, we deduce
1-5p<^> (x, y) * Sp>/r (x,y) . = : (x, y) . = XiV . ^ (x, y),
i.e. a relation is determined by its extension, and vice versa.
On the analogy of the definition of “arc 2,” we put
*(0*J(£»0)}y---0‘i(*,y) Df *-
This definition, like that of "x ! 2,” is not introduced for its own sake,
but in order to give a meaning to
x\&9<t>(x ,y)}y.
This meaning, in virtue of our definitions, is
(30) : <f> (x, y) . =*. v . i/r! (x, y) s * (0! (£, P)) y,
%e - (30):0(*,y)-=*.v0K*»y):0 ! ( a? .y)»
and this, in virtue of the axiom of reducibility
“(30*) : 0 (*. V) - v - 0* * (*. y) ”
18 equivalent to <f> ( x, y).
Thus we have always
h : {£p <f> (x, y)} y . = . <fi {x, y).
• This definition raises oertain questions as to the two senses of a relation, which are dealt
with in *21.
a&W i
6
82
INTRODUCTION
[CHAP.
Whenever the determining function of a relation is not relevant, we may
replace ay 0 (s, y) by a single capital letter. In virtue of the propositions given
above,
h R = S . s : xRy . = x „ . a.Sy,
V z.ll = fit) <f> (a. y). = : sRy . = . 0 (a, y).
and h . # = J.5 (sRy).
Classes of relations, and relations of relations, can be dealt with as classes
of classes were dealt with above.
Just as a class must not Ik* capable of being or not bring a member of itself,
so a relation must neither be nor not be referent or relatum with respect to
itself. This turns out to be equivalent to the assertion that 0! (.)*, i)) cannot
significantly be either of the arguments a or y in 0! (.c, y). This principle, again,
results from the limitation to the possible arguments to a function explained
at the beginning of Chapter II.
We may sum up this whole discussion on incomplete symbols as follows.
The use of the symbol "( ia)( 0r)” as if in "/(i.r)(0a)" it directly represented
an argument to the function fz is rendered |>ossiblc by the theorems
I- :. E! ( ix) ( 0 a) . D :( j )././ . D . /< is) ( 0 / >.
H : (l.r)( 0 .r) = <ia)( 0 a). D ./(la) ( 0 a) mf(Hc) ( 0 a).
h : 1C ! ( is) ( 0./ ) . D . (I.r) ( 0r) = (I.r) (0r),
h ; (I.r) ( <f>s) = ( is) < 0 a) . = . ( la) ( 0 a) = (la)( 0 r).
h : (la) (0a) = (i.r) (0a) . (la) (0r) = (la)( X a) . D . ( Is) (0a) - (l.r) (**)•
The use of the symbol "s (0a)” (or of a single letter, such as o. to represent
such a symbol) as if. in "/\s (<f>s)\," it directly represented an argument a to a
function fa, is rendered possible by the theorems
h:(a)./a.D./|*< 0 a)),
h : .r ( 0 -r) = a ( 0 a) . D ./|.*( 0 a)) ■/l$( 0 a)|,
I- ..i‘(0r)«5‘(0x),
h : .*• ( 0 a) = .S' ( 0 a) . = . ( 0 a) = a ( 0 a),
h :*( 0 O = ' ( 0 *) - *(00 = .* (*r) .D ..7 ( 0 r) = ^(*a).
Throughout these propositions the types must be supposed to be properly
adjusted, where ambiguity is possible.
The use of the symbol •*.#$ | 0 (a,y)} ” (or of a single letter, such as R, to
represent such a symbol) as if, in “ f {^y <f> (x, y)\” it directly represented an
argument R to a function fR, is rendered possible by the theorems
\-:(R).fR.D.fW<t>(x,y)\,
I- : st) 0 (a, y) = ^y 0 (a, y). D •f\$9 <t> (*, y)| =/l^9 0 (•**. !/))>
h . sy 0 (a. y) = ay 0 (a, y),
h : sy 0 (a, y > = sy 0 (a. y) . = . sy 0 (a, y) = a# 0 (a, y),
h : ay 0 (a, y) = ay 0 (a, y). u'y 0 (a, y) = *5 X (*» S') -
D.^ 0 (a,y) = ^ X (*,y).
Ill]
INCOMPLETE SYMBOLS
S3
Throughout these propositions the types must be supposed to be properly
adjusted where ambiguity is possible.
It follows from these three groups of theorems that these incomplete
symbols are obedient to the same formal rules of identity as symbols which
directly represent objects, so loug as we only consider the equivalence of the
resulting variable (or constant) values of proposit ional functions and not their
identity. This consideration of the identiti/ of propositions never enters into
our formal reasoning.
Similarly the limitations to the use of these symbols can be summed up
as follows. In the case of (ix)(<f>x), the chief way in which its incompleteness
is relevant is that we do not always have
(.r) mfx . D ./(tx) (<f>.v),
\-e. a function which is always true may nevertheless not lx* true of (u.) (<£.<).
This is possible because f(ix)(<px) is not a value of/.**, so that even when all
values of f$ are true,/(?x) (<£.r) may not be true. This happens when (i.r)(<£«)
does not exist. Thus for example we have (a). x = x, but we do not have
the round square =» the round square.
The inference ( x ) .fx . D ./(?.«•) (<£*)
is only valid when E ! (lt) As soon as we know E ! (?x) (</>#), the fact that
Ox)(if>x) is an incomplete symbol becomes irrelevant so long as we confine
ourselves to truth-functions* of whatever proposition is its scope. But even
when E ! (?&•) the incompleteness of (?ar)(</>x) may be relevant when we
pass outside truth-functions. For example, George IV wished to know whether
Scott was the author of Waverley, i.e. he wished to know whether a proposition
of the form “c = (?x) ( <f>x )” was true. But there was no proposition of the form
"c = y” concerning which he wished to know if it was true.
In regard to classes, the relevance of their incompleteness is somewhat
different. It may be illustrated by the fact that we may have
without having yfr ! 2 = x • *■
For, by a direct application of the definitions, we find that
= 2 . = .<£*=**!*•
Ihu9 we shall have
h : <f>x = x ^r lx. <f>x = X %1 ec. D . 2 (4> z ) = ^'.*.2 (<t> z ) = X •
but we shall not necessarily have yjr ! 2 = % ! 2 under these circumstances, for
two functions may well be formally equivalent without being identical; for
example,
x = Scott .= x .x = the author of Waverley,
but the function “2 = the author of Waverley” has the property that George IV
wished to know whether its value with the argument “Scott” was true, whereas
* Cf. p. 8.
6—2
34 INTRODUCTION [CHAP. Ill
the function “ 3 = Scott" has no such property, and therefore the two functions
are not identical. Hence there is a propositional function, namely
* = y ,x = z . D . y = *,
which holds without any exception, and yet does not hold when for * we
substitute a class, and for y and r we substitute functions. This is only
possible because a class is an incomplete symbol, and therefore =
is not a value of *' .r= y."
It will be observed that " 6 ! 3 = ^ ! 2 ” is not an extensional function of
\fs ! 3. Thus the scope of z(<f>z) is relevant in interpreting the product
3 (<t>z)= yff Iz .z (<£*) = x •
If we take the whole of the product as the scope of 3 (<£-), the product is
ipiivnleiit to
(%|0): </>.c = r 0 !.r.^!3 — >/'!3.d!3=»x*“ <
and this does imply
^!3 = X !3.
We may say generally that the fact that 3 (<f>z) is an incomplete symbol
is not relevant so long as wo confine ourselves to extensional functions of
functions, but is apt to become relevant for other functions of functions.
PART I
MATHEMATICAL LOGIC
SUMMARY OF FART I
lx this Part, wo shall don I with such topics as belong traditionally t«»
symbolic logic, or deserve to belong to it in virtue of their generality. We
shall, that is to say, establish such properties of propositions, propositional
functions,classes and relations as are likely to be roptired in any mathematical
reasoning, and not merely in this or that branch of mathematics.
The subjects treated in Part I may be viewed in two aspects: (1) as a
deductive chain depending on the primitive propositions, (2) as a formal calculus.
Taking the first view first: We begin, in *1, with certain axioms as to deduction
of one proposition or asserted propositional function from another. From these
primitive propositions, in Section A. we deduce various propositions which are
all concerned with four ways of obtaining new propositions from given proposi¬
tions, namely negation, disjunction, joint assertion and implication, of which
the last two can be defined in terms of the first two. Throughout this first
section, although, as will be shown at the beginning of Section B, our proposi¬
tions, symbolically unchanged, will apply to any propositions as values of our
variables, yet it will be supposed that our variable propositions are all what
we shall call elementary propositions, i.e. such as contain no reference, explicit
or implicit, to any totality. This restriction is imposed on account of the
distinction between different types of propositions, explained in Chapter II of
the Introduction. Its importance and purpose,however,are purely philosophical,
and so long as only mathematical purposes are considered, it is unnecessary to .
remember this preliminary restriction to elementary propositions, which is
symbolically removed at the beginning of the next section.
Section B deals, to begin with, with the relations of propositions containing
apparent variables (i.e. involving the notions of “all” or "some”) to each other
and to propositions not containing apparent variables. We show that, where
propositions containing apparent variables are concerned, we can define
negation, disjunction, joint assertion and implication in such away that their
properties shall be exactly analogous to the properties of the corresponding
ideas as applied to elementary propositions. We show also that formal im¬
plication, i.e. “(x) . <f>x D yfrx” considered as a relation of <f>x to \fr$, has many
properties analogous to those of material implication, i.e. "p D q" considered as
a relation of p and q. We then consider predicative functions and the axiom
of reducibility, which are vital in the employment of functions as apparent
variables. An example of such employment is afforded by identity, which
is the next topic considered in Section B. Finally, this section deals with
descriptions, i.e. phrases of the form “the so-and-so” (in the singular). It is
shown that the appearance of a grammatical subject “the so-and-so” is deceptive.
88 MATHEMATICAL LOGIC [PART
and that such propositions, fully stated, contain no such subject, but contain
instead an apparent variable.
Section C deals with classes, and with relations in so far as they are analogous
to classes. Classes and relations, like descriptions, are shown to be “incomplete
symbols" (cf. Introduction, Chapter III), and it is shown that a proposition
which is grammatically about a class is to be regarded as really concerned with
a propositional function and an apparent variable whose values are predicative
propositional functions (with a similar result for relations). The remainder of
.Section C deals with the calculus of classes, and with the calculus of relations
in so far as it is analogous to that of classes.
Section l>d« als with those properties of relations which have no analogues
for classes. In this section, n number of ideas and notations are introduced
which are constantly needed throughout the rest of the work. Most of the
properties of relations which have analogues in the theory of classes are compara¬
tively unimportant, while those that have no such analogues arc of the very
greatest utility. It i* partly for this reason that emphasis on the calculus-
aspect of symbolic logic has proved a hindrance, hitherto, to the proper develop¬
ment of the theory of relations.
Section 10. finally, extends the notions of the addition and midtiplication of
classes or relations to cases where the summands or factors are not individually
given, but are given as the members of some class. The advantage obtained
by this extension is that it enables us to deal with an infinite number of
summands or factors.
Considered as a formal calculus, mathematical logic has throe analogous
•branches, namely (1) the calculus of propositions, (2) the calculus of classes,
(3) the calculus of relations. Of these, (1) is dealt with in Section A, while
(2) and (3), in so far ns they are analogous, arc dealt with in Section C. We
have, for each of the three, the four analogous ideas of negation, addition,
multiplication, and implication or inclusion. Of these, negation is analogous
to the negative in ordinary algebra, and implication or inclusion is analogous
to the relation “ less than or equal to" in ordinary algebra. But the analogy
must not be pressed, as it has important limitations. The sum of two pro¬
positions is their disjunction, the sum of two classes is the class of terms
belonging to one or other, the sum of two relations is the relation consisting
in the fact that one or other of the two relations holds. The sum of a class
of classes is the class of all terms belonging to some one or other of tho
classes, and the sum of a class of relations is the relation consisting in the
fact that some one relation of the class holds. The product of two pro¬
positions is their joint assertion, the product of two classes is their common
part, the product of two relations is the relation consisting in the fact that
both the relations hold. The product of a class of classes is the part common
to all of them, and the product of a class of relations is the relation consisting
1 ]
TilK LOCIOAI. C.XUTLrs
8!»
in the fact that all relations of the class in i|iicstion hold. The inclusion «»f
one class in another consists in the fact that all members of the «»ne arc
members of the other, while the inclusion of one relation in another consists
in the fact that every pair of terms which has the one relation also has the
other relation. It is then shown that the properties of negation, addition,
multiplication and inclusion are exactly analogous for classes and relations,
and are, with certain exceptions, analogous to the properties of negation, ad¬
dition. multiplication and implication for propositions. (The exceptions arise
chiefly from the fact that “ p implies ij " is itself a proposition, ami can there¬
fore imply and be implied, while “a is contained in where a and f3 are
classes, is not a class, and can therefore neither contain nor be contained in
another class 7 .) But classes have certain properties not possessed by pro¬
positions: these arise from the fact that classes have not a two-told division
corresponding to the division of propositions into true and false, but a three¬
fold division, namely into ( 1 ) the universal class, which contains the whole of
a certain type, ( 2 ) the null-class, which has no members, (3) all other classes,
which neither contain nothing nor contain everything of the appropriate type.
The resulting properties of classes, which are not analogous to properties of
propositions, are dealt with in *24. And just as classes have properties not
analogous to any properties of propositions, so relations have properties not
analogous to any properties of classes, though all the properties of classes have
analogues among relations. The special properties of relations are much more
numerous and important than the properties belonging to classes but not to
propositions. These special properties of relations therefore occupy a whole
section, namely Section D.
SECTION A
THE THEORY OF DEDUCTION
The purpose of the present section is to set forth the first stage of the
deduction of pure mathematics from its logical foundations. This first stage
is necessarily concerned with deduction itself, i.e. with the principles by which
conclusions are inferred from premisses. If it is our purpose to make all our
assumptions explicit, and to effect the deduction of all our other propositions
from these assumptions, it is obvious that the first assumptions we need are
those* that are required to make deduction possible. Symbolic logic is often
regarded as consisting of two coordinate parts, the theory of classes and the
theory of propositions. But from our jn>int of view these two parts are not
coordinate; for in the theory of classes we deduce one proposition from another
by means of principles belonging to the theory of propositions, whereas in the
theory of propositions we nowhere require the theory of classes. Hence, in a
deductive system, the theory of propositions necessarily precedes the theory
of classes.
But the subject to be treated in what follows is not quite properly described
as tlie theory of propositions. It is in fact. the theory of how one proposition
can be* inferred from another. Now in order that one proposition may be
inferred from another, it is necessary that the two should have that relation
which makes the one a consequence of the other. When a proposition q is a
consequence of a proposition p. we say that p implies q. Thus deduction
depends upon the relation of implication, and every deductive system must
contain among its premisses as many of the properties of implication as arc
necessary to legitimate the ordinary procedure of deduction. In the present
section, certain propositions will be stated as premisses, and it will be shown
that they are sufficient for idl common forms of inference. It will not be shown
that they are all necessary , and it is possible that the number of them might
be diminished. All that is affirmed concerning the premisses is (1) that they
are true, (2) that they are sufficient for the theory of deduction, (3) that we
do not know how to diminish their number. But with regard to (2), there
must always be some element of doubt, since it is hard to be sure that one
never uses some principle unconsciously. The habit of being rigidly guided
by formal symbolic rules is a safeguard against unconscious assumptions; but
even this safeguard is not always adequate.
*1. PRIMITIVE IDEAS AND PROPOSITIONS
Since all definitions of* terms are effected by means of of her terms, every
system of definitions which is not circular must start from a certain apparatus
of undefined terms. It is to some extent optional what, ideas we take as
undefined in mathematics; the motives guiding our choice will be (1) t«*
make the number of undefined ideas as small as possible. (2) as between two
systems in which the number is equal, to choose the one which seems the
simpler and easier. We know no way of proving that such and such a system
of undefined ideas contains as few as will give such and such results*. Hence
we can only say that such and such ideas arc undefined in such and such
a system, not that the)’ are indefinable. Following Peano, we shall call the
undefined ideas and the undemonstrated propositions primitive ideas and
primitive propositions respectively. The primitive ideas are explained by means
of descriptions intended to point out to the reader what is meant; but the
explanations do not constitute definitions, because they really involve the ideas
they explain.
In the present number, we shall first enumerate the primitive ideas
required in this section; then we shall define implication ; and then we
shall enunciate the primitive propositions required in this section. Every
definition or proposition in the work has a number, for purposes of reference.
Following Peano, we use numbers having a decimal as well as an integral
part, in order to be able to insert new propositions between any two. A change
in the integral part of the number will be used to correspond to a new
chapter. Definitions will generally have numbers whose decimal part is less
than T, and will be usually put at the beginning of chapters. In references,
the integral parts of the numbers of propositions will be distinguished by
being preceded by a star; thus ••*101 ” will mean the definition or proposition
so numbered, and *•*1” will mean the chapter in which propositions have
numbers whose integral part is 1, t.e. the present chapter. Chapters will
generally be called “ numbers."
Primitive Ideas.
(1) Elementary propositions. By an “elementary” proposition we mean
one which does not involve any variables, or, in other language, one which
does not involve such words as " all," “ some,” " the " or equivalents for such
words. A proposition such as “ this is red," where “ this” is something given
in sensation, will be elementary. Any combination of given elementary
propositions by means of negation, disjunction or conjunction (see below) will
• The recognized methods of proving independence are not applicable, without reserve, to
fundamentals. Cf. Principle of Mathematic., § 17. What is there said concermng prim.l.vo
propositions applies with even greater force to primitive ideas.
92
MATHEMATICAL LOGIC
[PART I
be elementary. In the primitive propositions of the present number, and
therefore in the deductions from these primitive propositions in *2—*5, the
letters p, 7 . r, s will be used to denote elementary propositions.
(2) Elementary propositional functions. By an " elementary propositional
function" we shall mean an expression containing an undetermined consti¬
tuent, i.e. a variable, or several such constituents, and such that, when the
undetermined constituent or constituents are determined, i.e. when values are
assigned to the variable or variables, the resulting value of the expression
in tpicstion is an elementary proposition. Thus if p is an undetermined
elementary proposition, “ not-p is an elementary propositional function.
We shall show in *9 how to extend the results of this and the following
numbers (★!—*.>) to propositions which are not elementary.
(3) Assertion. Any proposition may la- either asserted or merely con¬
sidered. If I say "Caesar died,” I assert the proposition "Caesar died,”
if I say "'Caesar died ’ is a proposition," I make a different assertion, and
" Caesar died ’’ is no longer asserted, but merely considered. Similarly in a
hypothetical proposition, e.y. " if « = b, then b = a," we have two unassorted
propositions, namely "a = 6 * and " 6 — a," while what is asserted is that the
first of these implies the second. In language, we indicate when a proposition
is merely considered by " if so-and-so" or " that so-and-so" or merely by
inverted commas. In symbols, if p is a proposition, p by itself will stand
for the unassorted proposition, while the asserted proposition will be de¬
signated by
"I-.;,."
The sign "h" is called the assertion-sign*; it may be read "it is true that”
(although philosophically this is not exactly what it means). The dots after
the assertion-sign indicate its range; that is to say, everything following is
asserted until we reach either an equal number of dots preceding a sign
of implication or the end of the sentence. Thus " b : p . D . 7 " means " it is
true that p implies 7 ," whereas " b . p . D h . 7 ” means " p is true ; therefore
7 is true+." The first of these does not necessarily involve the truth either
of ]> or of 7 , while the second involves the truth of both.
(4) Assertion of a propositional function. Besides the assertion of
definite propositions, we need what we shall call "assertion of a propositional
function.” The general notion of asserting any propositional function is
not used until *9, but we use at once the notion of asserting various special
elementary propositional functions. Let ^rbea propositional function whose
argument is or; then we may assert <f>x without assigning a value to x.
This is done, for example, when the law of identity is asserted in the form
"A is A.” Here A is left undetermined, because, however A may be deter-
• We have adopted both the idea and the symbol of assertion from Frege,
t Cf. Principle* of Mathematics, § 38.
SECTION A]
PRIMITIVE IDEAS AND PROPOSITIONS
mined, the result will be true. Thus when we assert tj>.r, leaving.* und'-lt-rmiiu-d.
wo are asserting an ambiguous value of our function. 'I'his is only legitimate
it, however the ambiguity may be determined, the result, will be true. Thus
take, a\s an illustration, the primitive proposition *1*2 below, namelv
•• h : p v p . D . /»,"
%.e. p or p' implies p." Here p may be any elementary proposition: l»v
leaving p undetermined, we obtain an assertion which can be applied to any
particular elementary proposition. Such assertions are like the particular
enunciations in Euclid: when it is said “let A HO be an isosceles triangle;
then the angles at the base will be equal," what is said applies to any isosceles
triangle; it is stated concerning one triangle, but not concerning a definite
one. All the assert ions in the present, work, with a very few except ions, assert
propositional functions, not definite propositions.
As a matter of fact, no constant elementary proposition will occur in tin-
present work, or can occur in any work which employs only logical ideas.
The ideas and propositions of logic are all general: an assertion (for example)
which is true of Socrates but not of Plato, will not belong to logic*, and if an
assertion which is true of both is to occur in logic, it must not be made
concerning either, but concerning a variable .r. In order to obtain, in logic,
a definite proposition instead of a propositional function, it is necessary to
take some propositional function and assert that it is true always or some¬
times, i.e. with all possible values of the variable or with some possible value.
Thus, giving the name “individual” to whatever there is that is neither
a proposition nor a function, the proposition “every individual is identical
with itself” or the proposition “there are individuals" will be a proposition
belonging to logic. But these propositions arc not elementary.
(5) Negation. If p is any proposition, the proposition “not -p,” or “p is
false," will be represented by "~p.” For the present, p must be an elementary
proposition.
(6) Disjunction. If p and q are any propositions, the proposition “p or q,"
i.e. “ either p is true or q is true,” where the alternatives are to be not
mutually exclusive, will be represented by
“p v 9-’
This is called the disjunction or the logical sum of p and q. Thus “ q ”
will mean “p is false or q is true”; "~(pvg) " will mean “it is false that
either p or q is true,” which is equivalent to “p and q are both false”;
“»>-.(p v iv q) ” will mean “it is false that either p is false or q is false,” which
is equivalent to “p and q are both true”; and so on. For the present, p and
q must be elementary propositions.
• When wo Bay that a proposition “belongs to logic," we mean that it can be expressed in
terms of the primitive ideas of logic. We do not mean that logic appliet to it, for that would of
course be true of any proposition.
94
MATHEMATICAL LOGIC
[PART I
The above are all the primitive ideas required in the theory of deduction.
Other primitive ideas will be introduced in Section B.
Definition of Implication. When a proposition 7 follows from a proposition
/>. so that if p is true. 7 must also lx* true, we say that p implies 7. The idea
of implication, in the form in which we require it. can be defined. The mean¬
ing to be given to implication in what follows may at first sight appear some¬
what artificial; but although there are other legitimate meanings, the one here
adopted is very much more convenient for our purposes than any of its rivals.
The essential pnqierty that we require of implication is this: “What is
implied by a true pro|Hisitioii i> true." It is in virtue of this property that
implication yields proof**. But this property by no means determines whether
anything, and if so what, is implied by a false pro|iosition. What it does
determine is that, if p implies 7. then it cannot be the case that p is true and
The most
7 is false, i.e. it must be the case that either /> i> false or 7 is true,
convenient interpretation of implication is to say. conversely, that if either />
is false or 7 is true, then " p implies 7" is to be true. Hence " p implies 7
is to be defined to mean : '* Either p is false or 7 is true.” Hence we put:
*101. p D 7 . = . ^ p v 7 l )f.
Here the letters" Ilf" stand for " definition." They and the sign of equality
together are to be regarded as forming one symbol, standing for " is defined
to'mean*." Whatever comes to the left of the sign of equality is defined to
mean the same as what comes to the right of it. Definition is not among the
primitive ideas, because definitions are concerned solely with the symbolism,
not with what is symbolised; they are introduced for practical convenience,
and arc theoretically unnecessary.
In virtue of the above definition, when " p D 7" holds, then either p is false
or 7 is true; hence if /> is true, 7 must be true. Thus the above definition
preserves the essential characteristic of implication ; it gives, in fact, the most
general meaning compatible with the preservation of this characteristic.
Prim iti v e Propositions.
* 1 T. Anything implied by a true elementary proposition is true. Ppt-
The above principle will be extended in *9 to propositions which are not
elementary. It is not the same ns “ if p is true, then if p implies 7, 7 is true.
This is a true proposition, but it holds equally when p is not true and when p
does not imply 7. It does not, like the principle we are concerned with, enable
us to assert 7 simply, without any hypothesis. We cannot express the principle
symbolically, partly because any symbolism in which p is variable only gives
the hypothesis that p is true, not the fact that it is true*.
• The sign of equality not followed by the letters "Df" will have a different meaning, to be
defined later.
t The letters •• Pp” stand for "primitive proposition," ns with Peano.
* For further remarks on this principle, cf. Principles of Mathematict, § 38.
SECTION* A
PR1M1T1VK I OKAS AND PROPOSITIONS
The above principle is used whenever we have («» deduce a proposition
from a proposition. But the immense majority of the assertions in iIn-
present work are assertions of propositional functions. /.<*. they contain an
undetermined variable. Since the assertion of a propositional function is a
different primitive idea from the assertion of a proposition, we ivqiiiiv a
primitive proposition different from *11, though allied to it. to enable us
deduce the assertion of a propositional function "\Jr.r" from tin* assertions ..f
the two propositional functions **</>.#•" and " </>.»D \fr.r." This primitive pro¬
position is as follows:
* 1 T 1 . When <f>.v can be asserted, where .r is a real variable, and <£.0 yf/.r can
be asserted, where a- is a real variable, then yfr.r can be asserted, where .*• i* a
real variable. Pp.
This principle is also to be assumed for functions of several variables.
Part of the importance of the above primitive proposition is due to tin-
fact that it expresses in the symbolism a result following from the theory of
types, which requires symbolic recognition. Suppose we have t he two assertions
of propositional functions ,<f>x ” and then the "a” in </>./• is
not absolutely anything, but anything for which as argument the function * </>./■“
is significant; similarly in “ <f>x D '/rar'* the x is anything for which “ <f>.r D xj/.r "
is significant. Apart from some axiom, we do not know that the a s for which
“ <f>xD yfrx ” is significant are the same as those for which “ tf>.v " is significant.
The primitive proposition * 111 , by securing that, as the result of the assertions
of the propositional functions "<£a ” and “ <px D yjrx," the propositional function
u yfrx” can also be asserted, secures partial symbolic recognition, in the form most
useful in actual deductions, of an important principle which follows from tin*
theory of types, namely that, if there is any one argument a for which both
“ <f>ci ” and “ yfra ” arc significant, then the range of arguments for which "
is significant is the same as the range of arguments for which “ yfrx ” is sig¬
nificant. It is obvious that, if the propositional function “ <f>x D yfrx ” can be
asserted, there must be arguments a for which " </>n D " is significant, and
for which, therefore, and “yfra” must be significant. Hence, by our
principle, the values of x for which “ <px ” is significant are the same as those
for which “y^x” is significant, i.e. the type of possible arguments for <£.? (cf.
p. 15 ) is the same as that of possible arguments for yfrx. The primitive pro¬
position *1-11, since it states a practically important consequence of this fact,
is called the “axiom of identification of type.”
Another consequence of the principle that, if there is an argument a for
which both <f>a and yfra are significant, then <f>x is significant whenever yfrx is
significant, and vice versa, will be given in the “ axiom of identification of real
variables,” introduced in * 1 - 72 . These two propositions, *IT 1 and * 172 , give
what is symbolically essential to the conduct of demonstrations in accordance
with the theory of types.
06
MATHEMATICAL LOGIC
[PART I
The above proposition *111 is used in every inference from one asserted
propositional function to another. We will illustrate the use of this proposition
by setting forth at length the way in which it is tirst used, in the proof ol
*2 06 . That proposition is
“ h /O 7 . D : 7 D r . Z> . yO r."
We have already proved, in *2 ()">, the proposition
h 7 D »•. D : /> D 7 . D . /O r.
It is obvious that *2 06 results from *2 0 ."> by means of *2 04 , which is
I- p . D . 7 D #•: D : 7 . D • /O #•.
For if. in this proposition, we replace /> by 7 D r, 7 by /O7, and /• by pO r,
we obtain, as an instance of * 204 . tin- proposition
h::r/ 3 r. 3 :/) 37 . 3 ./ 0 >:. 3 :./ 0 </.D:«/ 3 i-. 3 ./ 0 i' (I).
and here tie* hy|M»thesis i> a>serted by *2 0o. TIiun «»ur primitive proposition
*111 enables us to assert the conclusion.
* 12 . H : p v p . D . p Pp.
This proposition states: H If either p is true or p is true, then p is true.”
It is called the “principle of tautology,” and will be «pioted by the abbreviated
title of'* Taut.” It is convenient, for purposes of reference, to give names to
a few of the more important pro|»ositions; in general, propositions will be
referred to by their numbers.
* 13 . h : 7 . D . /> v 7 Pp.
This principle stales: “If 7 is true, then 'p or 7’ is true." Thus e.g. it 7
"to-day is Wednesday” and p is “ to-day is Tuesday," the principle states:
" If to-day is Wednesday, then to-day is either Tuesday or Wednesday.’ It
is called the " principle of addition," because it states that if a proposition is
true, any alternative may be added without making it false. The principle
will be referred to as "Add.
* 14 . h : p v 7 . D . 7 v p Pp.
This principle states that “p or 7" implies “7 or p." It states the
permutativc law for logical addition of propositions, and will be called the
“ principle of permutation.” It will be referred to as " Perm.”
* 16 . h :v(7 v r). D . 7 v (p v r) Pp.
This principle states: " If either p is true, or 'q or r’ is true, then either
7 is true, or ‘ p or r* is true.” It is a form of the associative law for logical
addition, and will be called the "associative principle." It will be referred to
as "Assoc." The proposition
P v (7 v /•) . D . (p v 7) v r,
which would be the natural form for the associative law, has less deductive
power, and is therefore not taken as a primitive proposition.
SECTION A]
PRIMITIVE 11 »EAS ANI> PROPOSITIONS
97
*16. h:.(/D/'.D:^V(/.D./)Vr Pp.
This principle states: '* If q implies r. then ' p or •/' implies • /» «»r hi
other words, in an implication, an alternative may be added to both premiss
and conclusion without impairing the truth of the implication. The principle
will be called the “principle of summation," and will be referred t«> as "Sum."
*1*7. If p is an elementary proposition, ~p is an elementary proposition. Pp.
*171. If p and q are elementary propositions, p v »y is an elementary pro¬
position. Pp.
*1'72. If <f>p and y\rp arc elementary proj>ositional functions which take
elementary propositions as arguments, <f>p v yfrp is an elementary propositional
function. Pp.
This axiom is to apply also to functions of two or more variables. It is
called the “axiom of identification of real variables.” It will be observed that
if <f> and yfr are functions which take arguments of different types, there is no
such function as because <f> and yfr cannot significantly have the
same argument. A more general form of the above axiom will be given in *9.
The use of the above axioms *1 *7*71’72 will generally be tacit. It is only
through them and the axioms of *9 that the theory of types explained in the
Introduction becomes relevant, and any view of logic which justifies these
axioms justifies such subsequent reasoning as employs the theory of types.
This completes the list of primitive propositions required for the theory
of deduction as applied to elementary propositions.
R&W 1
4
*2. IMMEDIATE CONSEQUENCES OF THE
PRIMITIVE PROPOSITIONS
Summary of *2.
The proofs of the earlier of the propositions of this number consist simply
in noticing that they are instances of the general rules given in *1. In such
cases, these rules are not premisses, since they assert any instance of them¬
selves, not something other than their instances. Hence when a general rule
is adduced in early proofs, it will be adduced in brackets*, with indications,
when required, as to the changes of letters from those given in the rule to
those in the case considered. Thus “ Taut ” will mean what “Taut ” becomes
when is written in place of p. If “ Taut " is enclosed in square brackets
before an asserted proposition, that means that, in accordance with “Taut,
we are asserting what "Taut” becomes when is written in place of p.
The recognition that a certain proposition is an instance of some general
proposition previously proved or assumed is essential to the process of de¬
duction from general rules, but cannot itself be erected into a general rule,
since the application required is particular, and no general rule can explicitly
include a particular application.
Again, when two different sets of symbols express the same proposition in
virtue of a definition, say *101, and one of these, which we will call (1), has
been asserted, the assertion of the other is made by writing " [( 1 ).(* 1 ’ 01 )]
before it, meaning that, in virtue of *1 01, the new set of symbols asserts the
same proposition as was asserted in (1). A reference to a definition is dis¬
tinguished from a reference to a previous proposition by being enclosed in
round brackets.
The propositions in this number are all, or nearly all, actually needed in
deducing mathematics from our primitive propositions. Although certain
abbreviating processes will be gradually introduced, proofs will be given very
fully, because the importance of the present subject lies, not in the propo¬
sitions themselves, but (1) in the fact that they follow from the primitive
propositions, (2) in the fact that the subject is the easiest, simplest, and most
elementary example of the symbolic method of dealing with the principles of
mathematics generally. Later portions—the theories of classes, relations,
cardinal numbers, series, ordinal numbers, geometry, etc.—all employ the
same method, but with an increasing complexity in the entities and functions
considered.
• Later on we aboil cease to mark the distinction between a premiss and a rule according to
which an inference is conducted. It is only in early proofs that this distinction is important.
SECTION A]
IMMEI'IATK CONSEQl* KNOES
00
The most important, propositions proved in tin* present number are lhe
following:
*2 02. h:<f.D./0<f
/.e. q implies that p implies q. i.c. a true proposition is implied by any
proposition. This proposition is called the ** principle of simpliHoation " (re¬
ferred to as “Simp”), because, as will appear later, it enables us to pass from
the joint assertion of q and p to the assertion of q simply. When the special
meaning which we have given to implication is remembered, it will be seen
that this proposition is obvious.
*203. hip'D'^q.O.qD^-'p
*2T5. h: ~ p D q • D • ^ q D j>
*2 16. \-:pDq.D.~qO~p
*217. h : ~ qO ~p .D .pO q
These four analogous propositions constitute the “principle of transposition."
referred to as“Transp.” They lead to the rule that in an implication the two
sides maybe interchanged by turning negative into positive and positive into
negative. They are thus analogous to the algebraical rule that the two sides
of an equation may be interchanged by changing the signs.
*2 04. . 0 , q "D r : D : q . D . p D r
This is called the “ commutative principle ” and referred to as “ Comm."
It states that, if r follows from q provided p is true, then r follows from p
provided q is true.
*2 05. h z. q O r . D : p D q . D .p D r
*2 06. I -z.plq.DzqOr.O.pDr
These two propositions are the source of the syllogism in Barbara (as will
be shown later) and are therefore called the “ principle of the syllogism ”
(referred to as “ Syll ”). The first states that, if r follows from q, then if q
follows from p, r follows from p. The second states the same thing with the
premisses interchanged.
*2 08. l-./Op
I.e. any proposition implies itself. This is called the “ principle of identity "
and referred to as “ Id.” It is not the same as the “ law of identity ” (“ x is
identical with x ”), but the law of identity is inferred from it (cf. *1315).
*2 21. \-z~p.D.pDq
l.e. a false proposition implies any proposition.
The later propositions of the present number are mostly subsumed under
propositions in *3 or *4, which give the same results in more compendious
forms. We now proceed to formal deductions.
7—2
100
MATHEMATICAL LOGIC
[PART I
*2 01 . h zp D ~p • ^
This proposition states that, if p implies its own falsehood, then p is false. It
is called the "principle of the red net io ad absurdinn,” and will be referred to ns
• Abs."* The proof is as follows (where “ Dem." is short for “ demonstration ):
Dem.
[(1).(*1'01)] V : p D ~/>. 0 . ~p
*2 02. I- s q . D .p D 7
Dem.
Add h : 7 . D . ~p v q (I)
[(!).(♦ 1*01)]
*2 03 . h : p . D . 7 D^/>
Dem.
Perm I h : ^p v ^7 .0 . v ~p (*)
P> <1 J
(1).(*101)] 1- :/>D^7. D ,tjO~p
*2 04 . h . D . 7 D r s D s 7. D .p D r
Dem.
£ Assoc h :• v (^7 v r). D . ^7 v v r) ( 1 )
[(1 ).(*1 01)] I -p . D . 7 D r : D : 7 . O . p 3 r
*2 05 . h 7 D r . Z> : /O 7 . D . /O >•
Dem.
j^Sum fj O r. D : v 7 . D . v r (I)
[(1 ).(*1'01)] h 7 D r. >: /O 7 . D . p D r
*2 06 . H :./0 7 . D : 7 D r. D ./O r
Dew.
r C omm2^^A2lt£2r']h::,Dr.D:/,D9.3.p3r:.
P ’ ' J Oz.pOg.O-.qOr.O.pO
[*205] h:.ryDr.D:/07.D.p3r ( 2 )
[(1).(2).*111] h.p07.0:7^'--^-P :)r
In the last line of this proof, “(1) . (2) . *111” means that we are
inferring in accordance with * 111 , having before us a proposition, namely
pDq.DzqOr.O.pOr, which, by ( 1 ), is implied by 7 D r. D : pD 7 .3 • P? r »
which, by (2), is true. In general, in such cases, we shall omit the reference
to*l'U. .
• There is an interesting historical article on this principle by Vailati, "A proposito tin
passo del Teeteto c di una diraostrazione di Euclide," Rivista di Filotojia e *ei*nze affine,
SECTION A]
IMMEDIATE CONSEQUENCES
101
The above two propositions will both bo referred to as the “ principle ol
the syllogism ” (shortened to Syll "). because, as will appear later, the syllo¬
gism in Barbara is derived from them.
*2 07. h : p . D . p v p ^*1*3
Here we put nothing beyond “*l-3^,” because the proposition to be
proved is what *1*3 becomes when p is written in place of </.
*2 08. h . p D p
Dem.
jj*2'05 H ss p yp . O . p s D s. p . 3 - p vp : 3 *p Op
[Taut] h : p v p . D . p
[(1).(2).*111] h z.p.D .py/p:D .pOp
[*2 07] h zp.D.pvp
[(3).(4).*M1] h.pOp
* 21 . h.~pvp [*208. (*101)]
*211.
Dem.
( 1 )
( 2 )
(3)
W
Perm
~£iP]
P> 9J
h i~p v. D .pv~p
(1)
(1)
[(1).*2T.*1T1] h ,pv~p
This is the law of excluded middle.
*212. h.pDo-( p )
Dem.
j^*2T 1 h . v~(~;;)
[( 1).(*1 01 )] h . 7 0 -(~/>)
* 2 T 3 . I-. p {'^'(~|>)}
This proposition is a leinma for *2 14, which, with *2T2, constitutes the
principle of double negation.
Dem.
a, r
^p>lj
h ~p . D . ~ ('■^'(~p)I • ^ :
u z *
p *1
pv~p . D .p v~{~(~/>)}
(i)
[•«•?]
h : . D . ~[~(~p))
(2)
[<1).(2).*1*11]
h :pv~p . D ./> v~{~(~ < p)l
(3)
[(3).*2T 1 .*1 -11]
h . p v~(~(~p)}
102
MATHEMATICAL LOGIC
[part I
*214.
Dem.
[
Perm -
~( ~p)
J h : /> v — J — (—y>)J . D . — ! ~(^y>)] v
H.
(-/)>: vp
[(1).*213.*111]
(( 2 ).(* 101 )]
*215. f- : ~ p D 7 . D . ^7 D y>
Dem.
( 1 )
( 2 )
*0 Q", ~~
y>. —(— 7>"|
V 7 D '>.7). D : ~y> D 7. D . ~y> D ^7)
(i)
>, r
*212 2
PJ
h .7 D -M ^ 7)
(2)
[tl M2).* I'll]
H : <^y> D 7 . D . ^y> D ~(~7>
(3)
H : ^y> D^( ~7). D . ~7 3 ~(~p)
(+)
(5)
( 0 )
*203^-^l
P* 7 J
*2 05 ~ f /' ft ' ^ I K -w-p) D y>. D : ^7 D ~(~p). D . ^7 D y>
P» 7» r J
[<5).*214.*1*11) K :-v7D'v(~/')* 3 •~7^P
r.„n. 01 ~7°7. ^/>3M^7^7^^^) 1 h ..
L />• 7 . r J
~p D^*( ^ 7 ). D . ~7 D ~('■^y*): D :•
~p D 7 . D . ~y> : D : ~y> D 7 . D . ~7 D~( ~p) (7)
[<4).(7).*111 ] H :. ~p D 7 . D . ~p D ~< - 7 ): D :
~y> D 7 . D . ^7 D ~(^y>) ( 8 )
[(3).(8).*111] H : ~p D 7 . D . *wy D^(~p) ( 9 )
r «2 05 ~ P 3 '!■ ~'t 3 ~^ 1 h 3~(~j» ■ 3 ■ Op :
I 7'. 7. r J
D '-«y> D 7 . D . ~7 D ^(~y>) : D : ^y> D 7 . D . ^7 Dp (10)
[(G).(10).*l ll] H :. ~y> 3 7 . D • ~7 3 : 3 :
~p D 7 . D . ^7 Dp (11)
[(9).(U).*ril] h : ~y> D 7 . D .-wy Dy>
j\ r o/e o/i f/ie proof of *215. In the above proof, it will be seen that (3),
(4), (G) are respectively of the forms p x Dp., y> 2 Dy> s , p,Dp 4 , where y>| 3/>4 ,s
the proposition to be proved. From y>, Dy> 3 , p*Dy;„ y> s Dp 4 the proposition
y>, Dy > 4 results by repeated applications of *2 05 or *20G (both of which are
called “ Syll ”). It is tedious and unnecessary to repeat this process every
time it is used; it will therefore be abbreviated into
“[Syll] !-.(«).( 6 ).(c).Dh.(rf)”
where (a) is of the form p, D p., ( 6 ) of the form p, D p„ (c) of the form p, Dp 4 ,
and (d) of the form p, Dp 4 . The same abbreviation will be applied to a sorites
of any length.
SECTION A]
IMMEDIATE CONSKQUENDES
103
Also whore we have '*1- . />," ami "h ./>, D p. t " ami is the proposition to
be proved, it is convenient to write simply
•‘1
[etc.] K p,."
where “ etc." will be a reference to the previous propositions in virtue of which
the implication “p x Dp." holds. This form embodies the use of *1*1 1 or *11,
and makes many proofs axt once shorter and easier to follow. It is used in the
first two lines of the following proof.
*2T6. b zpD q .D • ~qD~p
Bern.
[*212] b
[*205] b zpDq.D .pD~ (~q) (1)
*2 03 b zp D ~(~q) . D . ~q D ~p (2)
;Syll] K(l).(2).Dh zpDq.D.~qD~p
Note. The proposition to be proved will be called “ Prop,” and when
a proof ends, like that of * 21 G, by an implication between asserted propo¬
sitions, of which the consequent is the proposition to be proved, we shall
write “ b . etc. D h . Prop”. Thus “ D h . Prop ” ends a proof, and more or less
corresponds to “Q.E.D.”
*217. h ! D . D .pD
Bern.
*2 03
b : ~q D . D »p D ~(~q)
(i)
>214]
h ; ~(~q) DqzD
[*205]
b z pD ~(~q) . D .pD q
( 2 )
[Syll]
h . (1) . (2) . D h . Prop
*2T5, *2’16 and *217 are forms of the principle of transposition, and will
be all referred to as “ Transp.”
*218. 1 -: rs^p
Bern.
[* 2 - 12 ]
[*205]
b .p D ~(~p ). D
b . ~p Dp . D . ~p D
( 1 )
[•*“ =*]
1- : ~p D ~(~p) . D . ~( ~p)
( 2 )
[Syll]
b . (1).(2). D 1- : ~p Dp . D . ~(~p)
(3)
[*214]
b.~(~p)Dp
W
[Syll]
h . (3) . (4) .DK Prop
This is the complement of the principle of the redactio ad absurdum. It
MATHEMATICAL LOGIC
104
(PART I
states that a proposition which follows from the hypothesis of its own false¬
hood is true.
* 22 . I- : y> . D . p v n
Dem.
h . Add . D h : p . D . #/ v p (1)
[ Perm] h : ij v y>. D . /> v q (2)
[Syll] h . (1). (2). D 1-. Prop
*2 21 . jj *22
The above two propositions are very frequently used.
*2 24. h : y>. D . ^ yO ry (*2 21. Comm]
*2 25. 1- p : v : p v q . D . q
Dem.
h . *21 . D h : ^(y> v 7 ). v . (y> v q) s
[Assoc] D 1-: y>. v . (~(/> v 7 ). v . q\ : D h . Prop
*2 26. I- :.~y>: v : yO 7 . D . 7 £*2 # 25 -J -J
*2 27. 1- y). D : yO 7 . D . 7 [*2’26]
*2 3. h s y> v (7 v r) • D . yj v (r v 7 )
Dem.
Permhsovr.D.rvo:
l. P>'l J
j^Sum ^ - ; r^ j D h : /> v (7 v r) . D . /> v (r v 7 )
*2 31. h : p v (7 v r). D . (p v 7 ) v r
This proposition and *232 together constitute the associative law for
logical addition of propositions. In the proof, the following abbreviation
(constantly used hereafter) will be employed*: When we have a series of
propositions of the form a D 6 , 6 Dc, cDrf, all asserted, and u aDd” is the
proposition to be proved, the proof in full is as follows:
[Syll] K:.aD 6 .D: 6 Dc.D.aDc (1)
f-sa.D .6 ( 2 )
C(l).(2).*lll] h: 6 Dc.D.aDc (3)
h : b . D . c W
[(3).(4).*1-11] h : a . D . c (5)
[Syll] h:.aDc.D:cDrf.D.aDrf ( 6 )
[(5).(G).*1T1] l-:cDrf.D.aDd (7)
hsc.D.d W
«7).(8).*111] hra.D.d
• This abbreviation applies to the same type of cases as those concerned in tho note to •2-15,
but is often more convenient than the abbreviation explained in that note.
SECTION A]
IM M EDIATE CONSKQ V ENTES
10 0
It is tedious to write out this process in full; we therefore write simply
h : a . D . b .
[etc.] D.c.
[etc.] D . c/ : D h . Prop,
where “aDd” is the proposition to be proved. We indicate on the left by
references in square brackets the propositions in virtue of which the successive
implications hold. We put one dot (not two) after •‘6," to show that it is l>,
not "a D b,” that implies c. But we put two dots after d, to show that now
the whole proposition “ a D d ” is concerned. If "<iD(T is not the proposition
to be proved, but is to be used subsequently in the proof, we put
h : a . D . b .
[etc.] D . c .
[etc.] D . d
and then “ (1) ” means “ a D d." The proof of *2 31 is as follows :
Devi.
>2 3] h : p v (q v r) . D . p v (r v q ).
AsSOC ^~r] O.rv(pvq).
Perm D . (p v q) v r : D h . Prop
*2 32. h : (p v q) v r . D .p v(<j v r)
Devi.
( 1 ).
Perm — h : (p v v r . D . r v (p v 7)
3 .?v(rv ? )
Assoc
D . p v (9 v r) : D h . Prop
[*2-3]
*233. pv qvr. = .(pv q)v 7 - Df
This definition serves only for the avoidance of brackets.
*236. hi.^Dr.Djpv^.D.rvp
Dem.
[Perm] h: pvr.D.rvp:
r Syll ^ - V V’ P v r ' rv P l Dh : .pv9.D.pvr:D:pv9.D.rvp ( 1 )
L P» T J
[Sum] l-t.pr.Dspvj.D.pvr (2)
h . (1). (2) . Syll .DK Prop
*237. h:.gDr.D:gvp.D.pvr
[Syll . Perm . Sum]
hr.^Dr.Dr^vp.D.rvp
[Syll. Perm . Sum]
*2-38.
MATHEMATICAL LOGIC
[PART I
100
The proofs of *2 37 38 are exactly analogous to that of *2 36. (We use
•• *2 37’38 " as an abbreviation for " *2 37 anil *2 38.” Such abbreviations will
be use-1 throughout.)
The use of a general principle of deduction, such as either form of" Syll,"
in a proof, is di tie rent from the use of the particular premisses to which the
principle of deduction is applied. The principle of deduction gives the general
rule according to which the inference is made, but is not itself a premiss in
tin- inference. If we treated it ns a premiss, we should need either it or some
other general rule to enable ns to infer the desired conclusion, and thus we
should gradually acquire an increasing accumulation of premisses without
ever bring able to make any inference. Thus when a general rule is adduced
in drawing an inference, as when we write " [Syll] h . (1) .(2) • D h . Prop,” the
mention of " Syll " is only required in order to remind the reader how the
inference is drawn.
The rule of inference may, however, also occur as one of the ordinary
premisses, that is to say, in the case of "Syll " for example, the proposition
"p D q . D : q D r . D .p D r ” may be one of those to which our rules of deduction
are applied, and it is then an on 1 inary premiss. The distinction between the
two uses of principles of deduction is of some philosophical importance, and
in the above proofs we have indicated it by putting the rule of inference in
square brackets. It is, however, practically inconvenient to continue to dis¬
tinguish in the manner of the reference. We shall therefore henceforth both
adduce ordinary premisses in square brackets where convenient, and adduce
rules of inference, along with other propositions, in asserted premisses, t.e. wc
shall write c.y.
" h • (1). (2) • Syll. D H . Prop M
rather than “ [Syll] K(l).(2).Dh. Prop ”
*2 4. h p . v . p v q : D . p v q
Dem.
I- . *2 31 . D h p . v\ p v q : D : p v p . v . q :
[Taut.*2*38] D : p v q D h . Prop
*2*41. V Z.q .v.pvqzD.pvq
Dem.
£ Assoc ^ ^ h q . v . p v q : D :p . v . q v q :
[Tnut.Sum] D : p v q D h . Prop
*242. h rsjp .v.p^qzO.pOq |^*2'4
*2 43. I- z.p . D ./O q : D .p D q [*2 42]
*2 45. I -:~(pvq).D . [*2 2 . Transp]
*2'46. h z~(p v q) . D . ~q [*T3 . Trausp]
107
SECTION
a] immediate consequences
*247.
1“ :~{p V q) . D ,~p V q
*2-45 . *2-2 -2. Syll]
*248.
1- :~(p V q) . D .p V~g
[*246. *13 -ji . Syll]
*249.
h q).D p v ~g
1*2-45. *2-2 7 .Syll
L P '/
*25.
1“ :~(p D g) . D . ~p D g
*251.
1- : ~ (p D q) . D . p D ~ g
*2-48
P J
*252.
K :*^(p 3 g) . ~p D ~g
*2-49 ^]
*2521.
H:~(pDg).D.gDp
[*2*52*17]
*253.
hpv^.D.^pD^
Dem.
*254.
*255.
*256.
H . *2-12 38 . D I-: p v q . D . ~(~p) v q : D h . Prop
pO q .D .pv q [*214*38]
-p . D :p v g . D . g [*2*53 . Comm]
q .D-.pvq.O .p
*2 6. Hz.^-pDg.DspDg.D.g
Dem.
[*2*38] h :.~p D g . D :~p V g . D . g V g
[Taut. Syll] V z.~p v q . D . q v g : D :«^p v g . D . g
h . (1) . (2) . Syll. D H :.~p D g. D :~p v g . D . q D I- . Prop
*261.
h :.p D q . D :~p D g . D . g
[*2*6 . Comm]
*262.
1- :.p v g . D : p D g . D . g
[*2*53*6 . Syll]
*2621.
1-:.pDg.D:pvg.D.g
[*2*62. Comm]
*263.
1-s.pvg.D :~p v g . D . g
[*2*62]
*264.
1- p v g . D : pv~q . D - p
[*2*63^. Perm]
L 7 J
*265.
1" :.p D g . D zp D • D.*^p
[•*«?]
*267.
h :.p vg . D . g : D .p D g
Dem.
•
[*2*54.Syll] h p v g . D
. g : D : ~p D g . D . q
[*2*24.Syll] H:.~pDg.
D . g : D .p D g
b . (1) . (2) . Syll. D h . Prop
( 1 )
( 2 )
( 1 )
( 2 )
108
MATHEMATICAL LOGIC
[PART I
68 . Perm . *2 62 ^1
P> <1 J
*2 68 . b p D q . D . q : 3 . p v 9
Dem.
j*2‘G7 J h /> D q . D . 7 : D . 3 7
h . ( 1 ). *2*54 .DK Prop
*2 69. b z. /> 0 q. 0 . q z 0 : q 0 p. 0 . p £*2*
*2 73. b p D q . D : /> v 7 v r . Z>. q v r [*2*621*38]
*274. f- q D. D :y* v q v r . D .v r J*2’73^ . Assoc . SyllJ
*2 75. h:/)vry.D:.;».v. 7 Dr:D.pvr £*2*74 ^ . *2*53*31 J
*2 76. b p . v . q D r z D : p v q . 0 . p v r
*2 77. h />. D . q D r: D : p D 7 . D . p D r
*2*75. Comm]
*2-70 ^1
P J
( 1 )
*2 8 . b z.qv r .D :^rv $ .0 .qv s
Dem.
b . *2*53 . Perm . D h 7 v r . D : ~>0 7 :
[*2*3S] D:^rvs.D.ryV 5 :.Dh.Prop
*2 81. b :: 7 . D . r D s : D p v q . D : p v r . D . p v 5
Dem.
h .Sum •Dh::f/.D.rD«:D:./)V 9 .D:p.v.rDf ( 1 )
b . *2*76 . Syll . D I-p v 7 . D : p . v . rD s D
jtvq.Dzpvr.D.pvs ( 2 )
h.(l).(2).DH. Prop
*282. l*:./Jvr/vr.D:;)V^rvs.D./)vgvs
L 7. >•. * J
*2 83. 1-:: p . D . 7 D »*: D p . D . r D s : D : p . D . 7 D s
f #2 . 82 ^L^Zl
L /». ?J
*2 85. b p v 7 . D .p v r : D : p . v . 7 D r
Dem.
[Add.Syll] bz.pvq.D.rzD.qDr (1)
I-. *2*55 . D b :: ~p .D:.j»vr.D.r:.
[Syll] D: .pvq.O.pvrzDzpvq.O-rz.
[<1).*2*83] . D z.pvq.D .pvrzOzqDr (2)
h . (2) . Comm .Db z. pvq.O.pvrzOz ~p . D . q D r:
[*2*54] Dsp.v.^Drs.DI*. Prop
*2 86. Hr.pDg.D.jOrOrp.D.gDr £*2*85-^ J
*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS
Summary of *3.
The logical product of two propositions p and j is practically the pro¬
position " p and q are both true." But this as it stands would have to hr a
new primitive idea. We therefore take as the logical product the proposition
i.e. “it is false that either p is false or q is false." which is
obviously true when and only when p and q are both true. Thus we put
*3 01. = Df
where “ p . q" is the logical product of p and q.
*3 02. p D q D r . = . p D q . q D /• Df
This definition serves merely to abbreviate proofs.
When we are given two asserted propositional functions " H . tf>.v ” and
" 1" • >/'■#,” we shall have “ h . <f>x . yfrx” whenever <t> and take arguments of
the same type. This will be proved for any functions in *9 ; for the present,
we are confined to elementary propositional functions of elementary pro¬
positions. In this case, the result is proved as follows :
By *17, ~(f>p and ~y}rp are elementary propositional functions, and there¬
fore, by *172, ~<t>pv~yfrp is an elementary propositional function. Hence
by *211,
h : '*-><f>p v ~y}rp . v . <f>p v '^'yfrp).
Hence by *2 32 and *101,
1 - <f>p . D : yfrp . D . ~(~<f>p v
i.e. by *3 01,
<f>p . D : yjrp ,D.<f>p. yjrp.
Hence by *111, when we have "K<*»/>” and “\-.+p” we have "H . <t>p*'l'p ”
This proposition is *3 03. It is to be understood, like *172, as applying also
to functions of two or more variables.
The above is the practically most useful form of the axiom of identification
of real variables (cf. *1 72). In practice, when the restriction to elementary
propositions and propositional functions has been removed, a convenient means
by which two functions can often be recognized as taking arguments of the
same type is the following:
If <px contains, in any way, a constituent x( x » V> z * •••) an< * y l rx contains,
in any way, a constituent %(ar, u, v ,...), then both <f>x and y\rx take arguments
of the type of the argument x in x^ x 'V' z > •••)> an< l therefore botli <f>x and yfrx
take arguments of the same type. Hence, in such a case, if both <f>x and yjrx
can be asserted, so can <f>x . yfrx.
110 MATHEMATICAL LOGIC [PARTI
As an example of ilie use of this proposition, take the proof of *3 47. We
there prove
h p D **. 7 D s . D : p . 7 . D . 7 . r (1)
and H y> D r . 7 D s . D : 7 . r . D . r . 5 ( 2 )
and what we wish to prove is
p D r. 7 D *. D : . 9 . D . r. s,
which is *3*47. Now in ( 1 ) and (2), p, 7 . r, s are elementary propositions
(as everywhere in Section A); hence by *1*7*71, applied repeatedly.
“ p D r . q D s . D : p . 7 . D . q . r ” and “ /> D r. 7 D s . D : 7 . r. D . r. 5 " are ele¬
mentary propositional functions. Hence by *3 03, we have
I-:: 7 O r . q D ft . D : /». 7 . D . 7 . r p D r. q D s . D : q . r . D . »*. s.
whence the result follows bv *3*43 ami *3 33.
The principal propositions of the present number are the following:
*3 2. I- p . D : q . D . p . q
I.e. " p implies that 7 implies p . q," i.e. if each of two propositions is true,
so is their logical product.
*3 26. bip.q.^.p
*3 27. 1* : p . q . D . q
I.e. if the logical product of two propositions is true, then each of the two
propositions severally is true.
*3 3. b i.p . q . D • r: D : p • D . q D r
/.e. if and 7 jointly imply r, then /> implies that 7 implies r. This
principle (following Peano) will be called "exportation,” because 7 is "exported
from the hypothesis. It will be referred to ns " Exp."
*3 31. b p . D . 7 D #•: D : p . 7 . D . r
This is the correlative of the above, and will be called (following Peano)
" importation ” (referred to as “ Imp ").
*3*35. I*: p . p D 7 . D • 7
I.e. "if p is true, and 7 follows from it, then 7 is true.” This will be called
the "principle of assertion” (referred to as "Ass”). It differs from *11 by
the fact that it does not apply only when p really is true, but requires merely
the hypothesis that p is true.
*343. b p D 7 .p D r . D zp • D. 7 . r
I.e. if a proposition implies each of two propositions, then it implies their
logical product. This is called by Peano the " principle of composition.” It
will be referred to as " Comp.”
*3*45. b p D 7 . D : p . r . D . 7 . r
I.e. both sides of an implication may be multiplied by a common factor.
This is called by Peano the “ principle of the factor.” It will be referred to
as " Fact.”
SECTION A]
THE LOGICAL PRODUCT OF TWO PROPOSITIONS
1 I 1
*3'47. h p D r. D s , D ; />. <y . D . r.,«
I.e. it ' p implies q and /• implies s. then p and q jointly imply #• and *
jointly. The law of contradiction. ** b . ~ (/> . ~ p)," is proved in this min.b. i
(#3 24); but in spite of its tame we have found few occasions for its use.
tl)
(2)
*3 01. p . q . = . ^ p v ~ q) Df
*302. pO<]Dr. = .pDq.qOr Df
*3 03. Given two asserted elementary propositional functions “\-.<f>p" and
“ • 'kP whose arguments are elementary propositions, we have . 4 >p . \pp.
Dem.
h • *17 72 • *211, D h ;<v (f>p v *—• \frp . v . ~ (<£>y> v •—> ^jt)
H . (1) . *2-32 . (*101). D I -:.<t>p.D:>lrp.D . ~ ~ <f>p v ~ \JfJ >)
h . (2) . (*3*01). D h <f>p . D : yfrp . D . <f>p . yfrp
h . (3) . *1*11 ,D h . Prop
*31. hp.9.D.^(^p V ^^) [Id.(*3*01)]
*311. b .D.p.q [Id. (*3*01)]
*312. h:~p.v.~*.v./*. 9 *211
*313. :~(p . q) ."2 .~p V [*3*1 1 . Transp]
~p v ~ q . D . ~ (p . q) [*3*1 . Transp]
\-:.p.3:q.5.p.q [*312]
*321. b i.q.O-.p.D.p.q [*3*2 . -Comm]
*3 22. hrp.f.D.fl.;,
This is one form of the commutative law for logical multiplication. A
more complete form is given in *4 3.
Dem.
[*313iiifj
[Perm]
*314.
*32.
b : ~ (fl . />) . D . ^ q v ~p
D . ~p v
[*314] D.~(p.q)
b . (1) . Transp . D b . Prop
Note that, in the above proof, “(1)” stands for the proposition
“~(q.p).D.~(p.q),’’
as was explained in the proof of *2 31.
*3 24. b.»^(p.o^p)
Dem.
~p
( 1 )
* 2*11
*314
P J
9 J
b . v ~ (~ jy) . D
b .~(p.~p)
The above is the law of contradiction.
112
MATHEMATICAL LOGIC
[PART I
*3*26. h : p . q . D . p
Dem.
r.202'^1
h ; » . D . q D />
( 1 )
L P* 7 J
[(1 >.( * 1 *01 )]
K:~y>.v.~gvy>:
*2*311
Dh:^ y> v ~ 7 . v . p :
D K :-v(^i)VM/>. D. »
( 2 )
L /*• 7 J
[(2).(*301)]
h -.p.ij.O .p
*3 27. h : p . 7 . D . 7
Dem.
*3*22] h : yj. q . D . 7 . /).
1 3.a:3h. Prop
7*- vJ
*3*26 27 will both be called the “ principle of simplification,’ like *2 02,
from which they arc deduced. They will be referred to as “Simp.”
*3 3. I- p . 7 .3 . r s D s /> . D . q D r
Dem.
[Id.(*3*01)] h :. y>.
. 7 . D . r : D : ( ^y> v ~ 7 ). D . r :
[Transp]
Di'vr.D.'vyjV'v^:
[Id.(*l* 01 )]
D:'v/*.D.yO^</:
[Comm]
D:y>.D.~rD~g:
[Transp.SvIl]
• D : y>. D . g D r :• D h . Prop
*3 31. I-/>. D . 7 D r 2 D : y>. 7 . D. r
Dem.
Id.(*l 01)] h :.y;.D.ry Dr: D 2 ~y> . v . v r s
*2*31] Di'v/jV'v^.v.r:
*2 53 ~ r l D : ~ ~ p v ~ 7 ). D . r:
/>> '/J
Id.(*3*01)] D :/>. q . D . r D b . Prop
*3 33. H; yOg.^Dr.D./Or [Syll. Imp]
*3*34. htgDr./Og.D./Or [Syll. Imp]
These two propositions will hereafter be referred to as “Syll"; they are
usually more convenient than either *2*05 or *2*00.
*3 35. h s p .p D q . D . q [*2*27 . Imp]
*3*37.
Dem.
h . Transp . D h :(/Dr. D
[Syll] DH:.yi.D.«/Dr:D:p.D.~rD ~<7 ( 1 >
h . Exp . D H yj . ry . D . r : D : y> . D . «y D r (2)
H . Imp . (3)
H . (2) . (1) . (3) . Syll .Dh. Prop
SECTION A]
THE LOGICAL PRODUCT OF TWO PROPOSITIONS
1 13
This is another form of transposition.
*3 4. I-:y».«y.3.yOy [*251 . Transp . <*1 01 .*301)1
*3 41. b :./> 3 r . 3 zp . <j . 3 . r [*3’2c». Syll]
*3 42. I-:. (/ D »•. D ;y>. <y. 3 . [*3 07 . Syll]
*3*43. b :. y> 3«/ . y> 3 r. 3 : p . 3 . . r
Dem.
(-.* 3-2.Dt-:. ? .D:i-.D. (/ .r ( 1 >
b . (1) . Syll . 3 h ::y>3 #/ . 3 :. y>. 3 : r. 3 . *y . r s.
[*277] 3:.yOr.3:/j.3.#y.r (2)
b. (2). Imp. 3 h. Prop
*344. h:.g3^.rD/i.D:<yvr.3./>
This principle is analogous to *3 43. The analogy between *3 43 and
*3 44 is of a sort which generally subsists between formulae concerning
products and formulae concerning sums.
Dem.
b . S)'ll . 3 b ^ q 3 r . r 3 yj
. 3 : q Dp z
[*2-6]
3 : q 3 p . 3 . p
(1)
b . (1) . Exp . 3 b :: ^q 3 »■ .
3 :. r 3p . 3 : q Dp . 3 . p :.
[Comm.Imp]
3 :. q Dp . r 3 p . 3 .p
( 2 )
b . (2) . Comm . 3 b :. q 3 />.
r 3 p . 3 : ~ q 3 r • 3 . y> :.
[*2*53.Syll] 3 b . Prop
*345. b:.y>3g.3:p.r.3.g.r
This principle shows that we may multiply both sides of an implication
by a common factor; hence it is called by Peano the “principle of the factor.”
We shall refer to it as “ Fact.” It is the analogue, for multiplication, of the
primitive proposition *16.
Dem.
b . Syll ~ .31-:.y>3fl.3:^3*^ r.3.y>3~r:
[Transp] 3 : ~ (p 3 ~ r) . 3 . ~(q 3 ~ r)
[Id.(*l'01.*3‘01>] 3 b . Prop
*347. !■ :.])3r. j3s.3 :p.^.3,r.«
This proposition, or rather its aualogue for classes, was proved by Leibniz,
and evidently pleased him, since he calls it “ prajclaruin theorema*.”
Dem.
b . *326.3h:.yOr.93s.3:yOr:
[Fact] Dzp.q.D.r.qz
[*322] Dzp.q.D.q.r (1)
• Philosophical works, Gerhardt’s edition, Vol. vn. p. 223.
R&W I
8
114
MATHEMATICAL LOGIC
[part 1
H . *327 .Dhs.jOr.gDj.DsgDf:
[Fact] D : 7 . r . D . s . r :
[*3*22] D : 7 . r . D . r . s
h . (1). (2). *3 03 . *2*83 . D
( 2 )
I -j>D r. qO s .0 z p . q ,0 • f' • s :.0 . Prop
*3 48. H ;.p D r . 7 D* . D :p v ry. D . r v*
This theorem is the analogue of *3 47.
Dcm.
H . *3*2G . D h :.y> D r . q D i. D : p D r:
[Sum] D: p v 7 . D . r v 7 :
[Perm] D : pv 7 . D . 7 V r (1)
h . #3 27 . D h p D r. 7 D s . D : 7 D a* :
[Sum] D : 7 v r. D . 5 v r:
[Perm] Difvr.D.rvs ( 2 )
h . (1). (2). #2*83 . D
h p D r . 7 D s. I> : p v 7 . D . r v s 3 h . Prop
*4. EQUIVALENCE AND FORMAL RULES
Sinn mart/ of ?*c4.
In this number, we shall be concerned with rules analogous, more or less,
to those of ordinary algebra. It is from these rules that the usual "calculus
of formal logic” starts. Treated as a “calculus,” the rules of deduction are
capable of many other interpretations. But all other interpretations depend
upon the one here considered, since in all of them we deduce consequences
from our rules, and thus presuppose the theory of deduction. One very
simple interpretation of the " calculus ” is as follows : The entities considered
are to be numbers which are all either 0 or 1 ; " p D«? ” is to have the value 0
if p is 1 and q is 0; otherwise it is to have the value 1 ; ~p is to be 1 if p
is 0, and 0 if p is 1 ; p . q is to be 1 if p and q are both 1 , and is to be 0 in
any other case; pvq is to be 0 if p and q are both 0, and is to be 1 in any
other case; and the assertion-sign is to mean that what follows has the
value 1. Symbolic logic considered as a calculus has undoubtedly much
interest on its own account; but in our opinion this aspect has hitherto been
too much emphasized, at the expense of the aspect in which symbolic logic
is merely the most elementary part of mathematics, and the logical pre¬
requisite of all the rest. For this reason, we shall only deal briefly with what
is required for the algebra of symbolic logic.
When each of two propositions implies the other, we say that the two are
equivalent , which we write u p = q” We put
*401. j)~q, = .pOq.qDp Df
It is obvious that two propositions are equivalent when, and only when,
both are true or both are false. Following Frege, we shall call the truth-
value of a proposition truth if it is true, and falsehood if it is false. Thus two
propositions are equivalent when they have the same truth-value.
It should be observed that, if p = q, q may be substituted for p without
altering the truth-value of any function of /> which involves no primitive
ideas except those enumerated in *1. This can be proved in each separate
case, but not generally, because we have no means of specifying (with our
apparatus of primitive ideas) that a function is one which can be built up out
of these ideas alone. We shall give the name of a truth-function to a fuuction
f(p) whose argument is a proposition, and whose truth-value depends only
upon the truth-value of its argument. All the functions of propositions with
which we shall be specially concerned will be truth-functions, i.e. we shall
have
p = q.D . f(p ) =f(q)-
8—2
116
MATHEMATICAL LOGIC
[PART I
'1 he reason of this is, that the functions of propositions with which we deal
are all built up by means of the primitive ideas of *1. But it is not a universal
characteristic of functions of propositions to be truth-functions. For example,
"A believes p” may be true for one true value of p and false for another.
1 he principal propositions of this number are the following:
*4 1. b : jt D <y . = . *>. q D ^ p
*411. b : yi s q . = .^p=^q
1 liese are both forms of the " principle of transposition."
*4 13. H s -— (•— p)
Ihis is the principle of double negation, i.e. a proposition is equivalent to
the falsehood of its negation.
*4 2. V .pmp
*4 21. : p = i/ . = . tj = p
*4 22. H : y* 2 (y . y s »•. D , s »•
1 hose propositions assert that
transitive.
equivalence is refle.vive, symmetrical and
*4 24 h : p . = m p .p
*4 25. I-: p . s . p v p
I.e. p is equivalent to “p and p" and to " p or which arc two forms of
the lair uj tautology, and are the source of the principal differences between
the algebra of symbolic logic and ordinary algebra.
*4 3. h ip . //. = . r/ . p
ihis is the commutative law for the product of propositions.
*4 31. b : p v q . ■ . ij v />
This is the commutative law for the sum of propositions.
1 he associative laws for multiplication and addition of propositions, namely
*4 32. b : (p . fj ). r . = . p . (q . r)
*4 33. b : (p v ij) v r • 5 ,p v(^ v r)
The distributive law in the two forms
*4 4. b :• P • *i v >* . = : p. q . v .p . r
*4 41. b i.p . v . q . /•: = .p vry .p v r
Ihe second of these forms has no aualogue iu ordinary algebra.
*4 71. b p D q . = ; p . = . p . q
I.e. p implies q when, and only when, p is equivalent to p . q. This pro¬
position is used constantly; it enables us to replace any implication by an
equivalence.
*4'73. b z. q . D z p . = . p . q
I.e. a true factor may be dropped from or added to a proposition without
altering the truth-value of the proposition.
SECTION A]
EQUIVALENCE AND FORMAL RULES
117
*4'01. /> = q . = . p D q . q D p Df
*4 02. J> = q = r . = . p = q . q = y Of
This definition serves merely to provide ;i convenient abbreviation.
*41.
b : P 9 • = • ~ q D ~ p
[*2*16*17]
*411.
b : p = q . = . p = q
[*216*17 .*3*47*22]
*412.
b : /» = ^ 7 . = . 7 = .— > p
[*2*03*13]
*413.
b .p = ~(~p)
[*2*12*14]
*414.
b . 7 . D . r: = : p . ^ i*. D . ~ 7
[*3*37 . *4*13]
*415.
b p . 7 . D . ^ #•: = : q . >• . D . ^ p
[*3*22. *4 13*14]
*42.
b -P=P
[Id . *3 2]
*421.
)-:p = q. = .q=p
[*3*22]
*4 22.
b ; p 3 7.7 s r. D . p = r
Dem.
b . #326 . Obipsq.qsr.^.psq.
[*3*26] D./O 7 ( 1 )
b . *3*27 .
[*3-26] D . 7 D ?• (2)
b . (1). (2). *2*83 . D h :/> s 7.7 ■ ?*. D ./> D r (3)
b . *3*27 . D b : = 7.7 = r . D . 7 = r.
[*3*27] D.rDg (4)
b . *326 . D b :p = q.q = r.D.p = q.
[*3*27] D^Dp (5)
b . (4) . (5) . *2*83 .Db :p=q.qsr.O,rOp ( 6 )
b . (3) . ( 6 ) . Comp. D b . Prop
iVo^c. The above three propositions show that the relation of equivalence
is reflexive (*4*2), symmetrical (*4*21), and transitive (*4*22). Implication
is reflexive and transitive, but not symmetrical. The properties of being
symmetrical, transitive, and (at least within a certain field) reflexive are
essential to any relation which is to have the formal characters of equality.
#4'24. b : p . s . p . p
Dem.
b . *3*26 .Db : p . p . D . p
(1)
b . *3*2 . Db :.p . D :/>. D .p .p :.
[*2*43] Db rp.D.p.p
(2)
b . (1) . (2) . *3*2 . D b . Prop
*4*26. b zp.m.pvp
Note. *4*24*25 are two forms of the law of tautology, which is what chiefly
distinguishes the algebra of symbolic logic from ordinary algebra.
118
MATHEMATICAL LOGIC
[PART I
*4 3. b : p . 7 . = . 7 . p [*3 22]
Note. Whenever we have, whatever values p and 7 may have,
<#»</>.7)• ^
we have also
<f> (/>, 71 . = .<f> ( 7 . />).
For !<£ (/>. 7 ) . 3. 4>('j.p) ^ . D : <f> ( 7 . />) . D . </> (y>, 7 ).
*4 31. b : /> v 7 . = . 7 v /> [ Perm]
*4 32. b s (71 . 7 ). r. ■ ./>. (7 . /•)
Dem.
1-. *4’15 . D 1-/». 7 . D . r : = : 7 . r . D . :
[*412] s:j».D.~(g.r) (1)
1- . (1). *4 - l 1 . D b s ^ (y>. 7 . D . ^ r) • 3 . |/> • D . ^( 7 . r)):
[(*101.*3 01)] D 1-. Prop
-Vo te. Hero “(1)” stands tor ” b p . 7 . D . ^ r s s : p . D . ~ ( 7 . r),” which
is obtained from the above steps by *4 22. The use of *4 22 will often be
tacit, as above. The principle is the same as that explained in respect of
implication in *2 31.
*4 33. b : (p v 7 ) v r . = .p v (7 v r) [*2 31 82]
The above are the associative laws for multiplication and additiou. To
avoid brackets, we introduce the following definition:
*4 34. p • 7 , r •« . (p. q). r Df
*4 36. b :./) h 7 . D :. r. = . 7 . /• [Fact. *3 47]
*4 37. b :.y> = 7 . D : yj v r. = . 7 v r [Sum.*3‘47]
*4 38. b p = r . 7 = s, D : y>. 7 . = . r . s [*3 47 . *4 32 . *3 22]
*4 39. b z.p = r . 7 = s. D : y> V 7 . = . r vs [*3‘48*47 . *4 32 . *3‘22]
*4 4. b s. p. q V r. = : y>. 7 . v . />. r
This is the firet form of the distributive law.
Dem.
b . *3-2 .
D b s: p. D : 7 . D .p 0 : r . D • • r s:
[Comp]
D b . D 9 . D . 7 : r. D ./) . r
[*348]
Dz.qv r.Dzp.q.v .p.r
(1)
b .( 1 ) . Imp .
Dhz.p.qvr.Dzp.q.v»p-r
(2)
b . *3 26 .
Db:./). 9 .D./):/).r.D.p:.
[*3-44]
D b y> . 7 . v . yi . r: D . p
(3)
b . *3 27 .
Db:.;). 7 .D. 9 :/).r.D.r:.
[*348]
Db:./). 7 .v./).r:D. 7 vr
(•*)
b. (3). (4). Comp
. D b z.p .q .v.p.rzO.p.qvr
(°)
b • (2) . (5) .
D b . Prop
SECTION A]
EQUIVALENCE AND FORMAL RULES
*4*41. b z.p . v . 7 . r : = . p v q . p v r
This is the second form of the distributive law—a form t«» which there
is nothing analogous in ordinary algebra. By the conventions as to dots.
/> • v -7
Dem.
means " p v (7 . r).*
*4-42. b
Dem.
*443. b
Dem.
*444. b
Dem.
b . *326 . Sum .
D b z.p . v . q . r: D . p v q
(i)
b . *3 27 . Sum .
Db:./).v.j/.r:D./)vr
C-2)
b . (1) . (2) . Comp
. D b y>. v . 7 . r : D . y> v 7 . p v r
(3)
b . *2-53 . *3 47 .
D b z.p vq .p v r . D :*>-/> D q . ~/>D r:
[Comp]
[*254]
D : />. v . . r
W
K<3).(4).
D b . Prop
••
•
III
• •
•
<
•
. ~ q
b.*3'21. D b z.q v ~q . D zp. D ./>.</ v ~ q
[*2T1] D b ip. D .p. qv~q
( 1 )
b . *326 . Db ip.qv^q.D.p
b . (1) . (2) . D b z.p . a zp . q v ~q :
( 2 )
[*+•4]
3zp.q.V.p.~qz.D b . Prop
Z.p . 3 zpv q .pv ~ q
b . * 2*2 .
Dbsp.D.pVfltp.D.pv^tf:
[Comp]
( 1 )
1- . *2-65 ^ .
P
[Imp]
D b .D.pz.
[*2o3.*3-47]
Db z.pv q .pv ~q .0 .p
( 2 )
b.(l).( 2 ).
D b . Prop
z.p.mzp.v.p.q
b . *2 2 .
1 T J
D b :.p . D zp . v . q
(1)
b . Id . *
.3 2b . J r z.p Jp z p . q • j • p z.
[*344]
Db z.p .V .p.qzD .p
( 2 )
b . (1) . (2) . Db.Prop
*446. \-zp. = .p.pvq [*3*26.*2*2]
The following formulae are due to De Morgan, or rather, are the propo¬
sitional analogues of formulae given by De Morgan for classes. The first
of them, it will be observed, merely embodies our definition of the logical
product.
120
MATHEMATICAL LOOIC
[PART I
*45.
b:
p . 7 . = . ~ ~ p v ~ 7 )
[*4 2. (*3 01)]
*451.
1 -:
—- (yj. 7 ) - = . — y> v ~ 7
[*4-512]
*452.
b:
y>. ~ 7 . = . ~ ^ y> v 7 )
^*45 ~ 7 . *4*13
*453.
b :
7 ). = .~y< v 7
[*4-5212]
*4 54
h:
^ y>. 7 . = . ~ (y> v ~ 7 )
|*4 5^.*413
*4 55.
1 - :
~(~/>-7> - = . y> v ~ 7
[*4-5412]
*456.
1 - :
~y>. ~7 . = . ~(y> v 7 )
1 *4 54 ^.*41. 4
*457.
h : ~
(~ y>. ~ 7 ). s.y>V 7
[*4-5612]
The following formulae are obtained immediately from the above. They
are important a.s showing how to transform implications into sums or into
denials of products, and vice versa. It will be observed that the first of them
merely embodies the definition *1 01.
*46. b
*461. b
*4 62. h
*4 63. I-
*4-64. b
*465. b
*4 66. b
*4 67. b
*47. b
Dent.
y>Diy .3 [*42 . (*101)]
s 7 (*4-6*1152]
y* D ~ 7 . =. ^ y> v ~ 7 |^*4G J
^ (y> D ^ 7 ). = . y>. 7 [*4-62*11*5]
^ p D 7 . = . p v 7 [*2*53*54]
~ ~ yO 7 ). = . ~ y». ~ 7 [*4 6411-56]
~ yO ~ 7 . = . y> v ~ 7 ^*4*04
~ (~ yO ~ 7 ). = . ~ y>. 7 [*4G6-11\54]
. p D (/ . = z ji. 0 . p . rj
h . *3 - 27 . Syll . D h y>. D . y>. 7 : D . yO 7
h . Comp. D b y> D y>. y> D 7 . D : y>. D . p . 7
[Exp] D h ::yO y> . D y> D 7 . D : y>. D .y>. 7 ::
[Id] D h p D 7 . D : y>. D .y>. 7
K(l).< 2 ). D h . Prop
*4 71. h ]) D 7 . = : p . = .y>. 7
Dem.
h .*3 21.
D I - :: y>. 7 . D . p : D y>. D . p . 7 : D : y;.
[*326]
D h y>. D . y>. 7 : D :p. = .p. 7
( 1 )
h . *3 2G .
Dh:.y>. = .y). 7 :D:p.D.y >.7
( 2 )
h.(l).( 2 ).
Dh:.p.D.y). 7 : = :p. = ./)-7
(3)
1- . (3). *4 7-22
. D h . Prop
SECTION A]
EQUIVALENCE ANI> FORMAL RULES
121
The above proposition is constantly used. It enables us to transform
every implication into an equivalence, which is an advantage if we wish to
assimilate symbolic logic as far as possible to ordinary algebra. Hut when
symbolic logic is regarded as an instrument of proof, we need implications,
and it is usually inconvenient to substitute equivalences. Similar remarks
apply to the following proposition.
*472. b z.p D 7 . = : 7 . = .p v 7
Dem.
b . *47 . D I *p Oq .
[
•4 71 ~ 9 '
’]
7 . = . ~ 7
[*412]
[*4-57]
[*431]
*473. b 7 . D z p . = . p
P :
"'P) :
~ ~ 7
qvpz
pvq • • D b . Prop
s:7.
= m -q-
= -q •
7 [Simp.*471]
This proposition is very useful, since it shows that a true factor may be
omitted from a product without altering its truth or falsehood, just as a true
hypothesis may be omitted from an implication.
*474. b ~p . D : 7 . = . v 7
*476. b p D 7 . p D r . = : p . D . q . r
*477. q D p . r 0 p . ^ z q v r . 0 . p
*2-21 . *472]
*441
^-'•(* 101 )
*3-44 . Add . *2 2]
*478. h:.pDg.v.yDrsa:p.D. 7 vr
Dem.
b . *42 . (*101) .Db:.|)D 7 .v.pDr
[*433]
[*431-37]
[*4-33]
[*4-2537]
[*4-2.(*101>]
= :~pv 7 , v.~p vrs
a.'vp.v.7v<vpvr:
= : . v . v 7 v »•:
= :'v/)v~p.v.7vr:
s : . v . 7 v r s
= :p.D. 7 vr:. Db. Prop
*479. b:. 7 Dp.v.rDp: = z q . r . • p
Dem.
b . *41-39 .Db:. 7 Dp.v.rDp: = :~/)D~ 9 .v.~pD~r:
[*478] = :~p. D.~ 7 v~r:
[*275] = :~(~7V~r).D./j;
[*4-2.(*301)] =: 7 .r.D.p:.Db. Prop
Aote. The analogues, for classes, of *47879 are false. Take, e.g. *478,
and put p = English people, 7 = men, r = women. Then p is contained in 7
or r, but is not contained in 7 and is not contained in r.
122
MATHEMATICAL LOGIC
[PART I
*4 8 b : /> D~j>. = [*201 .Simp]
*481. I- : /O/>. = ./> [*218. Simp]
*4 82. b z p D 7 . p D ~tj . = . [*2*05. Imp. *2 21 . Comp]
*4 83. b : y> D 7 . ~ y> D 7 . = . 7 [*2‘61 . Imp . Simp . Comp]
Note. *4’82‘83 may also be obtained from *443, of which they are virtu¬
ally other forms.
*484. 1- p 5 7 . D : ;)Dr. = .pr [*2u<>. *347]
*4 85. I -p = 7 . D : r D />. = . /O 7 [*2 U5 . *3 47]
*4 86 h.y»3y.D:/)5r.s,gsr (*4 2122]
*4 87. K :. y>. 7 . D . r: = : p. D . 7 D »•: = : 7 . D . yO r: s : 7 . />. D . r
[Exp . Comm . Imp]
*487 embodies in one proposition the principles of exportation and im¬
portation and the commutative principle.
*5. MISCELLANEOUS PROPOSITIONS
Summary of *5.
The present number consists chiefly of propositions of two sorts: (1) those
which will be required as lemmas in one or more subsequent proofs, (2) those
which are on their own account illustrative, or would be important in other
developments than those that we wish to make. A few of the propositions of
this number, however, will be used very frequently. These are:
*51. h : p . q . Z> . p = q
I.e. two propositions are equivalent if they are both true. (The statement
that two propositions are equivalent if they are both false is *5 21.)
*5 32. \-:.p.0.q=r: = :p.q. = .p.r
I.e. to say that, on the hypothesis p, q and r are equivalent, is equivalent
to saying that the joint assertion of p and q is equivalent to the joint assertion
of p and r. This is a very useful rule in inference.
*56. : p • O . q v r
I.e. " p and not-# imply r” is equivalent to “ p implies q or r.”
Among propositions never subsequently referred to, but inserted for their
intrinsic interest, are the following: *511121314., which state that, given
any two propositions p, q, either p or ~p must imply q, and p must imply
either q or not-g, and either p implies q or q implies p ; and given any third
proposition r, either p implies q or q implies r*.
Other propositions not subsequently referred to are *5’22*23*24; in these
it is shown that two propositions are not equivalent when, and only when,
one is true and the other false, and that two propositions are equivalent
when, and only when, both are true or both false. It follows (*524) that the
negation of "p . q .v . ~p . ~ q" is equivalent to “ p. ~ q .v . q . ~p" *5*5455
state that both the product and the sum of p and q are equivalent, respectively,
either top or to q.
The proofs of the following propositions are all easy, and we shall therefore
often merely indicate the propositions used in the proofs.
*51. b:p.q.D .p = q [*3*4*22]
*511. h:pDg.v. ~p D q [*2-5*54]
*5*12. [*2*51*54]
*613. h:pD 9 . v . 9 Dp [*2*521]
*614 \-zpDq .v .qDr [Simp . Transp . *2*21]
* Cf. Schrdder, VorUtungen Uber Algebra der Logik, Zweiter Band (Leipzig, 1891), pp. 270—
271, where the apparent oddity of the above proposition is explained.
124
MATHEMATICAL LOGIC
[PART 1
*515. b
Dem.
*516. b
Dem.
*517. b
Dem.
*518. b
*519. b
*621. b
*522. b
*523. b
*6 24. b
*5*25. b
:y>= 7 .v.y, = ~ 7
b . *4 61 . D b : (p D 7 ) . D . y>. ~ q .
[*•> 1 ] D . y> = ~ 7 :
[*2*.54] D b : y> D 7 . v . y> =
b . *4*61 . D I- : <n/D. D . ry. ,
[*•5*1 ] D . 7 = ~ y> .
[*4 12] D.y, = ~ 7 :
[*2 .54] D b : q D yj. v . y> = ~ 7
b.(1).(2). *4 41 . D b . Prop
. ~ ( y> = 7 . yj = ~~ ry)
( 1 )
( 2 )
b . *326 . D h : p 3 q . p ^ ^ q . D . p D q . p 5 ~ q •
[*482] D.~y> (1)
b . *3*27 • D b i p = q . q . qO p . pD ~ q ,
[Syll] D. 7 D^ 7 .
[Abs] D.~q (2)
b . (1). (2) . Comp . D b : p = q . p D ~7 . D . ^/y) . ^7 .
r*+(i5 9,/ -] D.~(~ 9 Dp) (3)
L P* 7 j
b • (3). Exp. D b p = 7 . D : p D ~7 . D . ~(~7 3 y>) ?
[ I«l.(* 1 01)] D:~(yO~7 ).v. ~(~q D p):
[*4*51.(*4*01)] D : ~(p = ^ 7 )DH. Prop
: p V 7 • ~(y> • 7 ) . = • ps ^q
b. *4-64-21. D I- :y)V 7 . = .~qDp (1)
b . *4 63 . Tmusp . D b : ~(p . 7 ) . = . p D ^ 7 (2)
b.(1).(2). *4*38*21. D b . Prop
:y> = 7 . = .~(y; = ~7)
. ~(y; = ~y>)
:^y;.^7.D.y> = 7
~(p = 7 ) . = :y>. ~7 . v . 7 .
:.p = 7. = :y>.7.v.~y>.*-'^7
~ (y> . 7 . v . ^y>. ~ 7 ). = : p
Z. p V q . = i p D q . D . q
[*5*15*16. *5*17 Z*±£*ZS]
P. q J
*5*18 • *4*2 J
[*5*1 .*4*11]
p [*4*61*51*39]
£*5*18 . *5*22 ^. *4*13*36J
~q.v.q.~p [*5*22*23]
[*2*62*68]
SECTION A]
MISCELLANEOUS PROPOSITIONS
12 .“>
From *5 *25 it appeal's that wo might have taken implication, instead of
disjunction, as a primitive idea, and have defined -/#vy” as meaning
pDy.D.y, This course, however, requires more primitive propositions
thau are required by the method we have adopted.
*53.
[Simp . Comp . SylIJ
*531.
h . p D y : D : p . D . y .
[•Simp. Comp]
*532.
h :. p . D . q 3 r ; = : />. y . = . p . r
[*4*76. *3*3*31 . *5*3]
This proposition is constantly required
in subsequent proofs.
*533.
[*4*73*84 . *5 32]
*535.
hs./Ofl./Or.D: p.D.ysr
[Comp. *51]
*536.
hp.p=q.a.<j.p = <j
| Ass. *4*38]
*54.
H.p.D.pDy : = .pDy
[Simp. *2*43]
*541.
I -p D g . D .p D #• s = : p . D . <j D r
[*277-86]
*542.
h 1 : p . D . g D r : 5 p . D : y . D .p
. r [*5*3 . *4*87]
*544.
H::pDy.:>:.pDr. = :p.D.y.r
[*4*76. *5-3-32]
*55.
h:.p.D:pDy. = .y
[Ass . Exp . Simp]
*5501.
I- :.p . D : y . = .p s y
[*51 . Exp . Ass]
*553.
h:.pvyvr.D.s: = :pDs.yDs.
rDs [*4*77]
*554.
[*4*73 . *4 44 . Transp .*51]
*555.
h:.pvy. = .p:v:pvy.a.y
[*1*3.*51 .*4 74]
*56.
h p . ^y . D . r : = sp.D.jvr
^*4*87 . *4 64 85 J
*561.
h:pvy.~y. = .p.«^y
[*4 74. *5*32]
*562.
h :.p . y . v . ~y : ■ . p V ~y
*563.
h :.p v y . = :p . v . ~p . y
[•«* Ti)
*57.
b:.pvr. = .yvr: = :r.v.p = y
[*4 74 . *1 3 . *51 .*4 37]
*571.
b:.yD^r.D:pvy.r. = .p.r
In the following proof, as always henceforth, "Hp" means the hypothesis
of the proposition to be proved.
Bern.
h.*4'4. Dhs.pvy.r.ssp.r.v.y.r (1)
h . *4*62'51 . DH:: Hp . D ~ (5 . r)
[*4'74] Dz.p.r.v .g.rz = zp.r (2)
h . (1) . (2) . *4-22 .DK Prop
120
MATHEMATICAL LOGIC
[PART I
*5*74. I *p . D . 7 = r : = : p D q . = . p D r
Dem.
b . *541 . D b :: /O 7 . D .;)Dr: = :/».D . 7 D r
yO r. D . /O 7 : = : p. D . >0 7 ( 1 )
b . ( 1 ). *+-.18 . D b :: p D q . = . p D r. = p . D . q D 7 *: y>. 0 . r D q
[*4*76] = :.y>. D . 7 = r :: D b . Prop
*5'75. b »0 ~ 7 : p . = . q v r: D : p . ~ 7 . = . r
Dem.
b . *5*6. D b Hp. D : p • ~ 7 . D . r (1)
b . *3*27 . D b Hp . D : 7 v »•. D . :
[*4 77] OzrDp (2)
b . *3’2<> . D b Hp . D : r D ~ 7 (3)
b . (2). (3). Comp . Db:. Hp .D:»0/).?0^g:
[Comp] D : r . D .p . 7 (4)
b . (1). (4). Comp . D b Hp . D : y>. ~ 7 . s . r D b . Prop
SECTION B
THEORY OF APPARENT VARIABLES
*9. EXTENSION OF THE THEORY OF DEDUCTION FROM
LOWER TO HIGHER TYPES OF PROPOSITIONS
Summary of *9.
In the present number, we introduce two new primitive ideas, which may
be expressed as "(fix is always* true” and "(fix is sometimes* true," or, more
correctly, as ,l <f>x always” and "(fix sometimes.” When we assert "(fix always,”
we are asserting all values of tp.T, where "(fix’’ means the function itself, as
opposed to an ambiguous value of the function (cf. pp. 15, 40); we are not
asserting that (fix is true for all values of x, because, in accordance with the
theory of types, there are values of x for which "(fix" is meaningless; for ex¬
ample, the function (fiut itself must be such a value. We shall denote “ <f>x
always” by the notation
(*) • <t>*>
where the “(a:)” will be followed by a sufficiently large number of dots to
cover the function of which “all values” are concerned. The form in which
such propositions most frequently occur is the “formal implication,” i.e. such
a proposition as
(x) : (fix . D . yfrx,
i.e. "(fix always implies yjrx." This is the form in which we express the
universal affirmative “all objectshaving the property«/» have the property^."
We shall denote “ (fix sometimes” by the notation
(g*) . (fix.
Here “a” stands for “there exists,” and the whole symbol may be read
“there exists an x such that (fix."
In a proposition of cither of the two forms (x). (fix, (a#). <fix, the x is
called an apparent variable. A proposition which contains no apparent
variables is called “elementary,” and a function, all whose values are elemen¬
tary propositions, is called an elementary function. For reasons explained in
Chapter II of the Introduction, it would seem that negation and disjunction
and their derivatives must have a different meaning when applied to elemen¬
tary propositions from that which they have when applied to such propositions
(«) • <fix or (ga?) . (fix . If (fi£ is an elementary function, we will in this number
call (x) . (fix and (g#) . (fix “first-order propositions.” Then in virtue of the fact
* We use “always” as meaning “in all cases,” not “at all times.” A similar remark applies
to “sometimes.”
128
MATHEMATICAL LOGIC
[l’ART I
that disjunction and negation do not have the same meanings as applied to
elementary «*r to first-order propositions, it follows that, in asserting the
primitive propositions of *l,we must either confine them, in their .application,
to projiositioiis of a single type, or we must regard them as the simultaneous
assertion of a number of different primitive propositions, corresponding to the
different meanings of “disjunction" and “negation.” Likewise in regard to
the primitive ideas of disjunction and negation, we must either, in the primi¬
tive propositions of * I, confine t hem to disjunctions and negations of elementary
propositions, or we must regard them as really each multiple, so that in regard
to each type of pro|iosition.s we shall need a new primitive idea of negation
and a new primitive idea of disjunction. In the present number, we shall
show how, when the primitive ideas of negation and disjunction arc restricted
t«« elementary propositions, and the p, <j. r of * 1 —*5 are therefore necessarily
elementary propositions.it is possible to obtain definitions of the negation and
disjunction «>f first-order propositions, and proofs of the analogues, for first-
order propositions, of the primitive propositions * 1 * 2 —*(>. (* 1’1 and * 1*11
have to be assumed afresh for first-order propositions, and the analogues of
*17 717*2 require a fresh treatment.) It follows that the analogues of the
propositions of *2—#•’> follow by merely repeating previous proofs. It follows
also that the theory of deduction can be extended from first-order propositions
to .vich as contain two apparent variables, by merely repeating the process
which extends the theory of deduction from elementary to first-order pro¬
positions. Thus by merely re|>eating the process set forth in the present
number, propositions of any order can Ik- reached. Hence negation and
disjunction may be treated in practice as if there were no difference in those
ideas as applied to different types; that is to say, when “ *>-* p" or “pvq
occurs, it is unnecessary in practice to know what is the type of p or 7 . since
the properties of negation and disjunction assumed in *1 (which are alone used
in proving other properties) can be asserted, without formal change, of pro¬
positions of any order or. iu the case of p v 7 , of any two orders. The limitation,
in practice, to the treatment of negation or disjunction as single ideas, the
same in all types, would only arise if we ever wished to assume that there is
some one function of yj whose value is always ~ p, whatever may be the order
of p, or that there is some one function of p and 7 whose value is always pv 7 ,
whatever may be the orders of p and 7 . Such an assumption is not involved
so long as p (and 7 ) remain real variables, since, in that case, there is no need
to give the same meaning to negation and disjunction for different values of
p (and 7 ), when these different values are of different types. But if p (or 7 )
is going to be turned into an apparent variable, then since our two primitive
ideas (x). <£x and (g.r) . <f>x both demand some definite function <f>, and restrict
the apparent variable to possible arguments for <p, it follows that negation
and disjunction must, wherever they occur in the expression in which p (or 7 )
is an apparent variable, be restricted to the kind of negation or disjunction
SECTION li) EXTENSION OF THE THEORY OF l>EIHVTIO.\' 12‘)
appropriate to a given type or pair of types. Thus, to take an instance, if uv
assert the law of excluded middle in the form
'* h . p v ^ p "
tliero is no need to place any restriction upon p: we mav give to p a value
ot any order, and then give to the negation and disjunction involved those
meanings which are appropriate to that order. But if we assert.
" *■ • (/>)-/>v ^ p "
it is necessary, if our symbol is to be significant, that "p v ~~ p" should be the
value, for the argument p, of a function <f>p\ and this is only possible if the
negation and disjunction involved have meanings fixed in advance, and if, there¬
fore, p is limited to one type. Thus the assertion of the law of excluded middle
in the form involving a real variable is more general than in the form involving
an apparent variable. Similar remarks apply generally where the variable is
the argument to a typically ambiguous function.
In what follows the single letters p and q will represent elementary pro¬
positions, and so will “yfra:," etc. We shall show how, assuming the
primitive ideas and propositions of *1 as applied to elementary propositions,
we can define and prove analogous ideas and propositions ns applied to pro¬
positions of the forms (a) . <f>x and (ax).<£x. By mere repetition of the analogous
process, it will then follow that analogous ideas and propositions can be defined
and proved for propositions of any order; whence, further, it follows that, in
all that concerns disjunction and negation, so long as propositions do not
appear as apparent variables, we may wholly ignore the distinction between
different types of propositions and between different meanings of negation
and disjunction. Since we never have occasion, in practice, to consider pro¬
positions as apparent variables, it follows that the hierarchy of propositions
(as opposed to the hierarchy of functions) will never be relevant in practice
after the present number.
The purpose and interest of the present number are purely philosophical,
namely to show how, by means of certain primitive propositions, we can
deduce the theory of deduction for propositions containing apparent variables
from the theory of deduction for elementary propositions. From the purely
technical point of view, the distinction between elementary and other propo¬
sitions may be ignored, so long as propositions do not appear as apparent
variables; we may then regard the primitive propositions of *1 as applying
to propositions of any type, and proceed as in *10, where the purely technical
development is resumed.
It should be observed that although, in the present number, we prove
that the analogues of the primitive propositions of *1, if they hold for propo¬
sitions containing n apparent variables, also hold for such as contain n + 1,
yet we must not suppose that mathematical induction may be used to infer
that the analogues of the primitive propositions of *1 hold for propositions
r& w i 9
130
MATHEMATICAL LOGIC
(PART I
containing any number of apparent variables. Mathematical induction is a
method of proof which is not yet applicable, and is (as will appear) incapable
of being used freely until the theory of propositions containing apparent
variables has been established. What we are enabled to do, by means of the
propositions in the present number, is to prove our desired result for any as¬
signed number of apparent variables—say ten—by ten applications of the same
proof. Thus we can prove, concerning any assigned proposition, that it obeys
the analogues of the primitive propositions of *1, but we can only do this by
proceeding step by step, not by any such compendious method as mathematical
induction would afford. The fact that higher types can only be reached step
by step is essential, since to proceed otherwise we should need an apparent
variable which would wander from type to type, which would contradict the
principle upon which types are built up.
Definition of Negation. We have first to define the negations of (or).
and (g* ).</>». We define the negation of (.*•) . <f>.r as (gj). ~ <£■**, "d 18
not the case that </>./ is always true" is to mean "it is the case that not -<ftx
is sometimes true.” Similarly t he negation of (gx). is to be defined os
(.*•). ^ <ftx. Thus we put
*9 01. ~ {(./■). <f> r\ . « . (gx). <f>.r Df
*9 02 . ~ |(g.#*). j. = . (•**) . ~ tf>x I)f
To avoid brackets, we shall write ~ (x). <ftx in place of ~ ((x). <f>x\, and
^ (g-c). </»/• in place of ~~ |(g-r) . <f>x }. Thus:
*9 011. ~ (a:) . tf>r . = . {(.r) . <f>.r\ Df
*9 021. ~ (gar) . <ftx . — . ~ |(g r) . <f>.c\ Df
Definition of Disjunction. To define disjunction when one or both of the
propositions concerned is of the first order, we have to distinguish six cases,
as follows:
*9 03 (a ) . <f>x . v . p : = . (x) . tf>r v p Df
*9 04. ft . v . (x) . <f>x : = . ( x ). ft v tf> J ' Df
*9 05. (a*) . <f>j -. v . p : = . (gx). tf*x v y> Df
*9 06. p . v . (g-r). <f>xz =. (gx). p v <f>.r Df
*9 07. (x) . <ftx . v . (gy) . yfrj : = : (x) : (gy) .(ftxvyfri/ Df
*9 08. (gy) . ^y . v . (.»•) . <f>.r : = : (x) : (gy) .yjryv<j>x Df
(The definitions *9 07 08 arc to apply also when <f> and are not both
elementary functions.)
In virtue of these definitions, the true scope of an apparent variable is
always the whole of the asserted proposition in which it occurs, even when,
typographically, its scope appears to be only part of the asserted proposition.
Thus when (g.c). <f>x or ( x ). tf>x appears as part of an asserted proposition, it
does not really occur, since the scope of the apparent variable really extends
SECTION B|
EXTENSION OF THE THEORY OK DEDUCTION
13!
to the whole asserted projmsitiou. It. will In* shown, however. that. s.> far as i h<>
theory of deduction is concerned. (g.r) . <f>x and (.»•> . <f>.r behave like propositions
not containing apparent variables.
The delinitions of implication, the logical product, and equivalence niv in
be transferred unchanged to (.r) . <f>.c and (g.r). </>»•.
The above definitions can be repeated for successive types,and thus reach
propositions of any type.
Pnmitive Propositions. The primitive propositions required are six in
number, and may be divided into three sots of two. We have first two
propositions, which effect the passage from elementary to first-order proposi¬
tions, namely
*91. I- : <f>.r . D . (g*) . <f>z Pp
*911. h : <f>x v </>//. D . (g*) . <f>z Pp
Of these, the first states that, if <f>x is true, then there is a value of </>?
which is true; i.e. if we can find an instance of a function which is true, then
the function is "sometimes true.” (When we speak of a function as "some¬
times” true, we do not mean to assert that there is mure than one argument
lor which it is true, but only that there is at least one.) Practically, the above
primitive proposition gives the only method of proving "existence-theorems”:
in order to prove such theorems, it is necessary (and sufficient) to find some
instance in which an object possesses the property in question. If we were to
assume what may be called "existence-axioms,” i.e. axioms stilting (g*) . <f>z for
some particular tf>, these axioms would give other methods of proving existence.
Instances of such axioms are the multiplicative axiom (*88) and the axiom of
infinity (defined in *12003). But we have not assumed any such axioms in
the present work.
The second of the above primitive propositions is only used once, in
proving (a*) •</>*. v . (g*) . <f>z : 0 . (gs) . <frz, which is the analogue of *12
(namely pvp.O.p) when p is replaced by (gx). <f>z. The effect of this
primitive proposition is to emphasize the ambiguity of the z required in order
to secure (gx) . tf>z. We have, of course, in virtue of *9T,
4>x . D . (g z) . <f>z and <f>y . D . (g z) . <f>z.
But if we try to infer from these that <f>x v <f>y . D . (gx). </>x, we must use the
proposition qDp.rOp.D.qyrDp, where p is (gx). <f>z. Now it will be
found, on referring to *477 and the propositions used in its proof, that this
proposition depends upon *1*2, i.e. py p.D . p. Hence it cannot be used by
us to prove (g^:) . (f>x . v . (ga;) . <f>x : D . (g#) . <f>x, and thus we are compelled
to assume the primitive proposition *9*11.
We have next two propositions concerned with inference to or from propo¬
sitions containing apparent variables, as opposed to implication. First, we have,
9—2
MATHEMATICAL LOGIC
132
[PART I
tor the new meaning of implication resulting from the above definitions of
negation and disjunction, the analogue of * 11 , namely
*9 12. What is implied by a true premiss is true. Pp.
That is to say, given “h .//' and "h . /O 7 .” we may proceed to"!-.?,”
even when the propositions /> and 7 are not elementary. Also,as in *1*11, we
may proceed from "h . <£./ and " h .</>./- D yjr.r '' to “ h . yfrx," where x is a real
variable, and </> and \f/ are not necessarily elementary functions. It is in this
latter form that the axiom is usually needed. It is to be assumed for functions
of several variables as well as for functions of one variable.
We have next tin- primitive proposition which permits the passage from a
real to an apparent variable, namely “when <f>y may be asserted, where y may
be any po**ible argument, then (.#•). «f>x may be asserted.” In other words, when
<f>y is true however y may be chosen among possible arguments, then ( x ). <f>x
is true, i.c. all values of <f> are true. That is to say, if we can assert a wholly
ambiguous value <f>y, that must be because all values are true. We may express
this primitive proposition by the words: “What is true in any case, however
the case may be selected, is true in alt cases.” We cannot symbolise this pro¬
position, because if we put
"\- :*t>y. D .(x). 4>u
that means: "However y may be chosen. <f>y implies (x). <f>x," which is in
general false. What we mean is: " If tf>y is true however y may be chosen, then
(x). (f>s is true." But we have not supplied a symbol for the mere hypothesis
of what is asserted in "h . 4>y. where y is a real variable, and it is not worth
while to supply such a symbol, because it would be very rarely required. If,
tor the moment, we use the symbol [<f>y] to ex pros this hypothesis, then our
primitive proposition is
h :[<f>y).D .(x).<t>x Pp.
In practice, this primitive proposition is only used for inference, not for impli¬
cation; that is to say, when we actually have an assertion containing a real
variable, it enables us to turn this real variable into an apparent variable by
placing it in brackets immediately after the assertion-sign, followed by enough
dots to reach to the end of the assertion. This process will be called “turning
a real variable into an apparent variable.” Thus we may assert our primitive
proposition, for technical use, in the lorin:
*9T3. In any assertion containing a real variable, this real variable may be
turned into an apparent variable of which all possible values are asserted to
satisfy the function in question. Pp.
We have next two primitive propositions concerned with types. These
require some preliminary explanations.
Primitive Idea: Individual. We say that x is “individual” if x is neither
a proposition nor a function (cf. p. 51).
SECTION b] EXTENSION OF T1IE THEORY OF DEDUCTION 133
*9131. Definition of "briny of the same type." The following is a stcp-by-st«-p
definition, the definition lor higher types presupposing that Ihr lower types.
We say that u and v “are of the same type" if (1) both are individuals, (2) hot li
are elementary functions taking arguments of the same type. (3) // is a function
and v is its negation, (4) u is <£7 or \fr.r, anil v is <f>.7 v \fr.r, where <f>.r and \ls.7
are elementary functions, (5) « is (y) . 4> (.7, y) and /• is (;> . \fr (.7 1 , s), where
<f> (•**. J), ^ (.r. y) are of the same type. (G) both are elementary propositions.
(7) u is a proposition and e is ~i#, or ( 8 ) u is (u ) . <f>.r and r is (//). ypy. where
</>•«* and are of the same type.
Our primitive propositions are:
*914. If u <f>x" is significant, then if x is of the same type as a, "<!>•' " i*
significant, and vice versa. l*p. (Cf. note on *10 1*21, p. 140.)
*915. If, for some a, there is a proposition <f>a, then there is a function <f>x,
and vice versa. Pp.
It will be seen that, in virtue of the definitions.
(*•) . 4>x . D . p means ~(ar) . <f>x . v . p, i.e. (g.c) ,~<f>x . v . p,
i.e. (gar) .-v^xv p. i.e. (gx) . <f>x D p
(gar) . <f>x . D . p means ~(gx). <f>x . v . p, i.e. (x) . fix . v . p,
i.e. (a:) .-v^vp, i.e. (x). <f>x D p
In order to prove that (x) . tf>x and (gar) . <f>x obey the same rules of deduction
as <f>x, we have to prove that propositions of the forms (x) . (f>x and (g.t) . <t>x
may replace one or more of the propositions p, q,r in *1*2—'G. When this has
been proved, the previous proofs of subsequent propositions in * 2 —*5 become
applicable. These proofs are given below. Certain other propositions, required
in the proofs, are also proved.
*9 2. h : (a:) . <f>x . D . <f>y
The above proposition states the principle of deduction from the general
to the particular, i.e. “what holds in all cases, holds in any one case.”
Dem.
h . * 2-1 . D h
. ^ <fty v <f>y
( 1 )
h . *91 . D h
:~<f>y v 4>y
■ ^ • (a*)
.~*'tf>x v <f>y
(2)
h . (1) . (2) . *1T1 . D h .
(a*)
<f>xv<f>y
(3)
[(3).<*905)]
h :
(a*) —
(ftx . v . <f>y
(4)
[(4).(*9 01 .*1 01)] H :
(x) . 4>x
■ ^ -4>y
In the second line of the above proof, “ ~ <f>y v <f>y ” is taken as the value,
for the argument y, of the function 11 ~ <f>xv 4>y,” where x is the argument.
A similar method of using *9T is employed in most of the following proofs.
* 1'11 is used, as in the third line of the above proof, in almost all steps
except such as are mere applications of definitions. Hence it will not be
MATHEMATICAL LOGIC
134
[PART I
further referred to, unless in cases where its employment is obscure or specially
important.
*9 21 h (./*). <f>> (.#•). <f>, . D . ( x). >\r.e
I.e. if <f>j- always implies \fr.r. then "<f>x always” implies "^.c always.” The
use of this proposition is constant throughout the remainder of this work.
I)em.
H . *2 08 . D h : <f>z D z . D . </>.0 y\rz ( 1 )
y . ( 1 ). *0 1 . D y : (gy): 4>z D yfr: . D . «/>// D >\rz ( 2 )
4.(2). *91. D 1- (gx)(g//): <f>.c D \Jrx . D . <£y D \frz (3)
y.(3).#!) 13. D h ::(*):: (gx):. (gy): 4>x D yfrx . D . D (4)
[<4).(*9’0(i)] y (gx):.<£/• D yfr.c. D : (g»/). <£// D \frz (5)
[(•">).(*I 01. *0 08>] y (g.« ) . D f r) ; v : : (g v). ^ v ((>)
ft (>).<*') 08)] y (gx). ^ (<£./ D yfr.r) : v : (gy) . ~ <f>y . V . (t). y/r* (7)
[< 7 >•(*» I *01)] y fO >/r a-. D:(y).0//. !>.(*). yfrz
This is tin- proposition to be proved, since *'(y). <f>y '' is the same propo¬
sition as ’*(. r ). <£./," and yf'z" is the same proposition as "(x). yjrx."
*9 22 y (x) . <f>xD yfrx . D : (gx) . </>x . D . (gx). ^rx
I.e. if <f>c always implies \frs, then if «/>x is sometimes true, so is yjr.c. This
proposition, like *0 21 . is constantly used in the scipiel.
Dem.
H . *2 08 . D y : <£y D y\nj . D . <£y D yjri/ ( 1 )
h. (1). *9*1 . (2)
y . (2) .*91. D h :. (ax) :. (gx) : <£.r D >/rx . D . </>y D (3)
h . (3). *013 . D 1- ::(y)::(gx):.(gx): <£x D >/rx. D . </>y Df: (4)
[(4).(*!)0G)] h::(y)::(gx):.<£xD^x.D:(g*).<£yD>/rx (5)
[(5).(* 1 01 .*0 08)] l-::(gx )—(tftc D yftx ): v : (y): (gx). <f>y D yfre (C)
[(0).(*1 01.*9-07)] H :: <gx) .~(4x D >/rx): v : (y) .~</>y . v . (gx). '/'x (7)
[(7).(* 1 01 .*9 0 1 02)] y :.(x) . «/».c D >/r.c. D : (gy) . «£y . D . (gx) . yfrz
This is the proposition to be proved, because (gy). </>y is the same pro¬
position as (gx) . (f>. r. and (gx) . yfrz is the same proposition as (g.r) . yfr.r.
*9 23. h : (x). <f>x . D . (x). <*>x [Id . *913 21]
*9 24. y : (gx). <f>x . D . (gx). 0 .r [Id . *91322]
*9 25. h :. (x) . p v <f>x . D : p . v . (x). ^x [*9 23 . (*9 04)]
We are now in a position to prove the analogues of *12—' 6 , replacing
one of the letters p, q, r in those propositions by (x) . <£x or (gx). <f>x. The
proofs are given below.
SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 135
*9 3. b (a ) . 0a . v . (a-) . 0a : D . (a:) . 0a
Dem.
h. *1 * 2 . Dh. 0 a v 0 a. D . 0 a (l)
h . (1) . *91 . D b : (gy) : 0a v 0y . D . 0a (2)
H . (2). *913 . D h (a)(gy) : 0a v 0y. D . 0 a (3)
[(3).(*9 05 01 04)] b (a*) 0a. v . (y). 0y : D . 0a (4)
h . (4) . *9 21 . D b (a*): 0 .v . v . (y). <f>y : D . (a*). 0a (5)
[(5).(*903)] h (a) . 0a . v . (y) . 0y : D . (a). 0 a DK Prop
*9 31. b (ga) . 0a . v . (gar) . 0 a : D . (ga) . 0a
This is the only proposition which employs *911.
Dem.
b. *91113. D b : (y) : 0a v 0y . D . (g*) . 02 ( 1 )
[(1).(*90302)] b : (gy) . 0 a v 0 y . D . (g*). 02 (2)
1- . (2). *913 .DP: (ar) s (gy) . 0 a v 0y . D . ( 32 ) . 02 (3)
[(3).(*9 03 02)] b (gar) : (gy). 0a v 0y : D . ( 32 ) . 02 (4)
[(4).(*90506)] b (gar) . 0a . v . (gy) . 0y : D . ( 32 ) . 02
*9 32. b 9 . D s (a) . 0a:. v . 9
Dem.
b.*l’3. D b 9 . D : 0a. v . 9 (1)
h . (1) - *913 . D b (a) 9 . D : 0a . v . 9
[*925] D h 9 . D : (a) : 0a . v . 9 (2)
[(2).(*903)] b 9 . D : (a) . 0a . v . 9
*9*33. b 9 . D : (ga) - 0a . v . 9 [Proof as above]
*9 34. b (a) . 0a. D : p . v . (a) . 0a
Dem.
b . *l - 3 . Db:0a.D.pv0a (1)
b . (1) . *913 . Db:(a):0a.D.pv0a (2)
1-. (2) . *9 21 . D b : (a) . 0a . D . (a) .p v 0a (3)
b . (3) . (*9 04) . D b . Prop
*9 36. b (ga) . 0a . D : p . v . (ga) . 0a [Proof sis above]
*9 36. b p . v . (a) . 0a : D : (a) . 0a . v . p
Dem.
b.*l*4. Db:pv0a.D.0avp (1)
b.(1). *9T3 21 . D b : (a) .p v 0a . D . (a) . 0a vp (2)
b . ( 2 ) . (*9 03 04) . D b . Prop
*9*361. b (a) . <f>x .v.p:D:p.v. (a) . 0a [Similar proof]
*9 37. b :.p . v . (ga) . 0a : D : (ga) . 0a . v .p [Similar proof]
*9371. b (ga) .0a.v.p:D:p.v. (ga) . 0a [Similar proof]
136
MATHEMATICAL LOGIC
[PART I
*9 4. b :: p : v : 7 . v . (x). £xD 7 : v :p . v . (x). <£x
Dem.
b . *1*5 . *9 21 . Dh. (x) zp . v . 7 v <f>x : D : (.r): 7 . v .y> v <£x (1)
I- . (1) .(*9-04). D b . Prop
*9 401. b :: y>: v : 7 . v . (go-). <£x D 7 : v : yi. v . (gx) . <f>x [As above]
*9 41. b ::y>: v : (x) .^r.v.rs.D:. (x) ,<f>rzvzpvr [As above]
*9 411. h :: y>: v : (gx). <f>r . v . #•D :.(gx). <f>x : v : j>v r [As above]
*9 42 I- :: (x). <f>x : v : 7 v rD :. 7 : v : (x). </>.r . v . r [As above)
*9 421. 1- :: (gx). Qxz v : 7 v rD 7 : v : (gx). <f>* . v . r [As above]
*9 5. b :: yO 7 .3 :./>. v . (x). <£./ : D : 7 . v . (x). ^>x .
l)em.
b.*l- 6 . D I-:. /> D 7 . D : /> v <^»v. D . 7 v <^>y ( 1 )
b . (1). *91 . (*9'06). D h /O 7 . D : (gx) : /»v 0./-. D . 7 v <£ ;/ (2)
I- . (2). *9 13 .(*9 04). D b :: /O 7 . D (»/)(gx): y> v <£x. D . 7 v <£// (3)
[(3).(*908)] h ::/> D 7 . D (gx) —(/> v $x). v . (y). 7 v <f>y (4)
[(4).(*9’01)] h ::/>D 7 . D (x)./) v . D.(»/). 7 v (/>// (5)
[(3).(*9 04)] 1- :: yO 7 . D . v . (x). (f>x : D : 7 . v . (y). <£.»/
*9 501. b :: /O 7 . D :. yi. v .(gx) . <£.» : D : 7 . v . (gx). «/>x [As above]
*9 51. b :: y>. D . (x). <£x : D p v r. D : (x). <£x . v . r
Dem.
b . #rc. D b y> D .Dspvr.D. ^>x v r 0 )
b . (1). *913 21 . D b :: (x). y> D 0x . D (x): /> v >•. D . <f>.v v r (2)
b . (2) . (*90304). D b . Prop
*9 511. b :: p . D . (gx). <f>.> : D p v r. D ; (gx). <£x. v . r [As above]
*9 52. b :: (x). <f>.r . D . 7 : D (x) .^x.v.rO.jvr
Dem.
b.* 10 . D b £x D 7 . D : <f>x v r. D . 7 v r ( 1 )
b .(1). *913-22. D b :: (gx) . <£x D 7 . D :.(gx) : <f>x v r. D . 7 v r (2)
b . (2). (*9 05 01). D b :: (x) . <f>.r . D . 7 s D s. (x). *x v r. 3 .7 v r (3)
b . (3). (*9 03). Db. Prop
*9 521. b :: (gx). <frx . D . 7 : D :. (gx). <£x . v . r: D . 7 v r [As above]
*9 6 . (x). </>x. ~(x). <£x, (gx). <f>r ami ~(gx). <f>x are of the same type.
[*9-131, (7) and ( 8 )]
*9 61. If <f>x and yfrf are elementary functions of the same type, there is a
function <f>x v
Dem.
By *91415, there is an a for which "yfra," and therefore “<£«." are
significant, and therefore so is “<f>a v y/ra,” by the primitive idea of disjunction.
Hence the result by *9'15.
The same proof holds for functions of any number of variables.
SECTION B]
EXTENSION OF THE THEORY OF DEDUCTION
137
*9 62. If <f> (.r, (/) ami are elementary functions, ami the .r-argumont. t o
<P is of the same type as the argument to yjr, there are functions
(y) • 4> !/) • V - yjr.r. (a*/) . <t> (.7. if) . V . ir.r.
Deni.
By *915. there are propositions <£(.r. b) ami yfra, where by hypothesis .»•
and a are of the same type. Hence by *9-14- there is a proposition <f>(a, b),
and therefore, by the primitive idea of disjunction, there is a proposition
<£ (<*, b) v \j/a, and therefore, by *915 and *9'03. there is a proposition
(y) • <f> (o, y> . v . yfra. Similarly there is a proposition • ♦ (<*. //) • v •
Hence the result, by *9 15.
*9'63. If <fr (.?, f)), yjr^, 5) are elementary functions of the same type, there
are functions ( i /). <t> (.2, y). v . (s) . \jr (.7, s), etc. [Proof ns above]
We have now completed the proof that, in the primitive propositions of
*1, any one of the propositions that occur may be replaced by (.r*). <f>.r or
(3 x). <f).v. It follows that, by merely repeating the proofs, we can show that
any other of the propositions that occur in these propositions can be simul¬
taneously replaced by (a). yfrx or (jpr). yfrx. Thus all the primitive propositions
of *1, and therefore all the propositions of *2—*5, hold equally when some
or all of the propositions concerned are of one of the forms (x ). <f>.r, (ft#) . </>.r,
which was to be proved.
It follows, by mere repetition of the proofs, that the propositions of *1—*5
hold when p, q, r arc replaced by propositions containing any number of
apparent variables.
*10. THEORY OF PROPOSITIONS CONTAINING
ONE APPARENT VARIABLE
Sii in mu ry of * 10.
The chief purpose of the propositions of this number is to extend to
formal implications (i.e. to propositions of the form (x) . ^D|.r) as many as
possible of the propositions proved previously for material implications, i.e.
for propositions of the form p D 7 . Thus e.y. we have proved in *3 33 that
p D <[ . #/ D /•. D . /O r.
Put /> = Socrates is a Greek,
tj m Socrates is a man.
/• = Socrates is a mortal.
Then we have "if 'Socrates is a Greek' implies 'Socrates is a man,' and
* Socrates is a man ' implies 'Socrates is a mortal,' it follows that 'Socrates is
a Greek implies 'Socrates is a mortal.’" But this does not of itself prove
that if all (Jreeks are men. and all men are mortals, then all Greeks are
mortals.
Putting <f>.r . = . .r is n Greek,
. » . x is a man.
\x . * . x is a mortal,
we have to prove
( x ) . tf>x D >frx : (x) . y/rx (x) . <f>x D yx.
It is such propositions that have to be proved in the present number. It will
be seen that formal implication ((j).fO|j) is a relation of two functions
4>x and y\rx. Many of the formal properties of this relation are analogous to
properties of the relation "pDt/" which expresses material implication; it is
such analogues that arc to be proved in this number.
We shall assume in this number, what has been proved in *9, that the
propositions of *1—*5 can be applied to such propositions as (x) . <px and
(gx). Instead of the method adopted in *9, it is possible to take negation
and disjunction as new primitive ideas, as applied to propositions containing
apparent variables, and to assume that, with the new meanings of negation
and disjunction, the primitive propositions of *1 still hold. If this method is
adopted, we need not take (gx). <f >.r as a primitive idea, but may put
*10 01. (gx). <f>x . = . ~(x) ,~<f>x Df
In order to make it clear how this alternative method can be developed,
we shall, in the present number, assume nothing of what has been proved in
*9 except certain propositions which, in the alternative method, will be
primitive propositions, and (what in part characterizes the alternative method)
SECTION B]
THEORY OF ONE APPARENT VARIABLE
130
the applicability to propositions containing apparent variables of analogues
of the primitive ideas and propositions of *1, and therefore of their conse¬
quences as set forth in *2—*5.
The two following definitions merely serve to introduce a notation which
is often more convenient than the notation (.#•) . <f>.r D yfrx or (.<*). </><• zi \fr.v.
*10*02. <f>x D x yfr.c . = . (.«•) . <fyv D yfrx Df
*10 03. <f>x = x yfrx . = . (.»’) . <f>x = yfrx Df
The first of these notations is due to Peano, who, however, lias no notation
for (x). <f>x except in the special case of a formal implicat ion.
The following propositions (*10*1*11*12*121*122) have already been given
in *9. *10 1 is *9*2, *10 11 is *9 13. *10 12 is *9*25. *10121 is *91+. and
*10122 is *9*15. These five propositions must all be taken as primitive
propositions in the alternative method; on the other hand, *9*1 and *911 are
not required as primitive propositions in the alternative method.
The propositions of the present number are very much used throughout
the rest of the work. The propositions most used are the following:
*101. h : (x) . <f>x . D . <f>y
I.e. what is true in all cases is true in any one case.
*10 11. If <f>y is true whatever possible argument y may be. then (x) . <pv is
true. In other words, whenever the propositional function <f>y can be asserted,
so can the proposition ( x ) . <f>x.
*10*21. h (a:) . p D <f>x . = : p . D . (x) . <f>x
*10 22. h (x). <f>x . yfrx . = : (x) . <f>x : (x) . yfrx
The conditions of significance in this proposition demand that <f> and yfr
should take arguments of the same type.
*10 23. h (x) . <f>x Dp • 3 : (gx) . 4>x .D.p
I.e. if <f>x always implies p, then if <j>x is ever true, p is true.
*10*24. h : <f>y . D . (gx). <f*x
I.e. if <f>y is true, then there is an x for which <f>x is true. This is the sole
method of proving existence-theorems.
*10 27. 1- :.(*). <f>z D yftg. D :(*).$*. D . (z) . yfrz
I.e. if <pz always implies yfrz, then “ tf>z always ” implies “ yfrz always.” The
three following propositions, which are equally useful, are analogous to *10*27.
*10*271. h {z) . <f>z = yfrz . D : (z) . <f>z . = . (*) . yfrz
*10*28. I-(x) . <fjx D yfrx . D : (gx) . <£x . D . (gx) . yfrx
*10*281. h (x) . <f>x = yfrx . D : (gx) . <f>x . = . (gx) . yfrx
*10 36. I-(gx) . p . <£x . = : p : (gx) . <f>x
*10*42. h (gx) . <£x . v . (gx) .yfrx: = . (gx) . tf>x v yfrx
*10*6. > (gx) . tf>x . yfrx . D : (gx) . <f>x : (gx) . yfrx
no
MATHEMATICAL LOGIC
[PART I
It should be noticed that whereas *10 42 expresses an equivalence, *105
only expresses an implication. This is the source of many subsequent
differences between formulae concerning addition and formulae concerning
multiplication.
*1051. h ~ |(g.r) . <f>x . yfrj -1 . = : $x . D x . ^ yfr.r
This proposition is analogous t• •
b : ^ </>.»/). = ./> D ~ 7
which results from *4<>3 by transposition.
Of the remaining propositions of this number, some are employed fairly
often, while others are lemmas which are used only once or twice, sometimes
at a much later stage.
*10 01. . 4>I . = . ^(.r) . Df
This definition is only to be used when we discard the method of *9 in
favour of the alternative method already explained. In either case we have
I- : (C*|*) • </>' • 2 . ~(x) • ^
*10 02. 4>.r D, \fr.r , m . (s) . <f>.r D \frx I )f
*10 03. </>./ =, >/r./•. = . (x) . <£.r 2 Df
*10 1. I- : (.#•). <f>.r . D . <fji/ [*!P2]
*1011. If <t»j is true whatever possible argument y may be, then (x),fa is
true. [*913]
This proposition is, in a sense, the converse of *101. *10 1’ may be stated:
" What is true of all is true of any," while *1011 may be stated : “ What is
true of any. however chosen, is true of all.”
* 1012 . b (.*•). p v <f>x . D : p . v . (a*). ^.r [*!b25]
According to the definitions in *0, this proposition is a mere example
of "7 3 7 ," since by definition the two sides of the implication are different
symbols for the same proposition. According to the alternative method, on
the contrary. *1012 is a substantial proposition.
*10121. If " <f)j ■" is significant, then if a is of the same type os x, u <f>a" is
significant, and vice versa. [*914]
It follows from this proposition that two arguments to the same function
must be of the same type ; for if x and a arc arguments to " <f>x ” and "</>«
are significant, ami therefore x and a are of the same type. Thus the above
primitive proposition embodies the outcome of our discussion of the vicious-
circle paradoxes in Chapter II of the Introduction.
*10 122. If, for some a, there is a proposition <f>a, then there is a function <f>$,
and vice versa. [*915]
*1013. If ffta 1 and yfrj 1 take arguments of the same type, and we have “b ,<f>x”
and “b . yfrx ,” we shall have "b . <px. ylrx."
SECTION 1*|
THEORY OK ONE APPARENT VARIABLE
II l
Dem.
By repeated use of *lH>rU2T>3131 (3). t here is a function «■'»' (fax V <—• \f/.r.
Hence by *211 and *3 01,
h : <p.v v <-**■ \fr.v . V . (far . yjr.r ( 1 )
h . f l) . **2 3*2 . (*101) .3 1-:. <*m . 3 : ^-x. 3 . </>.r . >/r.r (2)
1-. (2) . *012.3 h . Prop
*1014. h (.r) . (fax : (,r) . >Jr.r : D . <£y . \fay
This proposition is true whenever it is signiticant, but it is not always
significant when its hypothesis is significant. For the thesis demands that
0 and yfr should take arguments of the same type, while the hypothesis does
not demand this. Hence, if it is to be applied when (fa and \fr are given, or
when yjr is given as a function of (fa or vice versa, we must not argue from the
hypothesis to the thesis unless, in the supposed case, (fa and \fr take arguments
of the same type.
Dem.
K*10l. DK
: ( x) . (fax. D .(fay
(i)
K*10l. Dh
z(x).yfrx. O.yfry
(2)
(1). (2). *1013.3 1-
:{x).(fax.O .fay z(x) .yfrx ,D . -fry :
[*3 47] D h
:. (x). <fax : ( x ) . yfax : 3 . fay . yfry :. 3 H . Prop
*102. 1 -(x) . pv (fax .
s : p . v. ( x ). (fax
Dem.
h . *10*1 . *16.3 h p . v . (x) . (fax : 3 . p v (fay
[*1011]
3 1*:. ( y ):. p . v • (x) . (fax : 3 . pv fay :.
[*1012]
3 h :. />. v . (x). (fax : 3 . (y) . /> v <f>y
(1)
1 - .*1012.
3l*:.(y).pv^y.3:p.v. (x). (fax
(2)
h.(l).(2).
3 h . Prop
*10 21. h :. (x) . p 3 (fax
. = sjp.3.(x).^x 1*10 2^
w
This proposition is much more used than *10 2.
*10 22. h :. (x) . (fax . yfrx
:. = z(x).faxz(x).yfrx
Dem.
h.*101 .
3 1-: (x) . (fax . >/rx .D .fay. yfry .
(1)
[*326]
3.*y:
[*1011]
3 h (y) : (x) . (fax . ^x .D .fay z.
[*1021]
3 h :. (x). (fax . yfrx. 3 . (y) . <£y
(2)
h . (1). *3*27 .
3b:. (x) . (fax . yfrx . 3 . >p *2 :.
[*1011]
3 h :. (s) : (x) . (fax . yfrx . 3 . :.
[*10-21]
3 h :. (x) . (fax . yfrx . 3 . (*) . ^
(3)
h . (2) . (3) . Comp . 3 h :. (x) . (fax . yfrx . 3 : (y) . (fay : (z) . yfrz
(4)
h. *101411.
31-:. (y) (x) . (fax : (x) . yfrx : 3 . fay . yfry z.
[*10-21]
3 1*:. (x) . (fax : (x). yfrx : 3 . (y) . <£y . ^y
(5)
K(4).(5).
3 h . Prop
142
MATHEMATICAL LOGIC
[PART I
The above proposition is true whenever it is significant; but, as was
pointed out in connexion with *10 14, it is not always significant when
“ (./•) . <*>/•: (.*•). yjr.r " is significant.
*10 221. If 4>.r contains a constituent \ (x. y. z, ...) and yfrx contains a con¬
stituent x <'* M » r * •••)• "here x > s an elementary function and y, z ,... u, v, ...
are either constants or apparent variables, then <f>r and yjrx take arguments
of the same type. This can be proved in each particular case, though not
generally, provided that, in obtaining <£ and from x* X * s on b* submitted
to negations, disjunctions and generalizations. The process may be illustrated
by an example. Suppose </>./ is (//>• X<•'•.'/)• an! yjrx is fx . 0•(y)•x (*»y)*
By the d. tinit ions of *0, <£.#■ is (gy). ~ x (• r . y ) v & r . and '/'x is v X
Hence since tin* primitive ideas (./). Fx and (g.r) . Ftf only apply to functions,
there are functions //) v #' k - ~/a vx(•'*. y)- Hence there is a proposi¬
tion ^x^ # » b)vflii. Hence, since “pvtj" and are only significant
when jt and 7 are propositions. there is a proposition x^*> ^)* Similarly, for
some // and /*, there are propositions -v/« vx (m, «») and x( M * *')• Hcncc by
*014, // and «, v and are respectively of the same type, and (again by *914)
there is a proposition ~/u v x (". M- Hence (*015) there arc functions
^X (°> 5) v v X .V). and therefore there are propositions
<a//) • ~x <«. //) v (y ). V X (". y).
t.c. there are propositions <f>a. yfra, which was to be proved. This process can
be applied similarly in any other instance.
*10 23. b :. (x) . <f>.i Dy>. s : (gx). <f>x .D.ji
Dem.
b . *4 *2 . (*0 03). D b (x) . ~tf>x v p . = : (x) . . v . p :
[(*0 02 )] s-(a *).$*. 3 ./> W
h.(l).(*l'0l). UK Prop
In the above proof, we employ the definitions of *0. In the alternative
method, in which (g x). tf>.> is defined in accordance with *10 01 , the proof
proceeds as follows.
*10 23. b :. (x). <f>.i D p . = : (gx) . <f>x .D . p
Dem.
b . Transp .(*10 01). D b :.(gx). tf>x. D . p : = : . D . (x). ~<£x :
[*10-21] = : (x) : . D . ~ </>***: (0
[*10 1] D 3
[Transp] D z <f>x .0 . p z.
[*1011] Dbz. (x) (gx ). <px .0 . p zD z <f>x .D . p :•
SECTION ll]
THEORY OF ONE A 1*1*ARENT YAR1A lll.K
I 13
[*10*211
D h (g .r) . </>.,*. D . P : D : (.r>: </»./ . D . p {2)
1- . *10*1 .
D h (.r) : tf>.r tf>.r D /»:
[Transp]
D : . D . ^ </m*
[*10*11*21]
D h (a*) : </>./•. D . /»: D : (.•*) : . D . :
[U>]
D :(g.#>. «/>.r . D ./> (3)
1-. (2) . (3).
D H . Prop
\\ henever we have an assorted proposition of the form /> D <£.»-, we can
pass by *10*11*21 to an asserted proposition />. D .(.*•). This passage is
constantly required, as in the last line but one of the above proof. It will
be indicated merely by the reference M *10*11*21," and the two sli ps which it
requires will not be separately put down.
*10'24. h : <f>y . D . (gx) . <£.»•
This is *9*1. In the alteruative method, the proof is ns follows.
Dem.
h . *10*1 . D h : (a:) <f>.i
[Transp] . ~(.r) :
[(*10*01)] D h . Prop
*10-25. h : (x) . <f>x . D . (gar) . <f>.v [*10*1*24]
*10-251. h : (x) —<px . D — {(x) . <*>x) [*10*25 . Transp]
#10-252. h : ~ |(gx) . <f>x\ . = . (x) . ~ 4>x [#4 2 . (*9 02)]
*10-253. h : ~ |(x) . <*>x) . s . (gx) — <f>x [*4*2 . (#9*01)]
In the alternative method, in which (gx). <f>x is defined as in *10*01, the
proofs of *10*252*253 are as follows.
*10 252. h:~j(gx).<*>x). = .(x).~<*>x [*4*13 . (*10*01)]
*10 253. I- : ^ |(x) . <j>x\ . = . (gx) . ~ tf>x
Dem.
H • *10*1 . D I-: (x) . <f>x .D.tf*y.
[*212]
[*10*11*21] D h : (x) . <f>x . D . (y) —(~<*>y) s
[Transp] D h :~{(y) .~(~<f>y)j
.D.~|(x).*x):
[(*10 01)] D h : (gy) .~<£y •
D ,~|(x). <£x)
(i)
K . *10*1 . DH:(y).~(~^y).
D .~(~^2-) .
[*214]
D . <f>x :
[*10*11*21] D h : (y) —(~<*>y) -
D . (x) . <f>x :
[Transp] D t- : ~ ((x) . <f>x J .
^ — l(y) —(~0y» •
[(*1001)]
3 • (ay)-~£y
(2)
h . (1) . (2) .Dh. Prop
144
MATHEMATICAL LOGIC
[PART I
*10-26. h (z). <f>z ^ \frz : <f>.r : O . r [*101 . Imp]
This is one form of the syllogism in Barbara. E.g. put <f>z . = . z is a man,
>frz . = . z is mortal. .r = Socrates. Then the proposition becomes:
“If all men are mortal, and Socrates is a man, then Socrates is mortal.”
Another form of the syllogism in Barbara is given in *103. The two
forms, formerly wrongly identified, were first distinguished by Peano and
Frege.
*10 27. V U). 4> z 3 V r - • 3 : (z ). <frz . 3 . ( 2 ). y\rz
This is *9*21. in the alternative method, the proof is as follows.
Deni.
h . *10 14 . 3 h :.(*>• <t>z 3 y\rz : (r). <f>z : 3 . <f>y 3 . </>y .
[Ass] 3.>fry:.
[*101] 3 h (y)(i). <f>z 3 y/rz : (*). <f>z : 3 . yfrg
[*1021] Dh :.(z).<f>z'D\lrz:(z).<f>z .(g).yfrg (1)
h . (1). Exp • 3 h • Prop
*10 271. 1- :• (*). <f>z m yfrz . 3 s (z). <f>z . s . (*) . yfsz
Dem.
h. *10*22. 3 I-Hp . 3 : (z). </>* 3 yfrz :
[*10 27] 3: {*).+£ (1)
h. *10*22. Dh.H|i.D:(r).^:D</»::
[*10*27] D:(z).ylrz.D.(z).<t>z (*)
I- . (1) . (2). Comp . 3 . Prop
*10*28. h (x) . <t>f 3 >Jrx. 3 : <gx). </>x . 3 . (gx) . >frx
This is *0*22. In the alternative method, the proof is as follows.
Dem.
h . *10*1 . Dh.(x). <f>x 3 V'-r . 3 . <£y 3 >/ry .
[T rausp] 3 .^A/fy 3~<£y
[*10*11*21] 3 h :. (x). £x 3 ^rx. 3
[*10*27] 3 :(y).~>/ry . 3 .(y).~<£y :
[Transp] 3 : (gy). <py . 3 - (ay). ^y =• ^ *" • Pro P
*10 281. f-:.(x).^c=^x.3:(gx).</>x. = .(gx).i/rx [*10*22*28 . Comp]
*10 29. H ( x) . <f>x 3 yfrx : (x) .^x3xx: = :(x):</u.D.fx. X*.
Dem.
h . *10*22 . 3 h (x) . <f>x 3 >/rx : (x) . <f>xO x xz
= : (x) : 4>x 3 >/rx. <f>x 3 x* (*)
I-. *4*7G . 3 I-£x 3 >/rx . <f>x 3 x^ • = : <f> x • 3 • '/'“x • X a * : *
[*10*11] 3 h (x) <f>x 3 >Jrx. <f»x 3 x x • = : 4> x • ^ : *
[*10*271] 3h.(x): ^r3|x.^3xx:s:(x):^.D.fx.x^ (2)
h . (1) . (2) . 3 h . Prop
This is an extension of the principle of composition.
SECTION uj
THEORY OF ONE APPARENT VARIABLE
l ir,
*10 3. b (.r) . <f>x D yfrx : (x) . yfrx D yx : D . (.r) . </».r D y. r
This is the second form of the syllogism in Barbara.
Dem.
h . *10*22*221 . D b : Hp . D . (x) . <£x D yfrx . >/r.r D x .r.
[Syll.*10 27] D . (.r>. <*>., D **•: D b . Prop
*10 301. b (.r) . <£x = yfrx : (.r) . yfrx = *.i* : D . (x) . <f>x s yx
Dem.
h . *10*22 *221 . D b Hp . D : (.r) . <fix = >/r.r . yfrx = *.»• :
[*1*22.*10*27] D : (or). <f>x = x .r D b . Prop
In the second line of the proofs of *10*3 and *10*301. we abbreviate the
process of proof in a way which is often convenient. In *10*3. the full process
would be as follows:
H . Syll. D b : <f>x D yfrx . yfrx D yx . D . <£.* D ^.r:
[*1011] D b : (x) : <£x D yfrx . >/r.r D X x. D . <f».c D *x :
[* 10 *27 ] D b : (x) . <px D >/rx. >/rx D ^a*. D . (x). <f>x D ^x
The above two propositions show that formal implication and formal
equivalence are transitive relations between functions.
*10*31. I- (x). <f>x D yfrx . D : (x) s <ftx . yx • ^ ^
Dem.
H . Fact. *10*11 . D b (x) </>x D >/rx . D : <£x . *x . D . >/rx . ^x (1)
h • (1) . *10*27 . D b . Prop
*10 311. I- (x) . <f>x = >/rx . D : (x) : <f>x . yx . = . yfrx . *x
Dem.
b . *4*36 . *10*11 .D h
h • (1) . *10*27 . Db.
The above two propositions are extensions of the principle of the factor.
*10*32. b : <px = x yfrx . = . yfrx =* <f>x
Dem.
V . *10*22 . D b : (f>x = x yfrx . = . <f>x D x yfrx . >/rx D x <f>x .
[*4 3] = . yfrx D x <f>x . <t>x D x yfrx .
[*10*22] = . yfrx = x <f>x : D h . Prop
This proposition shows that formal equivalence is symmetrical.
*10*321. b : <f>x = x yfrx . <f>x = x yx . D . yfrx = x yx
Dem.
b . #10*32 . Fact . D b : Hp . D . yfrx = x <f>x . <f>x = x yx .
[*10 301] 0.yfrx= x yxzD b . Prop
*10*322. b : yfrx = x <f)X . yx = x tftx . D . \frx = x yx
Dem.
b . *10*32 . D b : Hp . D . yfrx = x <f>x . (f>x = x yx .
[*10*301] D . yfrx = x yx : D b . Prop
R&w i , n
(X) 0x = yfrx . 3 : <f>x . yx . = . yfrx . *x
Prop
146
MATHEMATICAL LOGIC
[PART I
*10 33. H (x) : <ftx . p : = : (x). <f>.r : p
Deni.
I-.*10*1. D 1-(.r) : <f>x ,p : D . <f>y . p . (1)
[*3 27] O.p (2)
h . (1). *3 26 . D h (.r) : <f>.r . p : Z) . <f>y z
[*101121] D h (x )z<f>x.pzD ,(y).<f>y (3)
h.(2).(3). D h. (x) z <f>.r . p : Z> : (y). <f>y z p (4)
h.*10*l. D \- :.(y) . <f>y . D.^.r:.
[Fact] D I- (y) . d>y : /#: D . <f>.r. p z.
[*10 11-21] D I- s. (»/). <f>y z p z 5 z (x) z <f>x. p (5)
H.<4).(5). D H. Prop
*10 34. I- (gx) . <f>.i D p . = : (a-) . ^>./-. D . p
This follows immediately from *90501 and *101. In the alternative
method, the proof is as follows.
Dem.
K. *4 2. (*1001). D
h (gx). <f>xDp . = :~ |(^> —(<#>x Op)):
[* 461.* 10-271] ~ :~|(x): <f>x .~p\ z
[*10 33] = z~[(x). <f>x z~ji\ z
[*+•53] = |(x) . ^x| . v ,p z
[*4 0] = : (.r). <J>.r . D . p
*10 35. I-(gx ). p. <f>x. = z p z (go-) .<f>x
Dem.
V . *3'26 . D I- : p . <f>s . D . p z
[*1011] Dl - z (x) z ]>. <f>x. D . p z
[*10-23] Dh:(a x).p.<f>x.D.p (1)
I- . *3*27 . D I*: p . <f>x . D . <f>x z
[*1011] D I-: (x) z p . fa .D . (f>x z
[* 10 28] D I-: (gx). p . £x. D . (gx) . </»x ( 2 )
I-. *32 .
[*1011*21] 0 z. p. D z (x) z <f>x. D . p. <f>x z
[*10 23] D : (gx). tfxc . D . (gx). p. <f>x (3)
I-. (1). (2) . (3). Imp.D h. Prop
*1036. I*(gx) . </>x v/>. s : (gx) . <£x. v ./>
This follows immediately from *9 05. In the alternative method, the
proof is as follows.
Dem.
h . *4*6+ . D h : <t>x v p . = .^ <f>x D p :
[*1011] Z> I-: (x) z <f>xvp .= .~<f>x D p z
[* 10281 ] D I-(gx) .<f>xvp. = z (gx) ,~<f>xDp z
[*10 34] =:(x).~<£x.D.i>:
[*4-6.(*10 01)] = : (a*). <j>x . v .p D H. Prop
SECTION B]
THEORY OF ONE APPARENT VARIABLE
The above proposition is only required in order to lead to the following:
*10 37. h (g.r) . p D (fix . = D . (gx) . <f>.r ^*1036
*10 39. h (fix D x y.e : \frx D x Ox : D : (fix . y/r.v . D x . yx . Ar
Dem.
h . *10‘22 . D h Hp . D : (x) : (fix D yx. yfrx D Ox :
[*3 +7 .*10 27 ] D : (.r) : </>.»•. yfix . D . yx . ArD h . Prop
This proposition is only true when the conclusion is significant; the
significance of the hypothesis does not insure that of the conclusion. On the
conditions of significance, see the remarks on *10 , 4, below.
*10 4. h (fix = x yx . y\rx = x Ox . D : (fij '. yfrx . s x . yx . Ar
Dem.
h . *10 22 . D h Hp . D : (fix D x ^x . \fix D x Ox:
[*1039] D : <f>x . yfrx, D x . yx. Ar (1)
Similarly h Hp . D : yx . Ar. D,. 0x. >/rx (2)
h . (1). (2) . Comp . D 1- Hp . D : <f>x . >/rx. D x . *x . 0x : yx. Ox . D x . <£x . >/rx :
[*10 22] 3 : <fix . +x. s, . yx. Ox : F D . Prop
In *10*4 and many later propositions, as in *1039. the conclusion may be not
significant when the hypothesis is true. Hence, in order that it may be legiti¬
mate to use *10 4 in inference, i.e. to pass from the assertion of the hypothesis
to the assertion of the conclusion, the functions <fi, yjr, y. 0 must be such as to
have overlapping ranges of significance. Iu virtue of *10 221, this is secured if
they are of the forms F {x, x (x,0.5,...)),/[x. x(*,$, 2. ...)), G {x, * (x. f). 2,...)),
9 \ x > X ( x » z > •••))• It * s also secured if <f> and yfi or <f> and 0 or % and yfi
or % and 0 are of such forms, for <f> and % must have overlapping ranges of
significance if the hypothesis is to be significant, and so must ^ and 0.
*10 41. h :. (x) . (fix . V . (x) . yfrx : D . (x) . (fix V yfrx
Dem.
h . *10T . D h : (x) . <fix . D . <fiy .
[*22] D.+yv+y (1)
h . *10'1 . D 1- S (x) . y\rx . D . yfiy .
[*13] ^•<fiyv>fiy (2)
h . (1) . (2) . *1013 . D h (x) . (fix .0 .(fiy y yfiy : (#) . yfrx .0.<fiyy \Jry
[* 3 ' 44 J D h (x) . (fix . v . (x) . yfrx : D . fiy y yfiy
[*1011-21] D h (x) . (fix . v . (x) . yfix : D . (y) . <fiy y yfiy D h . Prop
Observe that in the above proof the uses of *2 2 and *13 are only legitimate
if (fiy and yfiy have overlapping ranges of significance, for otherwise, if y is such
that there is a proposition (fiy , it is such that there is no proposition yfiy, and
conversely.
10—2
118
MATHEMATICAL LOGIC
[PART I
*10*411. b <f>x =, XX . yfr.r = r Ox . D : <f>.r v yfrx . = x . v #x
Dem.
H . *10*14 .Dh:. Hp.D: <f>x = ^.r. \fs.r = 0x :
[*4*39] D : <p.r v ^rx. = . x- r v Ox (1)
H . (1). *10*11*21 . D b . Prop
*10*412. b : <£x= x >frx.= [*4*11 . *10*11*271]
*10*413. I" 4>x = x xx • y l' r =x . D : 4>x D '/'.r . = x . ^.r D d.c
Dem.
1- . *10*411*412 . D H Hp . D :~ <f>.r v \/rx. = x • ~x x v &**
[(*1 *01)] D : 0X D >/r.r . = x . x-'* Dftrs.DK Prop
*10 414. H <f>x = x \X . = x Ox . D : <£./* = >/r.r . = x . X* = Ox
Dem.
H . *10*413 * 1 • *10-32 . D b :. Hp .D:^0 <*>x . = x . Ox D x-c 0)
1- . *10*413 . (1) . *10*4 . D b . Prop
The propositions *10*413*414 arc chiefly used in eases where cither x ,s
replaced by </> or 0 is replaced by >/r, in which ease hull* the hypothesis becomes
superfluous, beintf true by *4*2.
*10 42. b (ax). <f>x . v . (yj) .>frx:=. (gx). <f>x v >fr.c
Dem.
h . *10*22 . D b :. (x) .~<f>x : (x).^\/rx: ~ . (x) ,~<f>x .^yfrx
[*4*11] Ob :.~|(.r) : (.r) .~>/rxj . = .~{(x).~ <f>x .~ yfrx] !#
[*4*51*56.*10*271] 0 b j(x) .~<f>x\ . v . ~ |(x) . ~ >/r.r| :
•—((*) ■ ~ («/>.r v yjrx)]
[*10*253] Ob (gx). <£x. v . (gx). >/rx: = . (gx) . <f>x v yfrx
D h. Prop
This proposition is very frequently used. It should be contrasted with
*10*5, in which we have only an implication, uot an equivalence.
*10*43. b : <f>z =, >\rz . <f>x . = . <f>e =, \frc . >frx
Dem.
b . *10*1 ■ 0 b : <f >2 = x yjrz . D . <f>x = yfrx (I)
b . (1) . *5*32 . D I-. Prop
*10 6. b :. (gx) . <f>x . yfrx . 0 : (gx) . <f>x : (gx). >\rx
Dem.
b . *3*20 . *10*11 . D h : (x): <f>x . yfrx ,0.<f>xz
[*10*28] 0 h : (gx) . <f>x . yjrx . D . (gx) . <f>x (1)
b .*3*27 .*10*11 . D 1-(x) : <f>x . yfrx . D . yjrx :
[*io- 28 ] < 2 >
h .(1).(2).Comp. Dh:. Prop
SECTION B]
THEORY OF ONE APPARENT VARIABLE
The converse of the above proposition is false. The fact, that this
proposition states an implication, while *10 42 states an equivalence, is tin*
source of many subsequent differences between formulae concerning logical
addition and formulae concerning logical multiplication.
*10 51. h j(g.r) . <f>x . . = : <f>x . . ~>/r.r
Dem.
H . *10 252 . D h {(ax) . <f>x . yfrx] . = : (a ) -~(<^».»-. yfrx) :
[*4-5162.*l0-271] = : (x) : <f>x . D — yfr .r :. D I- . Prop
*10 52. h (gar) . <f>x . D : (x) . <f>x D p . = . p
Dem.
I-. *5 5 .Dh: Hp. D :.p. = : ( 3 a) .<f>x.D.p:
[*10 23] = : (x) . <f>.v Dp :s D h . Prop
*10 53. I-^*( 30 :) . <f>x . D : <f>x . D z . >/rx
Dem.
h .*2 21 .*1011 . D
h (x) z.~<fix . D : <£>x . D . yfrx
[* 10"27 ] D h :. (x) . ^ <f>x . D : (x) : tf>x . D . >Jrx :.
[*10252] D h :.~( 3 x) . <f>x . D : (x) z <f>x ,D . yfrx D h . Prop
*10541. h <f>y . D y .p v yfry : = :p . v . <£y D„ yfry
Dem.
h . *4 2 . (*101) . D I- <£y . D y .p v yfry z = : (y) .~$yvp v >/ry :
[Assoc.* 10-271]
[* 10 - 2 ]
[(* 101 )]
(y) .pv~<f>yv yfry :
V- V •(!/)-~<l>y v^y :
P . V . 0y yfry ;. D 4 . Prop
The above proposition is only needed in order to lead to the following:
10-542. h :. </>y . D„ . p D yfry : = : p . D . <f,y D y yfry |^*10-541
This proposition is a lemma for *84 43.
*10-66. h :. (3X) . <f>x . yfrx z <f>x D z yfrx z = z (3X) . <f>x z <f>x D z yfrx
Devi.
V . *4-71 . D I - Z. tf*xD yfrx .Dz<f>x. yfrx . = . <f>x (1)
4.(1). *101127. D
I- :. <f>x D x yfrx . D : (x) : <f>x . yfrx . = . <f>x z
[*10 281] D : (3#) - <t>* • ^x . = . ( 3 X) . <f>x (2)
h. (2). *5-32. D4. Prop
This proposition is a lemma for *11712121.
MATHEMATICAL LOGIC
[PART I
150
*10 56. h <t>j ■. . yfra -: (g.r) . <*>.r . yx z D . (gj-) . yjrx . yx
Deni.
H . *10*31 . D h <f>.r . D z . \J,.r: Z> z <f>x. yx. D x . \f/x . yxz
[* 10*28] D : (g.r) . <f>x. yx. D . (gx).f,r.^ (1)
h . (1). Imp . D h . Prop
This proposition and *10 57 arc used in the theory of series (Part V).
*1057. H <frx . D x . >/*.r V yx :D z (fixD x yfrx . v . (gj*) . (f>X . yx
Don.
I-. *10 51 • Fact. D
h <f>x . D x . \frx v x J ’ : '^'(y* r ) • 4>x . yx zO z <f>x . D x . v yx z tf>x . D x .~x* :
[*10 20] D z <f>x. "5 X . \jrxv yx ,~yx z
[*5«1] Oz<f>x.-D x .ylrx (1)
h . ( I ). *5 6 . D h . Prop
*11. THEORY OF TWO APPARENT VARIABLES
Summary of *11.
In this number, the propositions proved for one variable in *10 are to be
extended to two variables, with the addition of a few propositions having no
analogues for one variable, such as *11 , 2‘21*23 24 and *1 1 -.'>3-55'6-7. "<f> (.r, //)"
stands for a proposition containing x and containing y; when .r and y are un¬
assigned, <f> ( x, y) is a propositional function of x and y. The definition *1 101
shows that “ the truth of all values of <£(x, y)” does not need to be taken as a
new primitive idea, but is definable in terms of " the truth of all values of yjr.v. u
The reason is that, when x is assigned, <£ (x, y) becomes a function of one
variable, namely y, whence it follows that, for every possible value of .r,
(y) • <f> (x, y)" embodies merely the primitive idea introduced in *9. But
"(y) . <f) ( x, y)” is again only a function of one variable, namely x, since y has
here become an apparent variable. Hence the definition *1101 below is
legitimate. We put:
*11 01. (x, y).<f>(x,y). = : (x) : (y) . <f> (x, y)
*11 02 . ( x , y, z) . 4> (. x , y,z). = : ( x ) : (y, z) . <£ (x, y, z)
*11 03. (gx, y) . <f> (x, y) . = : (gx): (gy) . *f> (x, y)
*11 04. (gx, y,z) . <f> (x, y,z).~i (gx) : (gy, z) . <f> (x, y, z)
*11 05. <f> (x, y) . D«.v - ^ {x, y) : =- : (x, y) : <f> (x, y) . D . jr (x, y)
*11 06. *f> (x, y) . . yfr (x, y) : - : (x, y) : <f> (x. y) . s . -jr (x, y)
All the above definitions are supposed extended to any number of variables
that may occur.
The propositions of this section can all be extended to any finite number
of variables; as the analogy is exact, it is not necessary to carry the process
beyond two variables in our proofs.
In addition to the definition *11 01, we need the primitive proposition
that "whatever possible argument x may be, <f> (x, y) is true whatever possible
argument y may be” implies the corresponding statement with x and y inter¬
changed except in u <f>(x, y)’\ Either may be taken as the meaning of
“<£>(x, y) is true whatever possible arguments x and y may be.”
The propositions of the present number are somewhat less used than those
of *10, but some of them are used frequently. Such are the following:
* 111 . b : (x, y) . <f> (x, y) . D . <f> ( z . w)
*11-11. If <f>(z,w) is true whatever possible arguments z and w may be, then
(x, y) . <p (x, y) is true
These two propositions are the analogues of *10-1-11.
Df
Df
Df
Df
Df
Df
MATHEMATICAL LOGIC
[PART I
1V2
*11''2- J (x, y) • <*> (x, y). = • (y,x). <£ <x, y)
/.e. to say that “for all possible values of .r, <f>(x,y) is true for all possible
values of y is equivalent to saying “ for all possible values of y, <£(x,y) is
true for all possible values of x.”
*113. h :.y>. D .(x.y). <f> (x, y) : = : (x, y) :p . D . <f> (x, y)
This is the analogue of *10*21.
*H 32. h (x. y): <f> (x. y). D . ^ (a*, y) : D : (x, y). </>(x, y) . D . (.r, y). (x,y)
/.e. "if <f> (x, y) always implies >^(x. y). then '</>(x, y) always' implies
''frU.'j) always. " This is the analogue of *10 27. *11*33*34 341 are respec¬
tively the analogues of *10*271*28*281, and are also much used.
*11 35. h (x, y): <f> (x. y). D . p : a : (gx, y). 0 (.r, y). D . />
/x. if 0(x, y) always implies /#, then if 0 (.r. y) is ever true,/) is true, and
vice versa. This is tin- analogue of *10*23.
*11 45. h (gx. y): /•. 0 (x. y) : = :/>: (gx. y). </> (x, y)
This is the analogue of *10*35.
*11 54. h (gx, y). <£x .yfry.m: (gx). <f>x : (gy). yfry
This proposition is useful because it analyses a proposition containing
two apparent variables into two pm|>ositions which each contain only one.
yfry” is a function of two variables, but is compounded of two functions
of one variable each. Such a function is like a conic which is two straight
lines: it may be called an “ analysable ” function.
*11*55. h (gx. y). <f>, . yfr (x, y). s : (gx) : <f>.r : (gy). (x, y)
I.e. to say " there are values of x and y for which tf>.r. yfr(x, y) is true ” is
equivalent to saying " there is a value of x for which rf>.r is true and for which
there is a value of y such that yfr (x, y) is true."
*116. h :: (gx)(gy) . <f> (x. y). yfry : (gy) (gx). <f> (x, y) .
This gives a transformation which is useful in many proofs.
*11 62. h :: <f>> . ^ (x, y). D,.„ . \ (*■!/) ! s <t> c • ^ ^ (x, y). D ;/ . * (x, y)
This transformation also is often useful.
*1101. (x, y). <f> (x. y) . = : (x) : (y) . <f> (x, y) Df
*1102. (x, y, 2 ).<£(x,y. *). = :(x):(y. r).<£(x,y, r) Df
*11 03. (gx, y). <f> (x, y). = : (gx) : (gy). <f> (x. y) Df
*11 04. (gx. y, 2 ). tf> (x, y.z). = : (gx) : (gy. z) . <f> (x, y. *) Df
*11 05. 0 (x, y). D xv . yfr (x, y): = : (x. y) z <f> (x, y). D . yfr (x, y) Df
*11*06. <f> (x, y). = Xt!# . yjr (x, y): = : (x, y) : <f> (x, y). = . (x, y) Df
with similar definitions for any number of variables.
*1107. “Whatever possible argument x may be, <*>(x, y) is true whatever
possible argument y may be ” implies the corresponding statement with xand
y interchanged except in “4>(x, y)". Pp.
SECTION Bj
THEORY OF TWO APPARENT VARIABLES
153
*111. h : (a?, y) . <p (i\ y) . D . <f> ( -, «»)
Dem.
h . *10*1 . Dh:Hp.D. (y) . <£ (j, y).
[*10*1] D . <f> (j, «>) : D h . Prop
*1111. If rf>(z , u>) is true whatever possible arguments c and w mav l>c, then
(.r, y) . <f> (x, y) is true.
Dem.
By *10*11, the hypothesis implies that (y).<f>(z,y) is true whatever
possible argument * may be; and this, by *10*11. implies (x, y) . </> (.r, y).
*1112. h (x, y) .p v <f>(x, y) . D :p . v . (x, y) .<*>(•**, y)
Dem.
I- . *1012 - 3 b (y) .p v (or, y) . D : /> . v . (y) . <f> (x, y)
[*10*11*27] D h (a:, y) . p v 0 (x,y) . D : (x) : y . v . (y) . <f> (x, y ):
[*10*12] D . v .(a:, y) . </>(x, y)D h . Prop
This proposition is only used for proving *11*2.
*1113. If <f> (5, #), >/r (5, #) take their first and second arguments respectively
of the same type, and we have “h . <f>(x,y)" and “h . >jr(x, y),” we shall have
" K <*>(*, y) . * (a:, y). M [Proof as in *10*13]
*1114. h (x, y) . <£ (a:, y) : (a:, y) . yjr (x, y) : D : <f> (z, w) . yfr (z, tv)
Dem.
I-. *10*14 . D f-:. Hp . D : (y) . </> ( z , y) : (y) . yfr (z, y)
[*10 14] D s <f> (z, w) . \Jr {z, w):.Dh. Prop
This proposition, like *10*14, is not always significant when its hypothesis
is true.^ *11*13, on the contrary, is always significant when its hypothesis is
true. For this reason, *11*13 may always be safely used in inference, whereas
*11*14 can only be used in inference (i.e. for the actual assertion of the con¬
clusion when the hypothesis is asserted) if it is known that the conclusion is
significant.
*11*2. h : (x, y) . <f> (x, y) . = . (y, x) . <f> ( x, y)
Dem.
h . *11*1 .
^ h : (x, y) ■ <^» (x, y) . D . (*, w)
a)
h .(1) . *11*07*11 . D 1- :.(w, z) : (x, y) . <f> (x, y) . D . <f> ( z , w)
(2)
h. (2). *11*12
— y) - </> y)l D .
V
h (x, y) . <£ (x, y) . D . (w, 2 ) . <f> (z, w)
(3)
Similarly
h ( w , *) .<j>(z,w) . D . (x, y) . </>(x, y)
(4)
h.(3).(4). D h . Prop
Note that “( w , z) . <f> ( z , w )” is the same proposition as “ (y, x) . <f> (x, y)”;
a proposition is not a function of any apparent variable which occurs in it.
154
MATHEMATICAL LOGIC
[PART I
* 11 * 21 . : (x, y.z).<f> (x, y,z). = . (y, z, x). <£ (x, y, z)
Dent.
[(*1101 02 )]H:: (x, y, z) . <f> (x, y, z ). = (x):. (y) :(z).<f> (x, y, z)
[*' !'2] = (y) ( x) :(z).<f> (x, y, z)
I* 1 1*2.*10*271 ] = :.(y):.(z) :(x) . </» (x, y, z)
[(*11 01 02)] = (y, z. x). <£ (x, y, z) :: D h . Prop
*11 22. h : (3./-, y ). $ (. r , y ). = . ^ <(x, y).^<f> (x, y)|
Dem.
h . *10 *252 . Transp . (*1103). D
: !/) • <t> (*• y) • = ((•**): ~ (ay). (^. y)|.
[* 10 252*271 ] s . ~ |(x) :(y).~4> (x, y)\ .
[(*1101)] s . ~ |(x, y). ^ <t> (x, y)]: D b . Prop
*11 23. b : (yx, y).<f>(x,y). = . (ay, x) . <f> (x, y)
Dem.
h . *11-22 . D b : (ax, y). 0 (x. y). = . ~ |(x, y). ~ <*> (x, y)\ .
[*11‘2.Transp] = . |(y, x).~<f> ( x , y)| .
[*11-22] = . (ay, x) . </> (x, y) : D b . Prop
*11-24. 1- : (ax, y, z) . <f> (x, y, z). = . (ay, z. x). <f> (x, y, z)
Dem.
[(*11 03 04)] b:: (3X. y. z). <f> (x. y. z). = (3*) ( 3 y) : ( 3 z).</, (x, y, z)
[*11-23] = (3y)( 3 x): ( a z). <f> (x, y, z)
[*11 -23.*10 281 ] = :. (ay)(az) : (ax). <f> ( x , y, z)
[(*110304)] = (ay, z,x).<f> (x, y, z) :: D b - Prop
*11 25. b : ~ |(ax, y). <f> (x, y)) . = . (x, y) . ~ <f> (x, y) [*1122 . Transp]
*11 26. b (ax) : (y). <f> (x, y) : D : (y) : (ax) . <f> (x, y)
Dem.
b . *10-1-28.3 b (3*) : (y) • (*, y): 3: (a*) • </> (*. y) 0)
h • (1) • *10*11*21 .D1-. Prop
Note that the converse of this proposition is false. E.g. let <f> (x, y) be the
propositional function “ if y is a proper fraction, then x is a proper fraction
greater than y.” Then for all values of y we have (&x). <f> (x, y), so that
(y) : (3*) • .V) is satisfied. In fact " (y) : (a*) • 4> (*• y)” expresses the
proposition: “ If y is a proper fraction, then there is always a proper fraction
greater than y.” But “i^x) z (y). <f> (x, y)” expresses the proposition: “There
is a proper fraction which is greater than any proper fraction,” which is
false.
*11 27. b (ax, y): (az) . <f>(x,y,z) z = : (ax) : (ay, z) . <f> (x, y, z) :
= :(3 x,y,z).<t>(x,y,z)
SECTION B]
THEORY OF TWO APPARENT VARIABLES
155
Deni.
K *4*2. (*11*03). D
h " (a *.!/) : (3-) • 4> (•*'.(a«) (gy) : (g :) . «#» (.r. y. .') (l)
K*4*2.(*ll*03). D
*■(ay): (a s > • <t> (•*•• '/• -) : = : (3/A 2 ) • <t> (•**. y- (-)
K (2). *10*11*281 . D
h :: (g.r) (gy) : (gj) . <f> (x, y. z) (g.r) : (gy. 2 ). <f> (.r, y. 2 ) (3)
h . (1) . (3). (*l 1 *04) .Dh. Prop
All the propositions of *10 have analogues which hold for two or more
variables. The more important of these are proved in what follows.
*11-3. I- :.p . D . (x, y) . <f> (x, y) s = : (x, y) zp . D . </> (.r. y)
Dern.
h .*10*21 . D h ;.p. 3. (x,y). <£(x, y): m : (x) :p. D .(y) .<f>(x,y):
[*10*21*271] s : (« v y) :/> • D . ^ (ir t y) D h . Prop
*11*31. h (a-, y).<f>(x,y): (x, y) . + (x, y) z = : (.r, y) : 0 (x, y). yjr (.r, y)
Here the conditions of significance on the right-hand side require that
<f> and yjr should take arguments of the same types.
Dem.
*" . *10 22 . D h :: (x, y) . <f> (x, y) : ( x, y) . \Jr (x, y) :
= (x) s. (y) . <f> (x, y) : (y) . >/r (x, y)
[*10*22*271] 5 ( x , y) : <£ (x, y) . (x, y):: D h . Prop
The proofs of most of the following propositions are conducted exactly ns
those of *11*3*31 are conducted: the analogous proposition in *10 is used
twice, together with *10*27 or *10 *271 or *10*28 or *10 281 as the case may
be. When proofs conform to this pattern we shall merely give references to
the propositions used.
*11*311. If <£(£,#), '/'*(£,$) take arguments of the same type, and we have
• <f>(x,y)” and “ I- . yfr (x, y),” we shall have “ h . </> (x, y) . yjr (x, y).” [Proof
as in *10*13.]
*11 32. h («, y) : <f> (x, y). D . yfr (x. y) : D : ( x, y) . <f> (x, y) . D . ( x, y) . yfr (x, y)
[*10*27]
*11*33. h (x, y) : <f> (x,y). = . yfr (x, y) : D : (x, y). <f> (x, y) . = . (x, y) . yfr (x, y)
[*10*271]
*11*34. h (x, y) : <f> (x, y). D . ^ (x, y) : D :
(a*, y) •</>(*» y) • 3 • (a*, y) • ^ (*. y) [* 10 * 27 * 28 ]
*11*341. h (x, y) s <f> (x, y). = . ir (x, y): D :
(a*» y) • £ to y) • = • (a*, y) • ^ (*. y) [*10 271 * 28 i ]
*11*35. h :-(x,y) s <p(x,y). D ,j>: = : (gx.y) . <f> (x,y) .D . p [*10*23 271]
*11 36. h : <£ («, w) . D . (gx, y) . <f> (x, y)
Dem.
H . *11*1 . D h : (x, y) . *>•> <f> (x, y) . D . ~ <f> (z, w) (1)
h . (1) . Transp . D H . Prop
150
MATHEMATICAL LOGIC
[part I
*11-37. b :: (x, y): 4* (x. y). D . (x, y)(x, y) : \fr (x, y).D.x(^-i/) : *
D:(x,y):0(x,y).D.x(^.y)
Dem.
In the following demonstration. " Hp” means the hypothesis of the propo¬
sition to be proved. We shall employ this abbreviation, whenever convenient,
in all cases where the proposition to be proved is a hypothetical, i.e. is of the
form " p D fj." Similarly "Hp (1)” will mean "the hypothesis of (1),’’ and
so on.
b .*11-31 . D b :: Hp . D :.(x,y)<f>(x, y) . D . '/'■(•r.y) : yjr (.r,y) • 3 • x(x.y) 0)
b . Syll . *1111 . D b :.(x,y) </>(.r.y). D . yjs (x.y): ^ (x,y). D . x( x -y) :
3:<Mx,y).3.x(* r *y) : *
[*11-32] Dh:.(x.y): </,(.r.y).D.>/r(x,y):>/r(x.y).D.x(^y) :
D : (x. y) : </> (.r, y). 3 . x (*• y) ( 2 >
b.(1).(2). Syll. DK Prop
The above is a type of proof which recurs frequently in what follows.
Proofs conforming to this pattern will be indicated only by the numbers of
the propositions used.
*11371. b :: (x, y): <f> (.v, y). = . yfr (x. y)(x, y): yjr (x, y) . = >x( x '
D (x. y): <f> (x, y). = . x (x..»/) [*U'31 *11 *33]
*11 38. b :: (.r, y): <f> (x, y) . D • y/r (x, y) D
(.r, y): </» (x. y) . x (x. y). 3 . («, y). x (x. y) [ Fftct • #1111 ’ 32 1
*11 39. b :: (x, y) : <f> (x, y). D . ^ (x, y)(x. y): x (x. y) • D • 6 (*» y> 3 : *
(x, y): «/> (x, y). x (x. y). 3 . ^ (x. y). 0 (x, y) [*3 47 .*1111 32]
*11-391. b :: (x, y): <f> (x. y). D . yjr (x, y) (x. y): <f> (x. y) . D . x (x. y) ! -
= : (x, y) z <f> (x, t/). D . yfr (x, y). x (x, y)
Dem.
b . *476 . Dh.(/>(x,y).D.| (x, y) : <£ (x, y) . D . x (•*’. y) :
= : <f> (x. y). D . + (x, y) . x (x, y) :•
[*1111 -33] D h (x. y) : 0 (x, y) . D . >/r (x, y) : <^> (x. y> . D . x (x. y) :
= : (x, y): <f> (x, y). 3 . + (x, y). x (*» y> ::
[* 1 131] Dh: (x, y): 4> (x, y). D . ^ (x, y)(x, y): <f> (x, y). 3. x (*• y)
= : (x. y) : «^ (x, y) . D . (x, y) . X (*- y> ::
D 1-. Prop
*11-4. b :: (x, y): *t> (x, y). = . ^ (x. y) (x, y) : x (*» y) • - • y) 3
(x, y) : </> (x, y) . x (x. y). = . ^ (x, y) . 0 (x, y)
Deni.
h . *11 31 . D 1 -:: Hp. D :. (x, y) :. <f, (x. y ). s . + (*, y): * (®. y) • = • e (*• !-
[*438.*1 111-32J 3 :. (*, y) : <f> (x, y ). x (*. y) • = • + (*. y) • 6 ( x < V) "
D I-. Prop
SECTION B] THEORY OF TWO APPARENT VARIABLES 157
*11*401. H :: (.r, y) z 0 (.r, y) . = .yfr (.r % y) : D
{.v, y ): 0 (. 1 *. y). x (.r, y > . = . 0 * //) . * . y) U 11 -4 * . 1 < 11
‘ *11-41. h (g.r, y) . 0 (.r, y) : v : (g.r, y). >/r y) :
s : (a-*'- i/) : <t> //) • v . yfr {.r, y) [*10 42 *2S I |
*11 42. h :. (g.r, y) . 0 (a*. y) . >/r (.r, y) . D : (g.r. y) . 0 (.r, y): (gj\ y) . 0 (.r, y)
[*10*5]
*11*421. h (.*, y) . 0 (.r. y). v . (.r, y). yjr (.r. y): D : (.r, y): 0 (.r, y) . v . 0 (.r, y)
[*■ >'+- ^ . lWp . *4 5,i]
*11 43. I- (gx, y): </>(.r,y) . D ./>: s : (x, y). £(x,.»/) . D ./> [*10-3+'2Sl]
*1144. H :• (x, y): ^ (x, y) . v . p s s : (x, y). $ (x, y) . v . y [*102-271]
*11'45. l-:.(ar,y):p.*(x.y): = :p:(ax,y).^(x,y) [*10'33'281]
*1146. h (gx. y) :y . D . <*> (x, y) : = :y . D . (gx, y) . $ (x, y) [*10-37-281]
*11-47. >* i. (x, y) : y . £ (x, y) : = : y : (x, y) . <fi (x. y) [*10-33-271]
*115. 1 -(a*) :~l(y) •■#>(*. y)l: = :~|(x,y).^>(x,y)j: = :(gx,y)— <t> (x, y)
Dem.
h . *10 253 . D h (gar) : ~ {(y) . 0 (x, y)) : 3 :~ \(x) : (y) . 0 (a:, y)| :
[(* 1101 )] = s ~ {(or, y). 0 (a:, (1)
h . *10*253 . D h |(y) . 0 (*. y)). = . (gy) —0 (*, y) :
[*10*11-281] D h (ga:) :~ |(y) . 0 (x, y» : = : (ga-) : (gy) .^<f>(x,y) z
[(*1103)] = :(g *.!/) • ^ <P (*,!/) (2)
K(l).( 2 ). Dh.Prop
*11 51. h (gar) : (y) . 0 (a:, y) : s {(*) : (gy) —0 (a:, y)|
Dem.
h • * 10 ' 252 . Transp .Dh:. (ga:): (y) . 0 (a:, y) : = : ~[(a-) : ^(y) . 0 (a:, y)] (1)
1-. *10*253 . D h ~(y) . 0 («, y) . = : (gy) . ~0 (a:, y)
[*1011*271] D h :.(a:):~(y).0(a;,y): 3 : (a:) : (gy) .~0 (a:, y)
[Transp] 3 h :—[(a:) : — {(y) . 0 (ar, y)|] . =:• \(x) : (gy).~0(a:, y)) (2)
^ • (1) • (2) . D h . Prop
*11*62. h (ga:, y) . 0 (a:, y) . 0- (x, y) . = -~ {(ar, y) : 0 (a:, y) . --0- (a:, y)J
Dem.
h. *4*51*62. D
1 - :*~| 0 (a:,y).^r(a:,y)) . = : 0 (a:, y).D— 0 (a:,y) ( 1 )
1-. (1). *11-11*33. D
h (a:, y)- (0 (x, y) . yfr (x, y )| : = : {x, y) : 0 (a:, y) . D —(a:, y) ( 2 )
h : (2) . Transp . *11 22 . D h . Prop
*11-621. H :—(ga:, y) . 0 (a:, y) —^ (a:, y) . = s (x, y) z 0 (x t y) . D . -0 (a:, y)
r*ll-52. Transp.
MATHEMATICAL LOGIC
[PART I
*11*53. H (x, y). <f>x 3 x/ry • = : (3*) . </>x . 3 . (y) . x/ry
I Jem.
h • *10*21*271 . 3 h <x, y) . </>.» 3 x/^y . = : (x): «/>x. 3 . (y) . x/ry :
[*10-23] s : (3-*). «#>-r. D . (y). x/ry 3 H . Prop
*11*54. H (gx, y). <f>j- . x/ry . = : (3 c) . <f>x z (gy). x/ry
Deni.
h. *10*35. 3 h (gy) . $x . x/ry. s : <£x s (gy). x/ry s.
[*10*11*281] 3 I-(gx, y). <t>x. ^y. s : (gx): <*>x : (gy). x/ry:
[*10-35] = : (3-'*) • +*: (3'/) • s. D H . Prop
This proposition is very often used.
*11*55. h (gx,'/). $x. x/r (x, //). 2 : (gx): 0x : (gy) . x/r (x. y)
Dem.
h . *10*35 .31-:. (gy). <f>r . x/r (x, y). s : </>x: (gy). x/r (.r, y)
[*1011] 3 h (x)(gy) . </>x . ^ (x, y) . s s $x : (gy) . x/r (x. y)
[*10-2*1] 3 h :.(gx):<gy). $x. x/r(x,y).s:(g.i:):$x:(gy).x/r(x,y):.3H.Prop
This proposition is very often used.
*11-56. H (x) . </>x: (y). x/ry : s : (x, y). </>x. x/ry
Dem.
H . *10-33 . D h :: (/) . tf>x z (y) . x/ry: = (x) <f>xz (y) . x/ry ( J )
h .*10*33. 3h <*>x :(y). x/ry s » : (y). </>x. x/ry
[*1011] 3 h (x)<£x : (y) . x/ry : = : (y). *f>x . x/ry ;.
[* 10 271 ] 3 H :: (x)</>x: (y). x/ry:. ■ : (x) s (y). </>r. x/ry:
[(*1101)] s : (x, y). </>x . x/ry (2)
h.(l).(2).Dh. Prop
*11*57. h : (x). 4>x . 5 . (x. y). tf>r . </>y [*11’56 . *4*24]
The use of *4 *2+ here depends upon the fact that (x). <f>x ami (y). 4>'J arc
the same proposition.
*11 58. h : (gx) .</>./-. = . (gx. y) . <f>x . «/>y [*11*54 . *4 24]
*11-59. h :. <f>x . 3 X . x/rx: 3 : «#>x . </>y . 3 X .„. x/rx . x/ry
Dem.
h .*11*57 . 3 h </>x. 3 X . x/rx : s : (x. y)
[*3-47. *11-32] 3:(x.y)
h - 9*111 . 3 h (x, y) : </>x. </>y . 3 . x/rx
<f>x . 3 . x/rx : </>y. 3 . x/ry :
$x. </>y. 3 . x/rx. x/ry (1)
x/ry : D : <£x. </>y . 3 . x/rx. x/ry (-)
h . (2) j . *4 24 . 3 h Hp (2) . 3 : </>x. 3 . x/rx
I-. (3). *10-11*21.3
h :. (x, y) : </>x . </»y . 3 . x/rx . x/ry : 3 : <f>x . 3 X . x/rx
h . (1) . (4) . 3 h . Prop
SECTION B]
THEORY OF TWO APPARENT VARIABLES
IV.)
*116. h :: (gx) (gy) . </>(**’, #/) . yfri/ : *.r (gy)(g.r). «/>(.,-. y). Y .,- : ^ v
This proposition is very frequently employed in subset] i lent proofs.
Deni.
h . *10*85 . Dh. (ay) . <f> (. v, //). ^-y : x-r : = : (gy) : <f> (x, y) . ^ry . Y .r
[*10*11*281] D h :: (gx) (gy) . <£ (x, y) . >fry : x .v :
= - (a-*)(ay) • <t> (•*•» y). . x- r: -
[*11 *23] = S. (gy) (gx) . 0 (.r. y) . >/ry . Y x
[*11*341.Perm] = (gy)(g.r) .</> (x, y) . Y x . ^y
[*10*35*281] = (gy) (g.,) . </> (a:> ,,) . : oh. p rop
*11 61. h (gy) : 0 .r . D x . (x. y) : D : «*>x . D x . (gy). yfr (x, y)
Deni.
K*11*26. DI-:iHp.Ds.(*)s.(gy)s^r.D. V'(^.y) ( 1 )
h . *10 37 . D H :.(gy): </>x . D . >/r (x, y) : D : <£x . D . (gy) . >/r(x.y)
[*10*11*27] D h (gy) 3. ^r(ar,y) : .D:.(x)s^r.D.(gy).a.(a:,y) ( 2 )
K(l).(2).Dh. Prop
*11 62. h :: 0 x . * (x, y) . D x> v . *(x.y) z = :. <f>x. D x : yf, (x, y). D„. Y (x, y)
Dem.
h . *4*87 . *11*11*33 . D
h !! ** • * <*»y) • D *.y • X<*>y) = = :• (*. y) </>x. D : (x. y). D . v (x,y)
[*10 21*11*271] = :. ( X ) <£x. D : (y) : ^ (x, y). D . ^ (x, y) ::
D h. Prop
*11 63. h ~(g*. y). <*> (x, y) . D : 0 (a:, y) . D x . y . * (x. y)
Dem.
J-. * 2*21 .* 11*11 . DH :.(x,y):— <f> (x, y) . D :<*>(*, y). D . ^ (x, y)
r ! ! o !3 D h (x ’ — <t>(x,y).D: <*> y) : <t> (*. y) - ^ ^ (x, y) :.
[*1 l* 2 oJ D I- :—(gar, y) . 0 (x, y) . D : (x, y) '• <f> (x, y) . D . \fs (x, y):.
*11 7. h :. (gx, y) : <£ (x, y). v. </> (y, x) : a . (gx, y) . <f> (x, y) D ^ ’ P, ° P
Dem.
h . *11*41 . D I-:. (gx, y) : (x, y) . v . <f> (y, x) :
= * <3*. y) • 0 (*. y) . v . (gx, y) . 0 (y, x) :
ri s : <3** y) • <t> (*. y) • V . (gy, X). 0 (y, x) :
1*4 25J = . (g*, y) . <f> (x, y) :. D I- . Prop
In the last line of the above proof, use is made of the“fuct that
(a*> y) • <f> (x, y) and (gy, x) . <f> (y, x)
are the same proposition.
The first use of the following proposition occurs in the proof of *234 12.
Its utility lies in its enabling us to pass from a hypothesis
4>z.X w - D *. «> - ir* •
containing two apparent variables, to the product of two hypotheses each
containing only one.
1 G 0
MATHEMATICAL LOGIC
[PART I
*1171.
H::(g^)
. « 2 >j :
(gw). \iv : 3 :.
<f>z . 3*. 02 : x 11 ' • <• • 0 w
• ““ •
• — •
02 .
XW. 3*, w .
0 - 2 . 9w
Dem.
h . *101
. *3-47 .
3h:.
02 . 3,. 02 : • #w :
3 : <pz .
■ XW
.3.
02 . 0 W
a)
1- .(1). *11-11*3.
3h:.
02 . 3, . 02 : xw • 3.c • #w :
3 : <f>z,
XW
. 02 . 0 W
( 2 )
h .* 10-1
. 3 h ::0.
’•X w
. 3,.„.02. 0w: 3 :. 02 . xw •
x.
**
. 0 w:.
[*10-28] 3 (gw). 4>z . x w • ^ • (H"') • y \ rz ' Gw '•
[*10-351 3:. 02 : (gw) .xu»:D:^: (gw). 0w
(3)
H . (3). Comm . *3 2<>.31-:: (gw) . \ w : ^ :• 4> z • X w • :
3 : 02 . 3 . 0-2 (*1)
h . (4). *1011 *21 .Dl*:: (gw) . x w • ^ 4> z • X w * ^.^• y f rz • dw:
3 : 02.3,. 02 (5)
Similarly I*:: (g*) 3 <t>* • x w • 3/.* • ^ z • :
3 : x'» • 3.<* • #w ( fi )
h .(5). (6). *3-47 .Comp. 3
h :: Hp . 3 :. 02 . \ ,u • 3*.* . 0* • 0w: 3 :02 .3,. 02 : x*0.3* . ^
h .(2).(7). 3 H . Prop
*12. THE HIERARCHY OF TYPES AND THE AXIOM
OF REDUCTIBILITY
The primitive idea "(a :). has been explained to mean "</>.,• is always
true, i.e. “all values of <f>.v are true." Rut whatever function <f> may be, there
will be arguments x with which <£./• is meaningless, i.e. with which as argu¬
ments <f> does not have any value. The arguments with which <f>.r has values
form what we will call the “range of significance’' of </>.»•. A "type'' is defined
as the range of significance of some function. In virtue of *91-1, if <£.,•,
and \frx are significant, i.e. either true or false, so is yfry. From this it follows
that two types which have a common member coincide, and that two different
types are mutually exclusive. Any proposition of the form (.r). i.e. any
proposition containing an apparent variable, determines some type as the
range of the apparent variable, the type being fixed by the function <f>.
The division of objects into types is necessitated by the vicious-circle
fallacies which otherwise arise*. These fallacies show that there must be
no totalities which, if legitimate, would contain members defined in terms of
themselves. Hence any expression containing an apparent variable must not
be in the range of that variable, i.e. must belong to a different type. Thus
the apparent variables contained or presupposed in an expression are what
determines its type. This is the guiding principle in what follows.
As explained in *9, propositions containing variables are generated from
propositional functions which do not contain these apparent variables, by the
process of asserting all or some values of such functions. Suppose <f>u is a
proposition containing a; we will give the name of generalization to the
process which turns <f>a into (x). <f>x or <gx). <f>x, and we will give the name
of generalized jiropositions to all such as contain apparent variables. It is
plain that propositions containing apparent variables presuppose others not
containing apparent variables, from which they can be derived by generaliza¬
tion. Propositions which contain no apparent variables we call elementary
propositions +, and the terms of such propositions, other than functions, wc call
individuals. Then individuals form the first type.
It is unnecessary, in practice, to know what objects belong to the lowest
type, or even whether the lowest type of variable occurring in a given context
is that of individuals or some other. For in practice only the relative types
of variables are relevant; thus the lowest type occurring in a given context
may be called that of individuals, so far as that context is concerned. Accord¬
ingly the above account of individuals is not essential to the truth of what
• Cf. Introduction, Chapter II.
t Cf. pp. 91. 92.
R& W I
11
1G2 MATHEMATICAL LOGIC [PARTI
follows: all that is essential is the way in which other types are generated
from individuals, however the type of individuals may be constituted.
By applying the process of generalization to individuals occurring in
elementary propositions, we obtain new propositions. The legitimacy of this
process requires only that no individuals should be propositions. That this is
so, is to be secured by the meaning we give to the word individual. We may
explain an individual as something which exists on its own account; it is then
obviously not a proposition, since propositions, as explained in Chapter II ot
the Introduction (p. 43). are incomplete symbols, having no meaning except
in use. Hence in applying the process of generalization to individuals we run
no risk of incurring reflexive fallacies. We will give the name of first-order
prajmsitions to such as contain one or more apparent variables whose possible
values are individuals, but contain no other apparent variables. First-order
propositions arc not all of the same type, since, ns was explained in *9, two
propositions which do not contain the same number of apparent variables
cannot be of the same type. But owing to the systematic ambiguity of nega¬
tion and disjunction, their differences of type may usually be ignored in practice.
No reflexive fallacies will result, since no first-order proposition involves any
totality except that of individuals.
Let us denote by '* <f> ! x" or "</>! (?, y)" or etc. an elementary function whose
argument or arguments are individual. We will call such a function a predi-
cative fmiction of an individual. Such functions, together with those derived
from them by generalization, will be called first-order functions. In practice
we may without risk of reflexive fallacies treat first-order functions as a type,
since the only totality they involve is that of individuals, and, by means of the
systematic ambiguity of negation and disjunction, any function of a first-order
function which will concern us will be significant whatever first-order function
is taken as argument, provided the right meanings are given to the negations
and disjunctions involved.
For the sake of clearness, we will repeat in somewhat different terms our
account of what is meant by a first-order function. Let us give the name of
matrix to any function, of however many variables, which does not involve any
apparent variables. Then any possible function other than a matrix is derived
from a matrix by means of generalization, i.e. by considering the proposition
which asserts that the function in question is true with all possible values or
with some value of one of the arguments, the other argument or arguments
remaining undetermined. Thus e.g. from the function <f> ( x , y ) we shall be able
to derive the four functions
lx). <f> (x, y), ( 3 *) . <p Or, y), (y) . <f> (x, y), (ay) • <t> (*. !/)•
of which the two first are functions of y, while the two last are functions of x.
(All propositions, with the exception of such as are values of matrices, are also
derived from matrices by the above process of generalization. In order to obtain
SECTION B]
THE AXIOM OF KKIH'CI BII.ITY
\c,:\
a proposition from a matrix containing // variables, without assigning values
to any of the variables, it is necessary to turn all the variables into apparent
variables. Thus if <f> (.r, y) is a matrix. (.»•. //) y) is a proposition.) \\V
will give the name rirst-ori/er matrices to such as have only individuals for
their arguments, and we will give the name of first-order /'auctions (of anv
number of variables) to such as either are first-order matrices or are derived
Irom first-order matrices by generalization applied to some (not all) of the
arguments to such matrices. First-order propositions will be such as result
from applying generalization to alt the arguments to a first-oixler matrix.
As we have already stated, the notation “<*>! 2" is used for any elementary
function of one variable. Thus "<f >! . 1 ” represents any value of any elementary
function of one variable. It will be seen that “<*>!.r” is a function of two
variables, namely <f >! 3 and x. Since it contains no apparent variable, it is
a matrix, but since it contains a variable (namely <f> ! 2) which is not an in¬
dividual, it is not a first-order matrix. The same applies to <f> ! a, where a is
some definite constant. We can build up a number of new matrices, such as
~<£!a, ~<f>lx, <f> l x v </>! y, tfilxvyfrlx, (filxVyfrly,
<b l X . "5 . yfr l x, <f>lx .yfrlx, 4> l x V >fr l y v %'• *, and so on.
All these are matrices which involve first-order functions among their argu¬
ments. Such matrices we will call second-order matrices . From these matrices,
by applying generalization to their arguments, whether to such as are functions
or to such (if any) as are individuals, we obtain new functions and propositions.
Such functions (together with second-order matrices) will be called second-
order /unctions, and such propositions will be called second-order propositions.
Thus we are led to the following definitions:
A second-order matrix is one which has at least one first-order matrix
among its arguments, but has no arguments other than first-order matrices
and individuals.
A second-order function is one which cither is a second-order matrix or
results from one by applying generalization to some (not all) of the arguments
to a second-order matrix.
A second-order proposition is one which results from a second-order matrix
by applying generalization to all its arguments.
In addition to the above illustrations of second-order matrices, we may
give the following examples of second-order functions:
(1) Functions in which the argument is <f> ! 2 : (x) . <J> l x, ( 34 :) . 0 ! x ,
<f> l a . D . <£ l b, where a and 6 are constants, <f >! * . D x . g \ x , where o! 2 is a
constant function, and so on.
( 2 ) Functions in which the arguments are <f >! 2 and yfrl 2:
<plx yfrlx, 4>lx.= x .yfrlx, ( 3 *) . <f>x . yfrx, <f> l a . D . yfr l b,
where a and b are constants, and so on.
11—2
Ifi4 MATHEMATICAL LOGIC [PARTI
(3) Functions in which the- argument is an individual x: (<f>).<t>lx,
('•!</>). <J>! x, 4 >! .r . . <f >! a, where a is constant, and so on.
<4» Functions in which the arguments are $ ! 3 and .r: ! .r, <f >! x . D . <f >! a,
where a is constant, (3 : <f> !.r. = . \fc !.#•, and so on.
Examples of secnnd-ord«-r functions might, of course, be multiplied in¬
definitely, but the above seem sufficient for purposes of illustration.
A second-order matrix of one variable will be called a predicative second-
order fn net inn of one ca viable or a predicative function of a first-order matrix.
Thus </>! a, p ! a and <f >! n D <f >! h are predicative functions of <p ! 2 . Similarly
a function of several variables of which at least one is a first-order matrix,
while the rest are either individuals or first-order matrices, will be called
predicative if it is a matrix.
It will be seen, however, that a second-order function may have only
individuals for its arguments; instances were given just now under the
heading (3). Such functions we shall not call predicative, since predicative
functions of individuals have already been defined as being such ns are of the
first order. Thus the order of a function is not determined by the order of its
argument or arguments; indeed, the function may be of any order superior to
the order or orders of its arguments.
A variable matrix whose argument is <f >! * will be denoted by fl <f >! 5, and
generally, a matrix whose arguments are 0 ! 3, yfr ! 3,... x, //. ... (where there is
at least one function among the arguments) will be denoted by
/!(<*>! 3. yfr ! 3, ... j-. 1 /. ...).
Such a matrix is not of the first or second order, since it contains the new
variable whose values are second-order matrices. We proceed to construct
new matrices as we did with the matrix <t> ! .7; these constitute third-order
mat rices. These together with the functions derived from them by generali¬
zation are called third-order Junctions, ami the propositions derived from third-
order matrices by generalization are called third-order propositions.
In this way we can proceed indefinitely to matrices, functions and propo¬
sitions of higher and higher orders. We introduce the following definition:
A function is said to Ik* predicative when it is a matrix. It will be
observed that, in a hierarchy in which all the variables arc individuals or
matrices, a matrix is the same thing as an elementary function (cf. pp*
127, 128).
"Matrix” or “predicative function ” is a primitive idea.
The fact that a function is predicative is indicated, as above, by a note ot
exclamation after the functional letter.
The variables occurring in the present work, from this point onwards, will
all be either individuals or matrices of some order in the above hierarchy.
Propositions, which have occurred hitherto as variables, will no longer do so
SECTION B]
THE AXIOM OK REIH’CIBILITY
10 r>
except, in a few isolated cases of which no subsequent use is made. In practice,
for the reasons explained on p. 10*2. a function of a matrix may he regarded
as capable of any argument which is a function of the same order and takes
arguments of the same type.
In practice, we never need to know the absolute types of our variables, but
only their relative types. That is to say. if wo prove any proposition on the
assumption that one of our variables is an individual, and another is a function
of order n, the proof will still hold if. in place of an individual, we take a
function of order m, and in place of our function of order n we take a function
of order n + m, with corresponding changes for any other variables t hat may
be involved. This results from the assumption that our primitive propositions
are to apply to variables of any order.
We shall use small Latin letters (other than /). 7 , /•, s) for variables of the
lowest type concerned in any context. For functions, we shall use the letters
<t>> X> 0>f F (except that, at a later stage. F will be defined ns a constant
relation, and 6 will be defined ns the order-type of the continuum).
We shall explain later a different hierarchy, that of classes and relations,
which is derived from the functional hierarchy explained above, but is more
convenient in practice.
When any predicative function, say <f >! 2, occurs as apparent variable, it
would be strictly more correct to indicate the fact by placing '•(<*>! 2)’* before
what follows, as thus: "(<£! 2). /(<*>! 2). M Hut for the sake of brevity we
write simply "(<£)” instead of "(<*» ! 2). M Since what follows the <fi in brackets
must always contain <f> with arguments supplied, no confusion can result from
this practice.
It should be observed that, in virtue of the manner in which our hierarchy
of functions was generated, non-predicative functions always result from such
as arc predicative by means of generalization. Hence it is unnecessary to
introduce a special notation for non-predicative functions of a given order and
taking arguments of a given order. For example, second-order functions of an
individual x are always derived by generalization from a matrix
/!(<£! 2, yfr ! 2, ... x, y, 2 . ...),
where the functions /, <f>, \fr, ... are predicative. It is possible, therefore, without
loss of generality, to use no apparent variables except such as are predicative.
We require, however, a means of symbolizing a function whose order is not
assigned. We shall use u <f>x" or "fix ! *)” or etc- to express a function (<f> or/)
whose order, relatively to its argument, is not given. Such a function cannot
be made into an apparent variable, unless we suppose its order previously fixed.
As the only purpose of the notation is to avoid the necessity of fixing the order,
such a function will not be used as an apparent variable; the only functions
which will be so used will be predicative functions, because, as we have just
seen, this restriction involves no loss of generality.
MATHEMATICAL LOGIC
[PART I
l r,c>
\\ o have now to state and explain the axiom of red ucibility.
It is important to observe that, since there are various types of propositions
ami functions, ami since gon<-rn fixation can only be applied within someone
type (or. by means of systematic ambiguity, within some well-defined and
completed set of types), all phrases referring to "all propositions” or "all
functions,” or to “some (undetermined) proposition”or"somo (undetermined)
function,” are prim a facie meaningless, though in certaincases they are capable
of an unobjectionable interpretation. Contradictions arise from the use of
such phrases in cases where no innocent meaning can be found.
affecting the truth or falsehood of its values. This seems to be wlmt common-
sense effects by the admission of classes, (liven any propositional function
of whatever order, this is assumed to be equivalent, for all values of .r, to a
statement of tin- form **.r belongs to the class a." Now assuming that there
is such an entity as the class a. this statement is of the first order, since it
involves no allusion to a variable function. Indeed its only practical advantage
over the original statement \}r.c is that it is of the first order. There is no
advantage in assuming that there really are such things as classes, and the
contradiction about the classes which are not members of themselves shows
that, if there are classes, they must be something radically different from in¬
dividuals. It would seem that the sole purpose which classes serve, and one
main reason which makes them linguistically convenient, is that they provide
a method of reducing the order of a pmpositioual function. We shall, therefore,
not assume anything of what may seem to be involved in the common-sense
admission of classes, except this, that every propositional function is equivalent,
for all its values, to some predicative function of the same argument or argu-
men ts.
This assumption with regard to functions is to be made whatever may be
the type of their arguments. Let fu be a function, of any order, of an argument
ii, which may itself be either an individual or a function of any order. If/*s
a matrix, we write the function in the form flu; in such a case we call / a
predicative function. Thus a predicative function of an individual is a first-
order function; and for higher types of arguments, predicative functions take
the place that first-order functions take in respect of individuals. We assume,
then, that every function of one variable is equivalent, for all its values, to
some predicative function of the same argument. This assumption seems to
be the essence of the usual assumption of classes; at any rate, it retains as muo i
SECTION b]
THE AXIOM OF REIU'CI HILITV
U',7
of classes as we have any use for.and little enough to avoid ilm emit radiet ions
which a less grudging admission of classes is apt. to entail. We will call this
assumption the axiom of classes, or the axiom of reducibilit //.
We shall assume similarly that every function of two variables is equivalent,
for all its values, to a predicative function of those variables, i.c. to a matrix.
This assumption is what seems to be meant by saying that any statement about
two variables defines a relation between them. We will call this assumption
the axiom of relations or (like the previous axiom) the axiom of redncibilit •/.
In dealing with relations between more than two terms, similar assumpt ions
would be needed for three, four, ... variables. But. these assumptions are not
indispensable for our purpose, and are therefore not made in this work.
Stated in symbols, the two forms of the axiom of reducibility are as follows:
*121- Ma/> :**•■••/!* p P
*1211. h: (a/) : <f> (*, y) . =,, v ./! (x, y) Pp
We call two functions <f>x, \fr$ formally equivalent when <f>.v. =, . \Jr.c, and
similarly we call <f> (5, p) and yjr(£, p) formally equivalent when
</> (x, y) . . yfr (x. y).
Thus the above axioms state that any function of one or two variables is
formally equivalent to some predicative function of one or two variables, as
the case may be.
Of the above two axioms, the first is chiefly needed in the theory of classes
(#20), and the second in the theory of relations (*21). But the first is also
essential to the theory of identity, if identity is to be defined (as we have done,
in *13*01); its use in the theory of identity is embodied in the proof of *13101,
below.
We may sum up what has been said in the present number as follows:
(1) A function of the first order is one which involves no variables except
individuals, whether as apparent variables or as arguments.
(2) A function of the (a + l)th order is one which has at least one argument
or apparent variable of order n, and contains no argument or apparent variable
which is not either an individual or a first-order function or a second-order
function or ... or a function of order n.
(3) A predicative function is one which contains no apparent variables,
i.e. is a matrix. It is possible, without loss of generality, to use no variables
except matrices and individuals, so long as variable propositions are not
required.
(4) Any function of one argument or of two is formally equivalent to a
predicative function of the same argument or arguments.
*13. IDENTITY
Summary o f *13.
The propositional function ‘V is identical with y ” will be written “.r = y."
We shall find that this use of the sign of equality covers all the common uses
of equality that occur in mathematics. The definition is as follows:
*13 01. x *=;/. = : (<£): 0 ! x . D . <f> ! y Df
This definition states that .r and y are to be called identical when every
predicative function satisfied by x is also satisfied by y. We cannot state that
every function satisfied by x is to l>e satisfied by y, because x satisfies functions
of various orders, and these cannot all be covered by one apparent variable.
Hut in virtue of tin- axiom of roducibility it follows that, if x*= y and a*satisfies
\Jsx, where ^ is any function, predicative or non-predicative, then y also satisfies
yjry (cf. *13 101, below). Hence in effect the definition is as powerful as it
would be if it could be extended to cover all functions of .r.
Note that the second sign of equality in the above definition is combined
with “Df,” and thus is not really the same symbol as the sign of equality
which is defined. Thus the definition is not circular, although at first sight
it appears so.
The propositions of the present number are constantly referred to. Most
of them are self-evident, and the proofs offer no difficulty. The most important
of the propositions of this number are the following:
*13101. h : x = y . D . yJrxO >\ry
I.e. if x and y are identical, any property of x is a property of ;/.
*1312. h : x = y . D . \frx = yfry
This includes *13101 together with the fact that if x and y are identical
any property of y is a property of x.
*13T5T6T7, which state that identity is reflexive, symmetrical and transitive.
*13191. h y = .r. D„ . <f>y : s . tf>x
I.e. to state that everything that is identical with x has a certain property
is equivalent to stating that x has that property.
*13195. b:( 3 y).y = x.<t>y. = .<f>x
I.e. to state that something identical with x has a certain property is
equivalent to saying that x has that property.
*13*22. h : (a*. w).z = x.w = y.<p(z,w).==.4>(x,y)
This is the analogue of *13’195 for two variables.
IDENTITY
SECTION B]
160
*13 01. ;f = y . = : (<f>) : <f >! .«■. 3 .0 !y 1)1’
The following definitions embody abbreviations which are often convenient.
*13*02. # + y • = • = y) Pf
*13 03. a' = y = z . = . .«• — y. i/ = r 1)1
*131. h a? = y. = : 0! . 3* . 0! y [*4*2 . (*13 01). (*10 02)]
*13*101. h : a* = y . 3 . >/r.r 3 >/ry
Dem.
h . *12*1 . Dh (g0) y/r.r . = .</»! .r: >Jry . = . 0 ! y (1)
h . *13*1 . 3 h :: Hp. 3 0! .r. 3$ . 0! y
[*4*84*85.*10*27] 3 >/r.r . = . 0 ! .r : 0-y . = . 0 ! y : 3* : 0.r . 3 . 0y
[*10-23] 3 (a0): 0* . = . <f> lx: yjri/ . = . 0! y: 3: 0\r. 3 . 0-y (2)
h . (1) . (2) . 3 1-. Prop
In virtue of this proposition, if .r — y. y satisfies any function, whether
predicative or non-predicative, which is satisfied by .r. It will be observed
that the proof uses the axiom of reducibility (*12*1). But for this axiom, two
terms x and y might agree in respect of all predicative functions, but not in
respect of all non-predicative functions. We should thus be led to identities
of different degrees, according to the degree of the functions in respect of
which x and y agreed. Strict identity would, in this case, have to be taken as
a primitive idea, and *13*101 would have to be a primitive proposition, as would
also *13*15*16-17.
*1311. h x y . = :</>! x . =* . 0 ! y
Dem.
H. *10 22. 3t-:.0!e.s»
[*13*1]
h . *13*101 . 3h:.x
h . *13*101 . *1*7 . 3h:.x = y.3
[Transp]
h . (2) . (3) . Comp . 3 h : x = y .
[*10*11*21] 3H:.x =
h . (1) . (4) . 3 h . Prop
*1312. h : x = y . 3 . yfrx = yjri/
Dem.
h . *13*101 . Comp . 3 h s ft*
[Transp]
*1313. h : yfrx . x ■* y . 3 . 0y
*13*14. 1-: yfrx . ~>/ry . 3 . a: =$= y
*1316. \-.x = x
*13*16. V :x = y
. 0 ! y : 3 : 0 ! x . 3* . 0 ! y :
3 : .r = y
y . 3 . 0 ! x 3 0 ! y
^0 ! x 3 ^0 ! y .
3.0 ! y 3 0 ! x
3.0 ! x = 0 ! y :
y.3:0!x.=*.0!y
y . 3 . yjfx 3 0*y . ~ 0x 3 ~ 0y .
3 . i/rx = 0-y :3h. Prop
[*13*101 . Comm . Imp]
[*13*13. *4*14]
[Id. *10 11 .*13 1]
[*13*11 .*10*32]
o)
( 2 )
(3)
(4)
y = *
170
MATHEMATICAL LOGIC
[PART I
*1317. h : X = y.y = z.D.x = z
Dem.
h . *131 . D H :: Hp . D <f >! .r. . <£ ! y : 0 ! y . D* . <f >! 2
[*10 3] D <f >! x . . <f >! s :: D 1-. Prop
In the* above use of *10 3. <f >! .r, <f >! y. <f >! z are regarded as three different
functions of if>, and 0 replaces the x of *10*3.
The above three propositions show that identity is reflexive (#1315),
symmetrical (*1310). and transitive (*1317). These are the three marks of
relations having the formal properties which we associate commonly with the
sign of equality.
*13171. Ih:x-y.x«*. D.y-s [*1316 17]
*13 172. I- : y = x. * = x. D . y = * (*131617]
*1318. h:xay./ + r.D.// + ; [*13’17 .*4'14]
*13181. h:x-y.y + z.D.x+* [*13171 .*414]
*13182 h :..r- if . D :* = .r. = .r = y [*1317172 . Exp.Comp]
*13 183. h :. x = y . = : z = = y
Dem.
K *13182. *10 11
21 . D H:..r«y
.Dj x-y
a)
h . *10*1 . Dbz.;
: =* x . =,. z * y
: D : x = x . D . x = y :
[*1315]
D: x * y
(2)
h .(1).(2). D h . Pi
•op
*1319. h
•(g^).y-x [*13 15. *10 24]
*13191. h
:. y - x . . <£y : = .
4>x
Dem.
K*101. DH
y-ar.D„.^y:
D : x = x. D . <£x:
[*1315]
D: 0x
(1)
h .*13 12. Dh:.
y = x . D : 0.r.
3.<f>yz.
[Comm] D h :.
<f>x . D z y = x .
D.<f>y:.
(2)
[*10 11-21] Dh:.
$x.D:y-x.
-\-4>y
V .(1).(2). D h . Prop
This proposition is constantly used in subsequent proofs.
*13192. K (gc )z x = b. = x . x =• c z yfre z = . yfrb
Dem.
V . *4 2 . *3 2 . 0 \- zz yjrb . O z. x = b . = x . x = b z yfrb z.
[*10 24] D:.(gc):a- = 6.s,.a: = c:^c (1)
H . * 101 . D h * - 6. s,. x - c s yfre : D : 6 = 6. = . b = c s yfre s
[*5501 .*1315] Dzb = c.yfrcz
[*13T3] 3:
I-. (2) . *1011-23 .Dh. (gc) : x = b . =,. x = c : yfre z D . yfrb (3)
I-.(1).(3). D h . Prop
This proposition is useful in the theory of descriptions (*14).
SECTION B]
IDENTITY
171
*13193. b z
<f>.v
II
III
•
•
x = y
Dem.
*
b • Simp .
D b : <f>.r . x = y . D . .v = y
0)
b. *1313.
D b : <p.v . x = y . D . <f>y
(2)
b . (1) . (2) . Comp . D b: (f>.r . .r = y . D . <f>y ..v = y
(3)
b . *1316 . Fact
D b : (f>y .x = y . D . <f>y . y = .r .
~(3)^1
x > !/J
D . </>.<•. y = .c.
*13 16. Fact]
D . <f>.v . x = y
(4)
b . (3) . (4) . D b .
Prop
This proposition is very often used.
*13194. b :
<px . x = y .= . <f>x .
<f>y . x = y [*1313. *4-71]
This proposition is used in
*37 65 and *10114.
*13 195. b :
:(a y).y = x.<t>!/
. <f>x
Dem.
b. *3 2. *1315.
D b : <f>x . D . x — * . <px .
[*1024]
D • (a^) -!/~x-<t>y
(1)
b . *13*13 . *1011.3b:. ( y) z y = x . <f>y . D . <f>. v z
[*10-23]
3 •“ :• (ay). y = x . <f>y . D . <f>x
(2)
b.(l).(2).
D b . Prop
The use of
this proposition
in subsequent proofs is very frequent.
*13196. b
z. ~<f>x . = z <f>y . D v .
y^x [*13195 . Transp . *1051]
*1321. b
z
= x.w = y.D Z)U ,
.<t>(z,w)z = .<f>(x, y)
Dem.
b. *11-62. D
h :: t *- x . w -■ y . <f> (s t w) : a :• * — a:. D, 2 to — y. D w . 0 (*, w) s.
[*13-191] = u; =-y. D,„. <f) (x, w)
[*13T91] = :. <£ (x, y) :: D I-. Prop
This proposition is the analogue, for two variables, of *13191.
*13 22. b : ( 32 , w) . z = x . w = y . <}> (z, w) . = . <f> (x, y)
Dem.
1-. *1155 .Dbz. (3 z, w ). z = x . w = y . <f> (z, w) .
= : (a*) zz = x: (aw) .io = y .<f>(z,w)z
[*13195] = : (aw) . w = y . <f> (x, w) z
[*13T95] = : <f> (x, y) D b . Prop
This proposition is the analogue, for two variables, of *13195. It is fre¬
quently used, especially in the theory of couples (*54, *55, *56).
The following proposition is useful in the theory of types. Its purpose is
to show that, if a is any argument for which “ <f>a ” is significant, i.e. for which
we have (f>a v~ tf>a, then "<f>x” is significant when, and only when, x is either
172
MATHEMATICAL LOGIC
[PART I
identical with a or not identical with a. It follows (as will be proved in *20*81)
that, if and are both significant, the class of values of.r for which
is significant is the same as the class of those for which “ yjrx” is signi¬
ficant, i.e. two types which have a common member are identical.
In the following proof, the chief point to observe is the use of *10 221.
There are two variables, a and .«*, to be identified. In the first use, we depend
upon the fact that 4>o and .r = a both occur in both (4) and (5): the occurrence
of (f)ii in both justifies the identification of the two </'s, and when these have
been identified, the occurrence of x = n in both justifies the identification of
tin* two ./*. (1’nless the iTs had been already identified, this would not be
legitimate, because “.r = «' is typically ambiguous if neither .r nor a is of
given type.) The M-coiid use of *10*221 is justified by the fact that both <fm
ami 4 >r occur in both <2t and (G).
*13 3. b ::</>•» v ^ </>»/. D <t».r v . = : x = a . v . x + a
I fern.
b • * 2 * 11. D h . <f>.r v (f>.i (1)
b . (1). Simp . D b : <f>u v ^ <f>a . D . <f>x v ~(f>x (2)
b. *2* 11 . Db:j' = d.v.j + a (3)
b. (3). Simp. D b :. <f>.i . D —a . v .« + a (**)
b . *13 101 . Comm . D b <f>n v ^ <f>»i .D:x = o.D. <f>x v *>*<l>x (5)
b . (4) . (5) . *10*13*221 . D
b :: (f>ti v *— • 0// . D : .r = </ . v .. r + n :. tfxi . D s .r — a . D . <f>rv ~<f>r (C)
b.(2).(6).*10*13*221 . D
b :: <f)ii v^<fxi . D . <f>xv^<f>xz. <f)a .D:ar*»a.v..r :.
<f>a v~<j>a . D :.r=»o . D . <f>xv~<t>* (7)
b . (7). Simp . D
b :: <f)d v ~<f>a . D . :. <fxi .D:x = a.v..r+a (3)
b . (8). *5*35 . D b :: <fta v~<f>a . D s. ^.r v . =: x = a . v •a? + a ::
D b. Prop
#14. DESCRIPTIONS
Summary o f #14.
A description is a phrase of the form “ the term which etc.." or. more
explicitly. “ the term .r which satisfies where <p.7 is some function satisfied
by one and only one argument. For reasons explained in the Introduction
(Chapter III), we do not define “ the .r which satisfies hut we define any
proposition in which this phrase occurs. Thus when we say : “ The term .v
which satisfies <f>x satisfies yfrx,” we shall mean: “There is a term b such that
is true when, and only when, x is b, and yfrb is true.” That is. writing
" Ox) (<t>x) ” for “ the term x which satisfies <pj ," yfr (?.i) ( <f>.t ) is to mean
(3^) : <t>- c . . x = 6 : yfrb.
This, however, is not yet quite adequate as a definition, for when ( ix)(<f>,v)
occurs in a pro|>osition which is part of a larger proposition, there is doubt
whether the smaller or the larger proposition is to be taken as the “ yfr (/./•)(</>./•).”
Take, for example, yfr (ix) (<f>x) . D . p. This may be either
(31&) : 4>x • = x . x — b : yfrb : D . p
or (gfc) <fjx . s, . x = b : yfrb . D . p.
If “ (gt) : <f>x . =*. x — b ” is false, the first of these must be true, while the
second must be false. Thus it is very necessary to distinguish them.
The proposition which is to be treated as the “ yfr (ix)(<f>x) ” will be called
the scope of ( ix)(<f>x ). Thus in the first of the above two propositions, the
scope of (ix)(<f>x) is yfr (ix) (<f>x), while in the second it is yfr (ix) (<f>x) .Z). p.
In order to avoid ambiguities as to scope, we shall indicate the scope by
writing “ [(ix) (<£*•)] ” at the beginning of the scope, followed by enough dots
to extend to the end of the scope. Thus of the above two propositions the
first is
[Ox)(<t>x)].yfr(ix)(<f>x). D.p,
while the second is
[(**) (£*)] : ^ Ox) (<f>x) . D . p.
Thus we arrive at the following definition:
#14 01. [Ox) (<f>x)] . yfr (ix) (<f>x) . = : (^b) : <f>x . = x . x = b : yfrb Df
It will be found in practice that the scope usually required is the smallest
proposition enclosed in dots or brackets in which " (ix) (tf>x) ” occurs. Hence
when this scope is to be given to (ix)(<f>x), we shall usually omit explicit
mention of the scope. Thus e.g. we shall have
a + (ix) (<px) . = : ( a &) z<f>x .= x .x = b-.a^b,
~l a = Ox) (<f>x )}. = . ~ l(g6) : <f>x . = x . x = 6 : a = 6j.
171 MATHEMATICAL LOGIC [PARTI
Of these the first necessarily implies (yfc): <f>.r. = x . x = h, while the second
does not. We put
*14 02. E !( hr) (<f>r) . = : (%|& ):<f>r. = x . .#• = b Df
This defines: “The x satisfying <f >.7 exists.” which hohls when,
when, <f>7 is satisfied by one value «>f x and by no other value.
When two or more descriptions occur in the same proposition, there is
m-ed of avoiding ambiguity a* to which has the larger scope. For this purpose,
we put
*14 03. [( U’) (</>•'*), (l.r)( yjrx ■)]./(< tx)(<f>x). (l^)(^rjr)| . = :
(( U H</m )] : [(U)(>/r.r)] ./{( ix)(<f>.r), (l.r)(>/r.r)j Df
It will be shown (*14 113) that the truth-value of a proposition containing
two descriptions is unaffected by the question which has the larger scope.
Hence- we shall in general adopt the convention that the description occurring
first typographically i> t«. have the larger scope, unless the contrary is expressly
indicated. ' 1 ‘hus e.y.
(ix)(<^r)-(/x)(^x)
will mean ( 36 ): <f>x . = x . x — b z b = (lx)( yfr.r),
»>. t^):. <*m . ( 3 c): yfrx. m x .x-czb-c.
By this convention we are able almost always to avoid explicit indication of
the order of elimination of two or more descriptions. If, however, we require
a larger scope for the later description, we put
*14 04. [(lx)(^)]./Uir)(^r). (ix)(>/rx» . = .
[(lx)(^r.r), (l*)(£r)]./{(?*)(£.r), (u)(^x)| Df
Whenever we have E!(/.r)(</>.r),(i.r)(^>.r) behaves, formally, like an ordinary
argument to any function in which it may occur. This fact is embodied in
the following proposition:
*14 18. h E ! (ix)(tf>x ). D : (x). ^.r. 0 . \\r (ix){<f>x)
That is to say. when (ix)(<f>x) exists, it has any property which belongs to
everything. This does not hold when (ix) (<f>x) does not exist; for example,
the present King of France does not have the property of being either bald
or not bald.
If (J- 7 *) ( < t> x ) has any property whatever, it must exist. This fact is stated
in the proposition:
*14 21. h : yfr (ix)(<f>x). 3 . E l(ix)(<f>x)
This proposition is obvious, since “ Kl(ix)(<f>x)” is, by the definitions, part
of “ yjr (ix)(<ftx)." When, in ordinary language or in philosophy, something is
said to "exist," it is always something described, i.e. it is not something
immediately presented, like a taste or a ]>atch of colour, but something like
“ matter " or “ mind ” or “ Homer ” (meaning “ the author of the Homeric
and only
SECTION B]
DESCRIPTIONS
17 -')
poems "), which is known by description as “tin* so-and-so." and is thus of
the form ( 7 .r)(</>.»). Thus in all such cases, the existence of the (grammatical)
subject (ix) (<fxv) can be analytically inferred from any true proposition having
this grammatical subject. It. would seem that the word *'existence M cannot
be significantly applied to subjects immediately given ; i.e. not only does our
definition give no meaning to “ E !.r,” but there is no reason, in philosophy, to
suppose that a meaning of existence could be found which would be applicable
to immediately given subjects.
Besides the above, the following are among the more useful propositions
of the present number.
*14 202. b <f>x . = r . x ■* b : s : (tar) (<f>.v) = b : = : <f>x . = x .6»a:: = :& — ( ix) (<£.r)
From the first equivalence in the above, it follows that
*14 204 b : E! (ix) (<f>x) . = . ( a fc) . (?*) (<f>x) - b
I.e. (ix)(<f>x) exists when there is something which (ix)(<px) is.
We have
*14 205. b : \fr (ix) (<f>x) . m . ( 36 ) . b = (?x) ( <f>x ) . y\rb
I.e. (ix) ( <f>x) has the property ^ when there is something which is (ix)(<f>x)
and which has the property yjr.
We have to prove that such symbols as “ (ix) (<f>x) ” obey the same rules
with regard to identity as symbols which directly represent objects. To this,
however, there is one partial exception, for instead of having
we only have
*14 28. b : E ! (ix)(<f>x) . = . (ix) ( <f>x) = (ix) (<f>x)
I.e. " (ix) (<fix) ” only satisfies the reHexive property of identity if (ix) (<f>x)
exists.
The symmetrical property of identity holds for such symbols as (ix)(<f>x),
without the need of assuming existence, i.e. we have
*14 13. b : a = (ix) (<f)x) . = . (ix)(tf>x) = a
*14131. b : (ix) (tf>x) — (ix) (y^x) . = . (ix) (yfrx) = (lx) (<f>x)
Similarly the transitive property of identity holds without the need of
assuming existence. This is proved in *1414142144.
*14 01. [(?*)(4>x)] . yfr (ix)(tf>x) . = : ( 36 ) : <f>x . = x . x = b : yfrb Df
*14 02. E ! (ix) (<f>x) . = : ( 36 ) z<f>x .= x .x = b Df
*14 03. [(7a:) (<f>x), (ix) tyx)] ./{ (ix) (<f>x), (ix) (yfrx)} . = :
[(7x) (<f>x)] : [(?a:) (^rx)J ./{(ix) (<f>x), (ix) (yfrx) j Df
*14 04. [(7a:) (^a:)] . / ((7a:) (<f>x), (ix) tyx)} . = .
[(la:) (yfrx), (ix) (<f>x)] ./[(lx) (<f>x), (ix) (yfrx)\ Df
170
MATHEMATICAL LOGIC
[PART I
*141. !-:.[< ix) (<f>x )]. y\r ( is) ( <f>x) • = • : 4>‘ - s,. x- 6 s ^6
[*42.(*1401)]
In virtue of our conventions as to the scope intended when no scope is
explicitly indicated, the above proposition is the same as the following:
*14 101. h yfr ( ix)(<f>x ). = : (g&): <f>.r. =, . x = b : [*141]
*14 11. h E ! (u) (</w) . s : <3*0 : . 5* . x- b [*4‘2 . (*14-02)]
*14111. V :.[(ix)(^rx)]./{(fx)(^X • * :
(g*». c): <£•» . = x . x = b z yfrx. = x . x = c :/(*». c)
l)eni.
h. *4*2. < *140403).}
I- ::[(ix)(*x)] •/!<«)<$•*)• <»*)<*•*>! • s : -
[(, (f.. )]: [(i.. ) ($x)]./!(lx) (<f>x), (ix) (^x)|:.
[♦14--1 ] a :.[<ix)<*x)]:.<ai>: £r. s,.x-/.:/!Mlx)(*x)i:.
[*14 l] = :. (gc):. fx. s x .x = c:.< 36 ): $x. =,.x = b : f(l, c)
[*11-55] = :. < 36 , e) : <fix . s, .x- c s yfrx. 3 ,.x - 6 :/(<>. c) :: 0 h . Prop
*14112. H:./I(»x)<*x).(w)(*x)|.«:
( 36 . c)s *x. s ,.X = 6 : ^X. E,.X = c:/(».«)
[Proof as in *14111]
In the above proposition, we assume the convention explained on p. 1'4.
after the statement of *14 03.
*14-113. I-:[(w)(t*)]./|(w)(^(»)(^)l-«-/K w )(M(i*)(+ J? M
[*14111112]
This proposition shows that when two descriptions occur in the same pro¬
position, the truth-value of the proposition is unaffected by the question whic i
has the larger scope.
*14-12. ES(ix)( 4>x ). D : <f>x . <f>y . . x ».»/
JJem.
f- . *14 11 D y z. Hp . D :<3*0 z<f>x.= x .x = b
y. *4*38. *101 .*1111-3. D
h :. </>./•. = x . x = b z O z <f>x . <py. = x .y .x = b.y = b.
[*13172] O x , ¥ .x^y
h . (2). *1011-23 . D h (3*0 z <px. = x . x = b zD z <f>x. <f>y. 0 x , y •«
H.(l).(3). D h . Prop
*14121. y <f>x. = x . x = b z <f>x. = x . x = c z D . b =* c
( 1 )
( 2 )
3/(3)
h . *101. D h :. Hp . D : <f>b . = . 6 = 6 : <f>b . =
[*1315] D:<t>bz<t>b. = .b = cz
[Ass] D : 6 = c :. D h . Prop
*14122. yz. 4 >x.= x .x = bz = z<f>x.O x .x = bz<t>bz
= z<f>x.O x »x = bz (gx) • 4>x
. b = c
SECTION B]
descriptions
177
Bern.
K *10-22.
3 h :. <p.r . = x . .r = 5 : =
0.r . 3 X . .r = 5 : a- = 5.3. r . </>.r :
[*13-191]
s
0.»-. 3 X . .r = 5 : 05
(!)
h . *4 71 .
3 b:. 0.c .3 . .r = 5 : 3
0.r . = . 0.i-. .i- = 5 :.
[*1011-27]
3 h :. 0.r . 3 X . .r = 5 : 3
0.i-. = x . 0./-. .r = 5 :
[*10-281]
3
(a o • 0 » • = • (a-* ) • <t> • • •
[*13-195]
= . 05
C2)
h. (2). *5-32
Ml).(3).
. 3 h :. <^>a- .3 z . a- = 6 : (g.r) . </>.r : = : </>.r . 3 X . a: = 6 : 05
3 I-. Prop
(3)
The two following propositions (*14123124) are placed here because of
the analogy with *14122, but they are not used until we come to the theory
of couples (#55 and *56).
[*13-21]
h . *4 71 .
[*11-341]
[*13-22]
*14123. h 0 (z, w) . =, >IC . z =« x . w = y :
= i <f>(z, tv) . . * - « . w - y : 0 (.r, y) z
s:<f>(z, w) . 3,.*. z ■ x . w -y : ( 32 , w) . 0 (z, w)
Bern.
I- .*11-31 . 3 1 -<f>(z, w ). s ZtU> . z — x.w — yi
s ; <f> (z, w ). 3 f , w . z — x . to ™ y : x ■* a?. w « y. 3 ;> l 4 .. 0 ( 2 , w)z
■ : 0 (*, w). 3 f , „. 2 - a:. to - y : 0 (.r, y) ( 1 )
3 h :. 0 ( 2 , to) . 3 .2 = a:. to = y :
3 : 0 ( 2 , to) . = . 0 {z, to ). z *= x, w = y z.
[*11-11-32] DH tu) . Dg tW . z = x . vj = y z
D z <f>(z, to) 4>(z, tu) . z = x ,xu = y z
D : ( 5 ( 2 , to) . 0 (x, to) . = . ( 32 , to) . <f>(z, w). z = x . w = y .
= .<f>(x,y) ( 2 )
1-. (2) . *5 32 . 3 I- s. 0 < 2 , to) . 3,,„. x — x .to« y : ( 32 , to) . 0 ( 2 , to) :
, /n /ox ^ >> ss 0(x, to).3 x .„.*-#.to«y:0Or, y) (3)
r . (1) . (3) . 3 H . Prop
*14 124. h (a*, y) : 0 ( 2 , to) . . 2 = a:. to = y :
= : (a*, y) . 0 (x, y) z 0 (z, to). 0 (u, v) . 3,,,. z - u . to - v
Dem.
K *14 123. *3 27.3
** - (3*. y)z <f>(z, to) . = ( , „ . 2 — x . to « y : 3 • ( 3 a:, y) . 0 (x, y) ( 1 )
h . *11-1 . *3 47 .3 h z. 4>(z, w) . = z te . z = x. w = y :
3 : <f> ( 2 , to) . 0 (u, v) . 3 . z = a:. to = y . w = x . = y .
[*13172] D. z = u. w = v (2)
h. (2). *1111-35.3
^ !• (a 31 , y) : 0 (*, yj) - =x.t P .z = x.w = yz
3 : 0 ( 2 , to). 0 (u, o) . 3 .2 = u . to = V (3)
12
R&W X
178
MATHEMATICAL LOGIC
[PART I
K<3).*1111*3.3
*- :-( 3 x,y)i<f>(z, w). = ;ie .z = x. w=y :
3 : 4>(z. w) . <f> (•!, v) . 3>...,r.u.r • 2 = « • w =■ ti (4)
h . *111.3 h :. <*> (.r, y): <*> (*. w). <f> (//, »•). 3,. . * - u . w = v :
D:<t>(x.y):<t>(z, w).<t>(x,y).0,,„.z-x.w = y:
[*5 33] 3 : <f> (x, y):<f>(z,io).O r ,„.z = x.w=y:
[*14123]
0 :<f>(s, w).
Z = X. w = y
(5)
h . (5). * 11
11 34 45.3
h :.(g-'*.y)'
y) :</>(*. «’)
•</>(". V) . 3;
= W-t»:
(6)
3:(g.r,y):
</>(z.w).= :% „.z = x.w = y
h .< 1 ).(4).
(6). 3 h . Prop
*1413. h:
a ~(ix)(4>x). =
.0 x)(<f>x) =
a
Dem.
(1)
V . *141 .
3 h c/ =
(ix) (</>>). = •
: (g6 ): <frc. = x . x = b z a = b
h . *1310 . *4*36 . 0 4 >x . = x . x = b : a *= b : = : <f>x. = x . x =* b : b =* a :
[*1011-281] 3 H :.{zb): 4 >x.= J .x = b:« = b:
= : (gt ): tpx. = x . x = b : b = a :
[*U- 1 ] = :(ix)(<t>x)=*a ( 2 )
K(l).(2). 3 K Prop
This proposition is not an immediate consequence of *1316, because
•'a = (u)(<t>x)" is not a value of the function "x = y." Similar remarks
apply to the following propositions.
*14 131. h : (lx) (<t>x) - Ox) ( y/rx) . 5 . ( lx) (y/rx) - (lx) (4>x)
Dem.
h . *14-! . 3 y :: (u)(<*>.r) - (i.r) (y/rx) . = ( 36 ): <f>x . =,. * = b : b = (ix)(y/rx)
[*141] = ( 36 ) z.<f>x.= x .x = b:. ( 3 c): y/rx . =,. *- c s 6 - c
[* 11 * 6 ] a (gc) :.y/rx.= x .x = c:. ( 36 ): <f>x. = x . x = b : b = c
[*14 1 ] s (gc) y/rx. = x . x = c : (ix) (<f>x) = c
[*1413] = :.(g c):.y/rx .= x .x = c:c = (ix)(<f>x)
[*141] = (ix)(y/rx) = (ix)(4>x) :: 3 I-. Prop
In the above proposition, in accordance with our convention, the descriptive
expression (ix)(<f>x) is eliminated before (ix)(y/rx), because it occurs first in
"(»)(♦*)■(«)(♦')"; b,lt in U ( 1X ) (y/rx) — (lx)(<t>x), n (ix)(y/rx) is to be hrs
eliminated. The order of elimination makes no difference to the truth-vniu ,
as was proved in *14*113.
The above proposition may also be proved as follows:
H . *14*111 • 3 h S. (IX) (<f>x) = (lx) (y/rx) .
= :(3 c): <j>x . = x . x = b : y/rx. = x . x - c : b ■
[*4-3.*1316.*llll-341] =:( a 6,c):^*.=..* = c:^x. =,.x = b:c-
[*1 l-2.*14 lil] = : (J*)(**> = (>*)($*) =• 3 h • Pr0 P
SECTION B]
DESCRIPTIONS
*1414. h : a = b . b = (l.r) ( <f>.r ). D .« = (ix) (^.r) [*13*13]
*14142. I- : a = (7x) (<#>.»•) . (i.r)(<£.r) = (i.r> (>/r.r). D . a = (ix) (^.r)
Dem.
I" . *141 . D h :: Hp . D ( 36 ): <£.c . = x .x = b :<i = b
(ge) : </> r . = x . .r = c : c = ( u) (^.r):.
[*13195] D . = x . .r = a (gc) : </>.r. = x . .1 = c : c = (;./•) {\fr.r)
[*10 35] D (gc) <f>.v . = x . x « a s <£.r. = x . x = c : c - ( 1 ./) (^.r)
[*14 121 ] D (go) :4r.s x ..c = a:«-c:c = (i.r)(ifr.r)
[*3 27.*13195] D :.a~(ix)(\lrx):: D h . Prop
*14 144. t- : (?x) (<f>x) = ( 7 x) (>/rx) . (far) (y/rx) = (lx) ( X x) . D . (I.r) (<£.r) = (i.r) <*.*•)
Dem.
h . *14 111 . Dh:: Hp . D (ga,5) s <f>x . =,. x«= a : yfrx . = x . x - 5 : a - 6
(gc, rf): >//-x. = z . x = c : *x . = x . x = : c = <1
[*13195] D (ga) s <f>x . = x . x - a : yfrx . = x . x = a
(gc) : . » x . x - c : *x . = x . x - c
[*11 '54] D (ga, c) : <f>x . = x . x = a : yfrx. s x . x — a :
>jrx . b x . x - c ; x* • = x • * 155 c
[*14121.*11*42] D (ga.c) : <f>x . s x . x — a : ^x. - x .x — c : a = c
[*14 111] D (7x)(0x) * (ix)(^x) ::9K Prop
*14145. h : a = (7x)(<£x) . a » (?x) (^) . 3 . (ix) (<£x) = (?x) (^ x )
Dem.
. *14*1. D h s. a = (ix) (<*>x) . 3 : (g&) : </>.r. s r . x = 6 : a = 6 :
[*13195] = : <f>x . = x . x = a ( 1 )
H . (1). *141 . D H :s Hp. = <f>x . a,. x — a(g&): >/rx. 3 *. x = 5 : a — 5
[*10'35] = s. (g6) :.<f>x. = s .x=* a:\Jrx. = x .x = b:a = b:.
[*14*111] D :.(ix)(0x) = (ix)(>/rx):: D h . Prop
*14 15. h (?x) (tf>x) = b .D :yfr |(7x) ( <f>x)\ . = . yfrb
Dem.
K*14'l . D
K :: Hp . D (gc) :<£x.= x .x = c:c = 6:.
[*13'195] D <frx . = x . x = b ( 1 )
Ml).*14‘l •=>
h :: H p . Dz.^lr {{ix)(<f>x)] ■ = : (gc) : x = 6 . = x . x- c : yjrc :
[*13192] = : yjrb ::DK Prop
*1416. h (7x) (<f>x) = (7x) (y\rx) .D: X \0*) (<t>*)} - = -X (( 1 *) (^x)j
Dem.
h.*141. D I- Hp . D : (g6) : tf>x . = x . x = b : b = (jx) tyx) (1)
!■ • *14*1. D H s: tf>x . =*. x = b s D
X 10*) (0*)} • = : ( 3 c) • X = b . = x . X = c x c '
= : (2)
12—2
[*13192]
180
MATHEMATICAL LOGIC
[PART I
(3)
K . #141315 . D b :. 6 = (is)(yfrs ). D : X *> • * • X |(w) (+*>}
I- .(2). (3). D h :.<£r . = x . x = b : b = (lx) (yjrx):
3 : X | 0 - 0($*)1 • s • X 10 *) (^>1 ( 4 )
h .(1).<4).*10 1 23. D K . Prop
*1417. h :. (Mr)(<£.r) = 6 . = : yfr ! (ix)(<f>s) .=* .>//■! 6
Dent.
h. *1415. *1011-21 .D
I- :.(lx)(<px) = b . D : yjr ! (?.r)(<£j) . =+. \fr ! b (!)
h . *101 . *422 .Dh::^!x.= x .j-=5:>/r! (ix)(<px) .=+ ,>frl b:
0 : 0 x)(<f>x)-b.m.b-b:
[*1315] D :(?*)($*)-& (2)
h .(2). Exp. *10*11 *23. D
f- :: (gx) : X ' x • • r = ^ >lr ! (ix)(<f>s). = + ,y/r!b:D. (ix) (<f>x) = 6 (3)
P . *121 . Df-:(gx):x ! ** a ** x " 6 (4)
h . (3) . (4) ,Db:.y/rl (IX) (<f>x ). =+. f ! b : D . (fx) (<f>x) = b ( 5 >
K . (1). (5) . D H . Prop
It should bo observed that we do not have
(is)(<t>x) = b . = : yjr ! (tx)(<f>x) . D* . ^ ! b
for, if «—' E ! (ix)(tf>s), y\r ! (ix)(<f>x) is always false, and therefore
ylrl(ix)(<f>x).5+ .yfrlb
holds for all values of b. But we do have
*14171. I- :.(ix)(4>r) = b . = : yfr ! b . D* . yfr l(ix)(<f>s)
Dem.
h . *1417 . D h i.(lx) (<t>x) = b . D : >\r ! b . D* . * ! (lx) (<*>x) (1)
I-. *101 . *121 . D h + ! 6. D*. + ! (»*) (<^r) : D : b = 6 . D . (?*) (£r) = 6 :
[*1315] D:(Lr) (<£*) = 5 ( 2 )
I*. (1). (2). D h . Prop
*1418. h :. E ! (ix)(4>s) .0 :(x).+x .0 . + (ix)(<f>x)
Dem.
H . *101 • D h z(x).yfrx.D.ylrbz
[Fact] D <f>x. = x . x = b : (x). yfrx z D z <f>x . = x . x = b : >frb :
[*l(Hl-28]3l-:.(l[6):*x.=..*-6: (x).*x: 3 : (g&) = <#■*•=* ■ * = i : * hu
[*10-35] D h :: (gi) : .= z . x = 6 :.(x). :.0 : (gi): <#>x .=,.x = b: irbu
[*14111] D I-:. E! (?x) (tf>x ): (x). >frx: D : ^(Jx)(^x) :■ 3 • Pr°P
The above proposition shows that, provided (?x) (<£x) exists, it has (speaking
formally) all the logical properties of symbols which directly represent objects.
Hence when (jx)(<£x) exists, the fact that it is an incomplete symbol becomes
irrelevant to the truth-values of logical propositions in which it occurs.
SECTION B]
DESCRIPTIONS
181
*14 2. h . (7.r) (x = a) = a
Dem.
. *14*101 . D h (i.r)(.r = a) = a . = : (g5) : x = a . =, . x = b z b =
[*13*195] = : .r = a . = x . a* = a
h.(l).Id. D h . Prop
*14*201. h : E! (ix) (<J>.r) . D . (gar) . <*>.»•
Dem.
h . *14*11 . Dh. Hp . D : (g&) : <£.r . = x . .r * b z
[*10*1] D : (g6): «/>5 . = . 6 = 6 :
[*13*15] D : (g6) . </>6 D h . Prop
*14 202. h tf>x . = x . a: *» 6 : =: (;x) (<£.r) = b z = z <f>x. = r . b = x : = : b =
Dem.
h . *14*1 . D h (7a:) (<f>x) = b . = : (gc) : <£.r . = x . x = c : c = b :
[*13*195] = : <f>x. = x .x = b D I-. Prop
[The second half is proved in the same way as the first half.]
*14*203. h E ! (ix)(<f>x) . = z (ga:) . tf>x z <f>x . <f>y . D«, v .x = y
Dem.
D h E ! (ix)(<f>x) . D : (ga:) . <f>x z <j>x . <f>y . D XtJ/ . .r = i
D h <f)b z <px . . D f y . x y z D z <f)b z <f>x . tf>b . D x . i
D : <f>b z <f>x . D x . a: = b
D : a: = b . D z . <f>x :
<f>x. D x . x *=
[*10*22] D:<f>x.= x .x = b
K . (2) . *10*1*28 . D h :.(g6):</»6 z <f*x. <f>y . D«, y . a: = y:D:(g6) z <f>x. = x
[*10*35] D h :.(g6).«/>6 z <f>x. <f>y . D x#y . a: = y:D:(g6) z<f>x.= x
[*14*11] DzE!(ix)(<fix)
h . (1). (3) . D h . Prop
*14 204. h :. E ! (?x) (<*>a:). = : (g6) . (ix) (<f>x ) = 6
Dem.
h . *14*202 . *10*11 . D
h(6) <f>x . = x . x « b s 3 : (7a:) (<£ar) = 6D
[*10*281] h (g6) : <f>x . = x . x = 6 : = : (g6) . (7a:) (<£x) = 6
h. (1). *14*11. Dh. Prop
*14 206. h : yjr (ix) (<f>x ). = . (g6). 6 = (ix) (<f>x) . +b [*14*202*1]
*14*21. h : yfr (ix) (4>x) . D . E ! (7a:) (<fix)
Dem.
h . *14*1 . D
h ^ {( 7 a:) (<f>x )). D : (gfe) : <f>x . =*. x = 6 : yjrb :
[*10*5] D s (g b) z<f>x.= x .x=bz
[*14-11] D s E ! (ix) (*f>x) D h . Prop
h. *14*12*201
h . *10*1 .
[*5*33]
[*13191]
( 1 )
()x)(<f>x)
( 1 )
= 5:
bz
( 2 )
.x = bz.
• x~bz
(3)
(1)
182
MATHEMATICAL LOGIC
[PART I
This proposition shows that if any true statement can be made about
( 7 x)( 0 x), then (?x)(0.r) must exist. Its use throughout the remainder of the
work will be very frequent.
When (ix)(0x) does not exist, there are still true propositions in which
'‘(/x)( 0 .r)” occurs, but it has, in such propositions, a secondary occurrence,
in the sense explained in Chapter III of the Introduction, i.e. the asserted
proposition concerned is not of the form \jr (ix)( 0 x), but of the form
f\>\r (i.r)( 0 .r)}, in other words, the proposition which is the scope of ( 7 x)( 0 r)
is only part of the whole asserted proposition.
*14 22. h : E ! ( 7 x) (0x). = . 0 (>.r) (0x)
Dem.
h. *14122. D 1- <f>s. = x . x = 6.: D . 06 (1)
h . (1). *4*7 1 . D h 0x . = x . x = 6 : = : 0x. = x . x = 6: 06
[*1011281] D h M 36 ): 0 / . =, ,x = 6 : = : ( 36 ): 0 x. = z .x = 6 : 06
[*1411101] D h : E!(i.r)(0x). = . 0(?.r)(0x): D I- . Prop
As an instance of the above proposition, we may take the following: “The
proposition 'the author of Waverley existed’ is equivalent to ‘the man who
wrote Waverley wrote Waverley."’ Thus such a proposition ns "the man
who wrote Waverley wrote Waverley” does not embody a logically necessary
truth, since it would be false if Waverley had not been written, or had been
written by two men in collaboration. For example, " the man who squared
the circle squared the circle" is a false proposition.
*14 23. 1- : K ! (ix)(0x . yfrx) .= ,<$> J(ix)(0x. 0-x))
Dem.
h . *14-22 . D h E! (ix)(0.r . 0x) .
= : [(lx) (0x . 0\r)] : 0 [(7x) (0x . yjrx)} . 0* ((lx) (0x . 0*)l
[*10-5.*3-26] D : 0 {(7x)(0x. 0r.r)| (1)
h . *14 21 . D h : 0 ((ix) (0x. 0\c)| . D . E ! (/x)(0.r. >/rx) (2)
h.(l).(2).DH. Prop
Note that in the second line of the above proof *10*5, not only *3’26, is
required. For the scope of the descriptive symbol (?x)(0x. 0-x) is the whole
product 0 |(?x)( 0 x. 0 -x)| . |(ix)(0x . 0x)|, so that, applying *14-1, the
proposition on the right in the first line becomes
( 36 ): <f>x . yjrx . . x = 6 : 06 . yfrb
which, by *10 5 and *3 26, implies
( 36 ): 0 x. yfrx .=,.*= 6 : 06,
i.e. 0 [(ix)( 0 x. 0 >x)).
*14 24. h E ! (lx) (0x). = : [( 7 x) (0x)] : 0y. = v . y = ( 7 x) (0x)
Dem.
h .*14*1 .DP:. [(7x) (0x)] : 0y. . y = (7x) (0x):
= = ( 36 ) : 0 y. = y . y = 6 ; 0y. = y - y = ^ :
SECTION B]
DESCRIPTIONS
183
[*4‘24.*10-2S1] = : (g6) : <f>y . =,,. >/ = b z
[*1411] = : E! (#.*•) (<f>x) D h . Prop
This proposition should be compared with *14 241, whore, in virtue of the
smaller scope of (?x)(<£. v), we get an implication instead of an equivalence.
*14 241. h E! (ix)(</>.c) . D : <f>y . =,,. y = {lx) ( <p.r )
Dem.
b . *14 203 . D h :: Hp . D (f>y . <f>x . D . y = x :.
<f>x . D . y = .r ::
<£x. D x .y = x:.
<f>y z <f>x . D x . y — a :
y = x . D x . <f>x : <f>x . D, . y « a:
<f>x.= x .y = xz
y = (ix)(<f>x) Prop
[Exp] D <£y. D
[*1011-21] Dh::Hp.D:.<*>y.D
[*471] D:.*y.s
[*13191] =
[*1022] =
[*14202] s
*14242. h:.£x.= x .x = 6:D:^5. = .>/r(ix)(<f>x) [*14-202-15]
*14 26 . h :. E! (ix) (<£x) . D : </>x D x >/rx . = . >/r (>*) (<*>x)
Dem.
h . *4-84 . *10-27-271 . D h :: <*>x . = x . a = b : D <*>x D x >/rx . s : a = b . D x . >/rx :
[*13191] =
[*14-242] =.+(ix)(4>x) (1)
h . (1) . *1011-23 . D h :. (g&) : <*»x. = x . x « 5 :
D z <f>xD x \lrx . = . \lr(7x)(<bx) (2)
h. (2). *1411. Dh.Prop
*14 26. I-E ! (?x) (<f>x) . D : (gx). <f>x . >/rx. = . {(?x) (<£x)) . = . tf>x D x \^x
Dem.
h.*1411 . D
b :. Hp . D : (g&) : <f>x . = x . x = b (1)
b . *10-311. D b :: tf>x . = x . x = 6 : D <f>x . . = x . x = b . yfrx :.
[*10281] D (gx). <f>x . >/rx. = . (gx) . x = b . >^x .
[*13*195] = . yfrb ■
[*14-242] =. {(?x) (<£x)) (2)
h. (2). *1011-23. D
h s- (3&) : 4>x . = x . x = b z D : (gx) . <px . yfrx . = . yjr {(?x) (</»x)} (3)
h.(l). (3). *14 25. Dh.Prop
*14 27. h E ! (?x) (<£x) . D : <£x H x yfrx . = . (?x) (<£x) = (?x) (>/rx)
Dem.
h. *4*86-21. Dh::0x.= . x = 6 : D :. (f>x . = . ^rx : = : >/rx. = . x = 6 (1)
h . (1). *10*11-27 . D h :: <px . = x . x = 6 : D (x) :. <f>x . =. yfrx z = z yfrx . = . x = b z.
[*10 271 ] D (fix . = x . i/rx : = : >/rx . = x . x = 6 :
[*14-202] =: 6 = (**)(**):
[*14-242] = : (JX) ($x) = (lx) (+x) (2)
I-. (2). *1011-23 . *1411. D h . Prop
184
MATHEMATICAL LOGIC
[PART I
*14 271. K <f>x. = x . yfrx: D : E ! (i.r) (<p.r ). = . E ! ( )x) (yfrx)
Dem.
b . *4*8G . D H :: <£./ = yjrx . D <f>.r . = . x = 6 : = : yjrx. = . x = b :z
[*101127] Dh:: Hp . D (x) tf>x . = . .r = b : = : yfrx. = . x = b
[*10*271] D :.(.r) : <f>x . = . x = b : = : (x) : yjrx . = .x = b::
[*1011’21] D b :: Hp . D ^x • s,. x =6 : = : . = x . x = b
[*10 281 ] D (gi») : <f>x . = x . x = 6 : = : (g&) : = x . x = b::
D b. Prop
*14 272. b <f>x . = x . yfrx : D : ^ (#x)(</m) • = • X (*x)(>/rx)
J>em.
b . *4 80 . Dh:: <f>x = \^x. D <f>x . = . x = 6 : = : yfrx. = . x = b
[*1011414] D b :: Hp. D </>./ . = x ,x = b : = : yfr.r. = x . x = b
[Fact] D <f>x. = x . x = 6 : *5 : = : yjrx. = x . x = b : *6
[*1011-21] D b :: Hp. D :.(6 )<*>x . = x . x =b: X b:B lyjrx . = z .s=«&:x&:.
[*10 281] D (g&): <f>x . = x . x = h s xb • =
s(3&): • ■«.*-& s
[*14101] D ^(jx)(</>x). = . x(* x ) ('/ r * c ) :: ^ *" • P ro P
The above two propositions show that El(ix)(,<f>x) and x( 1X )(4> X ) arc
" cxtensioual " properties of ^.7, t,e. their truth-value is unchauged by the
substitution, for <£.7, of auy formally equivalent function yjr£.
*14 28. b : E l(tx)(<f>j ). = .<?x)(tf>x)=* (ix)((f>x)
Dem.
b . *13*15 . *4 73 . D b Qx . = x . x = b : = : <f>x . =, . x «= 6 : b = 6 (1)
h . (1) . *1011-281 . D
1 - :.(g6):<£x. = x .x = b: = :(^b):4>.c.= x .x=b:b = b (2)
h. (2). *14111 .DK Prop
This proposition states that ( tx)(<f>x ) is identical with itself whenever it
exists, but not otherwise. Thus for example the proposition “the present
King of France is the present King of France ” is false.
The purpose of the following propositions is to show that, when El(ix) (4>x),
the scope of ( ix)(<f>x ) does not matter to the truth-value of any proposition
in which (ix)(<f>x) occurs. This proposition cannot be proved generally, but
it can be proved in each particular case. The following propositions show
the method, which proceeds always by means of *14 242, *10 23 and *1411.
The proposition can be proved generally when (ix)(<f>x) occurs in the fonn
X (ix)(<f>x), and x ( 1x )(<t> x ) occure iu what we may call a “ truth-function, t.e.
a function whose truth or falsehood depends only upon the truth or falsehood
of its argument or arguments. This covers all the cases with which we are
ever concerned. That is to say, if x (7x)(<£x) occurs in any of the ways which
can be generated by the processes of *1—*11, then, provided E!(7x)(^x)>
the truth-value of /l[(7x) (<£x)] . x (7x) (<£x)J * 3 same “ tbafe of
[(?*) (<t>x)] .f ix (*r) («£*)).
SECTION B]
DESCRIPTIONS
185
This is proved in the following proposition. In this proposition, however, the
use of propositions as apparent variables iuvolves an apparatus not required
elsewhere, and we have therefore not used this proposition in subsequent
proofs.
*14 3. h up = q . . f(p) =f(q) : E ! : D :
^ /{ [(I*) (♦*)]-* (!*)(**)) • = (<#».«*)] ./| X (m)(<#>.!•)}
Dem.
H . *14 242 . D
H <f>x . = x . x - b : D : [(?*) (</>*)] . x (*•*)(</>.<•) . = . x&
K(l). Dh z.p = q . D,,., ./(p) =/(q)z 4>x.= x ..v = b
/|[(ix)(«^r)] . x (»•*) (^>*>! • = •
h . *14-242 . D
: D:
/(**>
( 1 )
h 4* .»•. «-6 : D : [(?a;)(<*>x)] ./1 x0*)(£*)1 • = •/(*&> (3)
K(2).(3).D
h s. p = <7 . D Pt9 ./(j>) =/(<?) :**.*,.«-6: D :
/IK 7 *) (4>x)] . x 0*) ($*)) . s.[(?x)(^ar)]./(x(».r)(^r)J (4)
h . (4) . *10-23 . *14*11 . D h . Prop
The following propositions are immediate applications of the above. They
are, however, independently proved, because *143 introduces propositions
(p, q namely) as apparent variables, which we have not done elsewhere, and
cannot do legitimately without the explicit introduction of the hierarchy of
propositions with a reducibility-axiom such as * 121 .
*14*31. h :: E ! (7x) (<f>x) . D [(?*) (fr)]. p v x (?*) (fc).
= s p - v . [(*r) (<f>x)] . x Ox) (<f>x)
Dem.
h .*14-242. D h z. <f>x .= x .x = bzO z[0x)(<f>x)].pv X 0x)(<f>x). = .pv x b (1)
h . *14*242 . D 1- :. (f>x . =*. x = 6 : D : [(ij;)(0j:)] . x( 1x )(<f>x) • - • X^ '•
[*4-37] D :p v [(»*)(<£*)] x(?*)(<£*) .= ,pv x b (2)
h . (1) . (2) . D h :. <*>x . = x . x = 6 : D : [(*r) (<£*)] . p v x Ox) ( <f>x ) .
= . p v [(?*) (<f»x)l x (»#) (<fre) (3)
h . (3) . *10 23 . *1411 . D h . Prop
The following propositions are proved in precisely the same way as *14 31;
hence we shall merely give references to the propositions used in the proofs.
*14-32. h s. E ! Ox) (<f>x) . = : [(ix) ( <f>x)) .
[*14-242 . *4*11 . *10-23 . *1411]
X 0x) (</>*).
= — ([(*«> (<px)] . x Ox) ( <f>x)\
The equivalence asserted here fails when ~E ! (?x) (<px). Thus, for example,
let <f>y be “ y is King of France.” Then O x )(<t>x) = the King of France. Let
XV b e “y * 8 bald.” Then [(?#) (<f>x)] .~x 0 X ) (<f>x) . = . the King of France
exists and is not bald ; but ~{[(m)($b)] . x (**)($*)). = . it is false that the
King of France exists and is bald. Of these the first is false, the second true.
MATHEMATICAL LOGIC
[PART I
180
Either might be meant by “the King of France is not bald,” which is am¬
biguous; but it would be more natural to take the first (false) interpretation
as the meaning of the words. If the King of France existed, the two would be
equivalent; thus as applied to the King of England, both are true or both false.
*14 33. b :: E ! (ix) (<£./-). D [(ix)(<£x)]. p D x (ix)(<f>x).
= :p.O. L(ix) (<*>x)] . x (»*) (4> x )
[•14-242 . *4 85 . *10 *23 . *14 11]
*14 331. b :: E! (?.#•) (<f>x ). D [(ix)($x)]. ^(ix)(</>x) D p .
= : [(?x) (<*>x)]. x 0*) (♦*) • D -P
[*4-84 . *14-242 . *10 23 . *1411 ]
*14 332. b :: E ! (is)(<f>x) . D [(ix)(</>x)] . p = xO*)(<t> x ) • =
:/>. = . [(?x) (</>x)] . x (**) (£ c )
(*4 8 G . *14 242 . *10 23 .*1411]
*14 34. b :. p i [(tx) (. x ( ,.c) (£r); a : [(ix) ( <f>x )]: p . x (»*) (♦*)
This proposition does not require the hypothesis E!(*r)($x).
Dent.
b . *14*1 . D
b p : [(ix) ( <f>x )]• x <*0 («M: ■ :/>! (gfr): $x. g„. x - 6 : * 6 :
[* 10 35] = : (gft) ; p • <f>x . = x . x = 6 : x^ !
[*141] = : [(ix)(0x)]: p . X ( tx )(4 >j: )^ ^ • P ro P
Propositions of the above type might be continued indefinitely, but as they
are proved on a uniform plan, it is unnecessary to go beyond the fundamental
cases of p v q, ~p, p D q and />. 7 .
It should be observed that the proposition in which (tx)(<f>x) has the
larger scope always implies the corresponding one in which it has the smaller
scope, but the converse implication only holds if either (a) we have E ! (ix)(<^)
or ( 6 ) the proposition in which (j x) (</>x) has the smaller scope implies
E! (?x) The second cose occurs in *14 34, and is the reason why we
get an equivalence without the hypothesis E!(ix)($x). The proposition in
which (ix)(<f>x) has the lurger scope always implies E!(ix)(<£x), in virtue of
*14-21.
SECTION C
CLASSES AND RELATIONS
*20. GENERAL THEORY OF CLASSES
Summary of *20.
The following theory of classes, although it provides a notat ion to represent
them, avoids the assumption that there are such things as classes. This it does
by merely defining propositions in whose expression the symbols representing
classes occur, just as, in *14, we defined propositions containing descriptions.
The characteristics of a class are that it consists of all the terms satisfying
some propositional function, so that every propositional function determines a
class, and two functions which are formally equivalent ( i.e. such that whenever
either is true, the other is true also) determine the same class, while conversely
two functions which determine the same class are formally equivalent. When
two functions are formally equivalent, we shall say that they have the same
extension. The incomplete symbols which take the place of classes serve the
purpose of technically providing something identical in the case of two functions
having the same extension ; without something to represent classes, we cannot,
for example, count the combinations that can be formed out of a given set of
objects.
Propositions in which a function <f> occurs may depend, for their truth-
value, upon the particular function </>, or they may depend only upon the
extension of <f>. In the former case, we will call the proposition concerned an
intensional function of <f >; in the latter case, an extensional function of <f>.
Thus, for example, (x). <f>x or (gx) . tf>x is an extensional function of <f>,
because, if <f> is formally equivalent to yfr, i.e. if <f>x . =,. yfrx, wc have
(*) . <f>x . s . (x) . yfrx and (ga?) . <f>x . = . (gar) . yfrx. But on the other hand
"I believe ( x) . <px ” is an intensional function, because, even if <ftx . = x . yfrx,
it by no means follows that I believe ( x ). yfrx provided I believe ( x ). <f>x. The
mark of an extensional function f of a function <f >! 2 is
<t>lx.= x .yfrlx:D^:f(<t>l2). = ./(ylr!n
(We write “<£ !$” when we wish to speak of the function itself as opposed to
its argument.) The functions of functions with which mathematics is specially
concerned are all extensional.
When a function of <f> ! 2 is extensional, it may be regarded as being
about the class determined by <f >! 2, since its truth-value remains unchanged
so long as the class is unchanged. Hence we require, for the theory of classes,
a method of obtaining an extensional function from any given function of a
function. This is effected by the following definition:
188
MATHEMATICAL LOGIC
[PART I
*20 01. / | 2 (yfrz )I . - : ( 3 *) 8 * ! x . s, . ifrx : / (* ! ?) Df
Here |3 (^r^)j is in reality a function of yfrS, which is*dcfined whenever
/•</>! 2| is significant for predicative functions <p ! 2. But it is convenient to
regard f z{y\r~)\ as though it had an argument z (yfrz). which we will call
"the class determined by the function yfrz." It will be proved shortly that
f \z (^r)| is always an erfensional function of yjfz, and that, applying the
definition of identity (*13*01) to the fictitious objects 2 ( <pz) and 2 (y^z), we
have
z(<f>z )=2 (yfrz) . = : (x) : tf>x. = . yfrx.
'l'liis last is the distinguishing characteristic of classes, and justifies us in
treating 2 (yfrz) as the class determined by \frz.
Wit h regard to the scope of 2 (y/rz), and to the order of elimination of two
such expressions, we shall adopt the same conventions as were explained in
*14 for (ix)(<f>x). The condition corresponding to
E ! (ir)(y/rx) is (g<£) :<p\x.= x . ^x,
which is always satisfied because of * 12 * 1 .
Following Peano, we shall use the notation
xcz (\Jrz)
to express "x is a member of the class determined by yjrz" We therefore
introduce the following definition:
*2002. xc(0!2). = .0!x Df
In this form, the definition is never used; it is introduced for the sake of the
proposition
h x ( 2 (+z ). = : (g</>): . a„. 0 ! y : <f> l x
which results from *20 02 and * 20 * 01 , and leads to
V : x € 2 . = . yfrx
by the help of * 12 * 1 .
We shall use small Greek letters (other than e, i, 7 r, </>, yjr.x- 0) to represent
classes, i.e. to stand for symbols of the form 2 (<f>z) or 2(0! z). When a small
Greek letter occurs as apparent variable, it is to be understood to stand for a
symbol of the form 2 (<f >! z ), where <f> is properly the apparent variable con¬
cerned. The use of single letters in place of such symbols ns 2(4>z) or 2 (<£!*)
i 9 practically almost indispensable,since otherwise the notation rapidly becomes
intolerably cumbrous. Thus "x € a” will mean "x is a member of the class a.
and may be used wherever no special defining function of the class a is in
question.
The following definition defines what is meant by a c/ass.
*20 03. Cls = a [(g<*>) . a - 2 (£!*)) Df
Note that the expression “a {(g<£). a = 2 (<f >! z)\” has no meaning m
isolation: we have merely defined (in * 20 * 01 ) certain uses of such expressions.
What the above definition decides is that the symbol “Cls” may replace the
symbol "a {(g<£). a = 2 (<f >! z)\" wherever the latter occurs, and that the
SECTION C]
GENERAL THEORY OF CLASSES
189
meaning of the combination of symbols concerned is to be unchanged thereby.
Thus “Cls," also, has no meaning in isolation, but merely in certain uses.
The above definition, like many future definitions, is ambiguous as to
type. The Latin letter z, according to our conventions, is to represent the
lowest type concerned; thus <f> is of the type next above this. It is convenient
to speak of a class as being of the same type as its defining function ; t hus a
is of the type next above that of z. and “Cls" is of the type next above that
ot a. Thus the type of “Cls" is fixed relatively to the lowest typo concerned;
but if, in two different contexts, different types are the lowest concerned, the
meaning of “Cls” will be different in these two contexts. The meaning of “ Cls"
only becomes definite when the lowest type concerned is specified.
Equality between classes is defined by applying *13 01, symbolically un¬
changed, to their defining functions, and then using *20 01.
The propositions of the present number may be divided into three sets.
First, we have those that deal with the fundamental properties of classes;
these end with *20 43. Then we have a set of propositions dealing with both
classes and descriptions; these extend from *205 to *2059 (with the ex¬
ception of *20*53*54). Lastly, we have a set of propositions designed to prove
that classes of classes have all the same formal properties as classes of in¬
dividuals.
In the first set, the principal propositions are the following.
*20 15. h yjrx . s,. yx : = . 2 (yfrz) = 2 ix z )
I.e. two classes are identical when, and only when, their defining functions
are formally equivalent. This is the principal proporty of classes.
*2031. H 2 (yjrz) = z (x z ) • = : x e 2 (yfrz) . = x .xez (x*)
I.e. two classes are identical when, and only when, they have the same
members.
*20 43. z. a = @ . = : x € a . = x . x e ft
This is the same proposition as *20*31, merely employing Greek letters
in place of 2(yfrz) and z (*z).
*2018. h 2(</>*) = 2 (irz) . D :/(2 (*«)). = -/{2«*»
I.e. if two classes are identical, any property of either belongs also to the
other. This is the analogue of *13*12.
*20*2*21*22, which prove that identity between classes is reflexive, symmetrical
and transitive.
*20*3. h : x e 2 (yfrz) . = . yfrx
I.e. a term belongs to a class when, and only when, it satisfies the defining
function of the class.
In the second set of propositions (*20*5—*59), we show that, under suitable
circumstances, expressions such as ( ix)(<f>x ) may be substituted for x in *20*3
100
MATHEMATICAL LOGIC
[PART I
and various other propositions of the first set, and we prove a few properties
of such expressions as < ia) ( fa),” i.e. “ the class which satisfies the function /.”
Here it is to be remembered that '* a ” stands for “5 (cf>z)," and that "fa"
therefore stands for M f{5($z) J.” This is, in reality, a function of <f>z, namely
the cxtensional function associated with f(\jrlz) by means of *2001. Thus
an expression containing a variable class is always an abbreviation for an
expression containing a variable function.
In the third set of propositions, we prove that variable classes satisfy all
the primitive propositions assumed for variable individuals or functions, whence
it follows, by merely repeating the proofs of the first set of propositions (*20*1
— '43), that classes of classes have all the formal properties of classes of in¬
dividuals or functions. We shall never have occasion explicitly to consider
classes of functions, but classes of classes will occur constantly—for example,
every cardinal number will be defined as a class of classes. Chisses of relations,
which will also frequently occur, will be considered in *21.
*20 01. f\t (^*)J . - : <f> lx. s r . yjrx:f\<f> ! 2| Df
*20 02. x€(<t>'.z).~ .<t>lx Df
*20 03. Cls«a Kg*).a« $($!*)) Df
The three following definitions serve merely for purposes of abbreviation.
*20 04. r, y € a . — . r € a . i /1 a Df
*20 05. x,y,z ta.-./,yta.:ea Df
*20 06. x~*c a. m . ^(xt a) Df
The following definitions merely extend to symbols representing classes
the definitions which have already been given for other symbols, with the
smallest possible modifications.
*20 07. («)./a.-.(0)./(3(*!«)| Df
*20 071. (3<*)./««- -(3$>•/!*<♦ **>l Df
*20 072. [(ia) (</>a)] ./(?a) (<f>a) . = : ( 37 ): <f>a . =. . a = 7 :fy Df
*20 08. /Ja(^a)| . = :( 3 ^):>/ra.=. . <f> l a z f(<f> l a) Df
*20081. acyfr!a. = .'l'la Df
The propositions which follow give the most general properties of classes.
*201. I- :./|3<**)! • = s <3 <f>) :</»! x. = x . :/{*/,! [*4*2 . (*20 01)]
*20 11. h -=z-X x:D: f I 2 (+ z )\ • = -f\ z ( X z )\
Dem.
h . *4-80 .D\-zz Hp. D <f> ! x. = x . ^x : =* : <p ! x. = z . *x
[*4 36] O z. <t>lx ,= x . yfrx zf\<t> ! 2) :=*:<£!*. = x . *x :/(</>! * 1
[*10*281] 3 (3 <t>) : <t >! x. = x . yfrx :/[</>! z\ :
= :(3 <f>) z <f> l x. = x • X x z f W 1
D :./[3(^» . = • P ™P
[*20-1]
SECTION C] GENERAL THEORY OF CLASSES 191
This proves that, every proposition about a class expresses an extensional
property of the determining function of the class, and therefore does not
depend for its truth or falsehood upon the particular function selected for
determining the class, but only upon the extension of the determining function.
*20111. h :./(<*,! . =*. g(4 >! ?) : D :/(£(</>! -)] . =,. <j |3 (0 ! z)|
Detn.
H . Fact. Dh:: Hp . 3 :. <filx . =, . ^ ! * :/(>/r!2): = :<£!,r.=,.>Jr!.r: ::
[*1011 21] DhstHp.D :.<*>!*. =,.^!*:/(ifr! 2): <7(^12):.
[*10-281] 3:.(a^-):<#.!*.=,.^!x:/(^!2):s:(a^):^!*.Sx.-f !..:j7(^!2):.
[*201] D :./(2(* l *)). = . g [2 (^!.r)] ‘ ( 1 )
H.(l). *1011-21. Dh. Prop
*20 112. I- ( a!7 ) :./(2(* ! ,)) . .,, ■ (2 ! *))
Dem.
h . *121 . D h < a!7 ) :/(<*>! 3) . =* . ^ ! («*>! 2) (1)
h . (1) . *20*111 ,DK Prop
Thus the axiom of reducibility still holds for classes as arguments.
*20 12. h : < a $) :<t>lx.= x .yf,x:f{3 <**)) . = •/{* (<*>! *)! [*2011 .*121 ]
*2013. I- yjrx . =, . X x z D . 2 (>/rz) - $ ( X z)
The meaning of u 2(\frz)** ^(\ 9 ) ” * s obtained by a double application of
*20 01 to *13 01, remembering the convention that z('Z'z) is to have a larger
scope than ^( X z) because it occurs first.
Dem.
l-.*20 1.DI-::2(^)=2( x *). = :.(a<#.):^x. Sl .^!x:^!2 = 2( x *):.
[*20-1] s-..(_' A 4,,e):.^x.s,.4>lx: x x.=,.0lx:4>Si = OlS (1)
h . *12-1 .*10-821 . D
h :: Hp . D ( a <£) : . = x . <f >! a:: • =* • <f> 1 x
[*13-195] D (g<£, 0) yfrx . = x . <f> l x z X* • =* • & '• x z $ l 2 = 01 2 (2)
h • (1) . (2) . D f-. Prop
*20 14. H 2 (\/rz) = 2 (^*) . 3 : yfrx .= z . x x
Dem.
I-. *201 . D h :: 2 (yjrz) = z ( x z ) • = (3 <f>) • y l rx . =, . <f >! x : <f >! 2 = 5 (^z)
[*201] s ( a <J>, 0)z.ylrx.= x .4>\xz x *-^*-O\xz<t>l2=0\$z.
[*13*195] = ( a <£) \Jrx . = x . <pl x z x x • = x • 4>' x
[*10322] D:.^.= I . XI ::Dh. Prop
This proposition is the converse of *2013.
*20 16. h yjrx . = x . X X z = . 2 (yfrz) = 2 ( X z) [*201314]
This proposition states that two functions determine the same class when,
and only when, they are formally equivalent, t.e. are satisfied by the same set
of values. This is the essential property of classes, and gives the justification
of the definition *20 01.
102
MATHEMATICAL LOGIC
[PART I
*20151. K(g<*>>.2(^) = 3<.*>!*)
Dem.
b .*2015. D b :. yjrx ,= x . <f>l x i D . z (^r*) = z (<f >! z)
[*10*11*28] D b :. (g<£): >frx . = x . 0 ! x : D . (g<£). z(y\rz) = z(<f >! z) (1)
b . (1). *1 21 . D b . Pmp
In virtue of this proposition, all classes can be obtained from predicative
functions. This fact is especially important when classes are used as apparent
variables. For in that ease, according to the definitions *2007 071, the ap¬
parent variable really involved is a predicative function. In virtue of *20151,
this places no limitation upon the classes concerned, except the limitation
which inevitably results from the nature of their membership. A class, there¬
fore, unlike a function, has its order completely determined by the order ot
its possible members, i.e. of the arguments which render its defining function
significant.
*2016. t-:<3*):/|3(*s)!.5./|}(*!x)| [*2012]
*2017. h :<*> 3./|J<**>| [.2016 .*101]
*2018. h:.2(*x) = 2<*x).3:/|2($x)|. = ./[2<*x)I [*201115]
*20 19. h :. 2 <*.-) = 2 ( X x). a : (/):/! 2 (*.-). 3 ./! 2 ( X x)
Dem.
V . *2018 . *101121.3 H :. 2(*r) = 2( x x) . 3 :
(/):/! »(**)-3. /! 2 (x*) <»
H . *201815.3 V :: <*>! .r. s, . 'f'x : 0 ! x. s,. :/! 2 (>Jrx). 3 ./! 2(x*) ■ 3 :
/!2(4>!x).3./!2(0!x) (2)
h. (2). *1011-27 33.3
h :: A ! x . s, . <tr* : 61 x . s,. X *:. (/):/! 2 (ifrx). 3 ./! 2( X x) :. 3 :•
(/):/! 2(^.1 x). 3./! 2(5! *):•
[*20112.*10-1 ] 3:.^!r.=,.*!x:3:*!x.= 1 .tf!x:.
[*4*2] D <t>lx.= x .0lx:.
[*10*301*32. Hp] Ds.ifrx.s,.**:.
[*2015] D 2 (**) «$<**) (3>
b.(3). *1011*23*35 . D
h :: (a*, 01: d>! x. =*. **: 0! *. s,. x *:. (/) s/12 (*x) • 3 ./! 2 ( X x) *•
3.2(tx) = 2(xx) W
I-. ( 4 ). * 121.3 I-:.(/):/! 2(*x). 3 ./! 2< x x): 3.2 (*x) = 2( x x) (5)
h . (1). (5). 3 h. Prop
*20191. I-:. 2 (<[rx) = 2 ( X 2 ) . = :(/):/! 2 (-fx ). s ./! 2( X x)
[*2018-19. *10-22]
*20 2. 1-. 2 (<*.;) = 2 (4>x)
Dem. , .
h. *2015.3 1-:. 2(^x) = 2(<^x). = :^x.= x .^.x
1-. (1) . *4 2 .*1011.31-. Prop
( 1 )
SECTION Cj
CGNKRAL THEORY OF CLASSES
103
*2021. I- = . 3(05) = 3(0:) 1*201”,. *IO-32|
*20 22. 1- : 3 (0.-) = 3 (0j) . 3 (0j) - 3 (*;> . D . 3 ( 0 j) - 3 (*;)
[*2015.*10:U)1]
The above propositions are not immediate consequences of *13*1",* hi* I 7.
for a reason analogous to that explained in the note to *14* 13. namely because
( <f>z)\ is not a value of f.r, anil therefore in particular " 3 (05) = - (0-» is
not a value of “ x = y."
*20 23. 1-: 3 (05) - 3 (05). 3 (05) = 3 ( x j) . D . 3 (0: ) - 3 (*5) [*20*21 -22)
*20 24. I-: 3(05) = 3(05) . 3(*5>- 3(05). D . 3 (05) = 3 (*5 > [*20*21*22]
*20 25. h a = 3 (05). =. . a-3(05) : = . 3 (05) - 3 (05)
Dem.
h . *10'1 . DI-:.a-J(f-).s..a.J(^)0 :
3 (05) = 3 (05). = . 3 (05) -3(05):
[*20-2] 3:3(05)-3 (05) (1)
h . *20 22 . 3 h : a = 3 (05) . 3 (05) = 3 (05) . D . ct = 3 (05):
[Exp.Comm] 3 h 3 (05) - 3 (05) . 3 : a = 3 (05) . 3 . a — 3 (0-) (2)
h . *20 24 . 3 h :. 3 (0-) - 3 (05) . a - 3 (05). 3 . a = 3 (05)
[Exp] Dh:.3(05)« 3(05). D:a = 3(05).D.a = 3(05) (3)
h . (2) . (3). 3 h :. 3 (05) - 3 (05) . 3 : a - 3 (05). a . a - 3 (05)
[*1011-21] 3 1- :. 3 (05) - 3 (05) . 3 : a - 3 (05). =„ . a - 3 (05) (4)
K(l).(4). 3 h . Prop
*20 3. h : x € 3 (05) . = . 0.r
Dem.
h . *201 . 3
h :: a:« 3 (05) . = (g0)0y . . 0 ! y : a; € (0 ! 3)
[(*20 02)] = (a0) 0y . =„ . 0 ! y : 0 ! a:
[*10-43] =:.(a0):.0y.= f/ .0!y:0a::.
[*10-35] = (30) : 0y - 3*. 0 ! y 0*
[*12 1] = :. 0a::: 3 h . Prop
This proposition shows that x is a member of the class determined by 0
when, and only when, x satisfies 0.
*20 31. h 3(05) = 3 (x^) • = :a;c3(05). = x .xez{x z ) [*20T5*3]
*20 32. h {a: € 3 (05)) =3(05) . [*20*315]
*20 33. h a = 3 (05) . = z x e a. = x . <f>x
Dem.
h. *20*31. D h a = 3(0z) . = : a: c a . = z . xc 3(05) (1)
I- . (1) . *20‘3 . 3 h . Prop
Here a is written in place of some expression of the form 3 (05). The
use of the single Greek letter is more convenient whenever the determining
function is irrelevant.
R&W i
13
194
MATHEMATICAL LOGIC
[PART I
*20 34. f- x = // . = : x € a . 3« . y c a
J)em.
H . *4*2 . (*2007). 3 h xt a . 3„ . y € a z = : x e z (tf> l z). Of y e 2 (<f> \ z) i
[*20*3] = z <t>l x .0* . (f> \ y :
[*I.‘H] = : x = y 3 h . Prop
The above proposition and *20 25 illustrate the use of Greek letters as
apparent variables.
*20 35. x = y . = z x t a . . y € a [*20*3 . *13*11]
*20 4 h:a f CIs . = . ( 3 *). a - z (</>! z) [*20*3 . (*2003)]
*20*41. h . 2 (**) < CIs [*20 4151]
*20 42 K;(:«a) = o
A Greek letter. sueh as a, is merely an abbreviation for an expression of
the form thus this proposition is *20*32 repeated.
Deni.
h . *20*3 . * I 0* 11 . 3 h : x t 3 ( \frz ). = x . yjrx :
[*20 15) 0 h . J* \x € z (^rr)l = ^(|i).DI*. Prop
*20 43. b z.a-0 .m zxt a.m, .x€0 [*20*31]
The following propositions deal with cases in which both classes and
descriptions occur. In such cases, we shall, in the absence of any indication
to the contrary, adopt the convention that the descriptions are to have a
larger scope than the classes, in applying the definitions *14*01 and *20 01.
*20 5. b (yfrz). = . ^ |(l.r) (</»./ ))
Dent.
I-. *14*1 .Oh zz (ix)(<f>.r)c z (y\rz ). = (yc): 4>x . =, . X- c z ce? (yjrz) :•
[*20 3] = (yc): <f>x . =,. x «= c : yjrc
[*14*1] = yfr |(»x)(<£x)J :: 3 b . Prop
*20 51. I- :. ( ix) (<f>r) = b . = : ( ix) (<f>x) ea.= 0 .bea
Deni.
b . *20-5-3. 3
h:.(w)(^r) f J(f !*). = .&«?(*!*): s :*!(>*)(£*)• = 3
[*10 11 ] t-:. (»x)($.r) (a.=..k<a: = :f ! (« )(£*)• =»• 'I'i b !
[*1417] = :(«)(£*) = &:. DK Prop
*20 52. b E ! (ix)(<£.r). = : (y5): (ix)(^r) € a . =« . 6 ea
Dent.
K *20 51 .*1011-281.3
b (y&). (ix) (<f>x) = b . = : (y 6 ) : (ix) (<f>x) c a . =«. b e a (0
h. ( 1 ). *14*204.3 h. Prop
*20*53. h Q = a . 3* . </>£ : = . <f>a
This is the analogue of *13*191.
SECTION C]
Dem.
GENERAL THEORY Ob* CLASSES
195
K*10 1.
D h fi = a . . <f>/3 : D : a = a . D . 4>a z
[*20-2]
D : <f>a
O)
h .*2018-21
.Dh./3 = a.D:</)a.D. </»/9:.
[Comm]
Dh:.0a.D:/5 = a.D. <f>@ z.
[*1011-21]
D h <pa . D : = a . . </>/$
(2)
K(l).(2).
D 1-. Prop
*20 54. h : (g/9) . /9 = a . <£/3 . = . <f>a
This proposition is the analogue of *13 195.
Dem.
h . *2018 . *1011 . D h : /3 = a . <J>£ . . <f, a :
*2055.
[*10-23]
h . *20-2 . *3-2 .
[*10-24]
I-. 2 (<£t) = (?a) (.rea.
3 h : (3/9) . /9 = a . <f>/3 . D . <f>a
Z > h : </>a . D . a = a . </>a .
D I-. Prop
-* • 4> x )
Dem.
(1)
( 2 )
h . *20 33 . D h xea. m x . <f>x : =« .a— 2(<£*)
[*20-54] D h (g/9) t e a . s, . <f>x : =. . a - /3 2 (</»j) =. /3 :.
[*141] D H . 2 (<f>z) = (?a)(Te a . =, . <f>x) . D h . Prop
*20 56. h.E!(?a)(^«a.= z .fc) [*20 55 . *14 21]
*20-57. Vz.2 (<f>z) = (7a) (/a) .0 ig{3 (<f>z)\ . = .g {(?a) (/a)|
Dem.
h.*141. Dhs:Hp. = s.(a/3):/a.s,.a«/3:2(^)-/3:.
[*20-54] s:./a.=..a = 2(<^) (1)
h . *141 . D h <7 ((7a) (/a)| . = : (3/9) :/a . =. . a = /3 : g/3 (2)
h . (1) . (2) . D H :: Hp . D £r ((7a) (/a)] . = : ( a /9): a = z (<f>z). = a . a =/3 z g/3 :
[*13183] = : ( 3 £) . a (**) = /9.<//9:
[*20*54] = : *7 (2 (<£*)} :: D h . Prop
*20-58. h . 2 (<f>z) = (70) (a = $ (<£*)}
Dem.
h. *4-2. *1011 . D h za = 2(<f>z). = m . a = z (<f>z) :
[*20 54] DV:. ( 3 /3) :. a = 2 (02). =. . a = 0 : 2 (02) = £
[*141] D H . 2 (0 2 ) - (ia) {a = 2 (02)) . D h . Prop
*20-69. I-: 2 (02) = (la) (/a). = . ( 70 ) (/a) = 2 (02)
Dem.
V . *20-1 . D I-a (<£r) = ( 7 a) (/a). = : ( 3 ^) z <f>x . = x . yfr l x z + l 2 = (?a) ( fa) z
[*14-13] = : ( 3 yfr) : <f>x . = x . yfr ! x z ( 7 a) (/a) = yjr ! 3 :
[* 201 ] = : (7a) (/a) = 5 (*») :.Dh. Prop
13—2
190
MATHEMATICAL LOGIC
[PART I
In the following propositions, we shall prove that classes have all the
formal properties of individuals, and have the same relations to classes of
classes as individuals have to classes of individuals. It is only necessary to
prove the analogues of our primitive propositions, and of our definitions in
cases where their analogues are not themselves definitions. We shall take
the propositions *10111 12121122, rather than those of *9, and we shall
prove the analogue of *10-01. As was pointed out in *10, we shall thus have
proved everything upon which subsequent proofs depend. The analogues of
*200102 and of *14*01 remain definitions, but those of *10*01 and *13*01
become propositions to be proved. *9*131 must be extended by the definition:
Two classes are "of the same type" when they have predicative defining
functions of the same type*. In addition to these, we have to prove the
analogues of *10 1 11 12 121 122. *11 07 and *12*1*11. When these have been
proved, the analogues of other propositions follow by merely repeating previous
proofs. These analogues will, therefore, be quoted by the numbers of the
original propositions whose analogues they are.
*20 6. h : (ga) ./a . b . -(a) .~ya|
Dan.
h. *4 2. (*20*071).}
: < 3 Q ) •/« • = • ( 3 <t>) •/{+ (<t> ■ *)| •
[(* 1001 )] ~/|2 (<*,!*)}].
[(*20 07)] = .~|(o) .~/a; Oh. Prop
This is the analogue of *10*01.
*20 61. h :(a).fa.D.f/3
Dan.
h . *10*1 . (*20 07). D h : (o) ./a . D ./(2 (<*>!*)): D h . Prop
This is the analogue of *10*1.
In practice we also need
This is *20*17.
h:(a)./a.D./|3(^».
We need further h . (ga). z (>/r*) = a.
This is *20*41.
*20 62. When //3 is true, whatever possible argument of the form z(<f>' 2 )
/9 may be, then (a).fa is true.
This is the analogue of *10*11.
Deni.
h . *10*11 . D . when f\z(<f >! z)\ is true, whatever possible argument (f> may
be, then (<f>) ./{2(<£ l z)) is true, i.e. (by *20 07), (a) ./a is true.
*20 63. h :. (a) . p vfa . D :p . v . (a) .fa
This is the analogue of *10*12.
SECTION C]
Dem.
GENERAL THEORY OF CLASSES
11)7
I-. *4*2 . (*20 07). 3
h :•(«>•/> v ./« • = :(</»)./» v/|2 (</>! *)j :
[*1012] = : • v . (</>) ./|? (<£! ^)j :
[(*2007)] s :/>. v . (o) ../a :.DF. Prop
*20 631. If “/a” is significant, then if 0 is of the same type as a, " f0" is
significant, and vice versa.
This is the analogue of *10121.
Dem.
By *20151, a is of the form 2 (<f> ! z), and therefore, by *20 01, fa is a
function ot <f >! 2. Similarly 0 is of the form z (yfr ! z), and f0 is a function of
! 2. Hence by applying *10121 to <f >! 2 and \fr ! 2 the result follows.
*20 632. If, for some a, there is a proposition fa, then there is a function fa,
and vice versa.
Dem.
By the definition in *20*01, f\z (\fr ! *)| is a function of \fr ! z. Hence the
proposition follows from *10*122.
*20*633. “ Whatever possible class a may bc,f(a, 0) is true whatever possible
class 0 may be” implies the corresponding statement with a and 0 inter¬
changed except in “/(a, 0 ).”
This is the analogue of *11*07, and follows at once from *11*07 because
/(a, 0) is a function of the defining functions of a and 0.
*20 64. h ( a ) .fa z(a).gazO .f0 . g0
Dem.
H . *4*2 . (*20*07) . D
h (a) .fa : (a) . ga s = : ( <f >) ./ (2 (<*>! z)} :(<f>).g (2 (<j >! z)} z
[*10*14] D :/{*<* t *)}'?(?(*!*)):. 3 h . Prop
Observe that " 0" is merely an abbreviation for any symbol of the form
2 (>/r ! z ). This is why nothing further is required in the above proof.
The above proposition is the analogue of *10*14. Like that proposition,
it requires, for the significance of the conclusion, that / and g should be
functions which take arguments of the same type. This is not required for
the significance of the hypothesis. Hence, though the above proposition is
true whenever it is significant, it is not true whenever its hypothesis is
significant.
*20*7. h :(3 g):fa.= m .gla [*20 112]
This is the analogue of *12*1.
*20*701. h : (apr) :/ {2 (<f> l z), x}.=+ iX .g ! )2 ( <f >! z), x\
[The proof proceeds as in *20 112, using *12*11 instead of *12*1.]
198
MATHEMATICAL LOGIC
[PART I
*20 702. h : (a if) :/|x, z (4 >! z )| . =*. z . <j ! Jo:, 2 (<£! z ))
[Proof as in *20 701.]
*20 703. H : ):/15 (.*>! *), 2 (*! *)} . =*.*. g ! (S(* ! *). 2 (*! *))
Dem.
h. *10311 . D h :./|x'.2, ^! 3J. = x . $ .gl \x'-*> #'• *\ ’ 3 s
0 !xE,x!x.Vr!xs x 0 !x./(x! 2 ( •=».••
<J> !x=,x!x. ^ ! ***# • *•&* lx 1
h.(l).*lMl‘3'3*1.3
1*Hp( 1). D : (ax- &) • <t> '• r =x X* x • 'l' • x -* &' x • f\x * 2. 6 ! 2) • =*.* •
(flX- 0). <t >! .r =x x ! x • ^ ! xs * 0 ! x • 0 ! lx ! 2, 0 * *) :
[*201 .*10-35] D :/|2(<£ ! *). 5<* ! *)) . =♦.*..? S «<*>! 2. *! 5) (2)
M2).*1011*281 . D
H :• (a!/) : IX * *! *1 • »*.•. 0 ! | X ! 2. 0 ! 3j: D :
(:•!'/> : /l- *</>! *). 5 <* ! *)| .=♦.*•<? t i*(*«*). n+ J *>! < 3)
h . (3) .*1211 . D h . Prop
*20*701*702*703 give the analogues, for classes, of *12*11.
*20 71. h :.a = /3. = : 7 ! a . D,. 7 !/3 [*2019]
This is the analogue of *1301.
This completes the proof that all propositions hitherto given apply to
classes as well as to individuals. Precisely similar reasoning extends this resu
to classes of classes, classes of classes of classes, etc.
From the above pro|>ositions it appears that, although expressions such as
z(<f>z) have no meaning in isolation, yet those of their formal properties wit
which wc have been hitherto concerned are the same as the corrcspon ing
properties of symbols which have a meaning in isolation. Hence nothing in
the apparatus hitherto introduced requires us to determine whether a g> ve jj
symbol stands for a class or not, unless the symbol occurs in a way in " 1C
only a class can occur significantly. This is an important result, which ena es
us to give much greater generality to our propositions than would otherwise
be possible.
The two following propositions (*20*8*31) are consequences of *13 3. The
•'type" of any object x will be defined in *63 as the class of terms either
identical with x or not identical with x. We may define the “type of t e
arguments to <f>z’’ os the class of arguments x for which ''<px is signi
*.e. the class £(<*>x v~£x). Then the first of the following propositions show
that if “fa" is significant, the type of the arguments to <f>2 is the type o a.
the second proposition shows that, if “<f>a ” and "yfra 9 are both signi can ,
the type of the arguments to <f>2 is the same as the type of the argumen
\fr2, because each is the type of a. *20 - 8 will be used in *63*11, whic is*
fundamental proposition in the theory of relative types.
SECTION C]
GENERAL THEORY OF CLASSES
190
*20 8. h : . D . .r (<f>.c v~<f>.r) =.?• (.r = «.v..r^u)
Dem.
K *13*3. *1011-21 . D
h :: Hp . D <f>.v v~<£.r .= x :.r = a.v..r + «i
[*2015] D «/>./) = .T‘(.t = (i.v./+(i)::Dh. Prop
*20 81. h : <f>a V ^<f>n . V >//“</ , D . .r (</>•'* V ~ <£>.r) = ,r (\Jr.r V ~ \^.r)
Dem.
h . *20 8 . D h : Hp . D . .** (<£.c v</>.t) * .2 (x = a . v . x + a) ( I)
*" • *20-8 . D h : Hp . D = .2(.r = a . v . .r * a) (2)
H . (1). (2) . *1012113 . Comp . D
H : Hp . D .£•(<£*• v~ <£.«•) ==:?(;r = a. v. * + «)..2 (>/r.cv~>/r.r ) = .2 (.r== a.V.,r + a).
[*20 24]D .i(^rv'v^c)»5^xv'v: D h . Prop
In the third line of the above proof, the use of *10121 depends upon the
fact that the "a” in both (1) and (2) must be such as to render the hypothesis
significant, i.e. such as to render
“ <t>u v~<t>u . yfra v yfra "
significant. Hence the “a" in (1) and the " a ” in (2) must be of the same
type, by *10121, and hence by *1013 we can assert the product of (l) and
(2), identifying the two “as.”
Since a type is the range of significance of a function, if <f>x is a function
which is always true, 2 (</>*) must be a type. For if a function is always true,
the arguments for which it is true are the same as the arguments for which
it is significant; hence 2 ( <f>z ) is the range of significance of <f>x, if (a) . <f>x holds.
Thus any class a is a type if (x).xea. It follows that, whatever function </>
may be, a! (<f>x v~<£*) is a type; and in particular, £ (x — a . v . a? + a) is a type.
Since a is a member of this class, this class is the type to which a belongs.
In virtue of *20*8, if <f>a is significant, the type to which a belongs is the class
of arguments for which <f>x is significant, i.e. 5 (<f>xv~<f>x). And if there is any
argument a for which <f>a and \fra are both significant, then <f>x and y]fx have
the same range of significance, in virtue of *20 81 .
*21. GENERAL THEORY OF RELATIONS
Snin milI'H «»/' * 21 .
I he definitions and propitious «»f this number are exactly analogous to
those of * 20 . from which they differ by being concerned with functions of two
variables instead of one. A relation, as we shall use the word, will be under¬
stood in extension: it may be regarded as the class of couples y) for which
. .. given function ^ (./•, y) is true. Its relation to the function >/r(2. 0) is
just, like that of the class to its determining function. We put
* 21 01 - A f !*.5 * < r - //>: •=:<:•!</» : 0 s (^. y>. ~x.., . * ^, y) :/{<#>! (u . V)} Df
II. re ".r#7^r(.r, //)” has no meaning in isolation, but only in certain of its uses.
In *2101 the alphabetical order of u and v corresponds to the typographical
order of 2 and y in /;.»? ^ (.»•. y)\, so that
/1 $*♦<*.!/)[ • - : (3+) : <t> '• <•*. //> • =x tJ , - * br. y) :/{* ! ($, fi)| Df
'I Ids is iiii|N>rtaut in relation to the substitution-convention below.
It will be shown that
y> - <- r * y) • s: ^ y) • =x. w • x <•*’. y).
/.c. that two relations, as above defined, are identical when, and only when,
they arc* satisfied by the same |»air of arguments.
For substitution in <£!(2. 0) and <£!(0.2), we adopt the convention that
when a function (as opposed to its values) is represented in a form involving
•' and 0, or any other two letters of the alphabet, the value of this function
for the arguments a and b is to be found by substituting a for 2 and b for 0,
while the value for the arguments b and a is to be found by substituting b
for .7 and a for 0. That is. the argument mentioned first is to be substituted
for the letter which comes first in the alphabet, and the argument mentioned
second for the later letter; thus the mode of substitution depends upon the
alphabetical order of the letters which have circumflexes and the typographical
order of the other letters.
The above convention us to order is presupposed in the following definition,
where a is the first argument mentioned and b the second:
*2102. « (</, ! <2, 0)) b . = . </>! (a, b) Df
Hence, following the convention,
& 1 <t >! (2, 0)| a . =». 4 >! (&. «) Df
a{4>l($,re)}b. = .4>l(b,a) Df
M<M(0.2))a. = .<*>!(«,&) Df
This definition is not used as it stands, but is introduced for the sake of
a (20^ (*, y» 6 . = : (g<£) : <£ ! (x, y ). . >/r (x, y ): <f >! (a, b)
SECTION C]
GENERAL THEORY OF RELATIONS
•201
winch results from *21 01 02. Wo shall use capital Latin letters to represent,
variable expressions of the form ■?-/<#.! (.r, y). just as we uses I Creek letters for
variable expressions of the form ?(i!-). If a capital Latin letter, say If. is
used as an apparent variable, it is supposed that the A* which occurs in the
form "(«)" or "(aA')" is to be replaced by •t*)"or' (a0)." while the H which
occurs later is to be replaced by “.?£«#,!(.r,y)." In fact we put
(R) . fit. = . (cp) .f\jy<f> ! (.r, y)| Df.
The use of single lettei-s for such expressions as .7y<f, ,/) is a practically
indispensable convenience.
The following is the definition of the class of relations:
*21 03. Rel = Ii (( 5 , 4 ,) . R =, .7^0 ! ( x , y)\ l)f
Similar remarks apply to it as to the definition of "Cls M (*20 03).
In virtue of the definitions *210102 and the convention as to capital
Datm letters, the notation “xRy" will mean has the relation R to y." This
notation is practically convenient, and will, after the preliminaries, wholly
replace the cumbrous notation x [2y<f>(x, y)\ y.
The proofs of the propositions of this number are usually omitted, since
ey are exactly analogous to those of * 20 . merely substituting *1211 for
*1~T, and propositions in *11 for propositions in *10.
T The propositions of this number, like those of * 20 . fall into three sections.
I hose 1 of the second section are seldom referred to. Those of the third section,
extending to relations the formal properties hitherto assumed or proved for
individuals and functions, are not explicitly referred to in the sequel, but arc
constantly relevant, namely whenever a proposition which has been assumed
or proved for individuals and functions is applied to relations. The principal
propositions of the first section are the following.
*2116. I"V' (*, y).=,.,. x (x, y) : = . St$+ (*. y) = $p x (x, y )
/.e. two relations are identical when, and only when, their defining functions
are formally equivalent.
*21-31. I- sp* (x, y) - $p x (x,y). = :x y) ] y x \xp x (x, y )) y
I.e. two relations are identical when, and only when, they hold between
the same pairs of terms. The same fact is expressed by the following
proposition: ”
*2143. h R = S . 5 : xRy . . xSy
* 21-22122 show that identity of relations is reflexive, symmetrical and
transitive.
*21-3. m 'X{&9'lr(x,y)\y . = .ylr(x,y)
I.e. two terms have a given relation when, and only when, they satisfy its
defining function. J
202
MATHEMATICAL LOGIC
[PART I
*21151. h . (g<£) . uyxfr (x, y) = .7y<f> ! (x, y)
I.e. every relation can be defined by a predicative function. Hence when,
using *21 07 or *21 071, we have a relation sis apparent variable, and are there¬
fore confined to predicative defining functions, there is no loss of generality.
*2101. /{.ry\fr(.c, y) . = : (g<£) : <f >! (x, y) . = x>y . >Jr (x, y) ! (m, v)| Df
On the convention as to order in *210102. cf. p. 200, ami thus relate «, v
to ,7\ y so that
/<•'*. y)| • - : (3 <f>) : <t >! ( r . If) • =x. v • ^ (•*. !f) -f\4> I («\ “))
*2102. u [<f >! (x. y)‘ 6. = . 4 >! (#/, b) Df
*21 03. Rcl = ft . Ji = .7ytf >! (x. y)\ Df
The following definitions merely extend to relations, with as little modifi¬
cation as }>ossible, the definitions already given for other symbols.
*21 07. (70 .//{ . = .<<*»)./( .7y<f> ! (.r, y)\ Df
*21071. (a/«)./7£.-.(ai)./PP*!(-r.y)| !->»'
*21 072. [(i If) ($/<)} . = : (g.S'): <f>/{ . s*. R = S : fS Df
*21 08. /{]&>!, (Ji, S )\. = : ( 3 <t>)' S ). s*. A -.*!(7f. S) :/(<*> !(7(, S)| Df
*21 081. 7* |«/>! <7f. S)| (P, Q) Df
The convention as to typographic and alphabetic order is here retained.
*21 082. /|7?(*7f)|. = : (g0): . = n . <f >! If :/(<*»! 70 Df
*21 083. I)f
*211. b *••/{?)'!'(s, y )| . = : < 3 </>): 4 >! (•*•. //>. s x .„ c. y) :/{<f >! (a, u)|
[*4*2.<*21 01)]
*2111. b + (x, y). . X <•*•. If ): ^ :/ !f)\ . = ./l-tyx ( r - .V)l
[*4-86-3G.*l0 2.xl .*211]
This proposition proves that every proposition about a relation expresses
an oxtcnsional property of the determining function.
*21 111. H :.f\4,U&,p)\.= t .9 |^!(x.y)|:D :/|^<-!(x. y)|. s, .yWH*,y))
[Fact .*1111-3. *10*281 .*211]
*21112. b :. (a y) :.f\*y<t >! <x. y)\ . =* . y ! 1 2y4> * <*. If )) [*121 . *21111]
It is *121, not *12*11, which is required in this proposition, because we
are concerned with a function (f) of one variable, namely <f>, although that
one variable is itself a function of two variables.
*2112. b :. (a <f>)<f >! (x. y ). = x . y . + (x, y) (x. y)} . = .f\?9<t >! (*, y)l
[*2111 .*12 11]
This is the first use of the primitive proposition *1211, except in
*20*701 702-703.
*21*13. b yfr (x,y). = x . y . * (x, y) : D . xp'/r (x, y) = ££* (*»
[*21*1 .*1211 .*13195]
SECTION C]
GENERAL THEORY OF RELATIONS
203
*2114. h 2f/yfr(a\ y) = xy X (.r. y) . D : yfr (x, y ). = x ., ; . v (./•, y)
[Proof as in *2014]
*2115. hz.yjr (x. y) . = ri , . x (*, y) : = . .7$+ (.r, y) = .77/ x ,/) [*21 13*14]
This proposition states that two double functions determine the same
relation when, and only when, they are formally equivalent, i.e. are satisfied
by the same pairs of arguments. This is a fundamental property of relations
as defined above (*2101).
*21-151. h . (a</,). 7y yfr (x, y) = ! (.r, y) [*21*15 . *12* 11]
*2116. h : (g</>) if [xp yjr (x, y)] . s ./ \.7y(f> ! ( .r, y)\ [*2112]
*2117. h : (<*>) ./(;?£</>! (x, y)} . D ./|.?y^(x, y)\ [* >l l<i. *101]
*2118. h :.;?£«/> (x, y) - £S}\fr (x, y) . D :/[^<f> (x. y)\ . S ./ \.77/ + (x, y )}
[*211115]
*2119. h *9+ (x, y) - 2$ x (x. y) . = : (/) :/! * (.r, y) . D ./! x (a:, y)
[*2118 . *101 T21 . *21*1 . *10-35 . (*13 01). *21 112 . *10 301]
*21191. h (x, y) - St$ x (x. y) . = : (/) :/! ty+(x, y ). s ./! (a-, y)
[*2118*19]
*21 2. h . (x, y) - (a:, y) [*21*15 . *4 2]
*21-21. h : 7$<f> (x, y) - y) . = .^y *(*.y)-*y *(*.y) [*21i 5 - *10*32]
*21 22. h : *0 0 (x, y) « 2$ * (x. y) . £0 * (x. y) = *0 x (x, y) . D .
*0 <t> (*. y) = *5x (*. y) [*21 15 . * 10-301 ]
*21 23. h : £0 <*> (x, y) - £0 * (x, y). £0 <f> (x, y) - *0 * (x, y). D .
*5'Hx,y) = $d X {x,y) [*21*21*22]
*21-24. h : x0>/r (x, y) - £0<f» (*, y) . £0* (x, y) — x0<£(x, y). D .
£9 + (x,y)-=$!) X (x, y) [*21 21 *22J
*21-3. h : x {£$^r(e,y)| y . = . y/r (x.y) [*21 102 . *10-43-35 . *12*11]
This shows that x has to y the relation determined by \fr when, and only
when, x and y satisfy \fr (x, y).
Note that the primitive proposition *12-11 is again required here.
*21-31. h £0^ (x, y) = 20 x (x, y). = : x {£0^ (x, y)j y. e x>v . x (£0* ( x , y» y
[*2115-3]
*21-32. K^[x{^0(x,y))y] = 5p0(x,y) [*21-315]
*2133. h R = ty <f> (x, y) . m zxRy (x.y) [*21*31*3]
Here R is written for some expression of the form £0-»/r (x, y). The use
of a single capital letter for a relation is convenient whenever the determining
function is irrelevant.
*21 4. h : R € Rel. = . ( a </>) . R - 20«*>! (x, y) [*20*3 . (*2103)]
*21-41. K&0<£(x, y)e Rel [*21*4*151]
*21-42. K£0(xtfy) = i* [*21*315]
*21-43. h R = 5. = : xRy . = x>v . xSy [*21*15*3]
*20 5 51-52 have no analogues in the theory of relations.
204
MATHEMATICAL LOGIC
[PART I
*21-53. h S= R . S': s . <f>R [*10 1 . *21-2*18-2! . Comm . *10*11-21]
*21*54. V (g S ). S = R . <bS . = .<£/? [*21*18. *1011*23 . *212 . *10 24]
*21-55. h . xy<f>U, y) = (tR) \xRy . = x .„. <f>(x, y)] [*21*33-54. *141]
*21*56. I - .El (iR) \.tRy . = x .„ . <f> (x, y)\ [*21*55 .*1421]
*21*57. K :. jy <f> (x. y) = uR){/R) .D:y \*y<t>(x. y )\. = . y {(iR )(/R)\
[*14*1 .*21-54. *13-183]
*2158. I- : .ry<f> (x, y) = uR )\R = xy<f>(x, y)\ [*4*2. *10*11 . *21*54 . *141]
The following propositions are the analogues of *206 If., and have a similar
purpose.
*21 6. H : < g R ). jR . = J( R ). ~~fR [ Proof as in *20 0]
*21-61. V OR) •fit - 3 -J s [Proof as in *20 01]
*21 62. When fR is true, whatever possible argument of the form xy4 >! {x, y)
R may be. (R) .fR is true. [Proof as in *20 62]
*21-63. I- :.(/?)•/> v/7? . D : p . v . (/?) . fR [ Proof as in *20 63]
*21631. If "fR is significant, then if N is of the same type ns R, "fS" is
significant, and vice versa. [Proof as in *20 631]
*21632. If. for some R, there is a proposition//?, then there is a function
fR, and vice versa. [Proof as in *20 632]
*21633. "Whatever possible relation R may be. f(R. S) is true whatever
possible relation 6* may be" implies "whatever possible relation S may be.
f(R, S) is true whatever possible relation R may be."
[Proof as in *20 633]
*21 64. h :. ( R). fR z(R) ,yR : D ,fS . gS [Proof as in *20*64]
*21*7. h : 0\y): fR .=*.*/! R [Proof as in *20 7]
*21701. h :(g g):f(R.x). = R x .y ! (R, x) (Proof ns in *20*701]
*21 702. h : (g«/): fix, R ). s Rx .y \ (R, x) [Proof as in *20*702]
*21703. H :(a'7):/(/?, S). a*.,. ffUR.S) [Proof as in *20 703]
*21704. h : (g#) :f(R. a ). = //>a . y ! (R, a) [Proof as in *20703]
*21705. h : (g y) if (a. R) ,= aN .yl (a, R) [Proof as in *20 703]
*21 71. h :. R = S . = : y ! R . D, . y ! .S' [Proof as in *2071 ]
From the above propositions it appears that relations, like classes, have
all the formal properties which the}' would have if they were symbols having
a meaning in isolation. Hence unless a symbol occurs in a way in which only
a relation can occur significantly, we do not need to decide whether it stands
for a relation or not. This result, like the corresponding result for classes
mentioned at the end of *20, is important as giving greater generality to our
propositions than they would otherwise possess. The results obtained in *20
and *21 for classes and relations whose members or terms are neither classes
nor relations can be extended, by mere repetition of the proofs, to classes of
classes, classes of relations, relations of classes, relations of relations, and so on.
*22. CALCULUS OF CLASSES
Summary of* 22.
In this number we reach what was historically the starting-point of
symbolic logic. The Greek letters used (except <f> . \Jr, y, 0) an* alwavs to
stand for expressions of the form ■?(*!■>). or, where the Greek letters are
not apparent variables. .?<«/,, ). The small Latin letters may either be sueh as
have a meaning in isolation, or may represent classes or relations; this is
possible in virtue of the notes at the ends of *20 and *21. Wo put:
*22 01. a C0 . = : .r € a . D t .arc>9 Df
This defines “ the class a is contained in the class 0 ." or “all as are 0's”
*2202. a*0-S>( xea . xe 0) Df
This defines the logical product or common part of two classes a and 0 .
*2203. ayj 0 = 2 ( X€ a.v.ae/9) Df
This defines the logical sum of two classes; it is the class consisting of all
the members of one together with all the members of the other.
*22 04. -a-a(*~€«) Df
This defines the negation of a class. It is read “not-a.” It does not
contain every object * concerning which “area” is not true, but only those
objects concerning which “area" is false; t.e. it excludes those objects for
which "area” is meaningless. Thus it consists of all objects, of the type next
below a, which are not members of a; but it docs not contain objects of any
other type but this.
*2205. a-/3 = an -0 Df
This definition gives an abbreviation which is often convenient.
The postulates required for the algebra of logic have been enumerated by
Huntington*. In our notation, they are as follows.
We assume a class K, with two rules of combination, namely u and ^ ;
and we then require the following ten postulates :
la. a vj b is in the class whenever a and 6 are in the class.
16. a r» b is in the class whenever a and 6 are in the class.
II a. There is an element A such that a \j A = a for every clement a.
II 6. There is an element V such that a rs V = a for every element a.
Ill a. a v 6 = 6 w a whenever a, b, a v b and 6 v/ a are in the class.
Ill 6. a r* b = b n a whenever a,b,anb and 6 n a are in the class.
• Tram. Amer. Math. Soe. Vol. 5, July 1904, p. 292.
206 MATHEMATICAL LOGIC [PART I
I\ a. a v (b r\ c) = (a b)r\(a v c) whenever a, b, c, a v b, a v c, b r\ c, a v(bnc),
ami (a c) are in the class.
IV b. a n (b \j c) — (o r\ b) \j {a r\ c) whenever n, b, c, a r* b, a r\ c,b \jc, an (hue),
and (</ r\ b)\j (.1 r\ c) are in the class.
V. If the elements A and V in postulates II a and II 6 exist and are
unique, then lor every element a there is an element — a such that
11 \j — (i — \ and a r> — a = A.
VI. There are at least two elements, u and */, in the class, such that -r + y.
The form of the above postulates is such that they are mutually inde¬
pendent, i.e. any nine of them are satisfied by interpretations of the symbols
which do not satisfy the remaining one.
For our pur|M)ses, “ A' " must be replaced by ,, C , ls." A and V will be the
null-class and the universal class, which are defined in *24. Then the above
ten postulates are proved below, as follows:
I a, in *22 37, namely .a v ft e CIs "
I b, in *22*36, namely **h . n r* ft e CIs"
II a, in *24 24. namely "K . « w A — a ”
II b, in *24 26, namely "I-.or* V - a"
III a, in *22*57, namely "Kov^^va"
III b, in *22*51, namely "h . a n ft = ft r% a "
IV a, in *22*69, namely 0 H . (u u ft) n (a w 7) = a v (ft 0 7) "
IN’ b, in *22*63, namely *‘h . (0 n/i)w(any| = an(^u 7 )''
V, in *24*21*22, namely " h . a n - a - A" and “Kau-fl-V"
VI, in *24*1. namely “h.A + V
Hence, assuming Huntington's analysis of the postulates for the formal
algebra of logic, the propositions proved in what follows suffice to establish
that this algebra holds for classes. The corresponding propositions of *23
and *25 prove that it holds for relations, substituting Rel, o, /S, A, V for
(Jls, \j, r\. A, V.
The principal propositions of the present number are the following:
(1) Those embodying the formal rules:
*22 51. b-.anft = ftr>a
*22 57. \-.uvft = ft'ja
These embody the commutative law.
*22*52. . (a r\ ft) r> y = a c (ft y)
*22 7. h .(a V ft) yjy = av(ftvy)
These embody the associative law.
*22*5. Kana = o
*22*56. h . a u a = a
These embody the law of tautology.
SECTION C]
CALCULUS OF CLASSES
207
*22 68. * (a r\ {3) u (a r\ y) = Q r\ v y)
*22 69. f-.(<.«^)n (I , U7 ) = ou(l8n7)
resuTtefmm'ttt 1'',° dU * rib " 1 tiw . ta "- U seen that t-ho second
Zhilin *"* b> ' e ' PrV ' Vhere *• »*"’ ... -d
*22 8. K-(-<,) = «
This is the principle of double negation.
*22^81. h:aC/9. = . — £C-a
This is the principle of transposition.
(2) Other useful propositions:
*2244. h:aCtf./9C 7 .:>. Q C 7
*22*441. haC^.xea.D.jf^
1 hese embody the two forms of the syllogism in Barbara.
*22 62. f- :aC£. = mQKJ @ = @
*22621. h:aC/3. = .ar*/9=sa
an equation™ pr ° p0sitions enable us 10 ‘™»aform any inclusion (oC/3) into
*22 91. h.aw^ = au(^_ Q )
from «»"“ ° r *" iS identicnl with ““ ‘he P«t of 0 which is excluded
* 2201 .
* 2202 .
*2203.
*2204.
*2206.
* 221 .
* 222 .
*223.
*2231.
*2232.
*2233.
*2234.
*2236.
*22351.
Dem.
C£.D x .xe>9 Df
*£(*€<*.*6/3) Df
v^/3 =»aOrca. v.x€0) Df
-a =5(x~ fa ) Df
- 0 =an-/3 Df
•••*C0,mzxea.D x .xe0
• ar 'fi c =£(xea.X€/3)
• — a = ^ e a)
• a — /3 = £(xea. x^e 0)
•X€ar>Q. = .x€a.xe@
m -X€aKj0. = iX€a.v.xe/3
i x e — a . = . a?<^e a
• —a=h a
[*4-2.(*2201)]
[* 20 - 2 . (*22 02 )]
[*20 *2. (*22 03)]
[*20-2. (*22 04)]
[*20-2 . (*22 05) . *22-2
[*20-3. *22-2]
[*20 3. *22*3]
[*20 3 .*22-31]
*20-32]
K *2235.*519. DH:~{* € _ a . = .* € ci) :
[*1011] Dh:(x):^{x€-a. = .areaJ :
[*10 251] :> h j(x) z X e — a . = .Xea] :
[*20-43.Transp] D H :~(- a = a) : D h . Prop
208
MATHEMATICAL LOGIC
[PART I
This proposition is used in proving that the null-class is not identical
with the class containing everything (*241), which is used to show that at
least two classes exist. Our axioms do not suffice to prove that more than
one individual exists, but they prove the existence of at least two classes and
at least two relations.
*22 36. h.a a/3c CIs [*20 41]
*22 37. CIs [*204 1 ]
*22 38 b. -at CIs [*20 41]
*22 39. b . 2 (</>.') n Uf,:) = 2 <0j . yfrz)
Deni.
h . *22*33 . D I-: xtz (Qt) r% z (yfrz) . = . xt z ( <f>z) . .r * z (\Jsz) .
[*20*3] = . <£.r. yjr.r (1)
H . (I). *20 33 . D h . Prop
*22 391. y . z i<f>z) z lyfrz) = z i<t>; v yfrz) [Similar proof]
*22 392. b . — z{<t>z) ■ 2 (^<t>z) [Similar proof]
*22 4. b :.aC0.&Ca. =
Dem.
b . *22*1 . D b :: a C & . = : .re a . D, . .#•« ft f) C a . i : ./* $ . D x . xe a :.
[*4 38] D b :: a C # . # C a . = z. x c a . 0 X . x e @ z x e fi . D x ,xea
[*10*22] DI-. Prop
*22 41. b:aCfS»f3Ca. = .a = fS [*224 . *20 43]
*22 42. b.aC* [Id. *1011]
*22 43. b:ar\f3Ca [*3*26 . *10*11]
*2244. b:aC{3.t3Cy.0.aCy [*10.3]
This is one form of the syllogism in Barbara. Another form is the following:
*22 441. haC^.Xfa.D.xe^ [*10*1 . Imp]
*22 45. b:aC/3.aCy. = .*Cfir\y
I)em.
h . *221 . D b a C & . a C y . = : X e a . 0 X . x e 0 z x e a . D x . x e y :
[*10*29] = zx€a.D r .x€/3.xcyz
[*22*33.* 10*413] Prop
*22 46. l-:.r6a.aC^.D.j6/3 [*22*441 . Perm]
*22 47. b:aCy.D.arsf3Cy [22*43*44]
*22 48 b:aCfi.D.ar\yCf3ny [*10*31]
*22*481. bza = &.D.any = finy
Dem.
h . *22*41 .D:.Hp.D:aC£.£Ca:
[*22*48] D:ar\yC0r\y.0nyCar\yz
[*22*41] D : a r\ y = 0 n y D b . Prop
SECTION Cl
CALCULUS OF CLASSES
*200
<l>
*22 49. ^:aC/3.yC8.D.QnyCiS^8 (*lo:io)
*22 5. h . q n a = a
Ban.
. **22 33 . D h z. .r e a r\ a . = : .<•«? a. .#• * a :
[«"»*] s:.r«,
H . (1). *10*11 . *20 43 . D h . p, op
The above is the law of tautology for ,ho logical multi,dieution of classes.
*22 51. h.a n 0-# na [*22:t8. *4 :!. *1011 . .2048]
*22-52. K(« n fj) n , = , n( ^ 7 | [*22-33. *4-32. *10 11. *20-48]
lnws l h R f° gical multi P lic,,tion of C| »SSM obeys the commutative and associative
.aw, References to *22 3MM5 and to .20 48 will i„ future often be omitted.
*22 53. an@ny=z(an&)r\y Df
This definition serves merely for the avoidance of brackets.
*2254. h:.«=/9.D:aC'r. = ./3C 7 [.2018]
*22 65. h:.o = /3.D: 7 Ca. = . 7 C/9 [.2018]
*22-551. h:a = /3.D.ou 7 = ^u 7 [.10-411]
*22156. h . a u a = a [*4-25 . .1011]
The above is the law of tautology for the logical addition of classes.
*22 57. h. aw /3-^ u a [.Ml . *1011]
*2258. KaCa^/S.tfCau# [.13..2-2]
*22 59. h:eiC 7 ./9C 7 . = .av / SC 7
Bern.
I':*® 1 . • 3 h :: H P • = * e a . D* .* e 7 : a e 0 . D,. * e s . ^^11'- v
. 0 - 22 ] S *««. D .«, 7 3
[*22-84.»10-413] =:.(x): It «v«. 3 . It7 . :3K ;' ^cc. J[- 0;
Ihe analogue of *478, x.e.
is false. We have only ° C * ‘ ^ 7 = H . a C 0 u 7 .
. “Cfl.v.aC 7 :D.aC,9u 7 . * V '*W '
A similar remark applies to the analogue of .4-79. Cf. *22-64-65 '
*226. h.o: ea ^.5:aC 7 ^C 7 .D y .x f7
Bern.
K.22-59.3l-,.«C 7 ./ l C 7 .D:.. a „^. 3 .«. 7! .
[Comm] Dh.j;6auj3.D:aC 7 .flC70
[.22-58]’ 3h "“ Cl '^ C ^ :, .- I '7 :: ):« c «v^. 5 Cav^.3. xel , ufl:
k • (1) . (2) .Dh. Prop (2)
R&W I
14
210
MATHEMATICAL LOGIC
lPART I
*22 61. H:oC/3.D.oC/3u 7 [*2244 58]
*22 62. H:aC/3. = .avy/3 = £
Dew.
I-. *4 72 . DJ*::A«a.D.ar€ < 8: = :.x«a.v.j'€/9: = .ar € /9:.
[*22*34] = x e a u /3 . = . a: e fi (1)
I-. (1) . *1027 l.DH:: oC5. = :.a , tawi3.= J . 1 re/3:.
[*2042] =:.au^ = /3::Dh. Prop
*22 621. h:aCi?. = .an/J-fl [*4*71]
The prool proceeds as in *22 62. The proposition *22 621 is one of the
most useful propositions in the present number.
*22 63. t'za\j{ar>fi) = a [*444]
I he process of obtaining *22*63 from *4 44 is of the same kind as the
process employed in the proofs that have been written out in this number.
Hence only *4 44 is referred to. We shall similarly restrict references for
later propositions in this number. The process is always roughly as follows:
/), ij. r are replaced by xea. xtfi, xty, then *1011 is applied, and such
further propositions of *10 as may be required, together with *22*33 34*35.
*22 631. h.ar»(aw/9)=ro [*22 58*021]
*22*632. = [*22*42*021 ]
*22 633. l-:QC^.D.flV7 = (on^)u7 [*22*551*621]
*22 64. H:.aC 7 .v./9C 7 :D.an/9C 7
Dew.
h. *22*47*51 .0\-:aCy.0.ar\/3Cy:/3Cy.5.ar\/3Cy (1)
K(1). *4*77. DK Prop
The converse of this proposition does not hold, because the converse of
*10*41 does not hold.
*22 65. H:.aC£.v.aC 7 .O.aC/9v 7 [*22*61*57 . *4*77]
Here again the converse is untrue.
*22*66. f*:aC/3.D.ou 7 C^v 7 [*2*38]
*22 68. K (a n ^3) w (o n 7 ) = o « (/3 w 7 )
Dew.
h . *22*34 .Dh::j*f[(aft^)u(ar\ 7 )|
[*22*33]
[*4*4]
[*22*34]
[*22*33]
I-. (1) . *10*11 . *20*43 .DK Prop
= z.xea.xeft.v.xea.xeyz.
= z. x e a z x c /3 . v . x e y z.
= z.xe a ,xe0 v y
= z.xea r\ (/3 v y) (1)
SECTION Cl
CALCULUS OF CLASSES
21 I
*22 69. • (a \J fi) n (a \J y) = a \J n y) [.Similar proof, bv *4 41 |
The above propositions *22 ««-69 are the two form* of the .listrilm.ive
low Note that e.ther results from the other by interchaining the sem-
addition and multiplication. ® n
*227. l-.i fl ^)u 7 = au( ^ 7 ) [*4:53]
*22 71. outfu 7 = ( 0u ^ U7 l_>f
*22 72. haC 7 .^C8.D.Qu/iC 7 ^5 (*3 4*J
*22 73. h:a = 7 .^ = 2.3.ou/^ = 7 u5 [*10 411]
*22 74. han^C 7 .an 7 C^. = .Qn/j = on 7
Dem.
h . *22 43 . *4 73 . Dh :an^C 7 . = .an/?Ca.an/JC 7 .
t* 22 45 J =.an^CoA 7
7. /3
A5*
DH:an 7 C/3.a.an 7 Car»/9
(1)
( 2 )
r*99d.i t 2> ’ * 4 ' 88 • 3 h! “"^ C T-«" 7 c ^.a.a<'^Ciiny.«nyCon^.
J = . f> ^ /Q _ ___ -V L I» _
*22 8 .
*2281.
*2281]
*2282.
*2283.
*22831
*2284.
*22 85.
*2286.
*2287.
*22
*2288.
*22 9.
*2291.
Dem.
h . *5*63
[*413]
haC^.E.-^C-a
[*41]
•• H:aC -/?.« ,/SC-a
[*41 . *22*8]
H:ar»/3C 7 . = .a— 7 C
-£ [*4 14]
1- : a « & . = . - a - - £
[*411]
• h:a«-£.s
[*412]
h.-(fl«/3) = -av-/3
04*51]
h.«r»^=n-(_ a v-/3)
[*22*84 831]
[*4 57]
H . - ct « _ £ « _ ( a w £>
[*22*86*8311
04*85*86*87 are De Morgan's formulae.
H . (ar) . a;« (a w — o) •
[.211]
1 18 a f orm of the law of excluded middle.
r . (a:) . a:~e(a — a)
[•3 24]
1 1S a ,orm «* the law of contradiction.
h.(au£)-/3 = a _ / 9
[*5-61]
h.aw/3 = aw(^_ a )
r ™ ^ z - X€a - y/ = :*€<*.v .xe/3 .x~ €a .
, ■* =i&ea\j(S — a)
h • (1) • *1011 . *20 43 . D h . Prop
(1)
14—2
212
MATHEMATICAL LOGIC
[PART I
*22 92. K:aC/3.D./9 = aw(£-a) [*22-9102]
*2293. l-.Q-i3 = a-(fln^)
Dtm.
h . *4"73 . Tr.msp . D H :. x* a. D : .r^c /3. = .~(xeo.xf/3).
[*22 33 ] = . x~ € ( a r\ £)
[*5-32] D b
• • •r c a . X'W >3 . = . X€ a . r» >9)
[*22*35-33] Dh:jffl-j9. = .j £ a-(aAj9):
[*1011. *20 43] DK a -/l = a-( fl n^).Dh. Prop
*22-94. 1- s (a) ./a . = . (a) ./(- a)
De/n.
K*101 . DH:(a).ya.D./(-a):
[*1011-21] Dh:(o)./a.D.(a)./(-fl) (1)
h . *101 . D h : <«)./(- a) . D ./{- (- a)) .
[*228.*2018] D./a:
[*1011-21] Dh(a)./(-a).D.(a)./ a (2)
K(l).(2). D h . Prop
'1'hi.s proposition is used in connection with mathematical induction, in
*90 102, which is required for the proof of *90132, which is one of the
fundamental propositions in the theory of mathematical induction.
*22 95. b : (go)./a . 5 . (ga) ./(-a)
Dent .
b .*22-94. D b : (a) .^fa. s . (a) ,~f(- a) (1)
h . (1) . Trnnsp . *20 6 . D b . Prop
ft
G8
. - : xRy . D x w . avS’y
Df
ft
nS
=» 5y (xRy . ^5y)
Df
ft
vS
=* .7# (xRy . v . a:Sy)
Df
•
ft*
2$ {~(x72y)J
Df
72
-=-5.
= ft
Df
*23. CALCULUS OF RELATIONS
Summary of *23.
The definitions and propositions of this number arc to be exact analogues
of .host ol *22 Properties of relations which have no analogues for classes
number ,s 'th " li " ° P,W ’ S b “ <""'tod h, the present
nurnbei as they are precisely analogous to those of analogous propositions in
’ . th,s number, as always in future, capital Latin letters stand for
expressions of the form ***!(.. y). or, where they are not being used al
arothTmT S ' ^ of tins number
aie the analogues of those of *22.
*2301.
*2302
*23 03.
*2304.
*23 05.
Simi
*231.
*232.
*233.
*2331.
*2332.
*2333.
*23 34.
*23 35.
*23351.
*23 36.
*2337.
*2338.
*2339.
*23 391.
*23392.
*234.
*2341.
*23 42.
*2343.
*23 44.
*23 441.
lar
h
K
h.
h .
h.
h :
h:
h:
h.
h.
h.
K
K.
K
h.
h:
h:
h.
h.
h:
h:
remarks apply to these definitions as to those of *22.
• RGS . = : xRy . v . xSy
R*S-mxRy.xSy)
R \y S *= xj) {xRy . v . xSy)
-R-m~(xRy)\
R-z-S^Sj) |xRy . (xS'y)J
x (R A S) y . a . xfty . ar&y
-x(RvS)y. = :xRy. y . xSy
x^-Ry. = - (xRy)
-ft*72
R r\ S € Rel
Rw Se Rel
— 72 e Rel
* (*. y> A 2 £* (x, y) _ 2 p (* (x, y, . * (ar> y) )
*9+ (x, y) w 25 ^ (x, y> - 2$ {* (x, y). v . + (x, y)|
(x, y) = 25 {~0 (x, y)(
./ees.seji.-.xRy.s .,.xs«
RCS.SCR.= .h = S
RGR
R*SGR
RGS.SGT.D.RC-T
RGS. xRy . D . ar/Sy
211
MATHEMATICAL LOGIC
[PART I
*23*45. b : R G S . R G T . D . R G 8 A T
*23-46. H : .r/ty ./«*(• .S'. D . ar.%
*23 47. h : y? G 7*. D . y» « N G 7*
*2348. h : /yG.S'. D . /y A 7G.S'n 7*
*23-481. h : 7? = ,V. D . /? A T = S r\ T
*23-49. I- : 7* G G .S'. D . 7* A 7y G Q * .S*
*23 5 I-. 77*77-/y
*23 51. h. /^a.S'-S'A 77
*23 52 h . < 77 * .S’) * 7’ = /?<S(.S'a y>
*23 53. 77*.S’* 7-</**£)* T I)f
*2354. I- :. 77 = .S’. D : RQT.s.SCT
*23-55. h 77 - .S'. D : 7’G 77 . = . T G .S’
*23551. l-:/7-S.D.77c/7=Sc/7
*23 56. h. /y c; yy = /y
*23 57. j-. /y^.V-.S'vy yy
*23 58. K. RGRvS.SGRvS
*23-59. K : 77 G 7’. S G 7\ = . 77 o ,S* G 7
*23 6. l-:.x(/yo.S , )»/. = :/yG T.SQT.^ T .xTy
*23 61. h//G5.D./iGSo7
*23 62. h : 77 G S. = . 77 o .S’ =* .S’
* 23621 . i-:77g.$'. = . 77 * 5=77
*23 63. Kyyo(7y/s.s’)=yy
*23 631. h.yyA(yyc/.v)-yy
*23632. l-:7y = .S , .D.yy= 77*5
*23633. H:77G5 .d.77ci7=(/7*5)c/ 7
*23-64. h yy G 7*. v . 5 G 7*: D . « * 5G T
*23 65. h:.7?G5.v. RGT-.D.RGRvT
*23 66. h: 77G5.D.77vy TGSv T
*23 68. h.(77 *5)o(77* T)=Rn(Sv T)
*23-69. h . (77 c/ 5) * (77 c; 7) = 77 c/(5 * 7)
*23 7. \-. (R v S) \j T = R v (S v T)
*23 71. RkjSsj T=(RvS)vT Df
*23-72. h: PQR. QGS.D.RvQGRvS
*23 73. h:/ , = 77.Q = 5.D..Pc/<? = 77c/5
*23-74. \-:P*QGR.Pr>R<iQ. = .P*Q = PnR
*23 8. h .—y?) = yy
*23-81. h:77G5. = .—5G —77
*23811. h: J7G-i-5. = .5G^/7
*23 82. h : R *5G T. = . R-^TQ-^S
*23 83. h:yy = s. = .-^yy=-5
CALCULI'S OF RELATIONS
215
SECTION C]
*23 831. : R = ^-S.= .S = -^R
*23 84. h .^(RnS) = -^Rv^.S
*23 85. . R r\ S = -^(-^ Rw ^-S)
*2386. I - .^(-^R*^-S) = Rv$
*23 87. b .-R*^-S = -^(Rv S)
*23 88. h . (ar, y) . x ( R \y -i. R) ,j
*23 89. b . (x, y) .~\x(R-^.R) y|
*23 9. V .(Rv S)^S=R^S
*23 91. . R w S = R w (S R)
*23 92. \-:RGS.D.S = Rv{S^-R)
*23 93. h.R^-S=R^(R*S)
*23 94. I- : (R) ./R . ~ . (R). R)
*23 95. h(zR)./R.m.(xR).f(^R)
*24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE
EXISTENCE OF CLASSES
Su in mu ry of *24.
The universal class, denoted by V, is the class of all objects of the type
which, in the given context, is being denoted by small Latin letters, i.e. of
the lowest type concerned. Thus V, like “CIs,” is ambiguous as to type. Its
definition is as follows:
*24 01. V - x (j = r) Df
Any other property |iosscssed by everything would do as well as = .r,”
but this is the only such property which we have hitherto studied.
The null-class, denoted by A. is the class which has no members. Like
V, it is ambiguous as to type. We use the same symbol, A, for null-classes
of various types; but these null-classes ditfer. The type of A is determined
by that of the terms .i concerning which "./•« A " is false: whatever x may be,
•' .ct A" will not represent a true proposition, but unless./• is of the appropriate
type, “ xt A" will be meaningless, not false. Thus A is of the type next above
that of an x concerning which ".rt A” is significant and false. The definition
of A is
*24 02. A = — V Df
When a class a is not null, so that it has one or more members, it is said
to exist. (This sense of ‘ existence" must not be confused with that defined
in *14 0*2.) We write "g !a” for " a exists." The definition is
*24 03. g la . = . (g.r)..rca Df
In the present number, we shall deal first with the properties of A and V,
then with those of existence. In comparing the algebra of symbolic logic with
ordinary algebra, A takes the place of 0. while V combines the properties of
1 and of x .
Among the more important properties of A and V which are proved in
this number are the following:
*241. h.A + V
I.e. “ nothing is not everything." This is useful as giving us the existence
of at least two classes. If the monistic philosophers were right in maintaining
that only one individual exists, there would be only two classes, A and V,
V being (in that case) the class whose only member is the one individual. Our
primitive propositions do not require the existence of more than one individual.
SECTION C]
217
THE EXISTENCE OF CLASSES
*24102103 show that any function which is always true determines the
° !,SS ' i,n<l any ru,,ctio " which “ always false determines the null.
*24 21-22 give forms of the laws of contra.liction ami excluded miiMIc. namely
notlung ,s both o, and not-a " (a n — a = A) and - everything is either a „Y
not-o (qvj —a = V).
*24 23-24 26-27 give the properties of A and V with respect to addition and
multiplication, namely : multiplication by V and addition of A make no change
(*24 07 S 2 Vr "n V ad : liti0n ? f V S-'os V.aml multiplication by A gives A
1 W ‘ L bC 0 ‘ >SerVed thnt the properties of A and V result from
each other by interchanging addition and multiplication.
*243. h:aC5.H.a-^A
I.e. a is contained in /9 ” is equivalent to “nothing is a but not ft"
*24 311. hoC-^.s.an^A
I.e. “ no a is a /3 ” is equivalent to “ nothing is both a and (3 "
*24 411. h:/9Ca.D. «-£«<«-£)
*24 43. ha-^C 7 . = .oC^u7
As a rule, propositions concerning V are much less used than the corre¬
lative propositions concerning A.
th, rr rt . ieS ° f thC cxistence of cesses result from those of A, owing to
the fact that a ! a , s the contradictory of a = A, as is proved in *24-54. Thus
we have, in virtue of *24 3,
*24 55. h :~(aC/3). = . a !a-/9
This t '.KnV" r 8 are &S ’’ U ‘ J<|U ' Valonl 10 " there are which are not 0\."
univcl m mr Pr T S ' U " n ° f f0rmal lo « ie - that contra,lictory of the
inivcisal affirmative is the particular negative.
We have
*24 56. h:.a!(au/3).= :a!a . v>g!/3
*24 561. h : a ! (a a /?) . D . H » a . g . ^
if a nrJL a , 8 " m . eX u tS Y th r n ° ne ° f the Smnman<ls exists.and vice versa; and
if a product exists, both the factors exist (but not vice versa).
The proofs of propositions in the present number offer no difficulty.
*24 01. V=5?(- r = ; r) Df
*24 02. A = — V Df
*24 03. g!a. = .(a x).x€<x Df
*241. H . A + V [*22-351 . (*2402)]
[*22-831 .(*24 02)]
*24 101. I- . V = — A
218
MATHEMATICAL LOGIC
[PART I
*24 102. I- : (x ). 4>x . = . 3 {4>z)=\
Dem.
h . *13*15 . *5*501 . D H <f>x . = : <f>x . = . x = x
[*10*11*271 ] D h (a *). <f>x . = : (jr ): <f>x. = . x = x:
[*20* 15] = : 3 (4>z) = 3 (.r =./ ):
[(*24*01)] = : 3(02)= V D h . Prop
Thus any function which is always true determines the universal class,
and vice versa.
*24*103. h : (x). ~0./-. = . 3 (0 j ) = A
Dem.
h . *24*102 . D h (.r) .^</>a*. = : 3 = V
[*22*302] = :-3(<*>j) = V
[*22*831] = :3(<M = - V
[(*24 02)] e A D h . Prop
*24 104. h . (x ). /f V
Dem.
H . *20*3 .DhucV.i.rs/ (1)
I- .(1). *13-15. *10*11*271.31*. Prop
*24*105. h . (**) . x~t A
Dem.
h . *22*35 .DhixeA.i. V :
[*4 12] Dhs/'vfA.s.xcV (1)
H. (1). *10*11*271 . *24* 104 . Z> h . Prop
*2411. !-.(«). «CV
Dem.
h. *24*104. *10 1 .Dh.artV.
[Simp] 31*:^«a.D.xi V :
[* 10 * 11 .*22*1] DhsoCV:
[*10*11] D h : (a) . a C V : D h . Prop
*2412. h . (a). A C a
Dem.
K *24*105. *10*1 . Dh.X'vfA.
[*2*21] 3l-:«cA.D.*ca (1)
h.(l).*1011 .*22*1 . D h. Prop
*2413. hta-A.s.aCA
Dem.
h . *24*12 . *4 73.3h:aCA. = .aCA.ACa.
[*22*41] = . a = A : D H . Prop
*24*14. h:(x).xea. = .a=V
Dem.
h . *24*102 . D I*: (x). x e a . = . £ (x € a) = V .
[*20*32] = . a = V : D h . Prop *
SECTION C] THE EXISTENCE OF CLASSES
*24 141. (■: VCa.s. V = a
Dem.
. *2+11 . *4*73 .Dh : VCa.s.aCY. V C a .
[*22-41] = . a = V : D (■. Prop
*2415. b : (.r). .r~ € a . = . a = A
J)em.
h . *24*103 . D b : (x) . a . s .(.r c a) «• A .
A : D h . Prop
219
[*20*32]
= . a =
*2417.
b:a = Y. = . — a = A
[*22*83 . (*24*02)]
*2421.
Kan-a = A
[*24-103 . *22*89]
*24 22.
h.au-a«V
[*22-88. *24-102]
*24 23.
b . a A = A
[*2412. *22*621]
*24 24.
b . a u A = a
[*24-12. *22*62]
»--- -— —j
The above two propositions (*24-23*24) exhibit the algebraical analogy of
A to zero. * b *
*24 26. h.anV-o [*22621 .*2411]
This exhibits the analogy of V to 1.
*24 27. KauV-V [*2262 . *2411 ]
This exhibits the analogy of V to oo.
*24-3. l-:aC£.3.a-/? = A
Dem.
b . *4*53*6 . D
h x e a . D . x e 0 : = e a . e /3) :
[*22*35]
[*22*33] s:~(xca-/9) ( 1 )
h . (1) . *10*11*271 . D
h : a C 0 . = . — #).
[*24*15] s.a — /9 = A:Db. Prop
The above proposition is very frequently used.
*24 31. h:aC/3. = .- a v/3 = V
Dem.
b . *4*6 . Db:.xea.D.xe/9:=: a . v . xe B
[»22-35] = :{x)-.xe-a.v.xe/3:
[.22-34J s:(i).ie(-ij u /3):
[•2414,] =:-ou£=V:.Dh. Prop
This proposition is the correlative of *243, but, unlike that proposition,
LZZr, ‘ h f Sequel - Ever y proposition concerning A has a corre-
' r ' 0 u '" ng V ;? Ut We 8hal ‘ ° ften DOt S* ve these correlatives, since they
are seldom required for subsequent proofs. y
220
MATHEMATICAL LOGIC
[PART I
*24-311. (-:aC-^. = . 0 ^=A
Dem.
^ . *22*35 .Dh:.jr«o.D,x€-/J:s:xfa.D. c £ :
[*4*5162] 5s-v(/ ( a.x«/3):
[*22-33] =Hrea^)
K . (I). *10*11 *271 .DhaC-^ = .|.r)./MflA/J.
|*24*1>) ■.an^«A:Dh. Prop
*24 312. h:-aC^.a.aw/J-V
H . *22*35 .Dh:.-oC/i. = : j-«w a . D z . .r c /i :
t* 4 ' 1 '-*] = :<x) sa-tfl.v.xe^:
[*22*34] = :(./*)..re a v£:
[*-+'H szayj/3- Vs.Dh.Piop
*24 313. b:an/3-.\. = .a-a-0 1*24-311 . *22021]
*24 32. l-:.ov/S = A. = .a-A./9-A
Deni.
H . *24* 13.Dh:.outf=A.= :aw^CA:
[*22 -.0] = : o C A . /S C A :
[*24-13] e : a = A . /S — A D K . Prop
*24 33. ha=V.D.Qw/J=\'
Dent.
H . *22*551 .DhHp.D.aw/9-Vw^
[*24*27.*22*57] = V : D h . Prop
*24*34. h:a = A.D. fl A/3=A [*22*481 . *24*23]
*24 35. ha-V.D.an^-/3 [*22481 . *24 26]
*24 36. ha = A.D.ou^ = j9 [*22 551 . *24 24]
*24*37. :• a n ft = A . = : x c a .•/ e fi. D; v . x ^ y
Deni.
(1)
I-. *24*15 ,DH:.an£=A. = : (x) . x~e (a n £):
[*22*33] = : (x) .~(xe a . xe£):
[*13*101] = :(x t y):x = y. D .~(xe a . y e£):
[Transp] = : (x, y) : xe a . y e B . D . x y D . Prop
*24*38. H:.an£ = A.D:a4 : £«v.a = A.£=A
Deni.
H . *22*481 . D H : a a £ — A • a «£. D . a a a«* A .
[*22*5] D . a = A .
[*20*23] D . a = A . £ = A (1)
h.(l). Exp .Dh.an£*=A. D:a = £.3.a = A.£=A:
[*4*6] D:a + |0.v.a = A./9 = A:.DK Prop
SECTION C]
THE EXISTENCE OF CLASSES
*24 39. h:.aod = A. = :j.,#. 3, ,3 [*>4 311 . *22 :»->]
*24 4. h:« ni }=A. = . l(IU( j)_ 1 , =((}i3i(8u( , ) _ /J = a
Dem.
H . *24*811 .Dh:an^=.\.= (/ jC-o
[*22!>] = .<au,3)-a = ,3 ,|)
*"' (1) a .’J ‘ 3 p :/3«a = A.».(/9va>-/3 = a :
[*22-51»7] 3h:an/3 = A . = . <o w £> _ S = o
. (1) . (2). 3 H. Prop
*24401. l-:/3Ca.3.(/3u 7 )-a = 7 - a
Dem.
H.*220S. 3H.,
K.24-3. 3h:Hp.3.e-«-.V
J - • (1) • (2). 3 K : Hp. 3. (/3 w 7 ) — a — A u ( 7 — a)
[*24 24] - 7 - a : 3 4 . Prop
*24402. l-:or./3-A.fCa.,C/9.3.fft,_A
Dem.
K*22-49.3h:Hp.D.f n „ C a«/9.
[*22-35] 3 . f n jj C A .
[*2413] 3.fn,-A:3H. Prop
*24 41. l-.a = (ar»/3)w(a-^)
Dem.
h . *2268 .DK(oft^) y ( a - i 3) = flft ( i 9 u _^)
[*24-22] =
„„„ [* 24 ' 2 «] =o.3h. Prop
*24-411. h:£Ca.3.a = /9u(a — /3)
Dem.
y . *22-633 ^ . 3 h : 0 C a . 3 . S « ( a - 0) =. (a r, 0 ) v, (a - 0)
[ * 24 ' 4,] =o: 3K Prop
*24412. l-:^Ca. 7 C/3.3.(a_ i 8) u ( / 8_ 7 ) = a _ 7
Dem.
[*24^3-23]" 3 h :Hp • 3 • = (« - 7 ) - (« - /9 - 7) v, (/ 9- 7 )
r*22-fi«l = (“ - £ - 7) w (/3 - 7 )
:ss,
• J ... =a — 7 :3I-. Prop
Th's proposition is used in *234181. in the theory of eontinuous functions.
*2442. l-:»n£C 7 .«-/3C 7 . = .«C 7
Dem.
[*24^411 9DH:aniSCT " a "’^ C ' y ’ S-(an ^ V/(a “^ C '> r -
L 1J =-aC 7 . 0 h. Prop
222
MATHEMATICAL LOGIC
[PART I
*24 43. h : a — (3Cy. = .aC£yjy
Dem.
H . *50 . Dh:/ta. j # . D . xe y : s z.orca.Ozrt&.v.xc y
[#22 35 83] D K :: j- € a — /3 . D . ./• e 7 : = .r € a . D : .r € >3 . v . .r e 7
[*22*34] =:./<a,D,/((^u 7 ) (1)
H .(I). *10-11*271 . DH . Prop
*24 431. Mow 7 )n(o'u- 7 ( = (Oft^)u(a- 7 )u(^n 7 )
This and the following proposition are lemmas for *24*44.
Deni.
H . *22*03.3 h . (a w 7 ) /M/3 w - 7 ) = ;(a u 7 ) r» /9( w J(a u 7 ) r\ - 7 I
[# 22 * 68 ] =(an n 8)\j(a-y)v{y-v)
[*24*21] = (a r\ /9) \j (7 r\ &) \j (a — 7 ) \j A
[*24 24] - (cr r» £) (7 r\ /3) v (a - 7 )
[*22*51 *57] =(ao^)u(a- 7 )u(/^A 7 ).Df*. Prop
*24 432. h .(a-y)\J 10 r\y) = (ar\ 0 )\j (* -y)v (0 r\y)
JJem.
h . *24*22*35 . D h . a = (a n /9) n (7 v - 7 )
[*22*08] = (a n/9 n 7 ) w (a r»/3- 7 )
[*22*51] = (a r\ 0 r\ y) sj (a r\ — y r\ 0).
[*22*551] Df-.(oo^)w(«- 7 ) = («Jft^A 7 )w(an -7 0/9)u(a-7)
[*22*03] — (a r\ 0 r* 7 ) (a — 7 )
[*22*57 ] «=(a-7)u(an/3n7>.
[*22*551] D h . (a /9) v (a — 7 ) v (/3 7 ) = (a— 7 ) u (a n /9 n 7 ) v (£ n 7 )
[*22*03] = (a — 7 ) v (£ n 7 ). D I-. Prop
*24 44. h . (a v y) r\ {0 v — y) = {a n - y) v {0 r> y) [*24*431 432]
*24*45. h|an 7 )u(/j- 7 ) = A. = .^C 7 . 7 C-a
Dem.
h . *24*32 . D h : (a r» 7 ) \j (0 — 7 ) = A . = . a n 7 = A . — 7 = A .
[*24*3*311] =. 7 C-a.^C 7 :Df* • Prop
*24*46. h : (a n 7 ) u (^ - 7 ) = A . D . a n /? = A
Dem.
h . *24*45 . *22 44 .Dl-: Hp.D.^C-a.
[*22 811] D.aC-0.
[*24 311] D.an/J-A: D H - Prop
9
The following propositions, down to *24*495 inclusive, ore lemmas inserted
for use in much later propositions, most of them being only used a few times.
SECTION C]
<n
D *• ' £ = f . V - V • 3. £ v> v - f ^>?'
Dh:.fu^-F uV.D:(fu^)n«-(f uV)a«:
( 1 )
THE EXISTENCE OF CLASSES .).)•»
wJO
*24 47. !-:««£= A.av/3 = 7 . = .aC 7 ./3 = 7 _ a
Dem.
H . *24*31 l.Dh:on/9 = A. = .^C-a
H . *22*41 . Dhou/3= 7 . 5 . ou ^c7.«yCau/i
[*22’59.*2*43] - . a C 7 .0 C 7 . 7 - a C * ( •>>
r# .>o-451 -aC 7 ./?C 7 ^C-o. 7 -aC^
[* 22-411 =.aC 7 ./9C 7 -a.'y-aC£.
L 1 E.oC 7 ^ = 7 - a:Dhi i. r(ip
*24 48. h f C or. £'C a . i; C/9. v f C£. a «£- A . D :
Bern. =
»■. *22-73.
K *22-481 .
[*22-08]
. *22-021 .
[*3-47]
I-. *22-48 .
[*22-55]
[*24-13]
Similarly
h.(3).(4) .
[*24-24]
^.(3).(5).
[*24 241
H.(2).(0).(* 1
Similarly
“ OOVe P ro P oslt| on, besides being used in the next two, is used in the
the h ° CO “ P eS < * 54 *')' ,n the lheor y of greater and less 0X17-682), and in
0170^8) ° n g ° f ° laSSeS by the Principlc ° f first di^renees
*24481. b:.«n/3 = A . a ~ 7 = A . D : a v, ,3 - a u 7 . s £ = y
Dem.
I- . *94. AQ g» ~ <*• «• «■ A y ^
hs.oCa.aCo.^C — a. 7 C — a.o —a = A.D:
h . *22-42 . *24-21 . D a u /3 = a u 7 . = . a = o . 0 = 7 (1)
P:.aC a . aCa .^C-a. 7 C-«.a-a = A. s .^ C -a. 7 C-a.
3 :(£ Aa)vy(7;^a)«=({:' n a)
W»/'rtcr) (2)
=>»-:fCa.fC a .D.fna = f.f' rta , f '
3H:*?C/?.an/9=-A.:). V naCA.
(3)
D . 7 ; r\ a = A
•■:>?'C/9. 0 n^ !3 A.D.,'na = A
^ I- Hp .D:(ffta)u(,ft 0 ) = fwA
(4)
(«)
3 H Hp . D t ((' r\ a) \j (rj'rs a) «= £' v A
<«>
s. Hp . D ; f ui| » D . f » j:'
(7)
(8)
*-:.Hp.D:fu, = f un'.D.„ = „'
3 I - . Prop
(y>
224
MATHEMATICAL LOGIC
[PART I
[*24 311] = .an0 = A.an 7 =A (2)
H . *20 2 . *473 .D^za = a.0==y. = .0 = y (3)
I- . (I). (2) . (3). D h . Prop
Tin* above proposition i> used in the theory of selections (*8374), in the
theory of greater and less (*l 17*582), and in the theory of transKnite induction
(*257).
*24 482. \-:.%Ca.riC{3.ar\ t 3 = A.D:t;'Ji) = ayj f j. = .}; = a.i)=t3
[*24 48 £ . *22-42]
The above pro|»ositioii is used in the theory of convergence (*232 34).
*24 49. a r\ 0 = A . D : a C 0 v y . = . a C y
Deni.
h . *22021 .Dh :aC(3vy. = .a = ar\((3\jy)
[*22118) =(an^)y(on 7 ) ( 1 )
h.*24 24. Dhsan/9 -A .Dt(#aj 8 )u(«a 7 )«an 7 (2)
h . (1) . (2). Dh:. Hp.D:aC/Ju7. = .asaA7.
[*22 02 1 ] = . a C 7 : D h . Prop
*24491. h:/9r»7"A.aC£w»7.
D.a-^»an y . a — y = a r\ 0 . a *= (a — 0) v {a — y)
Dan.
H. *22*621 .
[*22-481]
(*244)
Similarly
l-.(l).(2).
[*2208]
[* 22021 ]
D H : Hp.D.a-«a(/9v7).
D.a-y = an(0uy) — y
= a r\ 0 (1)
HsHp.D.a — /9 »oa7 • (2)
D h : Hp. D . (a — 0) v (a — 7 ) — (a n 7 ) v (a n /9)
= a r\ (7 u /?)
— a (3)
h.(l).(2).(3).DK Prop
The above proposition is used in the theory of selections (*83 03 05) and
in the theory of segments of a series (*211-84).
*24 492. h : 0Ca.a —0-y.D.a —y = 0
Dem.
h . *22481 . D h : Hp . D . a —7 = a — (a —£)
[*22 8-80] = a n (- a u 0 )
[*22-8*9] =an0
[*22-621] = £:DKProp
The above proposition is used fairly frequently, especially in the theory of
series. It is first used in *93 273, in the theory of “generations.”
SECTION C]
THE EXISTENCE OF CLASSES
*24 493. H : £ n 7 = A . D . a = (a - £) w (« - 7 )
Dem.
h . *22*84 . *24*17 .Dh: Hp .D.-£ W _ 7== V.
[*24*26] D.a = ar\(-/3u- 7 )
[*22*68] -ta- / 8)u(a- 7 ):3l-.Pm 1 ,
*24*494. t-:fC a .,C/3.«n^ = A.D.(?v,„)-« = ., / . (fu>>) _ ;8 _ {
Dem.
I;** 8 * 8 - 3 H : Hp . D . f — a = A (1)
t*.*24*311. D I- : Hp . D . /9 C — a .
[*22*44] D.ijC-a.
[*22*621] D. (*,
K.22C8. 3K(f u „.,. ( {-, )l/( ,.. ) ,3)
H * (1). (2). (8) . *24*24 . D h : Hp . D . (£ v. v ) - * = ( M
f imilar, y l-iHp.3.(fu 9 )-^.f (5)
H • W • (5) . D h. Prop
This proposition is used in the theory of selections (*83 63 and *88*43).
*24 495. I- : a a 7 = A . I) . (a u 7 ) - (£ v, 7 ) = a _ £
Dem.
I- .*22*87-68. D
h.(av 7 )-(/9u 7 ) = (a-,9- 7 )u( 7 -/3- 7 )
[*24*21] ««-*- 7 (])
K. *24*311 .*22 621 . D h : Hp . D . a - 7 = c (2 )
I-. (1) . (2). D h . Prop
(*20 T ^3*8l2 O *84) Pr ° POSitiOU " " Sed ^ thG the ° ry ° f l >oints
of T aind r u Ms " Umber we ahM be conceine< l with the existence
fact tW to y , PrOI>ert,e8 of the exigence of classes follow from the
to the n 11 I “ ^ CX,StS ,S C<|uivalc " t 10 “yi“8 that the class is not equal
to the null-class. This is proved in *24 . 54 . 1
*24*5. H : a ! a . = . (g*) . * e a [*4*2 . (*24 03)]
*24 51. :~g ! a . s . a = A
Dem.
h . *24*5 .Dh<vg!a. = {(3x).x e o|.
[*10*252] S .(*).x~ e «.
[*2415] = . a = A : D V . Prop
*24*62. I-. a ! V [*24*51*1 . Transp]
MATHEMATICAL LOGIC
[PART I
220
thing, which is equivalent to this proposition, is implicit in the proposition
*101. that what is true always is true in any instance. This would not hold
it there were no instances «.f anything; hence it implies the existence of
something. It will be observed that the above proposition (#2452) depends
on *241. which d.pondson *22 351. which depends on *10*251, which depends
on *10-24. which depends on *101 or on *91. The assumption that there is
something is involved in the use of the real variable, which would otherwise
bo meaningless. This is made explicit in *91, and in the proof of *9 2, which
is the same proposition as *10-1.
*24 53.
b • ~ g ! A
[*24-51 .*202)
*24 54.
b s g ! a . s . a + A
[*24’51 .Transp]
*2455.
b :^<a C fi ). s . g la -
a
[*24 3 . Transp . *24*54]
*24 56.
h :.g !(au/3).a:j|!o
•''•51 '-0
[*10-42. *22-34)
*24561.
1*: g ! (a a .£). D . g ! a,
[*10-5. *22-33]
*2457.
b a r\ 0 - A . D : g ! a
■D.a+tf
Dan.
b. *22481 . Db:o a £
- A . a = ff
. D . a n a ■■ A .
1*22-5]
D . a = A .
[*24-51]
0 .~g ! a
b . (1). ICxp. Transp . D
> b . Prop
*24571.
b:g!a.a®^.D.g!(
ar\0)
Dent.
b . *24 - 57 . Comm . D b
:• g ! a . D :
[Transp]
D:
a = /3.D.an/9 + A.
[*24-54]
D . g ! (a n £)
b . ( 1 ) . Imp . D b . Prop
*2458.
b:.aC^.D:g!a.D.
a !/3 [*10 28]
*24 6.
b!.aC/3D:a + /9.».g!/9 — a
Dan.
b . *22-41 . Transp.
Db:. Hp.
D:a*£.D.~(£C«).
[*24-55]
D.g!^-a
b. *24-21 .
D b : a = 0
. D . /9 — a = A
b . (2). Transp . *24 54
.Db:g!/9
— a . D . a +
b.(1) . (3) . 3b. Prop
*24 61. b:~g!/3.D.au£ = a [*24-51 24]
I- :~g ! £. D - a r\ 0 = A [*2451-23]
( 1 )
( 1 )
( 1 )
( 2 )
(3)
*24 62.
SECTION Cl
THE EXISTENCE OK CLASSES
227
*24*63. l-:.A~c/f. = :a e ^o a .g!a
bo P™P" sition ' *he conditions of significance require that * should
hvnothlsi S ‘ condition . 3. . a !«" is one required as
I vnoth m ,, i r ° p0S,t,0ns - 1,1 virtue of the above proposition, this
hypothesis may be replaced by “A~€k.” 1
Dem.
h .*13191 . DI-:.A~e*. = :j = A. 3 „.a~« l[ :
[Trnnsp] = : a e * . D a . a + A :
[*24-54] *:«e*.D,.gla:.3K Prop
This proposition is frequently used in later parts of the work. We often
5^ehto st exi r nt ciasscs ’ a,ui ti,e —* — l:?™
which to state that all the members of a class of classes exist is "
15—2
*25. THE UNIVERSAL RELATION, THE NULL RELATION, AND
THE EXISTENCE OF RELATIONS
Summary of *25.
This number contains the analogues, for relations, of the definitions and
propositions of *24. Proofs will not be given, as they proceed precisely as
in *24.
The universal relation, denoted by V, is the relation which holds between
any two terms whatever of the appropriate types, whatever these may be in
the given context. The null relation. A. is the relation which does not hold
between any pair of terms whatever, its type being fixed by the types of the
terms concerning which the denial that it holds is significant. A relation It
is said to exist when there is at least one pair of terms between which it holds;
“It exists" is written “<[ ! It."
The propositions of this number are much less often referred to than those
of #24, but for the sake of uniformity we have given the analogues of all
propositions in *24, with the same numeration (except for the integral part).
All the remarks made in #24 apply, mutatis mutandis, in the present
number.
*2501.
V
I)f
*2502.
A
=^.\-
l)f
*25 03.
<i
! Ji . = . f'q.r, y ). xliy
Df
*251.
b
A + v
*25101.
b
V—s. A
*25 102.
b
U \!/) -<t>(x, y). = .J
S <t> (* r . .
*25103.
b
(^, •/)• =
• <f> (.
*25104.
b
(x,y).xYy
*25105.
h
—(*Ay)
*2511.
b
(It). HCX
*2612.
b
(It ). A G Ji
*2513.
b
It = A. =. R <• A
*2514.
b
(x, y ). xRy . = . It =
v
*25141.
b
V G It . = . V = R
*2615.
b
• •
*5
I
i 11
R = A
*2517.
b
:Ii = Y . = .^R = A
*2521.
b
.R*^R=A
SECTION C]
THE EXISTENCE OF RELATIONS
*25 22. K Pc/-^P = V
*25 23. h . R o A = A
*25 24. I-. R \y A = R
*25 26. h . R A V = p
*25 27. M2oY a V
*253. h:iiG5. = ./?^ < S=A
*25 31. l-:PGS. 3 .^Pc/S«V
*25311. hRQ^S.~=.RnS = A
*25 312. H:-i.PG£.s..ftuS=V
*25 313. h :R nS = A. = . R^-S = It
*2532. h:PuS_A. 3 .P«A.S«A
*25 33. f-:P= V.D.Pc/S- V
*25 34. t-:R = A.D.R*S=A
*25 35. h:7i=»V.^.7i/S < S , = jS'
*25 36. h : P - A . D . P c; S =» 5
*25-37.
*2538.
*2539.
*25-4.
*25401.
*26402.
*2541.
*25411.
*25412.
*2542.
*2543.
*25431.
*25-432.
*26-44.
*2645.
*2646.
*2547.
*2548.
H:.«^S=.A.D:A + S.v.ie = A.S = A
h.iiAS.A.s: . 3 X y —(*,s y )
hPnQ = A.3.(Po(J)j.iJ = Q, a _^ l;j( jj i Q_p
I-: Q C R. D . (Q o R)^.p =. ^
H:PAQ-A./iCP.SC(3.3./e Al8 -_A
K .R = (« A S) o
I’iQCi'.SCQ.a. (.P—Q)<v(Q-^S) = P^-8
l-rRAQGR.P^QCR.s.RCR
t-iR-OGR.s.PGQoR
K(f» 8 )A(Q„ iS) = ( f. A g )o(f . s>o( ^ fl|
■(^fl)»(«Afl| = ( PA g )o(Pi/j)o(? . fl)
K(RoR)A(0v a ^R) = ( p A ^ iJ)l;((QA/J
!-:(8AB)o(Qiii) = A. = ,QGfl.flGiP
l-:(PAfl) 0 (Q ifi) = A_ 3iP AQ = A
l-:RAQ = A.R K ,Q-fl. = .p G7J .Q = s ^ p
l-:.Re/*.R'CP.SGg.S-G(3.i>A(3 = A.D:
*26-481. t-
*26 482. h
*2649. h
Rw S= R' \y S'
:.P«Q = A.P,sp = A.D:Pc/Q = Pc/P.=
:-i 2 GP.SGQ.PnQ = A.D:iic;S=? w Q 1
:.PoQ = A.D:PGQc/P. = .PcP
= .R = R'.S = S'
Q= R
=. R = P. S = Q
MATHEMATICAL LOGIC
[PART I
2:J0
*25 491. htQnR = \.PGQwR.2.
P-Q = PnR.P — R = Pr\Q. P = (P±Q) ci (P-P)
*25492. I :QQP.P^Q = R.D.P^R = Q
*25 493. I-: Qn R = A . D . P = (P-^Q)v [P — R)
*25 494. h : 7? G P . .S’ G Q. P A Q = A . D . (P c; tf )P - 5. (P v S) Q = R
*25 495. I-: P n R = A . D . (P c; P)^(Q o 77) = P-Q
*25 5. h : a ! 77 . s . <'.| r, »/). o:P^
*25-51. h:^a!P. = .P-A
*25 52. h • g ! V
*25 53. H.^aSA
*25 54. h : a ! P. a . P + A
*25 55. R-S
*25 56. h a ! (P kj »9). s : a ! R . v . a ! .9
*25 561. h : <\ ! (P A .9). D . a ! P • a ! * s '
*25 57. h P A .9= A . !> : 3 ! P. D . P * .9
*25 571. I-: a ! P. P - P. D . a UR A 8)
*25 58. h P G .9. D : a ! P . D . a ! 5
*25 6. h:.PGS.D:P + S.s.g! .9-^ P
*25 61. I- :~a ! 8 . D . P o .9 = 77
*25 62. h'v<|!,S’.D.PA6 , = A
*25 63. I* :• A~c * • ■ : P e «• D* • 3 ! P
SECTION D
LOGIC OF RELATIONS
am- SCCti ° n .- We Sl,al1 bc COUCCrne(l *“ch of the general
p.opert.es of relation, as have no analogues in the theory of classes The
rest TtUT W U SeCtiOD Wil1 be USed constantl y throughout the
rest of the work and the ideas expressed in the definitions will be found to
be of fundamental importance.
*30. DESCRIPTIVE FUNCTIONS
Summary of *30.
The functions hitherto considered, with the exception of a few particular
functions such as a have been propositional, i.e. have had propositions for
their values. But the ordinary functions of mathematics, such as .r a , sin#,
lng.r, are not propositional. Functions of this kind always mean “the term
having such and such a relation to For this reason they may be called
descriptive functions, because they describe a certain term by means of its
relation to their argument. Thus irj’2" describes the number 1; yet
propositions in which sin -r 2 occurs are not the same as they would be
it 1 were substituted for .siii7r,2. This appears e.y. from the proposition
‘ sin 7 r 2=1." which conveys valuable information, whereas “1 = 1 ” is trivial.
Descriptive functions, like descriptions in general, have no meaning by them¬
selves, but only as constituents of propositions 0 .
The general definition of a descriptive function is:
*30 01. R*ym(ix)(*Ry) Df
That is, “7f*y" is to mean “the term x which has the relatiou R to y."
If there are several terms or none having the relation R to y, all propositions
about ll‘y, i.e. all propositions of the form “£(/?‘y), M will be false. The
apostrophe in “ R*y " may be read “of." Thus if R is the relutiou of father
to son, “ R*y " means " the father of y." If R is the relation of son to father,
“ R\/" means “the son of y”; in this case, all propositions of the form
" <7> (/?‘y) ” will be false unless y has one son and no more.
All the functions that occur in ordinary mathematics are instances of the
above definition ; all are obtained in the above manner from some relation.
Thus in our notation “ R‘y takes the place of what would commonly be
"fu' tatter notation being reserved for propositional functions. We
should write “sin ‘y" in place of “siny” using “sin" to express the relatiou
of x to y when # = sin y.
A definition such as R*y = (tx)(xRy), where the meaning given to the
term defined is a description, must be understood to mean that the term
defined (in this case R‘y) and the description assigned as its meaning (in this
case (lx) (xRy)) are to be interchangeable in use: the definition is, in a sense,
more purely symbolic than other definitions, since the description assigned as
the meaning has itself no meaning except in use. It would perhaps be more
formally correct to write
f(R‘y) • = •/IO*)(*fly)J Df.
• Cf. *14, above.
SECTION D]
DESCRIPTIVE Functions
. e 7',‘i tHiS <l0f,1 ' it '? , ‘ would not bo co **>l>l*-*to. because it omits
“ete fLm ^ ^ Thu. the
[ Rt 'A - f (R*y) . = . [(M*)(-r7?^)]./l(i.r)(.r/fy)} l)f.
thiS f ° r,n ° f Prided it is under.
"0?UrB, V ; i"’ " M \ that “*V may bo written for
' **• , ,n ,nd,Cntio,,s of sc °p° as well as elsewhere. The
of the defin.tion occurs always in accordance with the proposition:
.... h : Wfl •/< R, 'J> • s • [(<*) (•' %)] ./( ,x) (.r liy),
which is *301, below.
It is to be observed that *30 01 does not necessarily involve
= (la) (i%).
For this, by the definition, is equivalent to
, , , («)(x%) = (,x)(.r%),
i'ii* 11 ' 2 , 8 ' ™ ] y ho “« EKtrlW, t>. when there is one term,
r Ve K nti ° nS n r l ° SCOpe CXpluincd in *M ate to be transferred to
fhe ° enCC any COntlar > , indication, the scope of R‘, is l„ be
in question occur ' * ° Se< ' d ° tS ° r ° ther bracketS ’ " hicl ‘ «« /*•*
We put
*30 02. R-S'!/ = Jt<(S-y) D f
«* b '“ k »“ '* » - *» -
In f., [ L R !f yl ■ /(R ‘ s 'y ) • “ • l R ‘(S‘y>] -/IX‘(S‘y)l Df.
forks''meaning • ° ften define a " ew expression as having a descriptive phrase
above “is an" 8 a CaSC ' the v efiniti ° n iS al "' a > S b « interpreted as
the proplsrt o„ whfeh rOPO K 1 0n i*\ Wh “* th ° ncw expression occurs is to be
» p ri£;:£!zi* 8u '" t,tuting the ° ,d f °" the
w e rf S’i" lhC t ab ° VC ' ! f 10 be interpreted by first treating S‘y as if it
R\S‘v) and bv C th P p T ’ r d appl - v ' ng * 30 01 an «i *14 01 or *14 02 to
-« y), and by then apply,ng *30 01 and *14 01 or *14 02 to S‘ij.
cons^uenci 0r o i f ty th 0 e f c t o h rre, Pr0, T iti0nS ° f preSeDt number “re immediate
*14-113 ‘he correspond 1 ng propos.t.ons in *14. Thus *14-31—34 and
whe"l^“ m th y t0 „n°I?- 16 - " hich show that, either always or
to thetu y tnaTues of su C c°h Pe nro " ° f ^ and ^ »» difference
*3(1 in t ° f 8“eh propositions as we are concerned with. We have
*3018. ^:.EiR‘y.(z ) . 4 , z; o . 4 ,( R -„)
MATHEMATICAL LOGIC [PART I
so that what holds of everything holds of R*y, provided R‘y exists. This
results immediately from *14 IS. and shows that, provided R l y exists, the fact
that " R*y is an incomplete symbol <loes not prevent its being substituted
as a value ol z whenever we have (j). <f>z, or an assertion of the propositional
function <f>:.
One of the most used propositions of this number is:
*30 3. b x= R*y . = : zRy . =,. z *=x
which results immediately from *14 202. The following analogous proposition
results from the above by means of *14122:
*30 31. V x = R‘y . = : xRy : zRy . D. . z =* x
I.e. "./ = R*y" involves, in addition to " xRy," the statement that what¬
ever has the relation R to y is identical with x.
A proposition constantly referred to is:
*30 37. I- s E! IV y .y = z.D. IV y = R‘z
In the hypothesis, E! R*y might be replace«l by E! R*z, but one or other
"f them i» essential. For, by *14 21. " R l y = R'z " implies E ! R‘y and E ! R‘z
(these are equivalent when y —r),and therefore cannot be true when R*y and
JVz do not exist.
The use of *30 37 is chiefly in cases where y or z or both are replaced by
descriptive functions. Suppose, for example, that z is replaced by S'to. By
*30 1.3, we may substitute S*w for r if 8*iv exists. By *1421, both sides of
the implication in *30*37 will become false if S‘w docs not exist, and there¬
fore the implication will still hold. Hence whether »S '‘w exists or not, we may
substitute it for z and obtain
b : E ! R*y. y = S‘u >. D . R'y = R'S'w.
In like manner, if we replace y by T*v % we obtain
b : E! R* Vv . Vv = S*w . D . R*T l v — R‘S‘w.
A very important proposition is:
*30 4. b E ! R‘y . D : a = R*y . = . aRy
This proposition states that, provided R*y exists, to say that a is the term
which has the relation R to y is equivalent to saying that a has the relation
R to y. Thus for example "a is the occupier of the house y ” is equivalent
to “« occupies the house y," "a is the writer of Waverley ” is equivalent to
“ a wrote Waverley,” " a is the father of y ” is equivalent to “ a begot y." But
we cannot argue from "John Smith inhabits London” to "John Smith is the
inhabitant of London.”
We shall introduce in this and subsequent sections many constant relations
for which E ! R‘y is always true. When R is such that E ! R‘y is always true,
we have, in virtue of *30*4,
a = R‘y . = . aRy
SECTION D]
DESCRIPTIVE FUNCTIONS
235
f"i e\ery possible value of y. The following proposition is useful in eases where
both R and S are such that R'y and S'y always exist:
*30 41. 1-( y ). R'y = S'y . = : (y) . E ! R'y s R = 8
.. , TI '" S Z e kn °' V that R ‘y aml S ‘y a,e always identical, we know not only
that R and S are identical, but also that R'y (and therefore S'y) always exists.
*30 01. R'y = (,x)(xRy) Df
*30 02. R'S'y = R'(S'y) Df
'hterpreting R‘{S‘y), S'y is to be treated as an ordinary symbol until
U y ' k “ S . e l |nilna ted by *30 01 an.l *14 01 or *14 02, and then the
above definitions are to be applied to S‘y.
*30 1. h : [R'y] ./(R'y) . = . [(,.*) ( x Ry)] ,/( 1x ) (xRy) [*4*2 . (*30 01)]
*3011. h>[R‘!/].f(R‘y).mz(ab):xRy m m x .x-b:fb [*30 1 .*141]
Ihc following propositions are immediate applications of *1431 ff, made
m accordance with *301.
*30 12. 1-:: E! R’y . D : . [fl‘y]. p v x (R ‘ y > .= :p . v . [_R* y ]. x
[*14-31] J
h :: E ! R'y. D :. [R'y j .~ x (R‘y). = j[7?‘y]. x(«‘y)) [*14 32]
• 3 ” [ ***I • » 3 * • ■ -p • ^ • [«‘y] • x (.R'y)
*30 141. h::EiR‘y.D:. [R‘y] . x (R<y) Op. = : [tf-y] . x (R'y ). ;> . p
[*14‘331]
*30142. I-:: E! R'y . D :.[Jfy] .pm x (R‘y) . = : p . = . [ft‘y] . x (R‘y)
[*14 332]
*3015. h :-p : [R‘y]. x (R‘y) : s : [R*ij] .p . x (R'y) [*14 .34]
The following two propositions are immediate consequences of *14113112.
*30 16. h : [R'y] ./(R'y, S‘z) . = . [S‘z] ./(R'y, S‘z) [*14113]
./(R'y,S‘z).m:
(a6, c) : xRy .= x . x = b: xSz .= x .x = c :/(b t c) [*14112]
I-E ! R'y z (z) . <f>z : D . <f> (R'y) [*1418]
h :. R*y = b.D:yfr (R'y) . = . yjrb [*1415]
f- s. E ! R'y. — : ( a 6) : x Ry & [*4*2 . *1411 . (*30*01 )J
F . , P J? V ‘ ng * 30 2 ' w L e have to use the definition *30 01, not *301, because
18 D ,°‘° thC form /(**) ($*)- This appears if we attempt to apply
the definition *14 01 to E! (,*) <**), which leads to an expression containing
the meaningless constituent E ! 6. But by the definition *30 01, every typo
graphical occurrence of the symbol "R'y" means what results when this
symbol is replaced by (,*) (xRy)," hence " E ! R‘y " means “ E ! (,*) (xRy)."
*3013.
*3014.
*3017.
*3018.
*3019.
*302.
MATHEMATICAL LOGIC
[PART I
230
*30-21. H :: E ! R‘y. = ( 3 x). */fy: .rrty . z Ry . ,. * = *
[*14 203. (*3001)]
*30 22. b : E ! R*y . = . R‘y = (ix)(xRy) [*14 23 . (*30 01)]
Note that we do not necessarily have
R‘y = (tx)(xRy),
which is only true when E! R‘y.
*30 3. I- :,x= R‘y. = : .=..* = .r [*14 202]
*30-31. h xwm R*y . = ; x Ry : z Ry . D,. z = a: [*14*122 . *30 3]
*3032. h : E ! ify. ■ . </<‘y) Ry [*14 22]
*30 33. b :: E ! R*y . D :. ^ (/f‘y) : = : (gar). xRy . ^x : = : xRy . D x . y\rx
[*14 26]
*3034. K%x/fy.* x .*Sy:D:E!rt‘y.5.E!S‘y [*14271]
*30-341. b */<y. ==,. xSy : D s E! R'y . s . R'y - S‘y
Dcm.
K *14-21 . 31-srt‘y-S‘y.D.El/f‘y (1)
I-. *14-27 . Comm . D h Hp • D : E! R'y . D . R'y = S‘y (2)
H.(l).(2). Db. Prop
*3035. I-:. R - S . D : E ! R'y . = . E ! S'y [*30 34 . *21 43]
*30 36. I-: E ! R'y, R = S,D . R'y = S‘y [*14 27 . Imp . *21*43]
*3037. H:E \ R'y.y- 1 . R'y R'z
Dcm.
h . *14*28 . Dh:E l R'y .0 . R'y-R'y (1)
b. *1312. R'y = R*y. = . R*y = R* z (2)
b . (1). (2) . Ass . D b . Prop
This proposition is very frequently used.
*30 4. I-:. E ! R*y . D : a = R'y . = . uRy [*14 241]
This is a very important proposition, of which the use is constant.
*30 41. b :. (y) . 7?‘y = S*y . = : (y). E ! 7?‘y: 7? = .9
Dew.
b. *14-21 .*1011-27
• 3 5 (y) • ^‘y = S‘y. D . (y) . E ! 72‘y
(1)
K *1413 142.
^ h :. (y). R'y = S‘y ,D:(x, y)zx = R‘y . = . x
•a
£
ii
f(l).*30"4]
D : (x, y) : xRy . = . xSy:
[*21-43]
D:R = S
(2)
b . *30-36.
D h : E ! R'y . R-S.D . R‘y = S‘yz
[*10*11*27*35]
Dh:.(y).E!/^y:# = S:D.(y)./fy = S‘y
(3)
h.(l).(2).(3).
D h . Prop
237
SECTION D] DESCRIPTIVE FUNCTIONS
*30A2. h (,). E ! . 3 : (,) .7?‘* = S'*. = = S j
*30 B. h : E ! . D . E ! Q‘_-
Dem.
r H :• 3 h E! • * = (at) : *P(Q‘,). 3 X . * = 6 :
, 3.(a*).6P(«‘,)...6.6:
[* 13Io] 3 : (at) • bP (Q- t ) :
, n „ [*14-21] D : E ! ;.3h. Prop
*30 50!. h : * (P‘Q‘t ). s . (a6 . c).c = Q‘,. b = P . c .
On the meaning of » *),•■ see note to the definition *30 02.
22 • 3 h !! ^ (P, «"> • »’•<**> • W* «*>: «P w.). a. •. - 5: *6 ,
[ * 22 2 ° 2] s *•<«». O • — Q‘* . 6 - P‘c. *6 .O H . Prop
Xbs l t i:*T- a ’ ( * 0) - b - P ' C - C - Q, ‘ [*30-501 . *13*195]
*30 52. t-. EI P-Q-. . h . (a6 , c). 6 — P‘c. c — Q«, [*30 51 . *14 204]
*31. CONVERSES OF RELATIONS
Summary of *31.
If R is a relation, the relation which y has to x when xRy is called the
converse of R. Thus greater is the converse of less, before of after, husband of
wife. The converse of identity is identity, and the converse of diversity is
diversity. The converse of R is written R (read U R- converse”). When
R — R, R is called a symmetrical relation, otherwise it is called not-symmetrical.
When R is incompatible with R, R is called asymmetrical. Thus "cousin" is
symmetrical, "brother” is not-sym metrical (because when x is the brother of
y, y may be either the brother or the sister of x), and "husband” is asym¬
metrical.
The relation of R to R is called "Cnv." It will be shown that every
relation has one, and only one, converse; hence, applying the notation of *30,
that one is Cnv‘/t. Thus R = Cnv'P. We have thus two notations for the
converse of R\ the second is more convenient for the converse of a relation
not denoted by a single letter.
The more important propositions of the present number arc the following:
*3113. b . E! Cnv‘P
I.e. any relation P has a converse. Hence the relation "Cnv” verifies the
hypothesis (y ). E ! R*y, i.e. we have (P). E ! Cn \ *P.
*3132. b:P«Q.s.P-Q
I.e. two relations arc identical when, and only when, their converses arc
identical.
*31 33. b . Cnv‘Cnv‘P = P
I.e. any relation is the converse of its converse.
Very many of the subsequent uses of the notion of the converse of a
relation require only the propositions which embody the definitions of P and
Cnv, namely
*3111. b zxPy. = .yPx
and
*31131. b : * (Cnv'P) y. = . yPx
SECTION D]
CONVERSES OF RELATIONS
*31-01. Cnv = Qp [ x Qy . = i fi . ^p x | Df
*3102. P = .?i?(yP,.) j) f
*311. H Q Cnv P. = : . yp^. [*213 . (#3101)1
*31101. H: Q Cnv P. R Cnv P.2.Q=R
Deni.
f.r^Tli 3 h ••• Hp • *<% ■ • 'J p * •• yP.r:
L*H371] ^■■■rQv rRy:
[*21-43] 0:Q = JI:. D I-. Prop
*3111. 1- : aPy . = . yP. c [*21-3. (*3102)]
*31111, h.PCnvP [*31111]
*3112. (-./’ = Cnv'P
Bern.
p • *®1'101 ■ 3 H : Q Cnv P . P Cnv P. 3. Q ™ p.
[*31-111] D 1- : QCnvP. 3 . Q = p
h •(!)• *10-11.*31-111 .D
I- : P Cnv P : Q Cnv P.Dq.Q — P;
[*30 31] DKP-Cnv'P
*3M3. I-. E ! Cnv'P [*14 21. .3112]
*31-131. * (Cnv'P) y . = ,yP x [*3111-12. *21-43]
*3 132. h:QCnvP. 3 . Q -Cnv'P. 3 . 0 .p [.30 4 . .31 1312 ]
*3114. h . Cnv‘(P A Q) = Cnv'P n Cnv'Q
Dem.
r.™ 31 ‘ 3 ^* |Cnv ‘ (P A s • y .
1*31-1 313 = -yPx.yQx.
.2 331 *(Cn VP) y. x (Cnv'Q) y
t nf ] s • * (Cnv'P A Cnv'Ql y
*3115. Cnv‘(P ci Q) — Cnv'P a Cnv'Q [Similar proof]
*31-16. I-. Cnv'= (Cnv'P)
Dem.
|-.*31-131.3t-.«(Cn v‘- P) y. = . yx. p x .
[*23-35] s.~(yPm).
[*23-35] ] H —(«(Cnv'P)yJ .
240
MATHEMATICAL LOGIC
[PART I
*31-17. b y = P‘x . = : jPj . =.. * = y [*303 . *3111]
*31 18 b E ! P‘.r. = : <gy) z.rPz.= t .z = y [*302 . *31 11]
*3121. KCnv‘A = A
Deni.
b .*31*131 . D b :x(Cnv*A) y . = . y\xz
[*25-105] D H .~jr(Cnv‘A) y
Ml). *1L’ll .*2515. Dh. Prop
*31-22. I-. Cnv'V = V [Similar proof]
*3123. I-:P= V.2.P-V
Dan.
*3124.
*3132.
Dan.
H.*25 14.Dh:P-
V . 5 . (X, y) . xPl/ .
[*31-11.* 11-33]
s - (^. y) • yP- r •
[*11*2]
■ • <//. *). yP* •
[*23*14]
= . P = V : D 1-. Prop
- A . b . P = A [Similar proof]
-Q.a.P-Q
1-. *2143 . D H :. P ->
Q . s : a Py. . .rQy :
[*4*86*21.*31*11]
= : yPx . a,., . yQ.r :
[*11-2]
V
= : yPx. . yQx z
[*21-43]
= z P = Q z. "D b . Prop
*3133. h.Cnv‘Cnv‘P-P
Dem.
1- .*31*131 . D H sx(Cnv'Cnv'P)y. = . y (Cnv‘P) x .
[*31*131] =-xPy
Ml).*11-11 .*21*43. Dh. Prop
*3134. bzP=Q. = .Q = P
Dem.
h. *31*32. D h:P = g. = .P«Cnv'$
[*3112*32] = Cnv'Cnv'Q
[*31*33] = Q : D h . Prop
*314. h : P G Q. = . P G Q [*3111. *11*33]
*3141. b:PGQ. = .PGQ [*31*4*33*12]
*31 5. b z a ! P . = . a ! P [*31*24. Transp . *25*54]
( 1 )
( 1 >
SECTION D]
CONVERSES OF RELATIONS
211
*31-51. h
Dem.
*31*52. H
:(P) ./P . = . (P) .fP
(-.*101. 3 I-: (P) ,/P. D .fp ■
[*1011-21] 3 I-: (P) ,/p . D . (P) ,/p
K *101 .*31-12.3
>•« (P) ■fP ■ 3 ./(Cuv'P) .
[*31-3312] 3./P:
[*1011-21] D h : (P) ./p . D . (P) ./P
H.(l).(2).3K.P rop
: -/ 7 " • s • (3i J ) •//» [*31-51. Trnnsp]
R&w i
16
*32. REFERENTS AND RELATA OF A GIVEN TERM WITH
RESPECT TO A GIVEN RELATION
Sit m ma ry of *32.
Given any relation /?. the class of terms which have the relation R to a
given term »/ are called the referents of y. and the class of terms to which a
given term x has the relation R arc called the relata of .r. We shall denote by
R the relation of the class of referents of y to y, and by R the relation of the
class of relata «»f x to x. It is convenient also to have a notation for the rela¬
tions of R and R to R. We shall denote the relation of R to R by “sg,” where
••sg” stands for “sagittn.” Similarly we shall denote by “gs" the relation of R
to R, to suggest an arrow running from right to left instead of from left to right.
1< and R are chiefly useful for the sake of the descriptive functions to which
they give rise; thus R‘y = x(sRy) and R‘x = j)(xRy). Thus e.g. if R is the
relation of parent to son. R'y = the parents of y, R‘x = the sons of x. If R is
the relation of less to greater among numbers of any kind, R‘y = numbers less
than y, and R*x = numbers greater than x. When R*y exists, R*y is the class
whose only member is R*y. But when there are many terms having the
relation R to y, R‘y, which is the class of those terms, supplies a notation
which cannot be supplied by R*y. And similarly if there are many terms to
which x has the relation R, R*x supplies the notation for these terms. Thus
for example let R be the relation “sin,” i.e. the relation which x has toy when
x = siny. Then *‘sin‘ar" represents all values of y such that x = siny, i.e. all
values of sii^'x or a resin x. Unlike the usual symbol, it is not ambiguous,
since instead of representing some one of these values, it represents the class
of them.
The definitions of li, R, sg, gs are as follows:
*32 01. It = aj) \a = 2 (xRy)\ Df
*32 02. /e’=/§^!/3 = P(x/?y)) Df
*32 03. sg = AR (A = R) Df
*32 04. gs = A R (A = R) Df
In virtue of the above definitions, we shall have s g*R = R, g$‘R = R. This
gives an alternative notation which is convenient in dealing with a relation
not represented by a single letter.
SECTION Dl
REFERENTS AND RELATA OF A GIVEN* TERM
243
It should be observed that if 7? is a homogeneous relation (,>. one in
wh,ch referents and relata are of the same type), then 1i and Tl are not
homogeneous, but relate a class toobjectsof the type of its members.
In virtue of the definitions of 7J and *R, we shall have
*3213. y.~R‘,j = $( x Ry)
*32131. I-
Thus by *14-21, we always have Zl~R‘y and E <R‘x. Thus whatever
relation R may be, we have (y).ElR'y and (x).E'.R‘.v. We do not in
general have (y). a ! Ry or (*) . a <*R< X . Thus taking R to be the relation
Of parent and child, Ry = the parents of y and R>x~ the children of*.
Thus R‘x = A, i.e ~ a ! R‘ x , when * is childless, and ~R‘y = A, x.e. ~ a !R< y ,
when y is Adam or Eve. The two sort^of existence, E l~R‘y and a \
can both be significantly predicated of li‘y, because ■R‘y" is a descriptive
function whose valuers a class; and the same applies to R‘x. It will be seen
hold in^general ^ 3 ^ ! 1>ut t * le converse implication docs not
We have
*3216. b :~R *=~S. = . 4 R'= 4 s m s .Rr=S
Aso by *3218181,
b : a: 6 R‘y . = . x R y . = . y c Ri^
be rcdnrc!?? he U , 3e . ° f ^ ° r ^‘ X ’ CVCry sta tcment of the form "xRy" can
the dra in , “ S r ° nl . asserlln « mem bership of a class. Since, however,
functions a hT Tk'" g ‘ V6n by “ descri P tivc faction, and descriptive
O reduc, n rthe fh n T? ° f r ° lati ° nS - WO do not thus a method
I reducing the theory of relations to the theory of classes.
*32 01. R=Sf/{ a =$ (*« y) J Df
*32 02. *R = fa{!3 = Q(xRy)) Df
*32 03. sg = AR(A=R) Df
*32 04. gs = a£(A = A) Df
*321. h ! ay . 3 . a = £(*%) [*213 . (*32 01)]
*32101. I-: fiRx •=.£=$ (xRy) [*213 . (*32 02)]
*3211. h .^(xRyy^fry [*321. *30 3]
16—2
244
MATHEMATICAL LOGIC
[PART I
*32111. h.f/(xRy)=R‘x [*32101 . *303]
*3212. I-.E lit*;/ [*3211. *1421]
*32121. KE!tf‘x [*32111 .*14-21]
“E! R*y" must not be confounded with " g ! R t y." The former means
that there is such a class as R*y, which, as we have just seen, is always true;
the latter means that R'y is not null, which is only true if y is a term to
which some other term has the relation R. Note that, by *14 21, both 3 ! R‘y
and ~3 ! R‘y imply E ! R‘y. The contradictory of 3 ! R‘y is not ~g ! R‘y,
but ~j[/*‘y] • H ! R*y\. This last would not imply E! R‘y, but for the fact
that. E! R*y is always true.
*32 13. h . 7?y - x (xRy) [*32T 1. *20 59]
*32131. \-.R‘x=f)(xRy) [*32111 .*2059]
*32 132. h : alty . = . a = R‘y . = . a = x(xi?y) [*32 1*13 . *20 57]
*32133. 1 1 0Rx. = .0-R‘x. = .0 =1)(xRy) [*32101131.*2057]
The use of *20*57 will in general be tacit. It happens constantly that we
have propositions such as *32*13, in which a descriptive expression is shown
to be identical with a class. In such cases, whenever the properties of the
class are asserted of the descriptive expression, *20*57 is relevant.
*3214. b:li=~S. = .R = S
Dem.
K *21*43.
[*321] =
[* 112 ]
[*2025] =
[*20*15] =
[* 11 * 2 ]
[*21-43] =
*32 15. h : 7F =*S. = .R = S
- w w
. aRy . =* iV . aSy :.
. a = £ ( xRy ). =«. v . a = $ ( xSy ):.
. (y):. a = 5 (xRy) . =. . a = 5 ( xSy)
. (y) : £ (xRy) = 5 (xSy) :.
.(y):-(z):xRy. = .xSy:.
. (x, y) : xRy . = . xSy :.
.7? = 5::Dh.Prop
*32*15. : R = S . = . R = S [Similar proof]
*32*16. h :~R=~S.= . < R = S.= .R = S [*32*1415]
*32 18. I-: x e R‘y . = . xRy [*32*13. *20 33]
*32 181. h : y e iF‘x . = . xRy • [*32*131. *20 33]
*32*182. h : x eR‘y . = . y €%x [*3218181]
SECTION D]
REFERENTS AND RELATA OF A GIVEN TERM
245
The transformation from “xRy" to "x e R‘y" is one commonly effected in
language. E.g. suppose " xRy » is - a- loves y," then - a eR'y" is " a is a lover
*32 19. I-: R C S. 3 . ~R‘y C ~S-y .*R‘xCS‘x
Dem.
h . *3218 . D f- Hp. D :
O 22 ' 1 ] 3:~R‘yC&y
1-. *32181.31-:. Hp. 3 : y e*R‘x. 3„ . y e*S‘x :
C* 22 ' 1 ] 3 : R‘.r CS'x
h . (1). (2) . 3 H . Prop
h:4sgii. = .^ = « [*21-3 . (*32-03)]
A=R [*21-3. (*3204)]
[*32-2. *30 3]
[*32 201 . *30-3]
032-21 .*14-21]
032-211 .*14-21]
0-32-21 .*21-2-57]
032211 .*21-2-57]
*322.
*32 201. I-: A gs R . = .
*32 21. h.7t~sg‘R
*32 211. H.*=.gs<7J
*32-22. 1-. E ! sg ‘R
*32 221. h.Elgs'B
*32 23. h.sg‘iJ = «
*32 231. I-. gs‘/i =
*32 24. h.sg‘/e = gs ‘it
Dem.
t- • *32-23 . (*32 01). 3 1-. sg -R = & 9 (« = £(xRy)j .
[*21-33] 3K'« <sg‘R) y. = . a = $ ( x Ry ).
03111. *2015]
O32101]
032-211]
h • 0) • *1111. *21-43.31-. Prop
*32 241. KgsOUsg‘.ft [Similar proof]
*32 26. h:^sgi7. = .^ = sg « ii [*30 4. *3222]
*32 261. I-: A gs R . = . A = gs‘R 030 4 . *32 221]
*32-3. 1-. (sg-(fl ft S)}‘y =t~R‘y n ~S‘y
Note that we do not have
Bg‘(R *S)= s g*R A s g‘S.
a = 5 ( yRx).
aRx .
a (gs‘R) x
( 1 )
( 2 )
(1)
246
MATHEMATICAL LOGIC
[PART I
Vein.
*• • *32*23 13 . D h . {sg‘(72 n S))‘y = .< [x(R n S)y)
= x(xRy .xSy)
= xlxRy) r> £ (xSy)
- IVy r\ S‘y . D h . Prop
[*23-33]
[*22*39]
[*3213]
*32 31. I- . (gs‘( RnS)\*j:= *R'x n S'*
*32 32. b . (sg‘(7* u S)\*y = R*y u
*32 33. h . |gs‘< R sj S)\‘x = tF‘.c u
* 32 34. h • ! s g*( — R)\*y m ~~ ~IVy
*32 35. b . jgs‘( R)[ ‘.c = - Wx
The proofs of the above propositions are similar to that of *32 3.
*32 4. b E! R't . m : g ! JVi z x, y c /?*. D AV . x - y [*30 21 . *3218]
*32 41. H E! £‘y . D : 7?y = S*y . = . R‘y «= S‘y
Dem.
b . *486 . Dh: xSy . = x . x ■ 6: D
. = x . .rSy : = : .r7?y ,= z .x = b (1)
h . (1). *5*32. D h j-5y .= x .x = b: xRy . = x . *Sy s = :
xSy .= x .x = b: xRy .= x .x = b (2)
b. (2). *10 11-281 .*3218181 .D
f-(a*>) : xSy .s x ,x — b: R*y = £‘y: =
[*30 3.* 14-13] =
[*14101] =
b . (3). *30-2 . D b E ! S‘y . 7?y = S*y. = . 7*‘y - S‘yD H . Prop
*32 42. b 7*‘y = S‘y . D : E ! R f y . = . E! S‘y [*30 34. *32 18]
(nb):xSy.= z .x = b:xRy.z= x .x = bi
(3 b ): xSy .3 x . x - 6 : 7?‘y = 6 :
7<‘y = S‘y (3)
*33. DOMAINS, CONVERSE DOMAINS. AND FIELDS
OF RELATIONS
Summary of* 33.
If R is any relation, the domain of R, which we denote by D‘R is the
tmat a‘7 7'"7 h T° th f relati ° n R l ° ,0mctl,in S " “‘her; the converse
ZZZ ». ’ ‘ ‘ he / I , n f S ° f terms to which something or other has the
elation R, and the field, C*R, is the sum of the domain and the converse
irZ'.) ( 6 ‘ hat thC fie ' d U ° nly 8i « nific ““t when It is a homogeneous
The above notations D‘/?. (J-R. C‘R are derivative from the notations
field rnsl 0r ( ? rel " t , ,0ns - to a rel “ tion . of 'ts domain, converse domain, and
neld respectively. We are to have
D‘J*-$((ay).*%)
Q‘It = f) j(g;r) . xRy J
v , „ * {(3y) : *% . v.y/J.c);
hence we define D, (I, C as follows:
*33 01. D « a}{ [a =. 2 {(gy). xRy j] Df
*33 02. <1 - [£■ p {(gar) . *Ky|] Df
*33 03. C = yR [ y « 5 {(gy) . xRy # v # j Df
The letter C is chosen as the initial of the word M campus ” We reouire
the" fiefd e oS nit Th n ' Tt ely ° f ‘ h Y Clati ° n ° f * tC R whe " - is “ member of
he held of R. Hm relation, which we will call F, is defined as follows:
*33 04. F=$R ((gy) : a :Ry . v . yRx] Df
We Shal1 find that <>-?. D will be the relation of a relation to its domain,
Z th ? C ' aSS nZ ( r !'Y° nS having “ f ° r their domain ' Si milar remarks
wdt/series. d & The ^ ° f * relat,on ,s 8 P ec ' all y important in connection
The propositions of this number are constantly used throughout the
remainder of the work. The ideas of the domain, converse domaZand fieW
kindZconakte' 'iT S ° me '! hat different uses for Nations of different
function X f ^ T ^'° n ^ ^ rise a descriptive
anvthZ hf - Y r. reqU ‘ re that R ‘V 8hould exist whenever there is
than^one term"! Yi? 41011 , * *° * that there should nevcr be more
tatuea of ‘foTwh V1 h g p relati ° n * *° a giVeD tCrm ^ In “>is ^e
.• “ a *R J7 fl y ei ; StS Wdl constitute ‘he ■■ converse domain » of R,
I.e. a R. and the values which R‘y assumes for various values of y will
MATHEMATICAL LOGIC [PART I
constitute the “domain” of J{, i.e. D‘7?. Thus the converse domain is the
class of possible arguments for the descriptive function R‘y, and the domain
is the class ot all values of the function. Thus, for example, if R is the relation
ot the square of an integer// to //, then R*y = the square of y, provided y is an
integer. In this case, <P7f is the class of iutegers, and D‘R is the class of
perfect squares. Or again, suppose R is the relation of wife to husband; then
R‘y= the wife of //, < I‘if = married men, T)‘R - married women. In such
cases, th eyield usually has little importance; and if the values of the function
R‘y are not of the same type as its arguments, i.e. if the relation R is not
homogeneous, the field is meaningless. Thus, for example, if R is a homo¬
geneous relation. R and R are not homogeneous.and therefore " C l li" and "C l R"
are meaningless.
Let us next Mip|>«>se that R is the sort of relation that generates a series,
say the relation of less to greater among integers. Then D‘/? —all integers
that are less than some other integer — all integers, — all integers that
are greater than some other integer = all integers except 0. In this case,
C‘R = all integers that are either greater or less than some other integer
= all integers. Generally, if R generates a series. Y)*R = all members of the
series except the last (if any), U‘/J - all members of the series except the first
(if any), and C*R — all members of the series. In this case, “ xFR" expresses
the fact that .r is a member of the series. Thus when R generates a series,
C‘R becomes important, and the relation F is likely to be useful.
We shall have occasion to deal with many relations having some of the
properties of series, and with many propositions which, though only important
in connection with serial relations, hold much more generally. In such cases,
the field of a relation is likely to be important. Thus in the section on
Induction (Part II, Section E), where we are preparing the way for the con¬
struction of serial relations by means of a certain kind of non-serial relation,
and throughout relation-arithmetic (Part IV), the fields of relations will occur
constantly. Hut in the earlier parts of the work, it is chiefly domains and
converse domains that occur.
Among the more important properties of domains, converse domains and
fields, which are proved in the present number, are the following.
We have always E ! D 'R, E ! (l‘R, E! C‘R (*33 12121 122). (The last of
these, however, is only significant when R is homogeneous.)
*3313. h : a- e D‘R . = . (gy) . xRy
*33131. h : y c G‘7? . = . (gx) . xRy
*33 132. h xeC'R . = : (gy) : xRy. v . yRx
*33 14. h : xRy . D .arc D ‘R .yed'R
*3316. H . C‘R = D <R u d'R
SECTION D]
•219
DOMAINS AND FIELDS OF RELATIONS
*33 2-2122. The converse domain of a relation is the domain of its converse,
the domain of a relation is the converse domain of its converse, and the field
of a relation is the field of its converse.
*3324. H: a !D‘i?. = . a! a‘ii. = . a !C‘iJ. = .a!iZ
*33 4. KD‘7? = .?| a !«‘,rl
with corresponding propositions (*33-4142) for <1‘R and OR.
*3343. \-iR.\R‘y .ytWR. R‘,jeT>‘R
*33-431. h : (y) . E ! R‘y . D . <£) . 0 C d‘R
*33-5. KC-J?
*33 51. h:* ( C‘R . m . xFR
The proofs of propositions concerning a and C are usually similar to those
for D, and are therefore often omitted.
*33 01. D - a.£ [a £ {(gy). xlfy)] Df
*33 02. d - fik \fi - $ ((gx). x Ry)] Df
*33 03. C « yft [ y = £ {(gy) : X Ry . v . yitr)] Df
*33 04. F~*ft Kgy) : x Ry . v.yRx] Df
*331. h s aDR . 3.0 «= £ ((gy). X R V J [*213 . (*33 01)]
*33101. h:/3d.R. = ./3 = £ {(g*) . *%)
*33102. h : yCR . = . y = £ ((gy) : x Ry . V . yR X )
*33103. h X FR . s : (gy) : x Ry . v . yRx
*3311. h . D *R => £ {(gy) . x Ry] [*331 . *30 3 . *20 59]
*33111. h. a<R = $ ((gar). x Ry]
*33112. h . C*R = 5 ((gy) : X Ry . v . yifcr}
*3312. KE!D‘/e [*3311 . *1421]
*33121. h.E!d‘.ft
*33122. h.E! C‘R
*33123. h s oD« . = . a = D *R [*30*4 . *3312]
*33124. h :0(IR. = .0 = (I‘R [*304 . *33121]
*33126. h :yCR. = . y = C‘R [*30 4 . *32123]
*3313. h : * e D‘.ft . = . (gy) . x Ry [*3311 . *20*3*57]
*33131. h : y e d ‘R . = . (gx) . xRy
*33132. h x c C‘R , = : (gy) : x Ry . v . yRx
*3314. h : xRy . D .xeD‘i* .y ed‘i2
Bern.
V . *10*24 . D h Hp . D : (gy) . xRy : (gx) . x Ry :
[*3313131] D : x e D‘ii . y « d‘2£ D 1-. Prop
200
MATHEMATICAL LOGIC
[PART 1
*3315. I- ,R‘y CD *R
Dem.
K . *3218 .Dhif R*y. D, . xRy .
[*10 24] ^x.(3y).x%.
J*33-13] O x .xe 1)*R :Dh. Prop
*33151. h./K*C(Rft
*33 152. b .~R , xyj 4 R‘x C C*R
*3316. b.C'R-D'RsjQ'R
Dem.
b . *33*132. *10 42. D
h s.xtC'R . 3 s <a«/). . v . (gy) . yfl.c ;
[*33'13‘131] = :xf D'R .v.xtCL'R:
[*22 34] (1)
H. (1). *1011. *20-43. DK Prop
*33161. y.D‘RCC'lt.Q‘RCC‘H [*33 16.*2258]
*3317. V :.r/fy. D . x.j/eC'R [*3314161]
*3318. y-.V , R = (\‘R.1.\> , R = C l R
Dem.
b . *22-56. D b : D‘R = (I‘R . D . D‘7? - D f R v d‘R
=C‘R:Db. Prop
*33 181. h : d‘/? C D*R . = . - C*R
Dem.
b . *2262 .Db: d*R C D‘7?. = . D‘R = D‘R v CI‘72
[*3316] = C‘R : D b. Prop
*33 182. b : D'R CG‘R. = . G‘R = C*R [Similar proof]
If /i is the sort of relation which generates a series, so that " xRy" may
be read "x precedes y then G*R C D‘R is the condition that the series may
have no last term, since it states that ever}' term which follows some term
precedes some other term, and is therefore not the last of the series.
*33 2. KCKA-D'A
Dem.
b .*3111 .*1011 .Db zxRy.= x .yRj>:
[*10 281] D I-: (gx). xRy . = . (gx). yRx :
[*3313131] D b: y € Oi‘R. s. y « D‘J*
h . (1) . *1011 . *20 43 . D b . Prop
*33 21. I-. D'R = d'R [Similar proof]
(1)
SECTION D]
DOMAINS AND FIELDS OF RELATIONS
*3322.
Dem.
*3324.
Dem.
V . C'R = C'R
H . *3316 2-21 . D h . C‘R = C l'R u D*R
[*3316] = C'R . D h . Prop
h : a ! D‘R . s . 3 ! . = . 3 ! C'R . = . 3 ! R
K *33-13. D H s. 3 ! D'R . =
[*25*5.(*11-03)] =
h • *33*131. D h 3 ! Q'R . =
[* 11 * 2 ]
[*2.V5] ~
*" • *33*132 . D H :: 3 ! C'R . s
[*11-7] =
[*255] =
^ • (1) • (2) • (3). D h . Prop
*33 241. h : D'R = A . = . (I'T* =» A . s .C'R
[*33 24 . Transp . *24 51 . *25 51]
f-. D‘(i2 A S) C D‘R r\ D'S
(S '*) : < 3 ^) • xRy :
•k'-R
(ay)--(a x).xRy:
(a r » y) • x Ry ••
a! r
• (a*) - (ay) •• *Ry • v. yRx
• (a*. y) • xRy ••
• &IR
A . = . R = A
*3325.
Dem
* :(ay)-*(# AS)y :
= : (ay) • *-Ry • *Sy:
D : (ay) • xRtj : (ay) . a:Sy :
D : a: € D‘/2 . a: c D'S :
D :a:c D*R D'S
*33251.
*33252.
*3326.
Dem.
*33261.
*33262.
>».*38*18.31-:.««D \R*S)
[*21*33.*10*281]
[*105]
[*3313]
[*21-33]
H. (1). *1011. Dh. Prop
Ka‘(Kn5)Ca‘fina‘S [Similar proof]
H . C‘(i2 AS) C C”7e o C‘S [Similar proof]
h.D<(/ZctS)-D<K uD‘S
H - *3313 . D h x c D \R c; S) . = : (ay) ,x(RvS)y:
[*23 34.*10'281] = ; (ay) : a:i?y . v . xSy :
[*10*42] = ; (gy). xRy . v . ( gy ) a lS
[*3313] = saifD'fl.v.ifD'S:
[*22*34] = : a: c D'R *j D'S
h • (1) • *1011 . *20 43 .Dh. Prop
Ka‘(i*uS) = a‘/2v(l‘S [Similar proof]
1-. C'(R oS)= C'R yj C'S [*33 26 261 16]
2.32
MATHEMATICAL LOGIC
[PART I
*33-263. h : R G S . 3. D‘7? C D'S
Dem.
*■ • *231 .0 t-Hp. D : ,r%. D, -y . xty :
[*10-28-27] 3:(*>: < 3i/ ) . xKy. . ( 3y ) . xty:
[*3313] D : (x): x e D'Tf . D . x e D'S:
[*231] D : l)‘J{ C D'SD h . Prop
*33 264. I- : if C S . D. d‘H C U'S [Similar proof]
*33 265. H : It G.' S. D. C" JtCC'S [•33-263 20416. *22 72]
*33 27. KC‘7f=D‘(fiort)
Dem.
H . *33*16*2 . D
KC.A-D'AvD'if
[*33*26]
j?).DKProp
*33271.
h.PA_a<(/Z
KjJi)
[Similar proof]
*33272.
KD‘(/ic/rt).
a*(/e c/ if) = C‘(/? o 7?) = C‘R
[*33-27-271 16]
*33 28.
KD't-d'V-C'V-V
Dem.
K*10 25.*25
104. D h :.(x):(gy).xVy:.(x):
(3y).yV^:.
[*3313131]
D h s. (or) . x e D'V s (.r) . x e
a*v:.
[*2414]
D h : D‘V = V . Q'V s» V
(1)
[*3316]
D h . C*V - VuV
[*22*56]
-V
(2)
H.(1).(2).DH
. Prop
*3329.
1- . D‘A = Q‘A = C'A = A [*33-241. *21-2]
*333.
h.aCD^.i
: x e a . Z> x . g ! if‘a-
Dem.
K *32181. Db:.x,
f a. D x . g ! R*x : = :xea.D x . (gy). xify :
[*33
13]
= : x e a. D x . x >
f D*R :. D H. Prop
*3331.
h./?ca^.=
:ye /9. D y . g ! R*y [Proof as in *33 3]
The three following propositions are used in the theory of selections (*80,
#83 and *85). The second of them is also used in the theory of greater and
less (*117) and in the theory of transitive relations (*201).
*33 32. h : D *R n D‘S = A . D . R n S - A
The converse of this proposition is not true.
Dem.
D f-: x (R r» S) y .0 . xRy. xSy .
D.*«D'AaD‘£.
1-. *23-33.
[*33*14.*2233]
DOMAINS AND FIELDS OF RELATIONS
SECTION D]
[* 10 ' 24 J 3 . a ! D‘if r\ D ‘8
t-. (1). Transp. D I-: D‘J? n D‘S = A.O.~{x{Rf\ S) «,)
H. (2). *11-11-3. D hr IV R „ D ‘S = A . D . y) . ~ |. E (R*S)y).
l * 25 ' lo J D.flr,,S=A:DP.Prop
*33 33. P : Q'R n Cl'S = A . D . R A S = A [Proof as in *33-32]
*33 34. V iC'Rn C‘S = A.3.flAS= A
Dem.
H . *33161 .*22-49 . D I-. D‘7? n D‘S CC'Rn C‘S.
[*2413] D I-: C‘R n C'S = A . 3 . D‘R „ D'S = A .
[ * 33 ' 32 } 3.iiAS — A:3h. Prop
*33 35. H:.D*iJC«.= :a -Ry.3 I , # . ire «
Dem.
K*33l3.DP:.D'.RC«.B:(ay).**y.D,.* e a !
[*10-23] = . x fty . 3 A1( .**a:. D P . p rop
*33 351. P d'rt C a . = : xRy . D,.„. y e a [p roo f as in *33 35]
*33 352. P C'ii Ci.s: xRy .O x „.x,yta
Dem.
^ . *33 16 . *22*59 . D
h C‘7* C a . = : D*R C a . (We C a :
[*33 35 351] = : xRy . D x , „ . * e a : xRy . D,.*. y e a :
[*11-391] = : ar/ty . D r> .*, y«a:.DK Prop
The two following propositions (*33-4-41) are very frequently used.
*334. P.D‘fi = 2[ a! flV x )
Dem.
I-. *3313 . D 1-: are D ‘R . = . (gy) . xRy .
[*32181] = . (gy). y c ST'ar.
[* 24 '5] = . g ! 5f‘ar
h . (1) . *1011 . *20 33 . D h . Prop
*33 41. h . G'iJ = P (g ! R‘y) [Similar proof]
*33 42. h . C*R = £ fg ! (R*x v*R‘x)}
Dem.
203
0 )
( 2 )
( 1 )
1-. *33-4-41-16.D P . C‘R = £ ( a ! R‘ x ) „ 2 ( a . % x]
[*22-391] = a (a ! R“x • v . a ! S‘*J
[*24-56.*2015] = * {g * (S'* « S'*)) .DP. Prop
234
MATHEMATICAL LOGIC
[PART I
*3343. hzEiR'y.O.yedW.R'yeD'R
Dem.
V . *30*32 . D f-: E ! R l y . D . ( R‘>/) Ry .
[*3314] ^.yed'R.R'yt D‘R :Dh. Prop
*33 431. h : (y). E ! . D . (£). £ C Cl'R
Dem.
H . *33*43 . D h :. Hp. D : y e d‘R .
[Simp] D: i/€/3.D.ye(3‘.K (1)
K (1). *10*11*21 .Dh: Hp. D . /9 C Q‘72 (2)
K(2).*10 11*21.3 h . Prop
*33 432. h :(//). E ! R*y . D . (I‘/J = V
Dem.
h . *33*43 . *10*11 27 . D h : Hp. D .(y).yc d‘R .
[*2414] D.d‘R = VrDKProp
*33 44. h : E ! R*.c . D .xeD'R . tf'* « Cl 4 /?
Dem.
w
h . *33 43 ^ . D h : Hp. D . * cd'tf . 7*‘.r e D‘7? .
[*33*2*21] D ,xcD*R . R‘xed*R : D K Prop
*33 45. h y € (Pif u d*8 . . R'y - S‘y: D . 7* = 5
Note that by our conventions ns to denoting expressions, the scope of
both R‘y and S*y in the above is " R*y ■■ S‘y” and R*y is to be first
eliminated.
Dem.
h . *30*11 . D b :: R*y = S‘y . = :. (g b ): xity. =,. x = b : b = 5‘y :.
[*3011 ] = :. (g b ):. x/ty .= x .x = b:. (gc): xSy .= x .x = c:b = c:'
[*13*195] = :. (g b) : xRy .= x .x = b: xSy . =,. x = b :.
[*10*322] D:.xRy.= x .xSy (1)
H.(l).Dhss Hp. Dz.y€d‘Rvd‘S.D:xRy. = .xSy:.
[*5*32] D :. y e d‘R w G‘S. ar/ty. = . y c G*i? u G‘5. xSy:.
[*33*14.*4*7l] 0:.xRy. = .xSy (2)
I- . (2). *11*11*3 . D f-:. Hp .D:(x,y): xRy .= . :
[*21*43] D :/£ = &:. Dh . Prop
*33 46. h:.ar C D^vD‘5.D x . J*‘a; = 5‘a:: I> . i* = 5 [Proof as in *33*45]
*33 47. Vi.yeQSR v d‘S .5 y . R‘y = S‘y iD. R~S
Dem.
h . *33*41 . Transp . D h : y~e d*R w 0*5. D . R*y = A . S*y = A
( 1 )
SECTION D]
DOMAINS AND FIELDS OF RELATIONS
255
*33-48.
*335.
Dem.
h • ■ *13172 . *483 . D h : Hp . D . („) . R‘ y =~S‘y.
t* 30 ' 41 J 3.^=5*.
t* 3214 J 3.;i=S:DKPro|,
H a: e'D'R u D'S . 3* . %x =S‘x : 3 . fl = S [Proof ns in *3347]
t -.C=~F
( 1 )
l-.*321.Dh:. < jffi. = . a = a(a; jp i j)
[*33103] = ® (( a y) : xRy . v . yRx J .
[*33102] s.aCR
h . <1) . *11-11 . *21 '43.3h. Prop
*33 51. I- !*« C‘R . s . xFR [*33132 103]
cenfjftrlT 1fUl °. rdina r l ari f metic - " ll0le we »ro concerned with a scries
of thTs stts a tk T ’ “ " XFP " eX P reSS0S the fact «>.t a is a member
Part TV K ! °, Ve *T P ro P° sltions (*33-5 51) will be much used in
of ordinal
*33 6. h ; R € E>‘ a . = . a « jy#
Dem.
I-. *32181. 3b:.R e D‘«. = .aD.R.
[*33123] = . or - D'R : D I-. p rop
*34. THE RELATIVE PRODUCT OF TWO RELATIONS
Summary of# 34.
The relative product of two relations R and S is the relation which holds
between x and z when there is an intermediate terra y such that x has the
relation li to y and y has the relation S to *. Thus e.g. the relative product
of brother and father is paternal uncle ; the relative product of father and
father is paternal grandfather ; and so on. The relative product of R and S
is denoted by “ 7*1 the definition is:
*34 01. 7* | £ - Zz j(gy). xRy . ySz] Df
This definition is only significant when Q‘7? and D‘S belong to the same
type.
The relative product of R and R is called the square of R ; we put
*34 02. 7*** = PjP Df
*34 03. R’-R'\R Df
The most useful propositions in the present number are the following:
*34 2 . h.Cnv‘(72j6*)-N|72
I.e. the converse of a relative product is obtained by turning each factor
into its converse and reversing the order of the factors.
*34 21. h . (/>, Q) j 72 = 7^|(Q| 70
I.e. the relative product obeys the associative law.
*3425. KP|((2c/P) = (P|Q)c/(P|P)
*3426. K(7 J uQ)|P = (P|P)u(Q|P)
I.e. the relative product obeys the distributive law with respect to the
logical addition of relations. (For logical multiplication instead of logical
addition, we only get inclusion instead of identity; cf. *34-23-24.)
*3434. \-zRGP.S(iQ.O.R\SGP\Q
*34 36. h . D‘(P | Q) C D‘P. d‘(P j Q)Cd‘Q
*34 41. I-: E! P'Q'z . D . P‘Q‘r = (P | Q)‘z
*3401. R\S=tt\(ny)-xRy-ySz) Df
*3402. R 1 = R\R Df
*34 03. 7*» = P S |P Df
*341. V z x (R | S)z . =. (g y) . xRy . ySz [*21*3 . (*34 01)]
SECTION D] THE RELATIVE PRODUCT OF TWO RELATIONS
*34 11. h Z x ( R | S) 2 . = . g ! (R*j; n ~s* 2 )
Dem.
K#;U’l .*3218181. D
l-:.r(«| S)z. = .(%>/). ye%ur.yt's‘z.
[*22 33] = . ( 3 ^). ,j e *Ri x n Jf j _
[*2+ 5] = . a ! (R‘.r r, ~S‘z) : 3 I- . p rop
*3412. h • (a [*2133 . *3411]
*34-2. t-.Cnv‘(R|S)_S|^
Dem.
K *31 131.3 H : * (Cnv‘(fi |fi)J * . = . , (R | S) x.
* 341 s -(a >j)-iR'j.i/Sx.
[ * 31 ' 11] s.(ay).y^.x5y.
[* 341 J s .*(S|ie)x (1
l-.(l).*1111.*21-43. 3K Prop '
*34 202. h./e|S = <Cov«/e)|S
Dem.
H . *31 131.3 I- ix(Cnv‘£) y . ySz . = . y/fcr . yS, .
[ * 31H] s ■ xliy . ySz ( 1 )
t’/ on’*!? 11 Ml •* 34 ’ 1 • 3 >-:« l(Cnv‘/e) | S) ,. s.x( J R|S , )« (2)
h - (2). *11-11. *21 48.3 h. Prop
*34 203. h . R 1 3 - R | (Cnv‘3) [Similar proof]
*34-21. f-.(P|Q)|fl./>|( (2 |i J)
Dem.
IX”-* 10 ' 281 • 3 h (a*) = (ay) • *?». y Qz , ,/e,
•(ay)” xPy : (a*) . yQz . zRi
(ay)-*Py -y(Q\R)w (l
[* 11 - 6 ]
[*341 .*10-281]
■ (1) . *1111 . *341 . *21-43.31-. Prop
*34-22. P|Q|fl-(P|Q)|K Df
This definition serves merely for the avoidance of brackets.
*34-23. l-.P|(QA/e)G(P|Q) A(P|jj)
Dem.
** * *341 . D
rj^JJ 1 (Q * y ■ s = (a*) • mP, ■ z (Q * R) y :
r*10-6? ] = = (3 D.xPz.zQy.zRy.
L D : (3^) • *Ps - zQy : (gz). xPz . zRy
R ScW I
257
17
258
MATHEMATICAL LOGIC
[PART I
[*341] D:x(P Q)y.x(P R)y:
[*23-33] Q)*(P,R)\y (1)
h . (1). *11-11.31-. Prop
The converse of the above is not true.
*34 24. K (/'*<?, R C,P R) r\(Q R) [Similar proof]
*34 25. V.P (QvR)m(p Q)v(P R)
Dem.
h .*23 84.*10-281 . D
h (gz). xPz . i (Q u R)y . = ; (gz) : xPz :zQy.v. zRy :
[*4'4.*1()'2M J = : <g.-) : x p z . z Q,j . v . xPz . z Ry :
[*10-42] = : (g,). x Pi . zQ,j : v : (gr). xPz. zRy :
[*3+ l ] s : *<P, Q) y. v '.x(P \R)y:
[*23'34] mixlP [ Q»P\R)y (1)
K(l).*1111 .*341.31-. Prop
*3426. I -.(P<j(J) /f-(y>,/f)o(0 R) [Similar proof]
The above two forms of the distributive law, and the associative law
(*34 21), are the only one* of the usual formal laws that hold for $he relative
product. The commutative law, in particular, does not hold in general.
*34 27. h: R=R.0.R P-K\P
Dem.
h • *21 -43 . D h Hp . D : (x. y): xRy . = . xRy :
[*11401 ] D : (x, y): xRy . yPz . =,. xR'y . yPz :
[*10 281] D : (.r): (gy). xRy . yPz . =,. (gy). xR'y . yPz :
[*2115] D : R\ P = R' | P D h . Prop
*34 28. bzR-K.O.P R = P\K [Similar proof]
*3429. \-:R=R'.0.P R\Q = P\R' Q
Dem.
h . *34 27 . D h : Hp. D . R | Q = R' | Q .
[*34 28] D.P i R\Q = PR , \Q:D\-. Prop
In proving the equality of two relations, say R and S, we usually establish
first an asserted proposition of the form
xRy . = . xSy
or Hp . D : xRy . = . xSy.
We then proceed by *1111 (together with *113 in the second case) to
(x, y) : xRy. = . xSy or Hp . D : (x, y): xRy . = . xSy,
whence the result follows by *2143. We shall in future omit these steps,
and write " D h . Prop ” after we have established
xRy. = . xSy or Hp. D : xRy . = . xSy.
A similar ellipsis will be made in proving the equality of classes.
SECTION D ] THE RELATIVE PRODUCT OF TWO RELATIONS
*34 3. t-:a!(^jQ)- = -a!(a‘/ 3 r.D‘0)
Dem.
I-. *25*5 . D
»■:: 3 HP\Q).ms. (g.r.,,). (]> Q) y
(3^..V):(3
[*34*1]
[*11*27]
[*11*24]
[*11*27]
[*11*54]
[*33*13*131]
[*22-33]
[*24*5]
(3 # ‘» 'J> •) • • zQy
O*' •**. y). A .
• (3*)( 3 -i*. y). xPs . .Qy
• (3*> :• (3* r ) • *P* : < 3 y> • *Qy
:-(3*):-*«a‘P.X£D‘Q:.
i.(3«):.i«a‘Prt D«y
• 3 ! (a*P n D'(J) ::Dh. Prop
*34 301. I-: (I‘P oD‘Q = A. S .P,Q=A [*343 . Trans,,]
*34302. h: C‘P n C<Q = A . 3 . P \ Q - \ . Q />_ A
Dem.
h . *3316 . D I-: Hp. D . CL‘P n D‘Q = A . CI'Qr, D‘P = A .
[*34-301] D.P|Q = A.y P. A OK Prop
*34 31. I-: a ! (.P | Q). D . g ! /'. 3 • y
Dem.
h . *34*3 . D K : Hp . D . g ! (d'P n 1) ‘Q) .
[*24*561] ^ • 3 ! d'P. g ! D‘Q .
[*33*24] Prop
*34-32. H:.P = A.v.Q = A:D.P|y,A [*34-31. Trnnsp . . 26 - 51 ]
*34 33. h :xe lVR.s .x(R\ R) x
Dem.
y . *33 13.3hi( D‘R . = . ( 3 y). xRy.
f * 4 ' 24 } s.(ay).a-fly.*Py.
[*31*11] s . (ay) . xR,j . yRx.
[ * 341 1 s .x(R | R)x: D H . Prop
*3434. y--RCP.SCQ.O.R\SCP\Q
Dem.
y . *231 . D y Hp . D : xRy . 3,.„ . xPy , y St. D„., . y Q z ,
U3-471* 10 1 41] 3 : x Ry • 3 . xPy : ySx. D . yQ*,
[*3 47] ^txRy.ySz.D.xPy.yQz m
^ • (1) • *10*11*21*28 . D )
M4 .n ^ : . H P-D ! (ay).*ny. yj a r o.(ay).*Py.yQ #2
[*34 1] 3 : X (R | 5) , . D . * (P | Q) , (2)
• (2) • *11*11-3 .Dh. Prop ( ’
259
17—2
260
MATHEMATICAL LOGIC
[PART I
*34 35. h : 3 ! /{. d‘i? C D 4 /*. D . g ! /J P
Deni.
I-.*33-24. Dt-tHp.D.gia'*
4 . *22021 .31- : Up. 3 .(l‘R= (l‘Ii n D‘P
K <1). (2). 3 I-: Hp. 3 . g ! (I 'll r> D ‘P.
D. g ! 7f P: 31-. Prop
*34 351 I- : 3 ! K. IV It C Q‘l>. 3 . 3 ■ P , It [Proof as in *34 35]
*34 36. I-. 1 )‘(P (D C D 'P . <J‘(P Q)C(VQ
Item.
y . *33 13 .Dh:.x,LV(p Q).0: (gr). x(P Q) :
l* 34 "'] 3:<g z, y). xPy. yQz:
[* 11 - 3 ] 3 : (g y. z). xPy. yQz :
l* 11 •■>.-..* 10-.»] 3 : (g ,j ). j-Py :
[*3313] D:««D‘P
•Similarly l-:.i<CI‘(/’ Q).0:ze(l‘P
H . (1). (2). *10-11.31-. Prop
Tim following proposition is n lemma for *!)5'31.
*34-361. I-: a ! It . D’H C (VP .(1‘ltC D ’Q. 3. g ! P | R | Q
Dem.
I-. *34-35. D I-: lip. 3. g • /? Q
I-. *34 36 . D I-: Hp. 3 . L)‘(/{ j Q) C (VP
y .(1).(2).*34-351.3 h. Prop
*34 37. y.fiP Q)CD‘Pv(VQ [*34-36.*33161 .*2272]
*34-38. h.C H (P Q)CC , PuC , Q [*3437 . *33161 . *22 72]
*34 4. y-.b = P-c .c = (fz. 3.6- (P | Qy t
Dem.
I-. *30-31. D h s Hp. 3. bPe . cQz .
[•341] 3.fc(/ J |Q>*
I-. *30.31 .Oh:. Hp.0:yQz.0,.y = e:
[ F " ct l 3: *1‘<J • yQ~ ■ 0,. y ■ xPy .y-c.
[*1313] 0 r y. xpc
I-. *30-31. 0 h :. Hp. 3 : xPc. 0,. * = b
-(2). (3).D h Hp. D zxPy . yQz. D XtV .x = bz
[*10*23] D : (gy). x Py . yQz ,D x ,x = bz
[*:**1] 3:*(P\Q)z.0 x .x=b
K(1).(4).*30*31 .DK Prop
*34 41. h : E ! P*Q‘z . D . P‘Q‘ Z = (p | Qy z
Dem.
K *30*52. D f-: Hp.D.(g6, c). 6- P‘c .c = Q‘z .
[*30-51 .*34 4] 3 . ( a fe). b = P‘Q‘z. b = (P | Q)‘z .
[*14145] 3 . P‘Q‘z = (P | Q)‘z ■ 3 |-. Prop
( 1 )
( 2 )
( 1 )
< 2 )
( 1 )
( 2 )
( 1 )
( 2 )
(3)
( 4 )
SECTION D] THE RELATIVE PRODUCT OF TWO RELATIONS 201
E ! 5|Q?r e s 1 nco7S iS ,0 " ger ,n ' e if W0 cha "g*-‘ ^ hypothwis into
th , ' Q \*- Sm * e (P 1 Q) * exlst "hen does not. Suppose, e.o..
; athe f Tht £*? ° f t hiW t0 fi ‘ thCT - P tho relanon of daughter to
the child of V ! ? ra d r" ghter ° f *• b,,t = ,hc 'laughter of
while the si ^ hrSt / X,8tS whenever * h«* only one granddaughter.
While the second requires further that r should have only one child.
For the same reason we do not have
L11 . 6 = <- P IO)‘^3-(3c).6 = / > 'c.c = 0‘r.
otherwise. P Qnte ° n °' raan ->’ relation8 (cf.*71), but not in general
*34 42. !-:(«). R‘ z = P'Q> Z ,^.R = P q
Dem.
*345.
*3451.
Dem.
j-. *14-21 . 31-:. Hp. 3 : (*). E! R‘t : (*). E! P‘Q‘ Z
h • (1) • * 34 ' 41 ■ 3 h s. Hp . D : (*) . R‘ z = ,p | ny z .
[*3042.( 1)] D : R - P \ Q D 1-. P rop
I-: xR'y . = . (g*). x Rz . z R y [*34. t . (*34 02)]
I-: xR'y . = . (as, w) . xRz . zRw . xuRy
y . *341 . (*34-03). D
h xR‘y . = : (gw) . x RHu . wRy :
[*34-5] = : (gw) : (gs) . xRz . zRw : wRy :
[*11-55] = : ( 310 , z) . xRz. zRw . wRy ■
[*112] = • (as, W) . xRz . zRw. wRy :. D h . p rop
I-. R’^R\If [*34-21]
l-ialfl’.s.gl D‘R n a‘R [*34-3]
.34 631. h : D‘R n(I'fi = A. = ,J(' = A [,3453 . Trausp]
*34 64. I- : xRx . D . x R?x
*3462.
*34 63.
Dem.
*3466.
*34 66.
*346.
h . *4 24 . 3 I-: xRx . D . xRx . xRx.
[*10-24] 3 ■ (ay) ■ *Ry. yRx .
[*34-5] 3 . xR , x . h Prop
V R‘Q S. = : x Ry . yRz. D Iy , . x Sz [*345 . *10 23]
H . D‘R’ C D‘R. <I‘R> C a-R . C‘R> C C‘R [,34-36-38]
I -.(.RnSy<ZR>r,S’
Dem.
H . *34*5 - D b a: (R A S)*y . =
[*23*33.*10-281] =
[*4*3.*10*281] =
(a*) - * (R A S) z . * (R * S) y
(3*) . xRz . xSz . zRy . zSy :
(3*). xRz . zRy . xSz . zSy :
( 1 )
202
MATHEMATICAL LOGIC
[PART I
[* 1 O' 5 ] D : (gj) . xRz . zRy z (g z) . xSz . zSy :
[*34*5] D : rR*y . xS-y :
[#2333] 0:.r(frnS'-)y (1)
H.(1).*1M1 .Dh.Prop
*34 62 .{RvS)--R-'v It SwS RvS s
Deni.
b . #34-26 .D\-.{RuS)■ =R (Rsj S) \jS (R v S )
[*34-25] - R* sj R , S c; S j R sv . D h . Prop
The above proposition is a lemma for *100-51, as is also *3473, which
employs the above proposition.
*34-63. h . Cnv*( R 1 ) - (Cnv-/?)*
Dem.
h .*31*131 . D
b a- |Cnv*(/f»)| y. = : y/ttr:
[*34*5] = :(gc).y/fj . zRxz
[*31131 .*10-281] = : <g*). xRz . zRy :
[*31-131.*34*5] = :x(Cnv-/0* y : D b . Prop
*34 7. b . Cnv‘(S 1 $) = .S’ 5
Dem.
I-. *34-2 . D b . Cnv'(&1S) «= (Cn v'S) 15
[*34-202] =S|5. Db. Prop
Thus S\S is always a symmetrical relation, i.e. one which is equal to its
converse.
*34 701. b.Cnv'(S|S) = S|S [*342-203]
*34 702. \-.C‘(S\S) = n*S
Dem.
b . *34-37 . D b . C‘(S t S) C D‘S w d'S
[*33-21] CD'S (1)
I-. *3313 .Dhxf D‘S . D . (g y) . .
[*31-11] 3. (g y).xSy.ySx.
[*341] D.x(S\S)x.
[*3317] D.xeC\S\S) (2)
I-. (1). (2). *1011 . D b . Prop
*34-703. b . C‘(S | S ) = d'S [Similar proof]
203
( 1 )
(2)
(3)
SECTION D] THE RELATIVE PRODUCT OP TWO RELATIONS
*3473. b : C*P rs C'Q = A . D . (P u Q? = p* c Q-*
Vein.
. *34-302 . D h : Hp . D. P J C = A . Q | p = /\ .
[*25 24] D . i>-' o = />-1, P | Q o Q PcQ !
[ * 34 ' 62] w = (-P ^ Q)=: D I-. Pro,,
*348. = R.0.R = 1V=R R
Vein.
V . *34 28 . D h : P-P. D . p* = p r
h . *34-33 . *3314 . D b : xRy . D . *<P p U .
h • (1) • (2) . D I-7e = P . Z> : *Py . D .
[* 47 !' * 23 ' 1 • 3 •*’•*-*• * c R. D: . D . ^,
[*10 24.*34-5] 3:^0.«Ar. *%•
K (4). *1171-3. 3t-:Hp.3.flC** ' *
H.*3-27. 3t-:Hp.D.J?*e« an
1 -. (5). (6) . *23 41 . D I-: Hp. D . R = IV
h.<l>.(7). D I-. Prop (7)
The hypothesis^ of the above proposition is the hypothesis that R is
ofTose iC “l' \ R “ “1“ u d transitive G «)• These are the formal properties
some respet be rC *“ r<led " “P"*"* equity in
*3481. i-:R = R.IVCR. = .R = R.ji, = J{ [*348.*471]
The following propositions are lemmas for *34 85, which is used in *72 64.
*3482. i--~R-R.R’<lR.3: X( D‘R. s . x R x
Dem.
*" • *3433 . Dhx< D*P . = . x (P | P )x
K.*34-8. 3b:.Hp.D:a:(.R|£) i c. = . x fl x
*".(!). (2) . D I-. Prop
*3483. I -:R=R.R‘QR. x Ry.O.% x ^R‘ y
Dem.
h . *31-11 . D H Hp . D : yRx :
P**] 3 : xRz . D . yRx . xRz .
[*3455.Hp] D.yRz
b . *3-2. D b Hp. D : y R z . d . xRy . y p^ #
[*34*55. Hp] D.arPs
I- - (1) * (2). D b :. Hp . D : x Rz . = , y p* :
[*10-11*21 .*20*15.*32*111] D : P‘* = P‘y D f-. Prop
( 1 )
( 2 )
( 1 )
( 2 )
264
MATHEMATICAL LOGIC
[PART I
*34 84. \-:R=R . R’CR.yclVR. *R‘x = *R‘,j . x R,j
j Vein.
H. *34*82. 3h:H| >.3.y% (1)
h . *321 >si . *20 31.31-:. Hp. 3 : xRz . =,. yRt :
O' 01 ) 3: . = . yRy (2)
H ■ < I). (2 >. 3 (-.Prop
*34 841. I-: R = R . R G It. .r ( D‘/f . /f« x ^*R‘y . 3 . xity
Dent.
h • * :U 84 7^ • 3 I- S Hp. 3. v Rx .
[*3111.1 Ip) D . xRy : D H. Prop
*34 85 huR-R. 77- G 7?. D : jr/ty. = . x € D‘7? ,%x -
[*34*83841 .*33*14]
*35. RELATIONS WITH LIMITED DOMAINS AND
CONVERSE DOMAINS
Summary of *35.
In this section, we have to consider the relation derived from a given
relation R by limiting either its domain or its converse domain to members
of some assigned class. A relation R with its domain limited to members of
a is written "o'] .ft”; with its converse domain limited to members ofit
is written “ftf*#”; with both limitations, it is written " a *] R f 0" Thus
e.y. "brother” and "sister” express the same relation (that of a common
parentage), with the domain limited in the first case to males, in the second
to females. "The relation of white employers to coloured employees” is a
relation limited both as to its domain and as to its converse domain. We put
*35 01. R^mxea.xRy) Df
with similar definitions for ftf*a and a“\ R[
A particularly important case is the case in which the same limitation is
imposed on the domain and on the converse domain, i.e. where we have a
relation of the form "o'] R [ a.” In this case, the limitation to members of a
may be more briefly stated as being imposed on the field . For this case, it is
convenient to adopt as an alternative notation. This case will be
considered in *36.
It is convenient to consider in the present connection the relation between
x and y which is constituted by x being a member of a and y being a member
of/9. This relation will be denoted by "at/3.” Thus we put
*3504. a| /3 = 2p0rca.y € £) Df
The chief importance of relations with limited fields arises in the theory of
senes. Given a series generated by a relation R, let a be a class consisting
of part of this series. Then a is the field of the relation al ftf*a or HI a, and
it is this relation which is the generating relation of the series of members of
« in the same order which they have as parts of the original series. Thus parts
of a series, considered not merely as classes but as series, are dealt with by
means of serial relations with limited fields.
ReJations with limited domains are not nearly so much used as relations
with limited converse domains. Relations with limited converse domains play
a great part in arithmetic, especially in establishing the formal laws. What
is wanted in such cases is a one-one relation correlating two classes or two
series, that is, we^want a relation such that not only does R‘y exist whenever
V C( f but al 1 8 ° R ‘ a exisfcs whenever xeD‘R. The kind of relation which is
most frequently found to effect such a correlation is some such relation as D
266
MATHEMATICAL LOGIC
[PART I
«»i (i or C, or some other constant relation for which we always have E! R‘y,
"itl* ' ,s con verse domain so limited that, subject to the limitation, only one
vahm of y gives any given value of R'y. Thus for example let X be a class of
relations no two of which have the same domain; then D[*X will give a one-
one correlation of these relations with their domains: if R, Se\, we shall have
TVR=D‘S.D.R = S.
We shall also have WR = (I )[\)‘R and D‘S = (D [\)‘S. Moreover the con-
‘ ,oma j n ot x * s ;u, d the domain of Df*X is the class of domains of
members of X. I bus J>f*X gives a one-one correlation of X with the domains
of members of X. It is chiefly in such ways that relations with limited converse
domains are useful.
I’oi purposes of reference, a great many propositions are given in the
present number, but the propositions that will be used frequently are com¬
paratively few. Among these are the following:
*35 21. Ka1/er^ = (a1/0r^ = a1(/?r/9)
* 3531 . h.</fr a )r£-/fr<«A£)
*35354. H.(/f[*a) .S -It a*\S
I.e. in a relative product it makes no difference whether we limit the
converse domain of the first factor, or the domain of the second.
*35 412. V.R\(&vff)=H\&KjR\ff
*35452 hWRCft.D.Rfft-R
*35 48. I-: (VP Q a . D . R) = P It
*35 52. h . Cnv*(/t f* £) = ft 1 R
*35 61. KI)‘(a1/?) = anD‘/?
*35 64. t-.(V(Rtft) = ft„(I<R
*35 65. h: £ C Q*R . D. ( V(R f ft) = ft
I he hypothesis ft C (I *R is fulfilled in the great majority of cases in which
we have occasion to use R f ft.
*3566. \-:a‘RCft. = .R[ft=R
*35 7. h :<f>\(R r /9)‘y(. a. .y e ft. <f> {R‘y)
Ihis proposition is used very frequently, owing to the fact that limitation
of the converse domain is chiefly applied to such relations as give rise to
descriptive functions ( e.g . D, a, C).
*35-71. h.yeft.^.R'y^S'jr.l.Rtft-Stft
This proposition is useful for a reason similar to that which makes *35'7
useful.
*35-82. Kat£ = a1Vr/9
Owing to this proposition, the properties of a f ft can be deduced from the
already proved properties of a*|i? f ft, by putting R=V.
SECTION D]
LIMITED DOMAINS AND CONVERSE DOMAINS 267
The relation “a f 0" is what may be called an “analysable" relation, i.e. it
holds between * and y when .re a and ye 0, i.e . when .r has a property inde¬
pendent ol y, and y has a property independent of .r.
*3585. h:a!/3.D.D‘(a1-/3) = a
*35-86. H:a!a.3.a*(at/3) = /3
If either a or 0 is null, so is a f /J (*35-88).
*3501.
a *] R = 5$ (x e a . xRy)
Df
*3502.
R T 0 = (xRy .ye 0)
Df
*3503.
a 1 I* T @ = $5 (* e a:. a:72y . y e 0)
Df
*3504.
0 = j}$(x e a. ye 0)
Df
*3505.
Df
I he last definition serves merely for the avoidance of brackets.
*351. H:*(a1./i)y.s.*«a.a:.Ky [*213. (*3501)]
*35101. \-:x(Rf0) y . = . x R 1/ . y( ff
*36102. >-:x(<i-\Rrff)y.^. xta . xR y , yt/8
*36103. h:*(at^)y.3.xea.y ( /3
*3511. h.«1«f/3-(a1^)A(iJf- j 8)
Dem.
y. *35102. Dh:*(«1fi^)y. a . X( a . x Ry.y ( 0.
[*4 24] =.xea.xRy.xRy.yef3.
[*35-1-101] — • * ( a 1 R) y. x(R f /9) y.
[ * 23 ' 33 J s . * ((a 1 R) A (R f /3)J y : D b . Prop
*3612. h . (a ] R) A ($ f" ,8) = a 1 (R f\S)f 0
Dem.
h . *23-33 . D h : a {(a ] R) A {S |- 0)) y. = . x (a 1 R) y . x (S |" 0) y .
[*35-1-101] 3 .xea.xRy .xSy .y e 0 .
[*2333] = .x ea. x(R A S)y . y e 0.
[ * 35 ' 102 ] 3 . * |° 1 (R A S) r /9| y : Z> I-. Prop
*3513. h.(a1JJ)A(/31S)_( art/ 9)1(fl,sS)
Dem.
J-.*23-33.Dh:«((a1«)A(/3-] S» y . = .x{a 1 R)y . x{0-\ S)y .
[*35*1] = .xea.xRy . x e 0 . xSy .
[*2233.*23-33] =.»<(«''ffl.*(SAS),.
[ * 36 ‘ 1J = . * ((a «>3) 1 {R A S)} y: O h . Prop
*36-14. h.(/if a) A(S[•£) = (« nS)f(an 0 ) [Similar proof to *3513]
208
MATHEMATICAL LOGIC
[PART I
*35-15. =
Dew.
H . *3511 . D
K( _ a 1 /f rA(a"|.s'f-/3') = (o1 .ft) A(.K|-/9) A(a'1 S) A(S[-/9')
[*3513-H] = |(a n o') ] (/f A S)| a | ( ft a S) f (ft a ft')j
[ * :i V11 J = !<a A „') 1 (ft A .S') r 03 A /?■)!. D H . Prop
*3516. K(a1«)A.S--a1(«AS) = ifA a 1S [Similar proof to *3513]
*3517. I-. (ft r ft) A ft- (ft A ft)|-ft = ft A Sf ft [Similar proof to *3513]
*3518. K(o] ftf- ft) Aft = a1<ft A,S)f ft=ft A a]ftfft
.oK.oi l [Similar proof to *3515]
*3521. K«1/fr-8-(«1fl,r/J-o1<«r/9> 1
Dew.
y. *35 102.31- :x(a1ft|-ft)j,. 3 .x«a.xfty.y*ft.
[ * 331 1 s.x«,-\R)y.y t 0.
[* ; J5101] H.x|< a 1ft)|-ft]y (1)
K *35102. Dl-:x< a 1ftf-ft),,. s.xta.xRy. yrft.
[*35101] s.x«B.x(ftf-ft)y.
t* 351 l =.x|a1(«r^))y (2)
I" • (1) • (2). D h . Prop
*35 22. K(a1/f> .$ = a1<K|S>
Dew.
H.*341 .DH:.x|( a -|ft) S| >J . = : (gr). x( a ] ft)r . zSy :
[*351] = :(g*) • a . rftz. zSy :
[*10-35] =:xca:(g s).xRi.zSy.
l* 341 J s:x (a .x(R\S)y:
C* 351 ) s : x (a 1 (ft 15)) y :. D h . Prop
*35-23. I- ..9;(rtf ft) = (S| ft)[ft [Similar proof to *35 22]
*3524. a1«,S=(a1/?)|S Df
*35 25. ft|ftfft = (S|ft)rft Df
*36-26. h . (a 1 /?) | (S’ r >9) = «1 («i ,S) [ >9 = [«1 («I -S)] r >9 = «1 I -SD r /3)
- ((«1«)| S] T/S = «1 [«|(-ST
= («1J£|S)[-/9-.a1<yi|Sr/9)
Dew.
K*341.Dh
[*351101]
[*10*35]
[*34*1]
[*35*102]
^!(a1«)|(^r^)ly. = :( a r).x(a1/?)^.r(5r/9)y:
= : (3*) •xea. xRz . zSy . y e 0 :
=zxea.yc0: (g*). xRz . zSyz
= : x e a. x (R \ S) y. y e 0 :
>:*{« 1 (A|S)rijy ( 1 )
I-. (1). *35-21-22-23. (*35’24-25) .DK Prop
SECTION P]
*3527.
*3531.
Dem.
LIMITED DOMAINS AND CONVERSE DOMAINS
a1«|Sr^ = (a1«|S)f- i S Df
K<.Rr<*)r£=flr(a«/9)
2i)9
r h • * 35101 • 3 *- ■■ * u* r <») r -si v . = (R r .>* s.
[*22-331 1
r L s..r%. years 8 .
*35 32 -.*|*r(-A»J,0(-.P Wp
*35 33 ' r V, iHnmR tI>r " 0f l ” tl >nt nr.35 3, ,
*35 33. H . 1 * r *) r 7 - |« 1 * r (* « *>1 l Proof nimihr to that, of .35 3. ]
a-](/i -] /f f 7 ) - {(ar,/9)1 flf 7 | [Proof similar to that of .35 311
*3535. h .o') R = ( a „ D*R)^\R J
Deni.
h.*351 .3 h :®(a1«)y . = .xea.xR,,.
* J!! l*™-*"*"** [Pmo/as in .35-35]
^" 1 ' B f - 0-(««D‘.R)1/if-(0 rt a‘.R) [Proof ns in .35*35]
*35-354. h.(i?(‘a)!S = ie;«-|S J
Dem.
*341. *35101. D
J- • * ((« r «) 15) * . s . (gy) . x Ry .yea.ySz.
f* 85 ’ 1 J s • <ay) • • y (* 1 *.
h f 1 r<*«*'>-*r*«*r*' [*35-101 .. 2 *L]
*36 413. h . (a w a ) 1* f- (0 u #) _ (a R f „ (a 1 R f
„ ^<«'1-« f £>o<« 1/i|-40 [.35 102..22-34]
11L J * «1 <*•*>-<■ 1 *>•<■ 1«) [.35-1 . .23 .34]
*35 42!. H.(fi 0 S) W = (RWo(S W [*35 101 . .2334]
*36-422. I-. a 1 (Ji o S) fff - (a R m 0 (a S m [t35 . W2 . „ 23 . 34
*35 43. h.-aC/3.D.a-]iiG/3-Ift
Dem.
LS5 1 ' = >| -*- c /»• 3 : «(«1 J*)jt....... .
[.2-1 3-
*35-431 u.o,- -N a. ^ *«(^1/i)y D H . Plop
*35 430 ■' 8 ' y ' :>- - R, ‘ 5Cfl r-y [Proof similar to that of .3543]
*36432.H:aC y .^CS.3.« 1flr)9GTli j rs
[Proof similar to that of *35 43]
270
MATHEMATICAL LOGIC
[PART I
*35-44. h.alPGP
Dem.
H . *35'1 . D b : .r(a ] R)y . D .xea . .r7?y.
[* :{ - >7 ] D.-rfly.OKProp
^35 441. b . R[0QR [Proof similar to that of *35 44]
*35 442. h . a ] R f* <Z 77 [Proof similar to that of *35 44]
*35 451. b : IV It C a.D.*\/t = It
Dem.
h . *4 7 l.3h:.Hp.D:xf D‘Ji . = .xt D‘i?. xea:
D : .»•« l)‘/f . xliy . = . xe D‘li . xRy. x e a (1)
h . *3314 .*471. 0 I- : .rl<y . = ..>•« JJ‘«. x/ty (2)
H.(l).(2).3h:. Up. 3:x%. s .x%.x«a.
f* 3v| ] s . x(a 1 7f)y :. D I-. Prop
*35452. H :(l‘7i Ci3.‘2.Hf/3 = K [Similar proof]
*35 453. (■: D‘/fC a .D.a1/fr/3=/fr/9 [Similar proof]
*35 454. H:(I‘/fC/3.D.o1/f[/3- a 1/f [Similar proof]
*35 46 I-: /f G S. D . o 1 /f C o 1
hem.
h . *23'1 . D h Hp. D: xRy . D . xSy :
[ ^ act ] 0 : x € a . xRy .D.xea. xSy:
[*35*1] D :jp(*1 R)y. D .ar(a]5 )y :. D b . Prop
*35 461. b : R (ZS, 5 , R[0 Q $S [ 0 [Similar proof]
*35462. h:/;GS.D.*17?r^C«1Sr/9 [Similar proof]
*35 471. F:a*Pna-A.D.P («17?)-A
Dem.
h . *34*1 . D I- j x \P j (a 1 R)l z . D . (gy). xPy . y (a ] 7?) r.
[*351] D . (gy). xPy .yea . y 7?£ .
[*33*14 . *10*5] D . (gy) . y 6 CP/* .yea.
[*22*33. *24*5] D.gia'Poa (1)
K(l).Transp.*24*51. D
b : (I*/* r» a = A . D . ~ x \P | (a ”] 7?)) z :
[*11*11*3] D(-:a‘Pna = A.D.(i, z).~x[P\{a'\ R)) z.
[*25*15] D. P j (a 172) - A : D h. Prop
*35-472. h : D‘P n a = A . D . (7? f* a) \ P = A
*35 473. h : <3‘7 J ^a=A.D.P|(a17?[‘/9) = A
*35-474. h:D‘Pn / 9 = A.D.(a‘17?|‘^)!P = A
SECTION D] LIMITED DOMAINS AND CONVERSE DOMAINS
*3548. h : (I‘P Za ."5 . P\( a '\R)-= P R
Bern.
K* 221 . Dh:.Hp.D:. ve a‘i>.D„. J , ea : -
7l J
[*10311] D:^.y e a-P.y € a.= v ..rPy.ye a < P (1)
• (1). (2) . D f-Hp . D : xPy .y e a . = v . x p tJ ;
fSJJJ' ] 3: ■ * « « • y** • • *l‘y . ytf*:
„ ^ = - (/* I a •, /<) , . s . (P , R) s D K Prop
*35 481. h : D«* C 0 . D . (/>|"/3) | R = />| « [Similar proof]
*35 51. h.Cnv‘(a1ii)_ ft[ a
Bern.
I”-* 3 !: 181 ■ 0h:x ICnv'fa 1«)) y. = . y («1 7f) a-.
^ S . y 6 a . yRx .
--*Jty.»...
[ * 85101 J w -^MyOKProp
h • Cnv‘(*|*).£ 1 P ^ [Proof similar to that of *3ool]
h . Cnv'(a1 Pf /?) = £ ] j{^ a [Proof similar to that of *35*51]
h . D‘(a1 P) = a ,> JXtt
271
*36 52.
*3663.
*3561.
Bern.
r'f,' 8 - Dh: - X( D ‘(“1 -«)• = : (ay) . *(« 1 R)y:
*i„ J, -:(ay). — .««y:
SS * : *««:(ay).<c*y:
[*22-33] s : xe(an D'R) D H . Prop
*36 62. h : a C D‘/J . D . D‘(a ] R) — a [.35 61 . .22 621]
*36 63. I- : D‘P Ca. = .a]fl = fl
Bern.
H . *35 61 -3l-:a1P = i?.3.ar, ]>/* = £)^ #
[*22-621] D.D'PCa
^ • (1) • *35-451 . D I-. Prop
*36 64. h . a«(* m = 5 (J‘R [Proof ^ in » 35 . 61]
*36 641. h : a „ D‘fl = A . D . a R _ A [.35-61. .33 241]
*36-642. h :an <1‘R =» A. 3. .R f- a — A [.35-64 ..33 241]
.36 643. H:«nD‘2e = A.3.a1(fi tt ,S) = a1S [.35-641-42]
( 1 )
MATHEMATICAL LOGIC
[PART I
*35644 h:anG‘/i = A.D.(/?c;.S , )|*a = .Sr« [*35G42*421]
(3y).*(/f S )'J :
<3y. *) • •
(3-• y>. r7?z. z£y:
<H*)s .**7fz:(gy).zSy:
(gz)..r7fz. c c D*S:
(gz)..r(/frD*&)z:
x t D‘( R [ IV.S'):. D I*. Prop
*35 65. h :/3C(I‘/f. D.(I‘<77 [*£) = £ [*35*64.*22621]
*35*66. htl'RCfi.s.Ktd-R [Proof as in *35*63]
*35*671. H . D‘( J< S) = D‘( 77 [ L)‘.N’)
Deni.
t*.*33'13.DI*:.xcD'</t .S'), s :<gy).*(/7 S)y:
[*341] =:(gy. t).*R:.xSyt
(*11 *23] = : (gj. y). .r7?z . z£y :
[*10-35] = : (gz): xRz : (gy). zSy :
[*33*13] = : (gz). .r77z . z € D*S:
[*35* 101] = : (gz). .r (77 |* D‘S) z:
[*3313] = : .rcD'(J?rD‘£)s. 3 H . Prop
*35*672. I- .({*(/( S>-CI‘(<I‘7?1tf> [Similar proof)
*35 68. h:an/3*A.D.(a1/? |W - A
Dem.
I-. *35*61 64*21 . D I-. D‘<a] 77 [*£) C a. <1‘<*1 77 [0)C0.
[*22 40.*24*13] Dh:an / i= A.D.D‘<«1 R [0) * (I‘(a1 7?^) = A.
(*34*531 ] D . (a] 77 |*£)’ - A : I> h . Prop
*35 7. bi$\(R[&)*y 1. = . y * . <j> (77*//)
This proposition is very often used in the Inter parts of the work.
Dan.
V . *14 21 . D I-: <*> J<7?>*y|. D. E ! <7?[*£)«//.
[*33*43] Z>.y<a‘<77[*£).
[*35 64] D.y«/9 0>
h . (1). *471 . Dh : <f> |(77 |*£)‘yj . = . y € £ . <*> |( 7? r£)‘y) ( 2 )
h .*4*73 .*35*101 . D h :.y*/9. D :x(7^I*/9)y . = x .xl 7y:
[*14*272] * D:<M<*r0)V)-s-*(*‘y> < 3)
I- .(3). *5*32 . D h :ye£. <£ ((/?f£)‘yj . = .ye &. <f>(R‘y) (*)
h . (2). (4). D h . Prop
*35*71. h :. y <• /9. D y . 77*y = £>*y . R[@ = S[/3
Dem.
h . *4*7 . D h :. Hp.D : ye 0. D v . ye 0. 77‘y = S*y :
[*35*7] O:y*0.D„.(R [*£)‘y - (S|*£>‘y:
[*35 04] D : ,j e CI‘(« p/3) « (I«(Sf/9). D„. (R \&)‘y = (S [fiYV !
[*33-45] 0 : R = S |-/9D h . Prop
*35 75. KA-]R = «rA = A"] «|\8 = a1 fl[A = A
Dem.
h.*35*61. DKD‘(Al/i) = A.
[*33-241] D h. A] iJ = A 0>
SECTION D]
*3576. h
Dem,
LIMITED DOMAINS AND CONVERSE DOMAINS
h. *35-64. 3 t- . U‘(ft|-A) = A .
[*33-241] DI-./J[-A = A
h • »3 > 441-21 . D I-. A 1ft 10 C A1 ft .
[(1).*25-13] D H . A1 W f/3 = A
K *35 44*21. 3Ka1«|-AC«r.V.
[(2).*2513] 3 t-. a ] ft p A = A
•" ■<!)• (2) . (3).(4). D h . Prop
V - ] ft = ft |- V = V] ft J-V = ft
273
( 2 )
(3)
(4)
S . X € V . .rliy
.alty
(1)
. . y c V .
. xRy
(2)
• x e V . a:/£y . y e V .
.xRy
(3)
h -*35l. DK :x(V]/?.)y.
[*24 104.*473]
H . *35*101 . Dh:.r (RfV)y.
[*24-104.*4-73]
*35102. DK:x(V1 RfV)y.
[*24104.*4'73]
^ • (1) • (2) • (3) .Dh. Prop
exce T p h tV35 « f 812 8 nU ’ nbCr - d0 ' Vn *° * 35 ' 93 CXC ' Usive - is C °— 1 wit » «TA
*35 81. h:*(«lV)y.«.*, a [*351 . *25104)
*35-812. l-:<r(Vf-/9) J ,. = . ye/3 [*35101 .*25 104]
*35-82. = Vf/3
Deni.
H.*85-103.3 (•:*(, T,8)y.
[*25104] s .t( a.xVt/ .ye/3.
= .*(«] Vf >9)y s D h . Prop
*35822. Ka-|ft[-/3=ft,s( a -]-/9)
Dem.
1- . *35*102 . D : a:(a ] ft f- 0)y. s . x t a . xRy . y e 0 .
^ = • *#y .xea.yt 0 .
OB „ 1 * 23 33] 3 . * (ft A (a -f ft)] y : D (-. Prop
*35 83. : D‘R C a . <3‘i* C £. = . i* G a f £
Dem.
H . *33 14 . D a;i*y .D:if D‘/f . y c d'/J :
L* 2 ^ 6 ] r. D:D‘ftCa.U‘ftC ft . D . x e a . y f ft (1)
U510? mm ‘ 3l ' ! ‘ D ^ Ca - a ^ C ^- 3! * iiy - :, - Xe “^^-
J D.x(af/?)y (2)
H.*35 103.
[*33-35-351]
h -00.(3).
R&W I
D H . Prop
'*.v
3 : C a . Q‘i? C 0
(3)
18
274
MATHEMATICAL LOGIC
[PART I
*35 831. K -Ma f/$) = < -of/3)iy(a f -j8)ia(-a f-/3)
Dem.
*" • . D h ::} -Ma f £>j y. = :.-(/( a | fi)y\
[*35103] = :. - (x e a . y e £):.
[*4 51] =:.i'v(a.v.y've3:.
f *4 42] = * a : »/ . v . y ~ €/9 :. v :. .r e a . v . c a : y ~ e 3 :.
[*4 4] = v .X'^ca.y<^>efS.v.xea.y~€p.v.x~ea.y~€&:'
1*4 25-31-37]
= :..r^<a . y <fi .v ,xea . y ~ e/9 . v . ea . y ~ e/9
[*22*35] = e-o.yeft.v.xea. ye — @.v.xe-a.ye — fi:.
[*35*103] = :..**(- o | /9>y. v .x(cr f - /9)y. v . x'- a \ - /9) y:.
[*23*34] = - a | tf)u(a f - £)o(- a t - £)| y :: D H . Prop
*35 832. h .-M* 1 t 3)c/(a T -/9)u(-a |-/9)c;-/f
[*35*822*831 . Tn.nsp. *23 84]
*35 834 h . (a t /3) A ( 7 | 5) = (a a 7 ) | (/9 n 3)
Dem.
h. *35-103.3
** :•/* |(a T >3) A( 7 T $)| y. = . j -ea.yefi..rey.ye& .
[*22 33.*35* 103] s . j* |(a « 7 ) f (/9 a S)| y : D b . Prop
*35 84. h . Cnv‘(a Ttf) = /9f a [*35103 . *31 131]
*3685. h :g !/9. D . LV(« | 5) = a
Dem.
K *35*103. *10*281 . D
H :.(y//)..r(a f £)y. = : (gy) .xea . ye/9 :
[*10*35] = :xea:(gy).ye/9:
[*24*5] = :xea.g!/9 (0
K (1). *33 13. *10*35 . D 1-. Prop
*35 86. : g ! a . 3 . (I‘(a f /9) = /9 [Similar proof]
*35 87 1-: g ! (a f /9). = . g ! a . g ! /9
Dem.
h .*35*103. D h :.g !(a f/9). = : (g x t y).xea.yef3z
[*1154] = :(gx).xea:(gy).ye/9:
[*24*5] .= : g ! a. g !/9 :. D I-. Prop
*35-88. K:.af/9 = A. = :a«A.v./9*A
[*35*87 . Transp . *24*51 . *25*51]
*36*881. V : (Pi* C a . D . R\ (a f /9) - D*R t /9
Dem.
h . *34 1 .*35103. D
h :x|.ft|(at£))y. = .(g*).:rik.*ca.y€£ ( ] )
SECTION D] LIMITED DOMAINS AND CONVERSE DOMAINS
275
( 1 )
K*3314. Dh:.a^Ca.D:.^. D .: fa :
r *10 351 (2) • 3 h :: H P ■ 3 =• - i«It >: // • H : (5r ,) .:
[.3313]
Log., oil a-.reD'Ji.gtfj:
*35 882. H:D‘/JC^. 3 .(at/3) « = a T a‘fl [Similar proof]
* 3 B 2L l ’ !a! ' , ‘ :> ‘ (aT/9)l(>3T ' y) " ( “ r7)! ~ a!/90 - (a ^> C-8t7)-A
h . *341 ■ 3 I-* ((a f £); </9 f 7) j *.
r.35-10,1 -‘(SMr).-(.T«jr.jrWTr)*:
; J s:<5Iy) - p,a • ye■ ye • **•/i
[*4 24] s:(ay)-^«a.y«/3.*e 7 !
[*10-35] 3 : 3 !/3:xfa.z ( y:
[*35103] = : a ! £: x (a f 7 ) *
l-.(l). 31- i: a !^.3«#|(.f i 8)| 0 8 ty))x. 3 .*(af 7 )*.-.
~ (3 1 0). 3 : ~ [* |(a f 0 ); { g f y) j ,] „ 3 p . Pl . op
*35-891. h a ! 0 . v . ~ a ! a : D . (a f >3 ) 1 <5 f a) = ( a t „)
/>em. .
*■ • * 35 ' 88 -
[SSS s.-T—A.cr^iwr^-A.
*36 892. I" : (o -f a)* = (a ^ o) [*35-891 |J
*35 895. l-:«o^ = A.D.(„t^). = A [*8508-82]
*35 9. I-. D‘( a ]■ a) - d‘( a f a) = C‘(o ]«) = ,
Dem.
h‘lS- 88 5 86 ' ^ J" ! 3 1«- 3 . D‘(o ]- a) = O. G‘(a ]«) = « („
r*33?91 3h: ~a!«-3-~3 ! («t«).
[*24-511 D.D«(« ra ).A.a'( a t«)- A .
®
*35-91. I-: .ft G a f a . = . C‘/£ C a
Dem.
[*33 3o2] =:C'iiC.:. 31 . Prop
*3692. l-:.(a«).P = a t a .3 = iecP. s .e«ftcC<P [,35991]
( 1 )
18-2
276
*3593.
I>e in.
*35931.
*35 932.
*3594
*35941.
*35942.
MATHEMATICAL LOGIC
H : (/{). 0 (D‘7t }. = . (a). 0a
h. *3312. *1418.
[* 101121 ]
H . *101 .
[*359]
[* 1011 - 21 ]
H.(1).(2).
!■:(/?). «/»(<!*/?). =
I- :</?). <^(C-/^). =
H:(y7f).0(D‘7O.
h:<y7?).0(Cl‘7O.
H:(y7O.0<C'‘7*).
D H : (a). 0 a . D . 0 (D*7?):
I>h:(a).0*.D.(7?).0(D‘7?)
DK:<J?).0(D‘7?).D.0{D‘(ata)]
D. 0 a:
DI-:(7O.0<D‘7?).D.(«).0a
D h . Prop
. (a). 0a [Proof ns in *35 93]
. (a). 0a [Proof jis in *35 93]
= . (ya). 0a [*35 93 . Transp]
= . (ya). 0a [*35 931 . Transp]
= • (fl a ) • 0 a [*35 932 . Transp]
[PART I
( 1 )
( 2 )
*36. RELATIONS WITH LIMITED FIELDS
Summary 0 f *36.
,■ i I ",-'“ 1 ‘ SnUmber " e me co " ce ™ ed ‘he special case in which the same
Um at,on .. imposed upon the domain and the convene domain of a relation.
In tins ease, the same result is achieved by imposing the limitation „„ the
«f . or VTr ‘° ab , le l ° rega,d as a descriptive function
7* t „ W t C f we Se “ urc hy U,e n °tation P[ a. whence, as will be ex-
and a C C‘* '/r C a a ' 1 , C ^,'1" ^7 menD P C K P is “ * eHal -'■•“•ion.
determined K for “ ,e ' CrmS of Q “"anged in the order
S2Z£: * “ ” -»■■ ” *• r t -»
*36 01. = Df
We thus have
*3613. h:*(PC«)y. = .x,ye a .*P y
so m e I at t |c 0 l th f,r 0| r iti °" 3 - COnCerning Pta dema,,d ,hat P should have
concerniuTp r- the . charact ‘'" st ' c * of a serial relation. Hence the propositions
part not tlfe D 1 71 ^ ‘ h ° P '' CSent numbe1 ' ara - *>r the most
part, not the most useful propositions concerning Ph a The most useful
propos,turns in the present number are the following- most useful
*3625. hC‘PCa.== .Pt a=a p
*3629. b.Pta = P*a fa
*36 3. h.Pta«Pf( an &P )
*36 33. V.Pt&P^P
*36 01. Pta-alPpa Df
*3611. KPta^alPfa [(*3601)]
*3613. K : x (P [ a) y. = . Xt y € a . xPy ^ 3 q. x x _ * 3 5 . 102 ]
*36^ e S wi a n , g it propoS H ions ar ! obtained frora those of * 35 b y meaus of
*00 II, which, as it is used in each case, is not referred to again
*362. KPCaAQ t>S = (P/.Q)t:( artiS) [, 3515 ]
*36201. h.PDaAP C> S = P t:(a „ i8) [.36-21
*36202. KPCaAgt: a = (PAQ)C a [*3621
*36 203. h.PC a c,g = (p A Q )t;a [.35181
*36 21. h.(PCa)t^ = Pt( a „ /9) [.35-33-341
27.S
MATHEMATICAL LOGIC
[PART I
*3622. M/’Ca) <<2Ca)G(P
Drill.
h . *3013 . *34*1 . D h : .#• '(l* [a) ((? £ o)| z . = . (gy). x, y, z e a . xPy . yQz .
[ #105 J 3* (3 y).x,zca.xPy.yQz (1)
Ml). *10*35. *34-1 .31". Prop
*36 23. I-. (P o Q) [ a = P £ a o Q £ a [*35*422]
*36 24. haCfS.D.PtaQPta [*35*432]
*36 241. I-: P G Q. D . P £ a G Q £ a [*35 462]
*36 25. h : ( U P Ca.*.P£a = P
Dem.
H . *:t(M3 . *V7 .DH:. = : xPy. D, , y .x, yea:
[*:«-352] = : C‘P Cai.DK Prop
*36 26 V : C‘P na-A.D.P (0t«> - A . (Q £ a) | P - A [*35*473*474]
*36 27. h:P [A-A [*35 75]
*36 28. h . P £ V = P [*35*76]
*36 29. KP£a-PAafa [*35*822]
*36 3. KP£a-P£(aAC‘P)
Dem.
V . *33*17 . *471 . 3 h : xPy . = . x, y c C*P . xPy :
[Fact] 3 I*: x, y t a . xPy . = . x, y * a . x, y e C‘P . xPy .
[*22 33) = . x,y e a r\ C*P. xPy.
[*•3613] =.*(/>[(« a C‘P))y (1)
1*. (1). *3613. 3K Prop
*36 31. H : a a C‘P = A . 3 . P £ a = A [*86*3*27]
*36 32. h : a a Q*P - £ a C*P . 3 . P £ a = P £ £ [*36*3]
*36 33. f-. P £ C'*/* - P [*36*25]
*36 34. h.Cnv‘P£a«(P)£a [*35*53]
*36 35. MPCo)’G(P*)Ca 1*36*22]
*36 4. h a a D‘P = A . v . a a Q‘P = A:D.(Pc/5 r )[a = jS , ta
Dem.
h . *35-643 . 3 h : a a D *R = A . 3 . a ] (P o S) = a *| S.
[*35-21] 3.(PuS)£a = £’£a (1)
Similarly h : a a Q f R = A . 3 .(P ci 5) £ «=* 5 £ a (2)
1-. (1) . (2) .3 h. Prop
R‘-”
*37. PLURAL DESCRIPTIVE FUNCTIONS
Summary of * 37 .
In this number, we introduce what may be regarded as the plural of R',,
<J was defined to mean "the term which has the relation R to \V e
ZJTrT ihe r M, z: R y " to mea '' “ ti,e **™» which t he
elation R to members of 0 Thus if 0 is .he class of great men. and R is
the relatmu of wife to husband. R-0 will mean "wives of great men.” If
0 s the class of fractions of the form 1 - 1/2- for integral values of,, and R
is the relat.on "less than,” R“0 will be the class of fractions each of which is
loss than some member of this class of fractions, i.e. R“R will be the class of
t T t,0nS - °f neral '/' R “& ** the class of those referents which have
reJata that are members of /9.
We require also a notation for the relation of R“0 to 0. This relation
r e an W d‘V2J ThUS f- “ ‘IT rclali< '" *“* h °«* between two cCs
membef of 0 “ C ° nS tCrmS which hove the relation R to some
case A arises When olwa - vs exists if V '0. In this
denote .! “ S ° * terms of the form R‘,j when y e 0. VVe will
denote the hypothesis that R‘y always exists if y ( 0 by the notation IS !! R“0
meaning “the Jis of/9*s exist.”
The definitions are as follows:
*37 01. R“0-Sti(ay).y e 0. x Ry) Df
*37 02. R, = fi/9 (a = R“0) Df
*37 03. R, = Cnv‘(iZ.) I)f
w Thl3 definition serves mcrel y for the avoidance of brackets. Without it,
" ft ” w ° u| d be ambiguous as between (R). and CnV(ft.), which are not equal.’
*• ■ h *" *• “»>«*■-. -
*37 04. R‘‘‘ K = R." K = Cnv ‘<«—)
Thus consists of all classes which have the relation R. to some
tTmcmh % '* ,S ° n ‘ y 8 ‘ gnifiCant When * is a c >“» classes relatively
to members of the converse domain of ft; in this case, R“‘ K is a class of classes
relatively to members of the domain of R.
*3706. EllR“0 .=:ye£ .D u .ElR‘y Df
the ay “ b °‘ “ E " ^ mU8t be treated “ * whole, .-«• we must not
regard it as making an assertion about R“0. If R-g _ we mu8t not suppoge
2«0 MATHEMATICAL LOGIC [PART I
that wo shall ho able to put “E!!a," which would bo nonsense,just as “E\x"
is nonsense even when ./ = R‘y and E! R*y.
I'he notation R**a. introduced in the present number, is extremely useful,
and embodies a very important idea. Its use is somewhat different according
to the kimi of relation concerned. Consider first the kind of relation which
loads to a descriptive function, say I). If X is a class of relations, D“X is the
class of the domains of these relations. In this case, L)“\ is a class each of
whose members is of the form L)‘P, where lie\. Again, let us denote by
*'xm" the relation of m to «ixn; then if we denote by "NC” the class of
cardinal numbers, xn“XC will denote all numbers that result from multi¬
plying a cardinal number by n. i.e. all multiples of n. Thus e.<j. x2 “NC will
be the class of even numbers. If 11 is a correlation between two classes a and
0, i.e. a relation such that, if yt0, R*y exists and is a member of a, while
conversely, if xta, R*x exists and is a member of 0. then a« 11**0, and we
may regard R ns a transformation applied to each member of 0 and giving
rise to a member of a. It is by means of such transformations that two classes
arc shown to bo similar, i.e. to have the same (cardinal) number of terms.
In the cast- of serial relations, the utility of the notation R**0 is somewhat
different. Suppose, for example, that R is the relation of less to greater among
real numbers. Then if 0 is any class of real numbers. R**0 will be the segment
of real numbers determined by 0. i.e. the class of real numbers which are less
than the limit or maximum of 0. Iii any series, if 0 is a class contained in
the series and R is the generating relation of the series, R**0 is the segment
determined by 0. If 0 has either a limit or a maximum, say x, R**0 will be
Ii*x. But if 0 has neither a limit nor a maximum, 11**0 will be what we may
call an ‘irrational segment of the series. We shall see at a later stage that
the real numbers may be identified with the segments of the series of rationale
i.e. if R is the relation of less to greater among rationale the real numbers
will be all classes such as 11**0, for different values of 0. The real numbers
which correspond to rationals will be those resulting from a 0 which has a
limit or maximum; the irrationals will be those resulting from a 0 which has
no limit or maximum.
The present number may be divided into various sections, as follows:
(1) First, we have various elementary properties of the terms defined at the
beginning of the number; this section ends with *37 *29. (2) We have next
a set of propositions dealing with relative products, and with such symbols as
R**Q** 7 , P**Q***k, and so on. The central proposition here i 9
*37*33. V.\1\Q)“y = P**Q**y
By the definition, Q***k = Q,**k. Thus P**Q* t *K = (P j Q t )** k. This connects
propositions concerning such symbols as P**Q***k with propositions concerning
SECTION 1>] PLURAL DESCRIPTIVE FUNCTIONS 281
relative products.^ This second section consists of the propositions from *37 :*
to *37 39. (3) We have next a set of propositions on relations with limited
domains and converse domains. The chief of these are
*37 401. h . D‘(72 [*£) = R“0
*37 412. =
*37 41. h . D‘( 7? [ a) = a ^ R“ a . <P (R [ a) = a r> R“ a
These propositions on relations with limited domains and converse domains,
together with certain others naturally connected with them, extend from *37 4
to *37-52. (4) We next have a number of very important propositions on the
consequences of the hypothesis EllR“&, U. the hypothesis that, for any
argument which is a member of £, R gives rise to a descriptive function 72‘y.
The chief proposition in this section is
*37 6 . h:E!S R“0 . D . R‘*/3 = 5 |( a y) . y eR . * = R'y]
Propositions with the hypothesis E !! 72“£ are applied to the cases o f~R
and /?, in which the hypothesis is verified. This section extends from *37 6
to *37-791. (5) Finally, wc have three propositions on the relative product
of w »th other relations. These propositions are useful iu relation-
arithmetic (Part IV).
T he propositions of the present number which are most used in the sequel,
apart from those already mentioned, are the following (omitting such as merely
embody definitions):
*37 15. h . R“a C D‘72
*37 16 h . R“a C d‘R
*37 2. h:aC/3.D. P“a C
*37 22. h . i>“(« « £) = P gi a v/ P“fi
*37 25. h . D‘72 = R**a*R . CI‘72 = R“D‘R
*37 26. h . R“0 = R“(0 „ d‘R) .
*37 265. h . R“a = R“(a C‘R ). R“a - R“{a * C'R)
*37 29. h . R“A = A . R“A = A
*37 32. h . D *(P | Q ) = P“D‘Q . Q‘(P | Q) = Q“(1‘P
*37 45. h (y). E ! R<y . D : a ! R“0 . = . 3 ! £
*37 46. h : x c R“a . = . g ! a n R*x
*37 61. h :: E !! R“0 . D R“0 C a . = : y e 0 . D u . R‘y € a
For example, let 72 be the relation of father to son, /3 the class of Etonians,
o the class of rich men; then “72“/9Ca” states “all fathers of Etonians are
rich, while R l y ea” states “ if a boy is an Etonian, his father
282 MATHEMATICAL LOGIC [PART I
must be rich." In virtue of the above proposition, these two statements are
equivalent.
*37 62.
b : E ! R*y . y * a. D . R*y c
*37 63.
b :: E!! R“a . D .re R“a .
lr.r: = :yca.D„.yJr(R‘y)
*37 01.
-»!<&).»
Dr
*37 02.
R. = a/3 (a = R“f3)
Dr
*3703.
R, = Ciiv‘(/?,)
Dr
*3704.
H‘“* =
Df
*37 05.
li* y
Df
*371.
b :./•€ /?“£• = .(3y).y«£
. .r/ty
[*20-3. (*37-01)]
*37101.
b :afc£.B. «-/<“£
[*21-3. (*3702)]
*37102.
b:a(70,/3.«.o-ii“/3
[*37101]
*37 103.
b : a < R ttl K . = . ($|/9). 6 *
. a —
R“8 . = .ae R“k
[*37 1 101 .(*37 04)]
*37104.
b E !! /£“# . = : y
. E! /e*y [*4-2 . (#37 05))
*37105.
b : a: 6 . a . (fly). #/ < £.
. yRx
[*371 . *3111]
*37106.
b E ! R*x .D:xe R^fS . =
. R‘.i
Dem.
b.
*37105 . *30 4 . D b :. Hp. I
"> zxe
. =. (ay) • y * & • y ■ ^ •
[*14205]
= . fj8:.Db. Prop
*3711.
b .RS0-R“0
[*37101 .*30 3]
*37111.
b . E ! R/0
[*37 11 .*14 21]
*3712.
b:(tf)./f“£=Q‘,9. = .R t
= Q
[*3042. *3711 1 11]
*3713.
b sP-Q.D.P“fi-Q“0
Dem.
b.
*21-43. Db:. Hp.D :*Py.
-x.v •
xQy :
[Fact] D s y c 0 . xPy . =,.„. y e /9 . xQy :
[*10 281] D :(gy) .y c/9.xPy. . (gy) . y e £ . :r<?y:
[*371] Dzxc P“/3 .= x .xe Q“8 :. D b . Prop
*37131. h:P = Q.D.P'= Q .
Dem.
b . *3713. D b :. Hp . D : a = P“/3 a = Q“/9 s
[*37101] D : aP,/9 . =.,* . aQ t 0 D b . Prop
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
283
*3714. f -iP=Q. = .l\ = Q,
Dem.
*37101 .*21 15. D
[•13183] s . (/3) . JM.0 = Q ..0 .
[•37 1 .•20-15] = : (£, x) : ( 3 y). y 1 0. xPy . = . ( 3 y). y e £ . xQy :
[•101] 3 : (*) : ( 3 y) .ju(j=ui). r/>y . = . < 3 y) .yti(z-w). xQy :
[•20-3] D : (*): ( 3 y). y = «•. xPy . s . ( 3 y). y = w. xQy :
1*13195] D : (.*•) : arPw. = . xQw ( 1 )
H .(1). *1011-21 .*112.D
P. = Q, . D z(x, w) : arPw. = . xQw :
[*21-43] D : P = Q (2 )
H.(2).*37131. Dh. Prop
*3715. h . R“a C D‘P
Dem.
H . *37-1 .Df-:xe P“a . D . (gy) .yea. xRy .
[*3313] D.xe D*R zDh. Prop
•3716. h.fl“aCa*7e [•3715 *332]
•3717. I- :. R“ff Ca.siy«/9. xRy . D, „ .xta
Dem.
. *371 .Dh:./i“^Co. = : (gy) .ye&. xRy .D x .xeaz
[*10-23] ■ : y e 8. xRy . Z> xy .xc a:.D . Prop
*37171. h R“a C >9 . = : x c a . xRy .O x , v .ye@
Dem.
K . *37105 .Dl-:. R“a C /9 . = : (gar) . * 6 a . xRy . D y . y e 0 :
[*10 23] = : x e a . xRy. D Xty .ye^ z. D H . Prop
*3718. h : y e £ . D . ~R*y C P“/9
Dem.
H . *32-18 .Dh:.Hp.D:xc R l y . D . xRy . y e .
[*371] D .arc P“/9 D h . Prop
*37 181. hxfa.D. R*x C R“a [Proof as in *3718]
*37 2. haC/9.D. P“a C P“0
Pern.
I". *22*1 • D h Hp • D : y c a . . y e f3 z
[*10-31] D zyea.xPy .D v .ye/3 . xPy z
[*10*28] D : (gy) .yea. xPy . D . (gy) .ye 0 . xPy :
[*371] 3 sarc P“a .D.xe P“/3 s.DH. Prop
284
MATHEMATICAL LOGIC
---- [PART I
The above proposition (#37 2) is one of the forms of asyllogisbic inference
due to Leibniz’s teacher Jungius. The instance given by Jungius is: “ Circulus
est figura; ergo qui circiilum describit, is figurant describit*.” Here the class
of circles is our a, the class of figures is our 0, and the relation of describing
is our P.
*37*201. H : P Q Q . D . P“a C </**a [Similar proof)
*37 202. zaC0. PGQ.5.P“aCQ u 0 [*37*2*201]
*37 21. h . P“(a *0) C P“* n P“0
Dent.
H . *371 . D h z.xe /'“(a n 0 ). = : (gy). y c a n 0 . xPy :
[*22*33] = : (gy). ye a . y € 0 . /Py:
(*1° •>] D : (gy). y * a . xPy : (gy) .ye0. a*/»y :
[*371] D :j eP“a.x€P“0:
[*22*33] D :xe P“a n P“0 :. D h . Prop
*37 211. .(P f\ Q)**a C /'“a Q“a [Similar proof]
*37212. \-.{PAQ)“{an0)CP“arxP“0nQ“anQ“0 [*37*21*211]
*37 22. h . 7'“<* u £)« /'“a v, ]'“0
This proposition is very frequently used. The fact that here we have
identity, while in *37*21 we only have inclusion, is due to the fact that
*10 42 states an equivalence, while *10*5 only states an implication.
Dent.
H . *37*1 P*\a \j 0) . = : (gy) . ye a\j 0 . xPy :
[*22*34] = : (gy) iyta.yf.ye0z xPy :
[*4 4] b : (gy) zyea. xPy . v . y e 0 . xPy :
[*10*42] = : (gy) .yea. xPy : v : (gy) .ye0. xPy :
[*37*1] = : * € P“a .y/.xeP 1 ^:
[*22*34] = : x c P“a v P“0 :. D I-. Prop
*37*221. M/'w Q)“a = P il a \j Q“a [Similar proof]
*37*222. h . (/»a Q)“(a v0) = P“a v P“0 » Q“a sj Q“0 [*37*22*221]
*37*23. I-. D‘7?, = a |(g/9) . a = R“0\ [*37*101 . *33 11)
*37 231. h.a‘7?, = Cls
The type of "Cls” here is that type whose members are of the same type
as Q*R. In the proof, use is made of the convention that a Greek letter
always stands for an expression of the form z(<f> l z).
Dem.
h . *37*101 . D h : *R t 2 (<*>! z). = . a = R“2 (</>!*):
[* 1011281 ] O h : (a«) - aKJ = . (g«) • <* = (-P
[*33131] Dh:J(^! Z )«a‘«.. = .(aa).o = B“2(^!r) (0
• Wo quote from Coutur&t. La Logique de Leibniz, Chapter m, § 15 (p. 75 n.).
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
--. ^. 285
H ■ * 20 ' 2 • (*3701) . D h : .?• [(gy) .ye3(0! 2 ). -. 1{“3 (<£•*);
[*1011-24] D I -1 <<#.): (go). a =K“? <■£»*)
h.<l).<2).*202. 3h5(Ji!j)f CIs . O . 3 (<{> l s) e R ,; t)
K *20-41 . *2 02 . 3l-:2(^!j)eG'/i,. D . 2(^ ! ») eCIs , 4 )
I- • (3) • (4) . 3I-. Prop
As appears in the above proof, it is necessary, when a proposition con-
taming Us is to be proved, to abandon the notation with Greek letters, and
revert to the explicit functional notation.
*37 24. h:« t D'K..D. J CD'ff
Dem.
h • * 33 ' 13 * *^7101 . D h :: a € . m <g£). a = R “0 ..
[*20-33.*371]
[*11-61]
[*11-23]
[*11-55]
[*105]
[*3313]
= :• (a£) : XC a . = z . (gy). ,j e £ . x R y
D x « a . D x : (g£, y). ye @ . xtiy :
: (ay. &)»ye@ . xRy :
3*:(ay):*%:(a£)-ye£ :
^* ! <ay)«*ffy s
D*:x«D‘/f :: D h . Prop
*37 25 . h. D*/e - A«a‘A. a <r *= /*“D‘/e
Dem.
h . *3313 . D I-: x e D‘/e . = . (gy). Xj R y .
[*3314.*4-71] ■ . (gy) . y * Cl Vi . xRy .
[*37 1] =.xeR“Q‘R
H. *33131 . Dh yeCl‘R. = . (gar) . xRy .
[*33 14.*4 7 I ] = . (gar) . a: * I)‘R . x Ry .
[*37105] = . y e R“ D‘/i!
^ • (1) • (2). D I-. Prop
*37 26. h . R“0 = ;<“(£ ^ d‘R)
Dem.
H.*371.3h:.a:«JJ“^. = :(gy). ye/ 3. a .^ y!
[*3314.*4-7l] = : (gy). y r 0 . y e a<_ft. ;
•* 22 ' 33 ] s :(ay)-y*/9na ‘R.xRy.
= :xtR“(0 n (I‘R) :.3K Prop
*37 261. 1-. R"0 = K‘*<0 n D-ft) [*37-26
*37 262. han CP.R = 0 n (J‘R . D . fi<« a = [, 37 - 26 ]
*37 263. = = [*37 261]
*37-264. h : g 1 a n R“£. = . (g*, y) .xea.y e 0 . xRy. s .El 0 n R“a
Dem.
h.*22-33.*371.DI-:.a l «r> R‘‘0. = t fa) *** «z (&,) • y < 0 - *Ry. ( 1 )
( 1 )
( 2 )
280
MATHEMATICAL LOGIC
[PART 1
( 2 )
[*11-55] y).x€Q.y € 0.xRy
h . (1). *116 . D I- 3 ! a /-» lV'fi . = : (ay) : y € /3 : (ax). x e a . xRy :
[*37 105] = : (gy) . y € 3 . y e R“a :
[*22-33] = :g!)9n7?“a (3)
h . (2) • (3) • D K • Prop
*37 265. h . R“a = R“ia n C*7f). P“a = P“(a n C‘R)
Dem .
h . *33101 . *22 621 .Dh. a*/? = C'R * (l‘R .
[*22-481] DKon (l‘R = a r\ C*R r* Q‘R .
[*37-262] DK«“a = r(anC‘i?) . (D
h .(1). *33*22. D h . Prop
*37 27. HsCT^Ctf .D.D‘7f-7P‘/9 [*22 621 .*3725 26]
*37 271. 1- : D‘77Ca. D.CWf «P“a [*22*621 .*37 *25 261]
*37 28. V . 71“ V - D 'R . V = U‘7e [*37 27-271 . *24 11]
*37 29. h . R“ A « A . P“A - A
JJem.
I-. *10 5 . D I-: (ay)• y « A . xRy . D . (ay). y c A
h . (1). Traiisp . *24-53 . D h .~(gy). y e A . xRy .
[*37 1] DK-a!^‘A.
[*24-51] D 1-. 7i“A * A
7 ( w
h . (2) . D 1-. 7i‘*A = A
h . (2). (3). D H . Prop
*37 3. h . |sg‘(P| V)«^ =
Dent.
K *32-23*13. D
H.|sg‘(P|Q)|^-^(x(7>|Q)x)
[*34ij -*K3y)-*Py.y«*l
[*32*18] = £ [(ay). xPy . y e Q‘z\
[(*3701)] = P‘*Q‘z . D h . Prop
*37 301. h . [gs‘(7' | Q)Yx = Q“P«x [Similar proof]
*37 302. I- s R = P | Q. D . 7?* = P“Q**. P*x = Q“P*x
[*37-3 301 .*32 23 231*16]
*37 31. Ksg‘(P|Q) = P c jl?
Pern.
(D
( 2 )
<•**)
h . *37*11*3 . D I-. (x). [sg*(P | Q)]‘z = P/Q‘z
H . (1). *34*42 .Dh. Prop
(1)
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
2*7
*37 311. . gs‘( P | (?) = ((?),; P [Similar proof]
*37 32. h . D‘(Pj (?)= P“D‘(? . a*(P • (?) = <?“d‘P
Dem.
. *33-13. *341 . D
h :• * D«(P| Q) . = . ( 3 s) : (51 y) . 3-Py . yQ z .
[*1123] = : (a^): (H-0 • -i-Py. yQz ;
[*11-55) = : ( ay ) s .,p, . ( 3i ). tJ Q z .
[*33 13] = : (ay) . ,r/>y . y c ])<<? .
[*371] = :.v f P“D‘Q (I)
*-■ (1). *1011 .*20-43 .0
KD*(Pi«)-P“D‘Q
1-. *33-2 . D 1-. CI‘(P j Q) = D‘Cnv«(P | (?)
[*34'2] = D‘(Q1P)
[(2)! -Q“D‘P
[*38 2] - Q“G‘P ( 3 )
^ • (2). (3) .Dh. Prop
.37 321. H : Q-P C D-Q . 0 . D‘(P j Q) = D ‘P [,37-32-27]
*37322. h:D«gca‘/».D.aV|Q,-a*Q [.37-32-271]
*37 323. 1-: CW» - T)‘Q . D . D'(P | Q) - D‘P. CI‘(P | Q) _ d‘Q [*87 321 322]
*37 33. K (P | <?)“ 7 = P “<?“ 7
Dem .
I-. *371 . D h xe (P| <?)“ 7 . ■ : (g*) . z e y . x(P\Q) z :
fin-231 11 55] B 8 <a *.»)•*« 7 • */*y . :
„ J = : (ay) ! xP ‘J■ (3*)• SQ* .*e 7 :
*:!. ==(a y).*Py.y'Q"y:
L* )71 J s:x f P“Q“ y; .0h. Prop
*37 34. H.(P|Q). = P.|0.
Dem.
h • * : * 71 1 • ^ H . (PI <?)/ 7 = (P| Q )« 7
[*37-33] = p**Q** y
[•37H] = P/0/7 ( 1 )
I-. (1). *1011 .*34-42. Dh. Prop '
* 37 341 >- • (Cnv*(P | Q)|. = (Q). | (P>, [, 34.2 . » 3 7 :)4]
*37 36. !■:<*). = P‘Q‘z .0.(y). R“y = P“Q‘< y
Dem.
h . *34 42 .Dh: Hp.D.P = P,Q.
[*3713] 3 . /e “ 7 = /p | Q )« 7
[*S7-33] = P*‘Q**y Of-. Prop
2*8 MATHEMATICAL LOGIC [PART I
*37'351. h : (a ). 77‘a = P*Q“a .D.(*). 77“* = P“Q<“«
[*37-35 ( j~ . *37 11 . (*37*04 /j
*37-352. f-: (a ). 77“a = P*Q“a . D . («). P*“« - P“Q'“k
£*37-351 . *37*11 . (*37*04)J
*37-353. !■:(*). 77‘S'* = . D . ( 7 ) . R“S“y = P“Q“y
Dem .
h . *14 21 . D f-: Hp. D . (s) . E ! 77‘S‘j .
[*34 41] }.(*). 7?‘S‘*-( 7? |
[*14-131-144] D m(t).(R\S) € g~ P'Q'z .
[*37-35] D . ( 7 ). (7? | S>“ 7 - 7 J “<?“ 7 .
[*37-33] D . ( 7 ). R“S“y = 7>“Q“ 7 : D H . Prop
*37-354. I-: (a ). 77‘.S*‘a - 7 > ‘Q“a .!>.(*). R“S“tc = P“Q*“* [*37 353
*37-355. h : (*) . 77‘S‘* - 7 , “Q‘* . D . ( 7 ). 77“.S*“ 7 - 7 M ”Q M 7 [*37-353
*37 36. h . D'7? J - 77“1)‘77 . d‘77* = 77“d‘77 [*37 32J
*37-37. H . (77*), -(77.)’ [*37-34J
*37 371. 77.’ = <77,)’ Df
This definition serves merely for the avoidance of brackets. Like *37 03,
this definition will be extended to all suffixes.
*37 38. h . li Jl x = 77“7?or [*37 3]
*37 39. b . 77 3 “a * R“R“a [*37*33]
*37 4. H . Cl 4 (a ] 77) = 77“a
Dem.
h . *33131 . *351 . D b : y e d‘(a ] 77) . = . (gar) . x € a . a;77y.
[*37 105] = . y e 77“a : D h . Prop
*37 401. b . I)‘(77 [•£) = 77"/9 [Similar proof]
*37-402. b . D*(a *| 77 f £) = a ^ 77“,8 . d‘(a ] 77 f £) = >9 n 77“a
Dem.
h . *33*13 . *35-102 . D
I-* e D‘(a 1 77 f* /3) . = : (gy). x e a . x77y . y e £ :
[*10-35] = :area:(gy).a-77y.y€/9:
[*371] = : are a .a*e 77“/9 :
[*22-33] = :x ( anR“/3 (!)
Similarly
I- : y e G‘(a *] 77[*>S). = .yey9n 77“a
h . (1) . (2) . D h . Prop
(2)
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
289
*37-41. = = [*37-403. *30-111
*37-411. Ma1-R)“ / 9 = D‘(a1.R[-/9) = a .We“tf
Dem.
f-. *37 401 . D I-. (a 1 R)“f3 = D‘(a 1 7i)f"/9
[*35-21] =D‘(o1 /?[£) (1
I-. (1) . *37 402 . Dh. Prop
*37 412. b .(Rta)“0-R“(an/3)
Dem.
h • * : * 7 ' 401 . D b . (It r a)“0 = D‘( It r* a) f* (3
[*37-401] - «“(a « 0 ,. 3 I-. P ro| ,
*37413. f-. (7J t «)<^ « a a R<‘( an &
Dem.
y . *37-411 . *35-21 . D I-. (ft [ a)“ft = an (Hfa)“0
[*37-412] —an l{“(a <-> £) . D I-. Prop
*37 42. H : K“(j C o . D . (a 1 R)“ff = ft“/9 [*37-411 . *22 621]
*37 421. y:0Ca.O.(R[ a y<3 = R“ff [*37-412 . *22 021]
*3743. H:. i 8C<Wl.D:j|!fl*. / 8 .«. a i / 8
Dem.
t: <i 8 ;:S- 2 : 38 ' 6s • i r r* 3: *•<* r«<»
*37 431.
*37 44.
*37 441.
*3745.
*37451.
*37 46.
*37461.
*37 462.
*3747.
Dem.
b
f“
b
b
b
h
b
b
. a C D ‘R . D : 3 ! R“ a . = . a ! o
*a-i2-V.D: a !fl“/9. = .a!0
. D‘/2 — V . D : 3 ! R“a . m . a ! a
• (y) • E ! R l y . D : a ! R“f3 . s . 3 ! /9
. (a:) . E ! .ft‘x . D : a ! £“a . = . 3 * a
x e R“ct . = .a lari R* x
x ~ ( Rt<a - = • « « R‘x = A . = . c
v/ t
x^e ii“a . = . a ^ R‘x = A . = . 7£‘x C
[(*37 04)]
ie
[Proof as in *37 +3]
f *37-43. *24*11]
[Proof as in *37*44j
[*33-431 .*37-43]
[Proof as in *37 45]
[*37 1 .*32 181]
[*37-46. *24-311]
[*37-461 .*32-241]
b.(l)
R‘
. 3 ! R “‘a
D P s 3 ! a . = . 3 ! R,“a .
s - a l Rt “a
( 1 )
R*“a
( 2 )
3 b . Prop
19
R&W I
200
MATHEMATICAL LOGIC
[PART I
*37 5. h : (/3). = Q‘/3•3.(*). 7"“* = Q“*
Dem.
K*37 12.DH: Hp.D./*,-Q.
[*37 13] . D.P“k = Q“k.
[(*37 04)] D . P ttl K = : D h . Prop
*37 501. h .0rsiVnCH“K“&
Deni.
H . *37 1 . *10 24 . D h: y e J . xRy .D.xe 7?“/9 :
(Exp.*10' 1121] D I-:.//€ /3 . D : xTfy . I> x . x « 7f“£ :
[*4 7] D : u7?y. D x . x7*y. x € P“0 :
[*10 *28] D : (a-r) . xRy . D . (gx). xRy .xeR“&:
[*33*131 .*37'105] Diy€(l‘R.O.y€R“R tt 0 (1)
H .(1). Imp. *22*33.3
t: ye&r\ <I‘7f . 3 . y c Ii“R“0 : 3 h . Prop
*37*502. . a r> ])*R C R tt R ,i a [Similar proof]
*37 51. \-:(3 C Q‘R . = . C !t“R“0
Deni.
h . *37\->01 . *22 021 . D h : /3 C (I‘if . D . /3 C Tc'IV'R (1)
H. *37-16. 3h:/9C/W'/S.D./3Ca'fl (2)
P . (1). (2). 3 h . Prop
*37 52. l-:aCD7f. = .aC/W‘a [Similar proof]
The following propositions, down to *37 7 exclusive, are concerned with
the special properties of R €t R which result from the hypothesis E!!7?“£, de¬
fined in *37 05. The hypothesis E!! R“fi is important, because it has many
consequences and is satisfied in many cases with which we wish to deal.
*37 6. V : E !! 7d“£. 3 . 7f“/9 = £ |(gy). y e p . x = R'y\
This proposition is very important, and is used constantly.
Deni.
V . *37 104 . 3 h :: Hp . 3 :. y e /9 . 3 V : E ! 7*‘y :
[*30*4] D y : x = R*y . = . xRy
[*5*32] 3 :. y € £ . x = R‘y.= v . ye &. xRy :.
[*10281 ] 3(gy) .ye 0.x = R‘y . = . (ay) • y « £ • •
[*371] s.x*R“R
h . (1 ). *10*11*21 . *20*33 . 3 b . Prop
*37 601. h : (x) . E! 7*‘x. 3 . R“V = £ ((ay) - * = R‘y\
Deni.
h . *2 02 . *10 11*27 . 3 I- :. Hp . 3 : x * V. 3 X . E ! R l x :
(X)
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
*25)1
[*37 104] D:E ll R“Y :
[*37 G] D : R" V = 3 {(gy) . y € V . = R<,,\
(I)
h .*3717 .
[*ll-262]
I- .*37104
[*30-33]
K*24-!04.*4-73.DI- :y « V . 4 —. a ]*-*•* :
[* 10 11-281 ] D h : (gy). jf * V .,- /?',/ . = . <gy). .* = /*<y :
M°> k <<>
*37 61. h :: E !! A“/9. D *“/9 C a . s :,, « t . R.y , „
Dem.
3 I-:: «“/3 C a . = :. v e ^. x /f (/ . y . .r f Q
= 0„zxRi/ m O x ..rea ( 1 )
3 r :s. Hp • 3 :: y «£. D* E! /?‘y
R*y e a . = : j/f// . D ,. .v e a /9\
• (l) • (2). D h :: Hp . D C«.B:ye^.D,. R‘yeat: 3 h , p rop
*37 62. h : E ! 7i*y . y c a . D . i(‘y e /J" a
Dem.
I-. *30 33 . D
H :: E ! . D «‘y « ie««„ . = : x7?y . . , £ R „ a
K*32. 3 l- :.y <«. 3 lafly.D.y ea.xllg.
[*10-24.*37 1] 0.xeH“ a
I-. (2).*10 11-21 . D h y e o. D : .rfty .O x .xe It“a
H . (1> . (3). D I-. Prop
••I inference concerning which Jevons says*:
cou d not .T ,, ! e .n r ’ ' ,rgaD ren,arkin « ‘hat all Aristotle’s logic
the hid r “TT “ h '"' SC is an ani "‘“‘. ‘he head of a horseis
Ariat X- i ° n,ma L mUSt be COnfess< - d ’hat this was a merit in
tern^ . E hh: r:, th r U ,f 0 P°* Kd infere,,CC U fallaciou8 w >‘hout the added
premiss E ! the head of the horse .n question." Eg. it does not hoi,I lor an
an imn 0, ', a / a ' V “ h the addition E ! ** ‘ h « above proposition gives
an important and common type of asyllogistic inference. *
*3763. H .t *11 3
Dem.
r*io- 23 i ‘ 3 h!! * ‘ H “ a (ay) .y«a.xjjy. a,.**..
r 10 Z3J = :. y € a . x/ty . D, „. dr X
3h H (!)
P . *37 104 . D J-Hp . D :: y e a . D,,E !
This proposition is very frequently used.
• Principles of Science, chap. i. (p. is of edition of 1887).
O)
(2)
(3)
19—2
MATHEMATICAL LOGIC
[PART I
*37 64. H E !! R“a . D : <3y) • y ea . ^ (P*y) . s . (gx) . art P“a . yjrx
Dem.
h . *30*33 .Dh: Hp . D y « a . D : x/** (P‘y) . = . (g.r). xPy . x/^x
[*5*32] D :.y*a . x/r(/*‘y). = : y €<* : (gx) .xPy. ^rx (1)
Ml). *10*11*21*281 . D
f- :: Hp . D (gy) . y e a . \fr (R*y) . = : (gy) z yea: (gx) . xRy . x/rx:
t* 1 *'■] =: (a^-): (gy) • '/*<*• = V r * e!
[*37-1] = : (g.r). j-e 7?“o . ^r.r:: D K Prop
*37 65. : E !! R“{3 . a C R"& . 0 . a - R“( R"a « 0)
Dem.
f-. *30 21 . *3 27 . D h :: H p . D y e Q . D v : zRy . .cRj . D . * - x (1)
h. *37*1.3 hi. Hp.D:
X € R“( Ji“a r\ 0). = . (gy) . y e R**a n & . xPy .
[*37 105.* 11*55] s . (gy. z). z e a . zRy. y e (3. .r Ry .
[(1 ).*4 7 1 ] = . (gy, z ). z e a . zRy . y e f3 . xRy • z = x
[ * 13* 194] s . (gy, z)*zea,ye $ . xPy. z “ x.
[* 13*195] = . (gy) . x e a . y € & . xPy .
[*10*35.*37*1] = .xea.xe R“/3.
[*4 a 71.Hp] a.xias.DK Prop
*37 66. h E !! P“£ . D : a C R“0 . = . (g 7 ). 7 C <3 . a - P“ 7
Dem.
h . *37*65 . Exp . *13*195 . *22*43 . D
h Hp . D : a C . D . (g 7 ) . 7 C£ . a = (1)
h . *37*2 . *1313 . D h : 7 C £. a = P“ 7 . D . a C P“£ :
[*1011*23) D h : (g 7 ). 7 C 0 . a = P“ 7 .D.aCR“0 (2)
Ml).(2). D h . Prop
*37 67. h i * 7 . D z . E! R‘&s : D . R‘ 'S“y = £ ’(gr). * * 7 . x = P‘S‘*)
Dem.
h. *34*41 . D h : Hp .zey.D g . R<S‘z - (R 1S)‘x (1)
Ml). *14*21.3 I- : Hp. 2 ey.D'.El(R\S)‘z (2)
h - (2). *37 6 . D h : Hp.D . (P jS)“ 7 = 3 {(g*). r c 7 . x = (P | S)*y)
[(1)] -2{<g z).Z€y.x = R‘S‘y) (3)
h . *37*33 . Dh. R“S“y = (P | S)“y (4)
h.(3).(4). 3 h. Prop
*37 68. h z.zey.D,. P*Q‘z = R*z : D . P“Q“y = P“ 7
Dem.
h. *14*21 . D h : Hp . z e y . D . E ! P‘Q*z . E ! R*z .
[*34*41 ] D . P‘Q‘* = (P | Q)‘z . E! R*z . (1)
[*14*21 *131 *144.Hp] D . E ! (P | . (P | Q)‘x = R‘z (2)
SECTION D] PLURAL DESCRIPTIVE FUNCTIONS 29;
I-. *37-33 . D t-. l'“Q“y = (P\Q)“y (S)
(2). (3). *37-6. D
h : H P ■ 3 • P“Q “7 = 2 !(a^). t « y . x = (P I Q)‘z j
[ (2) ]
[*37-6.(l)] = R" z .oh. Prop
*37 69. b :. y 6 0.O„.R‘y = S < y:O. R“$ = S“0
Pent.
I-. *14-21 .3K:: Hp. D :.y«/3. D . E! R‘y . E ! S‘i/:. (1)
[*30-4] D:.y«/9.D:* % . = .*-*V
t* 14142 ]
t* 304 -n)] = .„Sy:.
[*5-32] ^■■•yc&.xRy.s.yeB.xSy (2)
H. (2). *10 11-21-2810
h Hp O : (gy) . y e £. x Ry . = . (gy). y «/3 . a-Sy :
[*371] D :x t R “,3 . = .xeS“0 :0 h . Prop
A specially important case of is rt“/3 or *“/S. This case will be
further studied later (,n *70); for the present, we shall only give a few
preliminary propositions about it. It will be observed that the hypothesis
El! R“,3 or El! R “,9 is always verified, in virtue of *3212121. Hence the
following applications of *37 6 ff.:
* 377 . i-. r*‘& - a {( ay ). y e &. a -j^y)
*37 701. h . tf“a = y§ {(a*). K e a . 0 - }?*)
*37702 * . = : y e ,9 . . 7?‘y « «
*37 703. h C * . s : x c /9 . D r . /e** f *
*37704. 1-: yea . D .7?‘y ( ^“a
*37 705. h :arca.
*37-706. h a e . D« . : = : y 6 £ . ^
*37-707. h :. £ « . D* . • = • * € a . ^ ^
*37-708. h ( 3 a) . a * R“ 0 . . = . ( ay ) . y c £. ^ (7? y)
*37 709. h ( 30 )^ a e *R“0 . . = . ( 30 :)
*37-71. h : * ,D.tc = /?‘{(Cn« £}
*37-711. h : * C if“£ • Z) . * = ^ £}
*37-712. h : , C s . (a7> . y C ft. K ~jt„ y
*37-713. h s *C.s. ( a7 ). 7 C £. * = R« y
[*37 6. *32 12]
[*37-6. *32 121]
[*37-61]
[*37-61]
[*37-62. *3212]
[*37-62. *32121]
[*37-63]
[*37 63]
[*37-64]
[*37-64]
[*37-65]
[*37-65]
[*37-66]
[*37-66]
MATHEMATICAL LOGIC
[PART I
*3772. b : R = P Q . D . R“ y = P“‘Q“y
Dent.
I-. *37 11-302 . Z> b : Hp. D . (*). P/~Q‘z=ll‘z .
[*37-68] D . P“~Q“y = li“ y .
[(*37-04)] D . P‘“7j“ y =~R“y Prop
*37 721. \-:R=l> Q. D . /f“ 7 . §*“/** 7 (Proof as in *3772]
*37 73. 1-: a ! £. = . g ! 7?‘£. = . g l%‘0 [*37*45 . *32 12 121]
*37 731. b : /9 = A . = .7<“0 = A . = = A [*37 73 . Transp]
Observe that the As which occur in this proposition will not be all of the
same type. Ejj. if R relates individuals to individuals, the first A will be
l he class of no individuals, while the second and third will be the class of
no classes. Thus the ambiguity which attaches to the type of A must be
differently determined for different occurrences of A in this proposition. In
general, when this is the case with our ambiguous symbols, we shall adopt a
notation which indicates the fact. But when the ambiguous symbol is A, it
seems hardly worth while.
*37 74. I- >9 C (VR . = : a € 7?‘£ . D. . g ! a
Deni.
b . *37-706 . D b a € /?“/9 . D. . g ! a : = : y e/3 . . a \~R l y :
[*33*31] = : >9 C d'R D b. Prop
*37*75. bs.crCD ‘R. = : /9 €%‘a . . g ! £ [Proof as in *37*74]
*37*76. b.7*“£CCIs
Dem.
b . *37*7 .Dh.of R“0 . D : (gy) ,y(^.a= R‘y :
[*10*5]
[*32*13]
[*20*16]
[*20*4]
*37 761. V.*R“a C Cls
*37 77. b : a . D a . g ! a
*37 771. b : >9 e . g ! >9
*37*772. b.A~eJ?“CI‘.«
*37*773.
*37*78. b.D'^^'V
3 : (3y) • a = P*'J :
(3 y) • a = $ (xRy):
D:(g <*>). a = £(</» !*):
D z ac Cls D b . Prop
[Proof as in *37*76]
[*37-74. *22 42]
[Proof as in *37 77]
[*37-77 . *24*63]
[*37-771 .*24*63]
[*37*28]
SECTION D]
PLURAL DESCRIPTIVE FUNCTIONS
*37 781. KD *Jl = R“V [*37 *28]
*3779. . R“Y = £{('£'/). a = [*37*601 .*32 1*2]
*37791. h . R“V = /§ j(g.r). 0 - %x\ [*37*601 .*3*21*21]
*37-8. M«T0)|S-«T£“£
Devi.
h * **5103 . *341 .Dh:x((at£) | ( ay >. ., € a . y e 0 . ySz .
[* 10 - 35 .* 37105 ] m.xea.zeS“0.
[*35*103] = . . t . (a T : D I-. Prop
*37 81 . h./e|(at^)=-(/i-a)t/9 ^ [Proofas in *37*8]
*37 82. I-. R. <ct f 0) \S-(R“a ) T (S“/9) [*37*8*81]
*38. RELATIONS AND CLASSES DERIVED FROM A DOUBLE
DESCRIPTIVE FUNCTION
Sum mo ry of *38.
A double descriptive function is a non-propositional function of two
arguments, such as an&.av £. It A .S', It v S, It S, a 1 It, It [a, It l a. The
propositions of the present number apply to all such functions, assuming the
notation to be (as in the above instances) a functional sign placed between the
two arguments. In order to deal with all analogous cases at once, we shall in
this number adopt the notation
where stands for any such sign as n, \j, A, |, |, |*. [, or any functional
sign to be hereafter defined and satisfying the condition
(■*■.!/) • E !(x?y).
The derived relations and cln>scs with which we shall be concerned may be
illustrated by taking the case of ar\ft. The relation of a r» & to /9 will be
written a n, and the relation of a r\ to a will be written n /9. Thus we
shall have
h .a n fi = a = n /3‘a.
The utility of this notation is chiefly due to the possibility of 9uch notations
as ar\ li K and r\fS il K. For example, take such a phrase as "the foreign
members of English Clubs." Then if we put a = foreigners, k = English Clubs,
we have
a r\ tl /c = the classes of foreign members of the various English Clubs.
Or again, let a be a conic, and k a pencil of lines; then
an“< = thc various pairs of points in which members of k meet a.
In this case, since a r» a. we have a n = r\ a. But when the function
concerned is not commutative, this does not hold. Thus for example we do
not have It | = | R.
The notations of this number will be frequently applied hereafter to
In accordance with what was said above, we write It for the relation of It S
to S, and | S for the relation of It S to R. Hence we have
Hence will be the class of relations obtained by taking members of X •
and relatively multiplying them by S. Thus if X were the class of relations
first cousin, second cousin, etc., and S were the relation of parent to child,
| S“\ would be the class of relations first cousin once removed, second cousin
once removed, etc., taken in the sense which goes from the older to the younger
SECTION D]
OPERATIONS
25)7
It is often convenient to be able to exhibit \S“\ and kindred expressions
as descriptive functions of the first argument instead of the second. For this
purpose we put
X S=\S“\
ft
with similar notations for other descriptive double functions. We then have
just as in the case of R \S,
\\‘S= S*\ = \\R.
** tt tt
This enables us to form the class This class is chiefly useful because
the members of its members (i.e. as we shall define it in *40) con-
stitute the class of all products R | S that can be formed of a member of \ and
a member of
Thus we are led to three general definitions for descriptive double functions,
namely (if x% y be any such function)
x ? is the relation of x $ y to y for any y,
% y •• ^ a H „ x „ x,
a is the class of values of x%y when x is an a.
Since a?y is again a descriptive double function, the first two of the above
definitions can be applied to it. The third definition, for typographical reasons
cannot be applied conveniently, though theoretically it is of course applicable,
I he relations x ? and ?y represent the general idea contained in some of the
uses in mathematics of the term “operation," e.g. + 1 is the operation of
adding 1.
The uses of the notations introduced in the present number occur chiefly
iu arithmetic (Parts III and IV). Few propositions can be given at this stage
since most of the important uses of the notation here introduced depend upon
the substitution of some special function for the general function " £ ” here
used. In the present number, the propositions given are all immediate con¬
sequences of the definitions.
*38 01. x $ «* uy (u = i?y) Df
*38 02. ?y = fi£(u = a:¥y) Df
*38 03. = Df
*381. h :u(x$)y . = ,u = x%y
*38 101. m . u y) x . = . u = x % y
*3811. h . = =
*38 12. I" . E ! x% *y . E ! %y*x
*3813. h : u € x ? “a . = . (gy) . y ea . u = x%y
*38131. y“a . = . (gar) .xe a . u = x%y
[(*3801)]
[(*3802)]
[*381 101 .*30-3]
[*3811 .*14-21]
[*381 .*37 1]
[*38 101 .*37 1]
[PART I
298
MATHEMATICAL
LOGIC
*38 2.
77
[(*3803)]
*3821.
h . a J y = « |('.(.c) .x€a.M = .i-Jy|
[*38-2131]
*3822.
h.a?‘y = ?y‘* = a*y
77 77 77
[*3811]
*3823.
KK!a?‘y.E!¥/,‘«
77 77
[*3.v22. *14*21]
*3824.
JJem.
. *.382 . *37-29 . Transp .^hglo^.D.gla
K . *38-21 . Dhj-<a.D.(/}y)fa{y.
[*io-2+]
K.(]>.(2).Dh.Prop
(1)
( 2 )
*38 3. h .a £“£-7 |<gy).ye£.7 = a $y| = 7 |< 3 '/) • V * £ . 7 = ?y“a|
(*38 13 2]
*38 31 . i-. ?y“* - 7 1(3«) •«<*• 7 - « ? y) - 7 l(a a ) • a * * • 7 -S y“«l =* ¥ y‘“*
[*38 131 - 2 . *37 103 ]
NOTE TO SECTION D
General Observations on Relations. The notion of “relation” is so general
that it is important to realize the different sorts of relations to which the
notations defined in the preceding section may be applied. It often happens
that a proposition which holds for any relation is only important for relations
of certain kinds; hence it is desirable that the reader should have in mind
some of the principal kinds of relations. Of the various uses to which different
sorts of relations may be put, there are three which are specially important,
namely ( 1 ) to give rise to descriptive functions, (2) to establish correlations
between different classes, (3) to generate series. Let us consider these in
succession.
(1) In order that a relation R may give rise to a descriptive function,
it must be such that the referent is unique when the relatum is given.
Thus, for example, the relations Cnv. It, R, D. d. C, R t , defined above,
all give rise to descriptive functions. In general, if R gives rise to a
descriptive function, there will be a certain class, namely (I *R, to which
the argument of the function must belong in order that the function may
have a value for that argument. For example, taking the sine as an illustra¬
tion, and writing “sin‘y” instead of “sin y.” y must be a number in order
that sin‘y may exist. Then shi is the relation of y to x when .x-sin‘y. If
we put a = numbers between - tt/2 and tt/2, both included, sin [ a will be the
relation of * to y when * - sin'y and - tt /2 sy * tt / 2 . The converse of this
relation, which is a lain. will also give rise to a descriptive function; thus
(a ”1 sin)‘x =» that value of sin“»a; which lies between - tt /2 and tt/2. This
illustrates a case which arises very frequently, namely, that a relation R
does not, as it stands, give rise to a descriptive function, but does do so
when its domain or converse domain is suitably limited. Thus for example
the relation “parent” does not give rise to a descriptive function, but does
do so when its domain is limited to males or limited to females. The relation
“square root,” similarly, gives rise to a descriptive function when its domain
ih limited to positive numbers, or limited to negative numbers. The relation
“ wi *f gives rise 10 a descriptive function when its converse domain is limited
to Christian men, but not when Mohammedans are included. The domain
of a relation which gives rise to a descriptive function without limiting its
domain or converse domain consists of all possible values of the function; the
converse domain consists of all possible_arguments to the function. Again, if
R gives rise to a descriptive function, R<x will be_the class of those arguments
for which the value of the function is *. Thus Zhi‘x consists of all numbers
300
MATHEMATICAL LOGIC
[PART I
whose sine is ./•, i.e. all values of sin -1 x. Again, sin“a will be the sines of the
various members of a. If a is a class of numbers, then, by the notation of *38,
2 x “a will be the doubles of those numbers, 3 x “a the trebles of them, and
so on. To take another illustration, let a be a pencil of lines, and let R‘x be
the intersection of a line x with a given transversal. Then R il a will be the
intersections of lines belonging to the pencil with the transversal.
(2) Relations which establish a correlation between two classes are really
a particular case of relations giving rise to descriptive functions, namely the
case in which the converse relation also gives rise to a descriptive function.
In this case, the relation is 44 one-one," i.e. given the referent, the relatum is
determinate, and vice versa. A relation which is to be conceived as a correla¬
tion will generally be denoted by N or T. In such cases, we are ns a rule less
interested in the particular terms x and y for which xRy, than in classes of
such terms. We generally, in such cases, have some class fS contained in the
converse domain of our relation S. and we have a class a such that
In this case, the relation .S’ correlates the members of a and the members of
/$. We shall have also /$ = .S ,<4 a, so that, for such a relation, the correlation is
reciprocal. Such relations are fundamental in arithmetic, since they are used
in defining what is meant by saying that two classes (or series) have the same
cardinal (or ordinal) number of terms.
(3) Relations which give rise to series will in general be denoted by P
or (}, and in propositions whose chief importance lies in their application to
series we shall also, as a rule, denote a variable relation by P or Q. When
P is used, it may be read as 44 precedes." Then P may be read " follows,”
P*x may be read 44 predecessors of x." P*x may be read " followers of x.
1 VP will be all members of the series generated by P except the last (if any),
(I ‘P will be all members of the series except the first (if any), C‘P will be
all the members of the series. P“a will consist of all terms preceding sonic
member of a. Suppose, for example, that our series is the series of real numbers,
and that o is the class of members of an ascending series x l% .r a , ... x v .
Then P il a will be the segment of the real numbers defined by this series, i.e.
it will be all the predecessors of the limit of the series. (In the event of the
series x lt .r 2 , x s , ... x y , ... growing without limit, P tl a will be the whole series
of real numbers.)
It very often happens that a relation has more or less of a serial character,
without having all the characteristics necessary for generating series. Take,
for example, the relation of son to father. It is obvious that by means of
this relation series can be generated which start from any man and end with
Adam. But these series are not the field of the relation in question; more¬
over this relation is not transitive, i.e. a son of a son of x is not a son of x.
If, however, we substitute for 44 son ” the relation “descendant in the direct
SECTION D]
NOTE TO SECTION D
' 301
male line ” (which can be defined in terms of “ son ” by the method explained
in *5)0 and *91), and if we limit the converse domain of this relation to
ancestors of x in the direct male line, we obtain a new relation which /«
serial, and has for its field .r and all his ancestors in the direct male line
Again, one relation may generate a number of series, as for example the
relation “x is east of y .” If x and y are points on the earths surface, and in
the eastern hemisphere, this relation generates one series for every parallel
of latitude. By confining the Held of the relation further to one parallel of
latitude, we obtain a relation which generates a series. (The reason for
confining x and y to one hemisphere is to insure that the relation shall be
transitive, since otherwise we might have x east of y and y east of * but x
west of z.)
A relation may have the characteristics of all the three kinds of relations
provided we include in the third kind all those which lead to series by some
such limitations as those just described. For example, the relation + 1
< ,n v,rtue of the notation of *38) the relation ofx+1 to x, where x is
supposed to be a finite cardinal integer, has the characteristics of all three
kinds of relations. In the first place, it leads to the descriptive function
(-f I)x, %.e. x+l. In the second place, it correlates with any class a of
numbers the class obtained by adding 1 to each member of a, t.e. (+ l)« a .
This correlation may be used to prove that the number of finite integers is
infinite (in one of the two senses of the word “infinite”); for if we take ns
our class a all the natural numbers including 0, the class ( + 1 )“a consists of
all the natural numbers except 0 . so that the natural numbers can be corre-
ated with a proper part* of themselves. Again, the relation + 1 may be used,
like that of father to son, to generate a scries, namely the usual series of the
natural numbers in order of magnitude, in which each has to its immediate
predecessor the relation +1. Thus this relation partakes of the characteristics
ol all three kinds of relations.
l.e. a part not the whole. On this definition of infinity, *ce «l a J4.
SECTION E
PRODUCTS ANT) SUMS OF CLASSES
Sn minnry of Section R.
In tin* present section, we make an extension of a r\ /3, a v &, R r* S, R sj S.
(liven a class of classes, say k, tin- product of k (which is denoted by />**) is
the common part of all the members of k. i.e. the class consisting of those
terms which belong t*» every member uf «. The definition is
P*k = .?■ (a e « . D a .xta) Df.
If x has only two members, a and f3 say, p*x = a n (3. If k has three members.
a, f3. y. then p*x » a r\ (3 r\ y ; and so on. But this process can only be continued
to a finite number of terms, whereas the definition of p*x does not require
that k should be finite. This notion is chiefly important in connection with
the lower limits of series. For example, let \ be the class of rational numbers
whose square is greater than 2. and let *• xMy " mean "./•< //, where x and y
ationals.’ Then if j <X. M*x will be the class of rationals less than x.
are
Thus M**\ will be the class of such classes as M*.r, where xe\. Thus the
product of which we call will be the class of rationals which
are less than every member of X. i.e. the class of rationals whose squares are
less than 2. Each member of M tf \ is a segment of the series of rationals, and
y>‘J/“X is the lower limit of these segments. It is thus that we prove the
existence of lower limits of series of segments.
Similarly the sum of a class of classes k is defined as the class consisting
of all terms belonging to some member of k ; i.e.
£ Ka«) •« « * Df,
i.e. x belongs to the sum of k if x belongs to some k. This notion plays the
same part for upper limits of series of segments as p l K plays for lower limits.
It has, however, many more other uses than and is altogether a more im¬
portant conception. Thus in cardinal arithmetic, if no two members of k have
any term in common, the arithmetical sum of the numbers of members possessed
by the various members of k is the number of members possessed by s*/c.
The product of a class of relations (X say) is the relation which holds
between x and y when x and' y have every relation of the class X. 'I he
definition is
p*\ = ^(R € \.'D Jt .xRy) Df.
The properties of p‘\ are analogous to those of p g x, but its uses are fewer.
SECTION E]
PRODUCTS AND SUMS OP CLASSES ;$();{
The sum of a class of relations (X say) is the relation which holds between
•r and y whenever there is a relation of the class X which holds between v.
and y. The definition is
s'\ -= ZT} !(g.R) . /? e \ . .rIt//\ Df.
This conception, though less important than is more important than ,Y\.
I he summation of series and ordinal numbers depends upon it. though the
connection is less immediate than that of the summation of cardinal numbers
with s‘/c.
Instead of defining />'«. s'*. ,V\. i'x. it would be formally more correct to
define p, s, pnud s. which are the relations giving rise to the above descriptive
functions. 1 hus we should have
p = & (/? = £ (a « * . . . r , o)J Df>
whence we should proceed to
h : 0pK . 2 ./3 = 2(a€K.D a .xea),
and h . E ! p*K.
But in cases where the relation, as opposed to the descriptive function, is
very seldom required, it is simpler and easier to give the definition of the
descriptive function in the first instance. In such cases, the relation is always
tacitly assumed to be also defined; i.e. when we give a definition of the form
IVx * S*x Df,
where S is some previously defined relation, we always assume that this
definition is to be regarded as derived from
R = ti2(u-S*x) Df.
In addition to products and sums, we deal, in the present section, with
certain properties of the relations li , and | S. the meanings of which result
from the notation introduced in .38. Such relations are very useful in
arithmetic. 1 he reason for dealing with them in the present section is that
a large proportion of the propositions to be proved involve sums of classes of
classes or relations.
*40. PRODUCTS AND SUMS OF CLASSES OF CLASSES
Sum in tin/ o/' *40.
In this number, wo introduce the two notations (explained .above)
/>** =/(a€ * .D«./(o) Df
s l < = .7 '(%|a). a € * . xe aj Df
Both these notations will hi* (bund increasingly useful as we proceed, but s‘k
remains more useful than //* throughout. It is reipiired for the significance
of p l K and .*‘* that * should Ik- a class of classes.
In the present number, the most useful propositions are the following:
*40 12. b : a e * . D . />** C a
l.e. the product of * is contained in every member of*.
*40 13. b : a < * . D . a C *‘*
l.e. every member of* is contained in the sum of *.
*40 15. b (3 C p l K .= : 7 f < . D v . ^ C 7
l.e. & is contained in the product of* if /S is contained in every member
of *, and vice versa.
*40 151. b s*k C/3. = : 7 e * . D y . 7 C
l.e. the sum of* is contained in if every member of * is contained in
and vice versa.
*40 2. I-:« = A.D. p‘x = V
l.e. the product of the null-class of classes is the universal class. This may
seem paradoxical at first sight, but it is really not so. The fewer members *
has, the larger, speaking generally, //* becomes. If * has no members, then
* has no members to which a given term x does not belong, and therefore x
belongs to //*.
*40 23. b : a ! * . D . />** C .v‘*
l.e. unless * is null, its product is contained in its sum.
*40 38. b . R‘ V* =
This proposition is very often used in arithmetic. What it states is as
follows: Given a class of classes *. take its sum, s*k, and then consider all the
terms that have the relation R to some member of s‘tc ; this gives the class
R tt s , K\ next, take each separate member of *, say a. and form the class R“a,
consisting of all terms having the relation R to some member of a. The class
of all such classes sis R“a, for various as which are members of *, is R lit <\
the sum of this class, by the above proposition, is the same as R li s i K.
*40 4. b E !! . D . s*R“0 = £ {(fly) .ye&.xe R‘y)
This proposition requires, for significance, that R‘y should always be a
SECTION E]
PRODUCTS AND SUMS OF CLASSES OF CLASSES 305
class. The proposition states that, if R<y always exists when ye/3. then the
sum ot all classes which have the relation R to some member of 0 consists of
all members of such classes as i£‘y, where y e 0 .
*40 5. b .s‘ft“/9 = R “0
This proposition results from *404 by substituting 7? for R in that
proposition.
*40 51. b . p‘~R“f3 = £ |y € /9 . . xRy j
In virtue^of *40*5, p*R**0 is correlative to R“0. Thus if R is a serial
relation, p f R “0 consists of terms preceding the whole of 0, and R“0 consists
ot terms preceding part of >9. If >9 has a lower limit, it will be the upper limit or
maximum ofp'R 1 *^; if 0 has an upper limit, it will be the upper limit of R“0.
*40 61. b : g ! /9.3 C R“&.p*R“& C 11 “ 0
In this proposition the hypothesis is essential, since, if 0 = A, p‘R “0 = V
and R it 0 *= A.
*4001.
*4002.
*401.
*4011.
*4012.
Dem
*4013.
Dem.
*4014.
*40141.
*4016.
Dem.
p*K = £{a€K.^ a .xea) Df
*‘*-*f<3«)-«€*.x€a) Df
h * ep‘* . = : a e *. 3. . * < a [*20 3 . (*40 01)]
h : x e s‘k . 3 . (ga) .atK.xea [*20 3 . (*40 02)]
b : ae k . 3 C a
b . *401 . *101 . 3 b z.xtp'tc .3:ae*.3.a:ea:.
[Comm] 3h:.a«*.3:««p‘*.3.*€a
b . (1) . *1011-21 . *221 ,3b. Prop
b:a««.D.aC
b . *40*11. *10*24 .Db:a«*.*«a.D.#e s*k :
3b :.ae k.D zxea.D .X€s‘/c
b . (1) . *1011-21 . *221 .*3 b . Prop
braex.arepSc.D.arca [*40 i 2 .I rap ]
bsae^.aea.D.xe^ [*40 11 . *10 24]
b:.£Cp‘*. = zyeK.Dy.ffCy
b . *401 .Db::^C p* K
[*11-62]
[*43'84.*11 *33]
[*11-2-62]
[* 22 - 1 ]
R& w
x € 0 . 3 X : y € K . 3 V . x e 7
(#, y) z x e 0 .yex.D.xeyz.
*• (*, y): y e * . x € /3. D . x e y z.
y c * . z x e 0 . . x e y z.
7 e * . 3 y . £ C y :: 3 b . Prop
MATHEMATICAL LOGIC
[PART I
306
*40 151. \-z.8‘KC&. = :ycK.O y . y C/3
Dem.
h . *40-11 . D Y :: s*k C /3 . s ( 37 ) . y€K.xcy.O x .x€@:.
[*10 23] = ( 7 , x) 7 c k . x e y . D . xe fi
[*11-62] h:.( 7 ) ye k . D : (a*) :.r« 7 , D ..re/?:.
[*22* 1) =:. 7 e*c.D y . 7 C/i::DK Prop
This proposition is frequently used.
*40 16. hsxCX.D. C
Dem.
Y . *10*1 . D f-:: Hp • D 7 e k . D . 7 € X
[Syii] ^ • • y€\,0.xey:D:y€K,D.xty ( 1 )
t-.(l).* 10 * 11*21 . D
h :: Hp . D ( 7 ) :. 7 eX.D.j;€ 7 :D: 7 €/c.D..rc 7 :.
1*10 *27] D z. (y) z y e \ . D. x * y z D: (y) z y € #. D • x e y :•
[*401] D x e p*\. D . x ep*K (2)
K.(2).*10*11*21 .DK Prop
*40161. huCX.D.^CA
Dem.
t* • *10*1 .Dh. Hp.D:7i«.D.7e\:
[ Fact] D: 7 €/c.arc 7 .D. 7 cX.xe 7 :
[*10*11-28] D : ( 37 )-7« * • xey. D . ( 37 ). 7 « X - x ey z
[*10-11] D : ares** . D . xes'X
h. (1). *1011*21.3 h. Prop
*40-17. H . p‘* w p'X C p*(K n X)
Dem.
Y . *22-34 . D I- :: x e/*** w p*\ . = :.xep‘* . v . xep‘\
[*401] — • • 7€/c.D T .a*t7:v:7€X.D v .a:€7:.
[*1041] D:.( 7 ):. 7 «/c.D.xcy:v: 7 €X.D..re 7 :.
[*4*79] D ( 7 ) : 7 e « . 7 e X . D .x €7
[*22-33] D :.(y):ye * r\\.D .xty z.
[*401] D X€p*(tc n X)
h.(l). *10*11 .DK Prop
( 1 )
(1)
*40171. f-.s‘^s'X = sV u X)
Dem.
h . *22-34 . D h :: ares'* us‘X. = :.xes‘>c.v.xes‘X:.
[*40- 11] = :. (37) . 7 e * . x e 7 : v : (37) . 7 e X . ar e 7
[* 10 - 42 ] = (37) zy€K.xey.v.ye\.X€yz.
[*4*4] — :• ( 37 ) :• 7 € K • v • 7 6 X : xe 7
PRODUCTS AND SUMS OF CLASSES OF CLASSES
SECTION E]
[*22-34] =:-(37)-7**«X...c<r 7 :.
[*4011] s:.X£s‘(*v/X)::DI-.Prop
*40-18. f-. p\K yj \) — p* K r» p*\
Bern.
I-. *40-1 . D t-:: x £ /.<(* u X ). = y e K w x x e ..
[*22-34] = s. ( 7 )7 * «. v . 7 £ x: D. x £ 7
[*4-77] s ”<7):-7«*-3 .x£7: 7 £X.D.x£ 7 :.
[*10-22-221] 5 :-(7):7«*- 3.x£ 7 :.( 7 ) ! 7 e x. D.X£ 7 :.
[*401] =■. xep‘ K .xcp‘\:.
[*22-33] = z.xep'x n :: D I-. Prop
*40181. H.»‘(*«\)C*‘*na«X
Dem.
= : -(H7)-7* *-7«X.*«y:.
r f 1 ^ ( 37 ) •yctc.xey: ( 37 ) .yeX.xey:.
[*4° 11 -*^2-33] D:.ze5‘*As‘\::Dh. Prop
*4019. hs:* < ^. 2 : . 7 <ie . DT . 7 C^O l . a .^
lhis proposition is the extension of * 22 * 6 .
Dem.
h. *40-151 . D
H ::7« *- .**£:. 9 :.s‘*C/9. D $ .xeS
.*10-1 . Dh D : *«*C«‘*. D :
307
[*22-42]
3:xfs'
h . *22-46 . D h x t s'k . s*k C /3 . D . x e >9
[Exp] C£.I>.*«/9:.
[*1011-21]Dh:.xe^.D:^C,i.D a .,^
[* • (2) • (3) • D h Cft.D a .xcfi: = .xc s‘k
h.(l).(4).Dh.P r0 p
*40 2. H : * = A . D = V
Dem.
K *24-5-51. 3 I-Hp. D : ~ (ga). o £ «:
1 (°) 5«£*.3.xta:
t* 40 ’ 1 ] 3:x f/ ,V
l-.(l).*10-11-21. Dh:Hp. D.(x).x£ P ‘*.
[ * 24 ' 14 ] 3 . p‘K = V: 31. Prop
*4021. l-:* = A.3.*‘x = A
Dem.
r H ‘ * 24 ® 1 • 3t- = Hp.D.~ (a a).a£«.
[*10-6.Transp] 3 . ~(ga> . a ex . x e a .
( 1 )
( 2 )
(3)
(4)
( 1 )
20—2
308
MATHEMATICAL LOGIC
[PART I
( 1 )
[*40- 11 .Transp] D.x~€s*k
h . (1). *10*11*21 . D h : Hp . D . (x) . x~ cs'k .
[*2415] D . s** = A:Dh. Prop
In the above proposition, the two A’s are of different types, since k is of
the type next above that of s‘*. Tims it would be more correct to write
hjf-Aft CIs . D . s*k = A r\ V.
But in the case of A it is not very important to keep the types distinct.
*40 22. h : A e «. D. p*K = A
Dem.
h. *40-12. Dh : Hp . D . />** C A .
[*24‘13] D ,p*K = A : D H . Prop
In this projjosition, the two A’s are of the same type.
*40-221. h : V € * . D . - V
Dem.
h . *4013 . D h : Hp . D . V C s*k .
[*24141] D . s*k = V : D V . Prop
*40 23. h : g ! * . D . />‘* C
Dem.
H . *40- 12*13.DH:ae«r.D. p*K C a . a C s*k .
[*22-44]
[*10-11-23] D H : (ga) .ac«.D .//* Cs‘*:DI-. Prop
Observe that the hypothesis g ! * is essential to this proposition, since
when k = A, p*tc = V and s*k = A. Thus
h s a ! * . ■ . p*K C s'*.
*40 24. h.g!< :7f<.D y .i9C7:D./9C«‘«
-Dew.
h.*40-15. Dh:.7€*.D v ./3C 7 :D.£C/j‘* (!)
K *40-23. Dh:g!*.D./)‘«C#‘* ( 2 )
h . (1). (2). D I- : Hp . D . /9 C/>‘* .C .
[*22-44] D.j8 Cs‘/t:Dh. Prop
The above proposition is used in the proof of *215*25.
*40 25. h:x€5V. = .g!/<oa(a:ea)
Dem.
*4026.
Dem.
h . *22-33 .Dh:g!/tna(a;fa)
[*203]
[*4011]
V : g ! s‘k . = . (ga) . a c * . g ! a
S .(a7>-7«***«7*
= . x e s‘tc :Dh. Prop
h . *4011 .Dh:.g! s*k . = : (gx) s (ga) . a c * . x c a :
[*11-23-55] = : (ga) : a e * : (gx). x € a s
[*24-5] s : (ga) .ae/c. glas.DK Prop
SECTION E]
PRODUCTS AND SUMS OF CLASSES OF CLASSES
309
The following proposition is used in the proof of *210 51.
*40 27. h a s*/c ™A. = :y€<.D Y .on<y = A
Dem.
b. *24 311 . D
h :: a r\ s'k = A . = s‘tc C - a
[*22-1-35] = :.x e s*k . D z . e a
[*401] = (g 7 ). r€ 4 c. xey. D,
[*10 23] = 7 c * . *€7 . D x y . #*■'*' e a
[*11-2-62] s:.7 f *f.D Y : a:« 7 .D r .^<v f a:.
[*24-39] ss- 7 €/f.D y .or »7 — A::DH. Prop
The following propositions are only significant when R is a relation whose
domain consists of classes, for they concern or s‘R“a, and therefore
require that R“a should be a class of classes.
*40 3. h . p*R li (a w 0) =p*R“ a « [*37*22 . *40 18]
*40 31. 1-. s'R'^a u 0) = ^ [*37 22 . *40 171]
*40 32. h . p*R“a u C />'/*“(<* n 0)
Dem.
I-. *37-21.3 h . R“(a r> 0) C «“a « R“0 .
[•4016] Oy.p‘(R“ar>R“/3)Cp‘Ii“( an /3 ) ( 1 )
I-. *40-17 . D t-. p‘R" a u p'if# C p‘(R“a n R“/3) (2)
I- . (1) . (2) . *22-44 . D h . Prop
•40 33. h . s‘R“(a n R) C s‘R"a n s‘R“(3 [*37 21 . *40161 . *40181 ]
The following propositions no longer require that the domain of R should
be composed of classes.
*40 35. b. p'R'^K = £ \0 € k . D* . * < R“0\
Dem.
h . *401.31 - ■..X€ P -R‘"K. = iy ( R<"K.^ y .x ( y.
[*37 103] = -.(a/3)./3'*.y = R“0.O y . X( y:
[*10-23] =:0€K.y = R“R .^ 0 r . X(y:
[*13191] =:/9 €K.O 0 .x e R“0 (1)
K(l). *1011 .*20-3.31-. Prop
*40 36. * ■ s'R“‘ k=S [(■&&). $ eK . xe R" 0 \ [Similar proof]
*40 37. I- . R" p ‘k C p‘R‘“k
Dem.
V . *37*1 ,D\- zzxe R“p‘K . = (gy) . yep‘ K . xRy
[*401] = (gy) z 0 c k . . y € 0 : xRy
[*10-33] = (gy) z.(0)z0c*.D.y € 0z xRy ..
[*1126] ^:.(/9):*(a y)y e 0 z xRy z.
[*5-31] D (£) .. ( ay ) z0ck.D. y € 0. x Ry
310
MATHEMATICAL LOGIC
[PART I
[*1037] D (/3)£ e * . D . (gy) .y e/9 . xRy :.
[*371] D <£): # e *. D . x e /?“£
[*40*35] D x e y>‘7:: Z> k . Prop
*40-38. k . P‘V* = s*R“ € k
J)em.
k . *37*1 . D k :: x e R*‘s‘< . = (gy) . y € . xPy
[*40 11] = s. <gy):-(ga) - at* .yea i xRy z.
[*11*0] = :. tga )a e k : (gy) .yea. xRy :.
[*37 1] 5 (go). a e k . xe R“a z.
[*40-36] = .res*R t€t K ::Dh. Prop
This proposition is frequently used in the proofs of arithmetical pro¬
positions.
*40 4. k : E!! R“0 . D . t*R“R = .7 |<g y). y e /3 . x e R*y\
This proposition is only significant when D*R C Cls.
Vein.
h . *37*6 . D k : Hp .D.R“/3 = a |(g y) .yeff.a- R*y] (1)
h.(l). *4011 . D
k :: Hp. D :.x es‘P“/9 . = : (ga): (gy). y e f3 . a = R‘y z x e a z
[*11-6] = s (gy) :y «/9 : (ga). a - P‘y • :
1*14-205] = : (g y). ye 0 .xe R*yz: D I-. Prop
*40-41. k ; E !! R“/3. D . = 7 \y e £ . . or e R‘y\ [Similar proof]
*40-42. k : (x). R*x = I u .c \j Q*x . D . s*R“a =* *‘(P“a v Q“a) = ^P^a ^ s‘(?“a
Dem.
k . *14-21 . D k : H p . D . (x). E ! R‘x . E ! P l x. E ! Q'x (1)
k . (1). *40-4 . D k : Hp. D . s‘R u a = 7 {(gy) .yea.xe R*y\
[Hp] = 7 {(gy) .yea.xe P‘y u Q*y\
[*22-34] = 7 j(gy) :yea:xe P l y .v.xe Q‘y\
[*4-4.*10-42] = 7 |(gy) . y e a . x e P‘y . v . (gy) .yea.xe Q‘y )
[< 1 ).*40-4] = 7 {x e s‘P“a . v . x e s‘G“«i
[*20-42.*22-34] = s‘P“a u s*Q“a
[*40171] = s t (P“a » Q“a) : D k . Prop
This proposition is used in *40-57, where we take R = C, P = D, Q = H.
*40 43. h :: E !! P“/9 . D :. s'R^P C a . = : y e /9 . D y . P‘y C a
Dem.
k . *37-63 . DH:: Hp . D :. y e . R l y C a : = : 7 e P“£ . D y . 7 C a :
[*40151] = : C a :.0 k . Prop
*4044. f-::E!!ii“/9.D:.aC />‘P“£ . = : y e >9 . D y . a C P‘y
Dem.
k . *37-63 . D k :: Hp . D :. y e £ . D y . a C R*y z = : 7 e . D y . a C 7 :
[*4015] =zaC p‘R“& :: D k . Prop
SECTION E]
PRODUCTS AND SUMS OF CLASSES OF CLASSES
311
The following proposition is used in the proof of *8444.
•40 45. h y < 0 .3 y . R-y c S‘y : 3 . s‘R“0 C s‘S“0
Dem.
h • *14-21 . 3 h Hp . 3 : E !! S"0 . E !! R"0 : ( 1 )
[*37 «2.*4013] D:y f fl.D„.S‘jC s‘S "0 :
C H P] D:y^.3„.fi‘yCs'S“^:
[*40-43.(1)] 3 : *‘#“,8 c s‘S “0 3 I-. Prop
The following proposition is used in the proof of *94-402.
*40 451. h :.yf0. 3„. R‘yC S‘ y: 3 .p‘K“ 0Cp‘S“ 0
Dem.
I- . *14-21 . *37-62 . *40 12.3 1-Hp . 3 : y f 0.3 . p‘R“,9 C R‘,j.
[H Pl 2.p‘R“0CS‘y.
1*40-44]^ 3 : p‘R"0Cp‘S“0 Dh . Prop
*40 5. 1-. S >R“0 = R ",3
Dem.
h . *32 12 . *40-4 Oh. s‘R“0 _ 2 ;( a y). y ( 0 . * e R' y}
[*32-18] = * i( 3 y) ■ y e 0. x/iyj
[(*37 01)] - .31-. Prop
*40 51^ I- .p‘R"0 - 5) (y . 3 y . «.Ry) [*32 12 . *40 41 . *3218]
p‘R“0 is the class of terms each of which has the relation R to every
member of 0 , just as R “0 is the class of terms each of which has the relation
R to some member of 0. In the theory of series, p‘~R“0 plays an important
part, correlative to that played by R “0 (which is by *40 5). If 0 is
a class contained in a series whose generating relation is R, p‘ll “0 will be
the predecessors of all members of 0 , while R"0 will be the predecessors of
some /9.
*40 52. V.s‘R “0 = R ‘‘0 [Proof as in *40 5 ]
*40 63. h . p ‘%‘0 _$(*«£. 3.. xRy\ [Proof as in *40-51]
*40 64. t-.p 7 R‘‘0 = i(0CR-x) [*40-51. *32-181]
*40 65. h.p'ft“a.J(aCff,) [*40-53 . *3218]
From this point onwards to *40 69, the propositions are inserted on
account of their use in the theory of series.
*40-66. I-. ,‘C“\ - F“\ [*33-5 . *40 5]
In the above proposition, the conditions of significance require that X
should be a class of relations.
*40 67. h . «‘C“X = «‘(D“X u d“X) = *‘D“X w «‘d“X [*40-42 . *3316]
312
MATHEMATICAL LOGIC
[PART I
*40 6. h . p*R "A = V . p*R" A = V [*37 29 . *40 2]
*40 61. I-: a ! &. D . p‘“/9 C 7*“/9 . p‘/F“/9 C tf“/9
Dem.
h. *3773. Dh: Hp.D.g !/?‘/9.
[*40-23] D . y/"7e“/S C $‘7?‘/9.
[*40 5] D./>‘7?“/9C/e“/9 (1)
Similarly b s Hp. D . p*R“& CR“& (2)
h.(l).(2).Dh. Prop
*40 62. H : 3 ! /9. D . p‘/?‘/9 C C"/* . //v7“/9 C C*‘/*
[*40 61 .*3715 16. *33 161]
The two following propositions (*4063-64) are used in proving *4065,
which is used in *204*63.
*40 63. I* s a ! £ - (J'A . D • p<R“0 - A
Dem.
H . *33*41 • Transp • D h : x*w CP7?. D,R‘x= A (1)
K *37-704. D./?‘xe/?“/9 (2)
h . (1) . (2). *22-32 . D h :xe/9- a*/;. D . 7?‘x e/?“£. 7?‘x = A .
[*20*57] D. Aeli“0.
[*40*22] D.p‘7?“/9-A (3)
h. (3). *1011-23. D h . Prop
*40 64. h : 3 ! £ - D*R . D . p‘7F“/9 - A [Proof as in *40 63]
*40 65. b : a ! £ - C”/* . D . p‘~R“0 = A .p‘tf“/9 = A [*40 6364. *33*16]
*40 66. b aCp'R^fi . = :xea . ye/9. 3 x , y .x/fy
Dem.
h . *40-51 . Dh:aCp‘/*“/9. e :. a C2(y e 0 . !>„. xRy)
[*20*3] = :.xea. D x :y e/9. D„ .x/fy
[*1162] = :. (x, y):. xe a. y e/9. D . x/2y ::DK Prop
*40 67. b /9 C p<R“a . = : xe a . y e/9 . . x7?y : = . a C
[Proof as in *40 66]
*40 68. b .an p*P“a C T u< p t P it a
Dem.
h . *4053 . D b x e a n p*P lt a .D:xea:yea.D y . yPx:
[*10 26] D : xPx: y e a . D y . yPx :
SECTION E]
PRODUCTS AND SUMS OF CLASSES OF CLASSES
313
[*10-24]
[*40'58.*37‘105]
This proposition is used in
D zPr : y *a . D„ . yPz :
D:.r € ?yK:.Dh.Prop
the theory of series (*2062).
*40 681. h . a a p‘P“a C P“p*P“a [Proof as in *40 68]
The following proposition is used in *21156.
*40 682. h:g!an p‘P“& . D . /9 C P“a
Dem.
h . *40-53 . D h Hp . D : (gar) : a : y e/9 . ^ :
[*5 31 ] D : (go;) : y e & , D„ .xe a.yPx :
[*11-61] D :y e/3 . D v . (g#) .xca.yPx .
I> 37-1 ] 3 y .y«P“«:.Dh.Prop
*40 69. h : g ! C‘P a p^P^a . s . g ! P. g * p?P“a
Dem.
h . *33-24 . *24-561 . D h : g ! C‘P a p*P“a . D . g ! P. g ! p**P“a (1)
h . *40 62 . ^ : 3 * a • 3 ! p‘^P“a . D . g ! C*P a p*P f *a (2)
h . *40-6 . D h a = A . D : C‘7^ a p*P“a - C‘P:
[*33-24] DsgfP.D.glOPAj/P"* (3)
H . (2) . (3) . *4-83 . D h : g ! P. g ! p‘P“a . D . g ! C‘P a ;/P“a (4)
h.(l).(4). D 1-. Prop
The above propositions concerning p‘~R“/3 and p‘fr‘/3 of course have
analogues for s‘7P‘/3 and s‘/T“0. But owing to *40 5. these analogues are
more simply stated as properties of P“/9 and R**/3. Thus, for example,
*37-264 is the_analogue of *40 67. The above propositions concerning
p‘H“0 and p‘P“/3 will be used in the theory of series, but until we reach
that stage they will seldom be referred to.
*40 7. t-.8‘af“/3 = 2[(' 3i x,y).xc a .yt0.z = x%y}
Dem.
h- *40 11 .*38-3. D
Ks‘a£«/9~2((g 7 ,y).ye/3.y = ?y“a.*e 7 )
[*38131] =2((a%*. y)-y€^.7*?y“a.®ea.2 = a:?y]
[*13-19] =2 \('&x,y) . x e a . y e 0 . z = x %y) .O .Vrop
This proposition is of considerable importance, since it gives a compact
form for the class of all values of the function x%y obtained by taking * in
the class a and y in the class >9. Thus, for example, suppose a is the class
ot numbers which are multiples of 3. and >9 is the class of numbers which
are multiples of 5, and xxy represents the arithmetical product of a; and y,
MATHEMATICAL LOGIC [PART 1
then 6*ax“/3 will be the class of products of multiples of 3 and multiples
• »f •>, ?.e». the class of multiples of 15. Again suppose a and are both classes
of relations; then will be all relative products R' S obtained by
choosing R in the class a and .S' in the class /?.
*40*71. V . .*< ?y“* = (s*k)$ ,j = ? y‘V*
I Jem .
V . *40-38 . *38*31 y“* = ?y“s‘*
[*38*2) = (s*k) ? y. D f- . Prop
Ihe hypothesis R“a Co, which appears in *40*8*81, is one which plays
an important part at a later stage. In the theory of induction (Part II,
Section E) it characterizes a hereditary class, and in the theory of scries it
characterizes an upper .section (when combined with a C C‘R).
*40 8 h a e * . I>. . R“a C a : D . R* V* C
J)em.
h . *37*171 . D h :: Hp .D:.a««.D a :xea. .r/fy. .yea:.
[*11*62] D :. ae k .xt a .xRy . yea:.
[*4013]
[*4011 .* 10*23] D :.x c.s‘* . xRy . . yes**
[*37*171] D ft-V*Cs‘* :: D h . Prop
*40*81. I-:. a e * . D. . R“a C a : D . /{“/>'* C y‘*
Dem.
h . *37*171 . D h Hp . D :: a e * . D : x e a . x/fy. D . y e a ::
[Exp.Comm] D :: x/?y .D:.ae«.D:«ea. D.yea:.
[*2 77] D :. a e * . D . xe a : D : a e *. D . y e a (1)
h . (1). *1011*21*27 . D
h Hp.D zzxRy . D:.ae*.D«.xea:D:ae*.3«.yea:.
D:.ar e/>‘* ■ D . y e/>‘* ::
[Imp] D :: xcp‘* .xRy . D . y e y‘*
h. (2). *37*171 . DH. Prop
(2)
*41. THE PRODUCT AND SUM OF A CLASS OF RELATIONS
Summary of * 41.
Lhe propositions to be given in this number, down to *41 3 exclusive, are
the analogues of those of *40, excluding those from *403 onwards, which
have no analogues. Proofs will not be given, in this number, when they are
exactly analogous to those of propositions with the same decimal part in * 40 .
l he smaller importance of p*\ and *‘X. as compared with p‘\ and s‘X. is
illustrated by the smaller number of propositions in *41 as compared with
*40.
Our definitions are
*4101. Df
*4102. a‘X=££| (nR).Re\.xRy] Df
Of the propositions preceding *41 3, which are analogues of propositions
m *40, the only two that are frequently used are
*4113.
*41161. h.VXGS.miRcX.^ji.RGS
Of the remaining propositions of this number, which have no analogues
in *40, the most important are *4143 44 45, namely
D‘s‘X = s‘D“X, a‘i‘X = s‘CI“X, CVX=:^C“X.
These propositions are constantly required in the theory of selections (Part II,
Section D) and in relation-arithmetic. Most of the other propositions of this
number are used only once or not at all.
•4101. p‘\-H$(R e \.0 R .xRy) Df
•4102. i‘\~S${( a R).R f \. x Ry\ Df
•411. b:.x(J,‘\)j/.s :R t \.0 B .xR v
*41-11. I -:x(i‘\)y. = .( 3 R).R t x.xR v
*4112. I -:Re\.O.p‘\CR
*41-13. 1 -:Re\.3.RGi‘\
*41-14. \-:R e \.x(p‘\) v .O.xRy
*41141. ^:Re\.xRy.O .x(i‘\)y
*41-16. hi.Sep‘\.= -R t \.0 B .SCR
*41161. \-t.i‘\GS.s-.Re\.0 B .RGS
*4116, h:XC/i.3, P*P Gy'X
*41161. h:XC)i.D. s‘\ C i‘p
*4117. H . p‘\ im pV Gj‘(Xn p)
316
MATHEMATICAL LOGIC
[PART I
*41171. H.s‘Xo.v‘/i = «‘(X w /i)
*41 18. h .//(X ^ n) = //X rt j/fx
*41181. h.i‘(\ft/i)G4 , ‘Xrts‘/i
*4119. h :: r (*‘X) y. = R € X . D* . 7? G S : D*. x£y
*412. hsX- A.D.yVX- V
*4121. H : X = A . D. i*X = A
*4122. h:A<X.D./>‘X = A
*41221. H : V € X . D . .*‘X = V
*41 23. h : g ! X • D . yVX G *‘X
*41 24. 1-a ! X: 7f « X . D,...S’G7? : D . »S G i‘X
*41 25. h s .r(.v‘X)y . = .g!Xn R(j-Ry)
*41 26. h : g ! «‘X • s . (g/?) • 7? e X • g ! 7?
*41 27. h P A .v‘X ■ A . s : 7f « X. D* . P n 7J A
*41 3. h . Cnv'yVX = y>‘Cnv“X
Dein.
h . *31131 . Z>
h :.y (Cnv*yVX)x . = :x(/VX) y :
[*411] = : P c X . D/f. xPy :
[*31131] z : /?«X. D/?. y (Cnv‘7?)x:
[*37 63.*31*13] = : 7 J c Cnv“X . D,.. y/'x :
[*411] = : y (yVCnv“X)x :. D h . Prop
*4131. h . Cnv'^X — i‘Cnv“X [Proof as in *41*3]
*41 32. 1-. Cnv“//‘* - y>“Cnv‘“* [*41 3 . *37*354]
*41 33. h . Cnv* W = *"Cnv‘“* [*41*31 . *37*354)
*4134. h..v‘a1 “X = «1#‘X
I Jem.
h .*4111 .*38*13 . *13*195. D h x(*‘a "J “X)y . = : (gP). Pt X .x(a1 P)y :
[*35*1] = : (gP) .PeX.xea.xPy:
[*10*35] = : x € a: (gP). P c X. xPy :
[*41*11.*35*1] = : x(a 1 s‘X)y3 H . P™P
*41*341. H. i* fa“X = (s‘X)f*a [Proof as in *41*34]
*41342. Ks‘[a“X = (s‘X)Ca
jDent.
h.*36*11 .*35*21 .DK.i‘ta“X = 6*‘a1 “fa“X
[*41*34] = a] (i‘f* a“X)
[*41*341] =a1(i < X)f‘a
[*36*11] ' =» (s*\) ^a.Df*. Prop
SECTION E] THE PRODUCT AND SUM OF
A CLASS OF RELATIONS
317
The following proposition is used in *85 *22.
*41 35. h . s*M [“« = M fs*K
Dent.
h .*4111 .*3813 . D ..c(j!/[*a)y .
[*35101] =.(3 «)-aeK.!/€a.xMy.
[*40 ll.*3510l] = y:D(-. P rop
*41351. = [Proof as in *41-35]
*414. h . D‘/>‘X Cp‘D“X
Dem.
h. *3313. D
h ::a?«D‘/>‘X. =
[*411] =
[*11-61] D
[*3313] D
[*40-41.*33 12] D
May) :.
(3y) : -/i € X. D* . xRy :.
:. 7? e X . D* . (ay). */2y
:. X. D* . art D‘72 :.
:. xe p'D^X :: D h . Prop
*4141. f-.(l‘;V\Cp‘a“X [Proofas in *414]
*41-42. h . C‘p‘X Cp*C“\
Dem.
K *33 132 .D I-::
[*411]
[*10-41-221]
[*4-78]
[*11-61]
[*33132]
[*4041 .*33* 122]
. a: e C‘;VX . = :: (ay) : a: (p‘\) y . v . y (p‘X) x ::
= : : (ay): : R * X . Z) n • xRy :v:R € \.D K . y R x : :
^ :: (3y) :: W s. 7£cX.} .xRyz v z Re\. D.ylixzz
3 :: (ay) s: (R) J%«X. D : x Ry . v . y/fcr ::
D :: (/£) :: R e X . } : (ay) : xRy . v . yRx z
3 : xe C‘R zz
D :: xep‘C“\ D h . Prop
*4143. \- .T>*s t \ = s‘D“\
Dem.
*4144.
*4146.
Dem.
h . *33"13 . Dh.a:e D‘i‘X.
[*4111]
[*11-23-55]
[*3313]
[*40-4.*33 12]
(ay)-*(*‘*)y:
(ay) : (a^) - Re\. xRy z
(a R ) i Rc\: (ay) • xRy :
(aR).R€\.xeD‘R:
a€s‘D“X :. D H . Prop
h . CPi‘X = s‘(I“X [Proof as in *4143]
H . *3316 . D I-. C‘i‘X = D‘i‘X w d‘*‘X
[*4143 44] = *‘D“X w «‘<I“X
[*40-57] = *‘C“X. D I- . Prop
318
MATHEMATICAL LOGIC
[PART I
■*415. h . yVX lYfiG/Vs'X “u
»»
Dem.
f- .*34 1 . D
h :: x(//X /V/a)
2 . = :. (gy). x(/VX)y . y (/>»*
[*41*1]
= (gy) z.PeX.D?. xPy zQ (>a . D v . y<?*
[♦11*50]
= :• <3.'/> s. (P. (?): P « X. D . xPy: <? ey . D. yQz :.
[*11*37*39]
3 <3P) :• (P, (?): P € X . (? €/a . D . xPy. yQz
[♦11*61]
D (P, Q) s. P € X . Q e y . D . (gy) . .rPy . y(?* .
[*34l]
D..r(P|(?)2i.
[♦13191]
D (P. (?. P)P« X .(?«/*. P = Pj (?. D .xRz
(* 11*21-35]
D:.(P):(gP, (?). Pc X . (?* M . P = P| (?. D .xP;
[*40-7]
D s.(P): Pcs'X “y.’S.xRzz.
tt
[*411]
D:.x(/>VX #< /a) x :: D h . Prop
*4151. Ks‘X|*y
= iVXi‘V
■ I
Dem.
h . *34*1 . D
1- ::x(«*X | r
. = :.(gy).*(PX)y.y(*V)*:.
[*41*11]
5 ! * (ay) s - ( 3 P) • p «* • j Py : (3<?) • <? * p • y<?* ! *
[*11*54]
= (3y) •• OP* (?): p € X . xPy .Qcy. yQz
[*11*24-27]
= (gP, f?) :.*(gy ). PtX. xPy .Qey. yQz
[*10 35]
2 (gP, (?):. Pc X. (? c /a s (gy). xPy. y<?*
[*341]
a s.OP. (?) : P « X . (? € /a . x (P | (?) 2
[*13195]
a :.(3P* (?. P). P«X. (?€/a- P = P| Q.xRz
[*U-24.*40-7]
= :.(gP). Res'X “y . xP*
[*41*11]
= :. x (iVX *V) * ss ^ H • Prop
• a
The above proposition, which is used in *92 31, states that, if X and /a are
classes of relations, the relative product of the relational sum of X and the
relational sum of y. is the relational sum of all the relative products formed
of a member of X and a member of /a.
The following proposition is used in *96111.
*41-62. l-s.al^XGQ.srPeX.Dp.alPCQ
Dem.
K *351 .*4111. D
h :: a*] 5‘X GQ. = :.x<a: (gP). P c X. xPy : D*,,,. x(?y
[*10*35*23] = xea . P eX.xPy . Dj» tX>y . xQ)y
[*351] = :• P eX.x(a]P)y. Dp,*,* • xQy :•
[*11*62] = P c X. Dp. a] P G (? :: D h . Prop
SECTION E] THE PRODUCT AND SUM OF
A CLASS OF RELATIONS
310
The following proposition is used in *102 32 and in *160 401.
*41-6. 1 -,J 6 £ . D y . 0 #* t/ . D . ,s*p«0 = t‘Q«0 ^
Bern.
h .*37 6. *14-21 .*41 11 .*13 195. D
h :: Hp. D : . w(«‘P“£) v . = : (gy) . u € £. u ( P‘ y ) „ .
f 1Ip J = : (ay) •yffi.u {Q‘y KJ R*It) V :
3: <a*> • u e 0 •" W#)»• v • (ay) • y • e .«(/?<-/> v :
[*3/ G.*41 11] = . U (i 4 Q**0) v.v.u v :: D h . Prop
*42. MISCELLANEOUS PROPOSITIONS
Sit m inn ry of *4 2.
The present number contains various propositions concerning products and
sums of classes. They are concerned chiefly with classes of classes of classes,
or with relations of relations of relations. These .are required respectively in
cardinal and in ordinal arithmetic. Thus *421 is used in *112 and *113,
which are concerned with cardinal addition and multiplication, while *42*12*2
are used in *1(50 and *l(i2, which are concerned with ordinal addition. *42 22,
though not explicitly referred to, is useful in facilitating the comprehension of
propositions on series of series of series, or rather on relations between relations
between relations, which are required in connection with the associative law
of multiplication in relation-arithmetic.
*421. K*V‘*-*V*
Here k must, for significance, be a class of classes of classes. The proposi¬
tion states that if we take each member, a, of *, and form s f a, and then form
the sum of all the classes so obtained, the result is the same as if we form the
sum of the sum of *. This is the associative law for s, and is (as will appear
later) the source of the associative law of addition in cardinal arithmetic. The
way in which this proposition comes to be the associative law for 5 may be
seen as follows: Suppose k consists of two classes, a and suppose a in turn
consists of the two classes £ and y, and £ of the two classes £' and y. Then
6‘a ■* { v y . « £' v y. (This will be proved later.) Thus *•“* has two
members, one of which is f v y, while the other is f v y . Thus
s‘s“x = ((sjy)u(( , \jy).
But s*x has four members, namely £. y, g, y. Thus ***** = g v y u l; v y .
Thus our proposition leads to
which is obviously a case of the associative law.
Our proposition states the associative law generally, including the case
where the number of brackets, or of summands in any bracket, is infinite.
The proof is as follows.
Dem.
h . *404 . D h :: x € ****** . = (ga) .ae/c.xes'ai.
[*4011] = :.(ga):ae*:(g£).£ea.*e£:.
[*11*6] = :. (g|) (ga) .ae/c.fea:*ef:.
[*4011] = :-(gf)*f cs‘K.xe( 2 .
[*4011] = z.xt ***** :: D h. Prop
SECTION El
MISCELLANEOUS PROPOSITIONS
*4211. H . p t p t *x = p*s‘fc
Deni.
I - • *40 41 . D h x € p i p li K . =
[*40 1.*1162] =
[*11*2.*1023] =
[*4011] =
[*40 1] =
& € « • ^ .A'cp'Q :
.ye/3 . .xey:
<3£> • fi < k . 7 e (3 . D y . .i- € y :
y e s € k . D y . x e 7 :
X€p , s*K :.DK Prop
This is the associative law for products. Supposing again, for illustration,
that * consists of the two classes a, 0. while a consists of the two classes f. ,,
and /3 of the two classes f. then />“* consists of the two classes £ n ,, and
fn, ’So that />'/>“* - (f - V) « <f n V). while p‘s‘ K - f „ „ f' „ y. Thus
our proposition becomes
(£ A *?) ** (£' a V) — £ n *7 n £' r\ tf.
A descriptive function whose arguments are classes or classes of classes
may be said to obey the associative law provided
R‘R“* = R‘s‘k.
This equation may be interpreted as follows: Given a class «, divide it
into any number of subordinate classes, so that no member is left out, though
one member may belong to two or more classes. Let the classes into which
“ " d :r ,d0 ?. make “P. the class *■ 80 that * is a class of classes, and ,*« = a.
X hen the above equation asserts that if we first form the R ‘s of the various
sub-classes of -. and then the R of the resulting class, the result is the same
as if we formed the R of a directly.
In some cases-for example, that of arithmetical addition of cardinals-
the above equation holds only when no two members of * have a common
term, «. e . when the parts into which a is divided are mutually exclusive.
shalffi A de ’ C , r i ! PtiV ® fUn f ion L wh °se arguments are relations of relations, we
shall find another form for the associative law; this form plays in ordinal
arithmetic “ analo K ous 10 th »t played by the above form in cardinal
*4212. =
Dem.
h . *4111 . D h : * (s‘*“X) y . = . ( a „) . x(i‘p.)y .
r [ * 41 ' 11] =-(a
C* 40 ”] >.(H P).P"‘K.xPy.
[ * 4111 ] =-*(iVX)y:3l-.Prop
•4213. y .p , p“\ = p‘g‘\
Dem.
rirtriV • 3 h ■- 1 &‘p“V *■■ = ** ■ 3.•* (p» y:
.. w ^ xRy :
21
322
MATHEMATICAL LOGIC
[PART I
*42 2
[*U-2.*10 23]
[*4011]
[*+n]
= : (g/x ). /ie \ . R € ft .Or . xfty :
= : ft e s*\ .0/i. xfty :
= : x(/»VX)y :.0\-. Prop
. c*vop = . 9 * c“c<p = = /■*«/>
This proposition assumes that P is a relation between relations. For
example, suppose we have a series of series, whose generating relations are
ordered by the relation P. Then C*P is the class of these generating relations;
s'C'P is the relation "one or other of the generating relations which compose
C'P; and C‘s‘C*P is the class of all the terms occurring in any of the series.
C U C*P is the fields of the various series, and s i C t, C i P is again all the terms
occurring in any of the series. F tt C t P is all the terms belonging to fields of
series which are members of C*P, and F*‘P is all members of fields of members
of the field of P: each of these again is all the terms occurring in any of the
series. The proof is as follows:
Dem.
h . *4 1 45 . 0 h . WP - s*C“C‘P ( 1 )
h . *40-56 .Dh. s'C u C l P - F“C*P (2)
K*335. 0V.F“C*P
[*37-38] =~F 3t P (»)
h.(1).(2).(3).D h . Prop
The following propositions apply to a relation of relations of relations.
These propositions are useful for proving associative laws in ordinal arith¬
metic, since these laws deal with scries of scries of series, ami series of scries
of series are most simply constituted by supposing the generating relations of
the constituent series to be ordered by relations which are themselves ordered
by a relation P.
*42 21. h . = C‘= C“C‘s‘C < / > = C“F“C‘P = C^F^P
Dem.
h . *40-38. 0 h . s‘C‘“C“C‘P = C“s*C“C‘P (1)
K (1). *42-2. Ob. Prop
*42 22. h . sVC“‘C“C‘P = s‘C“s‘C“C*P = s‘C“C‘s‘C*P
= C‘s*C‘s*C‘P = s t C tt F ti C*P
= F^F^&P = F‘*F 2 *P =~F*‘P
[*42-21 . *41-45 . *40 56 . *42 2 . *37 3]
If P, in the above proposition, is a relation which generates a series of
series of series, the above gives various forms for the class of ultimate terms
of these series. Thus suppose Q e C‘P; then Q is a relation between generat ing
SECTION E]
MISCELLANEOUS PROPOSITIONS
323
relations of series. If now It e C‘Q, It is the generating relation of a series
Which we may regard as composed of individuals. The class of individuals so
obtainable may be expressed in any of the above forms, as well as in others
which are not given above.
*42 3. h . s‘s“?“o = s*R“a
Dem.
^ . *421 . D I- . s‘s“R“a =
[*4°-5] = s € R tf a . D I-. Prop
*42 31. h . = s *R“a [Proof as in *42 3]
21—2
*43. THE RELATIONS OF A RELATIVE PRODUCT
TO ITS FACTORS
Summary of *43.
The purpose of the present number is to give certain propositions on the
relation which holds between P and Q whenever P = Q | R, or whenever
R=R Q, or whenever P—R Q S, where R and S are fixed. In virtue of
the general definitions of *38, these relations are respectively | R, R |, and
< ft ) I (I &)• Such relations are of great utility both in cardinal and in ordinal
arithmetic; they are also much used in the theory of induction (Part II,
Section E). In place of the notation (7f |) ( S), which is cumbrous, we adopt
the more compact notation R S. If X is a class of relations, R |“X will be the
class of relations R P where P eX, | R**\ will be the class of relations P\R
where P * X, and ( R || «S')“X will lie the class of relations R | P |S where Pe\.
These classes of relations are often required in subsequent work.
In virtue of our definitions, we have
*43112. K(7i||S)‘Q-fl|Q|S
The propositions most used in the present number (except such as merely
embody definitions) are the following:
*43 302. K(P).P«(I‘(7J||S)
*43411. h. 7 P“(i“x = (i“|P“x
*43 421. H..v‘|7e << X = (s < X)|^
The remaining propositions arc used seldom, but their uses, when they are
used, are important.
*43 01. R\\S- (721)1(1 S) Df
At a later stage (in *150) we shall introduce a simpler notation for the
special case of P||72. The following propositions are for the most part
immediate consequences of the definitions, and proofs are therefore usually
omitted.
*431. h:P(7?!)Q. = .P = 72|Q
*43101. h:P(|P)Q. = .P=Q|7*
*43102. ^•:P(7^|| ^ $)Q. = .P=7^|Q|«S ,
*4311. V.R\*Q = R\Q
*43111. \-.\R l Q=Q\R
*43112. h.(P||S)‘Q = 7*|Q|S
*4312. h.E!7e|‘Q
SECTION E] RELATIONS OP A RELATIVE PRODUCT TO ITS FACTORS 325
*43121. h.E!|.ft‘Q
*43122. h.E!(.R||S)<Q
*432. t-.(R\)\(S\) = (R\S)\
Dem.
I-. *43-1 . D h : L {(«|) | (S|)J N . = . < a J /). I. = R | M . M = S I N.
[*13195.*34-21] = .L-R\S\N.
t* 43 ' 1 ] s.Z|(iJ l S)|)JV:Dh.P, 0 p
*43 201. I-.(|«)|(|S) = |(S|B) [Proofas in *432]
*43-202. H.(|ii)|(S|) = (S|)|(|ie) = S||ii [Proof as in *43 2 ]
*43-21. h.(P||Q)|(«|)-(P|U)||Q
*43 211. I- • (R |) | (P || Q) = (72 | P) || Q
*43-212. M-P||<3)|<|72)-P||(72|Q)
*43 213. l-.(|i2)|(P||Q)-P||(Q|72)
*43-22. h . (P || Q) | (R || S ) = (P | R) || (S | Q)
*43 3. H . (P) . P « a ‘R | [*4312 . *3343]
*43 301. K(P).P«<I‘j72
*43 302. K(P).P«a‘(fl||S)
*4331. h.Pfa‘ie| = Pf C‘R\ = P
Dem.
K *43-12. *33-431. D I-. d‘P C d‘721 (1)
[*33161] DKd‘PCC‘72| (2)
H.(l).(2).*35-452.31-. Prop
*43311. l-.Pra‘|P = PrC'|fi = P
*43 312. I- . P r d‘(/2 || S) = P r C‘(R || S) - P
*4334. h.P|‘P = |P‘P = iJ> [*4311111]
*43 4. I-.72««D‘P = D‘72|‘P [*37-32. *431]
*43 401. h . R“d‘P = a* | R-P [*37-32. *43101]
*43 41. K72“‘D*«\ = D“72|‘‘A. [*43-4. *37 355]
*43 411. h . J2‘“d“\ = d“ | R“\ [*43-401 . *37 355]
*43 42. t-.*‘«|“X = «|i‘x
Dem.
• *4111. *371. *43 1 . D
[*341] = . (aT) : T e X ; (ay). x R y . y Tz :
[*11-6] = : (gy) . xRy . ^ T) mT€Xm Tz .
[*4111 . *341] =zx(R\ i‘\) rr.DH. p rop
320
MATHEMATICAL LOGIC
[PART I
*43421. P.i‘ if“X = (i‘X) R [Proofas in *43 42]
*43 43 P . i‘< R |l .S')“X = (R || ,V)S‘X
lion.
H . *37 33 . D h . i‘< R || S)“\ - “ S“\
[*«•«] = /?(«* |S“X)
[•43-421] = R; s‘X 15
[•*3112] = (R||,S')‘i‘x Prop
*43 48. P : l)‘7*Ca.D.<? */*— (<?[■«) 1*7* [•35-481]
*43 481. P : <I‘/'C/3. D . 7f‘7’ = <>31 [*3548]
*4349. P : s‘E)‘*X C a . D . «/ )T X “ [(Qfo) 11 T x
Dew.
P. *4043. DP:. Hp. D : P t X . D . V‘P C a .
f* 43 ' 48 ] 3.$|'7 > -[(Qr«)l!‘^ > (!)
P.(l). *35-71 .DP.Prop
*43491. P:*«a“XC/9.D.(|R)|-X= [|(/91R)|fX [Proofns in *4349]
*43 5. P : IV/'C o . (PRCR. D .(Q|| R)‘P — (((?[• o)||(/91 R)}‘P
[*35-48*481 .*43112]
*43*51. h : s‘D“\ C a . *‘d“\ C 0 . D . (Q || R) f* \ - |(Q f «)|| (0 ] 7?)) f \
I)em.
h • *40-43 . D h Hp. D : P « \ . D . D*P C o . Cl ‘P C £.
t* +:i ' 5 ] ^ • <Q ||RyPm [(Qf-a)||(/9] R)\‘P (1)
P.(l). *35-71. DP. Prop
The above proposition is used in the proof of *74773.
PART II
PROLEGOMENA TO CARDINAL ARITHMETIC
SUMMARY OF PART II
The objects to be studied in this Part are not sharply distinguished from
those studied in Part I. The difference is one of degree, the objects in this
Part being of somewhat less general importance than those of Part I, and
being studied more on account of their bearing on cardinal arithmetic than
on their own account. Although cardinal arithmetic is the goal which
determines our course in Part II. all the objects studied will be found to be
also required in ordinal arithmetic and the theory of series. As this Part
advances, the approach to cardinal arithmetic becomes gradually more marked,
until at last nothing is lacking except the definition of cardinal numbers, with
which Part III opens.
Section A of this Part deals with unit classes and couples. A unit class
is the class of terms identical with a given term, i.e. the class whose only
member is the given term. (As explained in the Introduction, Chapter III,
pp. 76 to 79, the class whose only member is x is not identical with x.) We
define 1 as the class of all unit classes, leaving it to Part III to show that 1.
so defined, is a cardinal number. In like manner, we define a (cardinal or
ordinal) couple, and then define 2 as the class of all couples. The propositions
on couples will not be much referred to in the remainder of the present Part,
since their use belongs chiefly to arithmetic (Parts III and IV). On the other
hand, the properties of unit classes are constantly required in Sections C. D E
of this Part.
Section B deals, first, with the class of sub-classes of a given class, i.e. of
classes contained in a given class. The sub-classes of a given class are often
important in arithmetic. Next we consider the class of sub-relations of a
given relation, i.e. relations contained in a given relation. The propositions
on this subject arc analogous to those on sub-classes, but less important.
Next we consider the question of “relative types,” i.e. taking any object*, and
calling its type t‘x, we give a notation for expressing in terms of t l x the type
of classes of which * is a member, or of relations in which * may be either
referent or relatum. and so on. The notations introduced in this connection
are very useful in arithmetic, especially in connection with existence-theorems.
ut the propositions of Section B are very seldom required in the later sections
of the present Part.
Section C, which deals with one-many, many-one and one-one relations,
is very important, and is constantly relevant in the sequel. A relation is
one-many when no term has more than one referent, many-one if no term has
more than one relatum, and one-one if it is both one-many and many-one.
330
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
In this section, we define the notion of similarity, upon which all cardinal
arithmetic is based: two classes are said to be similar when there is a one-one
relation whose domain is the one and whose converse domain is the other.
We prove the elementary properties of similarity, including the Schroder-
Bernstein theorem, namely: If a is similar to part of /9, and /3 is similar to
part of a, then a is similar to
Section I) deals with the notion of selections, upon which both cardinal
and ordinal multiplication are based. A selection from a set of classes is
a class consisting of one member from each class of the set. Thus a selective
relation It may be defined as one which, for a given class of classes k, makes
Il'a a member of a whenever a is a member of k. More exactly, a selective
relation for a class of classes * is one which is one-many, which has k for its
converse domain, and is such that, if .rlia, then xea. Such a relation may
be called an c-sclector from k. More generally, we may define a /^-selector
from k as a relation which is one-many, which has * for its converse domain,
and which is contained in P. The theory of selectors is very important in
arithmetic. But until we come to cardinal multiplication in PartHI, Section B,
the propositions of this fourth section will seldom be relevant.
Section E deals with mathematical induction, not in the special form in
which it applies to finite integers (this is considered in Part III, Section C),
but in a general form in which it applies to all relations. The propositions
of this section are of very great importance, primarily in the theory of finite
and infinite (Part III, Section C, and Part V, Section E), but also in many
other subjects, and especially in the derivation of series from one-many,
many-one or one-one relations—for example, in ordering the "rational” points
of a projective space by means of successive constructions of harmonic points.
The ideas involved in this section arc somewhat complicated, and we must
refer the reader to the section itself for an account of them.
SECTION A
UNIT CLASSES AND COUPLES
Summary of Section A.
Iu this sectioD we begin (»50) by introducing a notation for the relation
of identity, as opposed to the function “.r-y”; that is, calling the relation of
identity I, we put
I = xy (x = y) Df.
Tlie purpose of this definition is chiefly convenience of notation. The
definition enables us to speak off D‘/, I\R, o']/, etc., which we could
not otherwise do.
At the same time we introduce diversity, which is defined as the negation
of identity, and denoted by the letter J. The properties of I and J result
immediately from *13, since
xly . = . x =* y.
We next introduce a very important notation, due to Peano, for the class
whose only member is *. If we took a strictly and purely extensional view of
classes, we should naturally suppose this class to be identical with x. But in
view of the theory of classes explained in *20. it is plain that * can never be
identical with a class of which it is a member, even when it is the only member
of that class. Peano uses the notation “ lx" for the class whose only member
is x; we shall alter this to “l‘x ” following our general notation for descriptive
functions. Thus we are to have
= P (y ~r) - p (ylx) =7‘x.
Hence we take as our definition
1=7 Df,
since this definition gives the desired value of i*x. The properties of i are
many and important.
It is important to observe that “7v means "the only member of a.” Thus
it exists when, and only when, a has one member and no more, in which case
a is of the form t‘<cjf * is its only member. Thus "7‘a" means the same as
"(J*) (*««).” and (<#>*)" means the same as " (>*) (^*).” What we call
"l*a is denoted, in Peano’s notation, by "ja.”
Classes of the form t*x are called unit classes, and the class of all such
classes is called 1. This is the cardinal number 1, according to the definition
of cardinal numbers which will be given in *100. The properties of 1, so far
as they do not depend upon other cardinals, or upon the fact that 1 is a
cardinal, will be studied in * 52 .
332
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
After a number (*53) containing various propositions involving 1 or t, we
pass to the consideration of cardinal couples (*54) and ordinal couples (*55).
A cardinal couple is a class i l x v i*y, where x\y. The class of such couples
is defined as 2, and will be shown at a later stage (*101) to be a cardinal
number. An ordinal couple, which, unlike a cardinal couple, involves an older
as between its members, is defined as a relation i‘x\ i*y (cf. *35 04), where
we may either add x % y or not. The properties of ordinal couples are in part
analogous to those of unit classes, in part to those of cardinal couples. In *56,
we define the ordinal number 2 (which we denote by 2 r , to distinguish it from
the cardinal 2) as the class of all ordinal couples i‘x f i*y, where x^y. It will
be shown at a later stage that this is an ordinal number according to our
definition of ordinal numbers (*153 and *251).
*50. IDENTITY AND DIVERSITY AS RELATIONS
Summary of# 50.
The purpose of the present number is primarily nofcationul. For notational
reasons, we must be able to express identity and diversity as relations, and not
merely as propositional functions, i.e. we require a notation for .71) (x = >,) and
xy (x ^ y). We therefore put
I = xj) (x = y) Df.
Df.
In spite of the fact that diversity is merely the negation of identity, the
kinds of propositions that employ diversity are quite different from the kinds
that employ identity. Identity as a relation is required, to begin with, in the
theory of unit classes, which is our reason for treating of it at this stage. It
is next required, constantly, in the theory of mathematical induction (Part II,
Section E). It is required also in showing that cardinal and ordinal similarity
are reflexive. These are its principal uses.
Diversity, on the other hand, is required almost exclusively in the theory
of scries (Part V). and the first number in that theory will be devoted to
diversity. Until that stage, diversity will seldom be referred to, with one
important exception, namely in proving the associative law of multiplication
m relation-arithmetic (#174).
follmving 1081 in,p ° rta,,t P ro P° sition s on identity in the present number are the
*6016. h . /“ a = a
*504. V .R\I = I\R = R
*50 5.
*50-51. I-. Cnv‘(a *| 7) «= a-] 7
*60 52. I-. D‘(a *] 7) - <3‘(a 17) - C‘(a *] 7) = a
*6062. h:a‘RCa.D.«|(/f*a)=/i
*60 63. h : D ‘R C a . 2.1 f a \R = R
the following lmP ° rtant propositions on divereit y in the present number are
*6023. hRGJ. = .R(zj
*6024. \-zRQ.J. = .( x) .~( xRx)
*6043. = A
*6046. h : R*GJ .D .RQJ
*5047.
334
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
It will be observed that all these propositions are concerned with RG J or
R-GJ, both of which are sati.-fied if R is a serial relation. The hypothesis
1< : G J or R r\ R — A characterizes an asymmetrical relation, i.e. one which, if
it holds between .<* and y, cannot hold between y and .r.
*50 01. 7 = .?y(x = y) Df
*50 02. J - - / Df
Most of the propositions of this number are obvious, and call for no
comment.
*501. b:x/y. = ..r = y
[*213. (*50*01)] *
*5011.
h : uJy . s . x + y
[*23*35 . *50* 1 . (*50 02)]
*5012.
h . 7 = 7y (x + y)
[*50*11 .*21*33]
*5013
h.g!/
[*13*19. *10 24 281 . *501)
*5014.
[*30*3. *50*1 .*10*11)
*5015.
K(y).E! I'.j
[*50*14.*1421 .*10 11]
*5016.
K/“a-a
Dem.
b . *37 l . D h zxe /“a . s . (gy). yca.x/y .
[*50*1 J = .(gy).y«? a.x = y
[*13*195] = . xe a : D h . Prop
*5017. h x ea . D x . 7?‘x»x: D . R
Dem.
h. *14-21 . D h : Hp. D . E !! R“a
h . *5014 . D h :. Hp.D:xca. D x • R tjc ™ I * x!
[*37*60.(1)] D: R“a = I“az
[*5016] D : R“a = a:.Dh. Prop
*502. h
Dem.
V
/ = /
h . *501 . D h : xly • s . x = y •
[*13*16] = . y = x.
[*501] = . y/x.
[*31*11] = . x/y Oh. Prop
*50 21. H../W
Dem.
h . *21 2 . (*50 02). D h . */= — /
[*50‘2.*23*83] = - /
[*31*16] =Cnv‘x_/
[(1).*31*32] = J. D h . Prop
( 1 )
(1)
SECTION A]
IDENTITY AND DIVERSITY AS RELATIONS
335
*50 22. h : Ii G /. = . it c / [*314. »50 ->]
*50 23. h-.RGJ.s.RGJ [*31-4. *50-21]
*50 24. I ■: RGJ.s. ~
Deni.
K *50-11= +
[Transp] = : a- = y . D ,, . ~ (.<%> :
[*13191] s : <*). ~ (xRi) :.3K Prop
*50 3. H . (x). xlx [*501 . *1315]
*50 31. t-. D‘/ - V . a*/ = V
Dent.
*5032.
*5033.
Bern ,
H . *50 3 . *10-24 . D H (*) ; (g/,) . x I,j ( x ). ( ay ) # y / >4 .
[*3313131] D t-: (*). * e D‘I : (x) .xe(l‘f:
[*2414] D P . D'/. V . Q‘I - V . D h . Prop
h . C‘I - V [*50-31 . *33 16 . *24-27]
ha!j.3.Dv=v.a‘/.v,w= v
P . *13-171 . Transp .SHt.y + z.Dix + y.v.ir + j:.
[*5011] DH:.y/r.D:r./,.v. *JV :
[*3314] O-.xcXVJ (1)
•"•(!)• *11-11-35. 3h jiy.D.te jyj :
[*1011-21] 9Ha!/.3.(*).«,DV.
t* 24,14 ] D.DV=V ( 2)
h . (2) . *50-21 .Dh. Prop
r “ the nb «'’e Proposition (.50-33), the hypothesis g ! J is equivalent to
the hypothesis that more than one object exists of the type in question. This
can be proved for all except the lowest type. For the lowest type, we can
only prove the existence of at least one object: this is proved in *24-52. Foi¬
led v X ^ tyPe ' WC T"- T ,h ° exUtencc of two objects, namely A
and V, these are distinct, by *241. For the next type, we can prove the
existence of 2- objects; for the next. 2*, etc. But for the class of individuals
we cannot prove, from our primitive propositions, that there is more than
one object ,n the universe, and therefore we cannot prove -j'.J. We might
UiaTmTwe H r ‘ a a T° n , g ° Ur P rimitive Propositions the assumption
wo^Uoi^eh"* md,V,dUal ° r S ° me *«***•
(a^*- x > y) • <t >! x . ~ <f >! y.
fhil T-° f th< a pr0p ? siti0ns which we mi g ht 'Vsh to prove depend upon
this assumption, and we have therefore excluded it. It should be observed
that many philosophers, being monists, deny this assumption
336
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*50-34. h . g !./ t Cls
Dem.
K . *20 41. *22 38 . (*2401 02). D h . A, V e Cls.
[*24’1] Dh.A + V. A, V«Cls.
[*3(>13.*5011] Dh.A {,/[ Clsj V .
[* 10'24] D H . Prop
*50 35. h . a ! J t Re I (Proof as in *50 34)
*50 4. h./?|/=/ R=R
Dem.
H . *341 . D h :.r(7? 1)2. = .(qy) .xRy. y/z.
[*50 1] s.(gy).j Ry.y = z.
[*13105] = .xRz (1)
y . *34-1 . D H : x (/1 72) «. = • (ay) • -r/y - yR* •
(#50-1 ] =. (ay) • * = y • yBx .
[*13195] m.xRz (2)
h.(l).(2).Dh. Prop
*50 41. y : R \ P G J. = . R \ P G J. = . R r> P = A
Dem.
h . *34*1. *501 l.Df-:. R\PGJ . = : (gy). xRy. yPz. D x> , .x + z:
= :(x):~ (gy). xRy. yPx :
= : ~ (gx, y) . ar% . y-P*:
» :~(3*. y).xRy.xPy:
w.RfsP- A: (1)
= :RhP- A:
= : R | Cnv'P G «/:
= :fl|PG./ (2)
[*13100]
[*10-252]
[*3111]
[*23-33.*25-51]
[*3114-24]
[*34-203]
H.(l).(2).Dh. Prop
*50-42. y . /*=/
Dem.
H . *34-5 .Dh: ar/*x. = . (gy). xly .ylz .
[*501] =-(g y).xly.y = z.
[*13195] = . xlz OK Prop
\s
*6043. t--.R > eJ. = .RnR = A [*50 41^]
This proposition is useful in the theory of series. “Rf\R = A is the
characteristic of an asymmetrical relation.
SECTION A]
IDENTITY AND DIVERSITY AS RELATIONS
337
*50-44. h : g ! (R « 7). D . g ! * /)
Vein.
h . *23*33 . *.">01 .Dha!(iEn/). = . (gar, y) . .cliy . x = y .
[*13*195] = . (gar) . xRx.
[*34 54] D.(g.r )..rR-\c.
[*‘3-196] D . (g*. ,j). xR‘y. x-y .
[*23’33.*50 1] D. g! (Rr n|):3h. Prop
*50 45 b ■. R-GJ.Z) . HGJ [*50 44. Trnnsp . *2V3 11 ]
*50 46. \-:R*R-A.0.RCJ [*50-43-45]
*50 47. h.R’GR.3:RGJ. s .R=GJ. E .Ji*R = \
Dem.
h . *23 44 .Dhs.Hp.Ds RGJ.^.R'QJ (1)
H . (1) . *50*45*43 . D h . Prop
This proposition is used in the theory of series. If R is a serial relation,
we shall have R 1 G R and RQJ.
*50*5. h .o']/ = /f*a = a']/f‘a
Dem.
*5051.
*5062.
Dem.
h .*35-1 . D h :x(a1 1)y .s.xea .xly.
[*50T] = . x e a . x y
[*13*193] = .yea.x<=y
[*50*1] = . xly .yea.
[*35-101] = . x(7 [ a)»/
h • (1) . *23*5 .DKa]/ -a] I * / [ a
[*35*11] =«1/r«
K.(l).(2).DKProp
h . Cnv*(a 1 /) * «1 / [*35*51 . *50-2*5]
h . D‘(a 1 /) = a*(o 1 /) = C‘(« 1 /) - a
( 1 )
( 2 )
I-. *35*61 . D h . D‘(a ] 1) = a rs D 4 1
[*50*31] =anV
[*24*26] = a
Similarly h . a‘(a 1 /) = a
h . (1) . (2) . *33*18.31-. Prop
*50*63. h.a1/r/3 = (a^/9)1/ = /r(«^/3)
Dem.
(-.*35-21 .*50-5.Dh. a -]/r^ = a1(fl1/)
1*35-32] = (ao/9)-|/
I- . (1) . *50-5 Oh. Prop
K&W I
(1)
( 2 )
(1)
22
[PART II
338 PROLEGOMENA TO CARDINAL ARITHMETIC
*50 54. h . (a 1 / y = a 1 /
Deni.
h . *50*5 . D h .<a1/) J = (a1/)iarc0
[*3512] =a1/ = r«
[*50*42] =a1/pa
[*50*5] = a 1 / . D h . Prop
*50 55. h:an0-A.s.«t#<&'S
Dem.
h. *24*37. *50* 11 . D
K a a £ ■= A . s : xe a . y e/3 • D x . y . x«/y :
[*35103] Prop
*50 56. I- s g ! (* a /J). « . a ![(a t /3) n /|
Dem.
I-. *50*55 . Trnosp . *24*54 . D
hgl(oA^). = . ~ [a |/3 G ./j.
[*25*55] =.a(«T
[*23*831.(*50*02)]■ .a ! |<at #) A/) x D I-. P,0 P
*50 57.
Dem.
K.*35*I6.DK
I rsa^R^a^I *R
[*50*5]
= l[ «A/i
(i)
[*35*17]
= I r\ R[ a
[*50*5]
= a]/[ a f\ R
(2)
[*35*16 17*21]
= Ifsa‘\R[a
h.(l).(2).DH
. Prop
*50 58. P:a1/?Cy. = .KraGJ. = .a1/ir«G^
Dem.
h.*50-57.3h:/Ao1/e = A.s./ARr“ = A.s./na1Rra = A (1)
1-. (1). *50*41.3 V . Prop
*50 69. h.(/fa)‘^ = an j 8
Dem.
V . *37*412 . D I-. (/1* a)“/9 = /“(<* a £) .
[*50*16] = an£.Dh.Prop
*606. \-.R\(I[a)=Rta
Dem.
h . *35 23 . D h . i? | (/ a) = (i? | /)[“ a
[*50*4] = /£ f* a. D H . Prop
*6061. h./M^ = «1^
Dem.
h.*35354.Dh./rai/e = /!(a1 R)
[*50 4] = a 1 i? . D P . Prop
SECTION A]
33!)
IDENTITY AND DIVERSITY AS RELATIONS
*50 62. CI‘.ft C a. D. ft !(/[<,)=« [.50-6 . *35-452]
*50 63. H : D‘«Co. D./Ca! « = [*5061 . *35-451]
*50 64. 1-. R | (/1- a<«) = R | (1 r C‘li) = R [*50-62 . *22 42 . *33101 ]
*5065. H . /f (D ‘R)\R=l[(C‘R)\R = R [,50-63 .*22 42 . *33-101]
*507. H:a‘/£CQ.D.fl|*/p a= 7f [*5002. *4311]
*50 71. \--.V‘RC a .Z>.\R‘I[ a = R [*50-03 . *43-111]
*50 72 K*j-(/rC‘fl) = |*< ( /|-C‘*) = « [*50-7-71]
*50 73. | «/-|«‘/-A [*50-4. *43 11-111]
*50 74. '
Dem.
h - *43112. Dh.(«,|/)‘Q-/e Q I
[*50-4] =R,Q
[*4311] =Rl . Q
h. (1). *30-41 . DK Prop
*50 75. H./||K_|71 [Proof as in * 50 - 74 ]
*5076.
Dem.
. *34-27 . *30-41.31 -:P=R.O.P\-R
h . *50-73 . *30-36 . 5 h : P \ = R\. 3 . p = R
*" • (1) • (2) .31-. Prop
*50 761. H : | P - | R . 3 . p _ R [Pl . oof ^ in » 50 . 76]
( 1 )
( 2 )
22—2
*51. UNIT CLASSES
Sn in mnnj of *51.
In this number we introduce a new descriptive function i*x, meaning
“the class of terms which are identical with .r,” which is the same thing as
"the class whose only member is We are thus to have
Hut fi(y - x) = I‘x. Hence we secure what we require by the following
definition:
*51 01. i =7 Df
As a matter of notation, it might be thought that / would do as well as f.and
that this definition is superfluous. But we need also the converse of this
relation, and "Cnv 1 / ” is not a sufficiently convenient symbol.
The propositions of this number are constantly used in what follows. It
should be observed that the class whose members are x and ij is i*x «-» i 1 }/, the
class whose members are x, »/, z is i*x \j (‘yv i*z , the class formed by adding
x to o isov i*x, and the class formed by taking x away from a is a — l l x. (If
x is not a member of a, this is equal to a.)
The distinction between x and i*x is one of the merits of Peano's symbolic
logic, as well as of Freges. On the basis of our theory of classes, the necessity
for the distinction is of course obvious. But apart from this, the following
consideration makes the necessity apparent. Let a be a class; then the class
whose only member is a has only one member, namely a, while a may have
many members. Hence the class whose only member is a cannot be identical
with a*.
The propositions of the present number which are most used are the
following:
*5115. h : \j e i*x. = . y = x
*5116. \-.xei‘x
*612. h:j<a. = .t‘jCa
This proposition is useful because it enables us to replace membership of
a class (.rco) by inclusion in the class (i‘iC a).
*51*211. h:x~£0. = .f‘xr>a = A
*51*221. h:ifa. = .(a-t‘i)vt‘x=a
• This argument is due to Frege. See bis article ••Kritische Belouclitung einiger Ponkte »n
E. Schroder’s Vorlcsungen liber die Algebra der Logik,” Archiv fur Syit. Phil., vol. l P-
(1S95).
SECTION A] UNIT CLASSES 34 j
*51222. h : x^e a . = . a — i*x = a
*51 23. h : i<x = L <y . s . ,j € i * x . = . x € 1 *y m s # r = y
*51 4. h : g ! a . a C l‘x . = .a = i*x
I.e. an existent class contained in a unit class must be identical with the
unit class. From this proposition it will follow that 0 is the only cardinal
which is less than 1.
*5151. : a = l*x . = . x = i‘a . = . x i a
For classes. i‘a has the same uses that <(x) <$*) has for functions; "7‘«"
means "the only member of a." We have
*51-59. h : >/r [t‘2(<f>2)\ . = .yfr (ix)(4>x)
*5101.
*511.
Deni,
i = I Df
h : aix . = . a - $ (y =» x )
h . *4-2 . <*51 Ol).D\-:aix, = .aTx.
• a * 9 (y =* *) : 3 h • Prop
0303. *51-1]
051*11 .*14-21J
[*20-57'-2. *51-11]
[*51113]
[*321]
O501J
*5111.
*5112. h . E ! i*x
*5113. h:aB(‘*.s.a»J(y a * x)
*51131. h : aix . = . a — i*x
*6114. h:.a = e‘x. = :y ea .= 4 ,. y = ar [*51 13 . *2033]
*51141. h :. a - i*x . s : a ! a : y * a . D,,. y = a:: = : x c a : y e a . D„ y =
[*51*14. *14122]
*51-15. h : y e i‘x . = . y = * [*5111. *20 33]
*6116. [*5115. *1315]
*51161. h . g ! i*x [*5116. *10-24]
*5117. h . = V
Dem.
h .» 51 -1 . *202.3 I-. {J (y = x) j t x.
[*10-24] DK( a a).Mx.
[*33131] PH. ttfl'i.
[* 1011 ] 3K(,). I( a'i.
[*2414] P h . d‘t = V
The above proposition is used .in the theory of selections (*83 71).
X
342
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*512. h/to.s./'xCa
Dem.
K*13-191 . Dl-sy-x.D,. yea;
l* 3l,5 3 sryei^.D^.yea:
[*22-13 = : i‘x Co :.D h . Prop
Thc nbove proposition shows how to replace membership of a class by
inclusion in a class; thus for example it gives:
Socrates is a man . = . the class of terms identical with Socrates is included
in the class of men.
Before Peano and Frege-, the relation of membership ( e ) was regarded as
merely a particular case of the relation of inclusion (C). For this reason, the
traditional formal logic treated such propositions as “Socrates is a man " as
instances of the universal affirmative A. 'All S is P," which is what we
express by "o C ff." This involved a confusion of fundamentally different
kinds of propositions, which greatly hindered the development and usefulness
o' symbolic logic. But by means of the above proposition (*51-2), we can
always obtain a proposition stating an inclusion (namely "p.tCa") which is
equivalent to a given proposition stating membership of a clnss (namely
*5121. h . ,r^€ a — i*x
Dem.
h . *2233 35 .Dh/ta-j'/.a.xca. x^e i*x .
[*3 27] D.xr*-'ti , x
h . (1). Transp . *51 16 . D b . Prop
*51211. h : x*>-€ a . = . i*x r\ a •= A
Dem.
h . *24-39 . D h f«r a a - A .
[*51*1 o] = :y = :
[*13 191] rSX'vfo:. DK Prop
*5122. h:flfw'/ = A.av i‘x =0. = .xc&.a = fi — t‘x
Deni.
h . *24-47 . D
O 51 * 2 ] Prop
*51 221. h : xca . = . (a - i‘x) v i ‘.r = a
Dem.
h.*51-2.Dh:x(a. = .f‘xCa.
[*22 G2] = . i‘x \j a = a .
[*22 91 ] = . (a — i*x) vj‘ic = fl:DK Prop
SECTION A]
UNIT CLASSES
343
0)
<->>
W
*51222. h :.r~ea. = . a-i‘.r = a [«5l *211 .*24*313]
*5123.
Dem.
h . *2031 .*5115.D
H I'ar - ity. s s * «ar y ;
[*13*183] = : x = y :
[*3115] = z x € l*y :
[(1).*13 16] = : y e i*x
*"•(!)• (2) • (3). D K . Prop
*51*231. h : t'arn py = A . = .x + y
Dem.
h. *24311 OI-i.i'i n i‘,= A . = : i‘xC-i‘«s
[*51-15] 3,!, + ,:
[*13-191] 3 (-.ft,*
*51-232. [*22 34 .*5115]
This proposition states that a member of i‘x u t‘,j must be either x or »
and vice vena. i.e. that i'x v,i, the class whose only members are « and V
*51 233. h a = i‘xu ^ «o. B : r — x. v . * — «
[*51-232. *1011. *2018]
*51 234. h :: a - t‘x w *« y . D.., e «. d,. ^ : = . ^^
Dem.
[*4*771 233 . :>h!: ' Hp . :>:: * t “. :> '.^ : f ■■■‘ = x.*.z-y:3 r .t*:.
[.10-22] ■ 3.*3.*..
[♦13-1911 - *. * - a . D,. <f>z : g-y . D,. 0*...
L * ldl91J DK Prop
*51235. h::i.t««wt‘y.D Sl ( !JI ) #<€a . #liB<i ^ iVt .
Dem.
K *51-233. D
h :: Wp. D (ge) •t€a.<t> 2 . = i (g*): * — x. v . * = y i <f>t:
f* 4 ’ 4 ] = : (a*) '.z = x.<t>z . v ,z = u .d> 2:
[*13195] = ! (a,) • (3*) ■ * - y • :
L*i.» 19o] aifr.v.^yiOKProp
*51236. h:.* et ‘ xw £. = :* = *. v .* c/3 [*22.34 . *51 15]
*51237. !;::«= «** w ^ . D (r)x f * . = : r = x. v . r e /3
[*51-236. *1011. *2018]
•“JJJ ►«««-»*#*»*.
[*10*221 - z — x . D . <f>z : z f (3. D.0 2: .
[*13191] ~ * = * * * tf * * € ^ ‘ D * ’ ^
J = :.*x:*e£.D,.<^: : .:>f-.Prop
344
PROLEGOMENA TO CARDINAL ARITHMETIC
-- - - *•' nituujiciiv, [PART II
*51 239. h :: a = t‘x \j 0 . D (gx ). z ( a . <f>z . = : <£x . v . (gx) . z e 0. <f>z
Dem.
K *51*237.3
I-:: Hp. 3 . z € a . <f>z . = : (%z): z = x. v . z € 0 : <f>z :
[* ++ ] =:(g z)zz =x.<f>z .v ,z€& ,<f>z:
[* 10 42] = : (a z).z = x.<f>z .v. (gx ). z c 0 . <f>z :
[*13-195] = : <f>x . v . (gx). z e 0 . <f >2 :: D b . Prop
*5124. b /* y C i‘j- 'J0. = :y = x.v.y€0
Dem.
K*51*236.3
1 -:: i*y C i*x sj 0. = :. z e i*y. 0,: z = x. v . z e 0 •,
[*5115] sr.x-y.D, :z = x.v.z€0:.
[*13191] =:.y = j-.v.yc/9::DI-. Prop
*51 25. h : a C i‘xv0 . a . 3 . a C £ [*51211 . *2+49]
*513. b z y € a . y * a-. = . y < a - r‘x [*51 15. *22 33 35]
*5131. F:g!a Af‘x.i.i‘xCo.*.aM‘x*t‘j.».xta
Dem,
b . *22 33 . *51*15 . 3 b :g ! a n t'x . = . (gy). y e a . y « x .
[*13*195] s.xca. (1)
[*512] 3.1‘xCa. (2)
[*22-621 ] s . i‘x - t*a- r\ a (3)
h . (1). (2) • (3) .DP. Prop
*5134. h:xfa. = .-aC-i‘x [*512. *2281]
*5136. h : ar ~ e a . = . i‘x C — a [*512 . *22 35]
*5136. hi'vfa.z.aC-j'i [*51*35 .*22*811]
*51*36 is frecpiently used.
*5137. Ka = J((‘iCd) [*512 . *2033]
*514. b : g ! a . a C i‘x. = . a = i*x
Dem.
h . *24*5 . *51*15 . 3 b :. g ! a . a C i‘x . = : (gy) . y c a : y e a . y = x:
[*1+122] =:yea.= v .y = xz
[*51*11.*20-33] =:a = i‘x:.Dh. Prop
*51 401. b :. a C i‘x . = : a = A . v . a = i*x
Dem.
b. *51’4. *56. DI-:.aCt ( x.D:a = A.v.a = i‘x (1)
b . *2+12 . *22-42 . 3 I-:. a = A . v . a = i‘x : 3 . a C i‘x (2)
b . (1) • (2) .31-. Prop
This proposition shows that unit classes are the smallest existent classes.
SECTION A]
UNIT CLASSES
:*1
*51 41. h ; t*.c yj i‘,j = t *, v yj I*- , = ,y = s
Deni.
[*13T6.*4'41] v.*-y:
[*13172.*2-621] 3.'!"
*" • (1) • (2) . 3 h . Prop <2 >
The two following propositions are lemmas for * 51 * 43 .
*5142.
Dem.
K *51*232. D
t«»i] —.—- ,,
..
Similarly I-: t‘« v t « y . ., _ „. ;> *. J = ” JJJ
h . (1). (2) . (3) . Dh. Prop 7 ( J)
*61421. H..*-*.y- u ,.v.x-u>.y-..O. l < 4:wt « y _ t « 4ut , w [. 5141 ]
*6143.
[*51-42 421] *
The following propositions are concerned with 7. U with the relation of
the only n,ember of a unit Cass to that Cass. If « is „ unit cla8s , is its
*51 51. h : a = t‘x. = . x . = a x 7a
Dem.
V ■ * 51131 • *3111.3 I- : a = ,«*. = . *7 a
(■ • (1) .3h:xia.yta. D. a = i‘x. a= i‘u.
fu<n t .do .
[*51*23**20 57*2] D-X = y
( 1 )
( 2 )
(" • (1) • (3). D h . Prop -:x- ( ‘a (3)
34 <3
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*51511. b . l*l*.r=x
[*51-51 '^.*20-2^
£*51-51 '\*1+**2M8~J
*5152. h : E ! i‘a . = . a = 1 * 1*0
v v
*51 53. hEli'a.s.i'ofo [*51*52 16 . *14 21*18]
v
*51 54. h : E 8 i*a . = .< 3 .r) . a = / ‘a [*51*51 . *14*204]
*51 55. h ! E ! i*a . = . E ! (i.r) (j < a)
Dem.
I" . *51*54*14 • D V E ! i*a . = : ( 3 .**): y e a . = v . y * a::
[*14*11] s : E!(i/)(x€ 0 > D h . Prop
*51 56. b :6«f *$(<t>y). = .y <<£//) = /‘5 . = . 6 = (i.r)(<£.r)
Dan.
b . *51 *51 . D4:.6»i‘5(</>»/).
[*20*15. *51*11 ]
[*14202]
h . (1). (2). D h • Prop
-i‘b:
H • s„. y - b s
b = (lx) (<f>.r)
*51 57. h : E ! f‘//(0y). = . i‘y (<*>y) = (i.c)(<*u). = . E ! (?.r)(0.r)
Dem.
( 1 )
( 2 )
( 1 )
b . *14*204 . *51*56. D H : E ! i*y (<f>y). = . E!(ix)(^r)
b . *14 205 . D 1- :(ix)(4>x) = 7‘#(<*>y). = .(g5). 6 = ( ix)(<f>x ). b = T^(0y).
[*51 *56.*4 71 ] = . ( 36 ). b = (ix)(<f>x) .
[*14*204*13] = . E ! (?.r) {<f>x) (2)
b . (1). (2) . D h . Prop
*51 58. b : E ! i*a . = ,7‘a - (ix) (x « a) [*51 *57 . *20 3 . *14*272]
*51 59. b : yfr \i*2(<f>2 )\. = . >lr(tx)(4>x) [*51*56 . *14*205]
*52 THE CARDINAL NUMBER I
Summary of *52.
of «!i unit d'.'ir t,,e car<ii,,ai ** ^
relevant at ! f t 1 “ <le ' i, "“ <l is 11 ««K««I number is not
has boon H P fi V“ r n " 0t ° f C0,,rse be P""* nntil "caHinal number '
the PrC ^ t ' tl,eref0re - 1 is * * Wed simply as
-t for "me J UU,t C,M ~ bein * SUch «*«•• « -e of the Ln
L.ke A and V. 1 is ambiguous ns to type: it means “all unit classes of
TZzxziZ ss;v»? ir- ;
jssks . s " v - *•*«« *
ca„l e rr ieS ° f i l ° ^ ?r ,Ved in the P reseut "«"*W are what we may
St lJS ; «ri</imeti'oul properties, they are not concerned
1 but with he? ? Hper “ t , ,0ns (add,t, o“- etc.) Which can be performed with
1 will be considered later,° in P^t 11?'““^ ^ nrith,,,etical Pities of
following Pr ° POSiti0nS ° f th ° PreSCnt numbcr which ■» most used are the
*52’16. f 1 !. afl.sj>ijj fl . ;r y (a 3 j. __
identical." “ Unit C,aSS if ' “ nd '*'* if '*‘ is null, and all its members are
*52 22. I*. i*x€ 1
*52 4. s.af 1 u(*A =• x y ( a ^ ^ _
h :a! a • 1 - ■ (ax. »>•«*.. • *+y
each^r^t^ fr0m •** * transposition, by negating
*5246. = a!(on)9)
* —
«02*01. 1 = 3 {(gar) . o = t‘x} Df
*521. ha f l. = . (g*) [#20 . 3 # ( * 52 0!)]
348
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*5211. h a 6 1 . = :(gx): yea. = y .y = x [*52*1 . *5114]
*5212. h :U<J> 2 )* 1 . = . E !(ix)(</>., )
Dent.
K . *52’ 11 . D H z (<f>z)€ 1 . = : (gx) :y ez (<f> 2 ) . = y . y = x:
[*20 3] = : (gx): </>y . = y . y = x:
[*1411] = : E ! (ix)(0.r)D 1-. Prop
*52 13. K . 1 = DO
Dem.
H . *51'131 . D H : a = i*x. = . aix:
[*10 11 281] D 1- :(gx) . a = f‘x. = . (gx) . ai.r :
[*52*1] D H : a € 1 . = . (gx) . oix
[*3313] s . o € DO : D K . Prop
*52 14. h . 1 -«“ V [*52 13 . *37 28]
*5215. \- : a e 1 . = . E ! /‘a [*5154 . *52*1 J
*52 16. h :. a e 1 . = : g ! a ix. yea. D,.„.x = y [*5215 . *51*55 . *14*203]
*5217. H : at 1 . = . i*a = (ix)(x<a) [*51*58.*52*15]
*52 171. I-: ae 1 . = . E! (ix)(xca)
*52172. h : ac 1. ■ . To>« a
*52173. 1- : a f 1 . = ,l‘ae a
[*51*55. *52*15]
[*51*52. *52 15]
[*51*53. *52*16]
*52 18. h a « 1 . s : (gx ): .r e a : y e a . 0,,. y = x
Dem.
V . *51*141 . D 1-:. (gx) . a = i‘x. = : (gx) : x € a : y « a . D,, . y = x (1)
I-. (1). *521 .Dh. Prop
*52181. H :. 1 . = : xt a . D, . (gy). y e a . y+ x [*52*18 . *10 51]
*52 2. h.lC CIs
Dem.
1*. *52*1
[*5111]
[*20*54]
[*10-5]
[*20-4]
D h : a € 1 . D . (gx). a = i‘x.
D . (gx). a = z (z = x).
D.(gx.*>.3(*!*> = 2<*»*).« = 2<*!*)
D.(g<*>).a = 2(<*>!*)-
D . a € CIs : D h . Prop
*52 21. h.A~*l
Dem.
h . *5216 .Dhat l.D,.g!a:
[*24 63] Dh:A~el
*52 22. KiOel [*51 12 . *14*28 . *10*24 . *52*1]
SECTION A]
THE CARDINAL NUMBER 1
( 1 )
(->)
*52 23. — i
Dem.
b. *52-22.,1024. 3h. (a . r) .,. rf l.
[*20-54] 3 H. (ax.
C*10-S] 3H.(aa).« 6 l
h • *52-21 . *22-35 . D b . A e — 1 .
[*10-24] DK (a «,. a< _,
*- *(i;.(2).
*52 24 Kl + AftC l..l + V A C.s [*52-23 . *24-54 . *2417 . Tninsp]
*52 3. h . t“ Q Cl 1 J
Dem.
K .52-22. *2-02. :
[*5112.*1011.*37-61] D K . £ “ a c 1
*5231. HuCl.s. (ga). * = i** c
Dem.
. *52-14 OhifCl.3.^C/ ,, V
[*:17 (iG.,51 J2] =. <a«). a cv.
[*->4 11] B • (3«) . k = i“o OH. Pr0 p
*524. h of 1 u / ‘a =*•<*. „. n
* . _ . x, // e a . j, v . x « y
Dem.
*■ • *52 10 .*24 54 . D
f.4'171 ^ = :« + A:*,y,«.
[.5-63] • * •«“ A : s ;. a — A ;. v:. a + A : «a. X.» .*»}!,
H • *24-51. .10-53 . ,11 62 . D h (,)
I- .(1).(2).*4-72 X.„.* = y (2)
b • (*). .51 -286 . ' Sl: ! p' p < 3 >
l ‘A T, th U 3 r the S i t h 0n U frC ‘ |UC " t, >’ useful - W « «»>all define the number 0 a«
when, and only th “ l “ f^ 01,6 “ en,ber or uone
1,. "’“'. ? CmberS “ re ,dent,cal - 11 wi» be .seen that
a having no members. ° ‘ m|> y 3 ! and thercfore “"<> w8 «-bc possibility of
*6241. b : a ! a . , . 3 . (a*„y). x.y ««.* + y
Dem.
b . *24 54 • D b :. g ! o . o ~e ! . =
349
[*4 56]
[*51-236]
[*52-4.Transp]
[*11'52]
<* + A . a1 ;
~|ael.v.o = A|:
« f 1 w «‘A) :
~|x,y €a . D x tl . x = y]
(3^y).x.y € a.a: + y:. D^-.Prop
300
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART
*52 42 l-:.flfl.D:g!an^.5.anjSc 1
Dem.
H . *31-31 . Dh.g!f‘x^. = . t *. c a 0 = t'x :.
[*20*53] Dh.a = /‘x.D:g!an^. = .on^= i‘.r
1*10 11*28] D b :.(a-r). a = «‘j*. D : (ax) : 3 !an^. = .aA^ = i‘i:
[ * 10 37 ] D : 3 ! a n £. D . (ax). a a £ = / ‘x ( 1 )
1- . (1). *52 1 .Df-:.a6l.D:g!an/9.D.aA^el (2)
h. *52‘16. Dh:aA/S<l.D.g!«A/j (3)
h.(2).(3). Ob . Prop
*52 43. h:a«l.g!oft/3.3.afl.an^«l [*52*42 . *5 32]
*52 44. 1*a e 1 .D:g an t 3. = .aCfi. = .ar\{3 = a
Dent.
I-. *51*31. DI-:g!/‘jr\^. = ./‘.rC/9:
[* 13* 13. Exp] 0 b a = /‘x .D:g!anj9, = .aC/3:.
[*1011*23] D K (ax) .a = /‘x.D:a!aA/3. = .aC/3:.
[*521] Dh.flf l.D:g!an^. = . fl C^ (1)
h.(l >.*22-621 .Ob. Prop
*52 45. b :: a .&el .D:.aC/iu7. = :a»/9. v.aC^
Dent.
K*51*236 ^£2.3
b x e t*i/ sj y . s : x — y . v . x e y
[*51-2*23] D h f'xC i*y v y . = : i*x **i‘y . v . i*xCy:.
[*13-21]
0 b
i‘x . £ = i ‘y . D :. a C /9 u 7 . = : a = /S . v . a C 7 ::
[*11*11 *35] D h ::(a^.y).a=i‘x.^»i‘y.D:. aCfivy. =: a=0.v .aCy (1)
H . (1). *521 .D b . Prop
*52 46. H :• a, £ e 1 .D:aC/9. = .a = /9. = .g!(oA/9)
Dent.
1-. *51-2-23 . D h : i*x C <‘y . * . t'x mi‘y (1)
l-.(l). *13-21 . Dh.a = i‘/.^ = < ‘y.D:aC^. = .a = ^(2)
I-. (2). *11*11-35 . *52 1 . 0 b a. >9 € 1 . D : a C 0 . = . a = 0 (3)
h . (3) . *52-44 . Db. Prop
*526. h.at 1 . D : x e a . = . i*x = a . = . x = l‘a
Dem.
1*. *51*23. Dl -zxe I*//. = . t‘x = i‘y z
[*13'13.Exp] 0 b :. a = f*y. 0 :xea. = . i*x = a :.
[*1011-23.*521 ] 0 1- s. a e 1. D : xe a . = . *‘x = a . (D
[*51-51] =.x=i‘a (2)
h.(l).(2).DKProp
SECTION A]
THE CARDINAL NUMBER 1
*S2601. hs: at 1 .3 3 :.r« a . 3,. : s : (a ,).
** . *5215 . Dh. Hp . D : E ! 7‘a :
[*30 4] 3:,-7 a .J.,. = 7‘ a .
f-. (1). *30-33 . 3
*52 602. I-2 (*,) .1.3:* (,x) ,*,). s . 3, *x. a . ^^
(*52 12 . *14-26]
*5261. h:.„1.3: t ‘ a «^. B . aC> 9. = . a!(an/9) L 5 2- 6 01 **-*)
*6262. h:.a,/9«1.3:a- / 9. s .:. a .r* / 9 ^ J
Dem.
UMi. 3 h :, .p.3j la ,^. i = I(0iil= ;, S;
U2-46] :: x< « 3 vrf :
*52 63. t-: a ,/9 e l. a + ^. D . an ^>A [.32-40 Mvansp]
*52 64. H:a € l.D. an ^ fl w £ < A
Dem.
K *52-43 . 3h:Hp. a ! an/ 9. 3 . a n^«l,
[*6-6.*24-64]3 l -:.Hp.3: an/3 . A . v . anj8 , l!
[*o 1-236] 3:on£el u «‘A :. 3 I-. Prop
*527. h:.fi- a< i.„cf.fC/9.3:f- a . v . f „a
Dem.
Mr 1 - 3 h : Hp .fC a .3.f = a
l ' *«o 5° " •" : ~(f C a ). 3. 3 ! f — a 2
r*isai <8> ' ;;;
■ * J 2 ; 3 : Hp . 3 . ( 3 .x ). 0 — a — t‘ x
^ . (1). ( 6 ) .Dh. Prop ’* ;
( 6 )
*53 MISCELLANEOUS PROPOSITIONS
INVOLVING UNIT CLASSES
Suwmart/ of f53.
'I'he propositions to be given in this number are mostly such as would
have come more naturally at an earlier stage, but could not be given sooner
because they involved unit classes. It is to be observed that /‘.ru i*y is the
class consisting of the members x and y. while t‘x\ i‘y is the relation which
holds only between .r and y. If a and fS are classes, f‘aut'£ is a class of
classes, its members being a and Q. If R and .S’ are relations, i* R f t*S is a
relation of relations; and so on.
'I'he present number begins by connecting products and sums //*, s'/c,
//X. a‘X. in cases where the members of * or X are specified, with the products
or sums a r\ a v R r% S, R o .S'. We have
*53 01. Ih ./>‘«'a-a
*531. I- v l l &) = a n £
*5314. I -./>*(* v l*a) = p*tc r\ a
with similar propositions for s, j> and s.
We have next a set of propositions on sums and products of classes of unit
classes. The most iuqiortunt of these is
*53 22 h.s‘i“a = a
We have next a proposition showing that the sum of k is null when, and
only when, k is either null or has the null-class for its only member, i.e.
*53 24. = = = CIs . v . k - i* A
(Here we write "A n CIs.” to show that- the “A" in question is of the next
type above that of the other two As.)
We have next various propositions on the relations of R*x and R‘x and
R tl a in various cases, first for a general relation R, and then for the particular
relation s defined in *40. Three of these propositions are very frequently
used, namely:
*53 3. hsEI/ftr.s.Jftrcl
*53 301. h . R“i t x = R*x
*53 31. V : E ! R*x . D . R lt i t x = i'R*. c = R*x
The remaining propositions of this number are of less importance, and are
seldom referred to.
SECTION A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES
353
*5301. \-.p t i t x = a
Dem.
*5302. h
Dem.
h . *401 . 0 h:.e p‘, ‘a . s : 0 ( i ‘a . 0„ . .r « a :
J* 51 ' 15 ! s: 0~a.O,.* e /3:
t* 13 l91 l Dh. Prop
S*l‘a
h . *4011 . D I- : x e s‘<‘a . = . < 3 /9) . (3 (
s-(a>3)./9-a.At^.
[•13*195] =.*(«:Dh. Prop
*53 03. h . pU‘R = R [Proof as in *53 01]
*5304. I -,i‘i‘R = R [Proof as in .5302]
*531. H ^ £‘/9) = a n /?
Dem.
t- . *4018 .Oh . p‘(i‘a u <‘/3) = />• I‘a n pU'B
[*5301] ~t.nff.0h. Prop
Ihis proposition can be extended to i'o e 1‘g «,<», e tc It shows the
srsfi ~ £
*5311. l-.i , (t , aw[^) B o Wi 9
Dem.
h . *40171.31-. s‘(t‘a o i‘/3) = s‘i‘a w s'l'B
c . f* 53 02 ] ■« u fl.3K Prop
Similar remarks apply to this proposition as to *53 1.
*5312. h.p‘(i ‘Rui‘S)-R*S [*41-18. *53 03]
This proposition shows the connection between the product for a class
*6313. H.i‘(i‘.ft ul ‘ l S) = rt 0 S [*41-171. •53-04]
Similar remarks apply to this proposition as to •5312.
*5314. h.p‘(*u t ‘«) = /><«„„
Dem.
^ • *40" 1S . 3 h , p*(fc u 1 * 0 ) = p *k a p*t ‘a
[* 5301 ] ~ P ‘ xna
*63 is. i-.,*(* u t . a)= ,., ua tProof M in „ 53 . 1+J
*5316. H.p‘(A UJ .rt )= ^ XA/f (Proofas in *33-14]
*5317. = [Proofas in *5314]
23
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
354
*5318. H. 5 *(a-«‘A) = «<a
Dent.
f - . *51 *221 .Dh:Ata. D. (a — t‘A) v c‘A = a .
[* >315] _ D . s*(a - i* A) \j A = s‘a .
[★-24 2+] D . 5*(a — / ‘A) = 6 *‘a (1)
• *51 22*2 . D h : A ^ f a . D . a — t*A = a .
[*30 37] D .s*(a -t‘A) = $‘a ( 2 )
h.(1).(2). D h . Prop
*53181. K.v‘(\-*‘A) = .y‘\ [Proof as in *5318]
*53-2. I- : k € 1 . D . / ‘k = p*K = 5*/f
This proposition requires, for significance, that * should be a class of
classes. It is used in *SN 47. in the number on the existence of selections
and the multiplicative axiom.
Dem.
I-. *52 <>01 • D h s: Hp. 3 t. x c c : a : a c«. D. . *«a: ■ s ( ga ). a e * . * e a (1)
Ml).*40 1-11 . DK Prop
*5321. h : X « 1 />‘X «.v‘X [Similar proof]
This proposition requires, for significance, that X should be a class of
relations.
*53 22. Ks‘i“a-a
Dem.
H . *40'11 .Dh:xf s i t il a . = . ( 37 ) . 7 c l tf a . x ty •
[*37*64.*51 12] = .( 3 y).y<a.x« f*y.
[★51*15] s. (ay), yea. a:-y.
[*13*195] = .a*c a : D h . Prop
*63 221. h . w t‘y) = iVx v c‘t‘y
Dem.
V . *371 . D h a e v «‘y). 5 : ( 3 *). x f (t‘x u t‘y). a t z :
[*51131] = : ( 3 *).; e (t‘x v c*y). a » c*£ 8
[*51*235] =za = i , x.v.a = i i yi
[*51*232] =:adtTxv«‘('y):.DK Prop
*53 222. h : * = i“a . D. a = 1l"k
D em.
h. *13*12 . *20 2 . D h : Hp. D . 7"sc = T“i“«
[*51 *511 .*14*21 .*37 *67 ] =5 [( 3 y) .yea.x = tTy|
[*51*511 ] =5 J(ay). y e a . x = y)
[*13*195] = a Oh. Prop
SECTION A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES 3;
(1)
( 2 )
( 1 )
( 2 )
*5323. h/cCl
Dem.
(•.*52-31 . D (• : Hp. = . (jj a ). „ — ,*< 0
I-. *53-22.31- :k = i“cl. 3..s‘* = a
1*53-2221 = :.. K
I-. (1). (2). *10-11-23. Oh. Prop
*53 231. (•:. Xe a . 3,. x = y : s : a - A . V . a = t ««
Dem.
^ . *51'141 . D h a !o:x(o, • x = y • = • a = / ‘/
(". *10-53 .
K2j
*"•(!)• (2) . *4-42-39 .31-. Prop
*53 24. I-:. s‘* = A . s : * = A n CIs. v . * = j«A
Dem.
H . *24 15 . *40 11 . D
h :.$<* = A . = s(ar)s^|( aa ).««/c.areo| s
[*10'51] = : (a:, a) : x e a . D . a ^ e k •
[*11 2..10-23] ■ : (g*) .« * « . 3. .« ~ !
[*24-54J =!*+A.D«.a~€«:
[Transp] s:oe*.3. . a = A:
[*53-231] s : * - A « Cls . v . « - t 'A :. 3 (- . Prop
we write*"* **",*«, tho * ost line of the proof of the above proposition,
next above That of the TLX 1“ 2" ^ thiS A mUSt ^ ° f tl,e
The following proposition is used in the theory of selections (*837:11).
*63 26 (• :. n t‘\ Cl*. v . x n X — j‘A
Dem.
K*40 181.DK:. Hp. D : 8*(k nX) = A:
[*53 24J D:<f>\«An Cls . v . k r\\ = t‘A DH. Prop
*53 3. b : E! € i
Dem.
h * * 30 ' 2 * D h E ! R ' x • = : (H&) : yrt*. = ,. y = l :
[*3218.*51 15]
[*20 31]
[*521]
The above proposition is very frequently used.
( 3 &) : ye R‘x .=„.ye i‘b :
(36). Ji‘ar-i‘6:
^**€1 DK Prop
23—2
356
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*53*301. h . R“i*.r = R*.r
Dem.
• * :i ~ 1 •*51-15. D h :ye R“i*x. = .(qz). z— x. yRz .
[*13-195] m.yMr.
[*32*18] ^ ^ = . y *R l x : D h. Prop
*53*302. I-. • \j i*!/) = R* x \j R‘,/ [*37*22 . *53 301]
The abovc proposition is used in the cardinal theory of exponentiation
(*11671).
*53 31. h : E! R‘x . D . R»u*x = t'lPx - /?.r
I he above proposition is one of which the subsequent use is frequent.
Dem.
h . *51-11 . *14*18 .DhsHp.D. i*R , x =.f)(y = R‘x)
[*30*4] -$(yRx)
[*32*13] JTi< x (1)
Ml). *53*301. D V . Prop
*53 32 H : E! R*s . E! R*y. D . i‘y)- i<R* r sj t ‘R‘y
Dem.
I-. *37*22 .Dh. 7i“(f‘x v i‘y) = w /e“c«y ( 1 )
Ml).*53-81 -3**- Prop
*63 33. H . *“f‘* = iV# £*53*31 ^j
*53 34. h . #"(«** w i‘\) = i t s t K w/V\ £*53*32 ^ |
*53 35. h w «‘X) — \js*\ = s‘(* v \)
Dem.
h . *53*34 . D h . s t s ,t (i , K kj i*\) = *«((«,«« v/ iVX)
[*5311] —
[*40* 171] = *•(* u\).DK Prop
'1‘he above proposition may also be proved as follows:
H . *42*1 . D h . s t s tl (i t K ui‘\) = v/ 1 ‘\)
[*5311] = *«(*vX)
[*40*171] =s‘«ws«\.Dh. Prop
*53*4. • H : x = . = . R*j fl R l y . = . i*x = R‘y. = .x= i*R i y
Dem.
H . *14*21 .*4*71 .31 -sx~R‘y.s. E ! R‘y . x = R*y .
[*30*4.*5*32] =.ElR‘y.xRy.
[*53*3.*32*18] = . € 1 ,xeR‘y .
[*52*6.*5*32] = . e1 . l '«-2Z*y.
(1)
SECT,ON A] MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES
357
= . tU = R‘y .
= . .r = / ‘
( 2 )
(«)
[*52*22]
[*51-51]
*■■(!)■ (2). (3). D K Prop
*53 5. K : a ! a . = . a e CIs — t‘A
Bern.
LtfJ' ■ 3 h: 3! * ( * 2> • £ ■ f ' CIs . a ! 3 (0.-).
T ,, L ' J = • J (4>z )« CIs — I'A OI-. Prop
*53 51. h : a ! R . s . ft « Rel - ,<A [Proof „s in *53 5]
*53 52 h«(«. 3 !a. E .a < ,- 1 .A
Bern.
54 ' 3 h 8 “ * *" 3 ! “ • s • “ * * • “ + A •
*■ ='«f<“(‘A:DK Prop
*53 53. h : ft e A . a ! ft . s . ft f X - i‘\ [Proof M in .53.52]
l-he follow,ng propositions arc inserted because of their connection with
the definition of a-./Sin *70. . ft»d‘ft and R .. V nre both ^
*536. h : ft-A .3 ! a . 3 . ft“« - «‘ A . ft“« - t « A
Bern.
H • *33 15 241 . *24 13 . D h : Hp . 0 . ~R‘x — A
I". (1) . *37 7.3h:Hp.D.f?‘a = £ ,(3 *). x « a . /9 — Aj
[ * ,035] -*ta...*-A)
t* 47 *! -^ce = A)
1*51*11] — I‘A
Similarly I- : Hp . D .*R"a - t*A
•-•(2).(3).3H.Prop
*53601. h:a!«-aoa‘A = A.3.^‘a=,- A
Bern.
*33"41 . Z>K-Hp.x« a . :>./?'* = A (J .
.*_ Aj
[*10-351 i .
f*473 *51-111 =/3{ a !a./9 = A}
l*4 73.*51 11] = *‘A OK Prop
*63 602. H : a ! a.« « D-JI _ A . D. *w A [Proof „ in * S3 . 60I]
*53 603. : 3 - U'Jl. 3. ft“(— G‘ft) =«. A [.24 21. .53 601]
0 )
( 2 )
(8)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*53 604. H : a ! - D‘R . D . /?“(- D*R) = f ‘A [*24 *21 . *53 602]
*53 61. b + D . 7?‘a = v /‘A
l)em.
K*22<)2. D h : Hp. D. o = <l‘/i w (a — d‘iJ) ( 1 )
t-.*24G. D I-: Hp. D .g ! a — (l‘R .
[*24-21.*53«01] D .~R“ (a — G‘if) = i‘A (2)
I-. (1). *37-22 . D h : Hp. D . /?‘o =7f“(I‘/f u li“{ a - d‘R)
[(2)] = li“d‘R v i‘A : D h . Prop
*53 611. b : V , RCa.D , R*a.O.*R“a = *R“lVR w i*A [Proof as in *5361]
*53 612. b : d'R + V . 3 ,~R“V = Jt“d'R \j i‘A [*53 61 . *2411]
*53613. h:D‘«+ V. D. 7?‘V - w i*A [*53 611 . *2411]
.53 614 I-. li ,, Q , R —7f“V — i‘A
Detn.
1* . *53 612 . *22-68 . *24 21 . D
*■ : CI‘/e * V . D . li “V - i‘A - 7?‘(I‘7e - i‘A (1)
H .*22 481 .Dh:<l*R~V -i<A = R“a‘R- l ‘A (2)
f-. *37-772 . *51 36 . *22 621 .Db. R“(I‘R - ,‘A = 7?‘CI‘7? (3)
b . (1). (2). (3). DK Prop
*53 615. h . 7£“I)*7? = 7f“V — i*A [Proof as in *53G14]
The two following propositions arc used in *70 12.
*53 62. b : li“a*R C 7 . = .rVC 7 v;i‘A
Dem.
h. *53-614. D b :W‘/{C 7 . = .rV-t‘AC 7 .
[*24*43] =.rVC 7 u[‘AOf-. Prop
*63 621. b : Ii“D‘R C y . = . 11 “V C 7 ut‘A [Proof as iu *5362]
*53 63. b : a*R * V . D . D‘/e - Il“<l‘R w i‘A [*37 78 . *53 612]
*53 631. H:D‘7**V.D.D‘!ft = ^‘‘D‘7?w‘A [*37 781 .*53613]
*53 64. b : d‘R = V . D . D‘7?= R“a*R [*37 78]
*53 641. H:D‘« = V.D.D ? fl=^“D‘iJ [*37-781]
*54. CARDINAL COUPLES
Summary of* 54.
Couples are of two kinds, namely (1) «'xw‘y. in which there is no enter
as between * and .j, and (2) *«* | <‘y. in which there is an order. We may
^nceTasw 11 h eSe h tW ° C ° UP ' eS “ C “" liDal and 0rdi ™' '-spectively
(Wheri ir \ Sh0Wn '‘ e,e “ fltr > tlle «='■«» of ••‘II couples of the form u Jy
form ,‘a “l " T' lbCr 2 ' While the C ' aSS ofaU ““P'-* of the
dulctll ' 6 ^ 18 6 0rdinal " U,nber 2 ' ,C which - foe the sake of
ds mctmn. we ass.gn the symbol ■• 2 ," where the suffix "r" stands for
the f H ’ eCa 'r ° ,dini11 2 U “ ClaSS ° f ri ' lat >ons. In the present and
o^dile ° W,D 1 nUI CrS '. We Sha " dcfine 2 and 2 ' “» the classes of cardinal an,I
so define°| ,p 8 rCSpeCt '. Ve lcnvin « !t 10 a lat er stage to show that 2 and 2,.
so defined are respect.vely a cardinal and an ordinal number. An ordinal
couple with len C “" ed 7 C ° UP ' e ° r 8 C °'‘ pU wM Se " se Th,,s «
couple with sense ,s a couple of wh.ch one comes first and the other second.
i a a introduce here the cardinal number 0, defined as i*A That 0 so
defined ,s a cardmal number, will be proved at a later stage: for the preset
^postpone the proof that 0 so defined has the arithmetical properties of
than C nwi| n “! C ° UP , |CS T mUCh le9S im Pottant, even in cardinal arithmetic
(.55 lud T56? U £!s Wi " C ° nSi, ‘ e,ed in the tw ° followin l? numbers
convenience of reference. The definitions of 0 and 2 are P
*54 01. 0 = £<A Df
*54 02. 2 = 5 !(a*, y) . a: =f y .a=-£‘xvi‘yj Df
*54 26. h s t*y «2 . = .ar + y
*64-3. h.2-a( (a
.6463, t-:««2. x .y«a.,r*y.3.a«i<x«t‘y
.6466. . +
360
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*54 01. 0 = e‘A Df
*54 02. 2 = a |(gx, y) . x+y. a = i*x w #‘y| Df
*541. = [(*5401)]
*54 101. h : a e 2 . a . (gx,y). a: * y. o = i‘x v *‘y [(*54 02)]
*54102. H : a e 0 . = . a = A |"*541]
The two following propositions have already occurred in *51, but are here
repeated, because they belong to the subject of the present number.
*5421.
*54 22.
*54 25.
\-:i*xu t‘y = i*x sj t*x . 5 .y = t [*51-41]
hM^u^y-^wi^.ijjr-i.y-w.v.flj-to.y-i [*5143]
b : l*X\J l*y < 1 . = . x = y
Deni.
h . *52*46'1 . *22 58 . D b : i‘x w f ‘y € 1 . D . / ‘a: u i‘y
[*20-23] D.i‘x=/‘y
b .*22*56. D 1* : (‘a; a t ‘y , D . (‘j; v t‘y = i‘j .
[*•’> 2 - 22 ] D.t‘xvi‘y € l
b .( 1 ) .(2 ). D h s i'x w |‘y c 1.3 . C*X« <*y.
[*51-23] =.x = y:Db. Prop
*54 26. b : i‘x v i*y e 2 . = .x + y
( 1 )
( 2 )
Dem.
b . *54101 . D b :: i*x wi'y? 2 .
= :. (gx. iv ). x<f w. ('su ('y a ( ( 2 u e‘ii;
[*5422] = (gx, w):x=J=?e:x = x.y = «/.v.x = u/.y = x:.
[*4 4.*1141] = :. (gx. w) . x 4= w. x = x . y = u;. v . (g* |M /) . x=f w • * = w.y = x :.
[*13*22] sr.x + y.v.y + x:.
[*1316] = i.x + y :: D b . Prop
*54 27. b . i‘x v 1*1/ € 1 U 2 [*54-25*26 J
*54 271. b . 1 \j 2 = a f(ga% y) . a = f *x v; f'yj
Dem.
b . *4 42 . D
*
b :. a = i ‘a u t ‘y . = : <T = y . a = / ‘a u i ‘y . v . x + y. a = I ‘x w i‘y ( 1 )
b . (1) . *11 11*341*41 . D b (gx.y) . a = i*x \j l*y .
= : < r A x ’ y) • x = y • a = t*x v i*y .v , (gx, y). x+y. a = i l x u i*y :
[*13195] -* = : (gx). a = i*x wt'x.v, (gx, y).x + y.a = t‘x w/ t*y :
[*22*56] = : (gx) . a = i*x. v . (gx.y). x + y. a = i‘x yj i*y :
[*521.*54T01] = : a € 1 . v . a € 2 :
[*22 34] =:ael v2:.Db. Prop
SECTION A]
CARDINAL COUPLES
361
*54*3. I-. 2 = a |(3*) .xca. a-1‘xel\
Dem.
>-.*52 1 .*10-35. D
H : (3*) ..Tea. a- i‘x «1. = . (3*, y). .r * a . a - i‘a = i*,j .
I *5i 22 * «y* ° 1
L' ~~ a,/3 J
[*51-231.*54I01] =.H(2:DK Prop
*544. .*-*«, .v .0-t‘xvi‘y
Dem.
h . *51-2 . Dhx,y € /3.D.(‘xwt‘yC/9:
[Fact]
l * 2241J 3.0-Sxsji‘u (,)
H . *51-25.3 h :./3C u ( *y. y^ t /9.3 : /3 C <‘x:
[*51 401] 3:£- A .v.£-t‘* (2)
Similarly I- :./9C <<* « «‘y .*~« ff . 3 . ff . A . v . 0- t‘u (;i)
I- • (2). (3). *3-48.3 9 W
f’:- / 9C ‘‘*»‘‘y.~(*.y»/3 ) .3 ! >9 - A .w./ 3 - ( ‘ iB . v . / s_ I <„ (4)
Ml).(4)..34-8.3
K^t C 12*^^^ (5 >
t d?:£•,, w
n 0 |Srr li : n | ,h0 " S * hat a clftss “"“‘ined in a couple is either the
“So . un,t class or the couple itself, whence it will follow that 0 and
1 are the only numbers which arc less than 2.
*5441. h::ae 2 .D :.£C® . D :0-A . v . 0. l . y . *« 2
Dem.
*52*1 . Dhs./9 = i^.v.^«t- y o. / 9 € i
h . *.>4-2G . D I- i.x^y . i &~ i*x \j i*y .0 . ft c 2
b • (1) * (2) . *54*4 . D
l-::x + y . D :. £ C t‘x w‘y. D : £ = A . v . £e 1 .v./? f 2 ::
[n 3 n 2 3^D ::a = ‘‘" U ‘‘ y ' x+y - 3: - /3Ca - 3: ^ = A - v - /9sl - v ^' 2::
'•:-(a*.y).«-«‘*w l «y. a: + y. 3 :.^Ce: / 9» A . v . /9e i.v. / 9 e 2 (3)
h. (3). *54101. Dh. Prop '
*54-411. h:.af2.D:5Ca.D.^f 0 ulu2 [*5441 102J
(1)
( 2 )
362
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*54 42. h::a e 2.D:./9Ca. a !/S./3+a.=./3« l «a
Dem.
^ . *54*4 . D h :: a = i*x \j t*y . D
, / Ca -3 ! ^ 5: ^A.v.4 = 1 ‘x.v.^,«y. v . i 9 = a:a!/i:
[*-4o3 ;>6.*ol*161] = z0=i*x. v . £ = i‘y . v . £ = a (j)
h • * 54 25 * Trans p . *52*22 + +
[*13*12] Dho = (‘/ U |‘/,.i-^ y .3. fl + / r ria: | 1 ^ (2 )
•■.(l).(2).Dh:a = i‘xu f‘y .x + y. D
0 = t*x. v .0 = 1*1/:
(g z). 2 € a. 0 = 1*2 z
£e/“a (3)
0 C a . 3 ! 0 . £ * a . =
[*>1*235]
[*37 6] _
K (3). *11*11*35. *54101 . DK Prop
*54 43. h:.a^fl.D:fln^ = A. = .au^ f 2
Dem.
K*54-20.DH:.a = ,^./$=^D:au£t 2 .s.x*y.
[*51*231]
[*1312]
K(1).*11*11*35. D
** v'£c 2 . = .an£=A (2)
h • (2) • *11*54 • *52*1 . D H . Prop
From this proposition it will follow, when arithmetical addition has been
defined, that 1 + 1=2.
= .l*x c\ i*y = A .
*.an£-A (1)
*54 44. H z.wtt'x u «‘y. D,.„.</>(*, «>):= • 4> {-r, x). <p (x, y). <p (y, x). <p(y,y)
Dem.
h . *51*234 .*11 *62 .Dh.r.wf i*x sj i*y . D, iir . </, ( z, w ) z = :
sel*xxj i*y .D z .<f>(z,x).<f> ( z t y) •
[*51 -234.* 10 29] = : <f> (x, x). <f> (x, y) . <f> (y, x) . <f> (y, y)z.Dh. Prop
*64 441. h :s *. w e i*x w i*y . **u/. . <f> (z, w) : = :.x=y: v : <f> (x, y) . <f> (y, x)
Dem.
h . *5-6 . D I- ss z, w c i*x u i*y. z + w . „. <f> (z, w)z = z.
z, w e i*x w i*y . D,„: z = w . v . 0 (z, w) z.
[*54-44] = : x = x. v . <f> (x, x) z x = y. v . <f> (x,y) z
y = x. v . <f* (y, x) z y = y. v . <p (y, y) s
[*1315] = zx = y. v . <f>(x,y) zy = x. v . <f>(y,x)z
[*13'16.*4*41] = zx = y . v. <f>(x,y). <f>(y, x)
This proposition is used in *163'42, in the theory of relations of mutually
exclusive relations.
*54 442. h :: x+y .Dz. z,w e i*x v i*y. .3 z , u .<t> (z, w)z = .<f> (x, y) . <f> (y, x)
[*54-441]
SECTION A]
CARDINAL COUPLES
363
*54-443. **:: .* + y : «£ (ar,y). = . £ (i/ t x ): D
*■ w e #‘<t v /‘y . 2 + w . D, ^ (* f . s . ^ |V»4-442]
*54-45. P (g*. w) ,;,» t /«.,• u /‘y . </> (*, „.).
= z <f> (•*". *)• V , <f> (.r, y) .V . <ft (y, ,v) . V . 0 (y, y) [*51-235]
*54-451. f- ::~^(.r,.r).^ 0 (y,y). } (g*. w»). *, W€ /** w t‘y. <£(*, w ).
= :<Hx,i/). v.(/,(y,.r) [*5445]
*54-452. I-:: - 0 (x. a). - 0 (y, y) ; <*> {x> tJ ). = . $ <r) . D .
(a*,*).* I *«l l *we‘y^(i |W ).s^( r| ^ [*54-451]
* 4-46. \-z(Z2,u^.t.wei'xu I'y.g + w.s.x^y [*54-452. *131516]
*54 5. H :. a € 2 . D : a C i‘z v «'
w
. a = /*z yj t*iv
Bern.
h . *54 4 . D
k ” tf.f* ” ‘* w • 3 : a - A . v . a - t‘t. v . a - iV. v . a - i‘t w i‘w
I". *54'3 . *2454. 3K:Hp.D.« + A
K * 54 ' 26 ^-* 13 ' 15 - 3 s Hp. 3 . a + 1 ‘»
M 3 )?.
D h : Hp. D . a + t*w
O)
( 2 )
(3)
<*)
4 * ^ oo’i? ^ ^ ' * 4) ‘ * 2 53 * ^ h f, !> ^:aC«‘:w i*w .D.a-t ( iw (5)
H • *22-42 .
^ • (5) • (6) .DP. Prop
*54 51. >-:.a<2./3elu2.D:aC£. = .a = £
Bern.
V . *54-5 .Dh.a«2.^ = t‘«u l‘w .Z>:aC/9. = .<* = #
P . (1). *11’11-35-45 . D
h: - af 2: (3*. w)- (3= i'zv i‘w :D:aC fl.s.a* 8
K(2).*54-271 . Dh. Prop
*54 52. ^:.o,/962.D:aC/9. = .a*^. s#/ 3Ca [*5451]
*54-53. H : a c 2 . a:, y c a . a- =f: y. D . a ■= «‘x w «‘y
Bern.
K »ol-2. 3 H : Hp . 3 . t‘x C a . i‘y C a .
[*22-59] 3 . i‘x u i'y C a
h . *54-26 . 3 h : Hp. 3 . l‘x U(‘w f 2
M1)-(2) . *54\52 . D K Prop
*54-531. h a e2.D:x ,yf«.i + y. = .a = t ^u ( ‘y
Bern.
p.‘^g- Exp -^l; ! - a ‘2-3 : *.y*«.*+y.3.«» t «xw t * y (l)
kS:*: IX(4>
(«)
(0
( 2 )
( 1 )
( 2 )
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*54*54. b:.ae2. = :x,y€a.x4=y. 3 x>y .a = t*x u i*y : (gx.y). x,y ea.x^y
Dem.
b . *54*531 . *11*11*3 . 3b:.a*2 . 3 : x,y « a . x=^y . 3 X#?/ . a = t‘x v «‘y (1)
H . *51*16 . *54*101 . 3 b : a e 2 . 3 . (gx.y). x,y e a . x4=y (2)
f-. *53 . *3*27 . Dhr.ar, y€a.x4 s y.3.a = t‘xu i*y z 3 :
x, y € a . x 4= y . 3 . x + y . a = £*x w t‘y :.
[*1111 *32*34] 3 I- s. x,y c a . x + y . 3 XtW . a — i*x sj i*y s 3 :
^3^' .V) • a*, y e a . x 4= y . 3 . (g.r, y) . a: 4= y . a = t‘x w/ i*y (3)
I- .(3). Imp . *34101 ,3b:.x, y«a.x4=y. 3 xy .a = i < xv t*y :
K,1). W .(«. DKP„.„ — » (*>
In the above proposition, ** x. y e a . x 4 s y. 3 x . y . a - e‘x v /‘y" secures that
a has not more than two members, while "(gx, y). x, y € a . x 4= y" secures
that a has not feiuei • than two members.
*54 55. KOwl v* 2 = 5 j.r,y « a .x4=y. 3,.„. a - i'x \j i‘y|
Dem.
H . *4 42 . 3 b ::x,y * a .x4=y . 3 X §( , .a = t‘/ui‘y: = j.
x,yca .x4 s y. 3 X ,„. a = e*x vi'y:~ (gx, y). x, y € a . x 4= y
v x,y « a . x + y. 3 x>y . « » i*x v t*y: (gx, y). x, y c a . x + y (1)
b.*11 63.3 b :• ^(g r.y) ,.r,y«a . .r + y. 3 :x,y«a . x + y. 3 X>|/ . a = i‘xv i‘y
[*4 71] 3 b :..r,yca . x4=y . 3 Xi „. a = £‘.r v £*y : ~(g.r,y) . x.y e a . x4=y : s :
~(gA-,y).x.y«a.x4-y:
[*11*521] = :x,y«a.3 Xt( ,.x = y:
[*52*4] =:a«0ul (2)
b . (1).(2). *54*54.3
H:.x, yca.x4 : y. 3 x>f/ . a = i‘x w c‘y :a:ae0vl.v.ac2t
[♦22-34] =:a£0ulu2:.3h. Prop
*54*56. h:a<v £ 0 ul \j 2. = . (gx, y, z). a*, y, a € a . x + y . x 4= *. y 4 * 2
Dem.
b. *54*55. *11 *52.3
b :. a ~ e 0 u 1 w 2 . = : (gx, y). x, y € a . x + y. a +1 ‘x u i‘y:
| *51 *2.*22*59] = : (gx, y) . i*x vAt < yCa.x4=y.a + *‘x w t‘y :
[*24*6] = : (gx, y).i‘xwt‘yCa.x4y.g !a-(t‘x w i*y) :
[*5 1 *232.Transp] = : (gx, y) : t‘x v i‘y C a . x 4= y : (g*) .zea.z^x.z^yi
[*51*2.*22*59] = : (gx. y,z).x.y,^ca.x4=y.x4=z.y4=A:. 3b. Prop
In virtue of this proposition, a class which is neither null nor a unit class
nor a couple contains at least three distinct members. Hence it will follow
that any cardinal number other than 0 or 1 or 2 is equal to or greater than 3.
The above proposition is used in *104*43, which is an existence-theorem of
considerable importance in cardinal arithmetic.
SECTION A]
CARDINAL COUPLES
365
*54 6. H a « £ = A . .r, x * a . y,y e/3 . D :
ZW ‘^^ = »Vu,y. 5 ., = ^ // = y .
r*o* S ioi ’ 3 h ‘'■ Hp ‘ ?' *1* c " • 1 ‘* C a ■ <,y C # • *y C 0 ■ a n 0 = A :
D:i
-**jr-i‘*w*y.».,.,_ (V . l . y _ ( y.
[*2448]
[ * 5 , 123 ^ ■ 3 k Prop
Ihe above propos.t.on .s useful in dealing with sets of couples formed of
one member of a class a and one member of n class 0. where a and 0 have no
(*113148)" COmm ° n ‘ U ^ USe<l ^ thC thC ° ry ° f ca,dinal multiplication
*55. ORDINAL COUPLES
Suminnry of *55.
Ordinal couples, which are now to be considered, are much more important,
even in cardinal arithmetic, than cardinal couples. Their properties are in
part analogous to those of cardinal couples, but in part also to those of unit
classes; for they arc the smallest existent relations, just as unit classes are the
smallest existent classes. The properties which are analogous to those of unit
classes do not demand that the two terms of the couple should be distinct,
i>. they hold for i*x | t*x as well as for i*.r f i‘y (where x + y); on the other
hand, the properties which are analogous to those of cardinal couples do in
general demand that the two terms of the ordinal couple should be distinct.
The notation is cumbrous, and does not readily enable us to
exhibit the couple as a descriptive function of .r for the argument y, or vice
versa. We therefore introduce a new symbol, " x ^ y," for the couple. In a
couple .r l y. we shall call x the referent of the couple, and y the relatum. In
virtue of the definitions in *38, this gives rise to two relations and ^ y;
hence we obtain the notations xi \ y“a, a 1 y, a 1 “8 and so on, which
will be much used in the sequel. It should be observed that.rl“/9 means
the class of ordinal couples in which .r is referent and a member of /9 is relatum,
while l y lt a or a l y denotes the chiss of couples having y as relatum and a
member of a as referent; a J, denotes all such classes of couples as l y“a.
where y is any member of /9; ami in virtue of *40 7, s*a X denotes all
ordinal couples of which the referent is a member of a, while the relatum is
a member of /9. This is a very important class, which will be used to define
the product of two cardinal numbers; for it is evident that the number of
members of s‘a J, is the product of the number of members of a and the
number of members of /9.
The first few propositions of the present number are immediate consequences
of the definition of x ^ y and the notations introduced in *38. We then pro¬
ceed to various elementary properties of the relation x \y, of which the most
used are the following:
*5513. h : z {x l y) ru . = . z = x . w = y
*5515. H . D‘(* l y) = i‘x . (I‘(a: | y) = i‘y. C‘(x l y) = i‘x v i*y
*5516. H : D ‘R = i*x . (Pi* = i‘y . = . R = x l y
*65202. \-:xly = zlw. = .x = z.y = w. = .ylx = iulz
SECTION Aj
ORDINAL COUPLES
30 7
This proposition should be contrasted with *54 22 is <ri v in<r
virt y ue^f n th e C0 Z l,S a,e m °' C " Sef " 1 “ ri,h '" etic than cardinal couiWeTln
referents ZtSSESt **
We proceed next to various properties of the relations a? I md l mm
sir s,*r; r v^- ■■ - srsr ,/r sr
•55232. =
This proposition is frequently useful.
are Ull.^loThe p^rt^’o} »jf “LT^ °' *- A » Whioh
of these properties are the following: g n,0re ,n, P ort “‘
* 66 ’ 3 ht * fi y-»-*iyea.».ai(*,iy ) A*
This is the analogue of *51-31.
* 8534 . I -■■alR.RG*ly. 5 .li =xl!/
This is the analogue of *514.
* 555 . h:.i4C*iy 0 , lw . = .
Th . . , R = A ' v - R=xi y- v - K= ‘i”-''-R-*i ! ,»ziw
I his is the analogue of *54 4 *
*5B 61. h : E! *.. E ! . 3 . <«„ S). (l | w) _ (ft.,) j
368
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*55 01. x\,y = i t x J \i t y Df
*55 02. R‘s = R‘(x l y) Df
This definition serves merely for the avoidance of brackets.
*55 1. b . x i y = (f‘x) | (i‘y) [(*55*01)]
*5511. b l y‘* = x l y = i*x J \ i‘y [*38*11 .*55*1]
*5512. b. Elxl‘y [*55*11. *14*21]
*55 121. h.E!|,y‘x
*55122. b : R(x l)y . = . R =x l y [*5511]
*55123 b:R(ly)x. = .R-xly [*5511]
*5513. l - :2(/|y)w. = .:=/.w = j/
Dem.
h . *35 103. *55 1 .Db : z(x l y)w . = . z c i*x .we i‘y .
[*51*15] = .* = .r.w = y:3K Prop
*55132. Kx(x|y)y [*5513]
*55 134. h.g!(x4,y) [*55*132]
*5514. b . x i y = Cnv'y l x [*55*13. *31*131]
*55*15. b . D‘x i y - i‘x. CI*r 1 y = i‘y. C*‘x | y - i*x w i‘y
[*35*85*86. *51 *161]
*5516. b : WR = i*x . (l*/£ ■= i‘y. ■. R - x ^y
Dem.
b .*33*13131 .*51 15.3
b :: WR = i *x. (I 'R = i‘y. = :. (gw). zRw. = z . z = x: (g z).zRw . = IP . w = y :•
[*14122] = :.(g*, w). zRw :(gw). zRw . 3,. * = x:
(gw, 2 ) . zRu ): (gx). *7fw. 3*. w = y:.
[*11 *23.*4*711 = :.(g*, w ). zRw : (gw). zRw . 3,. *= x: (g*). zRtv.D, 0 • w= y
= (gz, w ). r77w: zRw . 3, <IP . * = x: zRw . 3 X ., P . w = y :.
= :. (gj, w). *77w : *7?w. 3,., P . z = x. w * y:.
= :. zTfw . =,. IP . z = x. w = y :.
= :.*/fw.=,. IP .*(x,ly) w:.
= :. 7? = xly::3h. Prop
[*10*23]
[*11*31)1]
[*14*123]
[*5513]
[*21*43]
The above proposition is important, and will be frequently used.
*65161.
Dem.
b . X l y =7‘/e (D*R = f‘x. (I‘T? = i‘y)
h. *55*16. *2015.3
h . R (D‘/e = i‘x. CI‘77 = *‘y) = R (R = * l y)
[*51*11] -«‘(*ly)
K(l). *51*51.31*. Prop
(1)
SECTION Al
ORDINAL COUPLES
*5517. h..,ly = r‘(lT‘/Vntr‘#^) [*55-161 .*33<i<il|
*55 2. l-:ar|y = .rj2. = .y = *
Dem.
K *30-37. *5.Vii,2.DH:y= i .D.. rl ;
K *3037. *33121 . D
: i- i ^ . D . Q*..- i = U'.,. I z .
[*5315] 0.i‘ ;/ = i‘ s .
[*51-23] D.y- 2
*■•(!).(2).DI-. Prop
*55 201. I- „
*65 202 H : • 1i . s . = ..,, . ,, | .
Dan.
f-. *55-2-201 . D
[*1.317]
K *.30-37. *.3812121 . D
!" : x\,y -liw.D. DV I y - DV | *. (IV1 y - <J‘* I «..
[*5515]
[*51-2.3]
*--(l).(2).D
h * a; i y | m; . = . x = i . // « w
Similarly
*■ • . (4) . D I- . p ro p
'I’ho above proposition is important.
*55-21. h . Q'x i = V . G‘ i x - V [*33 432 . *55 12 121]
*55 22 I-. !)<*;-ft |( ay) . ft = x i y j [*55-122]
*55 221. H.n‘ix = fl[(ay)./e = yix | [, 53 ., 23 ]
*55 222. h : ft f D‘x i . = . D'ft - . Cl'ft ( 1
Dem.
h . *55-22 16 . D I-ft «D*«|. B : (g,,). V . R , ( « x . a</i = t , .
! : : D‘ft = : , 3 y). a-ft = t ' (/ :
1 * ’ 2 1 ] = : D'ft = t ‘x. a-ft »1 3 K Pro|
*65 223. hfi(D‘|j,=, a‘ft = e'x. D'ft . 1 [Proof as in *56-222]
*55 224. l-.D‘xir,D‘].y = ( <(xi„,
Dem.
I-. *55-222-223 . D
b : R € DV i n D‘,[y . = . D‘/£ = t'*. Q‘]{ f 1 . CE‘/* = l*v . D‘.K e 1
R 4c w i ^
370
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
[*52 22.*4 71] = . D*R = i*x. d‘R = i‘y .
[*•”>5*16] =. R = x l 1 / .
[*5115] = . R e i\x l y) : D I-. Prop
*55 23. b . x l “a = R {(gy). y * a . R = x i yj [*3813]
*55 231. b . I u “a = R {(gy) . y e a . R = y | x\ [*38131 ]
*55 232. b : g ! I ./“a n J, . = . x-. y . g larsfi
Deni.
b . *55*231 . *11*55 . D
H g ! 1 x u a r\ j y“0 . = : (g/f) : (g*. w) . z € a . R = z l x . w c £. R = w | y :
[*13195] = : (g», w). z € a . w e (3 . z l x = w ^ y i
[*55*202] = : (g z, w ). z € a . w e ft . x = y . z = w :
[*13*195] s : (g*). z *a r> 0 . x = y :
[*10*35] =:g!aA/9.j-=sy:. Dh. Prop
*55 233. b : * + y . D . I .r"a n i //“£ = A [*55*232 . Transp]
The above two propositions are frequently useful in arithmetic.
*55 24. b . l “a-iVrfa
Dent.
b. *41*11 . D
h 5 (ttc 1 “a) . = . (g R) • ft c « 4 “ a. zRw .
[*55*23] = . (g/t, y).y ca, R = x l y. zRw .
[*13*195] = . (gy). y € a . z (x | y) w .
[*55*13] = .(g y).yea.z = x.w-y.
[*13*195] =. z = x .w e a.
[*51 * 15.*35* 103] s . z (i‘x f a) w D H . Prop
*56*241. b . *"a - a |[Proof as in *55*24]
*66*26. h : g ! a . D. D“a: | “a = f‘«‘x
Deni.
b . *37*07 . *33*12 . *55*12 . D
h : £ e D“x l“a . = . (gy). y e a . >9 = D‘ar ^ y.
[*55*15] = .(gy).yca =
[*10*35] =.g!a./9 = £‘x (1)
b .(1). D b Hp. D : 0 e D“r J “a . = . £ = t‘.c.
[*51*15] = . /9 e tVxD h . Prop
*55*251. h : g ! a . D . (1“ i x“a = i*i‘x [Proof as in *55*25]
This proposition is used in the theory of cardinal multiplication (*113*142).
*55*26. b . d“x J, “a = i“a [*55*15 . *37*35]
*55*261. b . D“ l x“a = /“a [*55*15 . *37*35]
*55*262. b : l x“a = | y“/9 .D.a = f3 [*55*261 . *53*22]
SECTION A]
ORDINAL COUPLES
371
Z • !/ta 10
*55 27. h . C« i x“a = C“, V ‘ a ^ {(3y) ., « « .* = „ , y] [*55-15]
*55-28. h : d-x J y = d‘x | ,. s .y = , . s ., j = , , .
[*5515 .*51-23. *55 2]
*55281. =
*55-282. hC‘xl^C'4^.^,.3.,^..,^
[*5515-2. *5421]
*55-283. t-:&l,l — C‘,l*. m . 9 -, m3m9ljcm , ii
*55-29, K a | (*],) = i [*55-15 . *34-42)
*55 291. h . I>m «)_ , [*55 15 . *34-42[
*56-292. K . C|(x 4.) = C; (|,x) - 3?(a = i«.r M ,‘ y) [*55 15 . *34-41]
The following propositions, down to .55 51 inclusive, give properties of
ordinal couples winch arc analogous to the properties of unU cJJ
*563 Hxfty.s.xiyCA.s.al^i^A^ [»l 3 -21 -22 . *55 13]
I he first half of this proposition is the analogue of *51 2 - like that
propusitio " st0 thc *>"»
*55 31. = =
I his proposition is the analogue of * 51 * 23 .
Deni.
[.SMS 1 ]" • 3 *• ! * 1» - ■ ‘i • ■• ■ • n**! s - .•«. a-x i y - t v
[*5123]
[*55 13]
[(l).*1310j
[*5513]
I" • (1). (2). (3). D h . Prop
*5532. h:.x,ly,S,4 u , =! A. = :*+*. v .y*
Deni.
H . *55 3. D b :g Ixiy * z ± w. s .x(s 1 w)y.
[*55131 _ „ J
J J = .x = Z .y r=iu
. ^ • (1) • Transp . D H . Prop
*65 33. \--.xR !/ . = . xiyf . R = xiy [* 55 - 3 . *23621]
*6534. ha!fi.fiGxi,. = .fl,,u
Deni.
w™-* hu *'*-****'--'M.**'*.^ m .
[*5513] : : „ •=--■-* = *; •» - y «
- . Ztiw . =,. «, . z (x X y) w D I-. Prop
24—2
~ . x r= z . y = w .
s.x(z l w)y.
3 . z =* x . tv =* y .
= . z (x l y) w
w
( 1 )
( 2 )
(3)
0 )
X . W mm y :
372
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*55 341. t-:./ee a i J . = :i? = A.v.R = x|y
Dent.
h +42 ■ 3 h : • H G *1 'J . = : R G 1 y ■ 1< = A . v . if C x | y . R * A :
[* 2 . 5 - 54 ] = : if C*iy . R = A . v . RQ X \,y .g ! R:
[*55-34] = =
[*2512] e:/f — A.v.iJ-t.r^y:. Prop
*55 35. >':R*sl!/-A.Rvxl !/ -S.s.xS ! /.R=S
Deni.
h . *25 47 . D
h:/fr,.riy=A.«c;*|y = .Sf. = . a : 4 , J ,CS.i{-S^ a; | y .
t* 55 ' 3 ) Prop
*55 36. I-. = .(Itx l i/)sy.T X i/= It
Deni.
h . *55*3 . D h : xRy . = .x l yd R.
[*23(52] = .xly*R = R.
[*23 91] = Prop
*55 37. ^:xfo.yc/9. = .x|yGa|/9
Deni.
K . *35‘103 .Dh:;rca. ( yf/9. = ..r(a:J‘/9)y.
t* 55 *3] =-xlyQa\R:^\-.Vrop
'1’he following proposition is the Analogue of *51232.
*55 4. *-:-(i\xl!/vzlw\b. = :(i = x.b = !/.v.a = z.b = w
[*55*13. *23*34]
*56 41. h:/e=.r|yuzi w .D:. aRb . D fl . *. <f> (a, b ): = . <f> (x, y) . <f> (z, w)
Deni.
H . *55 4. D h Hp. D :: aRb . D a b . 0 (a, 5): =
a « a:. 6 - y. v . a = r . b - w: D a> 6 . <f> (a, 6 ):.
[*4-77] s (a, b )a * *. b = y . D . <f> ( a , 6 ): a - *. 6 « w . D. <f> (a, b)
[*ll-31]=s.(a.6):a-ar.5-y.D.^(a,6):.(«,6):a = ^. 6 »w.D.^(a, 6 ):.
[*13 21] = <fj(x,y ) . <f>(z,tu) D h . Prop
The above proposition is the analogue of *51 234. The following pro¬
position (*55-42) is the analogue of *51235.
*55 42. h: R = xlysvzlw.3 :.(y«, b) .. <£(«, b) . = : <£0r,y): v . <f>(z,w)
Deni.
h . *55-4.3 1-::. Hp . D :: (go, b ). ci/M . <f> (a, b ). =
(a«. b) a = X . 6 = y . v . a = z . b = w : <f> (a, b)
[*4 4] = (ya, b) : a = x. b = y. $ (a, b): v : a = z. b = w. <f> (a, b)
[*11 *41 ] = (ya, b) . a = x . b = y . 0 (a, 5). v . (ya, b). a = z. b = w. <f> (a,b) :•
[*1322] = <f>(x,y) .v.^(j,ti»)::.Dh. Prop
SECTION A]
ORDINAL COUPLES
*55 43. H : , I „ c i n. = , J y „ r i ,/. s . . = r . = . a . . ^ =; f/
I his proposition is the analogue of *5 1 41 .
Deni.
H . *55202 .Dhzrrc.j^f/.D.:! Wsf i f/t
H* 2 *2iH8 3(- | 3 • * I .* «* * i «• - l .'/vr 1,1 , |,
• 08 ' 3 h i "-.rijor 1 ./. D :
r*55-3 X3 ■> ^ M ' Ge A * I ‘ D' C.-i y o * J :
Si? 3 —=,-v.,=c.»-rf:c-«.rf.y. v . t ..e.—rf,
tl3 721 3-*-'.—jr.e-*.rf- y . v .,. e . „. rfl
L*lo 172J D:*=»c./p = rf
h. (3). *55-202. Dh. Prop ( {>
*55 431. h.a'^c/:j w=ia j^ c | f/|D;
Deni. *-a.y-6.,_ e . w _ d . v . x _ c y = (/ fi-(( w _ &
*554. D hs: Hp. a :.«-«.,..y. v .
r*ii ll v . « = c . v =*<!:.
1 1J 3:.ar-*.*-*.
[*13-15] 3 ,,-a.,.5.v.^r i - V - X ' C - y -' i! ; i)
[*55 43] 3*Hp.3.« 1 * w , il ,_ oAtl8 , eArf>
-- <;>
h -(l)-(3).(4). D h . Pr.,p P ' ' y “ =
*6644. l-:.*4,yK/*i w = ai 4 K , c . i(/
=:*««.y-4. .x-c. v .«- e .y = ,/ . j ,
Oem. = =
K.5S-M.
«-* <*>
H. (1). (2). *55-431-202.3 I-. p ro p 3 - * i y « * i '" = « | 6 c c 4, d (2)
The above proposition is the analogue of *5143
374
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*55 5. H :. RGx^ywz^tv.
= : R = A . v . I{ = j l y . v . R = * l w. v . R = .r l y v z l w
I)ein.
h.*25 12.*23 5K-42.D
h R = A . v . R = .r ^ y. v . R — z ^ w . v . R = x X y c; z ^ w :
'Z.RGxlysjzlw (1)
I - . *25 49 . D I- :./{ G x l y v z l w. I{ r\.r l y = A . 0 : It G z l ir :
[*55-341] Di/?-A.v./?-*4w ( 2 )
f- . *25 43 . D h C r ^ y \j z ,[ /#•. D : R — x ^ y G z j, w :
[*55-341 ] D : /U x X y = A . v . R ± x J, y = z | w :
[*2524.*23-551 ] D : (/f — *Ay)V'.*iy-x,J,y.v.
(3)
h. *55*3*3(3 .0 l (R * x l y) • 0 • (R — x l y) v .c l y = R (4)
H . (3). (4). D H It G x ^ y c/ j | 10 . y ! {R r\ j | y ). D ;
R =** i y . v . R = xj y v z X w (5)
h.(2).(5). D h:.
R = A.y.R-xly.y.R = zlw.y/.R^xlysjzlw (0)
K(l).<6). DK Prop
The above proposition is the analogue of *54*4.
*55 51. I- R G * X y ci S. D s xRy .v.RGS
Dem.
h . *55*3 . D V : y ! ( R A .r | y). D . xRy (1)
h. *25-49. Dh: Hp.-g ! (i* A.r i y). D . RQS (2)
h . (1) . (2) . D h . Prop
In the remainder of the present number, we arc concerned with properties
of ordinal couples which have no analogues for unit classes.
*55 62. h = wvylzvylw [*35*82-413]
*65521. \-:x*y. = .xlyGJ [*553 . *50 11 ]
*65 63. I-x * y . D : C*R - i‘x u i‘y. R Q J . s . 3 ! R. R G* J y vy J, a:
Dem.
V . *555 .Dh:.g!,ft.7?Gx,lyciy|x. = :
R = xXy.v. R = yXx. v . R = x l yw y l* (1)
. h. *5515. D h . C'xiy = v i‘y. 1 x=i‘a-v ( 2 )
h . (2). *33 262 . D h . C‘(x | y c; y A x) = t‘x w‘y (3)
H. *55521 . Dh:x + y.D.xiyG/.y|xC/. (4)
[*2359] D .x Xyv y ( 5 )
h.(l).(2).(3).(4).(5).Dh:.
x4=y.D:g!i2.#G^Ay^yi^-3- o*R = i‘x v i*y . r a J (C)
1-. *35-91 . D 1-: C‘R = i‘xv i‘y .0 . RG (t‘x v <‘y) t(«‘xv i‘y) .
[*55-52] D./?Gx|xc;x|yc;y|xc;y|y (7)
SECTION A]
ORDINAL COUPLES
t-. *50 24 . D 1-: ft C J. 3 . ~ ( x Hx ). ~ (y7ty).
[*55'3.Tr,insp] lK .
*55-54. h :: ,r +. D :. C-ft-,** u ,«„..R A 5 - A . ■ , A - , * y. y . * j ,
JJem.
H.*60-40.*471.Dh:HAS-A.«.i8cy.ilAji-A ( 1 )
h * (1) ‘ * 55 ’ 53 • ^ *■ :s ar + y. D :. C‘/e - i‘.r sJi*y.RAR~\.
= : g! i* . R Qx 1 y v y | x . ]{ * R * A :
ri 4]: ^ r ^' v ; ^' y 1 * • v ^ 1 y 0 ^ ^ ! ^ ^ - A < 2 >
K . *55 32.Dh:.tf + y.D:x|y*y 4 ,a:«=A:
[*5514] 3:ii = a-iy.3./enA-A:
ft“yi*-3-ftAft-A / 3)
K*5614.*31-15-33.3l-:iJ-x|y«, y |,o.*-fl.
[ * 235] 3,/(n« = fl.
[*55;134] D . 3 ! ft A ft (4)
H .(3). (4). *4-71 .*5-71 . D '
^::* + y.3:.ft- ; tiy.v.ft = y| x . v . / , = a . iyl; , yi;r:/f(S J J _ A:
K(2).(5). 3 K Prop = : ft-*4y.v. ft-y;* (5 ,
*55 67. H . ft |(x 1 y) = ft‘ x f ,-y [*37 81 . .551 . .53301 ]
*66 571. h .(*iy)|ft-«‘xtft‘y
*56672. l-.ft|(x4,y),V_lJ< z |S‘ y [*55-571 ..37-81]
*66-573. h . ft|(* l y )|S = ft«x fs<y [*5'572 ® j
* 6668 . t-:E!ft‘x.D.ft!(x 4 .y, = (ft^ )i y (.55-57 . .53.31 ..551]
*55 681. I- : E! S‘y. 3 . (* ; y)|,S' = * ( 5 < y)
*65-682. hE!fi. I .E,^y. 3 . Ji | (li y ) | S = wl( J,y ) [.55-58-581]
*66-583. h s Elft'x. Elft*y. 3 . ft|(*iy)|5= ( ft‘* ) ^ ( S‘y ) [.55 5821]
The above propositions are frequently useful in arithmetic. Their use
thHektion ft 8 ' 8 * V ’. 8 te Cla8SeS ° f Which ° 13 “Elated with y by
the relation ft, and 0 with 8 by the relation S. Then if 7 . y ,-S, the
376
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
couple consisting of the correlate of x ami the correlate of y is (R‘x) A (S‘y),
i.e., by the above. It (x l y) S, i.e. (R || S)‘(x A y ). Thus the relation rt||S
correlates the couples, in a ami composed of the correlates of terms in
7 and $. The most useful form, in practice, of *55 583, is that given below
in *55*61.
w * *
*55 6 b.(R S,U: A tu)= R'z^S'w [*55*573 . *43112]
*55 61. b : K! It*:. Ml S*,n .D .(R S)*<: l w) - (R‘z) A (S‘u>)
[#55*5X3. *43* 112]
*65 62. biz^m. S = x l .s 4 * = r. S*w - y
Dom.
I-. *55 13. Dhs: Hp.D:. uSz . = : u m .#•. t — z . v . u — y . z — w (l)
I-. (I). *13* 15 . D V s. Hp . D : uSz . = .h = x (2)
Similarly I*:. Hp.D: uSiv. =. u = y (3)
b . (2). (3). *30*3. Db. Prop
*55 621. b : x 4 = y . ,S' »= x A z kj y \ w . D . 5‘.c = 2 . S*y = w
[Proof as in *55 62]
I he four following propositions belong to *43, but are inserted here because
the proof uses *55*13.
*5663. b
Dent.
'AlQ*8.P\\Q-R\S. , Z.l>mR
b . *43*112 . D b s: Hp . D I* (y l 2 ) Q— R\(y A j)| 6 ’
[*341 ] D (gii, v) . xPt .u(yl*)v. vQw . = XttP .
(gw, v ). xRu . «(y |:) w. vSw
[*55*13.#13*22] D xPy . zQw . = Xi , r . xlty . zSw
[*4*73] D zQw. 2 Sw . D, c : xPy . =* . a: Tty (1)
h . (1). *10*11 . *11*35 . D h Hp . D :xPy . = x .*7ty (2)
b. (2). *10*11*21. DK Prop
*65 631. H:g!7 , n.R./ , ||Q=.7?||S.D.Q = £ [Proof as in *55*63]
*56 632. hjPUQ-^H^.gSP.gJC.D.glPA/f.glQA^
Dem.
b . *55*13 . D b : *7ty . . D . x (P | (y A *) | Q] w .
[*43112] =>.^l(/^||<2)'(i/4,^)J(1)
h . (1) . D H :• Hp . D : a*7ty . zQw .D.x {(7? || S)*(y A z)\ w .
[*43*112] 3 .*(/2|(y A *)jS) w.
[*34*1] *3 . (gw, v ). .s(yir)t>. ySw.
[*55*13.*13*22] D . */ty. zSw .
[*4*7] D ,x(P r\R)y . z(Q r\ S) w D h . Prop
*65 64. h:.g!P.g!g.v.g!/e.g! S: DzP\\Q = R\\S. = .P = R.Q = S
[*55*63*631 632]
*56. THE ORDINAL NUMBER 2, .
Summary of *56.
In this number, we have to consider the class of those relations which are
“ , { “ S ‘,“ gle CO,,pl '‘- 1,1 CnsC the tl ''° of this couple
a.e not identical the class ol such relations is (as will be shown Inter) the
rdinal number 2 , which, to distinguish it from the cardinal number 2 we
c ,r e A r • Hcre the S,lttix is i,,tL '" de ' 1 suggest “relational.") The
Cl^s of all relations consisting of a single couple, without the restriction that
the two members of the couple are to be distinct, will be denoted by
number 11 ! Z b ° ob8er ved t,lnt there is no ordinal
number 1, because ordinal numbers apply to series, and series must have
fuUv when ° nC '" em ' 1 ,r , they havc a "y This will appear more
fully when we come to deal with series. 1 *
of 2 T «r* P r P ° rtie8 r f 4 “ re lar p ly annlo 8 ous to those of 1 , while the properties
Of z r are more analogous to those of 2. 11
in the'r'TT ° f th ° ,,,eSCnt nu,nbel ' nre sc,dom '•‘■•ferred to
pronos, , ' : ,h 8 " rerere T 8 “ OCCUr nrc im P° r tant. The most useful
propositions in the present number are the following:
*56 111. h : R « 2,. = . D <li, (1‘Hc 1 . D ‘R = A
*56 112. K : R « 2 r . ■ . D*R, (l'R e 1 . C‘R e 2
*66113. K2 r «2nC“2
Observe that »C“ 2” means “relations whose fields have two terms "
•56T3. h.2-2, = ^(( aa ). fi = „ ia)
*6637. b: Re 2 r . = . C‘Re 2 . R * R = A
JZ 2 r is the class of asymmetrical relations whose fields have two terms
*66 381. h : C‘R = i‘x . = . R = x l x
*66 39. h.2-2 r =C“l
idemL| the r .u ati °? 8 Which are COUple8 wl,ose rcferen t and relatum are
dentical are the relations whose fields consist of a single term.
.66 01. 2 = “ Df
.6602. 2 r = ^(( 3a;> y ) . a . + J ,_ R = ;t . iyl Df
*66 03. 0 r =,‘A Df
.661. h:ii«2 . = .( ai r,y).ii = a : Aj , [.203. (.56 01)]
378
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*66101. = <Kft«l
Dem.
b . *55 16 .*11 11*341 . D
- (3*. y) • R = x J, y . = : (g*. y). D‘ft = i*x . (Fft = i*y s
t* 1154 1 s : (gx) . D‘ft = £‘.r: (gy). a*/? = t‘y :
[*521] = : D‘ft, d*R e 1 (1)
K(l).*56l . DK Prop
*66102. h.2=D u ud"l
Dem.
b. *56101 . *37*106 . D
H : Re 2. = . ft c D“l . ft«(J“l .
[*22-33] = . R c D“1 a CI“1 :DK Prop
*56103. Hft € 2.I>.g!ft
Dem.
b . *56101 . D b : R e 2 . D . D‘ft € 1 .
[*5216] D.glD ‘ft.
[*33-24] D . g ! R : D I-. Prop
*66104. b:R € 0 r . = .R = A [(*56 03))
*6611. b : R c 2 r . s . (g*. y).x + y./i=x|y [*203 . (*5602))
*56111. h : R e 2 r . s . D‘ft, d‘ft c 1 . D‘ft a CI‘ft . A
Dem.
b .*51-231 .*55 16. D
b z x + y . .ft = x l y . = . i*x r\ i‘y = A . D‘R « i‘x. d‘ft = e«y .
[*13193] = . D‘ft a d‘ft - A . D‘ft = . d‘ft - t‘y (1)
b .(1). *5611 .*1111-341 .D
b:.Re 2 r . = : (g*. y) . D‘ft a d‘ft = A . D ‘R = . d'ft - i‘y:
[*11-45] = : D‘ft a a *R = A : (g*. y) . D‘ft = t*x . d‘ft - i‘y :
[*11-54] = : D‘ft r\ d‘ft = A : (g*) . D‘ft = t‘x : (gy) . d‘ft - i‘y :
[*521] = : D‘ft a d‘ft = A . D*R, d‘ft cl:. Dh. Prop
*56-112. b : R e 2 r . = . D‘ft, d‘ft e 1 . C*R e 2
Dem.
b . *56*111. *64-43. D
H : ft e 2 r . = . D‘ft, d‘ft c 1 . D‘R u a*/? c 2 .
[*3316] s . D‘ft, d‘R e 1 . C‘R c2:Dh. Prop
*66 113. h . 2 r = 2 a C“2
Dem.
I-. *56112101 .Ob:Re2 r . = .ReZ.C‘Re2.
[*37 106.*33-122] = . R e 2 . R e C“2 .
[*22-33] . = . ft € 2 a C“ 2:Db. Prop •
SECTION a]
THE ORDINAL NUMRRR 2 r
*56114. h.2,-D“| nU“! n c ut '2 [*50-113-102]
*56 12. h : R € 2 r . = . R € i >. R Q J
Dem.
K*553. *50 11 . 3hu'+j, = . 3 . lyCJ:
[Fact] D Irt = .,■,[ yf y . = ./{ = I i/.x l yG J.
[* 13193 ] il.jiV!/ <i,
H .(1).*11-11-341.3
'•-(a*.}'). K = x iy.x^y. = i(^ x ,y).R = xly.RG.J-
5 * 11 ' 45 !
t* 56 ' 1 ] s-.Rti.RCJ (2 )
h. (2). *5611.3 h. Prop
*56121. K2,C2 [*56113]
*56122. h:/f«2 r .3.3l« [*56121 103]
*5613. b.2-2 r -Ji|( a a)./i_ 0 |a)
Dem.
K. *561 l.*l 1-52. Transp.D
H:JJ~f2 r . =:rt-*.J,y.3 *- y ,,,
Ml). *36-1.3
H:.JJ«S-2,.B:( a a,i)./J-ai6: ^-«4,y.3^ a: = y:
[*1145] ^■■O<‘.b):R.alb:R.xly.0 I , u ,x = y:
[*13-193] = : ( 3 a, 6):Ji-al6:o|6-*|y.3 jr ..«- V!
[*55 202] s:(3a,6)s« = a|6:a_ x .6 = y.3 I ' .x = y:
[*13-21] 3 :(a«,ft)./e = ai6.a = 5:
[*13'195] = : ( 3 a). « = a|a:.3l-.Prop
2-2 might be defined as the ordinal number 1, since it is what we shall
call a relation number (cf.,153). But we wish our ordinal numbers to be
classes of serial relations, and such relations have the property of being con-
tamed in diversity Hence if we were to define 2 - 2 r as the ordinal number
1. we should introduce a tiresome exception, from which trivial complications
this coursl lnt ° ° rdina ‘ arithmetic ' We have - therefore, not adopted
*6614. h . D‘(x = 2 n DVx
Dem.
h -*336.3h:D , R=,‘x. = .ReD‘i‘x
*" • (1) • *561.3 (1)
J" ! ^ " D ‘ 1 • 3 8 (3*. y) • A - r l y r D ‘« = r«x :
. rl!, J 3 ! (a *’») • D ‘ fl - • a‘R = I‘y: D‘R = «‘X:
L*i i 45 j = . (a * y) . D , R = t < 2 . d‘R = t‘y. D‘A = i‘x :
380
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
[•13*193] =:< a *,y). D 'R = i'z.(l‘R = i‘y.i‘z = i‘x-.
r*51 2 .*] = : ( 3 *.,/) . D«/f = /•*. a*/e = ,‘,J.Z = .r:
[•13*195] = : (gy). iYR = »«*. a*/? = »‘.y :
[•55 10] =:(ay). /? = ,r|y:
[•55*22] s : R e D‘(x J, )D h . Prop
*56141. 1*. D‘|.r = 2nfT‘/‘x [Proof ns in •56*14]
+56*15. h . !><(.. |)-iV]j*)-i«Wj*
Dew.
t* .*55*22*16. Dh R « ;I)<(.,• 4, )J - ;•(.<* ,[.r).
* ! (3^)• H‘/t = t‘x.Q‘Ii = f‘.y:~(D‘/?=l‘.r.Q‘J?=l‘ir):
[*10*35.*4*51 .*5*61 ] = : ( 3 //). IYR = l‘.r. <l‘li = i‘y .~((VR = i‘x) :
[•13*193] s : (a.y). D‘/f = t‘x AVR = Yy .~(»‘y = i‘x ):
[•51*23] 3 ■.(ny).Wli=t‘x.C\ , l< = i , '/.x$y:
[*131D5.*51*23] = : ( 32 . ,j). z * y. \YR = . a‘R - i‘y . i‘z - I’w t
[*13*103] = : (a*,y). z + y. iYR - t‘z. CI'/f - i‘y . D‘Vi = l‘x :
[•11*45] = :( 3 *,y).i + y. 1VR = i‘z .Q‘R = t‘i/: D‘R = l‘.e:
[*55*16.*33 0] 3 •Aqz,y).z$y. R = z l y: ReD‘t‘x:
[*50 11 .*22*3.3] s : R « 2, « W'r:. 3 1*. Prop
*66181. h.D'(ix)-i‘(.Tix)=2 r n < (i* ( ‘x [Proof ns in •56*15]
*56*16. \-.xiyti
Dew.
h.*21-2.Dh.xly = x|y.
[★11*36] DP.(gr.tt/).:r,[y=*,[i</.
[★.>6*1] DP.*iy«2.DP. Prop
*66*17. P s x l y «2 r . = . y ^ x c 2 r . = . .r + y
Dew.
P .*56*11. D
h x l y € 2 r . = : (g*, to ). z + w . x | y = z w :
[*55*202] = : (g z, w) . z ^ w . x = z . y = iu :
[*13*22] = :**y (1)
Similarly
P:y4,* € 2 r . = .x*y (2)
P.(1).(2).DP. Prop
*66 18. hjx-fa.E.i .[“a C 2 r . = . I x“a C 2 r
Dem.
P . *13*196 . D P :. a . = : y € a . D y . y =$= x :
[*56*17] = :y€a.D„.x^ye2 r :
[*37*61.#38*12*11] = :x,l“aC2 r (1)
Similarly h:a:^fa. = .|x“aC2 r (2)
P . (1). (2) .DP. Prop
SECTION A]
THE ORDINAL NUMBER 2.
*5619. : R e 2 r .x« D‘Ji. = . (gy)..*4y. R = x^ y. = . Hex - «.'*■
Deni.
K •SG ; 11 . .11 45.3 h , * * 2,., . D‘«. s . < aj , ,)., + *. « = vi ,. ,. e D .
* 5 , 1 ° » ■= <3*.*) ■!) * * • R = y l *. *«:
Si-JS 1 ■: (a*) •■* + *• * — i*« ( 1 )
f 38131 = :R(xl“-i‘x in
*56 191. J-: R .2,. * « Q‘R . = . (gy). *+ y . y< . , j ^. s . * , ; _ t<t
[Proof as in *5619]
*56 2. h Jl €$. = : (gar, ,,) : Z R W . = { to . 2 = x . w = y [# 55-13 . *561 ]
t'.Reb.s :g! /* •xRy.zRw.^ St9ttw .xtms . y- ur [*562.*14124]
*56 22. K A 2 [*56103 . *25 53]
*56 24. K 3 ! 2.3 ! - 2 [*56-22-16. *10 24 ]
*56 25. h . 2 + A n Rel. 2 + V /> Rol [*56 24 . *24 54 - 17 ]
*56-26; Hs./*«-WA
Ihis proposition is the analogue of *52 4 .
Dem.
^ . *51 236 . D b :: R € 2 \j i *A .
a:.R« 2 . v ./*-A
[*25-51 ] s 7i i . v . ^3 » 7?
fSS } 1 s *2 " : T J ■ tRv> ■ 3 -'- — y-«-v, ~g IR ,
h .11 ir T Vl‘ * •«-*•*-«.*.-SIA (!)
r .2 2 n ■ run * p • 3 h ~ a ! "• 3 •-<-%) • ~ i'Hu ,):
[*3 47] ?. ! XRv ■ ^ * V : tRw ■ 3 • '■> = w =
tasss. 8
*66 261. f-::«62.D:.,S-Gft. = : a = A.v.S= 7 i
Dem.
K*55-341.Dl::ie_ a , ly3: . lSC/e ._ :iV=A y s=/e
^.(1).*11H-35.*56 1. Dh. Prop
*66 262. K R e 2 . D : S C /* . 3 ! S. = . s = /*
Dem.
K.5C-22.3K : .A.i. 3 :«_/i.3.S + A (1)
h - (1) • *5-75. *56-261 . D ( '
H. (2). *25-54. DH. p ro p
( 2 )
382
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*56 27. b:. P € 2.3 : g ! R*S. = .R*Se2
Dent.
h . *55 34 . *23-43 . 3
h:.R = xl>/.D:Zl R*S. =.R*S=R.
[*5G1G] D.RnSe 2 (1)
b. *56 103. 5b: RnSe'2. 3 . a ! 7? a S (2)
h.(l).(2). >i-i-R = xl2/-0:alR*S. = .RrsSe$ (3)
H . (3). *11 11-35 . *56 1 . 3 b . Prop
*56 28. Pc2.3:a!7fA;s , . = .7?G6'. = .PnS=P
Dem.
K.*55 3.3H:. /? = x | y . D : y ! i? n 5. = . PGS. (1)
[*23*621] =. R r\S = R (2)
K(l). (2). *11 11-35. *56 1 . 3 h. Prop
*56 281. h:. 7te 2 , .DsgSRA&.s..tfn&a/Z. = ./?*&€ 2 r
Dem.
b. *56*121 .3h:. Hp.3:P«2:
[*56-28] 3 : &!PAS.«.PGtf.a.PAS-P (1)
b . *13*13 . 3 h Hp . 3 : P A 5 = R . 3 . R A £ € 2 r :
[( 1 )] Dz&lRfiS.D.RnSc 2 r ( 2 )
b . *56*122. 3 H : P a S « 2 r . 3 .3 ! P A S (3)
H . (2). (3). 3 »■ s. Hp. 3; a ! R * 8 . s . R A 8* 2 r (4)
h . (1) . (4) . 3 H . Prop
*5629. bzzP, QefcOs.PGQoP.ssP-Q.v.PGP
Dem.
b. *55*51 .3
h:.x|yG2iwc;7?.D:x(ziM;)y.v.iriyG/f:
[*55-31] D:xJ,y-^li(;.v.xiyG^ (1)
h .(1).*13*12 . 3
b P = x l >/ z: Q = z l w .0 PG<?oP.3:P = <2.v.PGP (2)
h. (2). *11*11*35. *56*1. 3
h::.P*2.3::Q = *.l.M'.3:.PGQc/P.3:P = Q.v.PGP • (3)
b • (3). *11*11*3*35 .*56*1. 3
h::.P*2.3::Q€2.3:.PGQc/P.3:P=Q.v.PGP (4)
b . *23-58 61.31-:. P = <?.v.PGP:3.PGQuP • (5)
h . (4) . Imp . (5). 3 b . Prop
*56 3. h:.P, Qe2.3:PGQ. = .P=Q. = .a!/ J nQ
Dem.
b. *55-3-31.3
b z x i y G r l w . = . x l i/ = z l w . = . a ! (x i y) a (z X w) (1)
SECTION Al
THE ORDINAL NUMBER I>
*13-12.3
h p = l!/ • Q = i l m: P <-Q . =. P = Q. = .* • p f> Q
-(2).*11 11-35.*56 1.3H. Prop
383
( 2 )
*5fr2\r?fc from , (2) t0 r hC C ° nclusio, ‘ »«■ analogous to those from (2) of
S'i merely ^ —*« pUrfs
*56 31. I -*.P.Q'i.l:Pi.Q. a .p /tQmA [*56-3 . Tmnsp]
*56 32. f- : P e 2 . D . J J r\Qei \j
D eni.
h . *56 27 .DH:. Hp.3:g!i>AQ.3.7->AQ«2:
[*2'54.*25'34J 3 : PAQ = A.v.PaO, i •
[*ol-236] 5 : P nQtiu (‘A 3 h. Prop
*5633.
H.*55-5.*13l2.3H::P, i r iy .Q = ilWi;):
0 )
*6634 ^.P.Q t i.P + Q^ ; . RePoQ ^ RR+pvtQsR ^ pvit ^
Dein.
K*56-33103.*5 75.*25-54. D
[*o®3] 31 -■■■P,Q'i.l:P = P aQ . s .p^ Q;
[ Transp] 3 : P* Q. D . P * p „ Q ■
* qV 3h ' ? 'e <2 -'' + «-3:fl^.3.«V 0 Q
K(2> ?Vq- 3h! -- P -e^-/ , + e.3:« = «.D.ft + p laQ
^: «>: s;: ^?v°.- c - 3 - K - ^ • ’ ■«- o==«
*5635. l-:C'‘l?e2.flAfi = A.D.« t 2,
Deni.
H . *55-54. D
*■. (1). *1111-35 .*54101 . D I-. Prop
( 1 )
( 2 )
(3)
(4)
( 1 )
384
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*56 36 K : ft € 2 r . D . C*R * 2.7? A ft = A
Deni.
H . *55 54 . D
y . R = x l y y . C U R = l*x \j i‘y . ft A ft = A (1)
h.(l).*llll-34.*5611.D
h /?€ 2 r . D :(gpr, y). .r% y ,C*R = i‘xvj . ft A ft = A :
[*54101.*11-45] D : f7‘ft e 2 . ft A ft = A Dh. Prop
The following proposition, in addition to being used in *5G\38, is used in
the elementary theory of series (*204*463).
*56 37. h : R € 2 r . = . C‘R € 2 . R A ft = A [*56-35 36]
*5638. h.2 r -C“2nft(ftAft«A>
Deni.
H . *37 106. *33-122 . D h : (7* ft * 2 . n . ft e C“2 (1)
H . *20-3 . D h : ft A ft - A . s . ft e ft(ft A ft - A) (2)
h . (1). (2). *56-37 . D h : ft € 2,. = . ft c C “2 . ft e ft (ft A ft - A) .
[*22*83] = . ft eC“2 nft(ftAft = A):DK Prop
This proposition is important jus establishing the connection between the
cardinal and ordinal 2. It shows that the ordinal 2 consists of those asym¬
metrical relations whose fields have (cardinal) 2 terms. It is used in the
theory of well-ordered scries (*250 44).
The following proposition, in addition to being used in *56 39, is used in
relatiou-ari thine tic (*16538) and in the theory of series (*2054).
*56 381. I-: 6“ ft =i'.r. = .^xj.r
Deni.
H . *33-24 161 .*51-161. D h : C‘R = i‘.c . D . g ! D‘ft . D'ft C l‘x.
[*51*4]
D . D*ft = l<x
(i)
Similarly
h : C l R = i*.r . D . G‘ft = i l x
(2)
h . (1) • (2). *55*16 .
Dh:C‘ft=t‘.r.D.ft = .rl.r
(3)
h . *5515 .
Dh:ft«=-/* ( |.r.D. C*R = l‘.c
(4)
\-. (3) . (4). D h . Prop
*6639. K2 —2 r =C“l
Dem.
~ K*5G-381.Dh:C‘ft€l. = .(a*).ft = *i«.
[*5613] = . ft e 2 — 2 r (1)
h.(l). *37106. DK Prop
THE ORDINAL NUMBER >
385
SECTION A]
,, , T i" S > >, ' opos j tio, i ' establishes the connection between 2 - 2 , and l.showine
“ 7 ‘. S th « class of those relations whose fields consist of a single term.
(*153 301 ) m d ‘ SCUSsio " of an<l ~ r as relation-numbers
*564.
Dem.
>■ . *41-11 . D h Hp . D : *(«*,.)*•■• lg »).««$. /f , M .
i*55- i'll a • (a*, w). * j, »•«^(, i ,,, .
| = • (3i, w) . s l w « „ . t = x. w - y.
o X nl l :::, P 7 OSiti0n ; Sth0 ana,og "° of * S3 ' 23 - II » «*«l in the number on
exponentiation in relation-arithmetic (*176*10).
h& w i
25
SECTION B
srB-OLASSES, .Sl'B-RELATIONS, AND RELATIVE TYPES
Summary of Section It.
In this section, we consider Hist the classes contained in a given class and
the relations contained in a given relation. If a is any class, the classes con¬
tained in a are the members of @(0Ca); those are also called the sub-classes
of a, or (sometimes) the " part* " of a. in this last usage, they arc called
"proper parts" when they are not coextensive with a, this phrase being formed
on the analogy of “proper fractions.” The sub-classes of a are all the classes
that can be formed from members of a; they are the same thing as the
"combinations" of members of a taken any number at a time. If n is the
number of members of a, 2“ is the number of sub-classes of a, whether n be
finite or inHnitc. The number of sub-classes of a is always greater than the
number of members of a. On account of these and other propositions, the
class of sub-classes of a given class is an important function of the class. If
the class is ct, we denote the class of its sub-classes by "CPa.” This is a
descriptive function, derived from the relation "Cl,” defined as follows:
CI = *a[« = /§(/9Ca)| Df.
The sub-relations of a given relation are all the relations contained in the
given relation, i.e. all relations which imply the given relation for all possible
arguments. That is, if l* is the given relation, li is a sub-relation of P if
It G P. Thus denoting the class of sub-relations of P by "RPP,” we are to
have
Wl> = tHRGl>):
hence we take ns the definition of "Rl” the following:
R1 = \P \ \ = R (R Q P)) Df.
Sub-relations have properties analogous to those of sub-classes, but they are
of somewhat less importance. It should, however, be observed that when one
series is contained in another, i.e. is obtained by selecting some of the terms
of the other series without changing their order, then the generating relation
of the one series is a sub-relation of the generating relation of the other series.
(It is not the case that a sub-relation of the generating relation of a series
must genemte a contained series, for its field may fall apart into detached
portions, or otherwise fail of being serial.)
SECTION B] SUB-CLASSES. SUB-RELATIONS. AND RELATIVE TYPES m
We shall also consider in this section (*62) tin- relation of membership of
a class, j.e. the relation which a- has to o when «». This relation bears the
same relation to V (a » as ••/" bears to -,r = y ." Strictly speaking, we ought
to introduce a new notation for it, putting (say)
A =.?$(*€ a) Df.
But as e, unlike is a letter, and capable of being conveniently used
alone, it seems more desirable, from the point of view of avoiding unnecessary
duplication of symbols, to put J
e = j-a (x € a) Df.
® t " Ct ,! y s P eakin g- this is faulty, since it gives two different meanings
e. But practically this does not matter, since the above definition gives
b :€ a .
xea,
where the first e has the meaning just defined, while the second has (he old
meaning. 1 hus all that is really required of the above definition, namely to
give a meaning to formulae in which r occurs without referent or relatuin. is
effected without the danger of any confusion that could load to errors.
The ch,ef ‘“Poitance of e as a relation arises from the fact that relations
contained ... r play a very important part in arithmetic. Take, for example,
the problem of selecting one term out of each member of a class of classes:
in this case we require a selecting relation Ii which is such that whenever
rr;* 18 * me, “ ber ® f •• s " ch (This condition is only part of
the definition of a selecting relation; the complete definition is given in * 80 .)
Three numbers in this section (*f.:i, .6+, *G5) are devoted to the discussion
of relative types. Given a variable x, we often want to define the relative
types of other variables, or of ambiguous symbols, occurring in the same con-
text, that n>, we wish to express the types of these other symbols in terms of
tained Th ‘ * f ° r thc of *• ** type in which « is con-
ta ned. Then fa = a w - «, f* - t‘x w - 1 «, _ and t‘a = tfCl‘a - Cl'tfa.
Also we introduce a notation (.65) for giving typical definiteness, relatively
to x, to typically ambiguous symbols. This notation is very useful in cardinal
and ordinal arithmetic, since numbers arc typically ambiguous, and the failure
take account of this fact has led to the contradictions concerning the greatest
cardinal and the greatest ordinal.
25—2
*60. TUB SUB-CLASSES OF A GIVEN CLASS
Sunt mini/ o/‘* »>0.
(,ur definitions in this number are as follows:
*60 01. Cl = *«(*'»/§(£<:«)! Df
Thi8 dofi,,es tllv re,nfio ” to a class a of the class of all its sub-classes.
*60 02. Cl ex ■£(*!* = ,3 <£ C a . g ! £)| Df
This'defines the relation to a class a of the class of all its existent sub¬
classes of all its sub-classes except A. This is often required, as, for
example, in the statement of Zerinclos axiom: “Given any class a, there is
a relation It such that, if £ is any existent sub-class of a. R*Q is a member
»f ft? i.c.
"<3 « Cl cx‘a . . R'0 e 0*
Tins axiom, or its equivalent the multiplicative axiom, plays (as will appear
hereafter) an important part as the hypothesis to many propositions in
cardinal arithmetic.
*60 03. CIs 3 = Cl‘CIs Df
A CIs 3 is a class whose members are classes.
*60 04. CIs 3 * Cl‘CIs 3 Df
A CIs 3 is a class whose members are classes whose members arc classes,
i.c. a CIs 1 is a class of classes of classes.
Apart from propositions which merely embody the definitions, the most
useful propositions in this number arc the following:
*60 3. I-. A € Cl‘tt
*60 32. h.CI*A-i<A
*60 34. KacCI'a
*60 362. I- . Cl‘t‘ar= t‘A sj i*i l x
I.e. A and i*x are the only sub classes of a unit class i*a\
*60 6. f". s‘CI‘a = a
*60 57. h.K CClV*
*60 - 6. h : x e a . D . i*x * Cl ex‘a
The propositions of this number are chiefly useful in cardinal and ordinal
arithmetic, but uses also occur in the theory of series; hardly any uses occur
before cardinal arithmetic.
SECTION nl
THE SUB CLASSES Op GIVEN CLASS
380
*60 01.
*6002.
*6003.
*60 04.
*601.
*6011.
*6012.
*6013.
*6014.
*6015.
*602.
*6021.
*6022.
*6023.
*6024.
*603.
*6031.
*6032.
Dem.
Cl = «« [* = /3(0Ca)|
Cl ex = ;S {* = /§ (j g c a . 3 ! tf)|
Cls- = CI‘CIs
Cls' = Cl'Cls 5
h: *Cla. = .* = / 5 (/3 Ca)
H : k Cl ex a , =. K s
. Cl‘a = c a)
*■•01 ox < a = 4(^Ca. 5I ! / 9)
KE'Cl'a
I". E! Cl ex'a
: & e Cl‘a . = ,/JCo
H:/3eCI ex‘a. s ./9Ca.y!^
t-:£eClex<«. = .£eCI‘a. a !£
l-:/9 € Cl ex‘a. = . /9 « Cl*a — £'A
h -Cl ex'a-Cl'a-i'A
K AcCl'a
Kg! Cl‘a
*■ . C1‘A = t‘A
Df
Df
Dl
Df
[*21-3. (*0001)]
fii <£ C a. g ! £) [*21 3 . (*00 02)]
1*303. *00-1]
[*30-3. *60-11]
1*6012. *14 21]
[*6013. *14-21]
[*0012. *20-33]
[*0013. *20 33]
[♦GO-2-21]
[* 60 - 22 . * 53 * 52 ]
[*00 23 . * 20 - 43 ]
[* 2412 . * 60 - 2 ]
[* 60 - 3 . * 10 - 24 ]
*60 321. H
Dem.
1" . *602 . *2413. 3 f- : a < Cl‘A . s . a ~ A.
C * 51 ' 15 ^ s . a c (‘A : D I-. Prop
A. = . Cl'
i‘a
Bs^Ca.s,.^
3 : A C a. = . A s
3 : A = a
= a
: a .
O)
( 2 )
H . *6032 . Dha = A.D. Cl‘a =* i l a
I-. *60-2. *51*15.3
H Cl‘a = t*a
[* 101 ]
[*2412]
Ml).(2). DK Prop
*60-33. I-. Cl ex*A = A« CIs
HigW S'o°;r, 0 ? K - iDdi - *"* *• A ^ -
Dem.
h . *60 22 32. D I-: 0 * Cl e,‘A. S . ff e t « A . 3 ! 0.
. 5 - 16 -. 24 - 54 ] .-/ 9 -A./+A
1 -. (X) .* 3 - 24 . D h ./ 9 ~eClex‘A
K ( 2 ). * 1011 . * 2415.3 K Prop
0>
( 2 )
390
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*60*34. l-.at Cl‘a [*22*42 . *00 2]
*60 35. h : g ! a . D . a * Cl ox‘a [*60 2234]
*60*36. h : 51 ! a . D . h ! Cl ox*a [*00 33 . *10*24]
*60 361. I-: g ! a . = . g ! Cl ox‘a [*60*36-33]
*60 362. h .CI < /V=/‘A u/V/ [*3I 401 .*60*2]
*60 37. h .dexV/r/'r'/
l h-m.
h . *I»0'2I . D h :/3 « Cl ex‘«‘.r. . ,i C ,‘x. 3 ! 0.
[*•141 = ./3-,‘x.
[•51’15] m •/3n‘i'x:D h. Prop
*60371. H : o« 1 . D.CI‘oCOu I
/><■
III.
t-.»51 401 .DK::a = ('.r. D :./9Ca. = :/9- A. v.
[•54-102.*5222] 3:£«0.v./9«l:.
[*60-2.*22-34] D :./3 ( Cl‘«. D . /3 « 0 w 1 (1)
I-. (1). *10-11 23 . *52-1.31-. Prop
*60 38. H : a « 1 . = . Cl ex‘a = 1 ‘a
Dem.
V . *00-37 . 31-: a - t‘x. D. Cl ox‘a - t‘a:
[*1011-231 3 I-: < 3 -r). a - i‘x. 0 . Cl ex‘a = / ‘a:
[*52 1 ] 3H:««1.3.C1 ex‘a = i‘a (1)
*-• *00-361 .*51-101. D h : Cl ex'a = <‘a. D . g ! a (2)
h. *60-21 . *10-1 . 3 I-:. Cl cx‘a = ,‘a . D : <‘.c C a . 3 I = . l‘x = a :
[*51161] 3 : C a. ■. i‘x— a:
[* 5I-2 J 0:x'a. = .i‘x=a (3)
H . (3) . *1011-21-281 . D h :. Cl ex‘a = l‘a. D :g 1 a . 3 . (gx). «‘.r= a .
[*521] g.ad:
[(2)] Z> : a e 1 (4)
h . (1). (4). D I-. Prop
*60 39. 1-. Cl‘(,‘x = t‘A «U i‘i‘y \J I‘{i‘x u I‘y) [*54 4 . *60 2]
*60*391. hae2.D.CI‘oC0wlu2 [*54411 .*60*2]
This proposition is used in the theory of the continuity of functions
(*234-202).
*60 4. h : £ € Cl‘a . y C /9 . D . y e Cl ‘a [*602 . *22*44]
*60 41. h : ^ e Cl‘a . D ./9 n 7€CI‘a [*60 4 . *22*43]
The following proposition is used in the theory of well-ordered series
(*250*14).
*60 42. h : /9 c Cl‘a .yCiS.gly.D.yeCl ex‘a [*60*422]
SECTION u]
THE SUB-CLASSES op A GIVEN CLASS
3**1
w
*6043. h:0,yt C'I'a - = •£<-> 7 e Cl'a [*22T.!». * 6 »- 2 j
*60 44. h : /}« O'a. yt Cl vx‘a. 0.0 « yf Cl cx‘« [*60-43 . ** 4 -r,(i. *«o-22
, 17 T A e ->°"° Wi " 8 pr0p0sition is »> the theory «.f "first difi'ei-onces'
v*i i U ba).
*60 45. V : p e Cl‘(a *- yS). = . (g 7 , S). y t Cl'n. S < CI7J. p = 7 u S
Dem.
^ . *60*2 . *22 (J 21 G 8 . D
h ; ? e C, ‘(® ” ^ - p = (p * a) \j (p n 0) ,, ,
h ■ * 60 ' 2 • *2243.3Kpn«« Cl‘a . p « /3 e Cl'fl ,,
*" • (1) • (2). *10 24 . D
h : p « Cl‘(a o £). D . (g 7 , £). y e Cl‘a . 5 < CI‘/9. p = 7 v 5 (3)
K *602.3 ' '
r oo 51 , 7 ' S) ' 7 <CI ‘°- SfCI ‘ / 3 -' , ”'''«S. 3 .(a 7 .S)- 7 Ca.SC i 8 .p = 7U «.
L*22-72] O.pCav/3.
* 60 ' 2 1 3. p« Cl'(««£)
h • (3). (1) . D K Prop
*60 5. I-. s‘Cl‘o = a
Dem.
H .*4011 . *60'2 . 3 H :a:es‘CI‘a . s . (g/9) . BCa.xe fj.
[*22'44] ] 3.* e a
H . *2242 . DHs«ca.D.aC«.ar<a.
[*\°: 24 J • (3^) • /3 C a . x e0 .
L (, )J l.X€s‘C\‘a
^ • (2) . (3) .Dh. Prop
*60 501. Ks‘Clex‘ a = a
Dem.
f • t*?, 1 .' • * 60 ' 21 • 3 h : ' ** CI ex ‘ a - s - <30> .0C«. a! 0.. r<< 8. ( 1 )
L*^**ij D.arta
K*2242. Dh: l( «.D. s Ca. I[a .
[*10’24.*24-5.*4-7] 3,.C ., 3
*‘° 24) 3 • <30) ./8C«. : .|!/9.x*,3.
L<1 >j 3. x es‘Cl ex‘a
K(2).(3). 3 I-.Prop
(*n r 5 h i° 7 ) abOVC pr0p08iti0n is uscd the lhcor >' of cardinal multiplication
*60 61. l-.p‘Cl‘a = A [*40-22. *603]
infin^ “ U9e<l in the «“*“■> th *ory of finite and
*60 62. H:«‘*C/9. = .*CC1‘£ [*40 151 . *60*2]
(1)
( 2 )
(3)
( 2 )
(3)
392
PROLEHOMEXA TO CARTUXAL ARITHMETIC
[PART II
*60 53. h ;jj C,,‘k . j ./3« ,,‘CI‘V
Dem.
. *40 15 . *00 2 . D h /3 C/>‘**. = : y €K . D v . CP 7 :
[*40*41.*00*l-S] = z/3*p<C\“*c D f-. Prop
*60 54. h . CT/>‘* = />‘CI“* [*00*53*2]
*60 55. I-: CPa = Cl*,3. = . a = /3
Dcm.
h . *30-37 . *00 U.DI-:a-£.D. CPa = CP/3
I-. *30 37 . 3 H : CPa =*OP£. D . 6*‘CPa = s‘CP£ .
[*60-5] D.a = /3
K(D.(2).DH. Prop
*60 56 H : Cl ex ‘a = Cl cx‘/3. = . a = /3 [Proof as in *60*55]
Hie following proposition is used frequently.
*60 57. h.K C CIV*
( 1 )
( 2 )
Dem.
h . *40 13 . *00*2 . Dh:of<.D.o f CIV* (1)
h . (1). *10*11 .*22 1 . DK Prop
*60 6. zxea.D. i t xeC\cx t a [*51*2*161 . *00*21]
The following proposition is used in connection with cardinal multiplication
and with greater and less (*115*17 and *117*66).
*60 61. h./“aCClex‘a [*37*61 . *51 12 . *60*6]
*60 62. h:a\yfa.D./ < .rw‘y € Clex‘a [*G0*6*44]
*60 7. KCTacCls*
Dem.
h . *60*2 . D h : /3 € CPa. = . /3 C a .
[*22*1.*20* 1*3] = . ( a </>. yfr) . a = ^(<f>l2).^ = ^(yJrlz).ylrlxD x <f>lx .
[*10 5] D . ( 3 ^). >9 = 3 (yfr ! z) .
[*20*4] D./3 f Cls ( 1 )
h . (1). *60*2 . (*60*03) . D h . Prop
*60 71. I-. CIs* = CPCIs [(*60 03)]
*60 72. h . CIs* = CPCIs* [(*60*04)]
*61* 1HE SUB-RELATIONS OF A GIVEN RELATION
Summary of * 61 .
Tl.e propositions of this number (except that *61-371 -372373 imperfectly
correspond to .60 371) are the analogues of those with the same decimal pan
in *00. Proofs are omitted, as they are exactly analogous to those in *00.
Ihcre are very few subsequent references to the propositions of this number.
*6101. Rl=>x£ (X-/}(ftG/')i Df
*6102. Rl ex *= \P {X = R(/i G /*. a ! /i>) ])f
*6103. Rel*=Rl‘(RelfRel) D(
*6104. ReP-RI'(Rel*fReP) Uf
*611. h:XRI/>. = .\_^(/j<:/>)
*6111. I-: X Rl ex P . = . \ = R (/< c P . a ! R)
*6112. KRl‘P>/f(ft G .P)
*6113. KRIex‘Z> = .£(/lGZ > .a!.R)
*6114. 1-. E ! Rl‘/>
*6115. I-. E ! Rl ex'P
*612. h: /2 « RPR .s.HCP
* 6121 . hRt'Rlex , P.n.RGP.£!R
*6122. h:ReRlex‘/'. = .fl t Rl<p. gijj
*6123. l-:ReRlex‘R. = ./i t Rp/>_,*A
*6124. K Rlex‘/'=R|«P_ 4 *A
*61-3. KAe Rl 1 /'
*6131. 1-. g ! Rl 1 /'
*6132. 1 -. RI-A = (‘A
*61321. h : /’ = A . = . Rl 1 /* = t*/>
*6133. I-. Rl cx‘A = A n Rel
*6134. K/> e Rp/>
*6135. 1-: a 1 /*•D./'eRl ex 1 /*
*61 36. h : a ! P. O . a ! Rl ex‘R
*61 361. I-: a 1 P . s . a ! Rl ex‘R
*61 362. I-. Rl‘(* | y) = ,<A „ ,«( x | y)
*6137. h . Rl ex‘(x iy) = t‘(x | y)
*61371. h : R e 2. D. RI‘R = t <A v t‘R
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
304
*61*372. \-zRei.D. Rl‘7* C 0, v 2
*61 373. : R c 2, . D . Rl‘77 C0,u 2,
*61*38. b : 7? € 2. = . Rl ex‘7? = t*/i
*61 39. K . Rl‘(.r lyvzl w) = i‘A ^ i‘(* | y ) ^ ,‘ ( * | w ) w | y ^ ^ w)
*61 391. I-: P, Q e 2 . D . RI‘(P vQ) = i*\ u /*/» ui‘(Jw /‘(T* vy Q)
*61 4. h : Qc RI‘P . R G Q . D . R € Rl‘7'
*61 41. t-: Q c RI*/ > . D . Q r\ H t R1 «/>
*61 42. h : V c R1‘7'. 7f G <?. g ! 7? . D . 7f e Rl ex‘7>
*61 43 h : V. 7.* e Rl-P . = . <? <y 7? « RI‘7 J
*61 44. h : y € Rl‘7*. /f < Rl ex‘7*. D . Q o 7? € Rl ex‘7'
*615. h . i'RI'P . 7>
*61501. h . #*RI ex‘7 > = P
*6151. H . /Yli[‘P = A
*61 52. I- : GQ.s.XC RI‘Q
*61 63. H : Q G . Q € />‘R|“X
*6154. h . Rl-yVX-p‘R|“x
*61 55. h . Rl 4 /* - Rl *Q . = . P=Q
*61 66. b . Rl cx‘7 J - Rl ex‘Q . = . P . Q
*61 6. h : xPy . D . x | i/ e Rl ex‘P
I he analogue ol *60 01 is not given, because we have no suitable notation
for expressing it.
*61 62. h : xPy . zPw -D.*,[yvy*jM/ € R| C x*P
*617. b . Rl‘7 3 « CRRel
*62. THE RELATION OK MEMBERSHIP OK A CLASS
Summary of *62.
When "x e a" was defined, in *20. it was defined as a propositional
unction; and this mode of definition was necessary, because we had to treat
of this function before treating of relations. But for many purposes it is
desirable to regard cos a relation, so that « M becomes an instance of the
notation “ uRv This requires, strictly speaking, a change in the meaning of
. x € *• ^ ut ' s a change which does not falsify any of the previous propositions
in which a" occurs; for if we call the new meaning «' a.*’ i.e. if we put
e' - ua (x c a) I)f,
wehave b:*«'*. =
Hence it is unnecessary in practice to have a new notation for the new
meaning, and we put simply
Df.
This definition, though strictly incorrect, is recommended by its convenience.
and by the fact that it cannot lead to any harmful confusions. The new
meaning of * may be taken as replacing the old throughout the remainder of
this work.
Ihe uses of the propositions of the present number occur almost ex-
c usively in the theory of selections from a class of classes (*83, * 84 , *85 and
*«8). Such selections are effected by means of selective relations, part of
w ose definition is that they are contained in *. Hence the uses of the present
number. If a: is the class of classes from which a selection is to be made, a
selective relation will in fact be contained in e f *; hence the properties of * f *
become important. Some of these properties arc given in *62 4 ff.
The most important propositions of the present number arc the following:
*62 2. b . e ‘a = a
*62 231. b : * C d‘« . = . A ~ e «
*62 26. b.ft = «|*ft
*62 3. b . t“ K = 8*<
*62 42. b : * . D . Q'ef* = *
*62 43. b.D‘«r
*6265. b:*Cl.:>.er*=:rr*
PROLEOOMENA TO CARDINAL ARITHMETIC
[PART II
*62 01. cs/a(*cft) Df
*62 1. h/ta. = .xea [*213 . (*0 2 0 1)]
In the above propositi,,,., the first t has the newly-defined meaning, while
the second has the old meaning. In virtue of the above proposition, the new
meaning may be substituted for the old in all propositions hitherto proved
concerning c, and may take the place of the old meaning in all that follows.
*62 2. h.t‘a = ct
Deni.
h . *3213 . D h ."?<* = .7 U € a)
[*2042] -a.DK Prop
*62 21. h . t V - a (x € c) [*32* 131]
I Inis e*.r consists of the classes of which .»■ is a member.
*62 22. K 1)‘€= V
Dem.
H. *24104. Dh. (x). xc V.
[* 10*24] D h ; ( x ): (go). .r c a :
[*3313] D »-.<*).*« D‘«:
[*2414] DKD‘«-V
*62 23. h . (Pc =* Cls — i*A
Dem.
h . *53*5 .Dh:af Cls - i* A . s . g ! a .
[*33*131] = .aeCI'f : D h . Prop
*62 231. huCd'e.B.A-vt* [*24 63 . *33*131]
*62 24. h . e T= V
Dem.
I-. *24104. *11*57 .Dh.(*,y).xcV.y€V.
[*31*11] D h .(a*,y).x€ V . Vey.
[*10 24] D h :(4r,y):(ga).x«a.aey :
[*341] ^ h : (x.y) :xc| ey :
[*25*14] DK*|€= V
*62 26 . H . e | € = a/9 |g ! (a n >S)|
Dem.
h . *341 .*31*11 . D h : a(c|e)£. = . (gx) . xe o . xc/9.
[*2233] - 3 !(an^):DK Prop
SECTION B]
THE RELATION OF MEMBERSHIP OF A CLASS
307
*62 26. h . ]{ = * i;
Vein.
^ . D I - : jrfiy . = . .c e lt*y .
[*30-33.*3212] = . (gra). a- € «. .
m.*{€ li)y:Db. [Vo,.
*62 3. h . € “/f = 6-V
Deni.
^ # * 371 • ^ • *“* = -^!<aa>. at k ..tea |
[(*4002)] =s<K.Dh. Prop
*62 31. h .7<*-«'*
Note that, since c is not a homogeneous relation, i.e. not one in which
referent and relatnm belong to the same type. is strictly meaningless
or.f we have ««.«», the two «■« have different meanings, and do not
therefore properly give ««=«. But it is convenient to allow e », on the under¬
standing that the ambiguity of , is to be differently determined for the two
■actors in the product r|c, namely the second e must make both referent ami
for the'first""^ *° ^ 116X1 tyl “ nb ° V0 tl,nt 10 "' l,ich ^ey respectively belong
Dc
III,
h . *32-13. D h. «y* . £ (x «• k)
[*34 5] m ^!(a«)- xta.atK]
[(*4002)] -*«*
*62 32. h . s = «, [*30-41 . *02-3-31 . *37 11J
*62 33. h.T./f* CIs
Deni.
K *022. *30-8.31-: >9 ?a . . 0 = a.
[*20 41] • j8 = a .« « Cl*.
[•6° I'* 35 ' 1 01] =, . 0(1 f CIs)a : D I-. Prop
The use of *20 41 in the above proof depends upon the fact that a is
merely an abbreviation for an expression of the form 2 (^ 2 ).
*62-34. !-./•« = sg‘(/' j ( )
Dem.
h . *37 101 . (*37 01) . D H a P t fi
[* 341 ]
[*321-23]
a =* Kay) *y<? &**Py\
= 2\x(P <)£} z
a \*&(P I e» >3D h . Prop
398
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*62 4. h . <?[** a .ra(j:€a . a ex) [*21*2 . (#3502)]
The relation e [ k is very important in cardinal arithmetic, in connection
with the problem of selection from the members of k, i.e. of extracting one
term out of each of the members of *. A relation which is to effect this
selection must be contained in ef *.
*6241. h.a<«r«-«-* c A
Dem.
h .*3.VI 01 . D 1 - zx(€[«)a . = .are a . a e « :
[*1011*281] D h (g.r) . ar(e [“*)<*. = : (gjr) .xe a . a e k :
[*10*35] =:(g/)..rfa:af<:
[*24 5] = : g l a . a e k :
[*53*52] szaex-i* A (l)
h . (1). *33*131 . Dh.Pn.p
*62 42. hsA~<*.D.U'cF*-*
Dent.
h . *31 *30 .DhHp.D./rC-i'A.
[*22 (521] O.k-k-i* A.
[*(52*41 ] D . CI‘e [**-*: D h. Prop
*62 43. KD‘« [**-*«*
Dem.
h . *3311.3 h . D'c T « « £ |(ga). .r (c p *) a|
[*35*101] =.^|(ga). j*€a . ae/r|
[(*40*02)] — «**. D h . Prop
*62 44. h : R G e . = . (a) . 7?a C a
Dem.
. *23*1 . D h C € . = : xRa . D x> . . .r e a :
[*32*18] = : ac R*a . D x> « . xc a :
[*11*2.*22 1] = : (a). 7?a C a D h . Prop
*62 45. z. li Ge. Ell R'WR . = : a * (T7* . D„. /*‘a e a
Dem.
h . *14*21 . *4*71 . D H R‘a e a . = : E! R‘a . R‘a c a :
[*30*33.*5*32] = : E ! 7?‘a : xRa .O x .xe a (1)
H . (1). *10*413 . D h :: a e (I‘R . D* . R‘a eer: =
a € Q‘R . D a : E ! R‘a : xRa . D x . x e a
[*10 29.*11 *62] = a e O'R . Z> a . E ! R‘a : a e (I‘R . xRa . D.,,. x € a z.
[*33*14.*4*71 ] = :. a c d'R . D.. E ! R*a z xRa . D m>x .xeaz.
[*37*104.*11*2] = :. E !! R“(l‘R .fiCe:OK Prop
This proposition is useful in the theory of selections. It is used in the
proof of *83*27, and thence of *83*28.
SECTION B]
THE RELATION OF MEMBERSHIP OF A CLASS
309
*62 5. I-. / G €
Dem. I-. *33 21 . *52*13.31". (1*7 — 1 .
[*52 173] 3 b : ae(J‘f. 3. . Paea:
[*62 45] 3 h.7 g *
*62 51. b : E ! 7*a . 3 . 7‘a = Pa
-Dem. h . *5215 172. 3 h Hp . 3 : PPa -a:
[*5115] 3 Pa. = x .a- f a :
[* 30 3 1 3 : Pa « Pa3 I-. Prop
*62 52. 1-: E! e*a. a . a e 1 . s . E ! 7*a
Dem. H.*80-2.3 h :.£!.*«•■ :(g6) :*€«.*,.
[*5211] 3 : a e 1 :
[*5215] = : E ! 7‘a 3 h . Prop
*62 53. H : E ! Pa . 3 . Pa - Pa [*62-51-52]
*62 54. h : a e 1.3 . Pa -Pa [*62 o 1 *52)
*6255. l-:*Cl.3.«r*-7r*
Dem. h. *62-54.3 b Hp. 3 :««*. 3.. Pa-Pa:
[*35-71] 3 ^ =7^^ : 3 h . Prop
*62 56. b . <[*p‘a- if* £“0 = 017
Dem.
•-. *52 3 . *62 55 . 3 h . f f* P‘ a = 7f* P‘a
H .*35101 .*37-6.3 b z.x(i[ i“a) P . =
[*51-51] s
[*10-35]
[*13193] =
[*51-23]
[*13195]
[*51-51]
[*35*1]
^ • (1) • (2). 3 h . Prop
*62 57. b . 7 =- e f 1
Dem.
< l >
HW) • y € a • P = l*y i
& = i*x: (yy). y e a. £ - py :
(ay) • £ ■ P^. y « a. £ - py :
(ay) • P = • y c a . I'x = I‘y :
• (ay) ■ P = i*x. y c a . X = y :
= : P = i‘x. x e a :
w
= : .xea:
= :x(ap )0
(2)
K *62-55.3H. f ri=7r 1
1*5213] =7r<T‘7
[*35-452] = 7 . 3 K Prop
*63 RELATIVE TYPES OF CLASSES
Sum mart/ o/ #63.
I he notations introduced in this and the two following numbers serve to
express the type of one variable in terms of the type of another. They are
very useful in arithmetic, where it is necessary to take account of types in
order to avoid con trad ict ions. The two chief notations are "tja? for the
type in which a is contained, and for the type of which .r is a member.
We put
*63 02. t,*a = aw — a Df
This defines "the type of members of a.” or "the type which is of the
same type as a." The characteristic of a type is tlmt.if r is a type, we have
(.r) . xe r,
and conversely, if (x) .xer, then r is a type. For in that case, “xtr” is true
whenever it is significant, i.e. whenever .r belongs to the type which is the
range of significance of a* in "xct." Consequently t is this range of signifi¬
cance, i.e. is a type.
Since we have (x).xe(a v — a), it follows that a \j - a is a type. It is
not "the type of a'' but "the type of the members of a." (In case a is null,
"the type of the members of a” may be interpreted as meaning "the type to
which x belongs when '.re a' is significant.'*) "The type of x," i.e. the type of
which x is a member, is defined ns follows:
*63 01. t‘x = i*x u - i‘x Df
By what was said above, "/,‘iV is the type of the members of i*x, i.e. the
type of x. By combining the definitions of t*x and t 9 a, we obtain
V . t*x = t 9 ‘l‘x.
Thus b . a: * f‘.r and b : y + x . D . y e t*x.
In short, t*x consists of everything either identical or not identical with x,
that is, every y for which there is such a proposition, whether true or false,
as " y = .r." We put "t‘x" here instead of because x need not be a class,
and is in fact subject to no limitation whatever, whereas "t/t r” is not signi¬
ficant unless x is a class, and therefore we write rather than “tjx.”
We put also
*63 Oil. t u x = t*x Df
This definition serves merely to bring t*x notation ally into line with t 9 x
and the types t : ‘x, t u x ,... t/x, tjx ,... defined below.
In virtue of *20'8, we have
I -: <pa v ~ <f>a . D , £ (<f>x v ^ <f>x) = t*a.
SECTION B]
RELATIVE TYPES OF CLASSES
101
jf if ■ is significant, then the- n.ngc of significance of the function 0 ? is
° f a U J°1 ,0 " S thi,t two ranges of significance which overlap are
dentic.d, and two different ranges of significance have no member in common.
(if jL W „ i 'l| be iS M "' ayS ° f th0 next ‘JT» aW ‘' of a', and .- V
(11 * IS a class of classes) is of the next type below that of *. We put
*63 03. Df
TciaHS'i* thC 777 bcl °' V that in wllich * is contained. Thus if , is
a class ot classes of individuals, t,‘ K is the class of individuals. We put also
*63 04. tr*x = t'tfx Df
*63041. e‘x = V0‘x Df and so on
*63 05. t,‘ K = tx > tl ‘ K Df
*63 051. (,«* - t,%‘ K Df and so on
Thus given any two objects which are members of any one of the follow-
*ng: the type of x, the type of the classes to which x belongs, the type of the
classes to which these classes belong, and so on, we can express the type of
her of our two objects by means of its relation to the other object.
The propositions of this and the two following numbers will hardly ever
the firs, UUt , We C0m ° J tO Cardinal arithmetic ' They are used constantly in
th! fi f ! ‘° n ° n Card,nal ar,th,nelic ' a "d fhey arc constantly relevant in
cardinal and 7" ™ lat,on - arith “*‘i c - Moreover they are usually required for
cardinal and ordinal existence-theorems.
folloting 8 lhC m ° St " SCf " 1 P r °P° sitions of the Present number are the
*63103. h.x<t‘x
*63 106. h .aC //ft
*6311. h :aet 9 ‘a.D.t‘x mma
which''If either ;i° r ’ 8 DOt a me,abe r Of «, then the type of x is the type
h contains a. This proposition uses *20 8.
*6313. h : fa :. fa . D . y e t * x
Of n !slZi S “* fUn ? ti0n satis6cd hy «**■> * and y, then y is of the type
ambiminn.f ? SSary l ° L the use of th,s proposition that, if 02 is a typically
for y For exan T’ “ " h ,““ ld T*"" the Same l fP ical determination for x and
these a! fafoef P of 6 ' h f aVe al " a f * = * a "d y = y; but we must not regard
ambiguous. On the°ofhl“hand ' '’ZF” ^ “ fUn0ti ° n “ typica " y
2=a because h^ro.k h hand, x = a and y = a are values of one function
, because here the presence of a renders the function typically determinate.
*6316. h. t 0 ‘t‘x = t‘x
*6319.
R & w i
26
102
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*6316. b : .r e t*y . = . y € t l x . = . g ! t*x r\ t'y . = . t*x = t*y
This proposition, which depends upon *6311, and thence upon *208 and
*13'3, and thence upon *9*14*15, is vital to the whole theory of types.
*63 32. I - .t l , /c = s‘t.. , ic
*63 371. bi0CtSa. = .0€t t a
*63 383 h.t'tSsc-tSK
We shall have generally t m *t ni K «■ t m + nt K, where we may count suffixes as
negative indices, so that t m *t n *K — t m ~ nt K or according as m or >1 is the
greater.
*63 5. h : x < t,*a . = .£*€ t u x . 3 . a C t*x . = . t*x « t 0 *a
This proposition is used constantly.
*63*51. b : a € U*k . = . a C . = . « C t‘a . = . t‘a = t 0 *K
*63 52. b : a e f,«X . = . a C /,‘X . = . XCf»‘a . = . t*a -1,‘X. = . f s ‘a - / 0 ‘X
*63 53. b : x c f„*a . = . = <‘a . = . * f/a
The above lour propositions, together with four similarones(*63*54*55*56 57),
give transformations which enable us to express any relation of type, as be¬
tween class and members or members of members or etc., that is likely to
occur in
practice.
*6364.
b .t*/3 = t 0 U**f3
This
proposition is often used in
the first section on cardinal arithmetic.
*6366.
b . CIV* - t u x
*6301.
t*x = t*.r u — i*x
Df
*63011.
t'*x = t*x
Df
*63*02.
to a = a yj — a
Df
*6303.
U*K — t 0 i s t K
Df
*6304.
Df
*63 041.
t 3 ‘x=t*t u x
Df
*6305.
tfK — tftflC
Df
*63051.
U*K = tftj*
Df
*631.
h . (a:) . x € to*a
[*2288]
*63101.
b ,t*x = t a ‘l‘x = i‘ x\j — l‘x
[*20-2. (*63-01*02)]
*63102.
1
[*631*101]
*63103.
h . X € t *x
[*63*101 .*51*16]
*63104.
h : 4>x . ~ <t>y . D .
yct*x
[*63101. *13*14J
*63105.
b.a C to a
[*22-58]
*63106.
h . t 0 *a = t 0 ‘ — a
[*22-8]
SECTION’ B]
RELATIVE TYPES OF CLASSES
*63 107. h (a) . <f>.c : f(<f>y) : D .
Dem.
K *2*11 . *10 11 . (Ij
*""(!)’ * 1013 ' 2 21 • D h :• (*) • <t>' - 3 : ./(<&/) v ~/( 4>y) :
D: ^* = -/(^y) v ~/(0y):
*■* “■* ^ : /(<£y) • ^ • </>yD I-. Prop
*63108. K:/(ye*‘a).D.y € *‘a [*63 107 102]
*63109. h:/(y€^a).D.y € //« [*631071]
*63'11. b : a c f 0 *a: . D . £‘a = a w — q = t^a
Dem.
. *22-34. (*6302). 31-:.Hp. 3 :#«a. v .*~«o:
3:P(y««.v.y~eo) = ^(y = a -. v.» + a ):
[*22'3'31.*5ri5] , n
l-.(l). (*630102). Dh. Prop
*6312. h :. 0* V~<£a:. D : <£y v~0y . s„.yel‘x
Dem.
h . *63 11 . ,20-8.3 h Up. D : <‘x- ?(*«) u _ ?(*,>.
[*20'81 .*22391 '392] 3 : y «,<*. =„. *y v~ *yD I-. Prop
*6313. I-: 4>x. <fty . D . y ( t'x [*6812. Imp. Add]
*6314. I-: (x) .xta.O. I.'a = a [*241417-24. (*63-02)]
*6316. h.t^'fx-fx [*6314102]
*63161. I-. t.'t.'a ■= t.'a [*63141]
*63162. h.xe l.'fx [*6310315]
*6316. hsxtfy.s.jItfx.s.^Ufxr.fy.m.t'x-Vu
Dem.
*63 101 .*51*2.3. Dh :x€t‘y. 3 . y*t‘x
h ' * 6313 * ^ h '• (3^) . z € t*x . s f t*y . D . y € t‘x
V • *63 103 . D V : y e t*x . D . y € *‘a. y * t‘y .
[*1024] D.g!<‘an<‘y
** • (2) . (3) . D b : y e t‘x . = . g ! t*x r\ t‘y
h • *63 103 . D I- : t‘x = t*y . 6 . y € t‘x
h . *6313 . 0 : y e t‘x. z € t*x. D . z e t‘y
^ . *63*13 . ^ : x e t*y . z e t*y . "5 . z e t*x •.
[(!)] 3 I- :y et'x.zct'y . D . zet'x
• (6). (7) . D b :.y « t‘x . D : * e t*x . = . z e t‘y
H . (5) . (8). D b :. y € **a. = . t‘x = t*y
^ • (1) • (4). (9) . DK Prop
(1)
(2)
( 3 )
( 4 )
( 5 )
( 6 )
(7)
( 8 )
(9)
26—2
104
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*6317. 1": y c P.r. * e Py. 3 . z e Px [*6316]
*6318. K a ! Ca
[*10-25. *631]
*63181. h:«Cpj5.s
Dem.
. /3 C Ca . = . a ! //a a C/3. = . t„‘a = C/3
K *63-105.
3 1- : C« = C/3.3 . a C C/3
a)
H . *24 6.
3 1- :. a C C/3.3 : a = C/3 . v . g ! C/3 - a
(2)
K. *63151 .
3 h :a = C/3.3.C* = C/3
(3)
H. *6311 .
3 H : x f C/3. x € - a . 3 . Px = C/3 . Px = C
— a.
[*63106]
3 . Ca = C0
W
h.(2).(3).(4).
3 h : a C C/3.3 . /./a = C/9
(5)
p-d).(o).
3h:aCC£.».Ca-C£
(6)
K(6 )J|.
a.p
3 H : /9 C Ca . = . Ca = C/9
(7)
H. *63-11 .
3 H : x e Ca n C/9.3 . Px = Ca . Px - C/9 •
[*13171]
3.Ca = C/3
(8)
K.*6318.
3h:Ca = C/3.3.a ! Ca n C/3
(0)
»■•(«).(9).
3 H : g ! Ca ^ C/8 . = . Ca - C/3
(10)
K(G).(7).(10).:>KProp
*63182. 1* : a C C/3 . /3 C C y . 3 . a C t.*y [*63181]
*6319. h . PCa - Pa
Dem.
h . *63105 . *2242 .Dh.aC Ca . Ca C Ca .
[*6313] Dh.a€t%‘a.
[*6.316] 3 K • Prop
*63191. h.CacPa [*6310.319]
*63 2. h : x € //a . a c C* • 3 • P‘x = Pa = C*
Dem.
h . *6311 . 3 b : Hp. 3. P« = Ca • Pa = C* (1)
h.(1). *63 19 . (*63 04). 3 I-: Hp. 3 . t u x «= Pa = C* : D h . Prop
*63-21. H : a C Par. = . t 0 ‘a = Px
Dem.
H . *6318115.3 H:aCPx. = .Ca = CP*
[*6315] = Px: 3 h . Prop
*63*22. h : a C Px. = . x * Ca . = . Px = Ca
Dem.
h . *63103 . 3 H : P*- Ca. D. *cC«
K(l). *63-11 .Dh:x€Ca. = .Px = Ca
h. (2). *63-21 .Dh. Prop
( 1 )
( 2 )
SECTION B]
405
RELATIVE TYPES OF CLASSES
*63-23. Y : a C t*x . * C t‘a . D. r*x = t‘ a = [*63-2*22]
P, ‘° p0Sltl0ns of the same ki,ld « s the above can obviously be extended to
l %V 9 Cl»C.
*633. Y :(a).a€K.D .(*)•**«<«
Dem.
b . *101 . D h : Hp ,D. Ye*.
[*40-221] V.
[*2414] D . (a) ..t€«V:Dh. Prop
*63 31.
Dem.
. *40 171 . D ^ v -«). = ; X€8 * K . v .*■«*« -
h • (1) • *22 88 . *63-3 .
K *22*88. D h : arc s‘*.v. #«-#«*
h * (2). (3). *10-22113. D
(1)
( 2 )
(3)
p h:.arec‘*. v . arcs'arcs'* . v . arc
l( ^ D I-. Prop
th», NOt V hat th \ USC ° f * 10 ' 221 in the above P*w>f Spends upon the fact
that *cs * occurs both in (2) and in (3). so that these are both of the form
J VA f 6' *).
*63 32. Y . t t ‘* ~ s‘f 0 '* [*63-31 . (*63 02 03)]
*63 321. I-. t,** - f/f.'* = W*
Dem.
Y . *2 0*2. (*63 03). Dh
[*63-32]
-W*
(l)
[*20-2.(*6303)J
= W8*K
[*63151]
= f 0 V*
[*20-2.(*6303)]
= t t ‘K
(2)
K(l).(2).DKProp
*63 33. P : - f«‘X . D . <,'* = f,'\ [*30 37 . *03 32]
*63 34. Y . t x i t t a = f 0 ‘« = 8*t‘a
Dem.
H . *63-32 . D Y . //*‘a = sV*‘«
[*6315] **s*(* a
[*63101] = 8*(i‘a v — i‘ a )
[*63 31] = s ‘t* a yj _ s i L t a
[*5302] = flu -a
[(*6302)] = f 0 *a
^ • (1) • (2) .DK Prop
O)
(2)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
106
*63 35. b : t‘a = t ‘$. D . t,‘a = [*30 37 . *63 34]
*63 36. I-: t 1 * = t‘\ .D . t y *< = /,‘\ [*63-3533]
*63 361. 1-: /.‘as f,‘0 - 3 • /‘a = /‘/3 [*30 37 . *6310]
*63 37. h: / n ‘a= /./£ . = . /‘a = (‘,3 [*6335-361]
*63 371. h:£C/„'«. = .£«/‘a
Dem.
b . *63*181 . D 1-: ^ C /.‘a . = . /„‘a - /„<£.
[*63-37] =.*‘a = *‘£.
[*6316] = . /9 «/‘a : D H . Prop
*63 38. haf //* . x * t.*a . D . f‘.r = t n *a =
Dem.
*■.*63-11 • >b : Hp.D.t'x-f # 'a.f‘a~«,*« (1)
f-. (1). *63-34 . D h : Hp. D. / 0 ‘a = /,%'*
[*63151-33] -t,<* (2)
h .(1).(2). D h • Prop
*63 381. 1- : .r e /,‘* . D . (‘.r = //*
Dem.
b . *6338105 . Dl-iflf /„** • x e a . D . /‘j- = t,‘/c:
[*1011 23.*40-11] D H sx€. D . *‘.r - //* (1)
K . (1). *63-32 .Db. Prop
*63 382. b . g ! /,«* [*6318 . (*63 03)]
*63 383. b . = / 0 ‘*
Don.
b . *63-3818. *1011-23-35 .Db: a c/ 0 ‘* .
. D . W* = W«
[*6319]
= t‘a
[*6311]
(i)
1-. (1). *1011-23 . *6318 . D b . Prop
*63-384. 1-: !/* - 1,‘X. D . / 0 ‘* = t n ‘\ . / V = [*63 383-37]
*63-39. b : </* = /,‘A.. = . tf* = U *\. s . Vk - e'A. [*63-33-384-37]
*63 391. 1-: *‘.r = t‘y . = . * 3 ‘.r = * 3 'y
Dem.
b . *63-39 .Dh: ( a< j «= ( 3 ‘y. = . t 9 ‘t‘x = U l t l y •
[*6315] = . t*x = /‘y : D 1-. Prop
*63-392. b : U*k = L‘\ . = . t* K = // X . = . u, € k = / 0 ‘X
Dem.
V . *63-39. D h : </* = /,‘X. = . = /,%‘X.
[*63-321] = (1)
K(l). *63-39. Dh. Prop
SECTION B]
107
RELATIVE TYPES OF CLASSES
*63 4. h : a € t a ‘*c . * € tj\ . Z> . t 0 ‘a = tf* = LS\
Dent.
.*63-38-18 .Dh: Hp.D .tja = £/*. = f,‘\ .
[*30*37.(*63*05)] D . V a - ^.
[*63 321] 3 . //a = /,**. = f 5 ‘\ : D I-. Prop
*63 41. = t,‘\
Deni.
H . *63-4*18 . *10*11‘23*35 . D b : * * t.‘\ . D . <,
[*63383] ' =t , K
[*63-38-18.*10 11*23*35]
K.(1).*6318. 3b. Prop
*63 42. I-. = t,‘\ [*30-37 . *63 41 -383]
*63 43. b . - t‘ x [*63-3415]
*63-44. I-. t/p'a = t.‘a [*63-43-34]
It is obvious that the analogues of the above propositions will hol.l for
arise, wl u CtC ' l. Ve Shal ‘ " 0t pr0ve thcse annlo 8»t'«. but if occasion
IMS we shall assume them, referring to the corresponding propositions for
*63 6. b: X c („<« . = .ae t“x .s.aCl'r.s. t‘x = („‘o
Dem.
h • *6315 . D h : a C . = . a C .
[*63-371] -.««<«*
l-.(l). *63-22.31-. Prop
*63 51. I-:«« v*o C C t‘« .*.««« -
Dem.
I-. *4-2 . (*63-03). 3 t-: a C t‘ K . = . « C I.‘s‘k .
[*63-371-19] m.aetWx.
[*4-2.(*6303)] *.«««,«*.
[*63-383] = . a ( (.«*
l-.(l).*635 22.3b. Prop
*63 62. b : a e t,‘\ . = . a C t,‘\. = . \ C Va. = . t‘a = t,‘\. s . <=<„ =
Dem.
( 1 )
(1)
H . *63-51 . (*63-03). D
I-: a c 1,‘X . = . a C .
[*63-321] s.aCaVX.
[(*63 03 05)] s.aCVX
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
K. *63-321 . D
H : a € //X . = . a e t+t/X .
[*03*22] = .fa = A/f/X
[*63-321] = /,‘X. (-2
[*03-391 -11 -42] =. t-'a = //X. (2
[*6315181] =.XC t 0 ‘t»a.
[*6315] = . X C t u a (4
K(l).(2).(3).(4).DKProp
*63 53 1-: x t t„‘a . = . t u x = t‘a . = . t*.c = t m ‘a
/Jem.
h . *30 37 . D : P‘x = Pa . D . - *,‘Pa .
[*63-43-34] D.Pa=A/a (1
H . *63 19. D h : t'x - A/a . D . f*c = Pa (2
K(l).(2). *035 . D H . Prop
*63 54 h : a e A/* . = . A/a = f,Vr. = . Pa =■ /.V . = . P‘a = Vk
D eni.
*■ • *30-37 . D h : t‘a = A/* . D . t x ‘t‘a - f, V* .
[*03-34-321) (i;
H . *30-37 . D h : A/a = A‘* . D . P//a * W* •
[*03-19-383] D.Pa-A/* ( 2 :
H . (1). (2). *03-51-53 . D h . Prop
*63 55. h : * * A/X . = . = A/X . = . A/* = t,‘X . = . P* - A/X . = . P‘* - PX
(Proof as in *03 54]
*63 56. H : a- € //* . = . Vx - A,** . s . P‘x «= A/*
Dem.
K . *G3 321 . Dh:xf /.V . = . .r e (/*,'* .
[*63-53]
■ .*“«-***/«
(i)
[*03-383]
(2)
h . (1) . *03-53. Dh:x€t, g «.
. s . t l x - A/A/*
[*63-321]
-</*
(»>
I- .(2). (3). D H . Prop
*63 57. h : a e A/X . = . A, 1 a = A/X . = . Pa = A/X . = . A»‘a = A/X
[Proof as in *63-56]
*63 61. (*6319101]
*6362. K : a € f/« .D.i'xe t'a . A'Par = Pa
Dem.
. *03-53 .DI-: Hp. D . PAr = /‘a .
[*63-61] D.PPar = Pa.
[*6316] D . i*xet‘a : D I-. Prop
SECTION B]
*63 621. h
*6363. h
Dem.
*63 64. f-.
Deni.
*63 65. .
*63 66. H.
*63 661. h.
*63 67. h.
*63 68. h .
RELATIVE TYPES OF CLASSES 10'J
: .r € a . D . /‘.r < [*63*62 . *63105]
: X€t,‘a . 3 . I't'xer'a . t'l'i *.,= /=‘ Q
*63101. =
[*6362] DhHp.D./‘ fl = W ,.
[*6319] 3.* s ‘a = *Ve‘.r (l>
Ml).*63-103.31-. Prop
>t € 0-t.‘i“0
h.*51-16. *37-62.3
h:a-€/9. 3 . x € i‘x. i‘xc t“0.
[*63*105-38] 3 . jc< . f.V* = f, V‘/9.
[*1313] 3 .**(,«<“£ (l)
H.(1).*63*51.3 h. Prop
Cl'l/a-f'a [*63*371 .*60-2]
CIV* - I"* [*63*5 . *60 2]
*‘Cl‘a = P'a [*60-34 . *63105-53]
Cl [*63-51 .*60*2]
ClV* = t x *K [*63-52 . *60-2]
*64. RELATIVE TYPES OF RELATIONS
Summary q/*(>4.
In the present number, we introduce notations defining the type of a
relation relatively to the types of its domain and converse domain, when
these types arc given relatively to some fixed class a. If R is any relation,
it is of the same type as D‘R t UWR. If U‘R and (I*/? are both of the
same type as a. U is of the same type as t.,‘a t tja. which is of the same type
as a | a. The type of Ca | t m *a we call U*a, and the type of t M ‘a | t n ‘a we call
r nnt a, and the type of tja | t./a we call t mn *a, and the type of t„,‘a 1 1"‘* we
call and the type oft m, a | t n *a we call We thus have a means of
expressing the type of any relation R in terms ot the type of a, provided the
types of the domain and converse domain of R are given relatively to a.
The most useful propositions of the present number are the following:
*64 16. I- s R G t„*a f t.‘0 . = .R* £•<£.*« T UP)
*64 201. h iRGS.D.Rc t‘S .t‘R = t'S
*64-231. h : R e t*Q . D . D*R e VWQ . (I‘R t . C H R * t'&Q
Here “ C‘R e t'C'Q" will only be significant if R and Q are homogeneous
relations, which is not required by the rest of the proposition. When R and
Q arc homogeneous relations we have
*64 24. V : R e t‘Q . s . C*R e C&Q . a . tjC'R = WQ
This proposition is useful in connecting ordinal and cardinal existence-
theorems.
*64 312. h . t ni x = V'H'x = Ut-*x
*64 6. H. RI‘(C« T UP) « t‘iU* T UP) - T 0)
This proposition is frequently used. It states that the class of relations
whose referents are of the type of members of a while their relata are of the
type of members of fS ( i.e . the class of all. relations contained in t 0 ‘a f UP) 19
the type of t 0 *a f t 0 ‘fi and is also the type of a t fl.
*64 65. K : C*P C t 0 ‘a . = . P e tja
*64 67. I-: C*P Q t f x . = • P e t ut x
The propositions of the present number arc mostly obvious, though forma
proofs are sometimes not very easily found. The use of the propositions of this
number occurs chiefly in the first section on relation-arithmetic and in the
proofs of existence-theorems in ordinal arithmetic and the theory of ratio.
SECTION B]
RELATIVE TYPES OF RELATIONS
111
*6401.
C‘o = tWa t Co)
Df
*64 011. t"‘x = t‘(t‘x 1* t‘.v)
Df
*64012.
t'-‘x = t‘(t‘x T tr‘x)
Df
*64013
t' u x = t‘(t?'x t t‘x)
Df
*64014.
t*‘x = t‘(L u x t t*x)
etc.
Df
*64 02.
C« = <‘(CatV«)
Df
*64 021.
C« - t‘(t ,‘a t C«)
Df
*64 022.
etc.
Df
*64 03.
C'a = <‘(C« T t*a)
Df
*64031.
f 1 u a = W«'T^)
etc.
Df
*64 04.
*C«-1‘(^ tc«)
Df
*64 041.
•C«- Wat Co)
Df
etc.
*641. Ka|a< t< n ‘a
Dem.
b . *212 . D h : a ■= Co .D.afa-C« | Co (1)
1-. *35*9 . Dh:a|a= Co f Co . D . a = Ca :
[Trnnsp] D I-: a + C«. 3 . a t « + Co T C« (2)
1- • (1)• (2). D H :.a=Ca -v . a+Ca:3: af a«C« f Co.v.a | o+C« t Co (3)
1-. (3). *5115 . *63101 101. D h : a t « - V« t Co C« T Co (4)
h . (4). *51 15 . *63 101 . (*64 01). D b . Prop
*64 11. h . C‘a = *‘(a f a) [*041 . *6316]
*6412. b . a t /9 e *‘(C« T C/9)
Dem.
h . *35*85*86 . *63*18 . D b : a T /9 = Co | C/9. = . a = Co . /3 = C/9 (1)
h . (1). Trnnsp. 3 h : a - t'a . £ - C/9. 3 . a T /9 = Co T C£ s
o = Co . £ + C/9. 3 . a T /9 + C« T C/9 :
[*63*101 .*51*15] D I-: a = Ca . D . a t /9 f l‘(C« T C/9) (2)
Ml) • Transp . Dh:a*Co.3.aT/9* (Co T C/9) .
[*63*101 .*51 * 15.Transp] D . a t£et*(C« T C/9) (3)
b • (2) . (3) . D 1- . Prop
*6413. h . t‘(t*‘a t C/9) = <‘(a f £) [*6412 . *63*16]
*6414. 1-. (x, y) . * (C« t C/9) y [*63*1. *35103]
*6416. b .(R).R G C« t C/9 [*6414 . *25 1411 ]
112
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART n
*64*16. 1-: R G tja t tjj . = - R e f‘(V« T C/9)
I Jem.
b . *211 . 3 b : R = C« t C/9. v . 7? + C« t Vi® s
[*26*42] 3 b : R = V* T '-‘a? . rt G C* T Vi® . v . if * C« t C/9 (1)
H.(l).*64*15.*10*221*13.3
biRCfa T C/9 : if = c« t C£. R g c« T C/9 • v. /e + v« T Vi® ( 2 )
I- .(2). *5*1.3
1 -K C Ca t Vi®. = s /f = Ca | C/9. if G C« t Vi®. v . J? + C« T C/3 :
[#23*42] s : R = //a | C/3. v . if * C« | C/9 :. 3 h . Prop
l*y putting tj'a (where i and s are some index and suftix which have been
defined) tor a and t/'a f«»r 0. the above propositions give results applicable
fo any of the types defined at the beginning of this number, because of
CC‘a = C‘a.
#64 2. I-: a S R * S . 3 . Se t'R . t'R = t'S [*63*13*16]
*64*201. b : R G S . 3 . R « t'S .t'R = t'S
Dem.
b . *2.5*6.3 I- s. Hp . 3 : R - S. v . 3 ! S±R:
[* 13* 14] 3 : R -* .S'. v . It + S :. 3 b . Prop
*64*21. h : .r/fy . 3.if c t'it'x | t'y)
Dem.
H . *63*103 . *35*103. 3 b .x(t‘x t f‘y)y (1 )
h.(l). 3HsHp.3.a!/2A(^xtf < y) ( 2 )
I-. (2). *64*2 . 3 h. Prop
*64*22. b . R « f‘(CD‘i? t CCI'i?) [*C4*1C . *63*105 . *35*83]
*64 23. b .t'R~t's't'R
h . *63*103 . *41*13.3 b . R G *‘f‘if (1)
h . (1) . *64*201 . 3 h. Prop
*64 231. b: Ret'Q.D. WRcl'D'Q Al'R c t'd'Q. C'R € t'C'Q
Dem.
b . *63*12.3 b Hp . 3 :: xRy . 3 x>y .4%. v . ~(.rQy) ::
[*10*28] 3 :: (ay) • x/fy. 3 X (ay) .xQy.v. (ay). ~ (*Qy) :•
[*5*63] 3 X :• (ay) - xQy v - (ay) • *Qy: (3y) o~*Qy*o
[*3*26] 3 X (ay). xQy . v . - (ay) • W
b . (1).*33*13.3 1-:. Hp . 3 : x€ D'R .D x .xe D‘Q w - D'Q :
[(*63 02)] 3 : D‘if C t 9 ‘D'Q:
[*63*371] DzD'liet'D'Q ( 2 >
Similarly b : Hp. 3.0 *R et'd'Q .C'Ret'C'Q < 3 >
b • (2). (3) . 3 b . Prop
SECTION B] RELATIVE TYPES OF RELATIONS 413
*64 24. h : R c t‘Q . = . C'R € t'C'Q . = . tjC'R = tJC'Q
This proposition is only significant when R and Q are homogeneous
relations.
Dent.
h . *64-22 . *63 181 . D h . R € t‘(tSC*R | .
[*13'12] DI-: t.‘C‘R = t,‘C‘Q.O.R e t‘«SC‘Q1t'‘C"Q) (1)
I-. *64-22 . *63 181 . 3 I-. Q e t‘(l 0 ‘C‘Q f t.‘C‘Q)
I-. (1). (2). *6316 .31-: t c ‘CR = t.'C'Q .O.Rtt'Q
I-. (3). *64-231 .*6316-37.3
I Ret‘Q. = . l c ‘C‘R = t,‘C'Q. = . C'R e t‘C‘Q : 3 I-. Prop
( 2 )
(3)
( 1 )
( 2 )
*64-3. I-: t„‘a = l m ‘0. =. a e t‘0 .a . t‘a = t‘&. a . t„‘a = C/3
Dem.
1-. *30-37 . (*64 01) . D h : t 0 ‘a - tSfi . D .
3l-:<.'« = 09.3.«T«f^.
[* 6416 ] 3 . a t o G C/3 | C/3.
[*35-9-91] 3. a C C/3.
[*63181] 3-C« = C/3
h . (1). (2). *63 16-37.31-. Prop
*64 31. t-. t"'x =■ tjt'x [*63 15. <*64-01 -Oil)]
*64 311. h . t„‘a = tCC« [*63 321 . (*64 022 01)]
*64 312. H . t”‘x = t"‘t‘x - tjfx [*6315 . (*6304). (*64014011 01)]
*64 313. h . t„‘a = = („*Ca [*63-321 . (*63 05)]
.64 32. t-: t a ‘a - <C/3. = . «„«a = („‘/9. = . („‘a = C‘/9. s . ("‘a _ t „. 0 .
0en( a • l”‘a = <"‘/3 . = .att‘0.3.fa = t‘/3
h . *64-313-3.3h.-y« = C‘/3. = . ft,‘a = f',‘$.
[*63-41-39] e . <‘a = t'/3
Similarly the other equivalences are proved.
*64 33. I": a e Cm . = . t„‘a = („•/*. = . U»‘a = t„>. = . t"‘o - •
Dem. Cm
H.*64-311-313.31-: (,,‘a = C‘m- = . «CC«-C‘Cm•
[* 64 ' 3 ] =. «‘C« = (‘Cm •
[*63-383-41-55] = . <‘o = Cm (1)
Similarly the other equivalences are proved.
*64-34. 1-: a e Cm • a. („<« = £=>. = . (“‘a = <„‘m . = . . = . («o=Cm
[Proof as in *64 33]
Ill
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*64 5. H . RlV.'a T tJ0) = f‘(C« f t,‘0) = t‘(a f £) [*641316 . *612]
*64 51. [*64 21 . *55 132]
*64 52. h:xc tja . >/ c tJ0 . D . * i y € /‘(C« f C£) [*63-11 . *64 51]
*64 53. K : x € //a . 5 C .<i‘x) l & c t‘(t‘a t
I)em.
h.*6451 . D h.(<V)i5e(1)
h. *63-62. Dh: Hp. D. tU'x-Va (2)
h. *63 181-37. Dh: Hp. D . t'S =<‘.8 (3)
I-.(1).(2).(3). D 1-. Prop
This proposition is used in connection with cardinal addition (*11018).
*64 54. h . Rl Wa t Va) = tja - *‘(a | a) = f.*Rl*(a T «)
[*64-5 . *61-34 . *63 105 11 . (*64 01)]
*64 55. H : C l l > C f„‘a . = . I* e f...‘a
Dcm.
h . *35-91 .Dl-jCPCCa.s.PG Co T •
[*64-54] s . P t tja : D K Prop
*64 56. h. RI‘U‘.rT f‘.r) = P“.r
Dcm.
h . *64-5 . *63 15 . DK Rl V* t *‘.r) = f
[(*64011 >] DH. Prop
*64 67. hsC‘PCfj-.s.Ptf u x [*64-56. *35-91 .*61 2]
*646. 1-. t‘P - t f.'CI*/’)
h . *35-83 . *63 105 .Oh. PC U‘D‘P T t.‘d‘P.
[*64-201 ] D h . t‘P = t V<r.P>
[*64-5] - R1‘(VD‘P T UWP) .31-. Prop
*64 61. V : D'P * fa . (l‘P tt‘0 ■ O . t‘P = f(a t 0)
I-. *63-16-35 .Oh : Hp. O . <„‘D ‘P = t.‘a . t.‘(l‘P = U‘0.
[*64*6] O.fP = f(t. t a‘[l. , 0)
[*64 5] = f(a 10):Oh. Prop
*64 62. I-: D‘P t t‘D‘Q. <I‘/> t fd'Q .m.PtfQ.m. t‘P = t'Q
h . *64-61.31-: Hp . O . fP = l‘(D‘Q f d‘Q)
[*64-5-22.*6316] = fQ <*>
h . (1). *64-231 .0 h . Prop
*64-63. I-: D ‘P e fa . d‘P e t‘0. = . fP = P(a 10). = .Pe f(a 1 0)
De ‘"' h . *64-5 .Oh :fP = f(a f /9 ).O.PP = f(t.‘a T t.‘0).
[*64 231.*35-85-86] O . T>‘P e Pt 0 ‘a . d‘P e t%‘0 ■
[*6319] O .T>‘Pefa.d‘Pef0 (!)
I-. (1). *64-61 . *63 16 . O h . Prop
*65. ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS
Summary of* 65.
In this number we are concerned with definitions and propositions in
which an ambiguous symbol is determined as belonging to some assigned
type. If “o' is an ambiguous symbol representing a class (such as A or V
for example), “ a x is to denote what a becomes when its members are deter¬
mined as belonging to the type of *, while “a(x)" denotes wlmt a becomes
when its members are determined as belonging to the type of t‘x. Thus
e '°: ". V| wil1 bc everything of the same type as .r, t.e. t*x\ V (*) will be t‘t‘x.
Similarly if ‘ R" stands for a relation of ambiguous type, such as A or V,
R x will denote what R becomes when its domain is confined within the type
of*; R lxu) will denote what R becomes when its domain and converse domain
are confined respectively within the types of a- and y; R(x,y) will have the
domain and con verse domain confined respectively to the types of Vx and t*y;
with analogous meanings for R (x) and R (x„). Throughout this number,
R and a do not stand for proper variables, but for typically ambiguous symbols.
i he notations of the present number are used in the elementary parts of
the theory of cardinals and ordinals, i.e. in Part III, Section A, and in Part IV.
Section A. The only proposition, however, which is much used, is
*65 13. I-: a - 0 X . s . a = t‘x rs £ . = . a C t € x. a = 0
Here /9 is supposed to be a typically ambiguous symbol. The first
equivalence. 0 X . = 1 3," merely embodies the definition of (3 X
(*65 01). It is the second equivalence that is important. Let us, for the
sake of illustration, put 1 in place of /9. Then we are to have
a = t‘x r, 1 . = . a C t*x . a = 1.
(Since 1 is a class of classes, we shall have to suppose that x is a class.)
Considerye*. If a - a l, y e a . s. y et*x . y e 1. But we have (y) . y e t*x.
Hence yea.s.ytl, whence a = 1. Also if of course a C t‘x.
1 hus a = t‘xn l.D.aC<‘*.a*l. The converse implication follows from
*22 621. The reason for the proposition is that a symbol such as “l,” if it
occurs in such a proposition as a-l‘ain 1, must, for significance, be deter¬
mined as meaning that 1 which is of the same type as a, i.e. the class of all
unit classes which are of the same type as members of a. And similarly,
when we put a = 1, that does not mean that a is the class of all unit classes,
but only that it is the class of all unit classes of the appropriate type, which,
PROLEGOMENA TO CARDINAL ARITHMETIC
41C,
[PART II
il a C t*x, will be t*x r\ 1. The proposition *7 *x r\ 1 = 1” is true whenever it
is significant., but t*.rr\ 1 is typically definite when x is given, whereas 1 is
typically ambiguous. The use of the above proposition lies in its enabling us
to substitute typically definite symbols for such as are typically ambiguous.
Another useful proposition is
*65 2 . K sg‘|/?„.„) = 7? (*»)
Here R is supposed to be a typically ambiguous symbol; the proposition
states that if It is typically defined as going from objects of type x to objects
of type y, then It must go from objects of type l*x to objects of type y. This
proposition is only used twice (*102 3 and *1542), but both uses are of great
importance, tin- one in cardinal and the other in ordinal arithmetic.
The only other proposition of this number which is subsequently used is
*65 3. I-. R $ “p - (H'Vfe * rt‘V a t € 0
This proposition is used in *102*84.
*6501.
a r =* a r\ t*x
Df
*6502.
a ( x) = a n Wx
Df
*6503.
R x = (t*x)Ut
Df
*6504.
R (x) - (fx) 1 R
Df
*651.
Df
*6511.
Df
*65 12. R (x, y) - (C-‘x) 1 R f (t"y) Df
*65 13. I- : a - 0, •« . a - t*x* 0 . ■ • a C tfx . a - 0
f-. *4*2 . (*65 01) .
Dh:a = /3x. = .a=f‘jn^
(i)
b. *22 621 .*1313
. D H : a C t*x. a = 0.0 . a = t*x r\ 0
(2)
b . *22 43.
D b : a = t*x r\ 0. D.aCl‘x.aC/9.
(3)
[*6313)
D.0et‘t‘x.
[*63-371 15]
O.0C t‘x.
[*22-621]
D . 0 = t*x n 0
(4)
K(3).(4).
D h : a = t*x n 0 . D . a C t*x . a = 0
(5)
b . (1) • (2). (5) .
D b . Prop
*6514. b : a: e t 0 ‘a . D . 7 (*) = 7- [*63 53 . (*65 01 02)]
*65 15. b:xe t 0 f a . D . R (x) = i?. . R (* y ) = R,.. v) [*63 53.(*65 03 04 1 11)]
*6516. b : X€t 0 ‘a.y€to‘@.D.R(x,y)=R(x fi )=R* t fi, [*63 53 .(*6511112)]
SECTION B] ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS
•117
*65-2. H . sg'iRir.y)}
Dem.
H. *32-1*23. (*65'1).D
h : a [sg‘j/* (x , y) |]w . = . a = z . w et*y . 2 72m/] .
[*22*39.*20*42] = . a = t*x r\z(we t‘y . zRw) .
[*65*13] = . a C t‘x. a = z (iv c t*y . zRw) (1)
I- . *20*33 .Dh:a = 3 (wet*y . zRiu) . = z ea . = t . w e t l y . 2/2?*;
[*63108] = z. w t t‘y z z e a . = t . w e t‘y . zRw :.
[*4*73] = :. t u e t‘y z z e a . = t . zRw z.
[*20*33.*32*1] m:.w€t‘y.<dlw (2)
H . (1) . (2) . *63*5 .Dha [sg‘(/£ (Xty ,)] w . s . a c c .
[*35*102.(*65*11)] = . a |/£(j; v )) w:Dh, Prop
*66*21. h . R lx , y ) = |/*«.„,) tr,i/i
Dem.
K *21*2 . (*651). D h . {i2 lx ,y») u . y , = ^1 ft‘*1 -R [ t‘y\ f t‘y
[*36*33 34] -t'xIRft'y
[(*65*1)] -72, x . y) .Dh.Prop
*66 22. h . R (x, y ) = (R (x, y)) ( x , y )
This and the following three propositions are proved as *65*21 is proved.
*65 23. V . R (x v ) - (/* (x„)| (x v )
*65 24. h. J* x = (/e x ) x
*65 26. !-..&<«)-{£<*)}<*')
*65 3. h. /w = iR“rb = « f/3
Dem.
h . *37*1 . (*65*03). D h . Rfi“fj = £ |(gy) • y * n . xRy . # € t‘fi j
[*22*39.(*37*01)]
[(*65*01)]
h . (1) . (2) . D h . Prop
R“p " t l &
( 1 )
( 2 )
R «c w I
27
SECTION C
ONE-MANY. MANY-ONE, AND ONE-ONE RELATIONS
Nummary of Section C.
In tin* present section we have to consider three very important classes of
relations, of which the use in arithmetic is constant. A one-many relation is
a relation li such that, if y is any member of Cl *R, there is one, and only one,
term x which has the relation li to y, i.e. IVy € 1. Thus the relation of father
to son is one-many, because every son has one father and no more. The
relation of husband to wife is one-many except in countries which practise
polyandry. (It is one-many in monogamous as well as in polygamous countries,
been use, according to the definition, nothing is fixed as to the number of relata
for a given referent, and there may be only one relatum for each given referent
without the relation ceasing to be one-many according to the definition.) The
relation in algebra of x 1 to ./• is one-many, but that of x to x* is not, because
there are two different values of* that give the same value of .r=.
When a relation li is one-many, IVy exists whenever yed'H, and vice
versa; i.e. we have
li € one-many . = : y e Cl*72 . D y . E! IVy.
Thus relations which give descriptive functions that are existent whenever
their arguments belong to the converse domains of the relations in question
are one-many relations. Hence Cnv, D, Q, C, R, R, sg, gs, R t .p. s, />.s, I, i, i.
Cl, Rl are all of them one-many relations.
When R is a one-many relation, IVy is a one-valued function; conversely,
every one-valued function is derivable from a one-many relation. A many -
valued function of y is a member of R*y, where R‘y is not a unit class, and
any one of its members is regarded as a value of the function for the argu¬
ment y\ but a one-valued function of y is the single term R‘y which is
obtained when R is one-many. Thus for example the sine would, in our
notation, appear as a relation, i.e. we should put
sin = 3# [x = y — y73! + y*/5 •••! ®f,
whence sin‘y = y — y*/3 ! + y*/5 ! — ....
so that "sin‘y” has the usual meaning of sin y. Then instead of sin" 1 *, we
should have sin'or, which would be the class of values of sin” 1 *; and instead
of “y - sin“' x," which is a misleading notation because y = sin- , ar and
3 - sin - ' x do not imply y = z, we should have yesin'x. Similar remarks
would apply to any of the other functions that occur in analysis.
SECTION C]
ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
•119
A relation R is called many-one when, if or is any member of 1) *R, there
is one, and only one, term y to which a; has the relation R, i.e. R'xe 1. Thus
many-one relations are the converses of one-many relations. When a relation
R is many-one. R‘x exists whenever xe D‘R.
A relation is called one-one when it is both one-many and many-one, or,
what comes to the same, when both it anil its converse arc one-many. Of the
one-many relations above enumerated, Cnv, sg, gs, /, i, 7. Cl, HI are one-one.
1 wo classes a, ft are said to be similar when there is a one-one relation R
such that D‘/£ = a . Cl 4 72 = ft, i.e. when their terms can be connected one to
one, so that no term of either is omitted or repeated. We write “asm ft" for
“ a * 8 similar to ft." When two classes are similar, the cardinal numbers of
their terms are the same; it is this fact chiefly that makes one-one relations
of fundamental importance in cardinal arithmetic.
According to the above, a relation is one-many when
y c (I* R . D v . R*y e 1,
i.e. when li“G‘ft Cl.
Similarly a relation is many-one when
R“D‘R C 1.
and a relation is one-one when both conditions are fulfilled. The classes
R“Cl t R, R^D'R, which appear here, are often important; some of their
properties have already been given in *8777*771*772-773 and in *">:J*61 to
*53*641.
It is convenient to regard one-many, many-one and one-one relations as
particular cases of relations which, for some given a and ft. have
m R“a.‘RC«. 4 R“TyRC0.
We put a-+ft=R\R“(l‘RCa. 4 R“D‘RCft\ Df.
Hence, without a new definition. " 1 -> 1" becomes the class of one-one
relations; also, as will be shown, “l-^Cls*’ becomes the class of one-many
relations, and “Cls—► 1“ becomes the class of many-one relations. Although
it is chiefly these three special values of a — ► ft that are important, we shall
begin by a general study of classes of relations of the form a-* ft.
27—2
*70. RELATIONS WHOSE CLASSES OF REFERENTS AND OF
RELATA BELONG TO GIVEN CLASSES
Summary of *70.
If a and 0 are two given classes of classes, a relation R is said to belong
to the class a-*/3 if R*yca whenever yeG‘/f. and R*xc0 whenever xeD*R.
If only one of these conditions is to be imposed, this result is secured by re¬
placing the class involved in the other condition by "CIs," since " R‘ycC\s"
always holds, and so does “ R‘x€ CIs.*' and therefore neither imposes any
limitation on R. In the most important cases, a and 0 arc either both cardinal
numbers, or one is a cardinal number while the other is CIs.
In virtue of *37702703. the conditions above mentioned as imposed upon
R by membership of a— >0 are equivalent to
~7i“WRCa.*R“Y) t RC0.
This form is used in the definition (*70-(>l).
The propositions of the present number are hardly ever used except in *71.
where a and 0 are both replaced by 1 or CIs. The most useful propositions are
*701. h : /£ c a —► £ • s . ft C a. C £
(This merely embodies the definition.)
h Re a -> 0 . = : (y ). ~R‘y * a » «‘A : (*) • « 0 ” «‘ A
K£-»a = Cn v“(a->£)
h. a -> CIs C a)
t-. CIs -» £ - R (R'WR c 0)
h . a — * 0 = (a — * CIs) n (CIs —» 0)
h: <KR n WS - A . R> S € a -¥ CIs . D. R v S e a CIs
with similar propositions for CIs —> 0 and a—>0.
*70 62. h : R e a -> CIs . D . R f y e a -» CIs
with a similar proposition for CIs —> 0.
*7013.
*7022.
*704.
*7041.
*7042.
*7054.
*70 01. a —* 0 = R (R“<1‘R C a . C 0) Df
*70 1. h s R e a -> 0 . s .~R“<I‘R C a . R“D‘R C 0
.70.1. h , J..— 0. = ■.. a-*. * ■ . • - ■ . „0U
[#203. (#7001)]
SECTION C]
CLASSES OF REFERENTS AND RELATA
421
*7012. h : R€a-*0. = .Ii“VCasjt t A.li“VC0sji‘A [*701 . *53-62*621]
*7013. . R € a -+ f3 . = z (y) . R*y eawi'A: (.r). R € x i*A
Deni.
h . *37*702 .Dh.rVCaw t‘A .arycV. 3., . ~R*y eav i‘A :
[*24104.*55] = :(y) .~R‘ycav l‘A (1)
Similarly h J?“V C/3 w e'A . = : («) ./2‘«6/8 w/ i‘A (2)
h . (1) . (2) . *70*12 .31-. Prop
*70 14. h:: R e a —> f3 . = (y ): e a .v.R*y = A :.(x)z R‘x e /3. v. = A
[*7018. *51*236]
*70 15. h:./eea-»/9. = :g! R‘y . 3 V . ea:g! . 3 X . 7?** c £
[*24 ol .*4-6. *70 14]
*70 16. h :Rea->/3.= . D‘~RCa sj i‘A.D*RC/3vi‘A [*37*78*781 . *70*12]
*70 17. h :: A « a . 3 R c a —»/9 . = : (y>. 72‘y € a : g ! . 3 X . c #
Dem.
I-. *51*2 . *22-62 .Dh:Hp.D.a-av( ( A (1)
Ml). *70*13.3
K :: Hp . 3 R c a —»/9 . s : (y) . /i«y c a : (x). R‘xc /9 ^ t‘A (2)
K . *51-236 .3 1-:. R*xe (3 \j i*A . = : R‘xt/3 . v . R*x = A :
[*24-51. *4-6] ssaJ/^.D.S^e/S (3)
H . (2) . (3) .31-. Prop
*70171. Hs: A«/9.3 :. J* . a-*£. = : g 1^‘y . D„ . fl’*y«o : (®).S'ae/S
[Proof as in *7017]
*70 18. h :: A t a . A e 0 . D :. R ( a 0 . = : (y) . c a : (x) . *R‘x 6 0
[Proof as in *7017]
*70 2. Ka-»^ = (ou i‘A) ->/9 = a -> (£ v, i*A) = (a w i‘A) -*(/?« t‘A)
Dem.
1 -. *22-58-62 . DH.(awi‘A)wt , A=aui‘A.(/3v;t‘A)u(‘A=^ut‘A (1)
1-. *7012.(1). 3t-:Aea—»£. = . V C(ow i‘A) w t‘A . 7?“V C £ u t‘A .
[*70-12] A)-*0. (2)
[*7012.(1)] = .‘fl“VC( awt ‘A)ui‘A.K»VC(^u t ‘A)u t ‘A.
[*70’12] =.R€(asji‘A)-+(0sji‘A). (3)
[*7012.(1)] = # 7?«V Caut'A. /*“V C (£ w i‘A) v t‘A .
[*7012] = . R e a—*(/3 u i*A) (4)
h.(2).(3).(4).3KProp
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
122
*70 21. = i‘ A) = (a - e‘A) -> (£ - i‘A)
Deni.
h . * 5 1 222 . D H : A € a . D . a - r ‘A = a : A ~ e /9 . D . 0 - I ‘ A = £ (1)
K *51 221 . Dh: A«a.D.(a- /‘A)w‘A=a: A e£. D . </3 - PA)= £ (2)
h.(l).D
H : A~€O.D.(a-i‘A)-»/3 = a-»/3.<a-i*A)->(£-i‘A)=a->(£-i‘A) (3)
K (2). *70*2.3
K : A c a . 3 . (a — f*A) —= a —► #3. (a — PA) —► (£ — PA) — a —►(/? — PA) (4)
h . (3). (4). *4 83 . 3
h .(a — i‘A)—>{3 = a—>j3 .(a — PA )—*(£ — PA) = a —* (/9 — PA) (5)
Similarly h . a -> (£ - PA) - a . (a - /‘A) ->(/3 - /‘A) = (a - PA) (0)
h . (5).((»). D K . Prop
*7022. h.(3->a =
Dent.
1-. *37 *6. *31*13.3
H :.g*Cnv“(a-*/9>
1*7012]
1*32*24 241]
[*13*103]
[*32*23*231.*10*35]
f*31*33.*10*24]
[*70*12]
P‘(a—►/$»
: (gi?). R € a —► &. Q = Cnv‘/f :
:(g/?).rVC«wi‘A ./?“ V C^w PA . Q - Cnv‘/f:
: (a K ). (gs'Cnv'/^p'V Coo PA .
<sg‘Cnv‘/?)“V Cfiyj PA . Cnv‘7? :
: (g/?).(gs‘(?)“V Caw p A .
(sg‘Q)“ V C £ w; i*A . <? - Cnv*R :
V“V Caw PA . Q*‘V C £ PA : (g/?). Q - Cnv'rt :
Q“V Caw PA . <?‘V C/9wi‘A:
Qe/3—>a:. D1-. Prop
*70 3. l-.aC7./3CS.D.a—*/9C7—
Deni.
(-. *701 . D h : Hp . R e a 0 . O .~R“Q‘R C a .*R“D‘R C0. aCy.0 C S.
[*22'44] 0.~R ,, a , RCy.R"D , RCS.
[*701] O.Re y-*B (1)
t- .(1). Exp. *1011-21 . D h. Prop
*70 31. h . (a -* 0) n (7 -♦ S) = (a r. 7 ) -»(/9 a S)
Dern.
h . *701 . D I-: R e(a—> 0) n ( 7 —»S). = .
7?“CI‘fl C a . ~R"a‘R C -y. J?“D ‘R C 0. R“D‘R C S .
[*22-45] = . R"C1‘R Cany. *R“X>‘R C0nS.
[*701] = .Re(any)-*(0nS):O h . Prop
SECTION C]
CLASSES OF REFERENTS AND RELATA
*7032.
Deni.
b . *701
| *3 ‘26 27
[*22-65J
[*701]
*704.
Deni.
*7041.
*7042.
*7043.
*70431.
*7044
*70441.
*7045.
*70451.
*7046.
K(a-»^)u(7-»5)C(au 7 )-»(^uS)
. D b s. R €(a — > w (y — > 3). s :
R“a*R C a .R“l)*R C (3 . V .R“a'R C y .*R“D l R C 5 :
•48] D : R“d‘R C a . v . R“(I‘R C y : *R“D'R Cg.v .*R“D*R C 5 :
D : R“(\‘R Cau 7 .C /3 w 8 :
D:fi<(a w 7 )->(/9w8):.Db. Prop
b . a -* Cls = R (R“a*R C a)
b . *701 . D b : R e a -> Cls . = .~R“a*R C a .*R“Y)<R C Cls .
[*37*761] = . Ca:Db. Prop
K . Cls -► 0 = ft (Ji“l)‘R C 0)
b . a —► /3 = (a —► Cls) (Cls —► /3)
b e a —* Cls . s s y e Q*R . . 72‘y e a
b : /e c a — ► Cls . = . J«‘V Caw t'A
b : /e e Cls /9 . = ./*“ V C £ w i‘A
b : c a —► Cls . = . (y) . 7< 4 y caw PA
b : R 6 Cls-> £. 3 . <*) ./*‘xc£ w i‘A
b 72 € a—>Cls . = : (y) : 7£‘y € a . v . 7£ 4 y = A
[Proof as in *70 4]
[*70-4-41]
[As in *7011]
[As in *7011]
[As in *7012]
[As in *7012]
[As in *7013]
[As in *7013]
[As in *7014]
*70 461. b A e Cls —» £ • b : (a?) : R*x «£. v . R*x — A [As in *70 14]
*70 47. b:. «ea-*Cls. 2 : a !J*‘y.D,./*‘yea [As in *70 1.5]
*70 471. b :.ReC\s->/3.= ! %x . D x .*R‘xe0 l As in *7015]
*70 48. b : R c a -* Cls. s . D‘7?Caw i‘A [As in *7016]
*70 481. b:/eeCls-*/9. = .D <4 Sc/9wt‘A [As in *7016]
*70 5. b . Cls -* a = Cnv“(a -* Cls) . a -► Cls = Cnv“(Cls -* a) [*70 22]
*70 51. b s. £ v « a . « v e a w i‘A : D : .R, S e a-»Cls. D . R A S <? a->Cls
Dein.
b.*323.D b Hp . D : R 4 ye a . S 4 y ca . D . {sg‘(R A &)) 4 y€ a w t‘A (1)
b . *32-3 . *51 15 . *24-34 . D
b : R‘y € a . 5 4 y e i‘A . D . {sg‘(7* nS)]'y = A.
D. {sg‘(72 *S)}‘t/ea\j i‘A (2)
[*51-236]
424
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
I".(1).(2).*4*40 h i.HpOs/Pyca.S'ycavpAO. [sg < (Rf\S)\‘t/€a\Ji‘A (3)
I-. *32 3 . *51 15 . *24 34 . *51*236 . D
h : . D . {sg‘(/2 A S)]‘y eow i ( \ (4)
h . (3). (4). *4*4 . R*y, S*y ea w i‘A . D . (sg‘(/? A 5)}*^ ea u i* A :
[*10 1121*27.*70*45] D : R, Sea —► Cls O.(y). jsg‘(7? A e a w t‘A .
| *70*45.*32*23] D. R A 5 e a —* Cls :0 h . Prop
*70 52. I-f. tj e 0 . D,.,. £ a 17 e 0 ^ i‘A : D: rt.Se Cls->£. D. 2? A SeCls-*£
[Proof as in *70*51]
*70 53. h :• £, 17 « a Of., • £ a ip ca v f*A : £. ip c /9 Of,, • £ a ip e/9 v PA : D :
R,Se a-+ 0 ,D . R r* 8 c a—> 0
Detn.
h . *70*5-31 Oh:. HpO: /?. 5«a-*Cls. i?, 6 * € Cls-► £ O .
/e A.Sea->Cls. /e a5€CIs->/3 (1)
h .(1).*70*42 Oh. Prop
*70 54. h : d'R a CPS c= A . R t S c a -> Cls O. R v S c a -» Cls
Dcm.
h . *24-15 . *22-33 . D
h CP/* a (PS = A O : <y): ~ |y e (Prt . y c CPS}:
[*33*41 ] 3 ! (y) s ~ la ! • a ! 5‘yi :
[*4-51 .*24*51] D : (y): 7*‘y - A . v . S'y - A :
[*24-36] D : (y ): 7?y w ~S‘y =~S‘y . v .li'y v&y = li*y ( 1 )
I-. *70 45 . D
h R.Sea — *CU. D « (y). R‘y « a w PA : (y ). S‘y € a sj i* A (2)
h .(1).(2)0 h HpO :(y). R'yvS'yeav PA :
[*32*32] D : (»/) . [sg ‘(R o S))‘y e a v PA :
[*70*45] D z RwSca—* Cls :OK Prop
*70 65. h : D‘ ft n D‘S = A . R, S € Cls -> 0 O . ft c; S c Cls —> £
[Proof as in *70*54]
*70 56. h : D‘ft n D‘S = A . CP ft a CPS = A. R.Sea-* 0.0 • Rv Sea-*0
[*70*54*55*42]
*70 57. h : C‘ft n C‘S = A . R,S € a—+ 0.0 . Rw S e a—+ 0
Dem.
h . *33*161 Oh. D‘ft n D‘S C C‘ft a C‘S . CPft a CPS C C‘ft a C‘S.
[*24*13] D h : C‘ft aC‘S= A O. D‘ft aD‘S- A . (Pi* n CPS = A
h .(1). *70*56 Oh. Prop
(1)
SECTION C]
CLASSES OF REFERENTS AND RELATA
■m
*70 6. h : S e a —> CIs .K‘“oCow t‘A . D . R ( «S'e a —* Cls
Dem.
V . *37-31. D I-. (sg‘(.K, S))“V = («< jl?)“ V
[*37-33] = R<*W‘V (1)
h . (1) . *70 44 . D h : 5€ a —► Cls . D . {sg‘(tf | S)|“V C ^ t‘A) *2)
h . *37-22 . D H . 7*«“(a v t<A)- /*«“a v, R ( “i *A
[*53-31] = RS'avi'RSA
[(*37-04).*37*11*29] = «“‘a ^ l *A (3)
h . (3) . *22-60 . D h : Cavt'A.D. i*«“(a u e«A) Cou i* A ^ t‘ A .
[*22-56] D.^awt^Caut'A (4)
H . (2) . (4). D h: Hp.D. (sg^/e 1S)}“ V Caw'A.
[*70 44] D . R | Se a -► CIs : D h . Prop
*70 61. h : ie e Cls C £ v, e‘A . D . R j S * Cls —» £ [As in *70 6]
*70 62. f- : R e a -> Cls . D . R f 7 € a -> Cls
Dem.
V . *35-64. Transp . D 1-: y ~ € 7 . D . y ~ f C1\R f* 7 ).
[*33*41 .*24 51] D . {s%\R [ 7 ))‘y = A .
[*51-236] D . (sg‘(.ft r 7 )l‘y e a v, i‘A (1)
H . *35-101 . *4 73.3 H y €7 . D : ar(/if* 7 )y . = x . ;r/ty :
[*20*15.*3213'23] D : {sg‘(rt f^J'y -l?‘y ( 2 )
H. *70*45. D I- : Hp . D . R‘y € a w i‘A (3)
h.(2).(3). D 1- Hp . D:yey.D. |sg‘(72 f* 7 »‘y c a u t‘A (4)
I- . (1) . (4) . *4*83 . D h : Hp. D . {sg‘(/J |* 7 »‘y € a v <‘A (5)
• (5). *1011-21 . *70-45 . D H . Prop
*70 63. h : /e * Cls -> 0 . D . 5 ] R * Cls -♦ £ [As in *70 02]
*71. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
Summary o f * 71 .
In this number we shall lx- concerned with the more elementary properties
of one-many, many-one, and one-one relations. These properties are very
numerous and very important. The properties of many-one relations (i.e. of
relations belonging to the class CIs —> I) result from those of one-many rela¬
tions by means of *70 >. whence it follows that many-one relations are the
converses of one-many relations. It is thus only necessary to interchange
R and /J. D and Cl, R and li in order to obtain a property of a many-one
relation from a property of a one-many relation. Or we may repeat the
various steps of any proof, making the above interchanges at. every step, and
tin- analogous proposition will result. For this reason, in what follows, we
shall omit all proofs of properties of many-one relations, confining ourselves to
proving the analogous properties of one-many relations.
In virtue of *70 42, one-one relations (i.e. relations belonging to the class
1 -> 1) are the relations which are both one-many and many-one; hence their
properties result from combining the properties of oue-inauy and many-one
relations. We shall omit the proofs when they consist merely in such
combinations.
A one-many relation gives rise* to a descriptive function which is existent
whenever its argument belongs to the converse domain of the relation. That
is. if li ( 1 —► CIs, we have E ! R*y whenever y « G‘7f. Conversely, if a descrip-
tive function IV y exists for the argument y, then li is one-many so far as that
argument is concerned, i.e. li*y € 1. Thus we find
/fel->Cls.= .E!!/f“(I‘/e.
The descriptive function R*y derived from a one-many relation R has thus
a definite value whenever yeCl‘77, and not otherwise. Thus the class of
arguments for which such a function exists is the converse domain of the
relation which gives rise to the function, i.e.
7? c 1 —* CIs. D . # (E ! R*y\ = G‘72,
and the converse implication also holds.
It often happens that a relation which is not in general one-many becomes
so when its domain, converse domain, or field is subjected to some limitation.
For example, let R be the relation of parent to child, a the class of males, and
/3 the class of females. Then R is not one-many, but a R and £ 1 R are one-
many, and in fact (a 1 R)*y = the father of y, (£ 1 R) ( y = the mother of y. We
shall often have occasion to deal with relations obtained by limitations imposed
on D or G; thus a(D \ X) R . = . R belongs to the class \ and has a for its
SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
•127
domain. The class X may be so constituted that onty one relation R fulfils
this condition; in that case. DfXeCls —* 1. Since D e 1 —> Cls, wo find
D f* X 6 Cls—» 1 . = . D f X € 1 —► 1. This typo of condition, D f* X e 1—>1 or
Cl T X e 1 —► 1 or Cf*Xt 1 —► 1, is one which frequently occurs in subsequent
work. Another condition which often occurs is Ft* ^ € Cls — 1 ► 1. When this
condition is realized, a term x which belongs to the field of one relation of the
class X does not belong to the field of any other relation of this class, i.e. the
fields of relations of this class are mutually exclusive.
For purposes of realizing imaginatively the properties of one-many
relations, it is often convenient to picture their structure as in the accom¬
panying figure. Here x t y, s t ... form the domain of R, and all the points
in the oval marked R*x are such that x has the relation R to each of them,
with similar conditions for y and z. What characterizes R as a 1 —► Cls
is the absence of overlapping in the ovals. For if R*x and R*y had a point
in common, this would be a relatum both to x and y, and both x and y
would be referents to it; whereas in a 1 —> Cls, no term has more than one
referent.
The above figure illustrates a very important property of one-many rela¬
tions, namely
R e 1 Cls . = . R | R = I\iyR.
128
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
In the above figure, /[* is the relation of identity confined to x,y, z ,....
If R were not a 1 —* Cls, we could sometimes go from x to some term of
4— 4— w
R'xr* R‘y by the relation R, and thence back to y by the relation R. But
when R € 1 —♦Cls, R R must bring us back to the point from which we
started.
When R € 1 —* 1, each of the ovals R*x % R'y, R'z, ... in the above figure
4— v/
shrinks to a single point, so that R*x = i* R'x. Thus when R is given as a
I—>Cls, it will be a 1 —♦ 1 if R'y = R'z . - y = z. This proposition is
constantly used, and so is the consequence that R[ fi is a 1 —> 1 if
//, z eft. R‘y = R'z . D, /t .. y = z. (These propositions arc *71'54*55 below.)
The hypothesis R « 1 —♦Cls is equivalent to the hypothesis
xRz . yRz . D x . y .,. x = y
(ef. *7IT7, below), and the hypothesis eCls—► 1 is etpiivalent to
rRy .xRz y = z.
These arc for many purposes the most convenient hypotheses to use.
The most useful propositions in the present number arc the following.
(NVe omit here propositions concerning Cls—*1 or 1 —♦ 1 which are mere
analogues of pro|>ositions concerning 1 —♦ Cls.)
*7116. I- : R € 1 —♦ Cls . = . K !! R"U'R
This gives the connection of one-many relations with descriptive functions.
We have also
*71163. h s. R e 1 -* Cls. s : y «<!•/* . s,. E! R'y
For many of the constant relations defined from time to time, such as Cnv
or 1). the following proposition is useful:
*71166. h : (y) • E! R'y . D . /( e 1 -* Cls
*7117. I -R € 1 -* Cls. = : xRz . yRz . -x = y
This might have been taken as the definition of one-many relations, if we
had not wished to derive them from the more general notion of a—>@. In
proving that a relation is one-many, *7IT7 is more often employed than any
other proposition.
*7122. h : R e 1 -* Cls . 5 G R . D . S e 1 -♦ Cls
*71 25. I-. R, Se 1 -► Cls. D . R | Se 1 —♦ Cls
*7136. H i? € 1 —♦ Cls .D :x = R'y . = - xRy
*71 381. h : R e Cls -♦ 1 . D . R"(a - £) = R"a - R"0
(This proposition is more useful than the corresponding property of
I Cls.)
SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
429
*7155. h :: R e 1 —► Cls . D R f* /3el —* 1 . = : y, se/3 .R*y = R l z . D„ >; . y = z
This proposition is constantly used. For example, putting Cl tor R, it
gives
V :. Cl \ 0 € 1 1 . = : P, Q e 0 . <3‘P = CFQ . D,. Q . P = Q.
Most of the relations used to establish correlations in arithmetic are
obtained from a one-many relation, such as Cl, by imposing some limitation
on the converse domain which makes the relation one-one.
*71571. h y e /9. D* . E ! R‘y z = ,R[fiel—> Cls . /3 C G‘R
Here "y e /9 . D v . E ! R*y " is E !! R“/3, which has already played a large
part as a hypothesis, e.g. in *37 6 ft'.
*717. h Qe 1 —> Cls . D : xP Qz. = .xP(Q‘z )
Thus for example we shall have x(P Cnv) R . = . xP(Cnv‘R).
*7101.
*7102.
*71 03.
*7104.
*711.
i -»cis = it (*R"ci‘R c 1 ) [* 70 +]
Cls -> 1 - St (R"D‘R C 1) [*70 41]
1-»1 =R(R“Cl*RC 1 .R“D‘RC1) 1*20 2. (*70 01)]
1 _>1 = (1 —» Cls) r\ (Cls—* 1) [*70-42]
Re 1 Cls . = .R“<PR C 1 [*20-33 .*71 01]
Re Cls -> 1 . = . r““D‘R C 1 [*20 33 . *71 02]
R«i -* i .m . R“ci‘R c i. R“D‘Rc 1 1 * 20 - 33 . * 7103 ]
R e 1 1 . = . R e 1 -> Cls . R e Cls 1 [*22 33 . *71 04]
Re 1 Cls . = . R“V Clut'A [*70 44]
Re Cls -> 1 . ■ . *R “V Clvt'A [*70441]
Re 1 -► 1.9 ."R“V Clut'A . R“V Clv(‘A [*7012]
R e 1 —> Cls . = . (y) . R*y flwt'A [*7045]
R e Cls —>1. = . (x). R‘xe 1 w i ‘A [*70*451]
. R« 1 —*1 . =: (y) . “r* y e 1 u i‘A : (x).*R‘xe 1 ^ i‘A [*70 13]
. R e 1 —► Cls . = : (y) : R‘y e 1 . v . R‘y = A [*70 40]
R e Cls —► 1 . = : (x) : R*x e 1 . v = A [*70 461]
: R e 1 —> 1 . = :. (y) : R‘y e 1 . v . R*y = A :. (x) : R*xe 1 . v . R‘x= A
[*7014]
R e 1 —* Cls. = : g ! R‘y. D v . R‘y c 1 [*7047]
:. ReCls—» 1 . = : a ! r"‘x. D x . R*x e 1 [*70 471 ]
R e 1 —► Cls . = . R“CI‘R C 1
*71101. h : R e Cls 1 . = . R“D‘R C 1
*71102.
*71103.
*7111.
*71111.
*71112.
*7112.
*71121.
*71122.
*7113.
*71131.
*71132
*7114.
*71141.
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
430
*71142.
b : . Re\ -*1 . = : 3 !/?‘y.D ¥ .7?‘y€l : 3 ! iT‘.r. D z .*R*xe 1
[*7015]
*7115.
b : R € 1 -> CIs . = . I)‘7?C 1 u i*\
[*7048]
*71151.
b : 2?eCls-» 1 . = . D‘/? C 1 ^ i *A
[*70481]
*71152.
b : Re 1 -* 1 . = . D‘7?C 1 we«A.D f J7C 1 ui‘A
[*7016]
*7116.
b : 7f € 1 —> CIs. = . E!! R**d*R
Dem.
b . *37702 . *7 l-l . D
b Re 1 —♦ CIs. = :yeCl‘22 . D„. 7?‘y « 1 :
[*53-3] b : y e G‘/£ . D„. E! 7?‘y :
[*37 104] = : E !! R“(l‘R :. D b . Prop
'Phis proposition is very important ; it exhibits the connection of descriptive
functions with one-many relations.
*71161. b : It e CIs -> 1 . = . E !! 11**1)*R
*71162. I-: R 1 1 1 . = . E!! R"Q*R . E !! R**D*R
*71163. V:.R* 1 -> CIs. = : y c d*R . =„ . E! R'y
Dem.
I- . *33-43 . D b : E ! R*y . D . y e Cl *R :
[*473] D b :• y • D . E! R*y : = :y6CI‘/i. = .E! R*y
[*1011 *271 .*37 104] D bE !! /{“CP/l Cl-/? . = y . E! R'y (1)
b. (1). *7110. DK Prop
*71164. b R c CIs —» 1 . s :*e l)‘/i . =, . E ! R‘x
*71165. I -R € 1 -> 1 . = : y c d*R . =„ . E ! R'y : .r f D*R . . E ! R*x
*71166. b : (y). E ! R l y . D . 2? c 1 —► CIs
Dem.
b . *2 02 . *10 1 . Dh. Hp. D :y eCI‘72 . D . E ! 7*‘y
[*1011-21.*37104] D H : Hp . D • E!! R**d*R.
[*71*16] D . R e 1 —* CIs : D b . Prop
*71167. b : (or). E ! 7i*x.D.Re CIs —* 1
*71168. I- s. (y) . E! R‘y : (*). E ! R‘x : D . 72 c 1 —► 1
*7117. b 7? e 1 -> CIs. = : xRz . yRz . . * = y
This proposition is constantly used in the sequel.
Dem.
—> —>
b.*524. D b R*ze 1 ^ c‘A . = : x,y e R*z ,3 XtV . x = y:
[*32-18] = = *R* • 'JR* • =>x. y • * = y :•
[*1011’271.*11’21] Dh. ( 2 ). R l z flv i*A . = : xRz . yifc .D XtVil .x = y (1)
b.(l). *7112. D b . Prop
SECTION Cj ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS 431
*71171. :. R e Cls —> 1 . = : xRy . xRz . D x . , />z . y = z
*71172. Y :. Re 1 —► 1 . = zxR z .yRz . ~5 x . y , e » x = y : xRy .xRz . D Xi „ iZ . y = z
*7118. Y R e 1 —> Cls . = : g ! R*x r\ R*y . D X J/ .x — y
Deni.
h. *32181 .*22-33. D
Y :. g! R‘ x ** R‘y • • x = y • = ' ( 3 *) • zcRz • yRz . D x>y . X = y:
[*10 23] = : xRz . yRz . 3 x , y> ,. x — y :
[*71-17] = : R e 1 —> Cls z.OY. Prop
*71181. Y :. R e Cls -* 1 . = : 3 ! R‘y r\ R*z . Z> y> , . y = z
*71182. H :: R e 1—>1 . = g ! ‘a- r* R*y . v . g ! R‘.c n R*y : D x y . x = y
*7119. 1- : 7* 6 1 -* Cls . s . i* | R « / f D *R
Dem.
Y . *341 • *31-11. ^Y . x(R\R)y .s ,(^z) .xRz .yRz ( 1 )
Y . *501. *35101 .OY ,x(T[ \}*R)y . = .x=*y.ye D*R (2)
l-.(l).(2).*21-43.D
H :: 7i | R - / f* D‘/2 . 3 :. (g*) . xRz . . s zx = y .ye D‘R :
[*33*13.*10*35] s x>y : ( 3 *) • x = y . yRz z
[*13104] s,: ( 3 *) .x~y. xRz . yRz :
[*10 35] =x.y : * — y : ( 3 *) . xRz . yRz z.
[*4 71] = :.(g*).* Rz.yRz . D XiV .x = yz.
[*10*23] = xRz . yRz . D x>y#x . x *» y z.
[*7117] = R € 1 -* Cls :: D Y . Prop
*71191. H:/e«Cls->l. = .«l^ = /ra < 72
* 71192 . y 1 Rci-ti. = .ftj« = /f*D‘/e./e|/e = /fa <R
*712. H . Cls —► 1 = Cnv“(l -► Cls) .
1 -» Cls = Cnv‘*(Cls—* 1). 1 -> 1 = Cnv“(l 1) [*70 22]
*7121. 1-: i2 c 1 —♦ Cls . = . Re Cls —* 1
Dem.
1-. *37-62 . *31 13 . D Y z R e 1 -♦ Cls . D . Cnv'i* e Cnv“(l -» Cls) .
[*3112.*7l-2] D . /£cCls —► 1 (1)
Y . *37-62 . *3113 .DYzRe Cls -> 1 . D . Cnv‘i? e Cnv“(Cls 1) .
[*31-33.*7l-2] D./*«l-»Cls (2)
h.(l).(2).DH.Prop
432
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*71'211. f- : i? e Cls 1 . = . i? e 1 -» Cls
*71*212. Hi?el-*l. = .i?el-»l
*71-22. b : i? e 1 -> Cls. S G li . 3 . S e 1 -* Cls
Dem.
H*231 .3
h.SC/i.D: xSx . ySz . 3 X . v . r . xRz . yRz (1)
H *7117.3
I -R e 1 —* Cls . 3 : xRz . yRz . 3 ZtWt . • y (2)
h.(l).(2).*U-37.3
h:. Hp.D: a:Sr. yS*.3 Zt •#=•/:
[*7117] 3 : Se 1 —► Cls Dh. Prop
*71-221. H i? e Cls —► 1 . S G i? . 3 . S e Cls —► 1
*71-222. I-! R e 1 -* 1 .SG /?. 3. Sc 1 -► 1
*71223. h : /e c 1 —> Cls . 3 . Rl‘i? C 1 -* Cls [*71-22 .*61*2]
*71-224. h : A < Cls -» 1 • 3 • HI 4 li C Cls -> 1
*71-225. b : R e 1 -» 1 . 3 . Rl‘i? Cl->1
•71-23. H J? e 1 —► Cls .D.WAStl-» Cls [*71-22 . *23 43]
*71-231. Hi?e Cls->1.3.i?AScCls-> 1
*71-232. b : i? e 1 -> 1.3 . R n Sc 1 -> 1
*71*233. h : i?. .S’ e 1 -* Cls . 3 . A n S e 1 -> 1
Dem.
H *71 23.3 H Hp . 3 . i? A S e 1 -* Cls (1)
K *71*21 . 3HHp.3. Setts-*1 .
[•71-231] 3 . i? n S e Cls —* 1 (2)
H (1).(2). *71103.3 H Prop
*71234. Hi?,SeCls-*1.3.i?ASel->l
*71236. H i? e 1 —► Cls . S e Cls —>1 .3.i?ASel—*1
*7124. H i?, S c 1 —» Cls . G‘i? « G‘S*= A . 3 . i? c* Sc 1 —» Cls [*7054]
*71*241. I- s R, S e Cls -* 1. D‘i? n D‘S = A.3.i?c/Se Cls-*1 [*7055]
*71 242. b : R, S e 1 -* 1 . D‘i? n D‘S = A. d‘R n (I‘S = A . 3 . R o Se 1 -> 1
[*70-56]
*71243. Hi?, Sel—>1 . C*R r\ C*S = A.3.i?oSel—»1 [*7057]
*71-244. H i?, S e 1 -* Cls . i? [* CPS G S. 3 . i? c; S e 1 -* Cls
Dem.
V . *23-34 . *4-4.3
t-:.x(RwS)z.y(RwS)z. = :xRz.yRz.v.xRz.ySz.v.xSz»yRz.v .xSz.ySz (1)
SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
433
I - . *71 "17 . D h R, S e 1 — >CIs . D : xllz . yRz . D . x = y : xSz . ySz . D. a:= y (2)
. *3314 . *4 7 . D h : a:/?s . ySj . D . xllz . ySfc . 2 e Cl ‘S .
[*35101] D.x(Rt<I‘S)z.ySz (3)
H - (3) . D h i*r CI‘S GS.D: . ySz . D . xSz . (4)
D h Rfa'SGS. 3 l.a-Si.ySt (.’,)
h . (2) . (4) . (5) . D h Hp . D : ar/Ss . ySs . D . x — y : aS's . . D . .t = y (6)
I" - (1) • (2) - (6). *4-77 . D H Hp . D : .r (R o S) z . y (R v S)z . D.x~y (7)
K (7). *1011-21 . *7117 . D h . Prop
*71 245. h : 72, & e CIs—► 1 . (D ‘S) R G S. O . R w S e CIs —► 1
*7125. hs/e.Scl-^Cls.D. «|5el->Cls
Deni.
h - *71-17 . Dh:. Hp . D : yjS». *S-e. D . y ™ x :
[Fact] D : a/ty . ySa;. . J&x .O.y = z.n Ry . vRz .
[* 13-13] D.uRy.vRy.
[*71-17] D.a=v (1)
h .(1). *1111-3-54. D
h :: Hp . D (gy) . a72y . ySx : (g*) . vRz . zSx O.h-ii:.
[*341] D :.u v(.R 5 )x.D.m-u (2)
h. (2). *7117. DH. Prop
*71 251. h : ii, S c CIs -> 1 . D . R | S c CIs -* 1
*71 252. b:R t S*l-+l .O.R\Sel-+l
*71 "25 may also be deduced from *70 6, as follows:
Alternative Deni, of *71-25.
h . *53 301 . *71 12 . D b s * c 1 -♦ CIs . D . R“i*x e lut‘A:
[*521] D 1-: lit 1 -*Cls.a e 1 . D./*“«*« 1 w e‘A :
[*37-6111103] DhiAcl-4 CIs . D . R*“ 1 Clu«‘A (1)
h . (1) . *70 6 . D H . Prop
Similarly *71251 may be deduced from *70 61.
*71 26. h : R e 1 -* CIs. D . R f* 7 * 1 CIs [*70 62]
*71 261. I-: R e CIs -> 1 . D . £ ] R c CIs —► 1 [*70 63]
*7127. h : R e 1 -► CIs . D . £ ] « c 1 -» CIs [*35 44 . *71 22]
*71-271. b:«€Cls-^l . D . « p 7 « CIs-* 1
*71 28. b:J2el->Cls.D./91/J[* 7 «l-»Cls [*35442 . *71 22]
*71-281. b:ie€Cl8-*l.D./9'|/e|‘ 7 cCls-»l
*7129. H:/26l->l.D.^'|i2,Rf‘ 7 , i S']22f 7 el-- > i
*71 31. \-zRel-+C\s.yc d‘R . D . ( R‘y ) i*y [*3032 . *71’163]
R 4c W I
28
434
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*71*311. bzRe Cls-*1 .x € D‘R .0 .xR(frx)
*71312. bzRe l-»l.xc D‘7* . y € d‘7* . D . xR(frx). (fry) Ry
*71 32. h :: /dc 1 —^Cls.yeCl*/?. 0:.\lr(R‘y).= :(’g L x).xRy.yfrx: = :xRy.D z .ylrx
[*30*33. *71*163]
*71321. I-:: R e CIs —> 1. x € D*R . D:. yj/ (72‘x). =: (gy) . xRy . yfry : =: xRy . D v . yfry
*71 33. h ::/?cl —*Cls. D :.\fr(R‘y): = z(^x).xRy. yjrxz = :yeQ‘7f zxRy.^ z .^x
Deni.
h . *71*32 . *5*32 . D
h :: Hp. D y € (1*7? . \fr (R l y ). = : y « G‘7£ : (g.r). .r7£y . yfrx z
= : y e C\ f R z xRy . D r . ^r.r (1)
h . *14 21 . D I- : >/r (7*‘y). D . E ! 7?‘y .
[*33*43] D.ycWRz
[*4 71] Dh.ye CI‘7? . (7?‘y). = . ^ (fry) (2)
h . *10*5 . Df*: (g.r). .vRy . >/r.r. D . (^ar) . a:7(y .
[*33*131] D.ycd‘7*:
[*4*71] DK y c(l*R z(T[x) .xRy .yjrxz = . (gar).a:7fy . yjrx (3)
h . (1) . (2) . (3) .DK Prop
*71*331. CIs —► 1 . D yfr(R‘x ). = : (gy) . xRy . yfry z = :
a: c D‘72 : a*7£y . . \fry
*71 332. h :• A e 1 —» CIs . D : R*y c a . = . g ! 7*‘y a a . = . y c G‘7? . R*y C a
[—w]
v 4— 4—
*71333. h:./?€ CIs —» 1 • D : 7?‘a: e a . = . g ! 72‘a: na.s.ifD'K. 7?‘a: C a
*71*34. I-: R e 1 -> CIs. R - S. y e G‘7? . 3.72‘y = fry [*30*36 . *71*163]
*71341. h: R* CIs-* 1 . R = S .x€D‘R .0 . R‘x=>S‘x
*7136. zz R el->C\s.O z.ycd'RyjCl'S.Oy. fry = fry i = .R=S
Dem.
H . *21*18 . Dh:. R = S.Ozyc d‘R w G‘S. = . y € G‘7* v G‘7? .
[*22*56] ' =.yed‘R (1)
. (1). *71 34 . D h z: Hp . R = S . D : y € d‘R w G'S. . fry = S‘y (2)
I-. (2) . *33 45 .DK Prop
*71 361. I-:: R € CIs -» 1 . D a? € D‘7? v D‘S . D,. .K'a: = S‘xz = .R = S
*71*352. l-::/2el-*l . D y €G‘« u G'S. D v . 7£‘y = £‘y : = : 7* = £: .
= zx€D*RyjI) ( S.D x .frx = frx
SECTION C]
ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
435
*7136.
1- R e 1 —> Cls . D : sc = 72*y . = . xRy
Dem.
h. *30-4. *71 163. D
h Hp . y e G‘72 . D : x = R‘y . = . xRy
V . *71-163 . Transp . D
(1)
h Hp . y ~ «? (3*72 . D . E ! 72‘y .
[*14 21.Transp] D . ^ (a: = 72‘y)
(2)
h . *33*14 . Transp . D h : y ^ e Cl*72 . D . ~ (.r72y)
h. (2). (3). *5-21 . D
(3)
K Hp . y ~ e G*72. D : ar =» 72*y. = . .r72y
h.(l).(4).*4 83.DH. Prop
(4)
*71361.
h 72 6 Cls —> 1 . D : y = 72*a;. = . xRy
*71362.
h :. 72 € 1 —> 1 . D : a: = 72*y. = . xRy . = . y = 72*x
*7137.
H 72 e 1 —► Cls . D : y e 72“a. h . 72*y « a
Dem.
1-. *71-33 . D h Hp . D : 72*y c a . s . (go:) . xRy . are a .
[*37 105] a . y * 72“* Dh. Prop
*71371.
Re Cls —> 1 . D : xe 72“a . s . R'xea
*7138.
h : 72 <? 1 —► Cls . D . 72**(« - /3) = £**a - 72“/3
Dem.
h . *71-37 . D h Hp . D : y e 72"(a - 0 ) . 3 . 72*y < a - /3 .
[*22*32.*14 21] 3 . R*y € a . ~(R‘y € y3).
[*7137]
= .y«72“a.~(y«72“£).
[*22-32]
■ . y € 72“a - 72“>9 :.Dh. Prop
*71381.
H : 72 c Cls —♦ 1 . D . 72*‘(a - £) = 72**a - 72**/3
*714.
H : 72 « 1 —¥ DU. .. , a ~ r-.oT.i
1 .ool
*71401.
I-: R < Cls -» 1.3 . H “8 = p [(ga:). * . 0 . y = «<*|
*7141.
h : 72 e 1 -» Cls . D . D‘72 = 5b {(gy) . a: - 72*y} [*3311 . *71 36]
*71411.
h : 72 e Cls —► 1 . D . 0*72 - £ {(gx) . y = 72**}
*7142.
H :: 72 e 1 —» Cls . £ C (3*72 . D :. 72“/9 Ca.ssye/9.3.,. R*y e a
[*37-61 .*71-16]
*71421.
1-:: 72 e Cls —♦ 1 . a C D‘72 . D R“aC/3. = : x e a .D x . Ii‘x c/3
*7143.
1- : 72 e 1 —» Cls .yean G*72 . D. 72*ye72**a [*37 62 . *71 16]
*71431.
U : 72 e Cls —* 1 . x e a n D*72 . D . R‘x c R**a
28—2
436
PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
*71*44. h :: Re 1 —» Cls . a C G‘7? . D z.xe R“a . D x . yfrx : = : y ea . 3 tf .^r(J2‘y)
(*37*63. *71 16]
*71441. h:: 7feCls->l .oCD‘/^. D y e R“a.D„ . yfry : = : x e a . D z . yjr (R‘x)
*71*45. I- R e 1 —* Cls . D : (gar). x c R*‘a . \frx . = . (gy). y € a . (R‘y)
IJeni.
h. *37*64. *71*16. D
h Hp . D : (gar).arc R‘*(a n Q*R ) . >/r.r. = . (gy) .yean (I‘/f . >jr(R*y) (1)
h. *37*26. Dh.7*“(an<I‘7?)=7(“a (2)
h . *1+ 21 . D H : y e a . >/r (7f‘y). D . E ! 7£‘y .
[*33*43] D.ycG'Tf:
[*4*71 .*22 33] D h : y c a . ^ (7f‘y). = . y c a a d'R . ^ (7*‘y):
[*10 11*281] D H : (gy) .yea. ^(Wy ). s . (gy) .yean d‘/e . yfr (R‘y) (3)
h . (I ).(2).(3). D h. Prop
*71 451. h R c Cls —* 1 . I> : (gy). y c 7?“a . yfry . = . (gar). ar ta.f (7f‘x)
*71*46. I- : 7f c 1 —> Cls . a C 7?“£ . D . a = R“(R“a n £)
Deni.
h . *37*26 . D h : 7f“/9 = . «“(^‘«aj 8) = R“(R“aK/3rsCl‘R) (1)
I-. *37*65. *71*16 . D
I-: /( c 1 -* Cls . a C 7f“(/9 n G‘7f). D . a - R“(R“a n £ a G‘7f) (2)
I- . (1). (2). D h . Prop
*71*461. H : 7? € Cls -> 1 .0 C 7f“a . D . £ = R“(R“0 * a)
*71*47. H s. /if e 1 —* Cls . D : a C 7f“/d . = . (g7). 7 C /? . a = 7f“7
Dem.
V .*71*46. *10-24. *22*43. Df-:. Hp.D :aC^.D.(a 7 )-7 C0.a-&‘y(l)
h . *37*2 . *10 11*23 . Dh: (g 7 ). 7 C £ . a = #‘7 • ^ • a C 7e“£ (2)
h . (1). (2). D I-. Prop
*71*471. I-:. R « Cls -» 1 . D : /3 C R‘ “a. = . (g 7 ) .7 C a . £ = £“7
*71*48. H : 7f c 1 —* Cls . D . D‘7?« = CI'D'A
Dem.
I-. *37-24. *60-2. Dh.D‘i?,CCl*D‘fl (1)
h . *37-25 . *71-47 . *60-2 .Dh: Hp. a« Cl‘D‘i«. D . (37) • 7 C <1*8. a = #“7 ■
[*10-5.*37-23] D.oeD'iJ,:
[Exp.*1011-21] 0 h: Hp. D. Cl‘D‘8 C D‘R. (2)
K(l).(2).3h. Prop
*71-481. h: R € Cls -» 1. D . D‘(.R), = Cl‘d'8
The following proposition is used in the theory of derivatives of a senes
(*216-411).
SECTION C] ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
437
*7149. b : 72c 1—►Cls.aCCl‘72. D. R tit CVa=0\ i R li a. R itt C\ ex‘a=Cl ex i R*‘a
Dem.
b . *71-47 . *00 2 . D h Hp . D : 7 € C\‘R“a . = . (g/3). 0 C a . y = R“/3 .
[*37 103] =.y€R‘“ Cl‘a (1)
h . *37 43 . D I- Hp . £ c Cl ‘a . D : g ! £ . = . g ! 72“/S (2)
h . (1) . (2) . D h . Prop
*71491. h:72cCls—*1 .aCD*72. D.72“ < CI < a = Cl < 72 << a.72 < “Cl ex‘«*CI ex‘R“a
This proposition is used in the theory of derivatives of a series (*216 4)
and in the theory of ordinal numbers (*251*11).
*71*5. b R c 1 —» CIs . D : xRy . = . x — 1‘72‘y
Dem.
I-. *71-36 . *301 . D I- Hp . D : xRy . = . x = (jx) ( xRy) .
[*51*56.*32*13] a .« -c‘7z‘y D h . Prop
*71 501. h 72 e CIs —> 1 . D : a:72y . = . y = i , R t x
*7151. h : 72 « 1 —► CIs . y e (3*72 . D . R‘y = 7‘72‘y
Dem.
h . *53 31 . *71*163 . D b s Hp. D . i‘/2*y-fl*y .
[*51*51] D . 72‘y -7‘7?y :Db. Prop
*71-511. h : 72 < CIs 1 . x e D*72 . D . R‘x = 7*32**
*71-52. h : R e 1 -* CIs . D . R“a-'i“li“a
Dem.
-SKauS).
- £ {(3^) . (3 c~R“a.x = 7*0}
= £ «a/3. y) . y c a . 0 = R‘y . * = i*0j
H . *371 . DH. t“72“a
[*51-51]
[*37-7]
[*1123.*13*195] = £ l(3y)-y «a .a: — i*Ji‘y\
I-. (1) . *71*5 . D b : Hp . D . {(gy) -yea- *72y}
[*371] = 72“a : D h . Prop
*71 621. H : 72 € CIs -» 1 . D . 72“a
*71 63. h : 72 c 1 —► CIs . 72‘x = 72'‘y . D . * = y
Dem.
b . *14 21 . D h : Hp . D . E ! 72‘*. E ! 72'y .
[*30-32] D . *72(.ft'*) . y R (ft'y) .
[*1416] D . *72 (ft'y) . yR (ft'y) .
[*71-17] D ,* = y : D H . Prop
( 1 )
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
438
*71 531. h : 77 c CIs 1 . 77‘y = 77‘s . D . y = 2
*71 532. I -R c 1 -» 1 . D : 77‘y = R'z . D .y = * : 1Vx-R'y . D .x = y
*71 54. K :: 77 € 1 —♦ CIs . D 77 c 1 —> l . a : 77‘y = 77*2 . D (/iJ . y = z
This proposition and the next (*71*55) are very often used.
Deni.
f- . *71*30 • D h :• Hp • D : (fl.r). xRy . xRz . = y> .. (^.r) . x = 77‘y . # = 77*2.
|*U*205] = y>r . 77‘y = 77*2 (1)
h . (1). D h :: H p . D 77‘y = R l z . D l# .,. y - 2 : = : (g.r). *7?y. *7 ?j . . y = z :
[*10*23] a: xRy . xRz . D x#r/> , • // = *:
[*71171] a: 77 c CIs —> 1 (2)
h . *71*103 . *4 73. D h Hp . D : 77 c CIs—> 1 . a . 77 c 1 —* 1 (3)
h .(2).(3). D h . Prop
*71 55. h :: 77 c 1 —»Cls. D 77[* &c 1 —> 1 . a : y, * c/9.77‘y = 77*2 . D y , z . y «= 2
Dem.
I-. *71 *20. D I*:: Hp. D :. 77 f /9 c 1 —> CIs
[*71*54] D:.7ir/9€l^l.-s(7?rWy-(«r« i *.^..Jf-*:
[*35*7] a :y.xe/9.7?‘y = 77*2 . D,,., .y = :::DK Prop
*71 56. h :. 77 c 1 —> 1 . y c 0*77 . D : 77‘y = 77*2 . s . y - x
Dem.
K*71*532. D h : Hp. 77‘y —77‘x. D .y — 2 (1)
h . *71*105 . *30*37 .Dh Hp. y = 2. D . 77‘y = 77‘x (2)
H . (1) . (2). D h . Prop
*71 561. h :• 7?« 1 —* 1 .xc D‘77 . D : 77*.r = 77‘y. a . * = y
*71 67. h 77‘y - 77*2 . a y .,. y = 2: a : 7? € 1 -» 1 : (y). E ! 77‘y
Dem.
K*10*l . D h:. 77‘y = 77*2 . a y>r . y = z zD : R*y — R*y . = y . y = y:
[*13*15] i-.W-R'y^R'ir-
[*14*28] D:(y).E!7?‘y (1)
[*71*166] D: 77 cl-* CIs (2)
H.(2).Dh:.Hp(2).D:77cl->Cls:77 < y=77'x.D J/ir .y = x:
[*71*54] D:77cl-*1 (3)
K (1). (3). *71*56. DK Prop
*71 571. h :. y c /9. . E ! 77‘y: a . 77 \ /9 c 1 -* CIs. £ C 0*77
Dem.
K *71*16. Dh:.77r/3el->Cls.= :yca‘(77ry9).^.E!(77r/9)‘y:
[*35-64 7] a : y e /9 n 0*77 . D y . y c ^ . E ! 77‘y :
[*22-33.*5-3] a : y c >9 « 0*77. D y . E ! 77‘y (1)
SECTION C]
ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS
439
*- .(1). *22-621 . D
h f £ e 1 —* Cls . 0 C <1*11 . = z y e 0 n d l R . D f/ . E ! ll‘i/ z 0 c\Q‘R = 0 :
[*13193] = : y c 0 . - E ! R'y : 0 * (I‘7? = 0 (2)
h . *33-43 .Dh.yf^.D v .E! 7*‘y : D . /3 C d‘R .
[*22-621] D. £*(1*72-1 (3)
K (2). (3). *471 .DK Prop
*71572. H i.ye&rs (1*72 . . E ! 72‘y : = . 72 f* /3 * 1 ->Cls
[*71-571 .*35351 .*2243]
*71-58. zz y, z c £ . D, JtZ : 72*y = 72‘* . = . y = 2 D . R f* £ € 1 1 .0 C d‘R
Dem.
K . *101 . D H :: Hp . D y c 0 . D y : 72‘y = -72‘y . » . y — y :
[*1315.*14-28] D y : E ! R*y :.
[*71-571] D:.72f-/9el—> Cls . 0 C Q*7? ( 1 )
t- . *3-26 . Imp . *1111-32 . D
h Hp . D : y, * e £. 72‘y = 72*$ . D (/i ,. y = s :
[*35 7] D : (ii r £)‘y - <72 [ £)*«. D.,. r . y = * :
[*71-54.(1)] D:i*r/3€l->l (2)
K(1).(2).DK Prop
*71 59. h :: y, 2 e 0 . D,,., : 72*y - 72*«. - . y - *s.72|*£«l->l.£C C1‘R
Dem.
K . *71-56 . D H ::72r£el^l.D:.y C a‘(72r£)0:(72r£)‘y«<72r£)‘*.s.y=s:.
[*35-64-7] Ds .y«£*a‘i2.D:y > *c£.72*y—72*«. = .y—* (1)
h.(l). *22621 /2|*£«1->1
[*4-73] D
h. (2). *1111-3. Dh::J*r/9cl
h. (3). *71-58. DK Prop
y c 0 . D z y, z e 0 . R*y = 72‘* . = . y — e :.
y,z c 0 . D z R*y = 72** . ■ . y — *
->1 .£C( 1*72.D:.
y, * c £ . D Vtl : R‘y = 72‘* . = . y = *
( 2 )
(3)
The following proposition is used in the theory of selections (*80-91).
*71 6. I- : 72 e 1 —* Cls . 3 • 72 — «*/* |(gy) . y e CI‘72 . P = (72*y) J, y)
Dem.
h. *41-11 .*13195. D
|(ay).y«a‘fl.p = (*«y)iy)] i . = .
(ay) • y * H‘72 . x {(/ 2 ‘y) i y} *.
[*5513] = • (ay) • y € d‘/2 . a: = 72‘y . * = y .
[*13195] = .ze d‘72 .x=R t z
h . *71 -36 . *33-43 . D h Hp . D : * e d‘72 .x = R‘z. = .
H.(l).(2).Dh.Prop
xRz
( 1 )
(2)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*71 61. h : T* 1 —> CIs. D . a a) = (2“T“a
Fent.
b . *37 103 07 111 .*32 12. D
I-: /3f a fl ). = . (gjr).x f QTn a . 0 = Q'T.r (1)
b . *53-31 . *7116 . D h ; Hp .xe(l‘Tf% a . D . Q‘7*r (2)
h . (1). (2). D h Hp. D : 0 c Q‘“r‘((I‘T«a) . = . (•ax).x € (I‘Tn a .f3=Q t T‘.v.
f *37 07.*71 16] = . 0 €Q“T“(a<T« a) .
[*37-26] s . /9 <;-Q“T“a D H . Prop
*71-611. h : 7*« CIs —► 1 . D. Q“ f f“(D‘T«*)- m Q“T“*
*71612. f-:7M -* CIs . D . r% a) = 4 Q“T“a
*71 613. h : Te CIs -> 1 . D . Q"‘^“(D'r a a) - Q' tt T lt a
*71’613 is used in the theory of series (*206*6), and in the theory of
‘•similarity of position ” (*272 131).
*717. I -Q € 1 -> CIs. D: xP \ Qt . = . xP ( Q‘z)
Dem.
H . *71 36 . D b Hp . Z> : yQg . = . y « Q'z :
[Fact] D : xPy . yQj . = . xPy .y = Q‘z:
[* 10 - 281 ] 3 : ( 3J/ ). xPy.yQz. = . (a y)-*Py-!/-Q‘‘-
[*341.*13195] D : */' | Qj . = . xP(Q'z) :.Dh. Prop
*71701. 1 -Q c CIs 1 . D : xQ Pi . = . (Q‘x) Pz
*72. MISCELLANEOUS PROPOSITIONS CONCERNING
ONE-MANY, MANY-ONE. AND ONE-ONE RELATIONS
Summary of *72.
In this number we shall prove various propositions involving 1 —►Cls,
Cls—► 1, or 1 —> 1, but not embodying fundamental properties of these classes
of relations.
The present number begins with various propositions (*721—191) show¬
ing that various special relations are one-many or one-one. The most useful
of these are
*72182. I- lye 1 -> 1
*72184. h.xl,
We have next a set of propositions concerning lt‘S ‘2 when It and S are
one-many, or R t R i z when 11 is one-one, and kindred matters. The most
useful of these is
*72 241. h « e 1 -> 1 . D : y c <3‘/e . = . y = R‘R‘y
We have next a set of propositions (*72 3—'341) concerning products and
sums of classes of relations; of these the one most used is
*72 32. h \ C 1 -> Cls s P, Q e X.. a ! il € R a d‘Q . D/.. Q . Q : D .i‘\< 1 -*Cls
which is an extension of *71 24.
We have next a set of propositions (*72*4—481) giving various relations
of R**a and when Rt 1-> Cls, or of It“a and lV'fr when .ftcCls->l.
The more useful propositions of this set are those that have the hypothesis
ReC\s —► 1 ; these are occasionally useful in arithmetic. We have
*72 401. V : R € Cls 1 . D . R“ a * = R*\ a ^ (3)
*72 411. H : li € Cls —> 1 . a n /3 = A . D . R“a n R“(3 = A
For example, the relation of son to father is many-one. Let a = Cabinet
Ministers, & = fools; then assuming a r\ R = A, it will follow that the sons of
Cabinet Ministers and the sons of (male) fools have no common member.
If we make R the relation of son to parent (which is not many-one), it no
longer follows that the sons of Cabinet Ministers and the sons of fools have
no common member.
We have
*72 451. h : R e Cls -► 1 . D . R t f* CPC VR e 1 -► 1
The effect of this proposition is that if a and /3 are both contained in
G'/i, and R“a = R“f3, then a = & (using Ri*a = R“a ).
442 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
We next have a set of propositions concerned with the relations of
lit and (/£)*, or, what comes to the same thing, with the circumstances under
which a = . = . & = R“a and under which R tl R il z — a. We have
*72 502. I-: R e 1 CIs. a C D *R . D . R“R“a = a
Thus for example the fathers of the children of wise fathers are the class
of wise fathers; but the fathers of the children of wise parents are not all
wise, and the parents of the children of wise parents are not all wise—the
first because "oC D t R’’ fails, the second because “Re 1 —>Cls" fails.
We have also
*72 52. h J? « 1 -» 1 . a C . £ C <I‘/*. D : a - R “&. = . £ - Ii u a
We have next a set of propositions (*7259—06) in which the relative
product R R occurs if R e 1 —»Cls, or R R if Re CIs—* 1. The most useful
propositions in this set arc*
*72591. R-SfWR
*72601. hs/ttCIs-tl.U'iSCCI'A.D.S,/! Ii = S
*72 66 . I-: S* C .S'. - S . = . (3 R ). R e CIs -* 1 . S - R ; li
This is the "principle of abstraction.” It shows that every relation which
has the formal properties of equality, i.e. which is transitive and symmetrical,
is equal to the relative product of a many-one relation into its converse; i.e.
whenever the relation S holds between .r and y, there is a term a such that
xRa.yRz, where It is a many-one relation; and *72*64 shows that this term
a may be taken to be S 1 *, which is equal to S‘y. This principle embodies
a great part of the reasons for our definitions of the various kinds of numbers;
in seeking these definitions, we always have, to begin with, some transitive
symmetrical relation which we regard ns sameness of number; thus by *72 64,
the desired properties of the numbers of the kind in question are secured by
taking the number of an object to be the class of objects to which the said
object has the transitive symmetrical relation in question. It is in this way
that we are led to define cardinal numbers as classes of classes, and ordinal
numbers as classes of relations.
The remaining propositions of this number are of less importance, with
the exception of
*72 92. b-.Rcl-tCls.SGR.O.S-Rta'S
This proposition shows that every relation contained in a one-many
relation is obtainable by a limitation of the converse domain. Thus e.g. every
relation contained in that of father to son can be specified by specifying the
class of sons who are to be its converse domain; for then all the fathers of
these sons must be included to provide referents. But if we take the relation
SECTION C]
MISCELLANEOUS PROPOSITIONS
143
of parent and child, which is not one-many or many-one, a contained relation
is not determinate even when both its domain and its converse domain are
given; for the relation may relate some of the children in any one family' (o
the father and some to the mother, and so long as all the children and both
parents are each related to some one by the relation, the domain and converse
domain remain unchanged by permutations within the family.
*721.
Dem.
*7211. h
Dem.
K Acl
h. *25105. D I-. ~{xKz . yAz) •
[*2'21] D 1-: xAe . yKz . D . x
j>llll.*7117J D V . A e 1 —► CIs
Similarly h. Ac CIs —>1
h • (1). (2). *71*103 . D H . Prop
Cnvc1 —► 1
J :
( 1 )
(2)
CIs
d)
h . *3113 .*71166. D h . Cnv c 1
H . (1) . *71-54 . *31-3212 . D h . Prop
*7212. b.JHjie l -> CIs [*3212121 .*71 106]
*72121. h . sg, gs c 1 —> 1
Dem.
1-.*32-22-221 .*71 106. D 1 -. sg, gs e 1 -♦ CIs (1)
h . (1) . *3214-15-21-211 . *71-54 . D h . Prop
*72 13. h . D c 1 —> CIs [*3312 . *71 166]
*72131. h . a c 1 —* CIs [*33 121 .*71 160]
*72 132. h : C c 1 —► CIs [*33 122 . *71 166]
*7214. h.x?,?a:«l-» CIs [*3812 . *71 • 166]
This proposition applies to a great many of the relations we have to deal
with, for example ‘\P t P\, Pt, P\ 9 \P t x l, l x, etc.
*7215. \-.I\el -*Cls [*37 111 .*71 166]
In *72-16 below, p has the meaning defined in *4001, and does not
represent a variable proposition. Similarly s in *72161 has the meaning
defined in *40 02.
*7216. h./>el->Cls
Dem.
V . *20-2 . (*40 01) . D h . P ‘k = £<a c * . D. . *ca) .
[*14-21] Dh.E lp*K (1)
h.(l). *71166. Dh. Prop
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
#72161. h . s e 1 —» CIs [Proof as in *7216]
*72162. H./>€ 1 —>Cls [Proof as in *72*16]
*72163. I" ..vc 1 —»Cls [Proof as in *72*16]
*7217. h . / c 1 -» 1
h. *52*22. <*51 01). Dh.(*)./‘a?el.
[*7112] DK/e l-*Cls (1)
h . (1). *71*21 . *50*2. D h • /c CIs—» 1 (2)
b . (1) .(2). D b . Prop
*7218. h . # € 1 —* 1 [*51 23. *71*57]
*72181. I-.7«1-+1 (*72*18. *71*212]
*72182. h.*Xy€l-»l
J)eni.
b. *55*13. D h : z (x ly)w.= ,z = x.w*=yi (1)
(#3*47 ] z z (r ly)w.z (x l y)w . "5 ,z = x.z' = x.
[*13*172] D.z = z' (2)
K . (1). *3 47 . D h : z (a* | y) v >. r (x y) w . 0 . w - y. iu = y.
[*13*172] D.w = w/ / (»)
h .(2). (3). *71*172 .DK Prop
*72184. b.xl, lxe\ -*1 [*55*2. *71*57]
*72 185. H . ( i x)< e 1 -> 1 [*55*262 . *37*11 . *72*15 . *71*54]
*72*19. b . Cl c 1 —► 1 [*60*55 . *71*57]
*72 191. h . Rl c 1 -» 1 [*61*55 . *71*57]
*72*192. b . Cl ext 1 -> 1 [*60 56 . *71*57]
*72193. H . Rl ox € 1 —► 1 [*61*56 . *71*57]
*72 2. b R, S € 1 -> CIs. D : * = R‘S‘z .s .x(R\S)z. = .x-(R\S)*z
Dem.
b . *71*36 . D b :. Hp. 0:x- R‘S‘z . = . xR(S‘z ).
[♦71*7] (1)
f- . *71*36*25 . D b :. Hp . 0 : | S) z. s . x = (/* | S)‘z (2)
1-. (1). (2) .DK Prop
*72 201. b R, Se CIs-> 1 .0 : * = S‘R‘x. = .x(R\S)z . = . z = (S\R)*x
*72*202. bz. R, Sel-+1.0:*= R‘S‘z . = .x (R\S)z . = .z=S‘R‘x [*72*2*201]
*72 21. b R, S e 1 -* CIs . D : * e S“(l‘R . = . E ! R<S‘z . = . E! (i? | £)'*
Dem.
b . *71*25*163 . D H :. Hp. D : ^ € CP(R | S) . = . E ! (R | S)*z (1)
h . (1) • *37*32 • D h :• Hp. D : z € . = - E! (1£ | S)‘z (2)
115
( 1 )
( 2 )
SECTION C] MISCELLANEOUS PROPOSITIONS
K .*72-2.*10 ir2r281 . D
h Hp . D : (ga:) . a: = R‘S‘z . = . (g.c) . .r = (R | :
[*14 204] D : E ! R<S*z . = . E ! (R \ S)‘z (3)
h .(2).(3). D K . Prop
*72 211. h :. R, S e Cls 1 . D : x € R“T>‘S . = . E ! S‘it*x . = . E ! (S j R)‘x
*72 22. h : R, Se 1 -> Cls. * « S“d‘R . D . /e‘S‘** = (7? S)*z
Dem.
. *72-21 .Dh: Hp . D . E ! R*S‘z .
[*34-41] D . = (R 1 £)<, : D h . Prop
*72-221. h : J?, S e Cls -> 1 . a: e . D . S‘R‘x = (51 ft)'*
*72 23. H : R,Se 1 —► Cls . D . T2“S “ 7 = £ {(gx). * * 7 . * - 72^ 7 )
Dem.
h . *37-33 . Dh. R“S“y = (/* j S )“ 7
h . *71 -25-4 . D h : Hp . Z> . (R | S )“ 7 = £ {(g*) . * e 7 . x = (R | S)‘y]
[*72 ‘ 2 ] =* £ {(C 4 2 ) • * € 7 . x *= /i < *S* 7 |
h . (1) . (2) . D I-. Prop
*72 24. H:. i2«l—»l.D:xc D*R . = . x = R‘R*.v
Dem.
h . *72-202 . *71-212 . D h Up . D : #= R‘R*x. m . x(R\R) x.
[*71192] s.xC/fD'/e)®.
[*85101 .*501] = . a:« a:. a: € D‘72 .
[*1315.*4-73] = .arcD‘72:. D h . Prop
*72 241. h :. /£« 1 -+ 1. D : y c (J*R . = . y = /kfl'y
*72 242. h:.I2cl-»l.D:0 {R‘R* Z ). m . Z€ D‘R . ^ : <t>(R*R< g ). = .zeCI'R .0*
Dem.
h . *30-501-51 . 3 >- : <f> {R'R'z) . = . (ga:) . a: = /?.£<* . <f,x (1)
H . (1) . *72-2 . D h Hp. D : 0 (i*‘/e‘s) . = . (gar) . a:(i* j /*) * . <f>x .
[*7 1 * 192 ] = .(gx) .x = z.ze D‘R . 0x .
[*13195] =.z<D'R.4>z (2)
72 ^
h • < 2 > 5 • * 71 ' 212 . D h Hp. D : 4 , (R‘R‘z). = . 2 e <3‘.ft . ^ (3)
H . (2) . (3). D I-. Prop
*72 243. >-::Rel-*1.0z.z e D‘R.4,z.=,. + (R‘z): = -.4,(R> w ).=„. we (I‘R.+w
Dem.
V . *72-242 . D I- :s Hp .D:.x« D‘.R. 0 z . = x . -0 (i2‘*) : D :
0 Cft‘ii‘*). =,.*(*'*):
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
[Fact]
[*14-15]
[*10-281]
[*71-411]
[*14*21.*71*163]
D : <f>(R t R‘z ). w = R*z . = z% * . \jr(R*z ). w = R‘z :
D : <f> ( R*w ). w = R‘z . = I>1C . yjnv . w = R*z :
D : ( 3 ^). <p(R‘w) . to = R‘z . = ir . ( 32 ). yfrw . w = R‘z:
D : <f>(R‘w ). WfQ‘7? . =„.. yjrw . w e d‘R :
D z4>(R‘tv) .= te .yjru'.W€(I i R (1)
b.(\)fi.l\-::Hp.3:.we(I t R.yl,u’.=„.>lr(R‘w):3:>lr(R‘z).= g .<t>z.Z€D‘R (2)
H.(l).(2).Dh. Prop
The above proposition is used in *272*4*41, which are used in the theory
of "rational series,” i.e. series ordinally similar to the series of rationals.
*72*26. h :. /? e 1 —> 1 : (y). E! R*y : D . (y). y = R'R'y
Dem.
h . *71*165 . D h R e 1 -> 1 . D s (y). E! R*y . s . (y) . y e <l‘R (1)
h . *72-241 . D h Re 1 1 . D : <y). y € <W* . 5 . (y). y - /?*J?‘y (2)
h . (1) . (2) . Imp . D h . Prop
V , .
The propositions Cnv'Cnv'P — Pand t'l'x — ar, which have been previously
proved, are particular cases of the above; the former is a particular case
because Cnv —Cnv'Cnv.
*72-26. h : (y). E! P‘y. D . R = *77?
In this proposition, the conditions of significance require that the domain
of R should consist of classes. This proposition is used in *72 27.
Dem.
h .*37-31 .Dh.e| /?-€, | ~R
[*62*32] =s\R (1)
V . *53*31. D h : Hp. D. (y) • **R‘y - «V/?‘y
[*53 02] = R*y •
[*34-42] D.s\~R = R (2)
H.(1).(2).DK Prop
*72 27. h.D = 7|D.a = e“|a [*72 26 . *33 12121]
*72-27 is used in *74-63*631 and again in *16315.
*72 3. h : 3 ! A. n (1 —> CIs). D . p**- el-* CIs
Dem.
H . *41*12 . Fact. D I-: R e X. R * 1 -* CIs. D . p‘\ G R . R e 1 -> CIs.
[*71-22] D. p‘\ e 1 —> CIs (1)
h . (1). *10*11*23 . D I-: ( 3 P). R * \. R e 1 CIs . D . e 1 -» CIs (2)
K (2). *22-33 . D K Prop
SECTION C]
MISCELLANEOUS PROPOSITIONS
4-17
*72*301. h : g ! X <*% (Cls —* 1) . D . p‘\ e CIs —♦ 1
*72*302. h:g!XA(l -* 1) . D .p'Xe 1 -* 1
*72 303. I" : g ! X n (1 -*CIs) . g ! X n (Cls—► 1). D ./>‘Xe 1 —* 1 (*72*3*301]
*72 31. h : s'X el—* Cls .D.XC1-+ Cls
Dem.
h . *41*13 . D h : s‘\ € 1 —► Cls . P e X . D . s*\ e 1 -> Cls . P G £‘X .
[*71*22] D. Pel-* CIs (I)
I- .<1). Exp. *10*11 21 . D h . Prop
*72 311. h : i'X e Cls -> 1 . D . X C CIs -* 1
*72 312. h : s'X el—*1.D.XC1—*1
*72 32. h X C 1 -*Cls:P, Q e X.g !Cl'Pr> d‘Q .D /t><? . p = Q. D . i‘X e 1 —>Cls
Dem.
H . *4111 . *11-54 . D h : x(6‘\)z . y (&‘K)z. s .
(a P,Q)-P.Q*\.xPt.yQt.
[*3314.*4'71] s.{^P.Q).P,Q e \.xPz . yQz . z td'P *CL‘Q (1)
h . (1). *471 . D h :. Hp. D : z . y (S‘\) z . s .
(3-P. Q).P,Qt\. xPz . yQz . z e d'P r. CI‘Q . P = Q .
[*13-19-5] D.(g P).Pe\.xPz.yPz.
[*7117.Hp] 0.x = y (2)
I- . (2) . *1111-3 . *7117 . D I- . Prop
*72 321. h :. X C Cls -♦ 1 : P, Q t \ . g ! D'P « D'Q. P - Q: D .i'XeCIs-* 1
[Proof as in *72*32]
*72 322. h:.XCl—*1:P, QcX.g! d‘P CI'Q .D pq .P=Q:
[.72-32-321] =
*72 323. h:.XCl-l:P,Q,x. a! C‘P«a'C.3,. 0 .P_«:3.^ e i_i
Dem.
I-. *33*161 . *22*49 .Dh. d'P « d'Q C C'P C*Q . D'P r* D'Q CC'Pn C‘Q .
[*24*58] D I-: g ! d'P « d'Q . D . a ! C'P C'Q :
g ! D'P D'Q . D . g ! C'P « C'Q (1)
H . (1) . Syll. D f- Hp . D : P, Q e X . g ! a'P « d'Q . D P Q . P = Q :
H ■ (2) . .72-322.31. Prop *■<?«*• 3 « ^ • =>« • * = Q <*>
*72*34. h : P e 1 —* Cls .g!*.D. p'P'''* = P''p'*
Dem.
I-. *40*35. Dht.yep'P'''*. = :/9e*.D* .yeP''£ (1)
h • (1) • *71*37 • DH::Hp.D:.y C p'P'''*. = :£ 6 *.D fl .P'ye/3 ( 2 )
448
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
h. *14*21. D h £ e k . D . 72‘y e £ : D : £ € * . D . E ! 72‘y
[*10*52] D I-:: Hp. D £ € #c. D*. 72‘y € £ ; D . E ! R*y (3)
1-. *14*28 . *401 . D h :: E! 72‘y. D ££ k . D* . 72‘y e/9 : = . 72‘y ey‘*::
[(2).(3 ).*5*32,*14*21 ] D h :. Hp . D : y ep‘R‘“tc. = . R'yep'/c.
[*71*37] =.^/f R“p‘k D K Prop
*72 341. h : R e CIs -* 1 . H * * • ^ • p € R“ € * -
This proposition should be compared with *40*37 and *40*38.
*72 4. h : /d € 1 —* CIs . D . 72“a a rt“£ - R“(a a £)
Dem.
H .*71*37 0 h Hp.D:y€ R“ar\ 72“£. = . R‘yea. R*ye @.
[*22*33] s . 72‘y r a a £.
[*71*37] s . y « 72“(a a £):. D h . Prop
When R is not a 1 —»Cls, we only have in general (cf. *37*21)
R“(a a £) C £“a a tf“£.
*72*401. h : R e CIs -♦ 1 . D . 72“a a 72“£ - 72“<a a £)
*72 41. H : 72 £ 1 —* CIs ,ao/5«A.D. £“« a 72“£ - A [*72*4. *37*29]
V72 411. h : R e CIs -► 1 . a a £ = A . D . 72“a a /e“/9 = A
*72 42. h : 72 £ l -*Cls.g ! 72“a a 72“£. D .g !a a£ [*72*41 . Transp]
*72*421. h : R eCls -* 1. g ! R“a a 72“£. D . g ! a a £
*72 43. h : R £ 1 -> CIs . 72“a = 72“£ . D . a a D‘72 - £ a D‘/2
Dem.
V . *71*37 . D Hp . D : 72‘y € a . =„. R‘y e £ :
[Fact] D : z = 72‘y . R*y e a. .z = R‘y . 72‘y e £ :
[*14*15] D:z = 72‘y. 2 c«. =„ • 2 = 72‘y. 2 e £ :
[*10*281] 3 : (3 y) . 2 = 72‘y . 2 € a . = . (gy) . 2 = 72‘y. 2 £ £ :
[*71*41 .*10*35] D : 2 e D‘72 . 26 a. = . 2 £D‘ft. 2 e^:
[*22*33] D s 2 e D‘72 a a. = . 2 € D‘72 a £ :. D h . Prop
*72*431. h : R £ CIs -> 1 . R“a = 72“£ . D . a a Q‘72 = £ a G‘72
*72 44. H : 72 € 1 —> CIs . a C D‘72 . £ C D‘72 . 72“a = 72“£ . D . 0 = £
[*72*43 . *22*621]
*72 441. h : 72 € CIs —» 1. a C G‘72. £ C G‘72.72“a = 72“£. D . a = £
*72*441 is used in the theory of cardinal exponentiation (*116*659).
SECTION C]
MISCELLANEOUS PROPOSITIONS
*7245.
Dem.
*72451.
*7246
*72461.
*7247.
Dem.
*72471.
*7248.
Dem.
*72481.
*7249.
Dem.
*72491.
*72492.
*725.
Dem.
n at w
I- : e 1 —♦ Cls . D . (R)< f* Cl‘D‘72 cl—*!
449
h.*602. Oh:aCD‘R.0CD‘R.m.a,0eC\ , D‘R (1)
I-. *3711 . 3 h : R“a = R“0. = . (R),‘a = ( R),‘0 (2)
K(l).<2).*72-44.3
h :. R e 1 -» Cls. 3 : a,,0 < Cl‘D‘7i . («),<« = («).‘/3.3.., . a = 0 :
[*71-55.*72-15] 3 : (.R), [ C1‘D‘R « 1 —♦ 1 :. 3 I-. Prop
1-: Ji e Cls —*1.3. «,|-CI‘a‘ie e 1-*1
I-:. R «1 -» Cls. 3 : R“a = R"0 . = . an D -R = 0 r> D ‘R
[*72-43. *37-263]
h :. R « Cls -> 1.3 : R“a = R“,3 . = . a /> d‘R = 0 a d‘R
h:.Re 1 -» Cls . 3 : R“a = d‘R . = . D‘R C a
h . *37-25 . *72-46.3
H :. Hp . 3 : R“a = d‘R . = . a n D ‘R = D‘R « D‘R.
[*22-5-621] s . D‘R C a : 3 H . Prop
h Re Cls—» 1.3: R“ a = D‘R . 3 . <1‘R C a
h R e 1 -» CS1» . a. 0 e Cl‘D‘ft . 3 : «“a = R‘*/9. s . a = 0
1-. *22-621.3h :. Hp. 3: a = ^. = .0 n D«R = 0 n D‘R .
[*72-46] s . /t“a ■= R"0 3 h . Prop
h :. « « Cls -» 1 . a, 0 e CI'CT'R . 3 : R“a -R“0 .3 . a - ,8
I-:. Qe 1 -»Cls. 3 : d‘(/» j Q) = d‘Q . = . D'QCa-P
h . *72-47.3 h :. Hp . 3 : Q“d‘P = d'Q . = . D‘Q C (I‘/> (1)
H . (1) . *37-32.31-. Prop
h PeCls-* 1.3: D‘(/»| Q) = D‘P. = . CPR C 11‘Q
H :. P « Cls —» 1 . Q « 1 —» Cls. 3 :
D‘CP I Q) *= D ‘P . a \P I Q) = d‘Q . 3 . a*P = D -Q [*7249 491 ]
K : R e 1 -* Cls. 3. R“R“a -a n D‘R
H.*37 33 . 3 \-.R"R“ a =(R\R)‘‘ a (X)
I-. (1). *71-19.3 h : Hp. 3 . R“R" a = (/ f D‘R)“a
t* 50 ' 59 ] = a « D«R : 3 (- . Prop
1 2 »
150
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*72 501. h:/?cCls-»l . D . Ii“R“a = a n (l*R
*72 502. h : /^ € 1 —► CIs. a C D‘R . D . R“R“a = a [*725 . *22 621]
*72 503. I-: R c CIs -► 1 . a C (VR . D . R“R“a = a
*72 504. hXC D‘/f<. D . R t “ii t “\ = \ [*72-50215]
V w
Note that R< means Cn v*R t , not ( R ) t . *72*504 is used in the theory of
segments of a series <*21164).
*72 51. h : R € 1 -> CIs . a C IVR . (3 = R“a . D . a = /[*72 502 . *2018]
*72 511. \-zRe Cls->1 . £ C <I‘/f . a = R“/3 . D . £ = 7*“a [*72503. *2018]
*72 512 h :.R < 1 -> 1 ./* C <I‘/e . D : y t &. = . R*y * R“&
Deni.
h . *71-37 . D h s. R « 1 -> CIs. D : y c R"R"0 . = .R*ye R“0 (1)
h . *72-503 . D h R e CIs -> 1 .(3 C iVR . D : y c R“R"fi . = . y e f3 (2)
K(l).(2).Dh. Prop
*72 513. h :. if e 1 —* 1 : (y).E!/7'yO:y</3. =. R‘yeR“& [* 72512 .*33431]
*72 52. h :. if* 1 -»1 .aCVH.0C(VR.O:a=R"0. = .0-R"a [*72-51-511 ]
*72 53. H :. if e 1 —» I . D : £ C (I‘if. o = R"0. = . a C D‘if . 0 = R“a
Dent.
V . *72-52. *5-32 . Z>
h uR*\ — \.'Zi*ClVR.&Ca'R.*-R“fi. = .*CTVR.RCa'R.R-R“* (1)
h . *3715 . D h : a - /*“£ O.aC D‘rt :
[*4 71] D h : a C I VR . & C (W* . a = R“/3 . = . /9 C d‘/i . a = R“P (2)
h . *3716 . D H : f3 - /*“« . D . /9 C d‘7* :
[*4. 71 ] D h : a C D‘/f . £ C d‘7* . /9 = £“a. = . a C D‘R . >9 = 7*“a (3)
K(1).(2).(3).DK Prop
*72 54. I-: R e 1 -» 1. D. Cnv‘(/*< T CI'dM*) = (R)< f C\‘D*R
Dem.
K *31131.3
h : £ |Cnv‘(/e« [* Cl‘d‘/?)j a . = . a (R< f Cl‘d ‘R) £ .
[*37101.*35101.*60 2] = . a = R“0 . £ C d‘R 0)
h . *37 102 . *35101 . *60-2 . D
H : £ l(/0< r CFD‘i?) <*. = .£ = £“a. a C D*R (2)
h . (1) • (2) • *72-53 . D h . Prop
*72 541. h:/*€l-*l.S-.R.D. Cnv‘(i?< f D‘,S<) = f D‘R<
[*71-48-481. *72-54]
SECTION C]
MISCELLANEOUS PROPOSITIONS
451
*72 55. H : ft e 1 -► CIs . D . a ] ft = ft |* ft“a = a ] ft f
Dem.
. *351 . *7136 . D h Hp . D : x(a] R) y . = .xea.x = R‘y .
[*14'15] = . R*y 6 a . x= R‘y .
[*71*37] = . y c R“a . x = R*y .
[*71-36.*35101] =.x(RfR“a)y (1)
h . (1) • *35*11 . D K . Prop
*72 551. h: J ReCls- > l.D./er/9 = (ft“£) 1 ft = (ft“ft) *] ft p £
*72 57. H : Q |* X e 1 CIs . X = .D./tn D'Q = Q“X
Dem.
t-.»37'42. Dh:X-Q“ # *.3.(XlQ)«V-§“/* (1)
H . *37'421. 3 h : X = Q“n . D.(<3f X)"Q“/* - Q“\ (2)
H.(l).(2). 3 h : X - Q* V • 3 • (Q r X)“(X 15)‘V - <3“X (3)
I-. *72-5 . *35-52 . D I-: Q [■ X < l -» CIs . 3 . (Q f X)“(X 1 Q )‘V - ^ „ D‘Q (4)
K(3).(4).3h. Prop
*7269. H:nel->Cls.D.S|ii|«=i&'|-r>‘^
Dem.
h . *71-19 . D V : Hp . D . S | R | R = S | (/ [• D‘i?)
[*50'6] - S r D‘« : D I-. Prop
*72691. H:ideCU-»l .D.S|«|* = S|-a*«
*72 6. h:«.l-*Cl8.a*SCD‘7i.D.S|B|ft = S [*7259. *35 452]
*72 601. I-: R , CIs -* 1 . (I‘S C d‘rt . 3 . S | R \ R - S
*7261. t-:ie e l-»ClB.a‘SCD‘*.D.S|7e|^|S = S|3 [*72-6 . *34-27]
*72 611. I-: ft e Cls-»l .a*SCa‘ie.D.Sj«|«|S = S|^
The following propositions lead up to the “ principle of abstraction ’’
(*72*66), which, though not explicitly referred to in the sequel, has a certain
intrinsic interest, and generalises a type of reasoning frequently employed
by us.
*72 62. H:flel-»Cls.S-«|fl.D.S’-S.S = S
Dem.
t*. *34-21 .
D H : ft = ft | ft . D . ft’ = ft | (ft | ft | ft >
(1)
h . *72-6 . *33-21 .Dh:ft€l->Cls.D.ft|ft|ft = ft
(2)
D h : Hp . D . ft* = ft | ft
[Hp]
-ft
(3)
H . (3) . *34 7
. D H . Prop
29—2
452
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*72621. \-:.R f l-*C\s.^:y(R\R)z. = .R‘y = R , e
Dem.
(-.*71-33.3 h Hp. D : R'y = R‘z. = . (a*) • • * = R ‘ z ■
[*71 -36] s . (g*) . xRy. xRz .
[*31-11] =.(3 x).yRx.xRz.
[*84-1] =.y(«|fl)j:.DI-.Prop
*72622. H:.«<Cls-»l .0:y(R\R)z. = .R , y = R , z
*72 63. h : ii e Cls —» 1 . S- R | R . 0 . S ! - S . S « S
Dem.
K*34-21. Z> \-:S=R\R.0.8'-=(R\R\R)\R 0)
K *72-601. Dh:««01s-»1 . D. R | R \ R = R (2)
K<1). (2). Dh:Hp.D. S>=/< Ji
[Hp] -S (3)
H . (3). *34"7 .DK Prop
*72 64. h : £»- S. S-S. R - Cnv‘(Sf D‘S). D. R * Cls -* 1 . S = R | R
Dem.
H . *7212. *71-26.3 V . Sf VSt\~* Cls.
[*71-21] 3 H : Hp. 3.7? f Cls —* 1 (1)
l-.(l).*72-622.3
1 - Hp. 3 : y <fl | R) z. = ■ R'y = R‘‘ •
[*31-34.Hp] =.{S [ D«S)‘y = (S f D*S)«*.
[*35 7] = .y,*(D‘.$ , .*S l y=*S , z.
[*34-85] = . zt VS. ySz (2)
1- .*31-11.3 1-:.Hp.3:yS2.3.2«y.
f *33-14] 3.**D‘S:
[*4-71] 3 : !/Sz . = . 2 e D‘S. ySz (3)
1-. (1). (2). (3) .31-. Prop
*72-65. h:S , = S.S = S. = .(a fi )-^ t Cls->l [*72-63-64]
*72-66. h:S>CS.S-S.s.(a*)-««Cls-»l. 1 S-A|B [*7265.*3481]
*72-7. 1-: JJ e 1 —* Cls. 3 •/Ff D'-R r 1 —* 1
Dem. 4 _
h . *33 4. *22-5 . D h : y, z € D l R. R‘y = R‘z .0 .&lR*y * R‘z (*)
h.(l).*71*18. Dh zy,zcD‘R. i R‘y=R‘z.D.y = z (2)
h . (2). *7212 . *71*55 .DK Prop
SECTION C] MISCELLANEOUS PROPOSITIONS
453
*7271.
h : R e Cls —* 1 . D . ~R[ d‘
i*el->l
*7272.
hRel-tl.D.Tlfa'R,
*RfD‘Re 1 ->1
*728.
h : \ C D‘x i . D . Cl r X e 1
—* 1 [*55*28*22
. *71*58]
The above proposition is used in *73*62.
*7281.
h:\CD'
-♦1 [*55*281*221. *71*58]
*729.
h R e 1 -> Cls . S G i* . D : E ! S‘y . = . 7*‘y
= S € U . = .!/€ d‘lS
Dem.
K*71-22. Dh:. Hp.
0:Sel-> Cls:
[*71*163]
D : E ! S‘y . = . y c
a <s
(i>
h . *14*21 . D h : /*‘y = S‘y . D . E ! S‘y
(2)
h. *30*82.(l).DH:.Hp,
O : y e (l‘S . D . ( S‘y) Sy .
[Hp]
D.(S
l y) % •
[*71*36]
O.S‘!/~ R‘y
(3)
1- .(1).(2).(3).DK Prop
*7291.
h : 72 e 1 -» Cls . S G R . D
.(I*(R-^S)~(I*R
-a <s
Dem.
h • *33*131 . *23*33*35. D
. xRy . ~ (xSy)
0)
I-. (1). *71*36. D
h Hp . D : y € ,
= • (3*) • a: = R‘y
■ = S'y).
[*14*15.*5*32]
= .(g x).x=R f y.
~<7*‘y = £‘y).
[*10*35.*14*204.*72*9]
= . E ! R*y .<N/(ye
ass).
[*71*163]
B-ycd^-a'iS
D h . Prop
*72911.
ViRe Cls-*1 .iS'G/i.D
.V‘(R^-S) = I>‘R
-D‘£
*7292.
h : /* c 1 -> Cls . S G R . D .
.s=R[a*s
Dem.
H.*23 1
. *33*14 . D K Hp . D : arSy . 0 XiV . x/£y. y t
(I ‘S.
[*35101]
3x. v .*(/*ra‘S)y:
[*2313
DsSGRf(I‘S
<D
H . *35*101 . *71*36 . D H Hp. D ::
z(R [ dSS)y . = . x
= R‘y.ye <1‘S.
[*72*9]
= . X
— R‘y . R‘y = S‘y
s
[*14*142]
D.x
-S‘y.
[*30*31]
D.xSy
(2)
h . (2) . *11-11*3 . D h : Hp . D . R f (I‘S GS
(3)
H . (1) . (3) . D I-. Prop
*72921.
\-zReC\B-tl.SGR D
.S = (D*S)'\R
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
434
*72 93. !-:./?< l-»Cls. AG&.s: y e <l € R . . (R l y) Sy
Item.
h . *14*21 . *4*71 . D H :: y e U ‘R . . (R‘y) Sy : =
y e Cl‘/e . D y . E ! R*y . (ii‘y) Sy
[*14 23] = y e Q‘R . D y . E ! R'y : xRy . D x . *Sy
[*10*29.* 11 *62] = y c <J‘ R . D y . E ! R‘y : y € iVR . *tfy . D Xt?/ . xSy
[*71*l(i.*33*14] =:./?€ 1 —► Cls . ft G £D 1*. Prop
*72 931. V :. ft « Cls -> 1. R G 8 . = s x e D‘R. D,. xS(ft‘.r)
*72 94. h :. ft. £ « 1 —► Cls .3:g !/2r*S.«. (ay). ft‘y = S‘y
Dem.
1- . *71 *30 . D h :. Hp . D : y ! ft A S. = . (a-»*. y). ;c = R*y . j* = S*y .
|*14*205] s .(gy). R‘y = S‘y :. D 1-. Prop
*73. SIMILARITY OF CLASSES
Summai'y of *73.
Two classes a and /3 are said to be similar when there is a one-one relation
whose domain is a and whose converse domain is /9. We express "a is similar
to /3” by the notation ‘‘asm/9.” When two classes are similar, they have
the same cardinal number of terms: it is this fact which gives importance to
the relation of similarity.
We have
a sm /9 . h .(g R). R e 1 -> 1 . a = D*Jl . = (W*.
The relation of similarity is that of the domain of a 1 —» 1 to the converse
domain, i.e. it is the relative product of L)f*(l —► 1) and < 1 —► 1)1 Cl, or, what
comes to the same thing, it is the relative product of Df*(l —> 1) and CT.
Most of the properties of similarity result immediately from those of
one-one relations and offer no difficulty of any kind.
When there are relations which correlate a's with /3‘s so as to make
a similar to £, we denote the class of such relations by “asIu/3.” Thus
we have
a sin /9 — (1 — ► 1) r\ D*a r\ Cl*# Df
an< l sm — 8/3 (a ! a sm /9) Df
When, as in this case, we have a descriptive double function closely
connected with a relation, we shall make it a practice to distinguish the
descriptive double function by a bar.
It is to be observed that “sm,” like A and V and 1 and 1 —* 1, is ambiguous
as to type, and only acquires a definite meaning when the types of its domain
and converse domain are specified. The domain and the converse domain
may or may not be of the same type, i.e. “sm” may or may not be a homo¬
geneous relation. This enables us to speak of two classes of different types
as having the same number of terms. We shall return to this point in
connection with cardinal numbers (cf. especially *102—*106).
The propositions of the present number are important, and are very
frequently referred to throughout cardinal arithmetic. In order to prove
that two classes a and /3 have the same cardinal number of terms, it is
generally necessary, in the fundamental arithmetical propositions with which
we are concerned, actually to construct a relation R such that Reasni/3.
Such a relation will be called a correlator of a and /9. It will usually be
obtained by taking some relation S for which we have (y). E! S*y, and
456
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
limiting the converse domain to ft, so that S[ft is the required correlator.
Very frequently we shall have Sc 1 —*Cls, not Sc 1 —► 1, but ft will be such
that S[ftc 1 -> 1.
Among the more important propositions of the present number are the
following:
*73 142. h:ftr/36aMn/3. = ./?[*£el-*l.£C <P/? . a = R“ft
l.e R[ ft is a correlator of a and ft if (1) R [ & is one-one, (2) ft is con¬
tained in the converse domain of R. (3) a is the class of those terms which
have the relation R to members of ft.
*73 2. h : R c 1 1 . D . D‘R sm (l‘R . (I‘T? sin D *R
This results immediately from the definition.
*73 22. h : Rc 1 -> 1 .ftC (l*R . D . R“ftsm ft . R [ ft c (R“ft) sin ft
*73 3. H.asma./pacasma
*73 31. h :asm/3. s ./Ssnia
*73 32. h : a sm ft . ft sm y . D . a sm 7
The above three propositions show that similarity is reflexive, symmetrical,
and transitive.
*73 36. h:. asm/3.D:a!a. = .a!/9
*7341. h . ("asm a . t fa c(<“a)siu a
Thus every class a is similar to a class i“a of higher type, and consisting
wholly of unit classes.
*73 45. h . 1 s£(£smt'x)
Thus 1 is the class of all classes similar to any unit class.
*73 48. h . 0 = /§(£sm A)
Thus 0 is the class of all classes similar to the null-class.
*73 611. H . I x“a sm a . ( l x) [ a c ( i x“a) sm a
This proposition is very often useful. For arithmetical purposes, we often
wish to obtain mutually exclusive classes. Now whether or not a and ft be
mutually exclusive, i x il a and | y il ft are mutually exclusive provided tf + y.
Thus by means of the above proposition we can always construct mutually
exclusive classes each similar to a given class, i.e. each having some assigned
number of members.
*73 71. V : a sm ft.y9m8.ar\y = A.ftf\8 = i\..'D.(a\jy) sm (ft v 3)
This proposition is fundamental in the theory of addition.
*73 88 . H : a sm 7 . ft sm B.yCft.SCa.O.asmft
l.e. “if a is similar to a part of ft, and ft is similar to a part of a, then
a is similar to ft." This is the Schroder-Bernstein theorem. The proof given
below is due to Zermelo.
SECTION C]
SIMILARITY OF CLASSES
457
*7301.
a sm £ = (1 —► 1) r\ D*a r\ d‘£
Df
*7302.
sm = o£ ( a ! a sm £)
Df
*7303.
^ : R e a sm £ . = . R e 1 —* 1 . a
= D‘77.£=d‘77
[*33-6-61 .(*7301)]
*7304.
H : a sm £ . = . a ! a sm £
[(*7302)]
*731.
H : a sm £ . = ,( a /7). Re 1 —-* 1 .a
= D‘/7.£ = d‘/7
[*730304]
*7311.
H : a sm £ . = .( a 7?). R e 1 —► 1
.aCD‘/i.£=K ,( a
Dem.
h . *22*42 . *37*25 . D
V : R e 1 -► 1 . a - D‘7* . £ - d f R . D . R « 1 -► 1. a C D‘i* . £ = R“a :
[*10 11 28]Dt-:( a «)./e c l -► l .a = D t R.0 = Cl‘R.D .
(a/e)./? € i i .aCD‘/e.£= i?“a:
[*731] D H : a sm £ . I> . ( a 7?) . /* e 1 -»l.aC D‘.R . 0 - i7“a ( 1 )
h . *71-29 . *37-4 . *35-62 . D
h : € 1 -> 1 . a C D‘* . £- £“a. D . «1 « 1 -> 1 . a - D‘(a1 7e).£-<3‘(«1 77).
[*10-24) D . ( a S). S € 1 -► 1 . a = D*S . 0 - dSS.
[*731] D.asmjS (2)
h. (2). *1011-23. D
I- : ( a /7) . R € 1 -> 1 . a C D *R . 0 = Ii“a . D . a sin 0 (3)
K(l).(3).Dh. Prop
*7312. \-:asm0.m .(g-R). 7*« 1 -* 1 .fiCa^.a = «“/3
[Proof as in *7311]
*7313. h : a sin £ . = . ( a 7?). 7* « 1 -► Cls . 77 f* £ e Cls-* 1 . £ C CP/d . a-Ji“0
Dem.
h .*71 103 271 . Dh:J2el-»l.D.i?c] —* Cls . /7 [* £ € CIs —* 1 :
[Fact] DHr/fe 1 -* 1 . £ C d‘77 . a = R**0 . D .
/7€l-*Cls.77r£«Cls-*l . £ C G.*R . a ■* 77“£ :
[*1011'28.*7312] D h : asm £ . D .
( a 77).7*€ 1 -*Cls. 77|*£€Cls-* l .£CCI‘/*.a=.ft"£ (1)
h.*71-26.Dl-: 77 6 l-*Cls. 77[*£*Cls-*l . D . 72 [*£« 1 -*Cls. R t0cC)s-> 1 .
[*71103] D.i*P£el-*l (2)
h. *35-65. *37-401 . D
H :£Cd‘7?.a= 77“£. D.£ = d < (77|‘£).a = D‘(7*r£) ( 3 )
h . (2) . (3) . D h : 1 -* Cls . R f* £ e Cls -> 1 . £ C d‘/7 . a = R“(3 . D .
7* r £ * 1 -* 1 . a = D‘(R [“£).£ = d‘(R r 0) •
[*10-24.*731] D. a sm £ (4)
h. (4). *101123. D
I- : ( a /e> . 77 c 1 -* Cls . R r £ € Cls -* 1 . £ C d‘/7 . a - R“0 . D . a sm £ (5)
I- . (1) . (5) .DH. Prop
458
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*73131. : a sin 0 . = . (g77). 77 c CIs -> 1 . a 1 77 e 1 -> CIs . a C D‘77 . £ « 77“a
[Proof ns in *7313]
*7314. h a sm /9. = : (g77): R e 1 — ♦ CIs . jS C G‘77. a = 77“/9 :
y,ze 0. 77‘y = R‘z . 0,,'Z • y = z
Dem.
K *71-55 .*5-32. 3
h:.Afl-»Cls.Ar0cl-»l.»:
/^el —♦CIs: y, 2 € 1 3.77‘y = 77‘* . ’5 v%: .y = z (1)
K *71*26. 3h :.77«1 -*01* . 3 : 77p#€ 1 -♦CIs:
[*4-73.*71 103] 3 : 77p/9el -♦ 1 . ■ . 77P/9€Cls-* 1 :.
[*5-32] 3 h :. 77 e 1 -♦CIs. R [ 0 * 1 -♦ 1 . = . 77 < 1 -♦CIs. R P/9cCls— ♦ 1 (2)
h.(l).(2).Dh. (g R ). R e 1 — ♦ CIs . R [ 0 e 1 -♦ 1 . 0 C G‘77 . a = 77“/9 . e :
(g77) : Re 1 -♦ CIs. 0 C G‘77 . a = 77“/9:
//,* e /9. 77‘y = R‘z . 3,,.,. y = * (3)
*73-13.31-. Prop
The use of this proposition in proving similarity is very frequent.
*73-141. H :. a sm 0 .= :(g77): 77cCls—♦ 1 . aCD‘77 ./9 = 77“a :
[Proof as in *7314]
*73142. h : 77 [“ /9 € a sm /9. = . 771* /3 e 1
//,c € a . 77‘y = R*z . 3 v>1 .y = ~
-♦1 .,9CG‘77.a = 77“/9
Dem.
h . *73 03 . 3
h : 77 p /9 6 a sm /9.s.7?P/9<l—♦!.<> = D‘(77 [ 0). 0 — G‘(77 P 0 ).
[*37-401 .*35 64] = . 77 [ /9 € 1 -♦ 1 . a = 77“/9 . 0 = /9 n G‘77 .
[*22-621] m . R[0€ l —*1 .a — 77“/9. /9 C G‘7? : 3 h. Prop
*7315. h : a sin /9 . = . (g77). 7? P /9 € 1 -♦ 1 . /9 C G‘77 . a = 77“/9
Dem.
V . *73*12 . *71*29 .Dh:a sm 0.0. (g77). 77 P/9« 1 —♦ 1 ./9CG‘77.a=77“/9 (1)
K *7314204. 3h:(g77). 77P£el-*l./9CG‘7?.a = 77“/9.3.asm£ (2)
h.(l).(2).DK Prop
*73 2. h : 77 c 1 —♦ 1.3 . D‘77 sm 0*77. G‘77 sm D‘77
Dem.
h. *202. *321 .D
h:il«l—♦l.D.jRel—*1. D‘77 = D‘77 . 0*77 = a*77.
[*10 24] 3 . (gS). S e 1 -♦ 1. D‘77 = D‘5. G‘77 = 0*5.
[*731] 3. D‘77 sm G‘77 (1)
h . ( 1 ) . *71 -212.3 h : 77 e 1 -♦ 1.3 . D‘R sm 0*77
[*33-2-21] 3 . a*77 sm D‘77
h ,(1).(2). 3 h . Prop
(2)
SECTION C]
SIMILARITY OF CLASSES
459
The following propositions, down to *73*241, are deduced from preceding
propositions of this number just as “ D*R sm Q‘7i ” was deduced in *73 2
from *731. The proofs are therefore merely indicated by references to the
previous propositions of this number which are used.
*73 21. I- : R c 1 —> 1 . a C D‘12 . 3 . asm R tf a. a R e asm (R“a) [*7311]
*7322. h: € 1->1 ./9C<I‘P. D . R“/3 sm/3. R 10 e (R“/3) sm 0 [*7312]
*73 23. h : .R « 1 —► Cls . /3 C CI'P . Rf 0e Cls —>1.3.
R“/3sm&. R [ f3*(R lt 0)sml3 [*7313]
*73 231. I- :Re Cls-> 1 .aCD‘P . a] R e 1 -*Cls . 3 .
asm R“a.a 1 «€asm(«»a) [*73131]
*73 24. h R € 1 Cls . £ C <1*R : y, * . fl‘y = . 3.,.,. y - * : 3 .
P“/3 sm £ . 7? r £ e(/?“/9) sm £ [*73 14 142]
*73 241. h /£ e Cls —> 1 . a C D‘P : y,z t a . 7?‘y = • 3 yiZ . // * z : 3 .
a sm P“a . a 1 R « a sm (R“a) [*7314103]
*73 25. h (y). E ! P'y : y, * « £ . P‘y - . 3 y .,. y - * : 3 . sm £
I) era.
h . *71166. 3 h : Hp • 3 . ft c 1 -» Cls ( 1 )
V . *33 431.31-: Hp . 3 . £ C CI‘/2 (2)
I-. (1) . (2). 3 V Hp . 3 : R « 1 -► Cls . £ C CP/* : y % z*$.R'y-R‘ty-zz
[*73-24] 3 : A“j9sm /3 3 h . Prop
This proposition will be convenient in such cases as the following: Let /3
be a class of relations whose domains are mutually exclusive, x.e. such that
no two members of /9 have domains which have a member in common, and
suppose we wish to prove that the class of these domains is similar to /3.
The class of domains is D“/9, and we have (P ). E! D € P. Hence we have
only to prove (putting D in place of the R of *73*25)
P,Q€0.D‘P= D‘Q.3,.<,.P = Q,
which, in the case supposed, is proved immediately.
*73 26. V <y) . E ! R*y : R c 1 —► 1 : 3 . R“0 sm fi.Rffi* (R“0) sm 0
Deni.
h . *33-431.3 I- : Hp .D.Rel—*l./3C G‘R .
[*73-22] 3 . P“/9 sm . P f £ e (R“0) sm £: 3 I-. Prop
*73-27. h R*y = R‘z . .y = z: 3 . R**t3 sm £ . R [ 0 e (R“/3) sm 0
[*73-26. *71 57]
*73-28. I-:: y, z e >9.3,,, : R*y=R*z . = .y=*zz. 3 .
sm £ . P r £ € (P“£) sm /3
Dem.
h . *71-58 . *7303 . *37-421.3h:Hp.3.Pf*/9 € (P“>9) 5m5:3K Prop
4 GO
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*73 3. b . a sm a . f [ ae a sm a
Dent.
b. *50-31 .*2411. Dh.aCd‘1 (1)
h . (1). *72*17 . *5016 .Dh./el—»l.aC Q‘I . 1“a = a (2)
H . (2). *73*142*04 .Dh. Prop
This is the reflexive property of similarity. The conditions of significance
rec|uire that a should be a class of some type, but impose no restriction as to
the type of class.
*73301. b : /feasm/3. = ./?€/9sma
Deni.
H . *73 03 . *71 212 .*33 2*21 . D
\-:Reasm{3.= .Rel -* 1 . D *R-0.a‘Rma.
[*73*03] =.7f€/9sma:Dh. Prop
*73 31. b : asm/9 . = . £sm a [*73*301*04 . *31*52]
'This proposition shows that similarity is a symmetrica! relation.
*73311. b : R e asm/3 . S €/3 suiy . D . R Sea stay
I Jem.
b . *73 03 . *71*252 . D b : Hp . D . /? 1 Se 1 -> 1 ( 1 )
H . *73 03 . *37*32 . D h : Hp. D . D‘(/? I S) = R“0 . (I‘(7*1 S) = S“/3 .
a *= D‘7? . /9 = a*/? . 0 = D'iS . 7 = Q*»S' .
[*37*25] D. l)‘(R\S) = a.a\R\S) = y (2)
h . < 1). (2). *73*03.3 b. Prop
*73 32. b : asm /9 . /9 sm 7 . D . a sm 7 [*73*311*04]
This proposition shows that similarity is a transitive relation. Thus we
have now proved that similarity is reflexive, symmetrical, and transitive.
*73*33. b . Cnv'sin = sm [*73*31. *31*131]
*73 34. b . sm’ = sm
Dem.
h . *34*55 . *73*32 . D b . sm* C sm (1)
H . (1). *73*33. *34 8 . D h . Prop
*73 35. b . D'sm = G‘sm = Cls
Dem.
h . *73*3 . D b . z ($ ! 2 )sm 2(<f> ! z ).
[*20*18] Ob :a = z(<pl z).D .asma:
[*10*11*23] 0 b : (g<#>) . a = 2 (<£ ! z) . D . asm a .
[*33*14j D . a € D'sm . a e G‘sm :
[*20*4] D h : ae Cls. D . aeD‘sni . ocQ'sm (1)
SIMILARITY OF CLASSES
SECTION Cj
•101
I- . *731 .*10*5.3
\- asm/9. D : ( 3 -ft). a = D'K . £ = C1*11 :
[*105.*33*11111] D :(a/2).a«*-{(a^).ar/ey):(a/J)./3 = *)\(&x).xRy\i
[*20*41*18] D: a e Cls. /9 c Cls (2)
h .(2). *10*11*23. D
*■ :• (3/9) .asm^.D.oe Cls : ( 3 a). asm >9. D . /9 e Cls
[*33*13*131] D h a c D‘sra . 3 . a 6 Cls : /9 e <3‘sm . D . /9 e Cls (3)
h .(1). (3) . D h . Prop
*73 36 h :. a sm £. D : 3 ! a . s . 3 ! £
Dem.
1- . *33*24 . D h a = D‘/2 . /9 = . D : 3 ! a . = . 3 ! &
[*3 42] I> t- J* e 1 -► 1 . a = D‘R . /9 = (1*R . D ; 3 ! a . s . 3 ! /9
[*10*11*23]D h ( 3 /d). 72 € 1 -> 1 . a = D‘i* . £ = (1‘R . D : 3 ! a . = . 3 ! (l)
H - (1) . *73-1 . D h . Prop
*73*37. h a sm /9 . D : 78 m a.s .7 sm /9
Dem.
1-. #73*32 . D H : a sm /9 .7 sm a . D . 7 sm /9 ( 1 )
H . *73*31 . D h : a sm >9 .7 sm /9 . D . /9 sin a . 7 sm /9.
[*73*32] D. 7 sn»a (2)
h . (1) . (2) . D H . Prop
*73 4. h . Cnv“X sm \ . Cnv f\c (Cnv“\.) sin \ [*73*26 . *72*11 . *31*13]
*73 41. h.t“a 8 ma.tf'a((t“a) 8 ma [*73*26 . *72*18 . *51 12 ]
This proposition is useful, because it gives a class ( 1 ** 0 ) similar to a but
of higher type. Thus if ft is a cardinal number, and it is known that in a
certain type there are classes having ft terms, it follows that there will be
classes having terms in the next higher type, and therefore in the next
type above that, and so on. No corresponding means exist for lowering the
type.
*73*42. h : a C 1 . D . a sm t“a
Dem.
K *52*13.3 h : Hp.D. a C D‘i ( 1 )
I- . (1) . *73*21 . *72*18 . D h . Prop
Ihis proposition gives a means of lowering the type without altering the
cardinal number, provided our class a is composed wholly of unit classes; for
i“a is of the type next below the type of a. But when a is not composed
wholly of unit classes, this construction fails.
*73*43. h . sm i‘y . « ^ y e (far) sm (t‘y) [*55*15 . *72*182 . *73*2]
462
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*73 44. h :. a € 1.3 : £sin a . = . 0 e 1
Dem.
b . *73 43.
[*1011-23]
[*10 11-21-23]
[*521]
b . *37-25.
D b :. a * i*y . 3 : /3 = i*x . 3 . 0 sm a
^ (3//). a = t'y . 3 : /9 = i*x . 3 . /3 sm a
3 b :• (ay) • a = i‘y . 3 : (gx) . 0 = i‘x .0.0 sm a :.
3 b :.ac 1.3 : 1.3 .£sm a ( 1 )
D b 7? € 1 -* 1 . D*R = i*x . 3 . d‘R = £“,‘.r
[*53 31.*71 1G5] = i*R*. c.
[*52-22] 3. d‘R e 1
[*20 18] 3b:.*«l-*l. D‘7? = f‘x. CI‘7? = .5. 3 . 0 e 1
[*10 11 23.*73 1] 3 b : i‘xsm £ . 3 . £ c 1 :
[*20 18] 3 b :.a« *‘.r. 3:asm£.3.£« 1
[*101123] 3 b:. (gx). a = i*.c. 3 : a sm 0 . 3 . 0 e 1
[*73-31 .*52 1 ] 3 1-a * 1 . 3 : £sm a . 3 . 0 c 1 (2)
b . (1). (2).3b. Prop
*73 45. b . 1 = 0 (£sm i‘x)
Dem.
b . *52-22 . *73-44 . 3 b : £ sm i‘x. = . 0 e 1 (1)
b.(l).*2033.3b.Prop
*73 46. b . A sm A [*721 . *33 29 . *73 *2]
*73 47. 1*: 0 sm A . s . £ = A
Dem.
1-. *73-46.3 b s £« A . 3 . £sin A (1)
b. *7312. *10-5.3
b : £ sin A . 3 . ( 3 /?). 0 = R “A .
[*37-29] 3.£ = A (2)
1- • (1).(2).3 b • Prop
*73 48. b . 0 = 0 (0 sm A ) [*73 46 .*51 11 . (*54 01)]
The following proposition is used in the theory of double similarity
(* 111111 ).
*73 5. CPO'/e C sm
Dem.
b . *35 101. *37 101 . *60 2.3
b :. 7?, r CI‘(I‘7< G sm . = : 0 C d‘R . a = R“0 .3. #/) . a sm 0 (1)
b . *73-22. Exp .3b:.22el—*1. 3 : £ C a *R .a = R“0 .3 . asm 0 :
[(1).*1111-3] 3 : R t [• C\‘(I‘R Gsm (2)
b. *3018. *51-12.3
b :. £ C (1‘R . a = 2*“£. 3.,* . a sm £ : 3 : t‘y C d*R . a = R“i*y . 3. . a sm t‘y :
[*51-2.*53‘301] 3 : ycd t R.a = R‘y. 3..asmi‘y:
[*20 53.*73-44] 3 : y c G‘/e . 3 . R‘y e 1 :
[*10T1-21.*37*702.*71 1 ] 3 : Re 1 -> Cls : (3)
SECTION C]
SIMILARITY OF CLASSES
463
[*72 51 .*3716] D : a C D‘ft . = R“a . D a(J . & C <3‘ft. a = R“(3 (4)
h . (4) . *47 . *11-37 . D h Hp(4) . D : a C D‘ft . 0 = R“a . D a « . a sm B :
w ■
[(8) ^ • *71-211.*73*31 J D: fteCIs-* 1 (5)
h . (1) . (3). (5). *71103.=> h : ft, f Cl'CI'ft G sm . D . ft e 1 —> 1 (6)
K(2).(6).Dh. Prop
*73 501. h : ft € 1 -► 1 . = . (ft), f* Cl'D'ft G sm
Dem.
h. *71-212. DH:ft c l->l # a.5il-»l.
[*73*5] = . (ft), r Cl'CT'ft G sm .
[*33 21] s . (ft), r Cl'D'ft G sm : D h . Prop
*73 51. h : ft € 1 -> Cls . a C D‘ft . D . %‘a sm a
Dem.
H . *72 7 . D h : Hp . D . ft f D‘ft « 1 -> 1 .
[<*35-481 .*71 ■ 5522] D . [• <*« 1-► 1
I- .*33-431.*32121 . D h . a C CI*!ft
I- .(1). (2). *72 12. D h : Hp.D .*Rc 1 -* Cls . /ifae 1 -» 1 .nCO'S.
[*73-23] D . *R»„ sm a : D h . Prop
*73 511. h : R e Cls -* 1 . a C (PA . D . 7?<a sm a
£*73*51 ^.*71 211 .*33 2 . *32 241J
*73 52. h : ft e 1 -> Cls . a C Cl‘D‘ft . D . (ft),“a sm a
Dem.
( 1 )
( 2 )
h . *72 45 . D h Hp . D : (ft), f* CI‘D‘ft « 1 1 :
[*7l*55.*72 15] D : ft if« Cl‘D‘ft . (ft),‘£ = (ft )«‘*7 . D #> , . f :
[ H P] => : ft*< «• - («V* 7 . D #t ,. £ -17 s
[*73-25.*37 111] D : (ft>“asm aD h . Prop
*73 621. H : ft c Cls —* 1.£ C Cl‘Q‘ft . D . ft,“j88m /3 [Proof as in *73 52]
*73 63. h : ft c 1 -> Cls . a C CPD'ft . D . ft‘“a sm a
*73 631. h : ft € Cls -► 1 . £ C Cl‘(I‘ft . D . ft‘“/9 sm 0
*73 61. f-.®j“a8ma.(a:|)f'af(ari “a) §m a
*73 611. K . j®“aBraa.(|a)[ ae(^a^*a)§ma
*73 62. H : X C D‘* . D . d“X sm X . a f X e (<I“X)smX
*73 621. h:XCD‘ji.D. D“X sm X . D p X e (D“A)smX
[*73-52. (#37 04)]
[*73*521 .(*3704)]
[*73 27 . *55-2]
[*73-27. *55*201]
[*73*23 .*72*131-8]
[*73-23. *7213-81]
164
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*73 63 h:S€asm0.rfa,Tt/3<:\->l.ayjf3Ca‘r.D.T S'.T€(T“a)§m(T“0)
Dent.
h . *73 03 . *35-452 453 . D b : Hp. D. T S T=T T
[*35-354] -rr«|£l£1?"
[*35-52.*71 252.*73 03] D. T S Tc 1 -¥ 1 (1)
I-. *37-32 . DK D‘(T S T)= T“S“a*T (2)
h . (2). *37-27 . *73 03 . D h : Hp . D . D 4 < T { S f)= T“a (3)
Similarly HsHp.D.d ‘(T S T)=T“/3 (4)
h . (1) . (3). (4). *73 03 . D h . Prop
The above proposition is used once in connection with cardinal addition
(*112 231), and once in connection with cardinal multiplication (*114-561).
The following proposition (*73*60) is a lemma for *73 7.
*73 69. h : /? e a Sfii # . a a 7 « A . 0 n y = A . D • ft c; / f* 7 6 (a u 7 ) sm (/? u 7 )
Dem.
h . *33-26 261 . *50-5-52 . }
h : I) 4 ft - a . (\*li -0.S=Rw/ty.O. D*8 « a u 7 . ( 1 * 6 ' = 0 v y ( 1 )
h. *7 l-242. *50-5-52.D
h : Hp (1). /? < 1 —* 1 . a a 7 ■ A . n 7 » A . D . 7? c; /7 c 1 —* 1 ( 2 )
H.(1).(2). *73 03 . D h . Prop
*737. h : a sin / 9 .a« 7 - A .^7 = A.D.(a V 7 ) sm (/$ w 7 ) [*736904]
*73 701. h : 7( t asm/9 .Sc 7 sm 6 .a ^7 = A./iA5 = A.D./^c/.S'c(a»-»7)sm(/3c/5)
Deni.
V . *73 03 .D h:Hp.D. D 4 ft n D 4 6 * = A . <J‘ft n Cl 4 tf = A . R,Se 1 -> 1.
|*71242] I>.ftc/tf<l-»l (1)
h .*33-26-261 .*7303. D h : Hp. D . D 4 (fto S) = avy.(l*(RvS)=/3vS (2)
h. (1). (2). *7303. Dh. Prop
*7371. h : asm/9.7sm5.ar»7=A./3r»$ = A .D.(a V 7 )sm(^v^) [*73-70104]
*73 72. I-: a v i‘a?sm @ sj i‘y ea . y ~€0. D. a sm f3
Dem.
h . *731 . }
h : Hp. } .{zR).Be 1 —> l.D‘/? = av i*x.(l*R=/3 \Jt‘y.x~€ a.y~€/9 (1)
h . *71 -381 .DhsAcl-tl.ae D 4 ft . y € a 4 /? . Z) . ft 44 (G 4 ft - i‘R*x - i‘y)
= R“a*R- R“i*R t x-R tt i t y
[*37-25.*53-31] = D 4 ft - i'R'R'x - < 4 ft‘y.
[*72-24] =D ‘R- Sx-i'R'y.
[*73 22] } - (D 4 ft - t‘x - £ 4 ft 4 y) sm (CI 4 ft - i*y - i‘R‘x) (2)
SECTION C] SIMILARITY OF CLASSES 165
h . *71*362 . *22-5 . D h : Hp(2) . x=R*y . D .
D‘ R - i*x - i<R*y = D *R - i*x . il'R -f'y- i'R'x = Q'R - t*y .
[(2)] D .(D*R — £*dr)sm(Q‘/i — i*y) (3)
V . *22-92 . *33-43 .DH:Hp(2).x+ . D .
(D‘ie - i*x - i*R‘y) v i‘R*y =D‘R - i‘ar (4)
H . *71-362 . D h : Hp (4). D . y 4 = R<x .
[*22-92.*33-44] D . (a*A - i*y - c«jftr) vf «£<.r-(I«A - #*// (5)
*■•(*). (5) . *73-71-43 . (2). D h : Hp (4). D . (D‘i£ - f‘.t) sm {iVR - t‘y) (6)
K(3).(G). D h : Hp(2) . D . (D‘7? - f*x)sm (d*R — f‘y) (7)
H .*51*211'22. D h : T)‘R = a ^ f *.c . (1*72 ™ f3 v i*y . .r e a . y *■>» c /9 .
D . D‘« -- a. d‘/e -i*y = /3 ( 8 )
h.(7).(8). D h : e 1 —■* 1 . Hp( 8 ) . D . asm£? ( 9 )
K(l).(9). D h . Prop
The following propositions give the proof of the Schroder-Bernstein
theorem, namely: If one class is similar to part of another, and the other is
similar to part of the one, then the two classes are similar. The proof here
given is due to Zermelo*. An explanation of the following proof is given in
connection with another proof in the summary of *94.
*73 8 . h : (I‘/e C 0 . >9 C D‘R . * - 5 (a C D‘R . £ - CI‘7* C a . R“ a C a). D .
Dem.
H. *22 42 43 44. D h : Hp . D . \YR C D* R . & - <l‘R C D*R (1)
h . *22 44 . *37-25 . D h : Hp . D . 72“D‘7* C D*R
h.(l).( 2 ). DI-:Hp.D.D‘/2c*
h. (3). *4012. Dh.Prop
*73 801. h:Hp*73-8.D. i S-a < /JCpV
Here “ Hp*73-8” means “the hypothesis of *73-8.”
Dem.
h . *20-33 Oh. Hp . D : a e * . D« . /3 — G‘R C a :. D h . Prop
*73-802. h : Hp *73 8 . D . R“ P ‘k C P *k
D em.
I-. *20-33 . D I- :. Hp. D s a c *. D. . /*“a C a ( 1 )
h • (1). *40-81.3 h. Prop
*73 81. I-: Hp *73-8 . D . P *k e k
D em.
H.*73-8-801-802.DH:Hp.D.p‘«CD‘ii.^-a‘«Cp‘*.fl‘‘p‘*Cp‘*OI-.Prop
Math. Annaltn. vol. vxr. Heft 2, February 1908.
< 2 )
<*>
R«CW I
30
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
m
*73811 b : Hp*738. D. j?“p*« C/j**-(£-CI'/?)
Dem.
h.*37 lG.DK/e'yxCCP/?
[*22s] C — (— CI‘/?>
[*22-8143] C-(£-<!<£) (l)
I-. (I >. *73802 .DK Prop
*73812. H : Hp *73 8. — <I*/?) u I{“p‘* . D . R ,t (p i x — f‘.r) C — f*.r
Dem.
H. *22-87. D h : Hp. D ,jc~~€ !{“/>**.
[*■>1-30) D./*>‘*C-i‘x (1)
I-. (I). *73 802 . D h : Hp. D . R ,t p , H C /*** — i‘x .
[*372] D . /?“(//* - i'x) C p t K -(‘.rjDh. Prop
*73 82. H : Hp *73*812 . D ./>** — i*.r = y>‘* .
Dem.
f-. *22 87 . *51 30 . D h s Hp•D.£- (1*7? C - /‘x.
[*73-801] D . (3 - (1 •R C p‘« - /‘.r (1)
h .*738 . Dhs Hp.D./P*— i*x C D‘/f (2)
H . (1). (2). #73’812 -DP: Hp. D . //* — «‘xc *.
[*4012]
[*51‘3I>. *22*43] D. x~« ;/*. p‘x Prop
*73821. h : Hp*73 8 .x€j>«*-(£-0*/?). D.xt ]?“//*
Dem.
I-. *73*82 . Transp. D H : Hp*73-8.x«/>‘* . D . x«(£ - CP A) u R tt p t K (1)
K . (l). *•'»•(». D H . Prop
*73 83. H : Hp*73 8. D . p*< - (£ - (1*11) = R“p*K ,p*K «(£- OPJl) ^ ii"/* 1 *
Dem.
V . *73-821 . D h : Hp. D . - (£ - d'R) C * V* (1)
V . (1). *73-811. D I-: Hp. D .p‘*-(£ - d‘R) = £“/><* (2)
h . (2). *24-47 . *73*801. D h : Hp. D . p** = (0 - (I‘R) v R“p‘* (3)
h . (2). (3). D h. Prop
*73 84. H : Hp*73 8 . D . £ =p‘/c u (<3‘i? -
Dem.
h.*2292.Dh : Hp. D .£ = (£-a*/?) ^ (P*
[*22-92.*37 16] = (£ - CP*) u R“ P *k u (d‘i? -
[*73 83] = p*K u (CP R - R“p‘k) : D h . Prop
SECTION C]
SIMILARITY OP CLASSES
467
*73-841. b : Hp *73 8. 7? 6 1 — > 1 . Z> . 0 sin (1*11 . 0 sm D*R
Dem.
b . *73-8-21
. D b : Hp . D . j)*k sin R**/)*tc
(1)
b. *24-21 .
D b . R“p‘K n((I*R- R“p*k) « A
(2)
b . *73-83 .
*24-492 . *73-801 . D
b : Hp . D
.p*K-R**p*K=/3-a*R.
[*24-21] D
. P *k r\ (G*R - R** p *k) — A
(3)
Ml).(2)
. (3) . *73-7 . D
b : Hp .
D.p**yj (CI*R-R**p*K) sm R** P *k u ((1*R -
[*73-84]
D . /9 sm ll**p*K v (CI‘7? — R**p*k) .
[*22‘92.*37’16]D . /3sm a *R (4)
b . (4) . *73 2 . D b . Prop
*7385. h:J2cl—»1.CI‘J2C/3 .j8C T)‘R . D . /9smQ‘/J./9smD‘7i [*73841]
*73 86. h :d‘.RCD'S.<3*5CD‘7J.:>.
„ D‘(/j | S) = d -it .(i‘(R\S)c a ‘s. a<s c i>‘(r i s>
Dem.
I-. *37-321 .Dh Hp . D . D‘(7?15) - D*R
b . *34-36 . D b : (l*(R | S) C <1‘S
Ml) • DhHp.D.a‘5C D‘(/e j S)
H • (1) . (2) . (3) .DK Prop
*73 87. b : 5 e 1 -* 1 . Ci*R C D*S . d‘S C D*R . D . D‘/£ sm D *S
Dem.
K . *71-252 . D h : Hp . D . 72 | £« 1 -» 1 .
[*73-86 85] D . d*S sm D‘72 .
[*73-2] D . D'Ssm D‘72 : D b. Prop
*7388. b : a sm y . /3 sm 8.7C/3.8Ca.D.asm/3
Dem. #
H . *731 . D h : Hp . D . (g/*, 5). R, Se 1 -► 1 . D *R - a . (J*R = 7 .
D‘£ = £ . d‘£ = 8. 7 C /3.5 C a .
[*73-87] D . (a«, 5) . D‘7? = a . D*S = £ . D‘72 sm D*S .
[*13 22] D. asm/Sob. Prop
This is the Schroder-Bernstein theorem.
( 1 )
( 2 )
(3)
30—2
*74. ON ONE-MANV AND MANY-ONE RELATIONS
WITH LIMITED FIELDS
N n in mnI'/ of' *74.
The purpose of the present number is to collect together various propo¬
sitions in which we have such hypotheses as
/{[ \( 1 —*CIs. * 1 lie I -> CIs. etc.
•»r in which such hypotheses are shown to he dcducible from others. Hypo¬
theses of this kind occur very frequently, and it is important to be able to
deal with (hem easily. For the sake of completeness, we shall here repeat
propositions previously proved on this subject.
The propositions of this number are mostly of the nature of lemmas, to be
used in the theory of selections (Part II. Section D), and in cardinal and
ordinal arithmetic. The most useful of them are *74772 , 773 , 774‘775. These
•• # w v
propositions are concerned with circumstances under which Q It or | R, with
or without some limitation of the converse domain, is a one-one relation. The
reason they are important is that the correlators by means of which many of
the fundamental theorems of cardinal and ordinal arithmetic are proved are
such relations as (] It (with the converse domain limited) for suitable values
of (] and It. The above-mentioned propositions are as follows:
*74 772. h (j ) . E ! Q*x : <//>. E ! /{<•/ : Q, R « CIs 1 .O . Q j| R e 1 -► 1
The hypothesis of this proposition will be verified if we put, for example,
Q = It ** i x. Thus (l«)||(Cnv‘ ^x)c 1 —> I. This proposition is used in
*1 Hi’531, which is used in proving one of the formal laws of exponentiation,
namely /z Txr x v™ — (/a x n) v .
*74 773. b:Q [a. Iif0€ CIs-* 1 .a C d‘<?. 0C d‘/f. s‘D“\Ca.s‘(I“\C/9.D.
(Q || H ) r x € 1 -♦ 1 . (Q B Z) r A € 1(0 II R)“\\ sm X
This proposition is used in connection with both cardinal and ordinal
multiplication and exponentiation. If Qf“a and Ii[ 0 correlate y with a
and 8 with 0, then if we take for X the class of all ordinal couples that
can be formed of an a and a 0, (Q R)“\ will be the class of all couples
that can be formed of a 7 and a 8. Thus in virtue of the above proposition,
if 7 is similar to a and 8 is similar to 0, the class of ordinal couples formed
of a 7 and a 8 is similar to the class of ordinal couples formed of an a and
a fS. This result is useful because we define the product of the number of
members of a and the number of members of 0 as the number of ordinal
couples formed of an a and a 0.
SECTION C] ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS 469
*74 774. h if e Cls -> 1 : (y) . E ! R‘y : 3 . | R e 1 —> 1
This proposition is useful when, for example. It is j.-.
*74 776. h : Q C s‘D“X, if [ s‘CI“X e Cls -* 1 . s‘D“X C CI‘Q . s‘(r“X ca‘K.3,
(Qll .(<21,B)|*Xe {(<3||«)“X) sm\
This is a particular case of *74 773, and has similar uses.
*741. b -.-.Rffft l-»Cls. 3 Rf/Be 1 -» 1 y.ztg .R‘y= li‘t j=z
Dem.
K *71-55. 3 I-:: Hp .
D (if r 0) f/9 e 1 -» 1 . h : y ,» 0 . (if [ 0)-y = (if f 0)‘z . 3„, ,. y=z:.
[*35-31-7] 3:. if |79« 1 —*l. = :y, t e 0. R‘y -- R'z . 3j,_,. y = « :: 3 f . Prop
*7411. h if [■ /9 f 1 —» Cls . /3 C (I‘if . = : E !! if“/9 [*71571 . (*37 05)]
*7412. b::R[R e l-»i .ffC CI‘if. s s. y, * < 0. 3,.,: R‘y - if‘». s . y = *
[*71-59]
*7413. 1-: if « 1 —» Cls . 3 . (if), [ Cl‘D‘if e 1 —» 1 [*72 45]
*74131. h : if e Cls —* 1.3 . if, [ CI‘Cl‘if r 1 —» 1 [*72 451]
*7414. l-:ff«l-*Cls.£ = if“a.D.a1/f = if[-/3-a1if|-/9 [*72 55]
*74141. h:if«Cls-»l.a=if‘‘/9.3.a1if = if r/3 = «1«r/3 [*72-551]
*7416. h:Q|-^«l-»Cls.X = Q‘‘*.3.*nD‘Q = Q“X [*72-57]
*74151. h:*1Q e Cls-»l .«=Q“X.3.X«CI‘<2 = <2“*
*7416. 1-: Qf X « 1 —» Cls. k C D‘Q . X — Q“k .3 . * = Q“X [*7415.*22 621]
*74-161. I-:«1 Q*Cls-»l.XCa‘Q.* = Q“X.D.X = y“«
*74-17. I-: Q \Q“k e 1 -* Cls . * C D ‘Q . 3 . * = Q“Q“k [*7416]
•74T71. 1-: (Q“X) 1 Q t Cls -» 1 . X C a-Q . 3 . X - Q“Q“\
*74-2. H:Q“aC/3.3.a1Q = a14>ri8
Dem.
*■ * *37 4 .Dh:Hp.D. a*(a *|Q)C/3.
[*35*454] D.a1Q = a1Qr/3:^ h - Prop
*74-201. b : Q“/3 Ca.3.<2[/9 = a'1Qf'/9 [Similar proof]
*74-21. h . a ] Q = a ] Q [• Q“a [*742]
*74-211. h . Q [ /3 = (Q“/3) 1 Q f" /9 [*74-201]
*74-22. 1-: D‘<2Ca.3.Q = a-|Q [*35-451]
*74-221. l-:(I‘<2C,8.3.Q = Q[-/9 [*35452]
170
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*7423. h:o = (?“y*‘o.D.c1V = Vr , <'“® = o1 , ?rV“« [*74-21-211]
*74 231. I- : /i = <rQ“(3 . D . V f -3 = < V‘‘/3> 1 Q = <Q"3) 3 [*7-4 21-211]
*7424. '■:*-Q"e.0=<'r*-'}-°'\Q = Qri3 = a-\Q[/3 [#7423]
*74 25. h : <?T>9« 1 -*Cls.oC D'y. 3-f>< 0 .3.«1 (? = Qf/9-al
[*741(i-24]
*74 251. t- : o 1 Q « CIs -* ! . 0 C««V. a - If'ff .3.«1<?-(?f/i = a 1Qf- / 9
[•74-Kil24]
*74 26. h : Qt £. 1 -» 1 . a C \VQ . (/“a . =. o] Q e 1 l ./SCd'Q.a- Q“0
Item.
I-. *7 425 . D h : (? f* /9 # 1 1 . a C D‘0. ^ - ^“ 0 .3. a 10 - Q T /9.
^ (1)
H . *37-16 . D K : - £"« .D.£C<I‘Q (2)
h . #7416 . D h : Qf £ « 1 -> 1 . a C D'Q. 0 - Q“a . D . a = (}“$ (3)
h.(l).(2).(3).D
H : QfSe 1 -> 1 . a C 1)‘V. ,3 - £“a . D . a 1 tjt I -♦ 1 ,£C<I •Q.a-Q"# (4)
Similarly
H : a 1 V € 1 —> 1 .0CQ , Q.amQ“0.5.Qf0 t 1 _» i . a C ]VQ. £ = Q“a (5)
h . (4).('»). D H . Prop
*74 27. h:Qr^cl-#l./9« £"tr V*. = . ( Q“0) 1 y e 1 -» 1 . £ C U‘(?
Dem.
Q“B
I- . *74-20 ‘ . D
a
I-: y r £ c 1 -* 1 . Q“0 C D‘l?. £ = </“Q“£. = .
(Q lt &)‘\Q*\-+\.fiC<l t Q.(r0 = Q“& (i)
h . (1). *37 15 . *20*2 . D h . Proj)
*74 271. hsalQcl-tl.a - <?“<?“« . =. Q f Q“a € 1 -> 1. a C D *Q
[,74-26 SgS]
*74 3. H gr e 1 -* CIs: (ga) ./3 = Q“a: D . #‘Q“/9 = £
Dem.
h . *7415 . D V : Q [ p € 1 -> CIs. & = $“a . D . QpQ“fi = <?“(a n D‘Q)
[*37-261] =Q“a
[Hp] = £
h . (1). *1011-23-35 . D K Prop
(1)
SECTION C] ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS 471
*74-301. h a 1 Q e Cls -> 1 : (g/9) . a = Q“0 : D . <?“Q“a = « [Similar proof]
*74*31. h : Q r £ e 1 -* Cls . £ e D‘( Q)< . D .
f3=Q“Q“0.0Ca‘Q.Qtf3 = (Q“/3)'\Q.(Q“/3)'\Q € l-+C\s
Bern.
V . *74-3 . *37-23 . D h : Hp . D . £ - Q“Q“i3 (1)
1-. *37-23 16 . Dh: Hp.D.£Cd‘Q (2)
h.(l).*74-231. Dh:Hp.D.Qr/9-(Q‘W1Q (3)
[*1312] D.(Q“^)1 Qcl-*Cls (4)
H . ( 1 ). (2) . (3) . (4) . D h . Prop
*74 311. 1- : a ] Q e Cls —> 1 . a e D‘Q«. D .
a = Q“f>‘a.aCD‘Q. a 1(2=Qr^ < a-^r5‘ < «^> 8 - > 1
[Similar proof]
*74-32. 1-: * C d‘P . R [ k c Cls -* 1 . D P [ « e 1 -♦ 1
Bern.
. *33 41 . D h Hp . D : y, z c k . R*y = TP* . D . (ga.). .tPy . xRz .
[*35101] D .(gar).* (7?|**)y. x (Pf* k)z ,
[*71171.Hp] Z>.y = * (1)
(1). *71-55. Dh. Prop
*74 4. \-:P\(Q[\)=P\Q. = . Q“(J‘P C X
Bern.
K *35-23. DbiP\(Qf X)«P|Q. = .(P| OTX-PjQ.
[*35-66] = .d‘(P!Q)C\.
[*37-32] = . Q“d‘P CX.Oh. Prop
*7441. H : G‘P n D‘Q C.k.'2.P\k‘\Q=‘P\Q
Bern.
K *3313 131 . *10-23. D
h :. H p . s : xPy . yQz . D,.,.,. y * * :
[*4-71] = : x Py . yQz . . xPy . yQz . y e *c :
# [*10-281] D : (gy). *Py. yQz . = x> ,. (gy) . xPy . yQs .ye«:
[*34-1.*35*1] "5 m .x(P\Q)z. = XfZ ,x(P j *”] Q)* :. D h . Prop
*74-42. h : d‘P C Q “\. D . D‘(P | Q f X) - D‘P [*37 321401]
*74-43. H : Q“X C d‘P. D . d‘(P |Qf*X) = d‘Q « X [*37-322401. *3564]
*74-44. h:d‘P = Q“X.D.D‘(P|Q[-X) = D‘P.d‘(P|Qrx) = d‘Qrt X
[*74-42-43]
172 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
*74 5. H : K ! ( P\ &)'y . = . //c 0 . E ! l u y . = . {P\&) t y=P t y
Dam.
h.*357. D b ir — iPf &)*!/• = •ye& •x = P i y (1)
H . (1 >. *10‘1 1*2*1 . D 1-:. (g.r) .x — {P[ &)*y . = : y c/8 : (gx) ..r =
[*14 >04) D 1-E! (P[ 0)‘y . = . y e £ . E! P*y (2)
K . *35 7 . D 1-: <7^ 0)‘y = P l y . = .y*&. P‘y = P*y.
[*U2H] = .yaQ . E! P*y (3)
K(2).(3).DKPiop
*74 51. V 7"y C o . D : E! (a 1 P)‘y . a . E! l u y . = . P*y = (a ] 7>)‘^
I)cm.
H . *32’1S . *351 . D h :. Hp. D : xPy . = x . x(a 1 P)y : (1)
[*30 34] D : E ! (a 1 P)'y . s . E ! P l y (2)
h . (1). *30341 . Dh Hp. D: E! P‘y . = . 7>‘y =>(a] P)*y (3)
H . (2). (3). D h . Prop
*74 511. K T 7 *.*- C £. D : E! (£] TV.* . = . E! 7~‘.r. = . l”.v = ] P)*x
[Proof ns ill *74-51]
*74 52. H : <6'"£) ] .S'c 1 —► Cl* . /9 C <I *&. ytfi .0 . ft#"#)] Sl'y-S^. E! S‘y
Dem.
V . *37 1H. D h : Hp. D . 8*y C S“/3
(1)
H.*37 1
. DhHp.D. (gx). xSy . x € .S'* 1 /?.
[*33-131] D . .y € (I‘ |(5“/9) 1.
[*71*l(j]
D . E ! |(.S'“/J) 1 A']‘i/
(2)
h. (1). (2). *74-51 .DK Prop
*74521.
h:.S'rN‘‘/**Cls->
1./3CI>‘S. y «/9. D. |(S“/9) 13|‘-/ = S‘y.
E!5\y
[* 74 ' 32 l]
*7453.
1-: (»S“/?) 1 .S'f 1 —*
1 ./3Ca‘S.y€/3.D.6’‘S‘y = y
Dam.
h.*37l .*33131 .
Dh:Hp.D.yja‘|(S“j8)1S|.
[*72 241.*35-51]
D.(3rS“/9)‘{(S‘W1S|‘.v = J /
(1)
h . *74-52.
:Hp.D.((S“/9)1S)‘y = S‘y
(2) .
K.(1).(2).
3 h : Hp . 3 . (3 r S“/8)‘S'y = ,/.
[*35*7]
D . S‘S‘y = y : D H . Prop
*74-531.
./9CD ‘S.yc0.O.S‘S‘y = y
*7453 ^
SECTION C] ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS 473
*74 6. h 7*6 1 —» 1 • \ C CPd*?'. * C CI‘D‘T . D : *• = T<“\ . = . X = ( T)<“k
D em.
h . *37 421 .Dh: Hp . D . T t “\ = ( T t f C)‘d ‘7’)“X .
&)<“*= hHrCYD'T]"* (1)
K. *72-451-52 . D
h Hp. D : * = (T t [ Cl‘d‘7*)“X . = . \ = {Cnv‘(Z\ f* CI'd'DI"* .
[*72-54] =.\ = {(2 ; ) < rCl < D‘y , ; << /c (2)
^ • (1) • (2). D h . Prop
*74 61. b:.Te 1-*1.D:XC CPd'r. * = F“X . = .^C Cl‘D‘7\ X = T"'k
D em.
H . *74 6 . *37 103 . D h Hp . D : « C Cl'D'y*. X C Cl ‘d'T.K = T“ l \ . = .
*cci‘D‘2 \xcci‘d‘y\x- F“* (i)
1-. *371516 .Dh* = T* u \ . D . « C CI‘D‘r: X = 2*“* . D.XC Cl‘d‘7’ (2)
•(!)• (2) . *4 71 . D h . Prop
*74 62. h y, z € £ . y * 2 . D y> , Ts'y rTs*z - A : b . S f & e CIs -► 1
Dem.
. Trnnsp .Dh:.y.«e/5.y + #. D y>/ . tf'y r» «= A : = :
—► —►
y, e e /9 . a ! S‘y o‘r . D y> ,. y — * :
[*3218] =:y,ze (3 . xSy . xSz . D z ,„, z - y =* * :
[*35101] = : *(S r>3)y .*(S - y- * =
[*71171] = : S [• /9 « CIs —* 1 :. 3 t-. Prop
*74 63. h:.P,Q < X.P + 4».D / .. (? .D‘P«D‘Q = A: = .e|Drx«Cls-»l
[*74-62. *72-27]
*74 631. Hs.P,Q e X.P + Q.D f . (? .a‘Pna‘Q = A: = .«|arx«Cls-» 1
[*74-62. *72-27]
*74 632. h:.P,Q f X.P*Q.D,.<,.C‘PnC‘Q = A: = .PrXeCls—»1
[*74-62. *33-5]
*74 7. h : Q « 1 —♦ CIs .P\Q = P‘\Q.O.P[ D‘Q = P / [D‘Q
Dem.
h.*34-27.Dh:Hp.D.FQ|Q = F|<2|Q.
[*72-59] D . P r D‘Q = F |* D‘Q : D 1-. Prop
*74-701. h : Q * CIs —► 1 . Q | P = Q | P'. D . (CPQ) "] P = (<I‘<2) 1P"
*74-71. H:.Qel—»cis. d‘P C D ‘Q . d'-PX D‘Q .3:P|Q-P'!Q. = .P=P'
[*74-7 . *35-66 . *34 28]
*74-711. I-:. Q « CIs—»1. D‘P C Q‘Q. D'P'C Q‘Q .3:Q|P=<2|P'. = .P = P'
474
PROLEGOMENA To CARDINAL ARITHMETIC
[PART II
*7472. h y c 1 —»(.Ms: 7*c X . Dj.. <1*7'C D*y: D .( Q)[\e( Q“\) srnX
l)em.
h .*74 71 . D H :: Hp.D P.P'eX . 7* Q = i > ' Q.= .P = P' (1)
H . < 1 >. *73’28 . D h . Pmp
*74 721. hs.yeCIs-* 1 : P € X. D,.. D‘7* C <I‘Q: D. <<? )|* X «<y| “X) sm X
*74 73 K : y « I —> Cl*..*.*< I“X C D*y . D. ( y>| k \€( Q“\)siiiX
| *74 72. *40 43]
*74731. Hy*n*-> 1 . A *i>‘*xc<i*y. :>.<y >rx€<y “x)srax
*7474. h:yt 1 -»ri«.(IWXCD'Q.D.< y>fXc( y*‘X)smX
(*74 73. *41 44]
*74741. l-:yffClN-*l .DVXC(I'Q.D.(Q Xc(Q “X)smX
*7475. H : a 1 y c 1 —> CIs . a C l)*y . s*<I‘*X Co. D .( y)f*X € ( Q“X)8U>X
I Jem.
h . *40-43 . D h :. Hp. D : PeX . D/>.0*7'C a .
[*43-481]
[*37 09] D: Q“X - |(a 1 Q)“X (1)
H. *43-491 . Dh:Hp.D.( y)TX-| (aiyi^X (2)
h . *74-73 . *3.VG2 . D K : Hp. D . » (a 1 Q )J [ X e [|(a 1 Q)“X] 5ni \ (3)
h.(1).(2).(3). D y . Prop
*74751. H : y p a e CIs —* 1 . a C H*y • s‘D“X C a . D . (Q ) f* X f (Q “X)smX
[Proof as in *74 75. using *74*731. *43 48 49]
*7476 hsQcCIs-*! .Jtfl-»CI*.Q P\R = Q\P'\li.D.
(U t Q)‘\P[D , R-(G‘Q)‘\P'tiyR [*74-7-701]
*74 761. y :. Hp *74*76. D‘P CWQ.Q'P C IVW.D'i" C Cl*y.<J‘7>' CD <R.D:
Q\P R-Q P'\Ji. = .P=P' [*7471-711]
*74 77. y : Q, Re 1 -* CIs. s‘D*‘X C D‘Q . $• CI“X C D*77. D .
( y II70 r X «1 -* 1. (QII«) r X e [(§ II5m X
Dem.
I-. *74*761 ^. *40-43 . D
hrsHp.Dr.P./'eX.Dsy P | 7? « Q j F \ R . = . 7^= P ':
[*43112] D : (Q || 70*7^ = ( Q II RYP' . = .P = P' (1)
H. (1). *73*28. DK Prop
*74 771. y : Q. R * CIs -* 1 . $‘D‘*X C (I‘Q. s‘d“X C 0*7?. D .
(Q||H)rXfl-*l.(Q|4)rx*{(Q||«) ## Xl5SBX
[*» n £4]
SECTION C] ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS 475
*74772 and its immediate successors are of very great use in cardinal and
ordinal arithmetic.
*74 772. f :.(*). E! Q'x : (y) . E ! R'y : Q, Re CIs —»l:D.Q||fiel—»1
[*74 771 .*33-431]
*74 773. H : Q [ a, R [(3s CIs -► 1. a C d‘Q,/9 C d‘«.*«D“X C a.s‘d“X C/3.D.
(Q||«)r\el-»1 .(Q„K)rx*[(gil«)“X!sm\
I-. *35-64 . D I-: Hp . D . s‘D“X C d‘(<2 1- a) . s‘d“X C d‘(7* |-/3) (1)
h.*43-51.DI-iHp.D.{(Q[-a),;(/3li<»rx = (g «>rx (2)
I- . (1) . (2) . *74771 .31-. Prop
*74 774. H R « CIs —» 1: (y). E! R'y : 3 . | R « 1 -♦ 1
Dein.
H. *71-166. D h : H p . D . /£ f CIs —> 1 (1)
h . *33-431 . D h : Hp . D . {P) . (T'P C D‘7* (2)
w
K . (1) . (2). *7471 ^. 3 H s. Hp. 3 : P \ R - F \ R . m r , r . P = P'(3)
K (3). *71-57. 3K Prop
*74 775. I-:Q[ s‘D"X, R f «‘d“X « CIs -♦ 1 . s*D“X C d 'Q. s' d“X C Q'R . 3 .
(Q || R) [ X <r 1 -♦ 1 . <Q i| R) [ X « |(Q ]| £)“X] sm X [*74773]
*74 8. 4 : (/3 u 7 ) e 1 -* CIs . = . 7? r A •« T 7 « 1 -* CIs
Dem.
H . *71-572 . 3 I-: R u y) , 1 -»Cls. e : y « d ‘R « </9 o 7 ). 3„ . E ! R'y :
[*22-68.*10-41] = : y e d‘72 «/3 . 3„. E! R'y : y e d‘7i o 7 . D„. E ! 7J‘y :
[*71-572] iJCyel—♦ CIs:. D K Prop
*74801. l-:(/9c 7 )1R e Cls-»l . = ./91ie, 7 1iieCls-»l
*74-81. I-: s'k e 1 —» CIs . = . R f “* C 1 -♦ CIs
Dem.
h . *71-572. 3 h Jlf#** e 1 —»Cls. = : y e Q'R n s'k . 3„. E! 7i‘y :
[*40-11.*10*35*23] = : a € k . y e Q'R r> a . D«. y . E! 7£‘y:
[*ll-62.*71-572] = : ae *. 3. . «r a« 1 -» CIs i
[*37-61] = 2 7J f “k C 1 —» CIs :. 3 I-. Prop
*74-811. I-: («■*) 1 R s CIs —♦ 1. = . 1 R“k C CIs -> 1
*74 82. h 2 09 « 7 )1 R «1 -> CIs.=. £1 R, 7 1 R e 1 -♦ CIs. R“(0 - 7 ) o R"y = A
Dem.
K *351. *7117.3
H w *y) 1 ^ e 1 —► CIs . = :.ar, y e & v y. xRt . y/te. * x = y
[*1312] Di.xeP .yey . xRz . y/k . 3,.*., . * Ts.YjJ-^C £
17 0
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
[Tmiisp]
[*10*21 '252]
[*10*28.*37*10.>]
[*24*39]
1* .<11. *71*22.3
3 .r e & — 7 . xRz . 3 ry> ,. ^(y e 7 . yRz)
D .r c (3 - 7 . xRz . 3 Z ... May) • y f 7 • '/#- s-
3 z c - 7 > • =>.* • *'^“7 : *
D:. 7i t ‘i0-y)*R“y = A
( 1 )
t-:(#vy) 1 /<« I -► CIs ./<“<£-7> * A“ 7 = A (2)
1*. *71*22.3 h i&^R* 1 ->CU.3.<0- 7 >1 /?e 1 ->Cls (8)
I- . *37*4 . 3 h : A*“<^ - 7 ) a R“ y = A . 3 . (I‘(£ - 7 ) J R r> <I‘< 7 "] A) = A (4)
H.(.S).(4).*7l-24.DI-:/2V^7l^< 1 -> CIs. R“(0 - y) n R“y - A . 3 .
(^- 7 )1 A c/ 7 |/*€l ->Cls.
[*35*41] 3.<£ w 7 )*|/fe 1 —>Cls (5)
1- . (2). (5). 3 H . Prop
*74 821. »- : /^i/jw 7 )*ris-» 1 . = .
Rf0. ^r 7 «CI*-*l .R“{/3-y)rs R“y - A
*74 822. l*:(£v 7 )1/f« I -♦ 1 . b .01 A. 7 I Ac 1-» 1 . R“(0-y)"R“ 7 -A
[*74 82*801]
*74 823. 1- : /*[><** v 7 )c 1 -♦ I . = . /<[*£■ 7^7* 1 -♦ * • /*“(£-7>" H u yA
[ *74*8*821]
*74*83. 1* /7“tf * /f“ 7 = A . 3 : (rf v 7 ) 1 Rt 1 —>Cls. = ./*] /A 7 1 R « 1 ->C1»
[*74*82]
*74 831. R“prs R“y= .\.0:R[(f3 v y)€C\s-+ l.m.Rfff, .RfyeCh-f 1
*74*832. I-/<“£ a £ “ 7 - A . 3: <£ v 7 ) 1 R1 1 -> 1 . 2 . & 1 R. 71 * « 1 “* 1
[*74*83*801]
*74*833. 1- R“f3 /% R“y - A . 3 s 7? f (£ * 7 ) « 1 -> 1 • = • R t A * T 7 « 1 1
[*74 8*831]
*74 84. h s.(«*#c) 1 Re 1 —>Cls. s :
1 R“«C 1 —» CIs : ( 8.7 «*. 3». T . — y)r> X“<y “ A
Dem.
h.*4013.*35*43.Dh:/3<^.D./91/JC(^)1i?:
[*71*22] 3 P:.(«St) 1 i?€l-*Cls. 3 :/ 3 €*. 3 .£*| ftfl ->Cls:
[*37*61] D: 11?“* C1 —»CIs (D
h. *72*41. *37*4*21.3 1-:. (s‘/c) 1 /* € 1 -» CIs. 3 :
/9. 7 **.3,. Y ..R‘‘<0- 7 )*iK==A (2)
K *37 *105. *24*39.3
1- :.f3,y€*.O li .y.R“(8-y)''R“y = Ai = i
/3,y€*.xf0-y. xRz . 3*. T . ~ (gy). y e y • yRz '
SECTION C] ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS 477
[Transp] D : 0, yezc.xe0.yey. xRz . yRz . Y . .<• e y .
[*4.-7] D fiy . .v, y ey. xRz . yRz.
[*351] ^ 3*.Y.*(7l<R>W(7lfl>* ( ; *>
H . (3).*7117 .Dh./9, 7 f /c. D„ >Y . - 7 ) *R“ y = A : *| C1-> CIs: D:
0,y e tc . x e 0 . y e y . xRz . yRz . Ofi.y.x.y.z • x = y :
[*10 23.*4011.*37*1] D : * ((s‘*) ] /if * . y ] rtj 2 . D,.,,.* . u: = y :
[*7117]
D : s‘/c 1 R e 1 -► Cls
(4)
1". (1) . (2). (4) .3h. Prop
*74841.
R[ s*k e Cls —♦ 1 . = :
R\“k CCls-*l i0,y eK.Dty. R“(0-
7 ) r* R if y = A
*74842.
Re 1 —* 1 . = :
] R €t K Cl—>1 : 0,y e k . Op,y • R“(0 — 7 ) r\ R €t y = A
[*74-84-811]
*74843.
1" R T e 1 -♦ 1 . = :
R [•“« Cl-»l:i 9 , 7 f<. D,. v . R l \0 - 7 )" /*“7 - A
[*74-81-841]
SECTION D
SELECTIONS
Summary of Section I).
Tin.* subject lo hi* considered in tliis section is important chieHy in
connection with multiplication, both cardinal and ordinal. In order to get
u definition of multiplication which is not confined to the case where the
number of factors is finite, we have to seek a construction by which, from
a given class of classes, k sav, we construct another class which, when k is
finite, has that number of terms which, in the usual elementary sense, is
the product of the numbers of terms in the various classes which are members
of k, mid which, whether * is finite or not. obeys as many as possible of the
formal laws of multiplication. The usual elementary sense of multiplication
is derived from addition; that is to say. p x v is to be the number of terms
in 8 , k. where * is a class of p mutually exclusive classes each having v members,
or vice versa. This sense can he extended to any finite number of factors,
but not to an infinite number of factors; hence for a number ol factors which
may be infinite we require a different definition, nud this is derived from the
theory of selections.
Selections are of two kinds, selections from classes of classes, and selections
from relations. The latter is the more general notion, from which the former
is derived. But as the former is an easier notion, we will begin by explaining
selections from classes of classes.
Given a class of classes *. a class /1 is called a selected class of * when
H is formed by choosing one term out of each member ol k. For example, if
k consists of two members, a and and if * € a and y c &, then i‘xu i y is
a selected class of k. If every constituency elects a local man. Parliament
is a selected class of the constituencies. If * is a class of mutually exclusive
classes, i.e. a class no two of whose members have any member in common,
then a selected class consists of only one term from each member of *; t.e. /i
is a selected class if
/l C s*k 1.
But if * is not a class of mutually exclusive classes, this does not hold
necessarily; for a term x which is a member of both a and B (where a,Be *)
may be chosen as the representative of a, while some other term may be
chosen as the representative of B. so that two members of B »“ a y belong
to the selected class. Again, if * is a class of mutually exclusive classes, the
relation of the representative to its class must be one-one, because, since no
term belongs to two classes which are members of *, no term can be the
SECTION D]
SELECTIONS
479
representative of two classes. But when k is not a class of mutually exclusive
classes, a term which belongs to two classes a and 0 may be chosen as Un¬
representative of both. Thus the relation of the representative to its class
may be only one-many, not one-one.
The relation of the representative to its class may be called a selective
relation. A selective relation of * is one which selects, from every class
a which is a member of k, a certain member x as the representative of a:
that is, we have, if R is the selective relation,
a € k . D« . R*a e a : (1*72 = k.
This condition is equivalent to
R e 1 —► Cls . R G € . Q* R = tc.
If R is a selective relation, D‘R is a selected class; and if p is a selected
class, there is a selective relation R such that Thus the study of
selections from classes of classes is wholly contained in the study of selective
relations.
The class of selective relations from a class k is called ( A ‘tc. Thus
R t € A ‘k . =. Re 1 —► Cls . RQe. Q *R = k,
- (1 -► Cls) Rl'c « <P*.
Then I)is the class of selected classes.
It will be seen that, if a etc, R‘a may be any member of a, and we get
a different R for each different member of a. Thus if we keep the repre¬
sentatives of all the other members of k unchanged, the number of selective
relations to be obtained by varying the representative of a is the number of
members of a. Hence the number of selective relations altogether may
be fitly defined as the product of the numbers of terms possessed by the
various members of k. In case k is finite, this agrees with the usual definition
of multiplication; and whether k is finite or infinite, the product so defined
obeys all the formal laws of multiplication.
To illustrate the notion of selective relations, let us take a very simple
case, the case where k consists of two classes a and 0, each of which has two
members. Let x and y be the members of a, z and w the members of / 3. We
assume a=$=£, x^y, z^w. Then the selective relations of k are the following:
x lav z l
x l av w l (3,
y lav z 10,
y l av w l 0.
Thus they are four in number, x.e. the number of members of is the
product of the number of members of a and the number of members of 0.
A similar process would show that our definition of the product agrees with
the usual definition in any case in which all the numbers concerned are finite.
18U
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
Selections from relations are an obvious generalization of selections from
classes of classes. We had above
CIs) a Rl‘e a
.CIs)A Rl«i* a a**.
€** = (l
We put, generally,
which we derive from the definition
= \ k IX = (I -♦ CIs) a HI *P All 1 *! Df.
This is the fundamental definition in the subject of selections. We have, in
virtue of this definition.
1-8
When *-<l‘/\ we may call /V* the class of selections from P. Thus
generally. /V* i* the class of se lections from /'[** provided and it
this condition is not fulfilled, PS* « A. We may call the class /V* the
class of ••/‘•selections from The class of "c-selections from *" will be
what we previously called the class of - selective relations of
It will be observed that we have
li € Pa* . i/t* .0 . lt'a « P'y-
Thus if -P“k is a class of mutually exclusive classes. D*R selects one Iroin
each of those classes, and is therefore a selective class of P“k\ hence in this
case
D“/V* - D“c A , y ,< ^.
In Cardinal Arithmetic. is the important notion, and the more general
notion /V* is seldom required. In Ordinal Arithmetic, Fa 1 * is the important
notion. It will be seen that
R € Fa** . s . /? « 1 —* CIs . HQF ,Q*R = *.
Thus Fa* is only significant when * is a class of relations; in this case wo
l,UVC RtFA'x.Qex.l-R'QcC'Q.
Thus li chooses a representative member of the field of every member of *.
The most important case is when k is of the form C‘P, where P is a serin
relation whose field consists of serial relations. Then F A ‘C*P becomes the
field of a relation which may be defined as the ordinal product of the relations
composing C‘P; in this way we get an infinite ordinal product analogous o
the infinite cardinal product. This will be explained at a later stage
Although it is chicHy and FS* that will be required in the sequel,
we shall treat IV* generally, because this introduces little extra complicate ,
and most of the theorems which hold for or FV* have exact analogues
for Pa 1 *-
SECTION D]
SELECTIONS
481
as above defined, is the class of one-many relations contained in P
and having k for their converse domain. We know of no proof that, there
always are such relations when * C Q*P. In fact, the proposition
* c a *p . ,. a!
is equivalent to the “ multiplicative axiom,” i.e. to the axiom that, given any
class of mutually exclusive classes, none of which is null, there is at least one
class formed of one member from each of these classes. (This equivalence is
proved in *88 36, below.) It is also equivalent to Zermelo’s axiom*, which is
(a) . a ! e A ‘Cl ex‘a ;
hence also it is equivalent to the proposition that every class can be well-
ordered. In the absence of evidence as to the truth or falsehood of these
various propositions, we shall not assume their truth, but shall explicitly
introduce them as hypotheses wherever they are relevant.
In the present section, we shall begin (*80) by considering such properties
of P a ‘k as do not depend upon any hypothesis as to P. We shall then
(*81) proceed to consider such further properties of P a *k as result from the
hypothesis /'f k « Cls —* 1. This hypothesis is important, because it is verified
>n many of the applications we wish to make, and because it leads to important
properties of /V* which are not true in general when P is not subject to
any hypothesis. These special properties are mostly due to the fact that
when P[k is a many-one relation, P a *k consists of one-one relations (not merely
of one-many relations, as it does in the general case). This is proved in *811.
We then (*82) proceed to consider the case of relative products, i.e. (P\Q) A ‘\.
It will appear that, with a suitable hypothesis, (P \ Q) A ‘\ = | Q“P A ‘Q“\ and
D“(P | Q) a ‘\ = D“P A *Q t *\. In the following number (*83) we apply the
results of *80 to the particular case where P is replaced by e, which is the
important case for cardinal arithmetic. In *84 we apply the propositions of
*81 to the case where P is replaced by c, and where, therefore, we have the
hypothesis <• [* * e Cls —► 1. This hypothesis is equivalent to the hypothesis
that no two members of k have any members in common, i.e. that
a, e k . a 4= /3 . D.,* . a r\ (3 «= A.
When k fulfils this hypothesis, it is a class of mutually exclusive classes.
For classes of mutually exclusive classes we adopt the notation “Cls’excl.”
It is shown in *84’14 that a Cls* excl is one for which we have c \ k e Cls —» 1.
When k is a Cls* excl, D f is a one-one relation, and D“€ A *k sm € A ‘k.
Also in this case T) lt € A f K consists of all classes formed of one member from
each member of *, i.e. all classes ft such that
ft C 8*/c zaetc.Da.ftnac 1.
Bee hU "Beweis, daea jede Menge woblgeordnet werden kaon,” Hath. Annaltn, Vol. Liz.
PP. 614—616.
BfcW i
31
182 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
In *85, wo prove various important propositions, of which the chief is a form
of the associative law*, namely
h : k € Cls’excl. D . sm
Finally, in *88, we consider the question of the existence of selections. This
cannot in general be proved when < is an infinite class. The assumption that
€ A ‘/c is never null unless one member of * is null is equivalent to various other
assumptions, for example to the assumption that every class can be well-
ordered. One of these equivalent assumptions is called the “ multiplicative
axiom.” This axiom is equivalent to the assumption that an arithmetical
product cannot be zero unless one of its factors is zero, and is regarded by
some mathematicians as a self-evident truth. This can be proved when the
number of factors is finite, i.e. when * is a finite class, but not when the
number of factors is infinite. We have not assumed its truth in the general
case where it cannot be proved, but have included it in the hypotheses of all
propositions which depend upon it.
• Cf. note# to •12*1*11.
*80. ELEMENTARY PROPERTIES OF SELECTIONS
Summary of * 80.
In this number, we shall give such properties of P A as follow most directly
from the definition, without any restrictive hypothesis as to P.
If ReP a ‘k, R selects one member of P‘y, whenever ye*, as the selected
referent of y . For, since R e 1 Cls . d*R = *, we have y e * . D . E ! R‘y ; and
since RQP, we have y e * . D . (R*y) Py, i.e. y e * . D . R‘y e~P‘y. Calling R*y
the selected referent of y, it is evident that we may replace R*y by any other
member of P*y, and still have a member of /V*. (This is proved in *80 4.)
Thus il /V* has any members at all, we can get as many members as there
are members of P'y by merely altering the selected referent of y, leaving the
other selected referents unchanged.
In the present section, we first prove various simple properties of iV*.
Most of these are almost immediate consequences of
*80 14. V : R e /V* . = . Re l—> Cls . RGP. d'R = *
The most useful of them are
*80 2. h : a ! /V* . D . * C C VP
*80291. ViR€P a *k.1.RG.P[k
*80 3. h : R e P a ‘k . y e *. D . E ! R*y
*80 33. \-iRe P a <k . D . D‘R C P‘‘k
We then have various propositions (*80 4—46) concerned with x J, y when
x Py- Of these the most important are the following :
*80 41. V : R e P A < K .yeK. x'Py . D . [\R^-{R‘y) l y\ v x'l y] e P a ‘k
I.e. given a selective relation R, the selected referent of y (where yeQ.*P)
may be replaced by any other term having the relation P to y, and we shall
still have a selective relation.
*80 45. h . P A Vy = l y“~P‘y
We then have a set of propositions (*80 5—*54) connecting (Pc/Q) 4 ‘(/fu\)
with P a 1 k and Q A ‘\. These are chiefly useful as leading to the next set
(*80 6—-69), connecting P a ‘(k v X) with P a *k and /VX. The most useful of
these are the following:
*«0 6. h-.RePSx.XCK.O.RfXePSX
*8065. h : * « X = A .Re P A ‘x .SeP A ‘\ .D.R oSe? 4 ‘(« u X)
*80 66. \-:.K*\=A.DzMeP A *(Kyj\). = .(&R,S).ReP A ‘K.SeP A ‘\.Af=RvS
31—2
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
184
We have next a set of propositions (*80'7—78) dealing with the relations
of if and il-R when (e.y.) J/ t P 4 ‘(««X) and RePS«. These propositions
are seldom used, but they would be useful in considering division.
We next have a set of ]>ropositions (*80-8—-84) dealing with the relations
of I\‘a and 7V/9. The most useful are
• 8081. h : g ! TV« . TV* = /V£ ■ 3 ■« - $
*8082. h : a =j= /3.3 . W« a 1\‘,3 = A
Finally, we have four propositions (*80 9—93) on ? 4 '(i‘j/vt‘t) and one
.... /V(£vt‘*)' The
most useful of these is
*80-9.
h :.//+ z . D :
McPS(Py v = .<3«m0-
uPy.vPzly
*8001.
(1-*CIs)a BPPntv*\ Df
•801.
1- : X P A * . 2
.\.(1-»CIs)aRI*/'aU<«
[*21-3.
(*80 01)]
*80 11.
h . PS* = (1
-*CI*)aRI«/ J aCT«*
[*801 .
*303]
*80 12.
H . E! /V*
[*80*11
.*14*21]
*8013.
1-: X P a k . =
. x = PS*
[*8012
. *30-4]
*8014.
b:R< PS*.
= K
[*80l1 .*20
•43 . *22-33. *61 2 . *33 61]
*8015.
ViPdQ.O
. PS* C QS*
[*8014]
*8016.
h : li <• PS* .
HQQ.D. lie QS*
I- .*80-14.31- : R tPS* . 3 . Rt l-»Cls.CI‘K- k :
[Fact] 3 I-: R'1\‘k.R CQ.3. fir 1—*Cls.CI*« = *
[*8014] O.Rt QSk : 3 I-. Prop
*80 17. h : Q C 1‘ . 3 . = /V* a RI*Q
RCQ-
h.*80-15. 3h:Hp.3.Q*‘*C7V* (1)
h. *80-11. 3 I- • Qa‘* C RI‘Q ( 2 >
h . (1). (2). 3 h : Hp . 3 . c /V* a R1*Q (3)
h . .80-10 . 3 h . TV* a RI*Q C <&** (+)
h. (3). (4). 3 h. Prop
This proposition is used in the theory of ordinal multiplication (*172162).
*80 2. 1-: a ! P* K • 3 - * c a</>
Dem.
h. *80-14.31 ■:R ( P^K.O.ReP.a i R = K.
[*33-264] 3.a‘flca‘P.a‘fl = *
[*13-13] 3.*Ca‘P
K(l).*10-11-23.3 1-. Prop
(1)
485
SECTION D] ELEMENTARY PROPERTIES OF SELECTIONS
*80 21. hj-^c d‘P) . D . PSk = A [*80*2 . Transp]
*80 22. V : P r /c = Q f* *. D . P A ‘* = Q A '<
Dem.
H . *3314 . D H :: Q.*R = k . D xRy . D • y e k
[*5-44] D a-Py . D . o-Z^y : = : xRy . D . xPy .ye/cz
[*35-101] =:.r Ry.D.x(Pt*)y (1)
H.(l). *1111-3-33. D
h d‘Zi = *.D:PGP. = .PGPr* (2)
l-.(2)^.Dh:.a«fi = *.D:iiCg. = .iJGQr* (3)
M2). (3). *1312 .Dh. CI‘.R = x . P{ * - « . D : J{ G P. = . IIG Q (4)
H . (4) . Comm . *5 32 . D
h Hp . D : P G P . d‘Z* x.s.ACQ.a'A-x:
[*8014] D : Zic P A ‘* . = . R c Q a ‘ac D h . Prop
*80 23. h . ZV* - (Pf *)*'«
Dem .
h. *35-31 . *22-5 .Dh.Pf*Ac = (P [•*)[** (1)
h. (1). *80-22. DI-. Prop
*80 24. h/cC d‘P. Q-Pf*#c. D . ZV* =. QSd'Q [*35-65 . *8023]
*80 25. V : g! iV* .Q-Pf**.D. P*‘* = Q A ‘d‘Q [*80-224]
*80 26. V . P a ‘A - t‘A
Dem.
V . *8014 . D h : P e ZVA •■•Pci—* CIs .PGP. d‘P = A .
[*33*241] ■.Pci-* CIs . RQP. R = A.
[*13*193] = . A c 1 —» CIs . A G P. P = A .
[*72*1.*25*12] s.P- A.
[*51*15] ■ . P c i‘A s D h . Prop
Note that P A ‘A is a unit class, not the null-class. It is owing to this fact
(as will appear later) that, if y is any cardinal, y° = 1. See the note to *83*15.
•80 27. h : a ! ac . D . A a ‘/c = A
Dem.
h . *8014 . D 1- s P c ASk . D . P G A . d‘P = * .
[*2513] D . P = A . d‘P = ac .
[*33-241] D.k = A (1)
I- .(1) . Transp. *1011-21. D
h^lx.D. (P). P ~ c A &K .
D . A a ‘ac = A : D I- . Prop
[*24-15]
480
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*8028.
I Jem.
*8029.
Deni.
*80-291.
Dem.
*803.
Dem.
•8031.
Dem.
*8032.
Dem.
*8033.
Dem.
*8034.
Dem.
P : a ! k . D . P±k
P . *8014 . Dh:.g!/r.D:/?€ /V* . D /; . g ! G‘P :
[*83-241 ] D : R *- 7 V* . D*. g ! R :
[*25 63] D : A^c P&tc D P . Prop
P:
P . *80’14 =
[*35-452] D . 77 = 7f|* * s D P . Prop
I -iRcPSsc.Z.HGPt*
P .*80*14.*33*14. D
P : • H p. D : jr Tty . D x§ y . rPy. y e k •
[*35*101] D XiV . D P . Prop
I" s 77 f 7V* . y € k . D . E ! IVy
1-. *8014 . D P : Hp. D . 7? * 1 —* Cl*. y € (I‘P.
[*71*163) D. E! R*y : D P . Prop
P: R€P A ‘K.ye*.0./Vy€P*y
P . *80*14. D P : Hp. D. R < 1 -* Cl*. R Q P . y e (l‘R .
[*71*31] D . 7? G 7'. ( IVy) Ry .
[*23-441] D . ( R*y) Py •
[*32*18) D . R‘y e P*y : D P . Prop
P R € TV* . D : y e k . = . E ! R*y . = . R*y c P*y
b . *80*14 .DPs. Hp. D s (I‘77 = *:
[*33*43] D:E!77‘y.D.ye*
P.*14*21.DP: J7‘y€?‘y.D.E!/*‘y:
[(!)] D b Hp. D : R'ycP'y . D .ye*
P.(1).(2). *80-3*31 .DP. Prop
P: J*6/V*.3.D‘flCP“*
P . *80*14. *37*25 .DPs Hp. D. D‘J7 = R“k .RQP.
[*37*201 ] D • D‘77 C : D P. Prop
P : P € P A ‘* - 3 . K !! R“k . /?“* = D‘77
P . *8014 .DP: Hp . D . 77 e 1 —* Cls. <P77 = * .
[*71 16.*37-25] D . E !! 77“*. 77“/c = D‘77 : D P . Prop
SECTION D]
ELEMENTARY PROPERTIES OK SELECTIONS
487
*80*35. h : R e P A ‘*. D . D *R = £ |(gy) . yeK .x= R‘y] [#376 . #80*34]
#80 36. V : R, S e P A ‘* . R[ av Sf - a e P*‘k
Dem.
H. #71*26. Dh: Hp . D . P [ a, Sf* - a e 1-> Cls (1)
h . #35*64 . D V . a\R [a)rs C l*(S [* - a) = A (*2)
H . (1). (2) . #71*24 .DI-: Hp — afl—> Cls (3)
K . #35*64 . #80*14 . D h : Hp . D . (I‘(P [ a) = * ^ a . d*(S f - a) = * - a .
[#24*41] D.a*(/erac;5[‘-a) = /f (4)
I-. #35*441 . #80*14 .DHrHp.D.i^raGP.iSr-aeP.
[#23*59] D. Pfac/Sr-aGP (5)
h . (3) . (4) . (5) . *80*14 .Dh. Prop
This proposition is used in dealing with greater and less among cardinals
(#117*68).
#80*4. : Re P a ‘k . y « * . xRy . x’Py . D . {(P -i- x J, y) v x J y) e P A ‘tc
This proposition is important. It shows that, if ReP^tc and x is the
selected referent of y (i.e. is R*y), then x may be replaced by any other
member of P*y without our ceasing to have a member of /V*.
Dem.
I-. *55*3 . D f-Hp . D : a* l y G R :
[*72 01] D : Cl \R^x J, y) - CI‘P - Cl*(x i y)
[#80*14.*55*15] = *-i‘y (1)
H . (1). *33*261 . D h: Hp. D . (I‘{(P-* i y) c; *' i y) = (* - i‘y) v OV | y
[*55*15] =(K-l t y)vi‘y
[*51221] -« (2)
h . (1) . *55*15 . D 1- : Hp . D . Q‘(R-^x |y)n Cl‘(a;' i y) = (k — i*y) n i*y
[#24*21] = A.
[*7l*24.*80*14] D . (R — xX y)wx' ^ y e 1 —* Cls (3)
h .#80*14 . *55*3.3 h -.Hp.D.R^xiyQP.x iyQP.
[*23*59] D.(flix|y)u«'|yGP (4)
h . (2). (3) . (4) . *80*14 ,DK Prop
*8041. h : R e P A ‘* . y e k . x'Py . D . [{P^(P‘y) ly}vxly]e P A ‘*
Dem.
V . *80*3 . *30*32 . D H : Hp . 3 . (P‘y) Py (1)
h . (1) . *80 4. 3 h . Prop
•8042.
Dem.
h .*41*11 . 3 h ix^pPS^y . = . (gP) . P € P A ‘* . xPy.
[*80*14] 3 . a:Py .ye*.
[#36*101] 3.<r(P|**)y (1)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
488
h . *80-4-1 .*35101 . D
b : P « /V* . x(P\k)ij . D . [|P-(P‘y) | yj ori(/]c /V* .
[*55 132]D.[;P.^(P‘y)iyI oxJ,y]eP A ‘*.x[[P-^(P‘yUy]c/xl,y]y.
[*41 *141 ] D . xijPP&K)y (2)
b . (2). Exp . *1M 13 . Z> H : R € P A ‘* . D . P f * G s‘P a ‘k (3)
I-. (3). *101123. D h : g ! P A ‘* i‘/V* (+)
H.(1).I4).DK. Prop
-80 43. K : rPy. = . .r | y € P^i *y
Jjem.
b . *72182 . *53 15 . D I-. * ly c 1 -» CIs. CP* 1 y = «‘y (1)
H . *35-3 . D H : xPy. = . x | y G P (2)
h.<I).(2). *4 73 . Ob: xPy. = .x | y G P . x J, .y € 1 CIs . d‘(x | y) = f‘y .
|*S 0 I4] 5 .x.|y€P A Vy :D H. Prop
‘•80 44. H : Rt P A ‘i‘y . D . P = (P‘y) | y
l)em.
b . *8014 . D b : Hp . D . P € 1 -> CIs . d‘P - i‘y.
[*37-25] D. P c 1 -> CIs. d‘P = i*y . D‘P = R“i‘,/
[*53-31.*711(»3] = i‘P‘y.
[*5510] D . P * (P*y) i y: D h . Prop
‘•8045. h. /V'V/ = iy“P‘//
JJem.
1-. *38131 . D b ; P < | y“P*y. = . (gx). * c P‘y. R = a; J y.
[*3218] = .(gx).xPy. R -x^y .
[*80-43] D.PtP A ‘i‘y (1)
h . *80-44-31 . D 1-: P e P A Vy. D . P = <P‘y) 1 y. P‘y f P‘y.
[* 14-205] D . (gx). P =* x l y . x e P‘y.
[*38131] D. Pc±y“P‘y (2)
I-. (1). (2) . D h . Prop
*80 46. b : g ! P A Vy . = . g ! P*y . = . y c Q‘P [*80 45 . *37 45 . *33 41]
*80 6. huo\=A.Pf P A ‘* . S e Q A ‘\ . D . P c/ S e (P c; Q) A ‘(* ^ X)
Dem.
1-. *8014 ,0b: Hp . D . P, S c 1 —> CIs . <PP = k . d f S = \. RG P. SGQ •
[Hp.*33-261.*23-72] D . P, Sc 1 -* CIs . d‘P n d‘S = A . d‘(P u S) = k w X .
Pc/SGPc/<2-
D.PuSel—♦Cls.d‘(Po5) = ^X.«o5GPaQ.
D . P c; Se(Pv Q) A ‘(x u X): D h . Prop
[*71-24]
[*80-14]
SECTION D]
ELEMENTARY PROPERTIES OF SELECTIONS
489
*80-51. h : X a (I <P = A .Re . Se Q±‘\ .O.RvSe(Pv Q)*‘(* o X)
Dem.
H . *10 24 . D h : Hp. D . g ! P^k .
[*80-2] D . * C (I‘P .
[*2248] D./rnX C CPP a X .
[Hp.*2413] D.*cnX = A (1)
H . (1) . *805 .Dh. Prop
*80511. H : k a G/Q = A . X a G*P = A . M e (P c/ wX).D.
M\k-M AP AQ
Dem.
.*8014. *23-621 . DH: Hp. D . M-MX(PwQ).
[*3517] D . M r ^ - Jl/ A (P wQ)f k
[*35 644] = M A P p k
[*35*642.*25’24] - M A(P\kv P\\)
[*35-41217] «A/[(/cwX)AP
[*80-29] -MAP
H . (1) . 0\-iHp.0.»rf\-M*Q
t y \£t *, A
( 1 )
(2)
h . (1). (2) .Dh. Prop
*80 62. H : * a a ‘Q - A . X a a*P = A . Me(P v Q)S(k w X). D .
MtKiPSK.MfXeQSX
Dem.
H . *8014 . *71-26 . D b : Hp . D . M f k. A/p X c 1 -* CIs (1)
h . *80-511 . D h : Hp . D . 3/ [** = M A P . M f X = M A Q .
[*23-43] O.M[kQ.P.M[\QQ (2)
1- . *8014 . *22-58 .Dh:Hp.D./cC(I *M . X C d‘M.
[*35 65] 0.a € MfK-K.<I*Mt\-\ (3)
I- . (1). (2) . (3) . *80 14 . D h . Prop
*80 63. h * a (l‘Q = A.Xo d*P = A . D :
M e (P u Q) A ‘(* V, X) . = . (3P. 5) . /e € /V* .SeQt‘\ .M-RvS
Dem.
*80-52 . DhHpJ/e(Po QV(* v \).O.M f kc PSk.M p X e Q*‘\ (1)
*80-29. Dh : Hp(l). "5 . M = (k \j \)
[*35412] (2)
I-. (1). (2). D h Hp . D . M e (P o Q) a ‘(k ^ X) . D .
(g«, S) . R e iV* . S e Q a ‘\ .M = RsjS (3)
K *80-51 . DH:.Hp. D: R e P*‘k . S e QS\. M = RvS. D .
Afe(PoQV(*uX):
[*11-11-3-35] D : ( a «, S). R e iV*. S € Q*‘\ .M-RvS.O.
M € (PkjQW(k»\) (4)
M3). (4). Dh. Prop
490
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*80 54 I -* n d*Q = A . X n Cl l P = A . D s
li e 1\ 1 k . S t Q A ‘X. = . (g.l/). M c (P u Q)*\k v\).R = M\k.S=M\\
l)em.
H . *80-51 . D 4: Hp .Re TV* • S * Q A ‘X .D. RsjS € (Pkj Q)*‘(k u X) (1)
I-. *8014. Dh: Hp(l) . D . * a d‘S = A.Xn d‘P = A .
[*35*644] D . (P vy S)f* = P [* * . (P u •
[*80-29] D . (P o .9) f** = P. (7? iy 5) |* X = 5 (2)
h . (1). (2). D1-: Hp. R e TV*. S f (? A ‘X . D .
RvSe(Pv Q) A ‘(Kyj\).(RvS)fK = R.(RvS)[\ = S.
[*10-24] D . (gif). M e (P v (?) A ‘(* w X). M f* * - R . M f X - S (3)
h . *80-52 . D h s. Hp. D : M e <P o Q) A ‘(* w X). P = M[ * . 5 = M[ X. D .
R € P&k . S € Q&‘\ :
[*1011-21 -23] D s (gif). A/ c (P « Q V(* * X). P = A/. S-A/r X.3•
PeP A ‘*.ScQ A ‘X (4)
H . (3). (4). D I-. Prop
*80 6. h : Ac TV*. X C * . D. P[* X e P A ‘X
Pern.
K. *80 14. *71*26. Dh: Hp.D. P|*X« 1 -»Cls (1)
h . *80 14 . *35-441 . D V : Hp . D . P [* X G P (2)
K *80 14. *35*65. D 4 : Hp. D . d‘P |* X * X (3)
h . (1) . (2). (3) . *80 14 . D K . Prop
*80-61. I-: M r * < PS* .M\\€ TVX . D . A/p (* u X) € TV(* w X)
Deni.
V . *800. D h : il/f* X f P A ‘X . D . .1/ f* (X - *) * P A ‘(X - *) :
[Fact] Dh:Hp.D.jl/[«c P A ‘* . M f (X — *) e P A ‘(X — *).
[*80-5.*24*21 ] D . A/f* * vy A/ f* (\ - *) c 7V(* v (X - *)».
[*35412.*22-91] DJ/f(*wX)« P A ‘(* wX):Dh. Prop
*80-62. h : 3/ € TV(* w X). Z> . A/ f* * € P A ‘* . AT f X « P A ‘X [*80 6 . *22 58]
*80-621. h : A/f(* v X)c TV(* vX). D. A/f *e P A ‘* . .1/ [* X e P A ‘X
Deni.
h . *35-31 . D V . [3/1* (* v X)) f k = A/ \ [(* u X) a *)
[*22-631] =M[k (1)
Similarly H . (AT f“(* v X)J f* X = A/ f* X (2)
H . (1). (2). *80-62 . D h . Prop
*80-63. 1-: M\k €P a ‘k . Mf\ c P A ‘X. = . AT [(* u X) c TV(* u X) [*80 61-621]
*80*64. h :. G‘Af = *uX.D:A/f‘*€ P A ‘*. A/f* X € P A ‘X. = . AT eP A ‘(* w X)
Deni.
h . *35-452 . D h : Hp. D . M= u X)
1-. (1). *80 63 . D H . Prop
(1)
ELEMENTARY PROPERTIES OF SELECTIONS
491
SECTION D]
*8065. h:«n\ = A. R e 1\‘k . S e P 4 ‘X . 3 . R vSeP.‘(« «X)
*80 o ^ . *23-56J
*80 651. h : R e . Sc P a ‘X . 3 . .ft u S [ (X — *) e P 4 ‘<k u X)
Dent.
K*80-6.Oh:Hp.O.Sr<*--*>«iV<X-*).
[*80 65] O . P o £ r (X - k) c P a ‘|* v (\ - *)) •
[*22-91] D.iiwS[(X-/f)cP A ‘(* v X):D h . Prop
*80 66. h:.*n\-A.D:
A/ c iV(* u X) . = . (gP. 6'). R € P*'k .SeP A ‘\. M =RsjS
Dem.
y .*80-62 . Oh: A/ € P A ‘(* wX). D.3/f*/ce P A ‘* . .1/p X€ P A ‘X (1)
h • *35-452 .Oh: A/c P A ‘(* v X). O . Af — A/ f (* w X)
[*35-412] =iV[*/coA/rx (2)
Ml).(2). ^b:jy e P^( IC yj\).D.M[K€P A i K.M[\€P A t \.iM^M[KsjM\'\.
[*1136] 0.<g R,S). R€P*‘k.S€P*‘\.M=RvS (3)
h • *3065 . 0 h :. Hp .3: Re P A ‘* . S e P A ‘X. A/ - P c; 5. O . A/« P A ‘(*uX) :
[*11 11-3-35] O : ( a P, 5). R < P A ‘*. S e P A ‘X. A/ - R c; £. O .
M3).(4). Oh. Prop
*80 661. h : * /> X = A .
Dem.
MePS{Kv\) (4)
ReP A *K.StP A ‘\.0. R = (RsvS)tK.S = (RvS)t\
*8014. O h : Hp . O . G‘P = * . il*S r* * = A . (1)
[*35-452] O .Pf** = P (2)
Ml) • (2) . *35-644 . O h : Hp. O. (P c* S) f k - R . (3)
Similarly h : Hp. O. (R o 5) T X -8 (4)
1-. (3) . (4) . O h . Prop
*80 67. h:.*nX = A.O: ReP*‘* . S € P A ‘\ . = .
(gA/). M e P a ‘(k yj X) . R = M f* * . 8 = M[ X
Dem.
y ■ *80-65-661.3h:.Hp.3:fic P 4 *«. Sc/> 4 ‘\. 3 .
b o s c p.‘(*« x). r =(j? c< s) r *. =(R * s> r X ■
[*10-24] 3.( a Jf). H C R a \k»\).R = M\-k.S = M^\ (1)
K *80-62. 3h:Af e P 4 ‘(*«\)..ft = .M|-*.S = A/fX. 3. fie P.‘*. S.P.'X:
[*10-11-23] 3 1: ( a M ). AT e JV(* wX).« = A/f*.S=A/|-X.D.
fieP„‘*.,SeP 4 ‘X (2)
*" • 0) • (2) .31. Prop
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*80 68 . b : R e /V< * — i*y). ye k . xPy. 0 . R kj x | y e P*k
Deni.
b . *8043 . 0 b : Hp. 0 .x^yePSi'y (1)
H. *24*21. 0 b . (k — I‘y) r\ (‘y = A (2)
I-. (1). (2). *80-65 . 0 b z Hp . 0 . Ii vy x ^ y e P* \(tc — ( l y) v pyj.
[*51*221 ] 0.Rvxlye P a *k :0 b. Prop
■‘80 69. I-: 3 ! P A ‘<* ^ X>. = . g ! /V* . g ! P A ‘X
Dem.
b. *80*62. DH:g!P A ‘U«X).D.g!P A ‘*.g!P A ‘X (1)
H . *80 6 . D b : g ! P A ‘X. D. g ! P A ‘(X - *):
(Fact] 0 H : g ! P A ‘* . g ! /VX . D . g ! P A ‘* . g ! P A ‘(X - *) (2)
b . *80*03 . Ob: Re /V* • S« 7V(X - «).0.R wSePSi* v X):
1*10 11 *23] D H :g ! /V* • 3 ! /V(X — *). 3 . g ! P A ‘(* v X) (3)
H. (2). (3). 0 1-: g ! P A ‘*. g 8 /VX. 3. g IP^w X) (4)
I-.(1).(4). D h . Prop
*807. b : (I*P aCPQ — A . *CG‘P. X Cd‘Q. i/ e(Po Q) A ‘(* «x).D.
M ^Q*PSk.M ± PtQS*
Dem.
b . *33*33 . *80*14. 0 H : Hp . 0 . P a Q = A . if G P v Q .
[*25*491] D.if^Q-ifAP. if^P-ifAQ (1)
b . *22*48 . *24*13 .Ob : Hp .0 . * r\ (l'Q = A. X a d‘P - A .
[*80-511 *52] O.AlAPe PS* . if a Q € Q A ‘X (2)
h.(l).(2).Dt*. Prop
*8071. h:d‘pAd‘<?« A.M^Q€p A ‘«.M^PeQ**\.O.Me(PvQ)S(Kv\)
Dem.
b . *33*33 .Ob: Hp. 0 . P A Q= A .
[*25*493] D.if«(if-=-P)o(if^Q) (1)
H . *80*2 . DH: Hp.D.XCd'Q.
[*22*48.*24*13] 0 . X a d‘P- A .
[*80*51 ] D . (ifQ) c; (if-s-P) € (P o Q) A ‘(* u X) (2)
h . (1). (2). DK Prop
*80*72. b d ‘P a d‘Q = A . * C d‘P. X C d‘Q. D :
if e(Pc/ <?) A ‘<* wX). = . if—Q€ P A ‘* .M-Pe <? A ‘X [*80*7*71]
*80*73. h:Q = Pr^-^ = ^rx.D. P A ‘(* «X) = (Qo P) A ‘(* w X)
Dem.
h . *35*412 .Dh: Hp . 0 . Q c; 7? = P f (* w X).
[*80-23] 0.{Qv P) A ‘(* u X) = P A ‘(* wX):DK Prop
*80 731. H:Q = Pr*.P = Prx.*v/XCd‘P.D./c = d‘<2.X = d‘P
Dem.
h . *22*59 .0 b : Hp . 0 . * C d‘P. X C d‘P.
[*35-65] 0 . k = d‘Q . X = d‘P :0 b . Prop
SECTION D]
ELEMENTARY PROPERTIES OF SELECTIONS
493
*80732. h:<2 = Pp*.P s =Ppx.*«X = A.D.CI‘<2na‘P = A
Bern.
H . *3564 .DF: Hp . D . (PQ
C*.d‘PCX.
[*22-49]
D . (l‘Q
Q‘P C * r» X .
[*2413]
3. Cl ‘Q.
r» CI'P ■A:D1-. Prop
*8074.
hun\ = A
. M € P 4 ‘(* \J X) .
D.
Bern.
A/p*=A/p-x=
A/ — Pp X . A/p X = A/p - * = A/
~p[
h . *24-4 .
D h : Hp . D . .1/ p* = M p J(« u X) - \|
[*35-31]
= {A/r<«-x)ir-x
[*80-29]
- A/p-X
a)
h . *80-732 .
D h : Hp . D . CI‘(P [ «) * CI‘(P r *) - A .
[*33-33]
3.pr* A -pr^-A
(2)
K *80-291 .
Dh: Hp.D.A/GPp<* v\).
[*35-412]
D. A/GPp*oPpX
(3)
M2).(3).
*25*491 . D H : lip . D . A/-5- P p X ■= A/ n P p *
[*35 17]
-<Af*P)p*
[*8014.*23-
621]
= A/p*
(4)
Mi). (4).
Dh:Hp
.D. Afp*-A/p-X- Jr-i-PpX
(5)
Similarly
H s Hp
. D . A/p X- A/p- * - AZ-^Pp *
(6)
M5).(6).
D h . Prop
*8075.
h«n\-A
..JkfcP A '(*v\).
D . A/-PpXe P±‘k . A/-Pp * €
P 4 ‘X
[*80-62-74]
*8076.
ViMe P A ‘p
. P e P A */c . R G A/ . D . Af — P c P A ‘(u — *»
Bern.
K*8014.
D h : Hp .
Z>.(I‘P = *.(I‘A/ = m
(1)
t-. *8014..72-91. Dh: Hp.
d . a'(A/^ P) = a*A/ - a*p
[(1)]
*/*"*
(2)
1-. *8014 . *71-22 . D 1-: Hp .
D.A/ — Pel— * Cls
(3)
K *8014. *23*47. Dh:Hp.
D.A/iPGP
(4)
^• (2) . (3) .
(4) . *8014 . D h .
Prop
*80761.
h : k r\ X = A
. Af € P A ‘(* w X) .
P e P A ‘* .PGil/.D. 3/iPc P A
‘X
Bern.
1-. *80-76 .
D h : Hp . D. A/ —
PeP A ‘|(* w X) — xr)
(1)
1". *24-4.
D h : Hp. D . (* u
X) — /c = X
(2)
*80-77.
Mi).(2).
D 1-. Prop
hMePSn
.M^-ReP A ‘(v-
k) . RGM. k C fj .0 . R € P a ‘k
Bern.
h . *80-76 . D h : Hp . D . * PS{p - (/* - *)) (1)
H.*25-411. DhsHp.D. Af=Po(Af-P) (2)
494
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
P. *2521 . 0\-.Rf%(M^R) = A
(3)
K (2). (3) . *25-4 .D P: Hp.D. J/-^(il/x- R) = R
(4)
^ . *24 411-21 4
P.<1).<4).(5). DP.Prop
(5)
*80771.
P : k n X = A . Me P a (k \j X). .1/x. R e P.‘X . P G M . D . P £
Dem.
P . *24"4 .DP: Hp .D.Xs^wX)-*
P . (1) - *80*77. DP. Prop
( 1 )
*8078.
P : M tf P A ‘n . xJ/y . D . A/ -s-a- P A *(/* — t‘y)
Dem.
P . *'».V3 . DP: Hp. D.x|yGil/
P .*8014. D P : Hp. D . xPy .
( 1 )
[*80*43] D . x | y e P&i'y
P.(l ).(2). *80-76 . DP. Prop
( 2 )
*808.
P:g!PA.D.aW*-«
Dem.
P . *80-42 . D P : Hp . D . *‘/V* = />[**
P . (1). *80-2 . *35 65 .DP. Prop
( 1 )
*8081.
P: 3 !/Va./V«-/V/3.D.«-0
Dem.
P . *30*37 . D P : Hp . D . (I= d's'/V/S .
[*80-8] D . a = 0 : D P . Prop
*8082.
P : a + £ . D . P A a rs iV£ - A
Dem.
P . *80 14 . D P : R e P A 'a . S€ P A ‘0 . D . CI‘7* = a. Cl'S = £:
[*13 13] DP:. Hp. D : /Je /Va. Se 7V/9 . D . Cl'/* 4= d‘tf.
[*3037.*33’ 121 .Transp] D . R + S
P.(l).*24-37.DP.Prop
(1)
The following proposition is used in *80 84 and in the theory of double
similarity (*111‘3).
*80 83.
Dem.
P . *8012 . *71 166 . D P . P A e 1 CIs .
[*7127] DP.(-i‘A)*|/> A el-*Cls (1)
P . *35*1 .*51*15. D
h : X !(- i*A) 1 P A | a. X |(- e‘A) 1 P A ) 0 .
= .X*A .\P A a.\Pj3.
[*2454.*80 13] = . 3 ! X. X = P A ‘a . X = P A ‘/9 .
[*8081] D.a = £ . (2)
P . (2). *71*171. D P. (- i‘A) 1P A c CIs —» 1 (3)
P.(1). (3) .DP. Prop
SECTION D]
ELEMENTARY PROPERTIES OF SELECTIONS
495
*80-84. b : A~e P A “* . D . P A “* sin *
Dem.
b. *51-36.
D b : Hp . D . P a “k C — i‘A .
( 1 )
[*37-42]
D.P a "*=((-i‘A)*|P a |“*
( 2 )
b. *8012. *33-431 .
DK/cCd'P*.
[*37-51]
Db.KC P A “P A “tC
(3)
K(l).*37-2.
D b : Hp . D . P A “P A “* C /V‘(- t‘A)
[*37-4]
Cd'|(-t‘A)1P A |
(4)
M3).(4).
D b : Hp . D . * C d‘((- t*A) 1P A )
(5)
b. (5). *80*83. *73*22
. D h : Hp . D . {(— «*A) 1 P A )“* sm k
(6;
H - (2) . ( 6 ) . D I-. Prop
The three following propositions are useful both in cardinal and in ordinal
multiplication (#113 and #172).
#80 9. b y 41 g. D s MeP A \i*y w i‘z). = .(g*/, t»). uPy . t»P*. 71/ = u l y w v l z
Dem.
b . #80 45-66 .Dh. Hp . D : Me P A *(i*y v t‘*) . = .
(3 R,S). Re l y €t P*y .Se l z'<P‘z .M-RvS.
[#38131.#32*18] = . (a u, t») . uPy . vPz . = Dh. Prop
#80 91. b: Me P*‘(l‘y v Pz) .D.M = (M‘y) l y o (M‘e) l z
Dem.
b. #71-6. #8014. D
b : Hp . D . M = s‘Q ((aw) . w e i*y v i‘z . Q =» ( M*w ) | w)
[#51-235] = (Q = (M‘y) ly.v.Q = (M‘z) | zj
[*51-232] = i € [i t {M*y) jyw l\M*z) | z\
[*53*13] *= (M‘y) i y& ( M*z ) ^ z : D h . Prop
*80-9-91 can be extended, by precisely similar proofs, to any finite number
of variables y, z, .... They will, on occasion, be assumed for three or four
variables, without fresh proofs.
#80 92. b : y 4= z . D . D“iV(£‘y \j i*z) = £ ((au, v ) . uPy . vPz . £ = l‘u u t'v)
Dem.
b . *5515 . *3326 . Dh. D‘(u |yoi;^) = t , uvt‘i; ( 1 )
h • (1) • *80-9 . *37 6 . D h Hp . D : ge D“P*‘(l‘y v l*z) . = .
(3^, v, M) . uPy . vPz . M = u^yw^z.t;=: i l u v/ i*v .
[*1319] = . (g u , v ) - uPy . t>Pz. £ = i*u Prop
*80 93. h : a ! P A ‘(t‘y u i*z ). = .y,ze d'P [*80 46 69]
#80-94. b : 3 ! P A ‘(£ ^ i‘*). = . a ! Pa*£ . * e d‘P [*80 46 69]
From this proposition, together with *80'26 (which gives a * Pa*A), we
shall obtain an inductive proof that Ps‘/3 exists whenever 0 is a finite class
contained in d‘P (cf. *120 011).
*81. SELECTIONS FROM MANY-ONE RELATIONS
N// /// urn ry of *81.
When 7 > f‘* is a many-one relation, P a *k has many important properties
which do not hold in the general case. In the first place, 7V* consists wholly
of onc-one relations. In the second place, if Iie 1 \‘k, IVR takes one term
and no more out of each member of P u k. Again, if 77 € 7V*. 77 is determinate
when D‘77 is given; i.e. 77. »S e 7V* . I) 4 77 = L) 4 &. D . 77 = .S'. It follows that
J)“7V* is similar to 7V*; hence the numl>cr of members of 7V* is the
—>
number of ways of choosing one member out of each class belonging to P li n.
It should be remembered that when P\ * is many-one, J Ui K is a class ol
mutually exclusive classes, i.e. no two different members of 7 ,4 ‘* have any
common member. This follows immediately from *71181.
As explained in the introduction to this section, the propositions of this
number are chiefly useful on account of their application to the case of e.
This application is made in *84. The most important propositions in this
number are:
*811. ViP r*cCls->l . I>.7V*C \-> 1
*8114. b : 7'f* * c CIs —» 1. i7c 7V*. 3.77 = (D‘77) 1 * = 1* f\ D 4 77 \ <
This proposition, by exhibiting 77 as a function of D‘77, leads immediately
to
*8121. I- :J j [k«C\s-¥ 1. D. Df 7V*e 1 -* 1. D“7V*am 7V*
This is the principal proposition of this number. The following also is
important:
*81*22. I*: P r * « CIs -> 1. D. L)“7V* - £ |y e *. D,. /i fl:/*C P“*\
( 1 )
*811. b : P f k e CIs 1. D. /V* C 1 -* 1
Dent.
b . *80*14 . D b : 77 c 7V* . D . 77 c 1 —> CIs
b . *80 291.I> b i.ReP^K . D : 77 G Pf k :
[*71*221] D : P * € CIs —* 1 . D . 77 * CIs —► 1 (2)
b . (1) • (2). D b . Prop
*81*11. b ; 7 J f * € CIs —> 1 . J7 cJV*.x€D‘/7.D.E! R'x .x{P\ k)R'x
Dem.
b . *71*165 . *81*1 . D b : Hp. D . E ! R‘x . (I)
[*30*32.*31 *11] D . xR (R'x) .
[*80 291] .
b . (1) . (2) . D b . Prop
(2)
SECTION D]
SELECTIONS FROM MANY-ONE RELATIONS
*8112. I- : P I* K € Cls —» 1 . R € 1\ <K . . 1 - e D‘/f . D .
R‘x = ((.</) lyf«. .<Py) = (<r 1 P)‘x
Deni.
I-. *71-361 . D I- Hp . 3 : j. (P r *) ■ = ■ R‘x = |Cnv '(F r *)):
[•8111] 3: R‘x= |Cnv‘(P|-*)|‘-r
[*35-52] = («1 P)‘x (1,
[*35-1] = (» y)(yex.xPy) (2)
I- • (1) • (2). 3 h . Prop
*8113 (-:..P|-*,Cls->l .ReP^K.^-.xRy.m.xtD'R.xPy.ytK
Dem.
H . *8112 .Dh: Hp . D :.xe D‘R . D : y = = . y = <* *]
[*71361] D :xRy . s . .
[*35101] ■. xPy, y € k (lj
h . (1) . *5 32 . D
h Hp. D : are D *R . ar/ty . = .x t D ‘R , xPy.ye*:
[*3314.*4 71] D : xRy . = . are D‘/£ . arPy . y e k I> f-. Prop
*8114. H : P r * e Cls -» 1 . R e P^k .D.R - ( D*R ) 1 P f* * = p * D'rt t *
[*8113. *35-102-822]
This proposition, by exhibiting R as a function of shows that
a member of /V* is determinate when its domain is given, provided
p r«eCls->l. 6 v
*8116. h : P f * e Cls —► 1 . R € P a ‘k . y c k . D . i‘R‘y = D*R n~P‘y
Dem.
*" • *®1 13 • 31:. Hp . 3 : xRy . =,. x c D‘/i . xPy :
[•32 18] 3 -.xe~R‘y.=,.xe D ‘R .xe'p-y.
[*20 43.*22 33] D:li‘y = D ‘R « ~F‘y :
[*53 31.*71 163.*8014] D : fjfy = D ‘R n ~P‘y 3 I-. Prop
*81 2. I-:. P f * { Cls -♦ 1 . ft, S e P A ‘« . 3 : D‘« = D‘£. = . R - R
Dem.
. *30 37 . *33 12.3 h : R = S. 3 . D‘R = D‘S (1)
I-. *81-14 . *1312. 3 (- :. Hp. 3 : D ‘R = D‘S. 3 . R - P r. D‘S | *
[•8114] =S (2)
Ml). (2). 3 (-.Prop
•81-21. (-: P f * ( Cls -» 1.3 . D f P„*« r 1 -» 1 . D“P„‘* sm /V*
[*81-2. *71-59. *73-28]
aacw i ao
498
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
This proposition wry important. The class D“/V*. when Pf*eCls-*1.
i' formed, as we shall prove later, by making every possible selection of one
term out of each member of P“*. each such selection giving us one member
«*| 1)“P A ‘*. The fact that, with the above hypothesis, the class of classes
l>“/V* has the same number «*f terms as P A *k (which results from the above
proposition), i> of great utility in the theory of cardinal multiplication and
exponentiation.
*81 211. b : P r * < Cl> —>1.3. D"P A ‘* C £ |y e * . 3 y . M a P‘i/ e 1 : p. C P“*J
Dew.
b . *8115. *52 1 . 3 b Hp. R * P A ‘* . M = D ‘R P*y e 1 :•
[*101123-35] 3 H Hp:(H/O.P€/^ < <./x-D < P:D:y^.D v . A xnP' < y€l:.
[*37G.*33- 12] 3 h s. Hp. m « D“ JV*. 3 s y c *. 3 y . ft n 7”y € 1 (1)
h. *80-21) I .*33-203.3
► S 71 € /V*. M = D‘fl .D./xC *).
[*87 401] D.mC/ w *j
[*I0 1 I 23-85] 3 h : < a /?>. /* « /V* . = D‘« . D./iC P“*:
[*37*G.*33* 12] Dh*i< D“P A ‘* . D./xC P“* (2)
h .(I).(2).3 h . Prop
-81212. I- ,D v ./«o P*y * 1 :/xC P**k : 3 . /* < D"/V* . /* *| P|* * € P A ‘*
Dew.
V . *35 442 . *37-402 . 3
I-: R = /i 1PT*. 3 . P G P. (J‘fl - * a P‘V . D‘P * ,1 rs P“ K (1)
b . *52*10 . Dh:.H|).D:y(«.D v .g!fin P‘//.
[*37-40.*32-241] 3„ . y € P‘> :
[*221] DuCPV (2)
h . (1). (2). *22 021 . 3 K : Hp .R = M 1 Pf* * . 3 . P G P. <3‘P = *. D‘P = /x (3)
h . *3218 . *35 102 . 3 b Hp (3). 3 : y c k . 3„. R*y = ft r* .
[Up] D y .7?y € l:
[*37-702] 3 : R“k C 1 :
[(3).*71 "I ] 3:Pcl->Cls (4)
b . (3). (4). *8014 . 3 I-: Hp. 3 . /* 1 Pfxe P±'k . D V1 P t *)-M • ( 5 >
[*37-6] 3. / xtD“P A ‘* (6)
I- . (5) . (G) .3b. Prop
*81 22. I-: P r * ^ CIs -* 1.3. D“P*‘* = [i\y e k .^ y . n k~P'ij d i ftC P“k\
[*81-211-212]
SELECTIONS FROM MANY-ONE RELATIONS
499
SECTION D]
*81 221. h : P [■ * « Cls -» 1 . D . P 4 V = 1 (Pf *)“L>“P 4 ‘«
Dem.
h. *81-14. *37-62. Z>
t-Hp . D : P £ P„‘« . D*. P = (D‘P) 1 P f * . D‘P £ D“P 4 ‘* .
[*10 -24] . ( a/1 ). P = M 1 P r * . M e D “I\‘ K .
[*38-131] . P £"] (Pf- *)“D“P 4 ‘« (1)
H . *81-22-212 .Dh:.Hp.D:^£ D“P 4 ‘* . D. . M ] P |- « e P a ‘« :
[*3701] 3:1(Pf-*)“D“P 4 VCP 4 ‘/c (2)
I". (1) . (2) . D h . Prop
*81 23. I- s P f- * £ Cls —» 1 . P £ PSk .-/£*. D . D‘P - ~P‘'j = D‘P - i‘P‘y
Deni.
t-. *22-93.3l-.D‘P-7”y = D‘P-(D‘Pr.P‘y) (1)
I-. *81-15 .Dh: Hp . D . D‘P — (D‘P ■-> P*y) ■= D‘P — «‘P‘y (2)
I-. (1). (2) .3h. Prop
*81 24. H : P f- * £ Cls —»1 . m < D “P 4 ‘* . rj * *. D . /x - P‘y £ D“P 4 ‘(* - f ‘y)
Dem.
h . *80-78 . D I- : P £ p 4 <* . y «K . D . P^-( P‘y) J, y £ P.'(* - t'y) .
[*37-62.*88-12J D. D‘(P J-(P‘y) 1 y| £ D“P 4 ‘(* — t'y) (1)
1-. *81-1 .*8014. D
: Pf * £ Cls —* 1 . P £ P 4 ‘* .y£x.3./2fl—»l.y€ G‘P .
[*72-911.*71-31 .* 55 - 3 ] D . D‘|P-(P‘y) J. y] - D‘P - t'P'y
[*8!-23] - i)i/ { _ p«y (2)
h • d) • (2). 3 H : Hp(2). D‘P = /i .0. p — P*// * D"P.'(* - t'y) (3)
I- • (3). *10-11-23-35 . *37-6 . *3312 . D (-. Prop
*81 26. I-: y £ « . x py . ^ f D“P„‘(* - t'y) .D.»ui‘if D“PP*
Dem.
h * * 80 68 ‘5l-:y€«. *Py . P € P A ‘(* - i‘y) . D . c * i y c P A ‘* .
[* 37 62 ] D . D‘(P c/4y) f D“7V* -
[*3326.*5515] D . D‘P v i‘x€ D“P*‘* (1)
h • (1) * D H : y e * . a:Py . P * P A ‘(* - py) . M = D‘«. D./iwt'if D“P A ‘* (2)
h -(2) • *10 11-23-35 . *37-6 . D f-. Prop
*8126. h:. P^cCls-* 1 .ye^./inPyel .3:
^ — P*y f D“P A ‘(* - i‘y) . = ./*€ D“P A ‘*
^ * *81-24. D h Hp. D :/* e D“P*‘* . D . /* - P^y € D“P A ‘(* - i‘y) (1)
h • * 81 ’ 25 - 3 h Hp . D ; M n ^‘y = Par. M - P*y € D“P A ‘(* - i‘y) . D .
0* — P*y) ^ i‘x 6 D“P A ‘* (2)
32—2
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
H . *22-551 . Dhj/irt P'y = i 4 .r. D . (/x — P‘y) sji t x = (p- /> 4 y) u o P‘y)
[*24 41] =,x (3)
l*. *52*1. D I-: Hp. D . (g.r). n n P*y = i*x (4)
y . (2). (3). (4). D I-Hp. D : M - 7”y € D 44 /V<* — f 4 y ). D . n e D“P*‘k (5)
I-. ( I). (5). D y . Prop
-81 3 !-:/'[ *€(>•-► 1 .\=/ ,4 ‘*. D. D 44 /V* = £ atls/tCA]
Deni.
y . *37-700 . D y z. i /1 k . D . t . n r\ P*i/€ 1 : = : a e P*'* .D ( ./tnacl (1)
H . *40-5 . D H s m C / ,<< <r. 3 . /x C * 4 P“* (2)
»-.(!>.<2).«81 22.D
h : « < Cl* —♦ I . D . I >“/V* = ? l« ‘ /"'* s‘P‘‘*| (3)
K (3). *1312.3H. Prop
*81-31. I- : 1‘[k.Q\k» ('U -* 1.7"‘« - V'« • 3 • 1 >“/V* - D“Q 4 ‘*
Dcm.
h . **1 *3 . D h : Hp . D • D“/V* = a* !« « Q“*. D, ./ioa«l:/iC **<?“*)
[*81*3] =* D 44 #* 4 * : D H . Prop
*82. SELECTIONS FROM RELATIVE PRODUCTS
Summary of *82.
The propositions contained in this number are not much used except in
connection with the associative law for cardinal multiplication, but they have
a certain intrinsic interest. We prove in this number that, with a suitable
hypothesis, (P| Q) a ‘\ results from by multiplying each member bv
Q, i.e.
*82 272. h:<2rXel->l.X € D‘(Q)<.D.(P Q) s ‘\ .j Q“P A <Q“\
Also under a suitable hypothesis the domains of (P Q)±‘\ are the domains
of i.e.
*82 32. h: Qf Xel—►l.X.C (I‘Q . Z> . D“(P Q) A ‘\ = D"/VQ“\
In the applications of propositions of the present number in *85, P and Q
are replaced by e and Q. By *G2'26, e Q = Q ; thus we obtain relations
between Q A ‘X and e**Q“\.
*82 2. ViMeP^K. AT € Q a ‘\
Dem.
h . *8014 . Dh
[*71-25]
h . *8014 . D h
[*34-34]
h. *8014. Dh
[*37-32]
h. *8014. Dh
[*37-201-25]
[Hp]
[*37 271]
H.(3).(4).(5).DI-
I-. (1 ). (2) . (6 ). *80
*8221.
Dem.
.Q“\Ck.O.MiV<(P\Q) a ‘>,
: Hp.3.iV,ATel->Cls.
D.Afi N* 1 -*Cls
: Hp .O.MGP.NGQ.
D.M NGP\Q
: Hp . D . d*M = k .
D.a^iV: N) = N t *K
: Hp. D . ATGQ.a'AT—X.
D . N“\ C Q‘*\ . N“\ = D*N
D.D'iVC*.
D . N“k - d‘A r
: Hp . D . a N) = \
-14 Oh. Prop
h • *80 29114 . D h Hp . D : R e Q±‘\ .D.RGQfX. Q*R = X .
j> 72 ' 92 ] D.R = (Q rx)ra^.a‘P = x.
[*35-31] D.i? = QfX
b. *35-441-660 h: Hp. D. Qf*X el -»Cls.Qr^GQ.a < «2rx) = X.
[*8014] D.Q[*XeQ A ‘\
h • (1) ■ (2). *61141 Oh. Prop
( 1 )
( 2 )
( 3 )
(4)
(5)
( 6 )
(1)
(2)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*82 22 1 -> Cl>. X = Q“*. .V* 7V*. 3. M Qc{P QU'X
Jh’in.
h . *80 14.*3732.Dh : Hp.D.U'tJ/ (/)-$"«•
jup) :>.<p<.i/ y>-x
[*35 452 23] D . -V y-J/ «?rM-
[*71-25.*8»14| D.-V y« 1 ->Cls (2)
h.*:H:U.#HOU.Dh: II,..D. A/ QG 7' Q (3)
t-. i I». < 2). (:*>. *ki» 14 . D H . Fr.*p
^ 82221 . i-syrxf i->ru.\c<i‘y. J/*/vy M x.D..i/ yrx«(P y> A *x
l/em.
K *7125. *80 14. D I* s Hp. D . .1/ yrXcl->CI* <D
K *34*34. *8014. DHsHp.D.J/ <?rxG7 > y (2)
I-. *37-32 . *3504 . *80 14 . D h : lip . D . < l‘< M yT X)- X a y“y“X
(♦37 51.*22i!2l] -X (3)
h . (1).<2).{3>.D H. Prop
*82 23. h:ypx« I-* I .*-y o \.7f*i7* y » A ‘X. D . R, Qt 7V*
I tl’III.
K*80I4. Dh: Hp.D.<P//-X.
[ *35-48) 0. R y-/f (Xiy>
[*35-51) -7f Cnv'cyfX). < 2 >
1*7125) D.ft y<l-»CI» < 3)
h.*37-32. D»-:Hp. :>.«!*<7? y)«y“(I‘7*
[(i» - y“x
[Hp] = * l 4 >
»- .*80*291 .DH: Hp.D./*G<P y>f*X.
[*35-23] O.RQP <Q[*X).
[*3434] Z>.7< Cnv‘(y [* X) G P Q[\ CnsUQ[\)-
[(2).*72-59) D . R QG 7'T D‘(Q[ X).
[*35-441] D. 7? J y G 7> < 5 >
h . (3). (4). (5). *80-14 . Z> h . Prop
*82*231. hsyrxcl-*! ./lc<P y^X.D.T? Q«/VQ“X. fl-P Q\Q T x
<i\
h. *80-14. Dh: Hp.D.U‘7f = X. I 1 '
[*74-41] D.7? Q = 77 X1Q
[*35*51] = R j Cnv‘(Q [ X).
[*34-27] 3>.7< y yrx = 7e|Cnv‘(yr x >:^r x
[*72-591] =77ra < (Qr>') (2)
SECTION D]
SELECTIONS FROM RELATIVE PRODUCTS
503
h . *80-2 . D h : Hp . D . X C <3‘(P i Q) .
[*34-36] D . X C <3 ‘Q .
[*35-65] D.d‘Qr\=\.
[(1).*7 4-221] D.Pra‘(Qrx)=7? (3)
l-.(2).(3).Dh: Hp.D. P^PIQIQfX (4)
. (4) . *82-23 .Dh. Prop
*82 24. l-:Qr\el-»l.«C D*Q . X = . R c (P Q)a‘X . Z>.
* = Q“\ .R Qe P a ‘k .R = R\Q\Q
Dent.
h . *7416 .DhHp.D./c = Q“X . (L)
[*82 23] (2)
[*8014] Z>.(I‘</e|Q) = *.
[Hp] D.Q“a'(R;Q)-X.^
[*74-4] D . R , Q j Qf X= R Q|Q.
[*82-231] D.R-RIQIQ (3)
H . (1). (2). (8). D (■. Prop
*82 241. D‘(Q),. R «(R | Q)S\ . D . R = R | Q , Q
Dem.
»- . *74-31 . D h : Hp . D . X = Q“Q“X
[*8014] =
[*37-32] -Q“a‘(i*|Q).
[*74*4] D.PlQiQrx^ R\Q'Q (1)
I- . (1). *82 231 . D I- . Prop
*8225. z Qf \ e l —+ l. * C D‘Q . X *■ Q if tc . R e(P Q) A ‘X. D .
(gA/) .Me P a *k .R = M\Q [*8224 . *1024]
*82 251. h : Q|*X«l-» 1 . R €{P\Q)^\ ,{^M) . M € P^Q^\. R = M \Q[\
[*82-231 .*10-24]
*82 26. h s. Qf\ € 1 -* 1. * C D‘Q . X = Q“* . D :
Re(P | Q)a‘X . = . (g A/). M € PSk .R = M\Q [*82-22-25]
*82261. hs. QfX«l-*l,XC <3‘Q. D:
[*82 221-251] ^ ' <P ' Q> ‘‘ X * ’" '■ (a "> * *' JV «" X ■* - "\ «r X
*82-27. D‘Q . X - Q“* .D.(P, Q) A ‘X = j Q“P+'k
[*82-26 . *43*121. *37*6]
PUOI.EOOMENA TO CAHIUXAL ARITHMETIC
[PART II
*82 271. h : VT v* \ -> I .\C(1‘^.D.(7' Q)S\= l<Jt X>“/VQ“X
[*82 201 .*43 121 .*37*l>]
*82 272. h: V[*\t l->l.Xel >, iV)f.^.(i > Q) A ‘X = ^“/VQ“X
hem.
1-. 47 2:i. D : HI*. D . IHM) • X »<J'v •
|*37'2*31) D.ia/i).X-y“(/i rx I VQ).
[*22 43) D. <a*>. X « . * C 1 )‘V
H . *v> 27 . *74* li>. 3
h:VrXc | _>| .*CD‘Q.X-y"*.D.|7' V^X= y«‘7V<T* (2)
h . 1 1 1 . < 'J». *l»*l I 23*35 . D K . Prop
v82 28 h:.*lV« • « . X CU«V.«-V - X . D s
/<€ (/* (7>a‘X . - . i;.| M). M€ /V *. li = M i Q
[*82*20. *74*20)
*8229 h:*1V« » —* I . X CU'V • * - V“X . 3.1 P V»a*X- V“/V*
[*82*27 .*74 213)
*82291. h:*lV* I ->l V“/V*
[Pimo! at in *8*2*272)
*82 3. h:.Vt7' A ‘V“\.D.D‘(.V Qrx)-l>‘.V
hem.
Y . *sn 14. 0 Y : H|). D . GM/ = (7“X .
[*74 42) D . D‘( M Q [ X) = D‘.1/ : D h. Prop
*82 31. Y :/;<(/' Q» A ‘X.D.D 4 </f
Pem.
h . *80 14 2 . D Y : Hp . Z>. Cl‘/f = X . X C (I‘( P Q ).
[*34*3<>] D.U'/fCG'Q.
(*37*321) 3 . L>‘(7? Q) - D‘7? : D Y . Prop
*82*32. h : vr * « 1 - 1 • * C U ‘V • => • D“(P VU'X = D“/VQ“X
Don.
Y . *82*271 . D
Hp. 3 : D“( <?> A ‘X = D“ t {Q [:
[*37*07] D : a e D“(7 > Qh‘X . = . (g.l/). if € . a = D‘(J/! GT •
[*82 3] => • (3^) • M e /VQ“X . a = D‘i/ .
[*37*6] D . a « D“P a ‘Q“X
SECTION D] SELECTIONS FROM RELATIVE PRODUCTS 505
h . *82-3-221 . D h Hp. D : M € P A ‘Q“\ . D . Y>*M - D‘(!f | f X).
[*3762] D.DM/fD^P (,>) A ‘X:
[*37 61] D : D“P A ‘Q“X C D“(P! QU‘X (2)
h . (1). (2). D h . Prop
*82 33. h:«1Qcl-»l.«e D‘£< . Z> . D“(P Q)SQ“k = I >“/V*
Vein.
I-. *37-23-26 .Dh/ff D <Q <. D . ( 3 X) .\Ca‘Q.<- Q“\ (i)
H . *74-26 . D
h : *1 (^el -> 1 ,\C(1*Q .k = Q“\ ,5.Qf\e 1 -* 1 .^CD‘C.X = ^ . (2)
[*82-32] D . D“(P j Q)±*\ - D“P A ‘<2“X .
[(2).Hp(2)] D.D“(P Q)SQ“k - V“P*‘k (3)
h. (3). *10 11-23-35. D
h:.**]Q«l-*l s(aX).XCa‘Q.<-Q«\:D. D“(P| - D“/V* (4)
H . (1). (4) .Dh. Prop
The following propositions (*82 4 41 41142) are lemmas for *82 43, which
is used in the proof of *114 5, in the theory of cardinal multiplication.
*82 4. h : Te 1 -♦ Cls . P“\ C CI‘7\ Z> . T { “PS\ C (T , P) A ‘X
Dm.
h. *8014.
*71 25.
DhsHp .Jit
JVx
. D . 7* P e 1 -> Cls
(1)
H .*8014.
*34-34.
D K s Hp .R e
/VX
.D.r\R<ZT\P
(2)
h . *80-33.
D h: Hp. R e
P*‘x
.D.D‘PC(I‘7\
[*37-322]
^.a\r R)-a f R.
[*8014]
d. a \r R) = \
(3)
. (3) . *80- 14 .
Dh:. Hp. D:
PeP
*-X.D.r|P€(2r|P) A ‘X:.DK
. Prop
*8241. h
:2r«Cl8-»l.
Me{T\P)S\
.D.
7i»/eP 4 ‘XJ/= P|P| A/
Dern.
h. *8014
. *71-25. Dh:
Hp.
3. PI 3/el -♦Cls
(1)
h .*8014
. *34 34 .Dh
Hp.
D-TiVcfiriP.
[*71-191.*342]
GP
(2)
h. *8014
. *34-36 . D h :
Hp.
D.D‘i/CD‘!r.
[*37-322]
D.a‘(r|3/)=aM/.
[*80*14]
D.a t (T\M) = \
(3)
h.(l).(2)
.(3). *8014.
Oh.
Prop
*82 411. h
: TcCls-* l .
0.(r|PV\
cri
“P A ‘X [*82-41]
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
VM» ..
+82-42. h:T<i-*1.1>“\C(-l‘T.0.(T P>S\-T “Pa‘\ (*824411]
*82-43. t-:rvr^<*-» l - 7 ’“ xCa ‘ :r - xC<1 ‘ < ?-' f = < 2“ X - 3 -
(T I>[\ (?) 4 ‘* = (7’ $)“/VX
l>r in.
h.*s2->7^.Dh:V«l-*l.XCn‘<,».* = ^“X.3.(i J <?>+'* = Q'Ts'X (1)
K % \
* .« I I x . 3 h : vr X C 1 -* 1 . x C 1 >‘.X 1 <Jt. « = i«?fX»“X . 3 .
(/' XI #>*** = <Xl#>“/YX ( >)
H . ( 21 . #3501 :».-.4. *37412. *43 4s 1 . •Nil-14 • 3
8: vrx«l -* 1 .XC<l‘V.«-VX.3.i/*rx vu‘«- 5“/VX < :i >
K|3> 7 ’/'. 3h:^rx. I-*1
.v /*rx vvy /‘>4,‘x i+)
h .(41.*82-42. 3KHp.3.«r /TX <?>*'«- V‘3T “'VX
| *43-202.*37-33] -• T tf)“iVX OK Prop
+8245. h : Q[ X < 1 -* I • X C<I'V• 3 •</*! Q'»‘X si.i /VV“X
Dan.
h . +K0* 14. *37-15 .3 H:K« /VfX . 3*. <l‘W - V“X . I?‘\ C D«<?.
[*14-15] 3 /; . <I‘/{ C I >*y,
[*7472] 3KHp.3. (Q[\rl^Q"\sm
[*82 - 271] 3.11* ^li'XMii 7VV“X OH. Prop
+82 5 I-: 7'[ <^“X t CIs—+ 1 . <?f X « 1 —* 1 . X C <I‘Q. 3 .
(/’ <?l i ‘X*inl>"iVG“X [*82-45. *81-21]
#82-51. H s f jc «Cl* —► 1 .«10«1 -*1 . X C ( l‘Q. k = Q“\ . 3 .
(/’ Qfe'XsmD“/V* [*82-5.*74-251]
#82-62. I- : P r * * 01s -+ I. * 1 Q « I -* I • * «I>‘Q< • 3 .< P »'■> D ‘ ,iV ‘
Deni.
h. *37-23 . 3 I-: Hp. 3 . (g/O. k = Q“n *
y . *37-20 . *22-43.3
KK = Q‘y.X = M«U*Q.3.* = tf“X.XC<rQ
K *74101 . 3 H : Hp. <e = Q‘‘X . X C (l‘<7.3 . X = Q‘‘k .
[*82-51] 3.(P 0) i '0“*s.n D»/V* •
[*10 11-28 : 35]3h:.Hp:(aX).* = Q“X.XCa‘Q:3.(7 > <2)+‘Q“*smD“7V* (3)
I-. (1). (2). 3 H : Hp . 3 . (gX). * = Q‘‘\ . X C (I‘Q (4)
V . (3) . (4). 3 1- • Prop
SECTION D]
SELECTIONS FROM RELATIVE PRODUCTS
507
*82 53. I-: P [ k, R f* * e Cls 1 . * 1 Q e 1 -> 1 . * € D‘Q« . P“k = R“k . D .
(P I Q)A‘Q“*sni (P | Q)SQ“* . ^
D “(P | Q)SQ“k =. D“(P I =
£ ja e P“*r .D,./iA8f 1 :/iC P“* j
= D“P A ‘* = D“P*‘*
Devi.
H . *82 52 . D h : Hp . D . (P j Q) a ‘Q“k am D“P A ‘* .
[*81-31] D . (P | Q) A ‘Q“* sin D“PP* .
[*82 52.*73*32] 3 • (P| QV$“*sm (P QWQ*‘« (1)
h . *82-33 . D K : Hp . D . D“(P| QVQ*‘« - D“PA (2)
[*81-31] -D“P*‘* (3)
[*81-8.*40-5] (4)
h . *82-33 . D h : Hp . D . D“(P j Q)SQ“k = D“P*‘* (5)
I- . (1) . (2) . (3) . (4). (5) .DK Prop
-83 sEUi« TION.S FROM « LASSES OF CLASSES
Sil III Hill l'f/ of *S3.
Ill ihi-. number. til.- general propitious which have been proved lor /V*
at., i.. he applied to lie- important s|Ktial case where V is e. In this case, we
have -elect ion^ ti..in . I:.--. - classes; if P picks out a re/>rc$e»tatirt
/{‘a from each elas* a which i' a nieiiiber of*; i.e. we have
a « h . . H*a * o-
The |»ni)i»«iti»iis «l this number result from those of previous numbers
eiiher immediately by »he submitmion -f < for P. or by the use of preposi¬
tion > of *(>-.' notably r.‘o=a t«i>2 2). and c"* * #V (*G2'3).
The pro|».^iiioiiH ..t the present number follow, in the main, the same
course as those o| *s<>. with e substituted for P (except that the special forms
of propositions lieloiv *st> 2 are not given). We have first a set of propositions
resulting immediately from early propositions of *80. Of these the most used
are :
*8311. h:A **.D.* A ‘*-A
This leads to the proposition that an arithmetical product is null il one
of its factors is null. (Wo cannot prove the converse universally without
assuming the multiplicative axiom.)
*8315. K*V.\ = /‘A
Thus «**A is a unit clas>. This is the source of the proposition m“= 1.
where p is a curdinal (ef. note t*» *83*15).
*83 2. H 1 <« . D : a « * . s . K ! li'a .*./?* at a
Here P*o is the " representative ot a.
*83 21. I- : 1< * . D . I>‘/f C s*k
We lmve next a set of propositions f**T4—44) on selections from unit-
classes and classes of unit classes. We have
*83 41. h .< A ‘f‘asma
This leads to the pn>|)osition that a product of one factor is equal to that
factor.
*83 43. H:*C 1 . D . = f‘H T *> = *'<« I* *>
This leads to
*83 44. h :* C 1 . D. 1
whence it follows that a product of factors, each of which is one. is one. This
holds even if the number of factors is infinite or zero.
SECTION L>]
SELECTIONS FROM CLASSES OF CLASSES
509
We have next a set of propositions (*S3*5—'58) on changing the repre¬
sentative of a class, and on selections from a class of classes some of which are
unit classes. These propositions are seldom referred to in the sequel.
We have next (*83 6—*74) a set of propositions «»n the domains of selec¬
tions, i.e. on the class We have
*83 66. h : g ! . D . s‘ D“*V* =
(The hypothesis here cannot be dispensed with unless we assume the
multiplicative axiom.)
*83 7. KD
*83 71. h . D“s A ‘t"a = • D‘o *]7- a
We have next two propositions (*83*8*81) on the types of e± l K and "D" €&,**.
The type of D “ca'k is the same as that of k (*83*81).
The last set of propositions in this number (*83*9—*904) deals with the
existence of selections. We have
*83 9. h . a ! e*‘A
*83 901. h : a ! 9 ** 1 *a . m . g ! a
*83*904. h : g ! € A ‘(* t‘/9) . 3 . g ! . g ! /9
From these propositions we shall deduce by mathematical induction that
whenever k is a finite class, e A ‘* exists unless Atk (cf. *120*62). Thus a
product consisting of a finite number of factors (which may themselves be
either finite or infinite) can only vanish if one of the factors vanishes.
*831. h : a ! . D . A*>*« k
Dem.
h . *80 2 .DI-:Hp.D.*Ca‘f.
[*62*2.31] D . A: D h . Prop
*83 11. I*:A«*.D. 9 A ‘k = A [*83T . Transp]
*8312. t-.€s‘K = (etK)±*K [*80*23]
*83 13. : A~€* .Q = c\- * .3 . €a ‘* = Q A ‘(I*Q [*80*24 . *62 231J
*83 14. h : g ! . Q = € f « . D . = Q A ‘CI‘Q 1*83*1 * 13]
*83*15. h . €a‘A = i‘A [*80*26]
In virtue of this proposition, the product of 0 cardinal numbers is 1—a
proposition of which a particular case, namely ^ = 1, is familiar. This arith¬
metical proposition results from the above as follows. We shall define the
product of the numbers of members of k as the number of members of €&‘k.
Thus when k = A, the number of members of is a product of 0 factors.
Now by the above proposition, « a ‘A has one member, namely A. Hence a
product of 0 factors is 1.
*83*16. h:g!«.D. A~ce**« [*80*28]
PUOl.F.coMKXA TO CARDINAL ARITHMETIC
[l'ART II
-83 2 4 . 3 : * «• * • s • K ! R*a . = . K'o < a (*80 3*2 . *62-2]
*83 21. 4 : «r tV < . D . I >‘ H C f** 0:W . *62 3]
>83 22 4 : li < . D . E !! R“* . = D‘7? [*80 34]
-83 23. 4: l>‘/»*«7 r.jo>. . •**« /?*« [*S035]
*83 24 »- : /f « *V* . a «* ..» e a . D . [! 1 <-t /f‘«» 1 a\ w.r l al < «a‘* [**0-41 ]
• 83 25 4 : ;.J ! tV* . 3 . *•**•* = *[« [*80*42]
-83 26 4 : if * «f * . $| ! V*‘* • 3 • = V l* s3 1
-83 27 I -»< 'Is . h : a « U‘/f . D. . H'a * a [*62*45 . * /1 * Hi)
■83 271. 1- K * «*•< |«/f. 3 : a e < I * /? • >.. /f‘a < a [*S3 27 . *86*14]
*83 28. H .iiot'f.D,.
[*83-27 .*80 14. *14 15)
• 83 29. 4 rt « •*•* . s : o € * . s* . /f‘a c a ; < l‘Jf - * l* 83 ” 2 28 J
*83 3. 4 s. * a X - A . D : .1/ f «r*V ^ X). ^ .
<//. N)./*’« tV* . *S' < <S\ . .1/ = /{ u .S [*S0 66]
*83-31. I-* a X - A . D : IU *S« . * < t^‘X . 5 .
i : .| .1/1. M t « A ‘<* sj X). If - M r * • $ - M r X 1*80 07 ]
*834 4 . tV'*a - j a“o (*80-45. *02*2)
-83 41 (• .tj‘»‘flMno |*83 4 . *73*ol 1]
Thin pro positi«*u shows that a cardinal product of one factor is e.pial to
t|, ;l | isicinr. K..I- tl»«- niaiiila-r of members of « A Va is the product of the
numbers of numbers ..f im-ml-rs of ,‘a. i.r. it is a product "hose only factor
is the number of members *•! o. By tlu- above proposition, this product is
fipml to tlu- munln-t of um-iiiIm-is of a.
*83 42 H . - i‘t« 1 M - »“«« T
lh '"' 4 .*83*12 . D 4. I*
(*62 56] w = (T[ i* t a)± i t ll a (D
4.*72181 .*7l-26.DK#r» M «« * 2)
4 . *37*15 . *33 21 . D4.f«*CU , ij
[*35 65] D4.« M *-a*(Tr* M «) ^ < 8)
4 . ( 2). (3> . *82-21 . D 4 . (7r «“ah‘l“a = i‘|0 t* '“«>r «“«!
[rtMl] =.‘(Tr-‘‘a) (+)
[*0-2oG] =l‘(«10 ( 5 >
4.(l).(4).t5).D4.Prop
This proposition shows that a cardinal product whose factors are all 1 is 1.
For i“a is a class whose members are all unit classes, and thus the numbei
SECTION D]
SELECTIONS FROM CLASSES OF CLASSES
511
of members of e±‘i“a is the product of a number of Is; and by the above
proposition, £ 4 , t L tt a is a unit class, its sole member being a ] 1 . This result is
rendered more explicit by *834344.
*83 43. = i‘( 7f* *) = t‘(« p *)
Dem.
1- . *83 42 . D h : * ■ i“a . D . e*** = i*(i f* *) (1)
K (1). *1011-23. D
h : (go) . * = i“a . D . = i*(i p k) :
[*52-31] D h : * C 1 . D . = e*(t p *)
[*(i2"55] = t*(e P x) : D 1-. Prop
*83 44. huCl.D. e 1 [*8343 . *5222]
*835. h : 72 c €&*k . a^e k .xe a . D./?o,r Jae ^ i*a)
Dem.
h . *80 43 . D 1-: Hp. D . x i a e « A Va (1)
h . *51-211 . D h- : Hp . D . * r» <‘a = A (2)
t*. (1) • (2). *80-65 . Z> I-. Prop
It follows from this proposition that if k is a class of classes for which
there are selections, and if one member (not null) be added to tc, there are still
selections from the resulting class of classes.
*83 51. [*80 78]
*83 52. h : R € e A ‘/e . a c k . area . D . J/2—(/2‘a) ^ a) c# x X a « [*80 41]
*83*54. h:<nX = A.XCl./if * . D . H 0 t p X e ^ X.)
Dem.
h . *80 65 . D h Hp . D : &c c A ‘X . D . R kj S e c A ‘(/c w\) (1)
H . *83-43 . D b : Hp . I> . « p X e £ A ‘X (2)
h.(l).(2).Dh. Prop
*83 66. A.XC1 • S e e A ‘(* u X) . D . S-w p X t f A */c
Dem.
h . *80 66 . D 1-: Hp . Z) . (gAf. A^) . A/e e A ‘* . N e c s ‘\ . S = M sv A r .
[*83-43.*5115] D . (gA/>. A/e e A ‘* . S = M v Tp A (1)
V . *80 14 . *35 64 . D 1- Hp . D : M e c±‘k . D . d‘A/ n CI‘(7p X) = A .
[*33-33] D . A/ A Tp X = A .
[*25-4] D . (A/c; 7p X) —Tp X = M.
[*1312] D-.AfecSic.S-Mw'ltX.D.S^irXceSx (2)
h. (2). *10 11-21-23. D
I- Hp . D : (gA/) • M « • S = Af o t p X. D . 5-2-t p X e cjtc
h . (1) . (3). D h • Prop
(3)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
:>1J
*83 56 K:*aX = A.XCI .
D . « v \) = j) <'.[ H >. K € c• M = H v 1 [ *•!
I>em
K . *s<MH>. D I-II|* -1>:
.1/ e *V< x yj \». a . 15 | /f. »Sl. A' < tiV . *S « <.\‘X . .V = /? ci 6* •
[**3*431 = .<>| H). He c A ‘*. .1/= o 1 1* X D h . Prop
The |i»l lowing pr.*p*»>ili'»n is used in the theory ««f cardinal multiplication
i*l 14-41 >.
0 )
( 2 )
• 83 57 hun\ = A.\Cl.D. e ^ \ > >"•
bem.
h . *83 r»i; .*3s i:;i . 3 h : lip. D. t*i« v \ i- (cm |*X >“«***
h . *NO I + . *3V34 . D h : 11 P . C ^ v . D . < I ‘ A A < I ‘(/ r X) -= A .
1*33-33] D./IAiTX-A.
D. A = (Ao7rX)^rx
I- . c2). *23*481 .*13172.D
K: Up. It.SetS*. /fci7rx-N«ii[*X.D./f-.S:
| Kxp.*Mil 3.*3* 11) D h Hp. D :
/A 6'e eV* . <o 7f X)‘/f - to 1 1* \YS. 0,:, s • Ji m s:
|*33 I >.*73 2:.) Dj.o/rXi'VAMnt^ (3)
h .(1) .(3). D h . Prop
*83 58. I* . <a‘* *•»» * — 11
Deni.
K *24-41 -21 . *22-43. D
!-.* = (*- 1)v(*a1).(*-1>oi*a 1>~ A **a 1 C 1 (I)
I- . (I). *83-57 . D h . Prop
Till* proposition shows that in a product nn.v number of factors each equal
to 1 may be omitted without altering the value of the product.
The following propositions, down to *83 74. are concerned with the domains
of selective relations, i.e. with the selected classes.
*83 6. V-.Itf <r.‘* . a ( k . O . a ! 0 n
Deni.
P.*83-2.3h:Hp .D.-ft'oca.
[*33-43] D.«‘oea«D‘/f.
[*10-24] 3.a!«nM:DKPhip
SECTION D]
SELECTIONS FROM CLASSES OF CLASSES
513
*83*61. b : R e e A ‘/c .a<<.an s‘(k — t‘a) = A . D . a r* D‘R = i*R*a
Vein.
b . *40*27 .Dh.oA s‘(* — t*a) = A . = : /3 € * — f‘a . 3„ . a r* /3 = A :
[Transp.*51'15] =:>3e/c.g!ar*^.D / ,./5=a ( 1 )
b . *83*23.3 1-:. Hp . D:xeD*R. = . ( 3 y9). /9 e * . * = /*‘/9 .
[*10*35.*14*15] ^ : x e a n D‘Ji . = . (g/Sj. /9 « * . x = R‘f3 . R*/3ea .
[*83*2] = . (3/9) . /3 c *. x = /2*/9 . ft‘/9 e a r\ {3 .
[(1).*471] =.(a/3).|S«.^^.l?‘^o^.a-/5.
[*13*195.*22*5] = . a € tc . x = R‘a . R*a e a .
[ H p.*4*7 3.*83*2] =.x=R*a (2)
I- .(2). *51*15.3 b . Prop
*83 62. h : € D“ f4 ‘/£ .3. M CsV [*83*21 . *37*63]
*83 63. b : s‘icr\s*\= A ./ifD‘V(/fwX). 3 . . /xr>«‘X«D“e A *X
De/n.
h . *80*62 . 3b: Me e A ‘(* u X). 3 . Mf* e e A ‘* . ;l/[*X e e A ‘\ . (1)
[*83*21] 3 . DM/ r * C s*k . DM/ |* X C s‘X (2)
I-. (2) . *24*494 . 3 b :. Hp . 3 : M e e A ‘<* v \).3.
dm/ r * - (dm/ r* ^ D'A/r m - . dm/ r x=(DM/r^ ^ dm/ r .
| *33 26.*35*412.*80 29] 3 . DM/r< - DM/ - .v‘X . DM/f X- DM/ .
[*24*491] 3 . DM/[V - DM/ n . DM/[*X = DM/ ^ s‘X (3)
b . (1) . (3) . *37*6.3 h Hp. 3 :
Af e €±\k v \) . 3 . DM/ n e D“* A ‘* . DM/ *‘X « D“c A ‘X :
[*37*63] D:pe D “* A ‘(* uX).3. M a«‘« D“€ A ‘* . M n a‘X * D“e A ‘X:. 3 b . Prop
*83*64. b * n \ = A . 3 :
M 6 D“# A *(* w» X) . 5 . ( 3 p, a). pe D “c^k . <r f D“e A *X . p = p \j <r
Observe that the hypothesis required here is * r» X = A, not v‘* ^ «‘X = A
jis in *83*63.
Deni.
I- . *80*60.3 b :. Hp . 3 : Me f A ‘(* v X). p = DM/ . = .
( 3 «. S) . R c c*‘k . ,S'« « A *X ,M=RvS.p = DM/.
[* 13* 193.*33*26] = .(^R,S). Re eS* . S c e A ‘\. M = R\y S. p = D‘R \j D‘,S* (1)
b . (1) . *10*11*21*281 . *37 6.3
h :: Hp. 3 :. pe D“* A ‘(* uX).= :
(3 J/, R,S). Re cSk . iST« « A ‘X . M = Rkj &. p = D‘/2 c# D‘&:
[*10*35] = : (3 R, S ): R e e A ‘* . S € e A ‘X. p = D *R v D *S : (3 M). M = R o ,9:
[*21*2] = : (3 R, S). Re c a ‘k . 6 ’* e A ‘X. p = D‘R v D‘ 6 ' :
[*13 22] = : (g ft, 5,p, a) . Ree^K . p = D‘Z2.6Ve A ‘X . <r = D‘*9 . p = p v a:
[*11*24*54] = : (3 p, a) : (3 R). Re e*‘* . p = D‘R : ( 36 ') . S e e A ‘X . <r = 1)‘S .
M = r o':
[*37*6.*10*35] = : ( 3 ^, a) . p e D“e a V . <r e D^c^'X * p = p.\J <r :. 3 b . Prop
R&W ! 33
prolegomena to cardinal arithmetic
[part II
The following proposition is user! in connection with cardinal multiplication
<*1 I.VUl.
v*83 641. h .n‘* a s*\ = A . 3 :
n e I >“***( * V X). = .<gp <T). p € . <r € I>“€a‘X . p = p ^ <r
/)r«(.
h . *53 2'>. 31-:. Hp . 3 : * a X = A a Hs. v . * a X « f‘A 0)
h . *83-64 . 3 I* :. * a X « A a <’ls. 3 : p c D“c A ‘<* v X). ze .
(flp. <r). p € D"« 4 ‘« • * * D“€*‘X .n=pv<T (2)
h . *51 Hi. D H # a X = i‘A . D : A < <f • A « X s
[*83*11] 3:< a ‘*-A .i 4 ‘X-A.< 4 , (* w M = A :
| 2!» | 3 : I >‘‘*V* = A . D"*VX - A. 1 >“«aV ^ X > = A :
|*24‘15] 3 : 0“«a‘(* * X) s (p). P~e :
| *l i:»5.Transp.*IO*2:>2| 3:p~* 1->“« A ‘<* « X):
-%,<j|p. <r) • P * D m «aV . <r el) l4 tVX . p = o v/<r :
[*.V2I] 3 : p w»X). = .
< : .|p. it) . p < . <r € D“«**X . p - p ^ <r (3)
I- .<1 ).<2>.<3>. 3 I-. Prop
‘-83 65 h :*«* ax«X - A ./if 1 ^ ^ X). 3 .
p - *•* € 1 )“ca < X . p - *‘X € r>“< A ‘*
Item.
h . *8.3*62 . 3 H:Hp.3 . p C s‘(k v X).
[*40171) 3.pCAu«'X (l)
h .(1). *24-401.3 h : Up. 3 . p — #** ■/lAi'X.p- <‘X « p a «** r2)
h .(2). *83*63. 3 H. Prop
*83*66. h :a !« A ‘*.3..t 4 D“<A‘*-«‘*
Deni. . ,i\
h . *4143 . 3 I-. = Wi'tt'* < 1 >
h . *83-25. D h : Hp. 3 . DV« a V = D‘e [ *
[*(>2'43] - *'* (2)
h . (1 ).(2). 3 I-. Prop
*837. h.l)"( 1 , i‘o = i“« [*83-4. *55-201]
w
*83*71. h • 1 )“< A ‘i“o= i‘a. 1)‘«1« = a
/>ew.
1- . *83 a 42.3 h . D“€ A ‘i
<«a*D“<‘(a1<)
[*53-31]
SlWlO
(1>
[*35*61]
= I*(a a D‘t)
[*33-2]
= i‘(a a Q‘t)
(2)
[*51*17-*24*26]
1-. (1). (2) .31*. Prop
= t‘a
515
SECTION D] SELECTIONS FROM CLASSES OF CLASSES
*83 72. I-:«C1.D. = iV*
Deni.
1- . *8343 .Dh Hp . D . D“f,V = D«i«(e f *)
[*53*31] • = t‘D‘( € p*)
[*02-43] = iV* : D »-. Prop
*83 73-731 are lemmas for *83 74.
*83-73. hunX-A.XCl.D.
KJ \)= a [(gp) . p € D . a = p\j $‘X|
Dem.
V . *83-56 . *37-6 . D h Hp. D :
€ D“€a‘(* ^ X) . = . (gie, S). Re e A ‘* . 6* - /f v f [* X . a = D‘.S'.
[*13193]
[*62-43-55]
[*10-35.*21-2]
[*37-04]
(g/*.£) . It € .S=Rvif\.<r = D ‘(It VI r X).
(g/e. ,$•>. ye * « A ‘*. s - /eot r x. <r - D‘it ^ *<x.
(g It) . It e € A *K . <r = D*R yj s*\ .
(Hp) • P € . <r — pv «‘X D f- . Prop
*83731. h:.XCl.D: r\ .*‘X = A.D.<nX«A
Dem.
h . *53*25 . *51*16 . D H s‘k r\ s*\ = A.3:<ftX = A.v.AfX (1)
h . *5216 . D I- X C 1 . D : a « X . D. . g ! a :
[*24 63] D : A~<X (2)
H . (1) . (2) . D h . Prop
*83 74. h : 8 *k r\ s‘X -A.XCl.D. D“c*‘(* v> X) sin
Dem.
h. *83-73-731 .*38131 . Dh: Hp . D . D“ €a ‘(* X) - (v s‘\)“D“c±‘k (1)
h. *83-62. *24 13. D
I- :: Hp . D :. p, v « • 3 : p n *‘X = A . */ r\ s*\ = A :
[*24-481] D : p ^ «‘X = v v «*X . = . p = i >:
[*38*11] D : (u &‘X)V = *‘X)‘i/. s . p = i/ (2)
I-. (2). *73-28 . D h : Hp . D . (v s‘X)“D“* A ‘* sm D“« A ‘* (3)
h • (1) . (3) . D h . Prop
*83 8. h . € A *ac C *,«** . € A ‘/c c
Dem.
1-. *8014 . *83-21 . *35 83 . D z It e e*'* . D . 22 G f « •
[*63105.(*63 03>] D . R G £,-* t C* .
[*64-201] D . ie e W* t «.'*) -
[(*64-021)] D.Ret*'* (1)
H . (1) - *63-371. Dh. Prop
33—2
;,10 PROLEGOMENA TO CARDINAL ARITHMETIC
-83 81 y . Ic K l K . I >“*V* € c<
g rw r#»,
D ./iC a-‘*.
(*<»3I05.< *0303)]
D .fiCt ,**.
[*03 51]
h . (1). *03 371 . D y . Prop
*83 9
y .g '• «a*a
[*8315]
*83 901.
H !€ A ‘/ 4 o.= .:•( !a
[*.80-4(1. *02 2]
• 83 902
1- v,\). r .a ! . >| !«*‘X
[*8009]
*83903.
1- : :•( ! e A V« s. #VJ>. = . M ! a • M '• &
[*83 901 902]
>83904
h : '.| ! f A ‘<* w i‘£>. - . H • 3 ! ^
[*83-901 -902]
*83 9-904 lr;nl to an in<luctivr proof (to be Riven later) of %| !
ever k is a tinit <- elans of classes nolle ol which is A.
[part ii
( 1 )
1 k when-
*84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES
Summary of *84.
A class k of mutually exclusive classes is one such that, if a and f3 are
two different members of k, a and have no common members; i.e. it is
a class composed of non-overlapping classes. Classes of mutually exclusive
classes have many important properties. They are important in cardinal
arithmetic, among other reasons, because if k is a class of mutually exclusive
classes, the cardinal number of $*k is the sum of the cardinal numbers of the
members of k. Also if * is a class of mutually exclusive classes, the number
of selected classes of k (i.e. Dis the same as the number of selective
relations (i.e. € A ‘/c).
"* is a class of mutually exclusive classes" is written "k e Cls* excl.”
An important case is when no member of k is null; in this case we write
k e Cls ex* excl.
For a Cls* excl which is contained in a class of classes 7 , we write
Cl excl‘ 7 ,
on the analogy of the notation Cl‘y.
The definitions are as follows:
*84 01. Cls* excl = *(<*, + D., a .an^ = A) Df
*84 02 . Clexcl ‘7 = Cls 9 exclr*Cl ‘7 Df
*84 03. Cl 8 ex* excl = Cls* excl — c ‘A Df
The propositions of this number begin (*841—-14) with various equivalent
forms for the definitions. Of these the most useful are:
*84 11. h k e Cls’ excl. = : a, /9 « * . 3 ! a . a = /3
*8413. h : k e Cls ex’ excl . = . k e Cls 1 excl. A~€ k
*84 14. z k e Cls’ excl . = . e f tc e Cls — ► I
The last of these is specially important, because it renders the propositions
of *81 applicable to e A ‘/c when tee Cls’ excl.
We have next (*842—-28) a set of propositions dealing with various
special cases, such as A and 1. The most useful of these are
*84*23. h.t'ae Cls* excl
*84*241. 1-. l“ct e Cle ex* excl
*84 25. H : k e Cls* excl .XCx.D.Xc Cls* excl
PROLEGOMENA TO CAR PINAL ARITHMETIC
[part II
w,. |)• • \i have a set ol propositions <*84*3—37) which are immediate
unices of propositions in *M. by means of *8414. The most uselnl
• •I t IlfM* IS
• 84 3. V : k t (‘Is* exel. D . *±‘« Cl —» 1
\\ .• next have a set of propositions ,*84 4—431 dealing with the domains
..| seleetions from a (’U-exel. The*.. :,»«• for tlie most part still immediate
. ..user, lienees of projM.sitions in *M. in virtue of **414. The most uselul are
• 84 41. H* : « e l 'ls s exel . Z>. l>r*V*t I -* I . I >“€.»** sm^**
84 412. h : * * < Is-exel . D . I >“*V* - A o « * . . n rs a e I : ^ C xV.
• 84 43. I -i.a.fSti Ms* exel. x*a * OsaCUV^.a^CI >“€ A ‘a
This proposition applies to such cjim-s as the relations of rows and columns.
Imagine miv set of terms arranged in rows and columns so as to form a
,.. r ,angle. Then each column is a selection from the rows, and each row is a
...lection from the . .. This is a particular ease of the above proposition.
We Iie.xl have a set of pio|n»sitions on /{‘V /»*“*, and /V*x (*84\> ’•>•>».
Tin- most important of these are
-84 51. I-: //[" «< 'Is—» I . D. It"" • Civ oxcl
84 53. t-: It . CIs -.!.*«< 'Is' vxcl .0. If"" , Civ ox,I
Filially wo l.avo a sot.,1 |m.|«Mliuni> (**+-59—UiMuwiiiR circunwtniifou
11,11 lor whioh ,«xi«a C'lv ox.-l. Tl.o ..nly oi.o ol thoxo which is used s„b-
seipiently is
-84 62. + i*& « Clvcxol. 2 .aA/Ja.\
*84 01. Cls’excl =x(a.tf«* + A )
*84 02. Cl exel ‘7 = CIs’exel ^ Cl ‘ 7 1){
*84 03 CIs ex’ exel = CIs 3 exel — e* ‘ A 1 M
-84 1. h k € Cls s exel . = :a.^€x.a=4^*^-.tf* an ^ = ^
[*20-3. (*8401)]
*84 11. V x e CIs 3 exel. = : a, 0 £ k . a ! a n • ° = ^
[*841 .Transp]
*84 12. h x f Cl exel* 7 . s : a, 0 € k . a * /3 - 3-.* • a r\ 0 * A : * C y : = :
* c CIs 3 exel. x C 7 (*20*3 . (*84 02). *22*33 . *84*1 ]
*84121. h x * Cl exel* 7 . = : a, & € *. a ! « n • a * ^ : * C 7
[*20*3 . (*84*02). *22*33 . *84*11 ]
519
SECTION D] CLASSES OF MUTUALLY EXCLUSIVE CLASSES
*8413. h : K € Cls ex 2 excl . = . k e Cls 2 excl . A k
Deni.
. *22-33 35 . (*8403). D
: k € Cls ex 2 excl . = . k c Cls 2 excl . k~€ c* A .
[*6221] = . k € Cls- excl . Prop
*84131. k e Cls ex 2 excl. = : a, /9 « k . « + ft . D*,# . a n *3 * A : A k
[*84131]
*84132. I- k e Cls ex 2 excl . = : a. /3 € /c . ftlar\ /3. D 0><i . a =*/9: A~e k
[*841311]
*84133. 1- k e Cls ex 3 excl . 5: a, . g ! a a /9. D a>p . a = /3 : a e * . D a . ft l a
[*84132 . *24-63]
*84134. H :: k e Cls ex 2 excl. = a, fie k . „ zftla.ftl/3:ftlar*t3.0.a=i/3
Dem.
h. *11-59. Dh:.ai/f.D..a!«sB:a tj 8€*.D, iJ .a!o.a! j fl (l)
h . *4 87 . *11-33 . Z) n a,(3 1 k . ft \ a r\ f3 . 0 . a = /3 : =
et,/9 :g! a a/ 9. D. a(2)
1- . (1) . (2). *84-133 . D 1- s: * « Cls ex* excl. = :.
or,£ e k . D.,* . g ! a . ft ! /9 ;. a, /9 e k . !>«.* :g!ar»/9.I>.a=«/9:.
[*11-391] = :. a,0 c k . D a> * lala.al^gJan^.D.a-^iOl-. Prop
*84 135. K :: /c c Cls ex 3 excl. = a. £ e k . D..* : g ! a a /9 . = . a — £
Deni.
h . *84133 . *22-5 . *13191 . D
1" :: * * Cls ex 3 excl. s:.a,/9«r*.3 !aA/9 . D a ^ . a = /9 :
a, /9 € *. a = /9 . D„. * . g ! a r» & :.
[*1131] = (cr,/9) o ,/9c* .g!ar*>9.D.a=»/9:
a, /3f/«.a = ^.D.g!ani9:.
[*4*87.Comp.*1 1*33] = (a, /9) a, /9 e *. D : g ! a a /9 . = . a = {3 :: D \-. Prop
*84 14. 1 - : k e Cls 2 excl . = . e f* e Cls —► 1
Dem.
h . *1023 . *84-11 .Dh:./ce Cls* excl . = : a, /9 e * . x c a . x e /9 . I>r. a .#» . a = /9 :
[*35 101 ] = : x <€ ftc) a . x (* [**) /9 . D Xi „ >ft . a = /9 :
[*71171] ssef/ce Cls -» 1 D I- . Prop
This proposition is important, since it enables us to apply the propositions
of *81 to when k e Cls 2 excl.
*842. l-.An Cls e Cls ex 2 excl
Dem.
h . *24105 . *11-57 . D h . (a, /9) . a, f3~c AnCls.
[*11-25-63] D h s. a, >9 c A « Cls . D.,* : g ! a « >9 . = . a = £ :.
[*84135] 3KAn Cls e Cls ex* excl
I’KOI.KOOMEXA TO 1'ARIUXAL ARITHMETIC
[PART II
•84 21. h.l„,C CIs-excl
.Vote. I,|. is iIk* class <> f all unit classes whose members are classes:
ibis results Iroiii *(>501. Thus "o* l,,.' is equivalent to “a consists ot
• •II'- cln>*»
lit m.
I- . #22-38 . (*(>50| >. D h a « l,,„. = : a € I . a CCU :
|*52 1l>) D : £.7 c a . . 8 = 7 :
| *:<•+!] 3:/J. 7 eo.a!3«7 - ->‘-y • <* - T *
| *s4 | | | D : a < I 'Is- exd 3 b . Prop
1*84 22. b . I f CI'fN rxd
l)e>ii.
b . *52 41 ;. D b a. £ < I . D : 3 ! a o /* . = . a ~ 3 (' >
b . ( I ) . *X4*ltt5 . D b . Prop
f.84'23. b . Pa€<'ls-. \cl [*84*21 . *52*221
*84 24. b : ;.| ! a . D . t‘a » 1 'Is ox= exd
Dew.
b . *13*101 . D b Hp .D:£-a.Drt.a!0s
[*5115) 3:0«fa.D,.:.|!0i
(*24I53) D:A-wi 4 a (1 >
b . (1). «*4'23*13 . D b . Prop
*84 241. b . i“o € t 'Is ex' exd
lie w.
b . *5*2 3 . D b £.7 c i“a . D„. r s £. 7 * 1 *
[*524(5) ^sa!^A 7 .«./}-7 (1)
b .(l).*H4135.Db. Prop
*84-242. b:<Cl . D . * « CIs ox* exd [*524(5. *84135]
*8425. b : * c CIs 3 exd .\C«f.D.X< CIs 3 excl
Dan.
b . *221 . *1 1 -5ft .Db:. \ 3-.* • a. 0 e k :
[*| 1 :iH) D : a. £ f X . a * 0 . D..* . a. 0 e k . a * 0 (D
b.*84-l. D b k c CIs* excl . D s a.&c k . a + • « a A (2)
b . (1) .(2). *11-37 . D b Hp . 3 : a./9cX.a*£. 3..* . a r» # = A :
[*841] D : X € CIs 3 exd :.Db. Prop
*84-26. b : * e CIs ex* excl . X C * . D . X € CIs ex 3 excl
Ih '"' b. *8413-25. Db: Hp. D . X € CIs* excl (D
b . *221 . *10 1 . D b Hp . D : A € X . D . A € * :
[Transp] D : A~e k . D . AX (2)
b. *84*13. DbsHp.D. A~c* ( 3 >
b. (2). (3). Db:Hp.3.A~€X (4)
b. (1). (4). *8413. Db. Prop
SECTION D]
CLASSES OF MUTUALLY EXCLUSIVE CLASSES
521
*8428. b : k e Cl excl*y .XC/c. 7C8.D.X«C1 excl‘8
Bern.
b . *8412-25 . D h : Hp . D . Cls*excl (1)
b . *8412 . Dh:Hp.D./cC7.\C*.7CS.
[*22-44] D.XC 8 (2)
b . (1) . (2) . *8412 . D b . Prop
The following propositions are concerned with selections from a CIs*excl.
In virtue of *8414, the propositions of *81 which have the hypothesis
iiT^fCls—>1 become applicable when R is e and k is a Cls-excl. Thus
€&‘/c has many important properties when k is a Cls ? excl which it does not
have in the general case.
*84 3. b :«cCls*excl.2.<A*«C 1 -► 1 [*8414.*81 1]
*84*31. b : k € Cls a excl . R e e A */c . xe D‘R . D . E ! R*x [*8414 . *8111]
*84 32. b z k € Cls a excl. R e . * c D‘R . D . ar c R'x . R l x e k
[*84 14. *81 11 .*35101]
*84 33. b : k e Cls a excl . R e f A ‘/c . xe D‘R . D . R*x^{ia) (a e <.xea )-(k *] e)‘x
[*84-14. *8112]
*8434. b :. k e Cls a excl . R e «*** . D : xRa . a . xe a . x e 1 )*R .aetc
[*81*18. *84-14]
*84 341. b : x € Cls a excl. R e ,0.R = D‘R *1 e f * - € A D‘7* f *
[*81-14. *8414]
*84 342. b z k e CIs a excl. R e c A ‘tc . a e k . D . l‘R‘a = a D‘/f
[*81*15 . *84 14 . *62*2]
*84*35. h k e Clsex a excl .DzRe e A */c. = .Re 1 —» 1 . /£ G € *.Q‘12 «■ [*
Dem.
b . *8413 . D b : Hp . D . *.
[*62-42] D.a*e\-K = K (1)
h . (1) . *71 103. *8014. D
b:. Hp.D:/lel—»1 .fiGtf * . (1*R = G*€ [ * . D . 72 e e±‘/c (2)
b . (1) . *8014 . D I-Hp . D : R e e A *K . D . O'fi = d‘« [* * (3)
b. (3). *80 291 .*84 3. D
b :. Hp . D : R e e A ‘* . D . R e 1-* 1 . R G e[ k . a *R = (!*€?* (4)
b . (2) . (4) . D b . Prop
*8437. b : * e Cls a excl. g ! e A ‘* . D . * eCls ex a excl [*83T.*84T3]
*84*4. b z .k eCls'excl.R,See A ‘*. DzD*R*=D‘S .= .R = S [*81-2.*84-14]
>22
PROI.ECOMKNA TO CARDINAL ARITHMETIC
[PART II
*84*41. :*cn*Vxcl . D . l>r« A ‘*« 1 —♦ 1 • !>“<*■* mu € A ‘* [*sl*21 .*8414]
This is :iii iii 11 >««rinii« |ir<>|x»>itioii. since il shows that, when k is a Cls-excl,
the numher ol classes ihat can he selected Iroiii * is the product of the numbers
• *l the various classes that are meiiilicis ••! k.
-84-411. h:.#Hf.D,.MAa«l [*81-212 . *62-2-8]
•84 412 h:*.CIs exel.D. I>“€ A V = 2 a < * . D« . y. n a « 1 : C***',
{ysl 2*2. *s4*14.*i»2*2-3|
I'his |»i..|i..siiion gives vvh.ii mil'll! he taken as the definition of the class
• >l selected elas-.es. namely
£ o * k . D* . n n a € 1 : /« C *•*}.
We might siarliiio with this as our di-finition. deal with the class ol
selected classes W it honi liis! considering selective relations. The disadvantages
• •I this method would he first, that it requires that k should he a ('Is-excl il
ii is in give ihe results desired in arithmetic: secondly, that it is much more
cutiihioiis technically than the method which proceeds by selective relations:
thiidlv. that it. does nol enable us to deal with selection from a class of classes
as ;i particular case of selection from a relation t namely from t [ ), nn«I there¬
fore does not yield theorems of such generality as those obtained by the
met hod adopted above.
‘84 42 h : * t < Ms- exd . a « * . /* < I>"«**< • ^ I< - ,ta)
|*M -24.*84 I4.*«i2-21
‘84 421. !-:«««..rca.p* WtSi* - i‘a>. 0 . ,< v /*.#•« 1>“*V* [*Sl-25]
‘84 422. h ««Ch»*oxcl «c I>“€aV - t‘a). s .n€ 1
|*S1-2G.*84 14.*«-2-2]
*84 43 h :• a./$ < Ols ? excl .x‘a-s‘/3. D :oC !>“<*'£. a
Dnn.
K *84*412 . D h Hp.D::
a C l>"c A 7?.s :.^a.D f : ^ ■ :fC
[*40*18. Hp] ~ :>?«/$. 3, : -
[*10-542-21] s »/c£. a. : -
[*40*13.Hp] = »; c £ - 3, : £ ««- • f « c 1 : C s*a
[*84-412] = s. ti € £. D,: * € V“e^a ss. D H • Prop
*845. h : CIs -♦ 1 . D .7?“< 1 4 /f €CIsex-exd
4 .*71181 . D I-Hp. D :g !
[*30-37] ^.^.r=7?‘y (1)
4 . *33*41 . *11 *•>?). D 4 : x,//c(I*/? . D,.„ . g ! R € * • 3 I < 2 >
523
SECTION Dj CLASSES OF MUTUALLY EXCLUSIVE CLASSES
h . (1) . (2). D h :: Hp . D :.ar,y € G.*R . D r „ :
3 ! R‘x . 3 ! ii'y : 3 ! R‘.i ^ R*y . D . 72‘a = 7?‘y
[*37*63] D a, e R“(1‘R . D a> * : 3 !a .3 !£ 13 \a r\ &.a =
[*84134.] D ~R“Q.‘R e Cls ex 9 excl Prop
It might be supposed that the converse of the above would also hold.
But this is not the case; for although /i“Q‘SeCl 8 e.\ ! excl secures that
R‘x and R‘y cannot overlap when the}* are unequal, yet we may have
R*x= R‘y without having x = y, so that if R‘.c = a—R‘y, we shall have
z € a . D . zRx . zRy , whence, if 3 ! a . x + y, it follows that R is not a Cls —► 1
even if R^&R c Cls ex*excl.
*84 51. h : R [* * e Cls -* 1 . D . ~R“ K € Cls 2 excl
Dent.
K *71*171 .*35-101.3
h Hp . D : xRy .ye*. xRz . z e k . D z ... y = z .
[*30*37] ^ D x .„' t .l?y = 'R‘z:
[*3218] D : y, z e * . x e /i‘.y r\ R*z . Vi , . R*y - :
[*10*23] D : y, ^ e * . 3 ! ~R*y r» ~R*z . . R<y-lt*t z
[*37*63] D : a, e /£“* .3 —
[*84*11] D:~R“kc Cls 2 excl D h . Prop
*84 52. H s R [ * e Cls -♦ 1 . * C <3‘rt . D . 7?‘* e Cls ex 2 excl
Dem.
h . *37*2 .Dh. Hp. D : a e ~R“k . D . a cli“(l‘R .
[*37*77] D. 3 ! a (1)
h . ( 1 ) . *84*51*13 . *24*63 . D I-. Prop
*84 521. h : R e 1 -* 1 . R“0 e Cls 2 excl. Z> . /* f/3 e Cls -* 1
Dem.
h. *71*55. *84*11 . D
h :. 72 [* £ e 1 —>1 . R“(3 e Cls 2 excl .Ozy.ze/3. R‘y = R*z . D, /f , . y = z :
y.x*&.^\R l y e\ R‘z. D gtX .R*y—ll‘x:
[*11*37] Dzy,zc&. 3 ! R'ynR'z. D,, %z .y=zz
[*74*62.Transp] D : R [* >9 e Cls -> 1 D I-. Prop
The above proposition is a lemma for *84*522, which is used in an
important proposition on relations of mutually exclusive relations (*163*17).
I’ROI. ROOMKX.\ TO CARIlIXAI. ARITHMETIC
•VJ 1
(PART II
-84522 h:./iC<!‘/,’.D: /ff^ e C|s-» I . = . /ff/J, 1 -> 1 . /{««,* «CV exd
/ frill.
L • H Il|i. D : •/.:*$. D . 3 • 7?//. 3 »7? *:
I *’--•») =>: // - e ^ . K'y = A*'-- . 3 . 3 ! 7?// A 7?.-:
[* 74,i -l => = /,*f*^ t r|s-» | .y.- t 3.7? // = 7?.-.D. v = ::
i * 71 v »i => = /-r.^cu^ I .D.7?r^€l -> I
H -< I >.*M5I . D
f-:. M |». D : // 1 * ,3 « CU —► I . D . /f T £ c 1 —» | . J{“J<C Is 5 exd
H .i2>. *84-521 . Dh.
^84 53 H :/,*«-* *K —» | . *, ('k>xd. D . /C“* <CI.s- .-x, I
ban
I- .*72 421 . D
H : H * < Is -♦ l . /,. t ie« . 3 ! /»"‘o a /C73. D . 3 ! a a /J
^ (I >. Sy II . D !■ !• //1 Cl> —♦ I !flA/3.Di
o. ,3 « * . 3 ! /C‘a a /f“,3. . . « - £.
[*30-37.*371l 111] D..„./f“a-/?“,3:
| *37 03.<*37U4>) D : p. «r « /f«“* . 3 ! ^ a <r. D„ ># . p - ,7
I- . (2). *M‘I I . DK1V..|.
(1)
( 2 )
( 1 )
( 2 )
*s4.v»;;
* 8082 ]
>8454. h: It< \ —* CIs. * < CIs 5 exd. D . J{ ,, *k t CIh* excl
-84 55. K/V‘*«Cls*cxd
*84 59. I-: * sj \ < i V « xcl. = . *, \ « CIs*exd . *••(* - X> a a ‘\ = A
be m.
V . *84 1 4 . D H : * o X « CIs- exd . s . € f*(* u X) e Cl» -* l .
[*74 821] s.<r^.*p\eCls-* 1 . t u {tc-\) A e“X- A .
[*84 14.*02 3] = . *\ X * CIs 5 exd . *'<* - X) a a ‘\ = A
*84 6 . h:.«A\-.\.D:«vXc CIs 5 exd . = . *. X t CIs 5 exd . g** a s‘\ = A
[*84 59 .*24 313]
*84 61. h :. £ * . D :k\j i*Jc CIs 5 oxcl. = . k e CIs 5 exd . /3 a *«* = A
[*51-211 . *53-02 . *84 23 6 ]
*84 62. f-:.a + ^.D:f‘av C/3 * CIs 5 exd . = . a a /3 = A
[*84 (51 . * >115 . * >3-02 . *84 23]
*85. MISCELLANEOUS PROPOSITIONS
Summary of * 85.
In this number certain important propositions are proved, and the other
propositions of this number are mainly lemmas. The most important propo¬
sitions are the following:
*85T and *8514, which show that if QfK is a Cls -* 1, then the domains
of Q**\ are the same as the domains of €**Q**\, and Q**\ is similar to e**Q**\,
thus reducing the problem of selections from many-one relations to that of
selections from classes of classes.
*85 27 and *85*43, which show that if * e Cls 5 excl, 1***8** consists of the
relational sums of the domains of 9**P***k and is similar to e**P****; i.e. the
class of P-selections from s** is similar to the class obtained as follows: take,
the members of * one by one, and form the 7^-selections of each; we thus
obtain a class of classes, each class being of the form J i **a, where a c *; we
then make a selection from this class of classes; this selection is a member
of e**P****\ the number of such selections is the same as the number of
P**8*K.
*85*28 and *85*44, which are special cases of *85*27 and *85*43, but more
useful than these. *85*44 is the source of the associative law in cardinal
multiplication; it states that, if * is a Cls*excl, ****** has the same number
of members as €**€****. (On associative laws in general, see the notes to
*42*1*11.) That is to say, if we form the class of selective relations (e^'a) for
every a which is a member of *, And then form the class of selective relations
for e****, we get the same number of terms as if we proceeded to form the
class of selective relations for c**s*k. The way in which this proposition
yields the associative law of multiplication may be explained as follows. We
shall define the product of the numbers of members of a as the number of
ee.*a. Thus e.y. if the numbers of the members of a are y al , y a . /t the
number of e**a is y al x y m3 x Suppose the other members of * are and
7 , and that ft and y again have three members each. Then the number of
c**€***k is the product of the numbers of e A ‘/9, e**y, i-e. it is the product
of m«, x m .2 x Hu x H* x and y^ x x x
But the numbers of the members of 8 ** are
/*«*» M«j.*M*i. Hfif Ho. Hr*-
Thus the number of ****** is
X ^ X ^ X X y fi9 X y fi , X y,, X y^ X y^.
I’KOI.ECOMEXA TO CARDINAL ARITHMETIC
[part II
llc’ii'v -^''•V44 eiial»!••** ii- to conclude (hat
*^o * x P* » x «/*/». x ft,, x /!„,) x (m», x fiy. x /x v ,)
= /*-. > A*., x /i., X fx a , X fi it x fi fi .x fi tl x fly. x fiy }■
wli m-Ii i- ;i ..I dm awH-iativc law. In fact *<S-V44 gives us this law in its
• 11 • • * 1 1 t■ • iiii wlii-ii flu* number of brackets, and of factors in each bracket,
may intimre or film** indifferent I v.
Anodmi iiii|»>i taut pan of |»it»|x»silion> is *S5 ’»:K>4. These enable us to
reduce the |»i• »bl«-in .»| s.-loetioiiH |br an>f relation to the problem of selections
1 1 • *|«» n da*s classes. I h«• method is a- follows: Given anv term ./*, form
dm class .»f ordi-ioil ..pies which j- is relatiun while the referent is a
term haviii}; the relation /' to .#. ( 'all this class of couples 1* \ r. Form
tin- class Ibi every .« which i- a member of a: we thus obtain a class of
cla-ses. namely 1’\"oi. Then dm number of selections from this class of
cla-ses is the same a- the number of J* A *a.
We have one otlmi important pair ofpropositious in this number, namely
*s.Vb! These show that what is called " Zcrmelo’s axiom " is equivalent
to what is called the "multiplicative axiom." Zennelo's axiom* is to tile
etiecl that it a is any class. « A ‘C*I ex'a is never null, #>. (a).y ! (±*L'\ cx‘a.
I lm 1 multiplicative axiom* is (o the elfeet that if * t (Jls ex*exel, there is at
least one class loinifd by tukiiit* one rvprcscntative from each member of k,
\\ liieh is equivalent to
* t ('Is ex- e.xel . . y ! tjV.
In *S.Vt».q. these two axioms are shown to Ik- equivalent. From Zennelo's
tlieo|-ein*f* it follows that l»otli are eipiivaleiit to the assumption that every
class can Ik- \\ cl bordered. This will be proved Inter (*2.*>N).
'I’lie above-mentioned propositions, stated symbolically, arc as follows:
•851. I .D.IV'fc'X-DvJ'X
#8514. H : ur\«CI»-> 1 . D . sin eVV“\
*85 27. h : * t C'ls'excl. D . ^“D'^'W 4 *
*85*28. H : xe Cb-excl. D .«*•*•*
*85 43. V : * * Chccxcl. D . /V«‘«saiicWV 4 *
*85*44. I- : k « CIs* cxcl. D . V* sin ca 4 ** 4 **
I'lie followin'* pnqtosi lions depend upon the definition
#85 5. 7'X.»/= 1//“/'•// 1)1
l.e. /' I // is the class of all couples whose re latum is y while the referent
has the relation /* to //. W’e then have
*85 53. I- . Wa =*“|)“tV/ > I“a
giving a construction for /Va by means of e*, And
* Sec Milth. AnnaUn, Vol. IJX.
t loc. cit.
SECTION D]
MISCELLANEOUS PROPOSITIONS
527
*85-54. h . P A *a sm e A *P l“a
which reduces the question of the existence of /^-selections to that of the
existence of e-selections.
*85 61. h . e I “k e CIs 2 excl . \ €€ k . e^sm €±‘e J “k
This proposition gives a construction for any e-selection in terms of an
e-selection from a CIs 2 excl, and reduces the question of the existence of the
former to that of the existence of the latter. A particularly important case
is when k = Cl cx*a. This is considered in
*85 63. h : e J"C1 cx*a e Cls ex 2 excl : g ! € A ‘CI ex* a . 3 . 3 ! e A ‘e J“CI ex*a
*85 1. h : Q r X e Cls 1 . D . D“QJ\ = D**e A *Q**\
Dem.
h . *81 -3 . D I-: Hp . D . D**Q A *\ - fi fa e ~Q**\ . D. . ,x « at 1 : C s*Q*‘\] ( 1)
h . *84 51 .DhHp.D ,~Q**\ e Cls 2 excl .
[*84-412] D . T>**e^*Q**\ -$|<n <?‘X .D a ./inacl :/xC s'Q^X] (2)
h.(l).(2).Dh. Prop
*85 11. H : Q [* X e 1 -> 1 . D . D“(/> Q)±‘\ - D“/VQ“X
Dem.
h .*33*431 .*3212. D h : Hp.D.\ca‘Q ( 1 )
h . (1) . *82-32 . D h : Hp . D . D“(P, Q) A ‘X - D“P A ‘4?“X OK Prop
*85 111. h:/lfee.i‘Q“X.D.D«(J/jQrX)-DM/ [*823]
*85-112. h : Me e±*Q**\ . "D . M ~Q [* * * Q*‘X £*82221 . *62 26 ]
*86-12. h : Q f X e 1 -» 1 . D . D“Q A ‘X = D M €*^‘X
jDem.
h . *62-26 .DK D“Q A ‘X - D**(e | Q)±*\ (I)
h . *82-32. D h : Hp . D . D“(c | Q).>‘X = D"« A ‘4?‘X (2)
I- . (1) . (2) . D h . Prop
This proposition is used in connection with ordinal multiplication (*17314).
*86 13. h :~Q r X e 1 1 . R e Q±*\ . D . R | Cnv‘4? c € A ‘Q“X
Dem.
h . *62-26 -Dh:Hp.D.grXel->l.P € (6| Q) A ‘X .
[*82-231] D.R | Cnv'Qe es*Q**\ : D h . Prop
The above proposition is used in connection with “families" (*9731).
52*
I'HULKliOMKXA TO CARIHNAI. ARITHMETIC
*85 14. I- : 7 r X * r l> I • > • V* 4 X Mil e.* 4 V“X
hr in.
b. •SI-21 .Dh: H|».D. V^Xmii1>“Va‘X.
|**5 1) D.V*‘Xmi. I >“«•.* *7“X
h . *84 5 I . D h : 11 |i. D . IJ**\ € C Is* i-xel .
|**44l] D.
l-.(l).l2).Dh. Prop
*8 5 21 ’22 ar»* I*’Iiiiiisin lor *n 524 which, with *85*20.
**5-27.
(part ii
(1)
( 2 )
is iv<|iiii-c<l for
*85 21. b :o€#f. M* I**'* 1 * ,M[ a* Pa‘o |*H0 0. *4013)
*85 22 h.1/t/VA.D..l/r <1 *1 Pa)-.)/
llcrc.l/f* * 1 Pa t €±*1**“* can also be written (.1/f) (* 1 Pa)! c (Ca'Pa 14 *).
The brackets arc omitted because no other meaning is possible.
hem.
h . •85-21 .
D h Hp. D : a « * . . g ! P±*a :
|*8() 81 J
D : a. fi t « . / Va * Pa‘3 -
3-.d . a - :
[*S0* 12.*71 * IOG‘55]
D : Pa T < * 1 -> • :
| *35-52 J
D : «1 /’* « 1 -» 1
(1)
b .(1). *72" 1 4. *71-25.
Dh: Hp.D.J/r < *) Pa « 1 —► CIs
(2)
b .*341 . *30*4 .
DhsP^/f* *1P*|\. = .
(ga). P-. 1 /ra.acff.
X — /*a*o (3)
h. (3). *85-21 .
Dh:Hp.D..l/f xj/^Ge
(4)
h . *37-322 . *33-431 .
Dh.cpj.l/r *1 Pa>U 4 («1Pa)
1*37 4)
= VV‘*
(5)
b .(2). (4). (5). *8014
. D h : Hp. D . JJ/T «‘\P*:< *a‘W‘*
(<i)
I- . *37 32 . *35-02 .
Dh.DT.ur *i/\>=j/r*v.
[*41-35]
D h . * 4 D*( J/ r * 1 Pa> = .v |V*
(7)
H . (7). *80 20 .
D h : Hp. D . **D‘( .1/ r <1 Pa) = .1/
<*>
b . ((»).(8).D h . Prop
*85 24 b . P,V* C *“I>“*a‘Pa“*
Dem.
h . *85-22 . D
b : McPSs'k . D . (g-Y). X e e A ‘/VV . M = A‘D‘X .
[*3707] D . J/*r «“D“eA‘P A “* : D h . Prop
SECTION D]
MISCELLANEOUS PROPOSITIONS
529
The following propositions are lemmas for *85 *26.
*85 241. h : X e e A ‘P A “/c X‘P±‘a e P s ‘a
Deni.
h . *83 2 . D 1- X e € A ‘P a “k ,Dz\e P±“k . D A . X*\e X :
[*37*63] D : a € * . D a . X^P^aeP^a :. D h . Prop
*85 243. h : /c e Cls 2 excl. AT e eWV‘* . D . i‘D‘JY € 1 Cls
Dem.
H .*83-21 .
Dh: Hp . D . D‘.Y C s*P± tt K
(1)
1- .*40151 .*8011 .
DI-.s‘PA Cl —> Cls
(2)
h.<1).(2).
Dh: Hp . D . D*AT C 1 -> Cls
(3)
^ . *80 35 . *11-45-55 . D h Hp . D : Af, N c I)‘,Y . g ! (IM/ r> C VN . D .
(ga. &). a, @ e k . At ~ X'PSa . iV= X'PSp . g ! QM/ n CT'iV.
[*85 241.*80-14] D . (ga./3) .«,£«*. A/- X‘P*‘a . X = X‘P*‘P,
g lawna^. a = a‘.i/./3 = ci‘iV.
[*13 22] D . CI‘A/,(I‘iVr * * . A/ * X‘PS(l*M .N-X'PSWN .
g ICl'A/^CPiV.
[*8411] D . (I*A/ — a*N . M = X‘P**(I‘M . N = X'PJCI‘N .
[*30-37] D.M = N (4)
h. (3). (4). *7232. Dh. Prop
*86 244. h : X e < A ‘P A “* . D . j‘D‘AT <• P
Pern.
I-. .83*21 . *40 4 . 3 h :. Hp . 3 : R e D‘X . 3„ . (g a ) . a * * . R e /Va .
Cm8014] 2 K .JiCP:
[*41*151] 3:WICP : . 3 I-. Prop
*86 246. I- : X t eSP s " K . 3 . a‘*'D‘^r = *•«
Devi.
H . *85*241 . *80 14. 3 I- s. Hp.3 :a«* . 3. . d'X'PSa = a :
[*50 17] 3 : a“X“PS‘ K - *:
[*80*34] 3 : a“D‘^ = * i
[*41*44] 3 : OVD 1 ? = s‘* 3 K Prop
*86 26. h : * e Cls J excl. X e t.‘P.“ic . 3. 6‘D‘X e P.‘s‘k
[*85*243-244*245 . *80*14]
*8626. h:«eCls*excl. 3 .«“D“ e „‘/V‘*C /VsV
Dem.
\-. *85 25 .DI-r.Hp.DsYe f4 ‘P 4 “/t. D x . i‘D‘AT f 7Vs‘* :
[*37-61-33] D s i“D“€.*‘P A “* C /V«‘* D h . Prop
*85-27. h : * c Cls 3 excl. D . P A V* = P'D“*PP A “* [*85-2426]
*85*28. H : * e Cls* excl. D . e A V* = £*85-27
H Ac W I
530
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
The following proposition is a lemma for *85*31.
>85 3. h : M * 1V« . s € a . D . M'z G * I>M/ . M'z G #P'z
I'll.* condition' • significance here and in *85*31 *32*33 34 require
\YPQ Rel.
ben*.
I- . *80 32 . *33 43 . D b : Hp. D . M*z e DM/. M*z € 7 ”z .
[*41*13] D . M*z G*‘PM/. M'tGVP'i : D h. Prop
The following proposition*, down to *85 +2 inclusive, deal with circum-
stanc. s under which we can infer M * A* from .v‘DM/ = .v‘I>‘A r . *85*32*33*34
an* not siihsoqiieutlx used: the remainder are used in proving *85 +3.
*85*31. y :. € a . : 4 s • 3*.*- • «‘l u : A s* D'm ■ A : D :
.V. .V t / v« . *‘l >M/ = *‘D‘.Y. D . .1/ = .V
Deni.
y . *25*5+ . D h : Hp. z. w * a . 3 ! x* J J, z A s 1 J t, m . D. „ «•;
(* 11 *351 DhjHp.;.w«o.« (s t l ,, z) 9 . u is*l n w) t . D. ir> , ( w ( 1 )
h . *85 3 . Dh::Hp.;*oJ/ l A r f iVa. *‘DM/ = x‘D‘A r . D
" t ;l/‘: )r,D::<a.« <x‘/ > 4 c) *•. m (.v 4 l) 4 A r ) r :
[* 80 * 35 1 D::ta.»(x‘/ ,4 .*) *•: (gw). w ta.u( A' 4 w) e :
(*85*3.*10*35] D : (gw/) . Z t w«r a . •• {VP* 2 ) r . u (.s , / u n)e . r :
[< I).*10*23] D : (gw) .z-w. n ( X*w) /•:
[*13*105] D:t/(iV < *)f (2)
y .(2). Kxp. *10*11*21 . Dh. Hp (2). D : r r a . D.. i1/ 4 r G X*z (3)
Similarly h Hp(2). D : c < a . D,. A ri 5 G ,l/ 4 j (+)
h . (3). < + >. D h Hp (2). D : r e a. 0. . M*z «= A’ 4 * :
[*33*+5.*30 l+] D : Jf- A r :. D h . Prop
*85 32. h 2 , w c «. r * w. D.. „• . s t C“P t z o *‘C u< / > *w = A : D :
il/. A' e /Va • *‘DM/ = x‘D‘iV. D. .1/ — A r
Dan.
y . *41*45 . D
h Hp.D::,«'tfl.: + tf/.D ;if . C‘s t P t z r\ C 4 * 4 / >4 w = A .
[*33*3+] D..., r . i‘7** A .i‘7“w = A ( 1 )
h . (1). *85*31 . D h . Prop
*85 33. y c. w e a . 2 * w. D,.,,. n «‘D“/^w = A : D :
il/, iY € /^‘a . i'DM/ = x‘D‘iY. D . .1/ = A' [*41 *43. *33*32. *85*31]
'Phe proof proceeds exactly as in *85*32.
*85*34. h w t a .:*//•. D-. „•. s'd^P'z r> s'd^PUe = A : D :
M. iVc P±a . s*\.YM = .v‘D‘iY. D . J/ = i Y [*41*44.*33*33.*85*31]
SECTION D]
MISCELLANEOUS PROPOSITIONS
531
The following propositions. *85*4*41*42, are lemmas for *85*43*44, which
latter are of fundamental importance, since they are the source of the
associative law in cardinal arithmetic.
*85*4. h r\ s'f* = A : D :
M, iV e c±‘k . i-'DM/ = *‘D‘AT .D.M—N £*8581 ~ . *62*2
*85 41. \~:.K€ CIs* excl .D:a,/3€*.a^/3.0. PP A ‘a A PP A ‘£ = A
Dem.
h . *8014 . DH :® (PP A ‘«)y . «(J*/V£) y . D x . v . y e a . y e 0 .
[*22*33.*10*24] D x , y .g! fl ^:
[Transp] Dh:an^ = A.D. PP/a A s‘iV/9 = A ( 1 )
h . (1) . *84*1 . D h . Prop
*85 42. \-ik* CIs* excl . M, iV € . i‘D*M = PD*A r . D . A/ = A r
Dem.
h . *30*37 . Transp . D h : 2Va * 7Vrf . . a + /3 :
[Fact] Dl*uf CIs* excl. a, /3 « * . /Vot * /V/3 . !>.*.
* * CIs* excl . a, 0 e k . a + # .
[*85-41] D. i# . *‘P A ‘« A *‘PS0 - A :
[*37*63] DFuf Cl s’*’ excl .\,^c /VVr. X + ^ . D A „ . A s*u = A (1)
h. (1). *85*4. Dh. Prop
*86 43. h : * e CIs* excl . D . P A V* sin « a ‘P A “*
Dem.
h . *34 41 .Dh. (Af) . - (* | D YM.
[*1312] Dl-:. M.Nt'JPS‘K.i‘VM^i‘D‘tr.0,, v .Af=JV:0:
H . (1) . *85*42. 0 Ml N * f *‘ P *“* • ( * 1 DrM = ( * 1 VyN < 1 >
h :• * « c,sS excl *SPS‘k . <# j D)‘A/ = (* j D)‘A . „ . A/ = JV :
[*73’25] D : (i | D)“« A ‘P A “* sm c A ‘P A "«r :
[*37*33] D : sm « A ‘/V‘*:
[*85*27] D : P A V* sm *SP*“k DH. Prop
*85 44. h : /c e CIs* excl. D . sm *k £*85*43
The following proposition is used in connection with cardinal multiplication
(*114*301).
*85 46. h : * « \ = A . D . € A *<* w \) sm eSii'cS* ^
Dem.
h . *85*44 . D
h : t‘/c w t<Xe CIs* excl. D . ejs‘(l**c v i‘\) am v t‘\) ( 1 )
h . *24*57 . Dh:. Hp .D:*4=\.v.* = A.X. = A:
[*84*62 23] D s i*k \j i*\ e CIs* excl (2)
h . *53*11*32 .Dh. *‘(i‘x o t‘\) = * w X.. e A “(l‘tc v l‘\) = l‘ej*c v (3)
h • (1) • (2) • (3) .Dh. Prop
34—2
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
The purpose of the follow it*rr proposition^, down to *85"55, is to show how
to o«-t from a class of classes a class of selections having the same number of
terms as ]\*k. For this purpose wv introduce a new notation, representing
a lather im|x>riant analysis ..| tin- couples contained in a given relation.
A couple .• I >/ is contained in a relation /' when jPy\ thus if, keeping y
fixed we form the class of couples l 7“7"// all these couples are contained
III /'. We pill
-85 5 P ly=ly“7”y 1>I
Then 7* J‘*0* P * CIsex*excl. Also .**7> J**< VP is the class of all
couples contained in /'. and *V/* J** ( l*/* * P. We shall now prove that
/'_ ‘o -r I >“«„*/'!“<». so that every member of I *.‘a can be derived from
a member <»f «r <i *7 , J“o. and the problem of the existence of l*\ l a is reduced
to that o| tin- existence of selection* from a class of mutually exclusive
existent classes.
*85 51. h . PSl'.r - l P‘7" r = 7'J .r [*80 4-5 .|*K5‘5>]
*85 52. h . 7V*'“a = PJ “a [*37*35 . *85*51]
*85 63. h . 7Ya = .v“ I )“«.»* 7* I “a
Pew.
I-. *84-241 . *53 22 . D h . /“a « < Is- excl. 0 = a .
[*85*27) D h . /V* -*“l>“e./7V‘i“a
[*8.5*52] = .i“D“<*‘P I “a . D h . Prop
-85 54. h. /VosnuV/'T'a
I)nn.
h . *84 241 . *53*22 . D h . /“a « CIs* excl. .v*t“a = a .
[*85-43] D h . P a *a sin «*‘P A “l"a .
[*85-52] D h . /Va sm € A ‘P J “a . D H . Prop
The following proposition is frc«|Ucntly useful.
*85-65. h . 7Va sin I>***,‘P J**a . l> J*‘a < CIs* excl
Pc m.
h . *85*51 . *80 14 . D h : 7? * 7 J J j- . D . 0*7? = /‘.r: 771 P J // . D . 0*77 = i‘y :
[*3 47)
[*13-171.*51 23]
[*30-37]
[*1011-23]
[*3-42.*l 111]
[*37t>3]
[*8411]
[*84-41]
[*8554]
DhsPePJaAPJy.D. 0*7? = i*.r . 0*7? = t‘y.
D = y.
D.PI-r-PJy:
D h : 3 ! 7* J .i- n 7* J 1 / . D . P J s = P J y :
D h : x.y (a.^ll'ljrsPly. D r . y . P J x = P J y :
D h : X./ifPI"o .3 ! X A/t. 3 a. m • X - y :
DH.PJ“ae CIs* excl.
D h . D“€ A *P J*‘a sin « A ‘P J“cc.
DKP^asm D'VPf'a
h . (1). (2). D H . Prop
SECTION D]
MISCELLANEOUS PROPOSITIONS
533
*85 56. K: P f - a e Cls —* 1 . D . e±‘P“a am e.‘P J“a [*85-14,-54]
*85-6. h . = £ |( 3/ 3 ). @ e k . p ^ 0“/3\ = <: 1“k
Deni.
h . *37-67 . D K . € A “c“* = £ j(a/3). 0 e k . ^
[*834J =A!(3/3).^e^. M = |/3“/3j (1)
h. (1). *85*52. DK Prop
The following proposition is frequently employed.
*85 601. h .<•!<*«! a“a . e J a sm a . e sin * . € J e 1 —► 1 . E ! * J'a
Dein.
K *8551 .*622. D h . e I a = i a“a (1)
[*73611] Dh.ejasma (2)
h. *3812. D K E! e I'a (3)
[*71-166] DKe J«l->Cls (4)
h . (2) . *73-47 . Dl-:a«A.€j as =fJ/3.D.6l/3=A.
[*73-47.(2)] D./3«A (5)
h . (1) . *38*131 . Dh:a;«a.«Ia»fJ/3.D.a;|a€i /3“/3 -
[*38-131] D . (g.y) . x i a = y l /3 .
[*55-202] D.a=/3 (6)
h . (6) . *1011-23-35 .Dh:a!a.cIa = eI/3.D.a = /3 (7)
h.(5).(7). Dl- :<I (8)
h. (4). (8). *71-54. Dh.«jcl->1 (9)
h . (9) . (3) . *73-26 . Dh.e J“* sm * (10)
h . (1) . (2) . (3) . (9). (10) . D I-. Prop
*85 61. h . e J“* e Cls* excl. \“k . sm € A ‘«? J“*
*85-53-54-55
*85 62. h : g ! . = . g ! c A ‘* J"* [*85 61 . *73*36]
*85 63. h : e J“C1 ex‘a c Cls ex* excl : g ! * A ‘C1 ex 4 a . = . g ! I“CI ex‘a
Dem.
h. *85 6. *60-21 . D
\-:\€€ J“C1 ex'a . = . (g/3) .0Ca.g!£.X-i /3“/3 ( 1 )
h . *73-611-36 . D h : g ! 0 . X - J, . D . g ! X :
[*3-42] Dh:^Ca.g!/3.\ = | /3“/3 . D . g ! X:
[*1011-23] D h : (g0) ./3Ca.g!^.\ = 4 r . D . g ! X (2)
h . (1) . (2) . Dh:X f ( f J“C1 ex‘a). D . g ! X :
[*1011 .*24-63] Dh.A«v<(f J“CI ex‘a) (3)
h . (3) . *85-61 . *8413 . D h . e J“C! ex‘a c Cls ex* excl (4)
h . (4) . *85-62 . D h . Prop
Note. (a) . g ! ca'CI ex‘a is "Zermelo’s axiom.” The above proposition shows
that this is true if
k e Cls ex* excl .D,.g! e A *K,
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
which again is true it’
x € (' 1 $ ex* excl. D : (1
hi virim* of *84*412. The last of these is the "multiplicative axiom," which
is thus shown to imply “Zcrmclo's axiom.
The following pro|H>sitions lead up to *85*72. which is used in the theory
of double similarity <*1 I 1 * 8 ).
*85 7. h 01 X . . H't3 C ^ : M € : D .
M It rx««j c X.D'(,l/ J<[\) = D*M
hem.
h . *14*21 . D h 11 p . D : & t X . D a . E l K*£ :
|*74 1 I | !>:/* [A* 1-»CI«.\CU</{ (1)
| *80* 14.*7 I -25] D : .»/ /f f X ( I -> Cl* (2)
h . < l). *71 -7.*857 . D K H p . D : a (.V li [ X . D . >9 < X . xM </?*£).
(*80*'ll.lip) D.£eX..«’«7* 4 £.
[Up] ( 8 )
I-. *80-14. *74*44.3
I- : U p. D . I > 4 ( il/ U r X> - 1>M/.<I 4 (.1/ rt^-X^d 4 /*
KD] -X (4)
K . (2). ( 3 ). (4). *80*14 . D h . Prop
*85 701. V \ . D* . /f 4 £C£: 3 . l> 44 € A 4 /f 44 X C 1> ,4 *VX [*85*7]
*85 702. H :. £ « X . D,t. /PCIVf € Cl 4 £ : 3 . I> 44 < A 4 tf 44 tT 4 X C D 44 * a 4 X
[—O' *Jr]
*85 71. h : 7*«€ A 4 CI 44 X . 3. ])“< A ‘D‘ftC D“<» 4 X [*85*702 . *83*2]
This proposition asserts that if we can select one sub-class out of each
member of X (where X is a class of classes), then selections from the sub-classes
so obtained arc selections from X.
*85 72. I- :.(.N'“/3)1 tfcl -> 1
L) 4 < «i 4 /i 4< X C D 44 €a‘*S' 44 X
Dem.
h . *14*21 . *38*43. 3 h Hp. 3 i&*\ . 3. ficd'S (1)
h.,85-701
If, \
h 7 « S“\ .0,.(R' S)‘y C 7 : 3. D “ ta ,‘R“S“8“\ C D“e 4 ‘S“X (2)
h. *37-03. *14-21 . 3
I- :: Hp. 3 7 « S“\ . 3, • (« S )‘ 7 C 7 : = :/3e\.3 s .(«i S)‘S‘£ C S‘5 :
[♦74-53.(1)] = :0€\.O,.R‘ffCS‘ff (3)
I-. ♦74-171.3 H : Hp. 3. S“S"X = X (4)
1-. (2). (3). (4) .31. Prop
SECTION D]
MISCELLANEOUS PROPOSITIONS
535
(1)
( 2 )
( 8 )
The following proposition is a lemma employed in the theory of double
similarity (*111-313).
*85 81. hr.Xf Cls’excl : /9e \ . *‘<3 “T*& C £ : R € €±‘T“\ : D :
/3eX. D,.(«<D<ft) f &-R € T*0
Dem.
(-.*14 21. D (-:. Hp . D : /9 e \ . D . E ! T‘0 :
[*83'2.*37'6] D : 0 e \ . O . R‘T-0 « T‘0 .
[*35'452.Hp] D . R-T‘0 = (R‘T‘0) f 0
(-. (1) . *83 22 . D I-:. Hp . D : /3 e \ . D . E ! R‘T‘0 .
[*33-43.*4113] D . R‘T‘0 G ,i‘D ‘R .
[*35-461] U . (R‘T‘0) [ 0 G(s‘D‘R) [ 0.
[(»)] O.R‘T‘0C(s‘D‘R) f0
I-. (1) . *37 6 . *83-23 . D (-:. Hp . D : D‘tf = M |( a -y) . y t \ . M = R‘T‘y\ :
[*41-11.*18-195] D : x(i‘D‘R)y . = . ( 37 ) . 7 f X . x(R‘T‘y)y :
[*35101] D:xl(i‘D‘«) t 0} y . = . (&y) .y f\ . x(R‘T‘y) y . y e 0 (5)
I- .(2). *33-14 . D (-:. Hp . 7 «X .3: x(R‘T‘y)y. D . y ed‘R‘T‘y. R‘T‘yt T‘y .
[•40 4] D . y t »'Q.“T , y.
t H P] O.yey (6)
(-.(5). (6). DI-::Hp.3:./9,X.D:
x |(*‘D‘/f) r/9|y. = . (37) -0,yt\ .x(R‘T‘y)y . y e 0 . y ty .
[*841 l.Hp] o . (37) -P.ytX.x (R‘T‘y)y. 0 = y.
[*13195] O.x(R‘T‘0)y (7)
I-. (4). (7). 3 (-.Prop
(+)
•88 CONDITIONS FOR THE EXISTENCE OF SELECTIONS
Su in uni rii of *-SH.
Tin- existence nl selection' cannot, so far as i> known a! present, be proved
in general, That is. \\i- cannot prove any of the following:
«/\#):#C<l‘/'.D. 3 | 8 /V*
</'.*>: /'«CI*-» I ^C(IT.D.;.| ! /V*
t/O-H ! /VO*/'
<*): k . 0 . a ! *±‘x
t k >: x t Cl*» **x* e\cl. D . 3 ! « 4 V
la). 3 !«.‘Clex‘o
<*>:.*«< *D..\5,-xel . D : (gp): au.D.^Aafl
These various propositions can be shown to be all equivalent inter se ; and
111 virtue of Zermelo* theorem tef. * 2 ">K). they are equivalent to the proposition
“every class ran hi* well-ordered.*’ In the present number we have to prove
the above equivalences. a> well as certain propositions giving the existence of
selections in various particular cases.
The most apparently obvious of the above propositions is the last, namely:
If k is a class of mutually exclusive clashes, no one of which is null, there is
at least one class p which takes one and only one member from each member
of#." This we shall define as the " multiplicative axiom."
Wo will call l* a multipliable relation (denoted by "RelMult") if
PjiVP exists, or. what is equivalent, if * C CVP . D. . 3 ! /V*. Thus we put
Rel Mult => P !3 ! PJQ'P\ Df.
We will call k a multi friable class of classes if c A ‘# exists, i.e. we put
Cla* Mult = * |g ! €±‘*\ Df.
The multiplicat ive axiom will be denoted by “ Mult ax.” Thus we put
Mult ax . = k e (-Is ex*excl. D. : (g/*) zaetc. ./* a a* 1 Df.
In the present number, we shall first give various equivalent forms of the
assumption that V is a multipliable relation (#881—To); we shall then do
the same for multipliable classes of classes (*88*2—*26); next we shall give
various equivalent forms of the multiplicative axiom (#88*8—*89). (Some
important equivalent forms cannot be given at this stage, as they depend
upon definitions not yet given, such as the definitions of cardinal multiplica¬
tion and of well-ordered series. Cf. *114*26 and *258*37.) Finally we shall
give propositions showing that various special classes of classes are multipliable.
Most of these propositions will not be used in the sequel, but they illustrate
SECTION D]
CONDITIONS FOR THE EXISTENCE OK SELECTIONS
537
the nature of the difficulties involved in proving that a class of classes is
multipliable, and some of them show that mere size does not prevent a class
from being multipliable. For example, *88 48 shows that, given any class of
classes k. if each member a is replaced by Pa, the result is a multipliable
class of classes; but the only effect of this change is to increase the number
of members of each member of our class of classes by one.
The chief propositions in this number which are afterwards referred to
are the following :
*88 22. h : * € Cls a Mult. X C * . D . X « CIs 3 M ult
*88 32. H Mult ax. s : k e CIs ex*excl. D„ . g !
*88-33. H : Mult ax. = . (a) . g ! €.>‘01 ex‘a
*88 361. h Multax . = :* = *.. .g ! R±‘k
*88 37. h Multax. = : A~e* . D, . g ! cP*c
The above is usually the most convenient form of the multiplicative axiom.
*88 372. h Mult ax . a : A <•*.=„. eP* — A
This proposition is used in *114, to prove that the multiplicative axiom
is equivalent to the proposition that a cardinal product vanishes when, and
only when, one of its factors vanishes.
*88 01. Rel Mult * P (g ! 7VU ‘P\ I)f
*88 02. Cls a Mult = k (g ! e A ‘*} Df
*88 03. Mult ax.-:./c« CIs ex 3 excl . D„ : (g/x) :of<.D«./ioael Df
*88 1. z P e Rel Mult. = . g ! P A ‘d‘P [*20‘3 . (*88 01)]
*88 11. h : P * Rel Mult. X C d‘P . D . g ! P A ‘\
Dein.
I-. *80 6 . Dh iR t /VCI'P. XCdT.D .RfXePPX .
[*10 24] D . g ! P A ‘X z
[*10*11‘23*35] D h : g ! P A ‘d‘P . X C d‘P . D . g ! P A ‘X (1)
I" . (1) . *88*1 . DI-. Prop
*88 12. I-P c Rel Mult. = : X C d‘P . D A . g ! Pp\
Dein.
I- .*88*11. Exp. *10 11 21. D
I-:. P € Rel Mult. D : X C d‘P . D A . g ! P A ‘X (1)
I- . *10 1. *22 42 . D
I-X C d‘P . . g ! P A ‘X : D . g ! P A ‘d‘P .
[*881] D. Pc Rel Mult (2)
I- . (1) . (2). D h . Prop
I-: P « Rel Mult. = . g ! e A ‘P J“d‘P [*85*54. *73*36 . *88*1]
*8813.
PROLEGOMENA TO CAR HINA I. ARITHMETIC
[PART II
'88 14. b :. * C <I‘7'. D : P [ x € Kd Mult. s . g ! iV*
Dl'in.
b . *80‘23. 0 h : a ! /V* • = • 3 ! (P [ * )i‘* (1)
I- . *35 05. D h : « C (l‘J J .DAlUPf x) = x (2)
I -.(1 ).(2). D b lip . D : 3 ! TV* . = . g ! (T'f * VIJ'U'r*> •
[* 88 l] = . 7V * * Kd Mult :.Dh. Prop
• 88 15. I- < I ‘J> = V . D : 7 '[ x *- Kd Mult. = . g ! TV* [*8814 . *24 11 ]
*88 2. I- : k t (.'Is 7 Mull . = . g ! V* [*203 . (*88 02)]
-88 21. b : Pt Kd Mull . 5 . /'J“CI‘7'« CU= Mult [*8813~2]
>88 22. h : * «r CV Mult. X C «. D . X « CIs- M ult.
hem.
V . *80*i». D b : /{ ( . X C * . 3 . 7J f X « *a‘X .
[*1024] D.g !c a ‘X:
|*I011-23-35)3 I- :g IfA.XC^.D.g !« A ‘X (I)
I-. (1). * 88-2 .31-. Prop
-88 23. b : x e CU- Mult. 3 . CP* C CIs* Mult [*88-22 . *60-2]
-8824 b 7't Cls-> 1 . 3 : 1\ Kd Mult. = . P“(\‘P e CIs 3 Mult
Dem.
b . *85-14 . *73-30 . 3 h lip . 3 : g ! 7V< I*/'. a . g ! € A < y , “(J < / > 11 )
b . { I). *881-2. 3h. Prop
-88 25. I- :.7V*<Cls-> I . * C iPP . 3 : Pf KdMult . = .7“* cCVMult
Deni.
b . *85-14 . *73-30 . 3
I- :. lip. 3:g ! TV*. * .g ! :
[*8814-2] 3 : 7*f* * * Kd Mult. = . 7^‘* * CIs 5 Mult:. 3 b . Prop
-88 26. P :s c Cls 2 exd . 3 :. * * CIs 5 Mult. = : (g/z ): a e x . 0 a . p r\ a e 1
Deni.
b . *88-2 . *37-45 . 3 I-: * « CIs 3 Mult. = . g ! D“ V* (1)
h. (1). *84-412. D
b :: Hp . 3 :. x e CIs* Mult. = : (g/*): <* €*.3. ./x^ael :/*C s*/c : (2)
[*10-5] 3:(g/*):o€*.3„ 1 ( ; *)
h . *40 13 . *22021 ,DI-:flu.D...v‘/frta = o.
[*22 481] 3 a ./z a = /z r» a :
[*2 77.*10‘27] Dhj.atff.D.^nael :D:acx.D,./tn s 1 k nail (4)
b . (4). *2243 . DP:.ac«.D a ./t/\acl:D:
a e x .D a . p r\ s‘k nofl :/ios‘< C s‘*:
3 : (gv) : ae*.3 a .i'r»acl tv C s‘/e (5)
[*10-24]
SECTION D]
CONDITIONS FOR THE EXISTENCE OF SELECTIONS
539
h. (5). *101123. D
•• (3/*) :ac*.D«./*nael:D: (gy) : a e k . D a . v n a e 1 : v C s*k (6)
K . (6) . (2) . D h :: Hp . D (g/*) :a€«.D../tnaf 1 O./te Cls 2 Mult (7)
h . (3) . (7) . D h . Prop
*88 3. 1-:: Mult ax . = k «Cls ex* excl. D K : (g/*) :a€*.D a .^^a<:l
[*4'2 . (*88 03)]
*88 31. h : Mult ax . = . Cls exaexcl C Cls 2 Mult
Detn.
h . *88 26 . *5‘74 . D h :: k e Cls ex 5 excl . D, . k e Cls* Mult: =
k e Cls ex 5 excl. D* : (g/*) :a««.D a ./tnad:.
[*88*3] = Mult ax :: DH . Prop
*88 32. I-Mult ax . s s *« Cls ex* excl. D, . g ! eS* [*88 312)
*88 33. I-: Mult ax . = . (a). g ! € A *C1 ex'a
Note that (a) . g ! e.»'CI ex'a is Zermelo’s axiom.
Deni.
h . *88 32 . *85-63 . D h : Mult ax . D . g ! € A '* J “Cl ex'a .
[*85*63] D . g ! c A *C\ ex'a (])
h . *60-57 . D h . * C ClV* .
[*60-24] D h . * - £' A C Cl ex's'* .
[*84-13] D H : k € Cls ex 3 excl. D . * C Cl ex's'* (2)
h .(2). *80-6.3 h : k c Cls ex 1 excl. R c e A ‘Cl ex's'* . D . R f * « *.*'* (3)
h . (3) .*1011-28-35 . 3 h : * « Cls ex* excl. g ! c A ‘Cl ex's'* . D, . g ! e.»'*:
[*10 1] D h (a) . g ! e A *C\ ex'a . D : * e Cls ex* excl. . g ! e*'* :
[*8832] DsMultax (4)
h . (1) . (4) . D h . Prop
*88 34. h : Mult ax . = . Cls -* 1 C Eel Mult
Detn.
h . *84-5 . *88-32 . D h :. Mult ax . D : R e Cls -> 1 . D
[*85 14.*73-36] D
[*881] D.Re Eel Mult ( 1 )
h . *84 14 . D H Cls -*1C Eel Mult. D :
* e CU ex* excl. D . e f * e Eel Mult.
[* 88 - 1 ] 3.g!(er*U‘a'«rr*.
[*8413.*62-42] D.g!(«f •
[*80 23] D. g! « A '* (2)
h . (2). *1011-21. *88-32 . D h s Cls —* 1 C Eel Mult. D . Mult ax (3)
h . (1) . (3) . 3 h . Prop
nlcSR“(I‘R.
g ! RSd'R .
R e Eel Mult
540
I'KOLKGO.'lKNA TO CARDINAL ARITHMETIC
[part II
#8835. h
1 >e m
: Mult ax . = .(/?». R € Rol Mult
H . *37*4') .
#55*121 .(*83*3). D h : a ! /' I x . ? . a 2 7“.r •
|*33-4I)
s.xfiVP
(1)
H . ( 1 ). *10*11 . *37*63 OhjafP J "<!*/'. D. . a 2 * :
[*24*63]
<*>
h . <2). *84 1 3. *8553 . Z> I- . /' I“< l‘ I' c CIs ex 3 excl.
[+S8-32] D H : Mult ax . D . a 2 c A ‘/ J
[*8.vr>4.#73 3u| D - a 2 .
|*ss- 1 ] D.)% HelMult
h.*IO 1 .*NVI .DH:(/^./MMM«iU.D.a !(4 Clex‘a^(l‘(€rciex'a).
[*H2’421 D . a ! <« r Cl cx'aU'CI ex‘a .
[*sO*23] 3. a ! <*‘C1 ex‘«
h . < 4 ». * 10 11 -21 . *8* 33 . D I-: ( 1 {). /{ c Rel Mult. D . M ult ax
h.<3)•(5).3 h• Prop
#88 36 H Mull ax . 3 s < C <Wf . D/:. . . a 2 [*88-35-12]
#88*361. H Mull ax . a : * C <l‘/f . E#r.. . a 2 /?■*'* [*88*3«. *8(>~>]
+88 37. H Mull ax «V*
(«)
(4)
(«)
I Jem.
h . *88 36 . # 62*231 . D b Mult ax . D : A*>*€ * . D« . a • *.»'* ( 1 )
I-. * 84*13 . * 88 * 32 . Dl-:. A<w«. >a 2 «*"« : D.Multnx ( 2 )
h . ( 1 ). < 2 ). 3 H . Prop
#88 371 . I-Mult ax . a : A~««.s fl -3 2 [* 88*37 .* 831 )
#88 372 . h Mult ax . a : A e * . =« . € A ‘k = A [* 88*371 . Transp]
This proposition shows that the multiplicative axiom is equivalent to
the assumption that a cardinal product is zero when, and only when, one
of its factors is zero.
*88 373 H : Mult ax . s . CPtCU — f*A) C CIs* Mult
Item.
I- . #24*63 . #53"5 . D H «c. = :a€K.3 J .a( CIs — t* A :
[*221 ] = : k CCIs — t*A s
[*60-2] =:*«CI‘(Cls-i‘A) (1)
I-. (1). *88*37 . D h Mult ax . = :«€ CP(Cls —1‘A). . g ! t^ie z
[*88*2] = : CP(Cls - l‘A) C CIs 3 MultD h . Prop
*88 38. h : Mult ax . = . CIs - t‘A c CIs 3 Mult [#88*23-373]
*88 39. h : Mult ax . = .(g/S). lie 1 -»Cls. RQe . D‘R = V . (I*R= CIs — t'A
Deni.
K #88*38*2 . *80*14. D
h : Mult ax . = .(gi?). R * 1 —► CIs . RQe. Q.‘R = CIs — t‘A
h . *31*161 . *53*5 . D h : Cl*/? = CIs - 1 ‘ A . D . I'xeQ'R
( 1 )
( 2 )
SECTION I>]
CONDITIONS FOR THE EXISTENCE OF SELECTIONS
541
I- .*23-621 .Dh:RGe.O.R = R* €
h . (2). (3) . D I :RGe. Cl ‘R = Cls - e‘A . D . i‘x € Cl \R A € ) .
[*33131] D • (ay) - !/Ri . y e i l x .
[*5115] D . (jjy) . t/Ri‘x. y = x.
[*13195] D.x Ri*x.
[*3314] O.xcD'R ( 4)
h . (4). *10 11 -21 . *24 14 .D\-:RGc. <1‘R = Cls - t *A . D . D‘rt = V (5)
h.(l).(5).Dh. Prop
The following propositions are concerned with certain cases in which a
construction exists by which the existence of selections can be proved.
*88 4. h./f 1Cl«c A ‘Cl“*
Deni.
H . *72 19 . *71-27 . D I-. * ] Cl e 1 -► Cls (1)
h . *35-52101 . Dh:a(<c*]Cl)\. = . ac<.\ = Cl‘a .
[*60*34] D.aA (2)
K (2). *1111. DK**|C1G« (3)
1- . *35-52 . D 1- . d‘<* 1 Cl) - D‘(C1 T *)
[*37-401] -Cl“* (4)
h • (1) . (3). (4) . *80 14 . D h . Prop
*88 41. h . CI“* c Cls 3 Mult [*88-4-2]
*88 411. h . * c D“* A ‘C1“*
Dent.
h . *35-52 . Dh. D‘(* 1 Cl) = d‘(Cl [ k)
[*3505.*33'431] = * (1^
h . (1). *88-4 . D h . (&R) . R € « A ‘CI“* . D‘/i = * .
[*37 6] D H . * e D“« A ‘C1“* . D H . Prop
*88 42. h : * e Cls 3 Mult ,g!a. = .«w i*a e CU 3 Mult [*83 904 . *882]
In virtue of this proposition, as will be proved later, every finite class
of existent classes is a Cls 3 Mult. For we have Ace^'A; aud, by the above,
a Cls 3 Mult remains a Cls 3 Mult when one existent class is added as an
additional member; hence the result follows by induction.
*88 43. h : e Cls 3 Mult. D . e Cls 3 Mult
Dh.a‘<*1CI)«D‘(Cir «)
Dem.
h . *88-2 . D I-: Hp . D . g ! .
[*85-24] 3-3! i“D“€je±“K .
[*37-45] D . 3 ! e*‘e±“K .
[*88 2] D . e Cls 3 Mult: D h . Prop
*88-431. h :. * « Cls* excl . D s e A “* e Cls 3 Mult. = . s*k e Cls 3 Mult
[*88-2. *85-28. *37-45]
PROLEGOMENA TO CARDINAL ARITHMETIC
•’>12 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
*88 44 H : ('I -xV* e Cls* Mult A e Cls 3 Mult [*60*57 . *88*22]
-88 441. f* : A vtv.Cl ex‘x‘#r e Cls- Mult .D.^f Cls 3 Mult [*8844]
*88 45 H : 1 >‘ /7 a < I ‘ /7 = A . 7' = .75 )x * <1 ‘ 7? . a = 7?., v, ,3 . I> €
hem.
h . *21*3 . D h Up . 0 : .rJ*a . =, 4 . x< (1*77 . a = R*.c w #*.r. ( 1 )
|*5I 10] 3*.« "•>’«« (•>)
l-.tl).*33*13.*51 2.D
H Up . D : j l*a . D,. a — /7‘x /«x. 7?x C 1)‘77 . tVCG‘7? .
1*24 194) D,.a-l)‘77-iV (8)
H . 1 3) . * 11 *59 . D I*H p . D : .r/’a . ///'a . D, „ . a — I)‘ 77 = t ‘.r .a — 1)‘ 77 = / * #/ .
(*2o*23.*51 23) 3*.*.
|*7I 171 D : 7*« I —>Cls ( 4)
I- . < 2). (4). *8(>-14 . D h . Prop
*88 46 H : 1 /7 n (1 ‘ 77 = A . \ = 5 ;i ^jx). e (I * /7 . a « 77‘x u (V) . D .
X € Cls- Mult
Dcm.
I- . *21 *3 . * 1 0*28 1 . *33 131 . D h :. - .75 |.r « < I ‘ 77 . a - 77V u / V| . D :
ac<J‘/V h. . < a .r). x «CI‘77 . a - 7?V v, # V (1)
h . ( I ) . *88*45 . D 1* : lip.D..75 ;x«(J‘/7.a- 77 V v iV) c € A ‘X .
|*10 24) 3. a 8t A «X.
[*88*2] D . X c Cl** Mult : D h . Prop
*88 47. K : /' = atf |a € *• . 0 - #“a u i«a|. D . /'« *V<I‘7 J
l>em.
h . *21 ‘3 . D H II|». D : a/’/3. = i># . fl « «r. ^ = i"a w f‘a . (1)
1*5116) :>*...«€£ (2)
h . ( 1 ). *11*5H . D h Hp . D : aP &. yl*0 . ^m.n.y • 0 - /“a ui'a./Ss /“y u /‘y .
[*40 1 7 1 .*53*22 02] 3..*. Y . *‘0 = a.s‘0 = y .
[*20*23] D« >1 i. > .a = 7
[*71*17] D : 7V 1 —► Cls (3)
h . (2). (3). *80*14 . D I-. Piop
*88 48. K.^:(ga).ac/f .0 = i“a wi'aj
The proof proceeds as in *88 46.
€ Cls 3 Mult [*88 47]
*885.
H . A a Cls € Cls 5 Mult
[*83 9 . *88-2]
*8851.
h : 3 ! a . D . e* Cls : Mult
[*83 901 . *88-2]
*8852.
1-. t“a € Cls* Mult
[*83-42]
*88 53.
h:/fC1.5 . Cls’ Mult
[*83*44]
SECTION E
INDUCTIVE RELATIONS
Summary of Section E.
The subjects to be treated in this section are certain general ideas of which
a particular instance is afforded by mathematical induction. Mathematical
induction is, in fact, the application to the number-series of a conception which
is applicable to all relations, and is often very important. The conception in
question is that which we shall call the ancestral relation with respect to a given
relation. If R is the given relation, we denote the corresponding ancestral
relation by "/£*"; the name is chosen because, if R is the relation of parent
and child, R+ will be the relation of ancestor and descendant_where, for
convenience of language, we include x among his own ancestors if a- is a parent
or a child of anything.
It would commonly be said that a has to z the relation of ancestor to
descendant if there are a certain number of intermediate people b, c, d, ...
such that in the series o, b t c, d, ... t each term has to the next the relation
of parent and child. But this is not an adequate definition, because the
represent an unanalysed idea. We may then try to amend this definition by
saying that there is a finite class a of intermediate terms such that one member
(6) of a is a child of a, one (y) is a parent of z, every member of a except b is
a child of one (and only one) member of a, and every member of a except y
is a parent of one (and only one) member of a. This definition is open to
several objections. In the first place, it is very complicated; in the second
place, there will, in regard to a general relation, be difficulty in securing the
uniqueness of the member of a which is to be a parent (or a child) of a given
member of a; in the third place (and this is the really fatal objection) the
proposed definition states that a is to be a finite class, and we shall find that
finitude, in the relevant sense, is only defined by means of the very conception
of the ancestral relation which we are here engaged in defioing. In fact, if N
denotes the relation of v to v + 1 , where v is a cardinal number, then a finite
cardinal (in the sense we require) is one to which 0 has the relation N+, i.e.
one of which 0 is an ancestor with respect to the relation
vfi 0* - + i).
544 PROLKttOMEKA TO CARliIXAl. ARITHMKTIC [PART II
IIflic** iini'l not use tin* notion of finit nde in defining the ancestral
relation. In tact, the ancestral relation i< defined as follows.
v/
Let us call p a licnt/iton/ rhiss with respect to It it li tt p C p. i.e. if successors
of n (with respect to A’lare//s Thus. For example, if p is the class of persons
named Smith p is hereditary with resj»ert to the relation of father to soil. If
p is the Peerage, p is hereditary with respect to the relation of father to sur¬
viving eldest son. If /a is numbers greater than KM), p i< hereditary with
respect to the relation ol i* in i> + I ; and so on. If now a is an ancestor of z.
and p is a hereditary class to which n be lout's, then z also belongs to this class.
i 'oiiver>cly. if z belongs to every hereditarv class to which a belongs, then (in
tlie sense in which a is one ot his own ancestors if a i-% anybody's parent or
child) u must be ail ancestor ,,f *. For to have a for one's ancestor is a
hereditary pr«»peity which belongs to a. and therefore, by hypothesis, to z.
Hence a i> an ancestor of j when, and only wheii.u belongs to the Held of the
relation in tpicstioii and z belongs to every hereditary class to which a belongs.
This property may be used to define the ancestral relation; i.c. since we have
a lt*z : It* *p C p . u € p . .: c p
we put
It* —!** ( C*/i s R**n C p . n e p . . • c p\ I >f.
We then have
h : a * t H R . D . /T*‘" = 3!/?*> C p . u e p . . z e p\.
Here It**" limy be called "the descendants of a.'' It is the class of terms of
which u is an ancestor.
To make plain the relation of the above to mathematical induction, put
0 for n. and a/5(/$ = a + 1)' for It. Then, since 1 = 0 + 1 , we have 0 e C‘lt.
Again
V/
/f‘V .« + \ e p.
Thus we find
li*‘0 = /§|ae/x.D*.a + 1 e p : 0 e p : . £ e ft.].
Thus if iB is a descendant of 0, belongs to every class to which 0 belongs
and to which a + 1 belongs whenever a belongs. Hence mathematical
induction, starting from 0. will prove properties of In elementary mathe¬
matics it is customary to speak as if this held of all integers, i.e. as if It* 0
(as above defined) included all integers; but in fact only finite integers (in
one of the two senses which the word finite may have) belong to the class
%‘O.nnd they belong to it &// definition, being defined as the class
/§ |a e p . D a . a + 1 € p : 0 e p : . $ e /xj,
i.e. as It* 0 in the above sense. To infinite numbers, inductive proofs of this
kind starting from 0 cannot be applied.
SECTION E]
INDUCTIVE RELATIONS
545
The study of R * will occupy *90. The relation R * holds between . 1 : and 1 /
if x(I [ C‘R)y or aRy or xRry or etc. The study of this “etc." occupies *91,
"on the powers of a relation." We may, for many technical purposes, regard
I [ C‘R as the 0th power of R: the other powers are R , R', etc. If S is a power
of R, so is R. Now 5 | R is | R‘S, according to the definition in *38. Thus
if we have
R € y : jS c y . D<y. 5 | R e y : D M . P e y,
P must be a power of R, because the class of powers of R is a value of y which
satisfies the hypothesis
R e y z S e y . Dg . S | R e y.
Conversely, if P is a power of R, then P is reached by repetitions of the pro¬
cess of turning .S' into 51 R, starting this process with R. Hence if P is a power
of R, we shall have
R e y ; S € y . D, . S | R € y : . P e y .
Consequently, if we denote the class of powers of R by Pot'72, we have
P e Pot ‘R . = R e y : S c y . . S \ R c y : Z> M . P c y.
We might use this as the definition of Pot'72; but we can get a somewhat
simpler form. For the above is shown, without much difficulty, to be equi¬
valent to
Pt Pot'7*. = . P(\ 72)* R,
that is, P belongs to the ancestry of R with respect to J R, in other words, P
is reached from R by proceeding along the series
R, \R‘R. | 72'j 72*72, etc.
which is the same as the series
R, R*, R\ etc.
The relation (| 72)* is important on its own account. We put
72 t . = (| 72)* Df,
and then we put
Pot ‘R = ItJR Df.
We often want to include I f* C‘R among the powers of 72; the class con¬
sisting of Pot'7* together with /[* C l R we call Potid‘72. The definition is
. Potid‘72 = 72^'(7 T C‘72),
whence we easily prove
Potid'72 = Pot'72 w t'(/r C‘72).
The relation of being related by some power of R (other than I \ C*R) is a
very important one. We denote it by R po , and put
Tip,, = *'Pot'72 Df.
Thus when xR^y, we have one of xRy, xR l y, xR*y, etc. It is easy to prove
that
R 4C W I
R* = R DO v If C‘72.
36
I’Knl.KisoMKN'A P> CARDINAL AHITHMRTIC
[PART II
:>u)
In a "'li's in wInch ••very term (except rlu- first, if* there is a first) lias an
muiM 'lintf |>i.. ami every term (except the last, if* there is a last) has
an immediate •tiicre-SMi' it I* i^ the relation of a term to its immediate
**ner , e-«.or / 1 * ( „ i» tin- iii«>n of any earlier term to anv later •me.
1 ‘lie next nuiiiher coiicitiis itself with some special properties of the
powers nf one-many. many-one and one-one relations.
The next iiiiiiiIh i analyses the field of a relation into successive
iit’iirmtiiw*. ni. it tie- relation is that of |uircnt and child, the first generation
will ei.nsisi o| Adam and live, the second of their children, the third of their
giandehildren. and >•» «m. taking always the longest route from Adam and Eve
when there have l»e.-n mteimarriages lietwecn generations. That is, taking
any relation /*, the tii^t generation is l)‘/ > — (I 4 /*, the second is <W J — (I*(/’•)
the third is (l*t 7**i —«|‘( !*•). and so on. (Senerally. if 7'is a power of /* (in¬
cluding / T 1 ''7*). ila* correspniiding generation is
< 1 4 7' - < 1 4 ( T P).
i.v. <1*7*—
In order to express this more conveniently, we iii(r«»diice a new symbol
min/., which is n-ipiired a Is.. «»n other grounds, esjiecinl I y in series. •• mill/***
may lie read " uiitiiinniii with respect to P." We regard "sPy" as ".r
preee.les #/ tlien iii a class a, tin- "minima of a will be those members of
a which belong to l H P and are not preceded by any other members of a,
/>. q rs( H P -/'“a We put therefore
x min /.a . = ./< a r\ C H P - l ut a.
min,, i-. ra (x t a r\C*P — P“a) L)f.
•nee we have
in in/.‘a = a n C*P — J Hl a.
i.e. miii/.‘a consists of those members of* a r\ ('* P which are not preceded by any
other members of a. (It a has a single first term, this term is min/.'a.) Thus
we have, when T is a power of P,
tmnr'CPT^aT-P^n^T.
Thus min/.‘G 4 7\ where T is any power of P (including I [ C‘P). is the
generation of P corresponding to 7’; thus the whole class of generations is
miii/. 4 ‘< I “Pot id 4 /'. Hence we put
gen 4 P = inin,.“(J“Potid‘P Df.
where "gen stands for "generation.”
The notation " mill/." will not be much used until we come to series, but
then it will be constantly used. At present, we shall only give such properties
of miii/> as are necessary for our immediate purposes, but in Part V (on series)
we shall devote a number (*205) to its properties.
SECTION E]
INDUCTIVE RELATIONS
547
In this number we also introduce the uotation “ xBP ” for “x e D‘P — (I‘P.”
ma y be rcad "* begins P.” If there is a single beginning of P, this
is B‘P\ otherwise the class of beginnings is B*P t which = D‘P — d‘P.
Thus if P is the relation of father and son, B‘P = Adam; if P is the relation
of parent and child, P‘P-Adam and Eve. B‘P will be the end of P if
I . -► V ’
there is one; generally, JJ‘P will be the class of ends, i.e. (1‘P-D‘P. The
first generation of P is P‘P. If p € 1 CIs. any generation of P is f“]?P,
where T is the corresponding power of P.
The field of a relation consists, in general, not only of the generations of
P, but also of another part, the part in which, however far we go backwards,
we never reach a beginning. This part is p‘CI“Pot‘P. The two parts
s‘gen‘P and p‘CI“Pot‘P are mutually exclusive, and together exhaust C l P.
I he two next numbers, *94 and *95, are hardly ever relevant in subsequent
propositions, and may therefore be omitted by any reader who is not interested
in their subject-matter. *94 deals with powers of relative products. It is
only used in the following number (*95), on " cqui-factor relations.” The
matter to be dealt with in this number (*95) may be explained as follows.
In dealing with correlations and similar topics, we often wish to consider
the series of relations
B.P\R\Q t P*\RiQ*,P»\R\Q>, etc.
Now we have not yet at our command a definition of P-, where v is any finite
number; thus we cannot define a general term of this series as P ¥ j R J Q*.
We need therefore a different method of definition. We have
P\R\Q = (P || QYR, P*\R\Q* = (P\\ Q)ur f
and so on. Thus if T is any power of (P |) I (! Q). a general term of our series
is 'PR. For convenience of notation, we put
P*Q = sg‘(P||Q)* Dft
Then our series consists of ( P*Q)*R . The sum of all relations of this class
is considered in this number.
The principal propositions proved in *94 and *95 are two which have the
same hypothesis as the Schroder-Bernstein theorem, namely
R f Sc 1-+1. a ‘S C D‘P . d‘R C D‘S.
These two propositions state that, with the above hypothesis,
^gen^P | S) sm «‘gen*(S | R)
and p'a^Pot'CPIPjsmp'a^Pot^J R).
The two combined reconstitute the Schroder-Bernstein theorem, since
*‘gen‘(P | S ) u p‘<3“Pot‘(P | S) = D *R •
and a'gen'OSI R) wp‘(3“Pot‘(S| R)= D‘S.
Thus they present, so to speak, an itemized account of the equality proved by
the Schr5der-Bern8tein theorem.
35—2
l’ROI.EOOMEN.\ TO CARDINAL ARITHMETIC
[PART II
IX
*fl(>, on tin- |M»*iU*rity of a icriii, i** concerned with the properties of
chieHv when If 1. In tln^ case, in general. R+ f .r consists of two parts,
first an open .series and then a cyclic series. Either of these may vanish, or
may reduce to a single term. If we call the two parts ,3 and 7 . the whole of
,3 precedes the whole of 7 . and & *1 If. 7 *] If € 1 —> 1. Thus if either j3 or 7
vanishes. R+r] /.*»!—> I. If 7 vanishes, the series never returns into itself,
that is. | If^.GJ. If 7 exists, there is a definite power of If. say T.
such that // € 7 . . //7//. If and 7 both exist, there is one term, namely
1 lie successor tin- last term of f3. which has just two immediate predecessors,
one in t3 and one in 7 ; every other term of lf^*.r has only one immediate
predecessor in Thus /»’**.'• ' s *haj>ed like a (J. with r at the tip of the tail.
*07 deals with the analysis of tin* field of n relation into families. Taking
jiiiv member .1 of < u /f. the family of x with respect- to If is If+*.v v If#‘j'. which
we write lf* , .r. 'finis the class of families is /{+“&!{. Those families which
—> 4 — —► ~ —►
contain a member of Il*/f are If^'H'lf. If we regard as arranged
in a rectangle, in which the generations are the successive rows, then 7f*“77‘77
will ho the columns. Thus the relation of gen 1 If to lf+‘*H‘/f may be regarded
as a generalized form of tin* relation of rows and columns. Under a suitable
hypothesis, each row is a selection from the columns, and each column a
.selection from the rows. This is expressed in the following proposition:
h ; 7? c 1 -> I . H‘Ii * gen *If u t* A . D .
77*“ 7? 77 C I >“<.»*( gen *77 - t *.\). gen 'If -i'AC D“*V77*“/?77
whence we derive existence-theorems for selections in the cases concerned.
The importance of the ideas dealt with in the present section is very great.
These ideas dominate the treatment of finite and infinite, the theory of pro¬
gressions ami N... and the transition from series generated by one-one or many-
one relations of consecutive terms to series generated by transitive relations
of before and after. Wherever, in short, mathematical induction is used the
ideas treated in this section are required. The portions of our subsequent
work in which this section is most referred to are the two sections on finite and
infinite cardinals and ordinals (Part III, Section 0 and Part V, Section E).
In the general theory of cardinals, t.c. in Part III, Sections A and B, before
the distinction of finite and infinite has been introduced, the present section
will be seldom if ever referred to # .
• The present section is based on the work of Frege, who first defined the ancestral relation.
See his Begrifftehri/l (Halle, 1879) Part in., pp. 55—87. Cf. also his OrundgettCe dtr Ariih-
ntftik, Vol. 1 . (Jcno, 1893), §§ 45. 46 (pp. 59. 60). In this work the ancestral relation is usod to
prove the properties of finite cardinals and N 0 .
*90. ON THE ANCESTRAL RELATION
Summary of *90.
If R is any relation, “ xR+y" is to mean “ x is an ancestor of y with
respect to R,” where a term counts as its own ancestor provided it belongs
to the field of R. The definition of R# is as follows:
*9001. R* = Zf){xeC t RiR“rQp.xcvL.'} l >.ye f L) Df
That is, xR+y is to hold when x belongs to the field of R, and y belongs to
every hereditary class to which * belongs; a hereditary class being a class m
such that R“/xCfx, i.e. such that all successors of ft’s are m’s.
*90 02. R * = Cnv'R* Df
This definition serves merely to decide the ambiguity between ( It )* and
Cnv‘R*. either of which might be meant of R*. It will be shown, however,
that the two are equal (#90-132).
The most important propositions of this number are the following:
*90112. h xR#y : <f>z . zRw . „ . <f>w : <f>x : D . <f>y
I.e. if xR#y and if <f>z is a hereditary property belonging to x, then it
belongs to y.
*9012. V ixe C*R . = . xli+x
I.e. R* is reflexive throughout the field of R, but not elsewhere.
*9014. K . D‘R* - d‘R* = C‘R+ = C‘R
*9015. h . / r C*R G Rtf,
*90161. KRGR*
*90 16. h . R* • R G R*
*90 163. f- . R“R*‘:rC R*‘*
I.e. R**x is a hereditary class.
*9017. h . R£ = R*
*90 21. h : a C C‘R . = . a C R m “a . = . a C R*“a
*90 22. h : R“a Ca. = . R*“a C a
I.e. the classes that are hereditary with respect to R are the same as those
that are hereditary with respect to R*.
*90 31. H . R* ~IfC‘R c/R*|R
*90 32. h.R|R* = Rc/ R|R*| R = R*|R
*9033. h . R+“a = (an C‘R) w R m “R“a = (a n C‘R) w R“R+“a
*90-4. h . (R*)* = R*
•Vi 11
PROLEGOMENA r«» CARDINAL ARITHMETIC
[PART II
'•90 01. /i’ # » ‘5 ! »t ('' //: /i ‘ V C f /i. D, . </ f /i 1)!‘
• 90 02 /7.* = rnv‘77* IM
*90 1 H .'7(*y . = :.. t (" 7/: 77*> C ^ ..*• // t ,z f^21 :J . <*90" I )]
*90 101. I- : /7“/z C //. = . /.*“ - u C - M
H . *:»7 17 1.31-:. //'V .»77v • X.., • //«/*:
| I ran*.|.| s ://« — n . .'77y.
| .*.'57’ 17 | 3 : 77" — /i C — D H . Pi*oj»
f.'MI' 102 I' a l« a liillisi !• il* *90" 1 1.
90 102. H 3 : /•** V C /*.//* m • ^#* • •*' *
Itr in.
h . *!MHII| . 3
I- a : 77‘* - /z C - /* ..rc/z. 3 . #/ € /a :
| Ti:iIin|> | S : 77** -/I C - /z . .»/€-/! . 3 ..r*-/z (ll
H .111. *10-11 271 .3
1* : * : 77“ — /< C — /i • // * — ^ . 3„ • .r « — /z:
|*22 941 = : 77* V Prop
♦ 90 11. 1- .*77+// . ^ : ./•« ("77 : 77‘V C/i.//c/i. [*90 1102]
*90 111. H 77+v . :../•« f "77« /z . * 77 //•. 3 :tr . »/ < /z
|*!>0 1 .*:17 171]
*90 112. H :. .# 77+// : 0: . c77«-. D. , r . 0//*: 0/ : 3 . 0//
hr in,
K *90111
M
1* s: • , '77 # // . 3 : e: [<f>z). cJhv. 3.. *.. /»• * 2 ( 0j ): ./ 6 2 (0-1: 3 : // e ? (0: >:.
| *2(K1] 3 :. 0: . r77w. D.„. ibw :0r:D. 0// (1)
H . (1). 1 ni|>. 3 h . Pmp
*9012. V -..rtC'li . = .jR+t
V ein.
h . *90 1 . 3 h : -i77*.t . 3 . .* € t"77 (1)
J- . *:P27 . * 10 11.3 h : 77*‘/z C /z . j* * /z . 3„ . .c * /z :
[*3 21 ] 3 h X t C"/7.3 : x € ("77 : 77“/z C /z . x e /z. 3 M . x e n :
1*901] 3:x/7** (2)
h.(l).(2). Dh. Prop
SECTION Ej
ON THE ANCESTRAL RELATION
551
*9013. h : xR^y . D . x, y e C*R . xR^x . y7£*y
Dem.
h .*37 16. *33161 . DK R“C‘RCC‘R (1)
h.*901. D h : xR^y .D.xe C‘R ( 2 )
K . *oo i Dh:. xR*y . D : R“C*R C C‘R ..ceC H R, D . y € C‘R :
[(1).(2)] Dsy«C*K (3)
*"•(-)• (•*) • *0012 . D h : xR^y . D . xR^x . y7?*y ( 4 )
h . (2) . (3) . (4). D h . Prop
The following proposition is a lemma for *90132.
*90131. H :.xR*y .= z y € C‘R : R“ p C p . y t ^
Dem.
H .*901118. D
h *7f#y . D : y e C‘7* : 7*“ M C/i.ye/t. D M . e /x (1)
h . *37 15 . *33161 .Dh. 7*“C‘7? C C‘7f (2)
f-. *101 . D h :.y e C*R : R“fx C /*.yc/i.D M ./f/*:D:
y c C‘/i : /*“C‘/e C C‘7* . y <• CVx‘. Z> . ,‘ f C*R :
[*5-33] D : R“&R C C‘7< .O.xeC'R:
L(2)] DzxeC'R (; j)
H . (3) . *53 . D h y e C*R : 7?‘V C /x.ye M .D M .xc^:D:
# f (J*R : R“fj. C ft. y e fx . . x e /x z
[*9011] D: xlt+y (4)
K<1).<4). Dh.Prop '
*90132. h.(7i)*=7<*
Dem.
h . *3133 . *33 22 . *90 1 . D
h s. y (/*)*«. a : y « C‘7* : 7?‘V C /* . y . a; c M :
[*90131] = zxR+y :
[*3111] = zyR+xz. D h . prop
In accordance with^our general convention as regards suffixes, and with
the definition *90 02, R+ means Cnv‘7f*. not (Ii)+.
*90 14. h . D‘7?* = d‘7?* = C‘7<* = C‘R
Dem.
I-. *9012 . *33 1417 .DH:k C‘R .Z>. xt V>‘R m , Ie a'/(, .xt&R* (1)
V . *33-13. D h : x t D'R *. = . (gy). xR*y.
[*9013] 0.xeC‘R (2)
Similarly h :xe a‘R 0 . 2 .xe C'R (3)
I-.(2).(3).*3316. Dl -:xeC‘R+.O.xeC‘R (4)
I-. (1) . (2) . (3) . (4) . 3 I-. Prop
I'ROI.ROOMRXA TO CARDINAL ARITHMETIC
(part II
*90 141 h : ;.| ! A*. = . 3 ! R [*9014 . *33 24]
*9015 K /fr-AC A*
l)e>n.
I- . *50 I . *33101 . D y : x(/ [ C'R)y . = ../= y. y € C‘A .
[*90 12] * =.x = y. yR+y .
(*1313] D . x A*y: D h . Prop
Noir that I [( U R may be conveniently regarded as the 0th power of R.
l»y *5O 04'6’». when multiplied by R it gives A; also it is contained in R \ R,
w
R- A\ etc 1 ha*» propm ties. as regards relational multiplication, analogous
to those o| | m Midinaiy multiplication: thus to regard /['C , ‘A > as the 0th
powei ol R is analogous to regarding 1 as the 0th power of n, where n is a
miuibei.
*90 151. y . R G A*
Dem.
y . * 11 * 1 . D *■ :: z t n • zRw • D.. *. • IU € f* ! D '• x t f* , x Ay . D . y t m :•
| Kxp.f.'omm] D :.xAy. D :x€/x. D .yep (I)
h . ( I ). ('•>miu . Imp . D
I- :: xAy . D z */x . zRm . D f<M .. w e y : x c y : D . y c y (2)
y .( 2 ). *1011 21 . D
y :: x Ry . D * /x . : A//*. . y e y s.
|*90l I I.*33 171 D :.xA*y ::Dh. Prop
*90 16 h . A* R G A*
Dem.
h . *1 I ‘ I .DH:.:</x. zRw . ^ I ir . w e y : 0 : y € y . yAi». D . w e y (1)
K *90111 .*101 . Fact . D
y :: xA*y . yRv . D : f /x . zRm . w e y : x * /x : 0 . y e y. yRe (2)
K(l).( 2 >. D
I- :: xA*y. »/ R r ,D:.;e/x> *Am» . D,.*.. we yzxe yz 0 . ve y (3)
h . (3). *10*11 '21 .*90111 . D
h : xA*y . yAe . D . xA # r (4)
h. (4). *1011 23. *34 1 . Dh. Prop
*90161. h 5 G R+.O.S R G A*
Dem.
y . *3434. I> I-: Hp. D . Sj A G A*J A (1)
I*.(1).*90*16.DK Prop
*90162. H . A= G A* [*90151-161]
*90 163. I-. «“fi.‘xcS*‘i [*37-301 . *3219 . *901(>]
This proposition is important, since it proves that R*‘.c is a hereditary class.
SECTION E]
ON THE ANCESTRAL RELATION
553
*90 164 . h . R“R*“a C R*“a [*37 33 201 . *90*16]
v
This proposition shows that R#“a is a hereditary class.
* 9017 . \-.Ri = R#
Note that R& means (R+y, not ( R 2 ) # .
De/n.
b . *9013 .
D b : xR+y . 3 . ar/**y . yR#y .
[*34*5.*10*24]
D . xRly
a)
b . *90*163*1 —
ft
. D h yR#z . D : y c 7^‘ar . D . s c R*‘x :
[*32181]
D : xR+y . D . xR#2
(2)
b . (2) . Imp .
D 1- : xR+y . yR+z . D . xR +2 :
[*11 11.*34*55]
Db-.R^CL R+
(3)
h.(l).(3).DK
Prop
*90171. b .R+“R+“a- R m “a [*90 17 . *3733]
*90172. b .R\R m G R m
Deni.
b. *90151 . D b . R\ R m d R£ (1)
b. (1). *9017. Dh. Prop
*90 18. b s P G Q . D . P* <2 Q*
Dem.
b . *33265 .Dh:.Hp.D:x« C'P .D.xe C‘Q (1)
h . *37*201 . D h :: Hp . D C Q‘V :•
[*22*44] D Q‘V C /* . D . C M :.
[P act] D s. Q ,4 ^x C/i.xf/ii 3. P ft ft C. /t. x e ft
V V
[ S y n ] ^*”P“fACfA'X€ft.5.i/cft:D:Q t ‘ftCft.x€ft.'D.yeft (2)
h. (2). *10*11*21*27. D
h:: Hp.D:. P*‘ftC ft. x * ft. . y e ft: D :Q“y. C ft. x e ft. . y e ft (3)
h . (1) . (3) . *90*1 . D h Hp . D : xP+y . D . a:Q*y D h . Prop
*90 21. b:aCC‘R. = .aC R+“a . = . a C R+“a
Dem.
I-. *4*7 . D b :. a C C*R .Dsarca.D.arca.arc C l R .
[*90"12] D . x e a . xR#x .
[*10*24.*371 105] D .arc . a:c R*“a (1)
b . *37*16 .DP:aC . D . a C <3‘.R*.
[*9014] D.aC C‘R
h . *37*15 . *90*14. D b s a C . D . a C C‘/i
h . (1). (2). (3). D h . Prop
( 2 )
(3)
l'RO|.F.<;n\|EXA TO CARMXAI. ARITHMETIC
| PART II
501
►90 22. h/kco.E.VoCft
I.h'iii,
^ . *90* I . 0 h .»7 i* 4,'/ . D, , : /.’“o C o ..#•«& . D . // t o
|»*•• 111111 1 D f- /.’“o C o . D : .«7? # v ..« t a . D, . y < a :
[*:iM7l| D:/VoCo <1>
h . *90 151 . *37->Ol . D I-. /7“o C //*“« •
(♦22-4+| D V : //*“» Ca.D. /7“o C a ( J >
H .(I).<2>. D K . IV..,.
‘90 23. K : a C C*lt . /7“o C a .-.<* = /7 v “a [+90-2122]
vImi J'l i> iis.-,ii| in the iI m'..i v *.f section* «.f a scries (*21 I). A section of
:li.- v,^••n.'iai.'.l l»y /«’ i* 'li'Hiuil a> a cla**- q Miti'fvinj'
o C'" /.’. /.’“a Co.
•90 24 I- : /> V C/i. a C /i. D . AV‘ft C „
/>»/«.
H . *37 2 . D H : 11. D . /.’*“«. C //*> (11
»-.*90>2. Dh II,.. D. ZfwC/i (2)
H .< I >,<2>. Z) h . Prop
I lii^ |H'o|»>Niiioii hIm.xvs dial if n i> a li.-i*•• lit;ii*y rlas*. which contains ft. then
H oiiraiiiN all tin* ili'M'cnilants *.| o <*.
+90 25. h : o C( H R . /7*“o C/*. D.oCp
//e##i.
K*9021 . Dh: Hp.D.ftC/.V’ft.
(II|»J D.oC/.:DK Prop
*90 26. H o C r*/,*. yj‘> . D : a C ^ . s . /7*“o C ^
//*•«#. ^
K*90 24. Dh. H|».D:aC/#.D. /^"oC*t (I)
H . *5)0*25 . D h 11 p. D : /? # “o C^.D.oC/i (2)
K . (1) .(2). D h . Prop
*90 27. I-a C C/f • D : a w //“/* C M . = . //*“« v 7/‘V C M
Item.
h . *90-26 . Exp . *5'32 . D
h a C 6*‘7? . D : 5-V C/i.oC/i. = . X - V C M . //*"« C ^ :
[*22-59] Dsawi'V C /i. = . /7*“o \j R**ft C /iD h . Prop
*90 31. H .//*=/ T ^7?
Dem.
h . *901510 . I) h . / r C*‘J? u //* // G //*
(1)
SECTION E]
ON THE ANCESTRAL RELATION
[Fact] D H : .r (/ I* C‘72 c; 72* | R) z . zRw . D . xR*z . *72«-.
[*10-24.*341] D.*(72* R) W .
[*23 58] > .x(I f C‘Rv R* R)w (2)
h. *9013 . *50 3. D b: .r72*y . D . xlx . x € C‘R .
[*35101] D-xilfC^x.
[*23*58] D . x (/ r C‘72 w 72* ; R) x .
[*4-7] ^ .xR*y .x(I [C'Rsv R*\R).r (3)
I-. (2). (3). *90112 r S. r -L^' t, i^ R *J R >*. o
<f>z
b : xR+y . D . a: (7 [* C“72 o 72* 72) y < 4)
b.(1).(4).D b . Prop
In the last line of the above proof, the process is as follows. Writing <f >2
for x(I f C*R «/ 72* | R)z t (2) becomes tf>z . zRw . D . <f>w, while (3) becomes
xR#y . D . xR+y . <f>x. Hence, by (2) and (3),
^ • xR*y ’• <\>z • tRw . D 7>tr . <£?«/: <f>x.
Hence, by *90112, xR^y . D . <£y, which is the proposition to be proved.
*90 311. b. 72*-/[* 0*72 0 72 72*
Dem.
b. *90 31 ^. *90 132. D
b . /2* = / r C‘72 « 72* i R
[*33‘22.*34*2] = / f C‘72 o Cnv‘(72 ; 72*)
[*50-5-51] = Cnv‘(7 r C‘72) o Cnv‘(72 j R *)
[*3115] = Cnv‘(7 r C‘72 o 7? i 7?*) ( 1 )
b. (1). *31-32. DI-. Prop
*90 32. b . R | 72* = 72 o R | 72* j R = 72*1 R (2)
Dem.
I-. *90-31 . D l- . R I 72* - 72 | / r C‘72 o 72 72*j 72
[*50*64] = 72 o 72 j 72* 72 (1)
[*50-65] - (7 r C‘72) I 72 o 72 j 72* 72
[*90-311.*34-26] = 72* j 72 (2)
b.(l).(2).Db.Prop
*90 33. I-. «»“a = (a « C'R) xj R m “R“a = (a r. C‘R) xj R“R„" a
Dem.
K . *90-31. *37-221. D
h . 72*“a = (7 r G“72)“a yj (72* 1 72)“a
[*37-412-33] = 7“(C‘T2 n a) v 72*“72“a
[*5016] = (C‘72 r> a) /2*“72“a
(1)
Similarly, by *90 311,
H . 72**‘a = (C‘72 « a) 72“72*“a
(2)
b . (1) . (2) .DK Prop
Ci
PROLEGOMENA TO CARDINAL ARITHMETIC
(PART II
-90 331 b . Va = (anC‘R) w 7?*“7?“« = <a o C‘R)vR“R*“a
( Proof as in *90*33J
*90 34 b :a C ' H D.O. R+“a = a ^ 7f*“7f“a = a u R“R*“a
[*9o33. *22«>2IJ
*90 341 haC C*R . D . /7*“a ^ a w ]{**']$“a = a v R“R+“a
(*9033I .*221121 |
90 35 b : ..,R
!<*: . 3 s 7i‘V C M . ^7‘.i C M . * c ft
Dem.
K *32 181
. D 1- .i-Rif ,D:yc /f *.#•;
[*22 4i»)
D : /f.r C ft.Z> . ye ft:
( Pad |
0:R , 't*Cn. 4 R*.rCfi.D.R“ftCn., € f X
(1)
1- . *90 1 .
^ !/K+z • ^ •* R‘ V c fi . •/€ fi.D . z € fj
(2)
K<1).<2>
[* 10*11*2.3.
. D H :. xRi/ . yR+z . D : R tl ftQ ft . R*.r C ft. 0 . z e ft
.*:U 1J D 1- Wf, 7f*: . D : 7f« > C M .%.r C ft.D.ztft
(*>
h.(.*<).*1011-21 .Dh.lVop
*90 351. b :. /?‘V C M . 7f‘.r C fi .D M . z € ft:D .xR 7f**
Dem.
V . *!!()• 172. Pad. D 1- : .r/f 7f*; .zRm.D. j-R+z . ;/?w .
[♦3+i J D..//^l/e w *.
1*90-32) 0..rR\R^w
K<1). *37*171 . Dh.D“2(.,R R m z)C z (sR l R+z)
h . *90 32 . Dh: .#•/*„. D . W* 7?*y :
[*321.31 .*20-3] Dhr^c/Tv.D.y^SCr/; R^zU
(*1011 .*221] D b . /<V C?(x/; | 7?*r)
b .(2).(3).*10-1 . D
H :. R lt ft C /z . 7tf*.r C ft. D,. • z e n i 0 .2 e 2 (.r7£ | 7 ?*j).
[*20-3] D. *7? | R*z z.Db. Prop
*90 36. b :. .cR . R^z . = : /f“/i C ft . 7?‘.r C ft. . z e ft [*90‘35'351 ]
*90 4. Mtt*)* = tt*
*904.
Dem.
1-. *90151*18 . D K . 7?* G ( 7f*)* (1)
h.*90 1l2 W *' X f* Z .D
IX, <p:
I-:. .r (7?#)* y : xR # z . zR+tu . D f>tr . j-7? # i<» : xR&v : D . .r7?#y (2)
SECTION Ej
ON THE ANCESTRAL RELATION
557
\- . *9013 . D h : *(/?*)* y . D . xe C‘R * .
[*9014] O.xeC'R.
[*9012] D.xR+x (3)
f- . *9017 .Dbz xR#2 . zR+vj . D /iM , . xR^w (4)
K(2).(3).(4 ).D\-:x(R*)*y.D.xR* I/ (5)
h . (1) . (5) . DK Prop
*9041. h . C‘P* [a = an C‘P
Dem.
I-. *37-41 . 3 K C‘P* [ a = a n (P*“a ^ .P*“a) (1)
h . (1) . *371510 . *90 14.31-. C*l\ [ a C a ^ C‘P (2)
I-. *90-33-331 . DKqa C*P C />*“<* w; P # “a (3)
H.(3).(l). DKa«C‘/ J CC‘P*t« (4)
h.(2).(4) . 3 K Prop
*90-42. I- . (Q* [ a)* = Q* C a
Dem.
K *9018. 3h.(Q*[a)*G<(?*)*
[*90-4] G<?* (1)
K. *9013. 3h:*(Q* [ a) # y . 3 . x,y « C‘Q* [a.
[*90 41] D.x.yea (2)
h.(l).<2). 3 !-.(«* C«)*GQ*C« (3)
K . (3) . *90151 .Dh. Prop
*91 ON I’nWKHS OF A RELATION
Sum mor>f of *!l|.
In f In* |»l**s«-lit IiiiiiiIht. we consider the r|;i>i of relation'*
It. If. If .
I‘«:n*1 1 "l these has to it** |inilm , »sor the relation 77: hv have
It ■* If It. R»m It* If. etc.
I I*'I s • ■ very term of tin- series ha** I! tin* relation ( 77)*: hence the powers
m! It may I"* defined a** those relation^ which have to It the relation ( 70*.
I It** serfs "I |N»\\er> starting with l[(' t /t instead ul' with It is similarlv
composed "I those relations which have to / fC‘77 the relation ( 70*. (This
'•lass i*MiiM't> of the pre\ ioii> class together with / [ C u It.) To say that the
relation 77* holds hetweeii .* and y turns out to he equivalent to saying that
one ol t In* relations
IfC-lt. It. If. It. ...
holds heiween .i and //; and to say that the relation 77 77* holds between
and >/ turns out to Ik* equivalent to saying that one of the relations
77. If. If. ...
holds between and if. Thus we might have begun hv defining power* of 77.
and jiroi.tied to fletine 77* a** their sum.
For tftational convenience we put
77,. — ( 77)* Df.
Then the definition of powers of 77 excluding / f* C*It is
Pot 1 77 = 77, ,* 77 Df.
and the definition of j lowers of 77 including I f" 7**77 is
Pot id* 77 ■*"/?„*( 7 r C*77) Df.
(Here the letters "id” an* added to suggest that identity is to be added to
Pot* 77.)
We put also
77 |l0 = v* Pot ‘77 Df.
Many of the propositions in this number are very often used. Among the
more important propositions are the following:
*9117. I- P € Pot id *77: <f>S . D,. <f>(S\ 77): (7 f C* 77): Z>. «/,7»
*91171. I- 7* c Pot* 77 : <f>S . D. s . <t> (*’ i : ^77-: D . 07*
*91*373. h P c Pot*77 . D r . <f>P : ^ z 4>Ii : S e Pot*77.0.S*. D,. <f> (S, It)
SECTION E]
ON POWERS OF A RELATION
559
These are formulae of induction. The first two state that if the property
<t> is hereditary with respect to , R, then if <f> belongs to I T C‘R it belongs to
any member of Potid‘7?, while if <f> belongs to R it belongs to any member of
Pot‘i2. The third gives a form of induction which is sometimes more powerful
than the second. It states that if <p is hereditary provided its argument is a
power of R, and if <f>R, then every power of R satisfies <f>. and vice versa.
*9123. b . Potid‘22 = P(1 f C*R) vj Pot ‘R
*91 24. b . Pot ‘R = | P“Potid‘P
These two propositions are very useful as giving relations of Pot‘if and
Potid'iS.
*91 27. b : P e Potid'P . D . C*P C C*R
*91 271. b : P e Pot ‘R . Z> . D <P C D ‘R . (VP C(l‘R
We do not have in general Pe Pot‘R . D . D*P = D‘R . (I‘P = (PR. If
R is the sort of relation which generates a series (i.e. is either itself serial, or
such that R vo is serial), the above would characterize a series without a first
or last term. To illustrate the matter, consider a series of four terms, x, xj, z, tv,
and let R be the relation of immediately preceding in this series. Thus R
holds between x and y, xj and z, z and w. Then R 3 holds between x and z, //
and w\ thus z, which belongs to D‘R, does not belong to V*R 3 . R 3 holds only
between x and w\ thus neither xj nor z belongs to D‘/£ a . All powers of R
beyond the third are null. On the other hand, if we take a cyclic relation,
such as that of left-hand neighbour at a dinner-table, we shall always have
D‘P« T)*R . (PP = (\‘R, whatever power of R P may be.
*91 282. b : P € Pot ‘R . D . P | R € Pot ‘R
This proposition shows that Pot*/7 is a hereditary class with respect to ' K.
*91 34. b:P,Qe Potid *R . D .P\Q=Q\P
This proposition states that the relative product is commutative when
each factor is / f* C*R or a power of R.
We come next to propositions concerning R lM . We have
*91602. b.RGR^
*91 604. b . V'R^ = D‘R . C PR„ = C PR . C‘R % „ = C‘R
*91611. b.R^RGR^
*91-62. b . R^ = R* | R = R R*
*91 64. b . R+ = 1 \ C*R kj R^
*91-52-54 are fundamental in the theory of inductive relations.
*9T642. b : xR+y . x + y . = . xR^ .x±y
This proposition is particularly useful when (as often happens) we have
7£po G J • In that case, it gives R^ = R * A J.
"iOO
PROLEGOMENA TO CARDINAL ARITHMETIC
[ PART II
*9155. b . /{# = x* Pot id 4 R
*91 56 b . /i,;, G /i„.
Thus A',*, i> always transitive, which is one of the three characteristics
• »f serial relations (cf. *20+). We shall find that is often serial when II is
not so.
*91 574 b. K* R„=R*. R+ = R„=R
*91602. K(/i,„,>*«/<*
*9101.
/*.,-(* >*
l>f
*9102
/{,. = t R >*
I)f
*9103
Df
*9104
V»t\i\‘R-l< u t (/[C t R)
Di¬
*9105
R^-V Pot* R
or
The first two of the above definitions are introduced merely for notational
convenience. The other three represent ideas of great importance. The last
is especially useful when a series is given as the field of a one-one relation
between consecutive* terms—as. e.y., when the series of natural numbers is
given as the field of the relation of n to a + I. Then ii |M , is the relation of
any earlier term to any later term— e.ff., in the above case of the natural
numbers, the relation of a less integer to a greater.
*911. b :: PR^Q . a 8* n . 0 S . R S t n : Q € n : . I* i fx
Drm.
b . *4 *2 . (*9101) . D
b
:zPR mX Q.rnz.P(R\) 0 Qz.
*
[*9011]
[*+3-3.*33*161) s (R \ Y‘n C y .. Q € M
. D* . i J €M
[*3761]
= St m • • R *R«t
x : Q * fit: . I* e
|*+:M1)
= :. .S’ e n . D s . R \ S * f,
i: Q e fx:0 H . I* t fizz D H . Prop
*9111.
1- :: J*R t Jj • s •• S e ft • 0 S • 8 ! R *
? M : Q € /x : . P € y.
*9112.
bzPcPot*R.s.PR u R
[*3218. (*9103)]
*9113.
b zz P € Pot *R . = St /* . D,. S
R t ft i R € fX Z m P € fX
[*1)11112]
*9114.
b : R « Potid *R . = . l’RuVX C € R) [*3218 . (*910+)]
*9115.
b zz 1* c Potid*ii . s S « /* . 0 S .
S j /i € /a : / [* C‘ R e y z . P r y
[*91111+]
*9116.
1- :: xR^y . = (g/*) S e /a . D s
. S R e y z R c y z . P e yz. xPy
[*+111 .(*9105). *9113]
SECTION E]
ON POWERS OF A REI.ATTOV
*9117. bft € Potid'ft : . 3 S . *(-S| ft) : $ (/[• C*ft): 3 . ,£ft
*91171. b :. ft « Pot'ft : . 3, .<f>(S\R): <pR .<t>P
[*91-13 *<M>]
These propositions are of great importance, because they enable us to
prove that a property <f, belongs to every power of ft if it belongs to ft
(or I [• C‘Il) and also belongs to £ | ft whenever it belongs to -S.
*91 2. 1- : Qft„ft . 3 . (Q | ft) ft, .ft
Dem.
h.*43101.(*91-02). 3I- :Hp.D.(4»|ft)(|ft) Q.Q{\R) m p.
[*90 172] 3.(0|ft)(jft)*ft.
[Id.(*91 02)] 3 • (Q! ft) ft,.ft: 3 I-. Prop
*91 201. I-: Qft.,ft. 3 . (ft j Q) ft„ft [Proof as in *912]
*91 204. I- : ft (ft,. | ( | ft)] Q . = . ftft,. (Q | ft)
Dem.
1-. *34-1 . 3 b : ft |ft„; (, ft)| Q . - . ( a 7*) . ftft,.ft. ft(| ft) y .
[*43101] -.(aft), ftft,.ft. ft=Q(/e.
[* 13195 1 = • (Q | ft) : 3 b . Prop
*91 205. b : ft (ft., | (ft |)] Q . = . ftft., (ft | Q)
*9121. h.ft,.-/oft„|(|ft)
Dem.
I-. *90 31 . (*9102) .31-. ft,, = 7|" C‘(l ft) w ft,. |(( ft)
. [**3-311] -= / w ft,. | (| ft). D f . p rop
*91211. I-. ft.,=./o ft.,|(ft|)
*91 212. b :. ftft„Q. = : ft = Q. v . ftft,. (Q|ft)
Dem.
1-. *91-21. *501.3 I- :.ftft,.e. = :ft=Q.v. ft (ft, .|(|ft))Q :
■>” 1 ' 204 J = : ft = 0. v . ftft,. (Q | ft):. 3b. Prop
*91 213. b :. ftft. t Q . = : ft = Q . v . ftft., (ft | Q)
*91-22. b . R„‘Q — t‘4> w ft„‘(Q | ft) [*91-212. *3218. *5115]
*91221. *-.~R. l ‘Q = i‘QuR. t ‘(R\Q)
*91 23. b . Potid-ft = t‘(7 f- C‘ft) vPot'ft
Dem.
h • *91-22 . (*9104) .DK Potid'-ft = i\I f* C"ie) w /£/((/ f C‘R) j /2}
[*50-65.(*9103>] = e-(/r « Pofc'A. D K Prop
36
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART I!
*91231. b = vPot‘/f [*91 22.<*91 <>3).*n0-4]
*9124 h.lW/t- /ir“Poticl‘/f
lk'in.
p . * 91-12 . d p : /*« Pot*/;. =. .
[*50G5] = /?>.
(*91 204) 2. t /?)!(/rcvo.
l*9o-32.<*9i 02i] s./'K /e> /;„!(/pc*/?).
1*373] =./'* /;“/;„*(/pc*/?).
[*4 2.<*91 04>] = ./*€ /;“Potid‘7? : D P . Prop
*91241 . p://;,</'. d.«/ 7 ? )/;,„(<? /')
J fcin.
P.*9I 2I2.DP.((/ //,.!<? /') (1)
P.*9P2. DP:«/ 5) /;„<<? 1>).3.(Q S li) Ii ln {Q\ 1>) <2)
Till- last line of thr above proof is obtained ns follows: writing fi for
S|(V *>/;,.«/ /*)!.( 1) becomes
/*«/* (1).
while (2) becomes 6’e fx . D . 6' /? c (2).
Hut bv *01 11. writ ing T for the /' of *9111, and l* for the Q. we have
TH t J> . D 5 < Ai • 3s - S H e p : /'«/*: D . 7’«
Hence, by < 1) and (2). TR t J *. D . 7*€ /i. t.c.
mj’.O.iQ T)Jt l% (Q\P),
which is the proposition to be proved.
*91242 biSIitAV PUD.S*Q\“l? t SP
Deni.
P. *9122. *4311 .DP.y ^!“/?,//' (D
P.*371 .*431 . D
h : .S'« y “«„'/*. s . O'n. T<Tr,.‘P .S-Q\T.
[»!)l-2] 3.(3^- T\Rt %.‘P .S\R<=Q\T\R.
(»:t71.*431] O.S\R ( Q\"li,.‘P (2)
h. (1) .(2). *9111 Q .Ob. Prop
*9126. b .R U ‘(Q P) = Q ,"R„‘P
Dent.
P . *91 242 .DP. 7V(0 |P)CQ
SECTION Ej
ON POWERS OF A RELATION
5G3
H.*91-241. Dt-: T t R„‘P .J! = Q\T .0 . S e~R„‘(Q\P) :
[*1011 -23] D f-: (a T). Tj~R„‘P .S=Q\T.O.S e ~U te ‘(Q | P) :
[*37 1 .*431] D I- :SeQ\“~R„‘P. D . S\ P)
H.(l).(2). DKProp
( 2 )
*91251. f-.R M ‘(Q[P) = \P“R, t ‘Q
*91-26. H ■~R„‘Q=Q\ , ‘R t ,‘T
[Proof as in *91 25)
* 91-25 1
*91261. I-. R„‘Q = | Q“P„‘7
P
.9. 25! £«]
*91 262. h : CI‘Q C C*R . D Jr^Q = Q j “Pot id‘.ft
[*91*26 7| tp* . *50*62 . (*9104)J
[
ft
*91 '25 -p . (*91 03) j
*91*22*263^1
*91263. Kft u ‘(Q|ft)-Q|‘‘Pot‘ft
*91 264. h . Pot‘ft = i*R u ft |“Pot‘ft [
*91 27. h: P e Potid'ft . D . 0*1* C C* R
Dem.
1-. *50-5-52 . D h . C‘(/ r C'ft) - C‘ft .
[*22*42] D I-. C*(I rC‘ft)C C‘ft
H . *34*38 . D 1-: C‘S C C‘ft . D . ft) C 0*11
0*SQ0*li
( 1 )
( 2 )
K(l).(2).*91*17
<p&
.DK Prop
( 1 )
( 2 )
*91 271. I -:Pe Pot ‘ft . D . D‘ft C D‘ft . a‘ft C <3‘ft
Z>cm.
H . *22*42 .DK D‘ft C D‘ft . d‘ft C CI‘ft
H . *34*36 . D h : D*S C D‘ft . D . D‘(5| ft) C D'ft . Cl*(S\ ft) C d‘ft
H ■ (1) ■ (2). ,91171 weyyc w ■ 3 h. Prop
*91 28. h : ft e Potid‘ft . D . ft | ft c Pot'ft [*91*24]
*91-281. I- : Pot'ft C Potid'ft . | ft“Potid'ft C Potid'ft [*91*23*24]
*91-282. h:ftePot‘ft.D.ft|ft e Pot‘ft [*91*28*281]
*91-283. h : | ft “Pot'ft C Pot‘ft [*91 282]
The following propositions show that the relative product of two powers
of ft is commutative, i.e. (cf. *9134)
P,Qe Potid‘ft . D . ft j Q = Q | ft.
We also have (cf. *91*341)
ft, Q e Potid'ft . D . ft ( Q e Potid ‘ft.
36—2
PROLKf50MKNA TO C\R1>1NAL ARITHMETIC
( PART II
II i«v iIh-m- propositions (as will appear in the sequel) which arc the source
• >f f ht* commutative law for the addition of finite ordinals. Ordinals in general
an- in it. com mutative, just as relative products in general arc not commutative;
l»ni owing to the fact that relative products whose factors are powers of a
given ielation are commutative, finite ordinals are commutative.
*913. h:/’*P»tiil<!?.D./t P = P R
hem.
K*:>0i;+I >;.Ob.R Ifi-lt-tt&R R d)
h.*342l. Ob.R (S /0-(/f .S) R (2)
l-.*3427 . 0\-:R S-S R . O . (R S) /*-(.* R)\R.
|(2)| 3-R (S [ R)-{S\R))R <3)
K.,U7 A ' N - S ' ".3
If}.''
H:.7M , »tid-/f:/f S-S R.O^.RiS R)-(S /f) /* : RI [ C'R - / [ C*R,R:
O.R P-P R <4)
b .( 1 ).(•!). (4). D b . Prop
*91301. biPiJtSHtWUi'l-tt P-P\R (Proof as in *9 13]
>91302. b . /{“l*otid‘/f — R “PotuP/f
liem.
I- . *91*3 .*13*182 . D 1- /*«s Potid* R . D : N- R ; . 2 . Sm P\R :
[*431101] D : N(/f|) 7*. s . N(| //) 1* (1)
b . (I). *V32 . D H : /* < Potid* . .V(/M /'. e . 7'c Pot-id*/? . .S*(j /f) :
1*10-11 2*1] D 1- s(a/ , )« /*« Potif VR.S{R ) P. = .
. /*« Potid */f . S( \ R) J* z
1*37-1 ] 0b : Sc R “ Potid* 7f. h . Sc /i“ Potid* A :0b. Prop
*91 303. b .| R'UtJilX&R)- R “1<«V \ t H R) (Proof as in *91302]
*91304 I- . /f“Pnt*/f — R ,**Pot*/f (Proof as in *91*302]
*91 31. b . Pot 'R = R i**Potid *R [*91*24*301]
*9133 I-. PotifPA =/?.,*(/ T C*R)
hem.
I- . *! 1 1 *23 ,0b . J [ C*R c Pot id* R { 1 )
H .*91*3 . Ob : Pc Potid *R .O.R\P = P\R.
[*91*281] 0 . R ; P c Potid‘7? (2)
V . (1). (2). *911 Potl,l ‘ A . 3 h : />/?„, (/ r C‘R). 3 . P ( Potid* ft (3)
I-. ,91 301.31-: PR„ (/ r CPR) . 3. ft | ft = ft | ft.
[,91201] 3. (P | «>/*„</rC*) (+)
H . ,91213.3 I-.(/ r C‘R) R* (/ r C‘R) (5)
I-. (4). (5). *0117 .Oh:Pf Potid'ft. 3 . ftft„ (/ p C‘R) (6)
h . <:)). (6). 3 I- • Prop
SECTION E]
ON POWERS OF A RELATION
565
*91331. h . Pot*P - P sl *P
Dem.
1- . *9124 33 . D K Pot*P = | P**P Ht *(/ f C*R)
[*91*251 .*50-65] =~R tt ‘R . D h . Prop
*9134. h:P,Q€Potid‘P.D.P|Q=:Q P
Dem.
*50-62 . *91-27 .DhPe Potid*P . D . P j (J f C*R) = P
[*50 (»3.*91-27] = (/ [ C*R) \ P (1)
K *34-27. D h:P€Potid‘P.P|5=-5|P.:>.P|£ R = S\P\R
[*91*3] = 5|P|P (2)
H • U) • (2) . *9117 J> ‘ P .Dh. Prop
This is the commutative law for the relative product of two powers of R.
*91-341. h : P,Qe Potid*P . D . P | Q e Potid* P
Dem.
I- . *50-62 . *91-27 . D h : P « Potid*/* . D . P | (/ f C'P) - P .
[*13-12] D.P|(7 rC‘P)«Potid‘P (1)
h . *91-281 . D h : P15< Potid*P . D . P | S\ R c Potid *P (2)
. P| Se Potid*P p
■ .(1).(2)• *91 17 J-. D I-. Prop
*91 342. H : P c Potid *P . Q e Pot*P .D.P\Qe Pot‘P
Dem.
h . *91-28 . Dh:Pe Potid*P . D . P | R e Pot ‘P (1)
h . *91 -282 • D I-: P | Q « Pot* R. O. P\Q\ Re Pot ‘P (2)
K (1). (2) . *91*171 . DH. Prop
*91-343. h : P, Q e Pot*P . D . P | Q e Pot‘P [*91 -342 23]
*91-36. h./rC*PcPotid‘P
*91351. KP«Pot‘P
*91352. h.P’ € Pot*P
*91 36. h : P e Pot*P . D . P | P, P | P e Pot‘P
[*91-23]
[*91-264]
[*91-282-351]
[*91-343 351]
*91-37. 1-Potid‘P C M . = : / r C*P c ^ : Se Potid*P. Re/x. D s . 5| P €/*
Dem.
h. *91 281 35. D
h /[* C*P€/x:56Potid*P.«S*€/x. D 5 .5|Pc/*:s :
/f* C*P € Potid*P . 1 [* C*R e ft : Se Potid* R . S e /x. O s .S J Pe Potid*P . S \ Re fx:
[*9117] D: P c Potid*P.D.P €/ * (1)
h . *91-35 . Dh: Potid'P C ^ . D . / f C*R e fx (2)
h . *91-281 . D h Potid*P C/* . D s 5 c Potid*P. D 5 . 5| P e/x :
[*3-41] D:5ePotid*P.5c^.D 5 .S\Rcp • (3)
h . (1). (2) • (3). D h . Prop
I’ROI.KCOMENA TO CARDINAL ARITHMETIC
[PART II
50 It
*91 371 b P< Potid 4 /? . D/.. 4>P : = :
</> (/ T < H /?»: * s ’ € Pot id 11 /? . <f>S. D s: . <f> (.S' 1 /?) (*91 *37]
*91 372. H:.P»t 4 /?Cji.s:/?cM:£cP»t</?.tfc#t.D<..? /? €/x
[Proof as in *9137]
*91 373. h :. /'* Pot 4 /? . D,-. <*>/': = : <*>/? : tfc Pot 4 /? . <f>S . D,. 0 (,S* /?>
[*91*372]
*9141. f-.7?,. 4 </ , |rt>«/ > 44 Pnt 4 /?
*91411. h ."/?.,*( /? /') = P“ Pot*/?
*9142 I-./?„•/' = i 4 P w /' 44 Pui 4 /? [*91*22*41 ]
*91421. b.liJP-PPs* P“ Pot 4 /? [*91*221*411]
*91 43. 1- : 7't Pot 4 /?. ?//?,./'. D . Pot 4 /?
I trm.
b .*91*42. Dh:. Hp.D:Q- P.v.QtP 44 Pot 4 /?:
[*37 1 .*+3*1) D : ?/ - /'. v . <g '/*). T e Pot 4 /?. - 7' /’:
(*1312.*91*343 J D : f/c Pot 4 /? :.Dh. Prop
*91 431 bi P< Potid 4 /? . <//?„/*. D. Q« Potid 4 /? (Proof ns in *91*43]
*91 44. b :. /'. Potid 4 /? . D : <//?„/>. v . /'/?,.?/
Item.
b .*91 14. Dh : /*€ Potid 4 /?. D . /'/?„</ T ?*'/?> (1)
H . *91*2. 0bzQR tn P.2.((J R) R tn P <2>
h . *91*212. ^ h s. PR t ,Q . D : Q. v • PR t% (Q\ /?) (3)
h* . *91*212. D b : /*-=?/. D . <//?,./'.
[*91 *2] D.<f/|/?)/?„/' . (4)
h . (3). (4). D h />/?„<?. D : (Q i /?) /?„/*. v . PR U (Q j /?) (5)
h . (2). (5) .Db:. <}li x J> . v . /'/?„</ : D : (Q /?) J? u />. v . PR tn {Q , /?) (0)
h.( I).(«). *91 *17. DK Prop
*91 45. h :. P. Q * Potid 4 /? . D : (gT): Tc Potid 4 /? : Q = /' T.v.P=Q T
J)em.
h . *‘11*202*27 . D b :. Hp. D : /?„*/>- /^“Potid 4 /? . /?„ 4 (/ = Q“ Potid 4 /? :
f *37 *l .*43*1 ] D : QR U P . = . (g7’). Tt Potid 4 /? . - 7 J | T:
PR U Q . h . (g T ). 7’ * Potid 4 /? .P=Q\T (1)
h . (1). *91*44 .*10 42 . D h . Prop
*91 46. h :. P. Q r Potid 4 /? . D : (giT): T< Potid 4 /? iQ=T P.v.P=T\Q
[*91*45*34]
The remainder of this number is concerned with /? |lQ and its relations
to /?*.
*91502. b ./?G/? I>0 [*91*351 .(*91 05). *41 13]
*91503. b.R-dR^ [*91*352. (*91 05). *41*13]
PR 1
*91 25 f j ^.(* 9103)1
SECTION E]
ON POWERS OF A RELATION
567
*91*604. b . D‘72 1MJ = D ‘R . d‘R lMi = d‘22 . &R£ = C‘R
Dem.
b. *91*502. D b. D‘R C D‘72 |K> (1)
b . *91-271 . *40*43 . D b . s‘D**Pot‘R C D‘R .
[*41-43] D b. D*R lHt C 1V72 (2)
K.(l).(2). Db.D‘72 = D‘72 |K) (3)
Similarly h. Cl*R = Cl*/?,*. C‘72 = C‘R lHt (4)
b . (3) . (4) . D b . Prop
The following propositions are concerned mainly with the relations of R lnt
and R These relations are embodied in the propositions
R ll0 = 7?* ! R = R j 72* (*91 -52)
72* = / |* C‘72 w 72 1>0 (*91-54)
and 72* = *‘Potid‘T2 (*9155)
*91*51. b.72 |(0 |72 = 72|72 1(0
Dem.
b . *43*421 .(*9105). Db. 72^172 = ** 72“Pot‘72
[*91-304] = 6‘72|“Pot‘72
[*43*42.(*91 05)] = R | 72 l>0 . D b . Prop
*91-611. b - 72 |K> I R G 72 1io [*43-421 .*91-283. *41-161]
*91-512. b . 72 |)0 G 72* | 72
Dem.
b . *90-32 . D b . R G 72* | R (1)
b . *90*16 . D b tS G 72 *|R . D . SO. 72* .
[*34-34] D. S! 72 G 72* | 72 (2)
b. (1). (2). *91171' SrG ^-^.Db :7>ePot‘72.:>..PG72*|72:
[*41151.(*9105)] D b . 72 |K) G 72* j 72 . D b . Prop
*91*513. b . 72* G i‘Potid‘72
.Dem.
K *90112
<t > 2
b xR’+y : x (£‘Potid‘72) z . ^72m; . D ZtW . a:(s‘Potid‘72) w :
a; (*‘Potid‘72) x : D . a:(i‘Potid‘72)y (1)
b . *43-421 . D b . (*‘Potid *R) | 72 = i‘| 72“Potid‘72
[*91*281.*41-161] G s‘Potid*72 .
[*341.*10*23] D b : a:(i‘Potid‘72) * . zRw . D r>u ,. a:(*‘Potid‘72) w (2)
b . *9013 . Db: a;72*y . D .xeC'R .
[*50-3.*36101] D.a(7[ C‘72) a:.
[*91-35.*41-13] D. a:(i‘Potid‘72)a; (3)
b . (2). (3). *4 71-73 . D b : Hp (1) . = . xR+y (4)
b.(l).(4).Dbs a»72*y .D.x (7‘Potid‘72) y : D b . Prop
I’ROI.EOOMENA TO CARDINAL ARITHMETIC
(PART II
-91514 I-.77* HQ It,.,
hem.
K *91*513. DI-. 77* Ii G (.v‘Potid‘7?) R
[*43 421J G x* 77"l\>ticl‘77
1*91-2+) G*‘Pof‘77
[(*9105)) G 77,„ . D h . Prop
-9152. I-. 77,.., = 77* 77=77 77* [*91*512-514. *90-32)
*91 521. h : /' t PotiiP77. = . h ( Poti.l •%
hrm.
h • *91 15 1 MX ,4 .Dh:: /*< Potid‘77 . D
7 [* C‘77 t < *iiv 4 > : N € (,'uv‘V . D % . .5* 7.’ « Cnv“/i: D . 7* e Cnv'V (I >
H . *72 5131 I . Dh: ]> ( Cnv‘V . = . 7' f ,x (9)
h .(2). *505-51 .Dh: 7 |* (’*77« 1’nv‘V . s . 7 f* 7'*77 f/x (.-{>
h . *3 151. Dh .S' € < •„v“/x . > N . N 77 «• Cnv'V : = :
S«Cm'V.X..S' ^cCnv-V:
|<2).*34‘2] “:N</i,D».7/.Nc/i (4)
H . (I ).(2).<:t>.<+>. D
H :: 7* e Pol ii I* 77. D 7 f* 7"77 «/*: A7«/x. D,. 77 Nc /x: D . P « M (5)
H .(5). *1011-21 .*!H i :W.D
h : 7*« Poiiil‘77. D . 7*< PotiiP77 ((})
:/*« I VUWf.D. 7'< l*..ti<l-/? ( 7)
K(li),("),Dh. Prop
*91 522. h : /* € Pol*77.3.7** Pot‘77 [Proof as in *91-521)
*9153. K77 l ,, = (70„.
hem. « ^ w
h.*91-52.Dh .77 ll0 =77 77*
[*!>0 1»21 =77 (7?
[♦9P52] =(77),... Dh. Prop
*9154. h . 77* = I T <'*‘77 77,.. [*90-31 . *91 -.52]
*91-541. H . 77* n./ = 77 , mi *./ [*25*401 . (*50 02). *35 +41 . *91 54]
*91542. h :./77*»/ ..»• + y . = . aR lHl y [*91 541 . *50*11]
*91 543. h . 77*“£ = (^n C‘77) u 77,„“£
hem.
K *91-5+. *37 *221 . D h . 77*“£= (7 f C‘R)“/3 o 77,
[*50-59] = ^ (7*77) w R^‘0 . D I-. Prop
SECTION E]
ON POWERS OF A RELATION
569
*91-544.
*91*545.
*91-546. b
*9155. h
Dem.
*91*56. t
Dent.
*91561. b
*91-562. b
*91-57. b
*91-571. b
*91-572. b
*91-573. b
*91-574. b
Dem.
*91-576. b
Dem.
*91-68. h
*91-581. b
*9169 b
Dem.
: 0 C C‘R . D . /?*“£ = 0 v, R XH> “0 [*91-543 . *22 621 ]
: 0 C C*R . D . R*“0 = 0yj R ik ,“0
. 7?* = 6‘Potid‘7?
h . *91-23 .DK 2* Pot id M2 = f* CM?) u Pot*/?]
[*53-17 .(*91 *05)] - / r C‘R v R lto
[*91-54] = 7» # . D f-. Prop
• R£> C R lto
h.*91-52.Dh .R*,-R+ R\R*\R
[*90*16]
[*90-17]
[*91-52]
G 7?* /?*1 R
GR+\R
G R l>0 . D h . Prop
•.SCR^.TGR^.D.S TQR
' 1*0
: S G 72 |IO . D . S| 22 6 22*,. 221 SGR lM>
. R lto = Rv R lto \R-Rv R\R lKt
.R lto \R~R\R lo
.22 1io ^(/2 po |22)G22
[*34-34. *91-56]
[*91-561-502]
[*90*32. *91*52]
[*91-52]
[*91-57 .*22-9-43]
[*91-571-572]
7?*|P 1K) = 7? 1 . 0 |7?*=P I , 0 =
it
H. *91*52. DK.7?*
.1*
[*90*17]
= -R.I-R
(i)
h. *91-52. Dh.P (IO
| if* = it i /e.
1*.
[*9017]
= «!•«*
(2)
b . (1). (2) . *91*52 .Dh. Prop
R,io=R
Ruo = Rpo\R*=R : \ R* = R*\R 2 = R\R*\R
h . *91-574-52 . D b . R£ = P J P 1K> = 7? IK> J R (1 )
b . (1) . *91-52 .Dh. Prop
P e Potid <R . D . P G P* [*9155 . *4113]
P 6 Pot‘P .D.PG Ppo [*4113 . (*91-05)]
PG-S.D.PpoG^
I-. *90 18 .Dh:Hp.D.fl*G5*.
[*34-34] D . P* | P G P* | P.
[*91-52] O . Ppo G Ppo : D h . Prop
•’*7*1
PROLEGOMENA TO CARDINAL ARITHMETIC
(PART II
*91 6. h : <Je P..C/7 . 3 . C Pot«77.0 w G77 H ,
Dem.
h /' < : .V « l’ol«/7 . 3, . .S' Pr,t«77 : Q, P„t‘77 : 3 . 1‘e Pot‘77
h . *!>r:m. 3 h P..t‘/7. 3 : .yf Pot 1 /?. Pot‘77
h •' I l ■ (• 3 H : /** P»I‘V. Q « Pot‘77 . 3 . /' t P..t‘77 :
I lv\p.*IO I I 21]3 h : y* Put* 77. 3. Pot‘VC Pot‘77.
I* +Il,i 'l 3 -Vh. C«...
H <4).3K Pr..|»
*91601. /f...
Dan.
U)
( 2 )
( 3 )
( 4 >
H.*!»l.-,02.3K/7 1 „G(/f l „ >1 „ (I)
h /*« I*-" * />•,-.: * e /7... .3, . .s 77,., G 77,.,: /7,.„ G /7,„ : 3. 1‘ G /7,., (2)
H . »:1+ :I4 . . 3 h : .s' G 77,„. 3, . S 77,,, C 77,.. (»)
H .< 2 >.(.•»>. *2:142.3 H : 7’t Pot* 77,.,. 3. 1‘ G /7,„ :
[•*'■131] 3 h. (77,.,),., G 77,,. (4.
K(h.(4).Dh. Prop
*91602. I-. (//,..)» = 77*
Deni.
b . *91-54 . D I- . (/<„.)* « / r t?*/*,,, c; (7* li0 ) |iw
[*9 1 *504 601 ] = 7 r C H liv R IM
[*91*54] = ft* . D 1*. Prop
*91*603. H.(7?*U-7^
Dem.
b . *91-52 .Dh. (R m ) %nt - (7f*)* | /?*
[*90*4] = 7?* 17?*
[*9017] = 7f*.DH. Prop
*9162. b xR IMt y. = z R ,t fi Q ft. R*x C jx • . y €/i [*91*52 . *90*36]
This formula should be compared with *901, in which an analogous
formula is given for 7?*. It will bo observed that here we do not require to
add x € C*R, for if R l x = A, the above formula leads to xR xto y . D . y e A, i.e. to
~(x7f,*,//). Hence xR lHJ y . D . 3 ! R*.r, i.e. J*7? 1 KV y. D . xe D‘R. It will be ob¬
served that xR^y holds whenever y belongs to every hereditary class which
contains the immediate successors of a*, whereas xR+y holds whenever y belongs
to every hereditary class to which x itself belongs.
*91-7. b . R^'Q'R = V‘R . R^WR = (l‘R [*91*504 . *37*25]
ON POWERS OF A RELATION
571
SECTION E]
*91-71. : R“n Cp. = . R, v “p. C M > C M
Dem.
h . *90-22*132 .Dh R“ft C M . = . P*‘V C /x . (1 )
C*S>l-602] =.(U'VC M .
[< ] > %] -VVC, (2)
h.(l).(2).Dh. Prop
* 91 * 711 . =
Dem.
I-. *91-71*52 . *37 2 . D h : Hp. D . R C P“/* (1)
h. *91 502. Dh.R“pCR lM> “p (2)
K(1).(2).DK Prop
The above proposition is used in the theory of minimum points in a
series (*205 68).
*91 72. h . R“(a u R^'a) =» P I>0 “a
Dem.
y . *37 22-33.51-. P“(a u R t „“a) = P“a w (R \ P 1K> )“«
[*37-221] = (Pc/P|P |K) )“a
[*91-57] = /e i(0 “a . D y . Prop
*91721. y.R“(a»R V0 “ a )=R l>o “ a [*9172 j*. *91*58 J
*91 73. y :. P, Qc Potid ‘R .P + Q . D: ( a P): P e Pot‘P :Q-P|P.v.P-Q|P
Dem.
y. *91-45. D
h:.Hp.D:( a P):PePotid‘P:Q = P| P. PjP+ P. v.P- Q|P. Q| P *Q (1)
y . *91-27 . *50-62 . D h : Pc Potid‘22 . D. P17|* C‘P - P :
[Transp] D h : P, P c Potid'P . P J P* P . D . P+ If C‘R (2)
K(1).(2).D
h Hp . D : ( a P) : Pc Potid'P . T+If&R : Q = P | P. v . P = Q | T (3)
K . * 91-23 . D h : Pe Potid'P . P+ / f C"P -O .Te Pot ‘P (4)
h . (3) . (4) .5 h. Prop
*91 731. h P, Q c Potid'P .P+Q.D:( a P):Pc Pot‘P :Q=P|P.v.P-P|Q
[*91-73-34]
By means of *91-73 or *91-731, the powers of R can often be arranged in
a series, the rule of arrangement being that P comes earlier than Q if
Q = P | P, and later in the converse case. But we shall only get an open series
from this arrangement if P c Potid'P . Pc Pot‘P .D ?ir .P| P+P; otherwise
the powers from a certain point onwards form a cyclic series.
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*91732 P, Q € Potid 'K.P + Q.O:
(hS>: Sc Potid‘7* :« = .$• R P.v.P = S\jR Q
Dem.
K*!H 731-24. D
h:.Hp. 3:( ? |N.7'>:.S'€l > «tid^.r-.S|/?:Q-2 > P.v.I^T^Q:
[*13P»5] D : (jjS) : Sc P«»ticl‘77 : (J=S R P.v.P=S R Q D h . Prop
•9174 h . 7t“ 4 R m ''rm 4 fi i J+ m R"!?^ =~R t J.r (*01*52. *37 :302]
*9175. K /f*o A>* = /f*u A,..- o if, = H,„vl[C‘Itu I{,„
hem.
h.*50'.V.>l . ^b.Cu\\IfC t R)=/[C t R.
l*!M .341 D 1- . 7t m m I f* t*‘77 u iJ |i0 .
(i)
[*0P54,*2*.7lip»-.
(2)
[*01.74] = R* v 7<„
(3>
KD]
(■*)
H .(21.(3).(4). D H . IVo|»
#92. POWERS OF ONE-MANY AND MANY-ONE RELATIONS
Summary of *92.
If #eCls-> 1 , it follows that, starting from a given term x, there is only
one series of terms x Xt x 2 , x 2 , ... such that
xRx x . x x Rx 2 . x 2 Rx t .
Thus for example the relation of son to father is a Cls—► 1; and starting
from a given man, the series of ancestors in the direct male line (which is the
above series x x ,x 2 , x 3 , ...) is unique and determinate. A result of this property
of many-one relations is that if, starting from a term y, we go backwards a
certain number of steps to a term x, and then forward a greater number of
steps to a term z, we must pass through y in going from x to z\ while if the
number of steps from x to z is less than that from x to y, z must lie on the
road from x to y. These facts are expressed by the proposition:
R e Cls —► 1. D . R# | Q R+ sy R +.
In the present number, we have to establish various propositions of this
kind.
We prove in this number various propositions which are used in the dis¬
cussion of “families'’ in *96 and *97, and some which are used in the theory
of finite and infinite. But on the whole the propositions of this number arc
not much used. The most important of them are the following:
*9211. b : R e 1 —> Cls . D . R„ | It <• R* . R^ | R = R* f* D*R
with a similar proposition (*92-111) for Cls—> 1.
*92 132. b : R e 1 -» Cls . Q, Te Potid ‘R .D.Q\T\QdT
with a similar proposition (*92133) for Cls—► 1.
*92 14. b : <1‘R C D*R . Q c Pot'R . D . D‘Q = D 'R
On this proposition, compare the remarks on *91271 in the introduction
to *91. If R is a serial relation, d*R C D‘R is the condition that the series
may have no last tern*.
*92 31. b : R c 1 —♦ Cls. D . R+ | = R* o R#
*92311. b:.R€Cls-*l.D.J?*|/?* = /?*w.fl*
*921. 1*: R e 1 Cls. D . Potid *R C 1 -> Cls
Dem.
b . *7217 . *71-26 . D b . / f* C*R e 1 —* Cls (1.)
b . *71-25 . D b Hp . D : Se 1 -» Cls . D . £ | R e 1 Cls (2)
b . (1) . (2). *9117 . D b . Prop
57 I
I'ROI.KCOMKXA TO CARDINAL ARITHMETIC
[ PART II
*92101. h: /fet'ls-* I . D . P»ti«l'/f C Cls 1 [Proof as in *921]
*92102. H : /f € 1 —» 1 . D . Potid'tfC 1 1 [Proof as in *921]
*9211. h:/f«l->CU.D.tf Il0 7?-7?*rD‘7f
Hem.
*•.*91-52. D h. R„ R = 7f* 7f £ (1)
h . *71*19 . D I-: Hp .3.7? 7? = /|-l>‘7f (2)
H . (1). (2). *50 G. D h : H p. D . R„ R - 7f * |* 1 VJt (3 )
h.(.l).*:i5 441 . D h . Prop
*92111. h:/f«eiH->l .O.R R^QR+.R /^-((p/01 77*
[Proof as in *92 11 ]
*92112 h:/tc l-»('U.D./f 7f (- . R - R„ f* D‘7f [*9211 . *9I-.V2]
*92113. h:/f€CI»-»l.D./7 /f,* 77 = <U‘7f) 1 77,., [*92111 . *91-52]
*92 12. h : 77 c 1 -* Cls. cP77 C I >‘77 . D . 1?,,. if - 77* [*92-11 . *35GG]
*92121. h: /f<Cls-> 1 . 1>‘77C<1‘77.D.77 77 |l ,,«=77* [*92111 . *3503]
*9213. I": /f € I —► Cl- . if. T € Puti«l*77 . D . 7' \)*Q
Deni.
h .*92 I . Dh: Hp.D.Qc 1 -*Cls.
[*7119] D.Q
1*50-6] D . T Q| Q - 7* r 1VQ s D I-. Prop
*92 131. h : /7 f Cls-* l . Q. Tt Potid‘7? . D.Q Q\T=(i\ t Q)yr
In this iiiiidIkt, when proofs have been given for R * 1 —►Cls. we shall omit
the proofs of corresponding propositions for 77 cCIs-* 1. as these are always
exactly analogous to the proofs for 77 € 1 —*Cls.
*92132. H : /t e 1 -» Cls. Q. Tt Potid‘77. D . Q t T | Q C T [*92 13 . *91 34]
*92-133. h : 7 7 c Cls -*l .Q.Tt Potid ‘77. Z> . Q , T QQT
*92-14. h : (I‘77 C D*77 . Q c Pol‘77 . D . I >‘(7 = D‘77
bem.
h . *91-271 . D h :. Hp . D : U‘Q C D‘77 :
[*37*321] D : D \Q I 77) = D'Q :
[*13182] D : l)*Q = D‘77 . D . D‘(Q R) = D‘77 (1)
H. *1315. Dh. D‘77 = D‘77 (2)
K.(l).(2).*91171 D<S ^ ,VI? -. D 1-. Prop
*92141. h : D‘77 Cil'R.Qe Pot ‘77. D . (I‘Q = C1‘77
575
SECTION E] POWERS OF ONE-MANY AND MANY-ONE RELATIONS
*92142. h : <1*11 C D'/e . Q c Potid ‘R . D . D'Q = D'/e
Rein.
h . *50-5*52 . D h : Q = / f C‘R . D . D *Q = C‘R
h . *33181 . D h : Hp . D . C‘R = D*R
h.(1).(2). D h : Hp. Q «/ T C'/e . D . D'Q = D'/e
H . *91-28 . I) h :. Hp . D : Q = / f* C‘R . v . Q e Pot ‘ft
h. (3). (4). *9214. DK Prop
*92-143. K : D'/e C a*R . Q e Potid'/e . D . d'Q - Cl *R
*92-144. h : Cl'/e C D'/e. Q c Potid'/e . D . d'Q C D'/e . d'Q C D‘Q
Dem.
I-. *91-271 . D h : Hp . Q e Pot‘/e . D . d'Q C D'/e
»-. *50-5-52 . D K : Q = / f C'/e . D . d'Q = C‘R
h . *33181 . Dh:Hp.D. C'/e = D *R
h . (2) . (3) . *23-42 . D h : Hp . Q- / f C*R . D . d'Q C D'/e
H . *91-23 . D I-Hp . D : Q ~ / [* C‘R . v . QePot'R
h . (1).(4). (5) . *92-142 .Dh. Prop
*92145. I- : D'/e C d'/e.Q« Potid'/e . D . D'Q C d'/e . D'Q C d'Q
*92146. h : d'/e C D'/e . 0, 7*6 Potid'/e . D . 7* r D'Q - T
Dem.
H . *92142144 . D h : Hp . D . D'Q = D'/e . O'/CD'/i .
[*13*13] D . Q'f C D'Q .
[*35-66] Z> . 7* p D'Q = 7 1 : D h . Prop
*92-147. h : D'/e C (I‘/e . Q, ft Potid'/e . D . (CI'Q) ] T - T
*92*15. V : /e 6 1 -> CIs . CI'/e C D'/e . Q, Tc Potid'/e . D . T\Q | Q - T
[*9213*146]
*92 151. V : R € CIs -> 1 . D'/e C d'/e . Q, Te Potid'/e . D . Q | QI 7 1 = T
*92-152. I-: # e 1 —* CIs . d'/e C D'/e . Q, 7*6 Potid'/e. D . Q| 7*| Q — 7*
[*9215. *91-34]
*92163. h : /e e CIs —* 1 . D'/e C d'/e . Q, 7 T t Potid'/e . D.Qf7\IQ = 7’
*92*16. hz.Re 1 -> CIs . P, Q € Potid'/e . D :
( 1 )
( 2 )
(3)
(4)
( 1 )
( 2 )
(3)
(4)
(5)
( 3 T) : Te Potid'/e :P\Q=Tf D'Q . v . P | Q - Cnv'CT* f D'P)
Dem.
I- . *91-46 . 3 I- Hp . 3 : (gF): 5", Potid'ft : Q = T| i>. v . P = T\ Q (1)
*■. *9213. 3 i-: Hp. T e Potid ‘R .P=T\Q.O ,P\Q=T\-D‘Q (2)
h • *9213.3 h : Hp . T « Potid'fi . Q = T | P. 3 . Q | P - 7 1 f D'P.
[*34-2] 3 . PI Q - Cn v‘( T [ D'P) (3)
H . (1) . (2) . (3) . 3 I-. Prop
5 7<»
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*92161. I-7? « CIs -► 1 . R. Q e Potid‘7? . D :
•a*’): Tt Poti.l*«:Q /< = ( <|.y,-| T.v.Q P-Cn\'\((l‘P)-\ T\
*92 17. H : /ft I —» C’ls . /’. Q e P.»tid*/f . D . <g T). Te l\>li<l</f . P Q G Tv T
Dem.
441. D!-:/' $= 7* [* I>‘<7 . D . 7' $ <• 7’.
[*2.T:>.S| D.RIQQTsjT (1)
^ . *35 52 44 . D f-: 7' $ - < *nv‘< 7* |* I>‘7*) .3./',^G
[*23*58] D . 7 J J £ G T yj T ( 2)
K.<1>.<2>.*92 lO.Dh. Prop
*92171. h : /?«<'Ih-» I . 7\<?c Pot id* 7?. D . (g 7’). T e Potid‘7?. Q RGTv T
*92 18. K : 7?r I ->Cls.tl‘7?C I >‘7?. Pj)e Pot id‘7?.:>.
R t}< Potid‘/?o PoticP/7
Dan.
h.*02*l<>14(». Z>
Hi. Up. D : (%(7*): Te Potid‘7 ?: R (J-T.v.R D=T:
|*I042) D:t:.|7 f ). Te Potid‘7?. R Q - 7*. v . t^T*). 7’c Potid‘7?. R £■?:
| *!U 521P s < a T). T e Pol i.P R.R Q * T . v . (a 2*). 7’« Pot id‘7?.7' 7*:
[*13*195] D : R ij t Potid‘7?. v . R Qe Pot id‘A*:. D h . Prop
*92 181. h : 7? * < Ls -> 1 .1>‘7? C (P R . 7'. V € Pot id* A*. D .
V 7't Putid‘7?u Potid*/?
*92 19. I*: 7? « 1 -> CIs. (I‘7? C 0*7? . R. (Je Potid‘7? . D :
/' £ c Potid‘7?. v . Q \ R e Potid‘7?
Dan.
h. *92*18. DH:. llp.D:7> Potid‘7?. v . P\Qe Potid‘7? (I)
I" • *01*521 . *84*2 .Z)bzR Potid‘7?. = . (J R € Potid‘7? (2)
h . (I). (2). D h . Prop
*92191. b:R f CIs -> I . I V R C < I‘7? . R. (j < Potid‘7?. D :
R\Qe Potid‘7?. v . Q R t Potid‘7?
*92 3. b :?77 e 1 -► CIs . R. Q e Potid* H .D.R QGR*v 7?*
Dem.
b . *01*58 . DbzTe Potid‘7? . D . 7’u TO. R * v 7?* :
[*23-44] D I-: Te Potid‘7? . R | £ C 7'o r. D . R , $ G 7?* o 7? # :
[ *10-11 *23] Dh:(g7').7'f Potid‘7?. 7> j Q Q T c; T . D . 7^: Q C 7?* u (1)
l-.(l). *92*17. Db. Prop
SECTION E] POWERS OF ONE-MANY AND MANY-ONE RELATIONS
577
*92 301. b : R e Cls -* 1 . P, Q * Potid‘/£ . 3 . P \ Q C R* w R*
*92 31. b : R e 1 -* Cls . 3 . R m | .ft, = R m o if.
Bern.
I-. *90-14 . *50-64 . 3 I-. R* = R^\/fC‘H
b . *9015132 . *33-22.3 I- .IfC‘R C R m .
[*34-34] 3 I- . R* 1 1 [" C‘R G /{,j R+ .
[(1)] 3 b . R t G R m | R 0
Similarly 1-. R m G R^ I
1-. *91 -55 . *90132 . 3 b . R +1 R 0 = i'Potid'R) i‘Potid‘fl
[*41-51] = i‘T K a i», Q).Pt Potid<« . Q ( Potid'P . T= P \ Q\
[*91-521] - i‘r KgP, Q).P.Qc Potid'P .T~P\Q)
b . *92-3.3 I- Hp. 3 : ( 3 P, Q) . P, Q e Potid ‘R . T- P | Q. 3 r . TG R m w j?„ :
[*41-151] 3 : i‘f{( 3 P, Q,. P. Qe 1‘otid‘ft . JT = P\Q\ C 77* w R m (5)
• (4) • (5) .31-: Hp. 3. R +1 R m G R m tu 0+ (6)
b . (2) . (3). (6) .31-. Prop
*92-311. h: PfCls-»l . 3 . if*, «„ = ft. c; i<*
*92 312. 1- : R e 1 —► 1.3 . ft* \ ft* = ft*1 ft* = ft* kj ft* [*92 31 -311]
U)
( 2 )
(3)
(4)
1.3 . (ft* w ft*) I (ft o ft) C ft* w ft*
*92 32. I-: R e 1
Deni.
b. *34-25-26. 3h.(ft*wft*)|(ftwft)=ft*|ft w .R, l |.ft K , R^RvR^R (1)
I-. *9016132. 31-.ft»|flCft*.ft*|ftGft* (2)
K *90-151. 3 I" • R» | ft G ft* | ft* . ft* | R G ft* | ft* (3)
h • (3) • *92-312.3 1-: Hp. 3 . ft*; ft G ft* w ft* . ft*1 R C ft* a ft* (4)
. (1). (2) . (4) .31-. Prop
*9233. H:ie e l-»l.D.(/{o«V-fl )(( w«„
Dem.
H.*9018.
D h . R+ G (R c/ R) 0 .R m G(R c; R ) m .
[*23-59]
D h . -ft* c; R m G(Rw j?) #
a)
1- - *33-272.
D h . / f* C\R c; .ft) = / f C*R .
[*9015.*23-68]
D K . / f- C*(R c; ft) G R m sy R 0
(2)
K. *92-32. *34-34.
. D H : . Hp . D : S G R+ u R+. D . S | (R c; R) Q R m c; R #
(3)
R Ac W I
37
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
57S
h . (2). (3). 7 *” ,S ' c **"«* . ;>
h :• Hp. 3 : /'« H..ti<l*(/f
(•4-I-1.5IJ D : *‘Puti<l‘( /fo«)G ff # o :
t* !,| ' 55 J 3 ff.ojL (4)
P. (1 ).(■*)• 3 KPi.>|>
<92 34. . 2) . < R c /?),.. - R+ o /?„
Dem.
K*!I2'3;}.«!»1 *.>2.D
1 -sHp.D.
(7?o/7>
[•34-25-2CJ -77* 77^77* JlvR 0 ‘it w K+\ K
[*np52 54 57] = 70 77c/(/|‘C‘/7u77u77*|77) /7 vy 77
[*50 «5.*71-l!)2.*72 r>f»-5f»l ] 1,0
= 77,... o /? ci /fn-/r o X* /?o/ M VJiv 77* l*D‘77c/77
[*3.V412.»9l-502]
- 7?,„ wlfC'RvjitfiVRw 77* f ly/i u /J
[*° 1 7 r, l - 77* v 77* V 77* r < I‘77 V 74 M>‘77
[*35*441 ] = 77* o 77* : D h . Prop
*93. INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION
Summary of *93.
For this number, we introduce three new notations, of which the first two
will be used constantly, especially in the theory of series, while the third will
be seldom used except in the present section. The two which are constantly
used are
xBP, meaning x e D‘P — G‘P
ai *d x min/. a, meaning xear\ C‘P — P“a,
i.e. x is a member of a and of C‘P, and no member of a precedes x in C l P.
The letter B may be regarded as standing for "begins.” Thus if we take
any member y of C‘P, and proceed backwards and forwards as far as possible
by P-steps, we obtain a scries which may be called the “family” of y: this
series, if it has a first term, has one which is a member of D‘P — (I‘P; thus
the members of D*P — G‘P are the beginners of families. For example, if P
is the relation of a peer to his heir, " xBP " will mean " x is a peer who is not
the heir of a peer”; thus x is the first of his family. If P is the relation of
parent and child, “ xBP ” will be satisfied only by Adam and Eve; and so for
other relations.
The definition of B is
B^SPixeD'P-a-P) Df.
Hence P‘P~ D‘P — G‘P. If P is the generating relation of a series which
has a first term, that first term is B*P\ if there is a last term it is B*P.
If a is any class, we may call a term x a minimum of a with respect to P
if it is a member of a and of C‘P, but does not follow any member of a, i.e. is
not a member of P“a. We denote this relation of x to a by “ min/."; thus
we have
x min/, a . = . x c a n C*P — P“a,
and the definition of min/» is
min Df.
We shall also, when convenient, write "min ( P )” in place of " min/.."
We have minp'a = a r\ C*P — P“a.
If P is serial, minp'a reduces to a single term if it is not null; thus if a
class a has a first term, this term is min/»'«- We also put
*-*
max/. = min (P) Df,
and then maxp'a, if it exists, is the last term of a in the P-series. Thus if a
37—2
I'ROLECOMENA TO CARDINAL ARITHMETIC
I'ROLECOMENA TO CARDINAL ARITHMETIC (l»\RT II
is the class »f peers, and P is the relation of father to son. uThi,.‘a consists of
tho,o |>«cTs who are the first of their line, while n^x,.‘a consists of those
peers who are the last of their line. If a is a class of numbers, and P is
the relation of less to greater, min/a is the smallest member of a (if it exists),
ami max,.‘a is the largest (if it exists).
H and “ max,. ‘ and min,." will be used constantly in connection with
senes, where the two latter will be considered in detail, but the present number
is more specially concerned with a less general idea, namely that of genera-
fions. Take. e .</, the relation of parent and child; let us call it P. Then
the^first genenttion consists of those who are parents but not children.
M '- ,,K ‘ st ‘ co,,<1 consists «f those who are children but not grandchildren.
''' l\‘p-tl‘i» i.e. <I‘P-P“<J‘P. U. ‘P: the third consists of those
who aiv grandchildren but nut great-grandchildren, i.e. <I‘P-_(I‘P\ i.e.
■" P ii ^l t P J , t.e. iniii ,.‘(\*P*\ and so on. Also we have
IPP = iniu,.'<J*(/ f C*P);
hence the generations of P are miii / .“(| << | > otid < / > . Thus we put
gcn‘P = min,."(l“Potid < P Dr.
when* “gen" stands for "generation."
When P is a one-manyj-olation, such as that ..f father and son. every
generat ion is of the form P'li'P, where T is a power of P (including /[* C 1 P).
When P is not a one-many relation, this is not in general the case.
The generations of P do not in general exhaust the field of P. For a- will
only belong to a generation of P if .r can be reached by successive P-steps
starting from a member of li'P. If some of the families constituting the
field n! P have no beginning, the members of these families will not belong
to any generation of P. Such terms together constitute the class
/>‘U“Pot‘P.
or />‘(I“Potid‘7 > ,
which is the same class.
Thus the field of P may be divided into two mutually exclusive portion*
s'gci i‘P and p‘U“Pot‘P.
The present number begins with some elementary properties of B and
min,, and max*. We then (*93*2—275) consider such properties of genera¬
tions as do not demand any hypothesis as to P. We prove
*93 25. P . gen‘P e Cls ? exc!
*93*261. P ./>‘CI“Pot‘P = p‘d“Potid‘P.p‘(I“Pot‘PCG'P
SECTION E] INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION 581
And we prove (*03474475) that s‘gen‘Pand p‘<I“Pot</> are mutually exclu-
sive and together constitute C‘P. We then proceed to a set of propositions
Hr ~' 41) d0mand ' n » that P should bo one-many or many-one or one-one.
We prove
*93-32. h p € l Cls. D : a e gen'P . = .(^T).Te Potid'P . « = T"~B'P
*93 36. h : P € 1 —> Cls . D . s'gcn'P = P^"~B'P
*93 381. h P € Cls — > 1 . D : x c p'G"l ) ot'l i . = . *P m 'x CD 'P.xeC'P
and various other properties of gen‘P and //CI“Pot‘P when Pe 1 -> Cls.
The propositions of this number are used throughout the rest of this
section; they are also used in the cardinal theory of finite and infinite. The
early propositions, down to *9312 inclusive, are also used in the theory of
series.
*93 01. B = &P(xeD'P-G'P) Df
*93 02. miny, = min (P) = (* f a n C'P - P"a) Df
*93 021. maxy> = max (P) = min (P) Df
*93 03. gen 'P - in?n i »“a“Potid‘/ > Df
*931. f- : X BP . s . a: e D'P - G'P [*21*3 . (*93 01)]
*93101. \-.~B'P=D'P-G'P [*931 .*32 18]
*93102. V:x= B'P . s . a: = 7‘(D‘/> - G ( P ). =. D'P - C VP cl.xe D‘P - CL‘P
[*93101 .*53-4]
*93 103. 1- 7b*P = C'P - G'P
Bern.
h . *22-9 . *3316 .DK C'P - G'P = D'P - G'P (1)
H.(l). *93101 . Dh. Prop
*93 104. h : xBR . D . ll+'x = i'x . R^'a: = A
Bern.
h. *931 .
DhHp.D.x £C‘R.
[*9012]
D.xc~R#'x
(i)
K *91-504
[Transp.*931] D h : a:RR . D . Rpo‘* = A
(2)
h. *91-542.
D h : y R*a:. y + a;. D . yRpo* :
[*3218]
D 1-yR*a^ .D:y = x.y.yc Rpo'a:
(3)
h.(2).(3).
D h Hp . D :yR*ar .D.y=x
(4)
h.(l).(4).
DhsHp .D.R#'x=i'x
(5)
K(2).<5).
D h. Prop
■
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*9311.
*93111.
*93112.
I >em
*93113.
*93114.
*93115
: .#• niin/. a. = . a-e a CP - P"a ((*<13 02)]
H . inui/ ‘a = a n C'P - l Ht a [*93*11 . *32*18]
H . 7P7' = nun/.<6* < / >
f-. *93*111.3 h . iiim # .<I)</* = D‘7> - P“ 1 >‘7^
[*37 25] = D^-CIV'
(*93101] = 2?»7>
Similarly h . miiVC^ =7*7'
K(l).<2). Dh . Pn.p
—►
h . mm,.‘a Cart 6“ 7* [*93111]
w
I-. max*/. = min ( P) [(*93 021)]
h :max/.a .c.xcan C'P — P'*a [*93*11 114]
*93 116 h . mux/.‘a -an C'P - 7'“a [*93*115 . *32*18]
—► « —► —»
I-. 77*7'™ nmx / .‘(l<7 > — maX|.*C'<7** (*93*112 114]
H . max/.‘a Can C‘7* [*93*116]
h . 7? 7* = CI ‘P - 1 >'P = C'P - 1 >‘ 7' [*93* 101*103 . *33*2*21 22]
V . miii|»*( !<(/[• CP) - 7P7> [*50*5*52 . *93*112]
K i.T?n,/< I‘7'= <I‘7'- <I‘7>-
I-. *93*111 . D b . lnin / . , a < 7 , = <1*7* - P“d‘P
[*37*36] = (I<7^ - (i<7». D I-. Prop
i-. min/zcpr = c'p n d'T — a*( t \ p)
h . *93 1 11 . D K imn r ‘<l‘r= C'P n Q'T- P"(S'T
[*37*32] = C'P n a'T-a'(T\P) . D h . Prop
H : a € gen‘P . = . (g7) . 7* Poti(P7 > . a = mm,,‘CP 7
[*37*67 . (*<13*03)]
1- : a € gon<P. s . (g7). 7 c Potid 'P. a = Cl<7- (I‘(7| 7 J )
[*93*2*132. *91*27]-
1-. 77‘7 J c gen<7 > [*93*2*13 . *91*35]
!■. GPP — (l'P : e gcn'P [*93*2*131 .*91*351*23]
*932.
*9321.
*9322.
*93221.
*93117.
*93118. I*. mnx/.‘a Can C‘7 >
*9312.
*9313.
*93131.
Dem.
*93*132.
Dem.
SECTION E] INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION
583
*93-23. b . gen'P = u mint'd" Pot'P
Bern.
K *91*23 . *37 22 . D
b . gen'P = miDj»“a«c<(/ f C'P) w i^f n/ .“(I“Pot *P
[*53-31] = e'mtnya^/ f* C‘P) v miiij»“a M Pot‘P
[*9313] = i<B‘P sj rnni*“(I“Pot‘P
*93-231. b S,Te Potid'P . £+ 2\ D : a‘SC/ ,,, a < 7'.v .
Bern.
b. *91*732.3
I-:. Hp. D : (giV) : S = ilf | .P | 2" . v . 2 1 = J/ j P | S:
[•91 *8] 3:(ail/):S=il/|r|P.v.r=iViS|/ >
I-. *34-36 . 3 I-: S- if j 2*jP. D . d‘S C d‘(T \ P) .
[*37-32] 0 .(l‘SCP"a‘T
Similarly h : r=jl/|S|P.:>. d'TCP-d'S
!■. (1) . (2) . (3) .31-. Prop
( 1 )
( 2 )
(3)
*93 24. 1-: S, T « Potid'P . S * T. D . imn r ‘Q‘S r> iri^'CP ‘ T - A
Bern.
I-. *24-3 . 3 1- : d‘S C P"Q‘T. 3 . d'S-P-Q-T- A .
[*24-34] 3 . Q‘S r\ (l‘T — P‘‘Q‘T — A .
[*24-34] 3 . (d‘S- P“Cl‘S) « (d ‘T - P“d‘T) = A .
[*93111] 3. imn/a , Snmii) f , a , 7’=A (1)
*■ • w • 3 *■ ■■ a'TCP“a‘S . 3 . min/a'Sn n^n,<a*r= A (2)
I-. (1). (2). *93-231.31-. Prop
*93-26. I- . gen ‘P t CIs’ excl
Bern.
b . *30 37 . Transp . D
b S, Te Pofcid'P . a = rainp‘Q‘5. £ = min^G'P .a + /3.D:
S,Te Potid'P .S^=Ti
[*93’24] D:an/9 = A (1)
H - (1). *11*11*35-54. D
b (a S) . S e Potid'P. a = r^np'd'S : (a T) . Te Potid'P. 0 = :
a^:D.an5 = A (2)
^ - (2) . *93'2 .Dha^f gen‘P ,a + ^.D.an^= A (3)
• (3) • *84*1 .Dh. Prop
PROLEGOMENA TO CARDINAL ARITHMETIC
:>s|
[part II
*93 26. h : N. T t Potid‘7*. TV \ S“VoVP. D . min/d*# a iuin / /Cl < 7 , *= A
hem
I-. *01*24 . D I-: Hp. D . 7 V |.s'“ 7 J “Potid‘/\
[*4311 l.*37'G7] D . (gJ7). T= M\ P S.
D.fg At). T - M s p.
(*34 30.*37*32] D . (VTC P'WS .
[*24-3] 0 P“(1*S= A .
[*24 34] D. - P^iVS) r* (Q‘7*-/*“<!*7*) - A .
[*031 1 1 .*01-27] D . A : D f* . Prop
*93 261. H . //< |“ Poi*/'» p , Cl“Poiid‘7 J . //<l“Pot‘7 > C (I 4 / 1
Dem.
*■ • *01*23 . D h .<I“Potici‘7 > -([“Pot 4 /' w i‘( V(I f C‘P)
[*50-5*32] -CI“Pot«/> v PC U P (1)
H . (I) • *33*14 . DH • />‘(I“|*otwl‘/'-/j'(I“Pot‘7 > A C‘P (2)
h . *40*12 . *01*351 . D K . |»‘C| ,i P«t‘7' C <l 4 7' (,*{)
H.(2).(3).*22621 .DH. Prop
*93 27. H :./f f’*7'. D : .**~ t j'gcn^. = . .#■«/j < (l* , Pot‘7*
hern.
K*40*ll .*10*51 . D
f- .#•**#€ *‘g»*ii‘7*. 3 : a € gen‘7** .D..,i>v(a:
[*03-21] s : 7V Polid‘7' . D r . * <I‘7’- <|‘( T P ):
[*4*53.*.V0J = : Te Potid‘7\ x e < l‘7\ D r . jfd^ p) (i)
H . *50-5*52 . Dh z.reC H P. D.xc <!*(/[* C*7 J ) (2)
h • (I). (2) • D h :: xeC*P . D •r«w«*gcn*/ > . = :
.r«Cl‘(/r C*P ): 7V Potid‘7*..r«(1*7*. D r .4t <P(2’| 7"):
[ *91 -371) = : TV Potid‘7 > . D r . x e <I* T:
[*40 41] = :4-€7> < <l“Potid < 7':
[*93 261]= : .r c//(J“Pot *P ::Db. Prop
*93 271. J-. C H P - 4‘geii‘P = //CP'Pot'P
Dem.
I- . *5-32. *03-27 . D H : .r € C‘P - s'gcn'P . = .xc C‘P . x e pW'Fot'P .
[*93-261.*4-71] = .xepWPotfP : D h . Prop
*93 272. h . s‘gcn*P C C*P
Dem.
1- . *93-2113 . D H : a c gen‘P . D . (g T).Te Potid l P . a C d'T .
[*9127] D.aCC'P (1)
I-. (1). *40151 . D H . Prop
: T“D‘P-T“J*“D*P
T“Y)*P-P**r«T)‘P
\r\\np < f ii D i P
SECTION E] INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION 58
*93 273. I-. C‘P-/)‘(I“P 0 t‘P = s‘gen‘P [*93 271*272 . *24 492 ]
*93 274. h . C*P = s'gen'P v/> < Q“Pot < P [*24 411 . *93-271*272]
*93-275. h . s'gen'P n p‘<I“Pot‘P = A [*93 271 . *24 21]
*93-3. H : P e 1 —> Cls . T c Potid‘P . D . mfn i »‘a‘2 , = T“13‘p
Dem.
H .*71-38 . *93101 . D h : Hp. D . T“li‘P= T“D‘P- r“(I‘P
[*37-25]
[*37-33.*91-3]
[*93*111.*91*27] - mi'n / »‘7 , “D‘P ( 1 )
H .*91-271 .*37-271 . D f-: 7*« Pot'P. D . P“D‘P = Cl‘T ( 2 )
H . *50-5-51 *59 . D h : T= If C‘P . D . T“D*P = D‘P.
[*93112] D . imn/.T'D'P =Z?‘p
[*9313] * nTfny/CPP (3)
h • (2). (3). *91-23 . Dh:Te PoticPP. D . Jn/^'D'P -■= nmiPCPT 1 (4)
^•(l).(4).DH.Prop
*9331. hP«l-» Cls . D . P“nrni/.‘a‘r = m?n/a*( Tj P)
Dem.
K *71 38 . *93 111 . *37*265 . Z>
h : Hp . D . P << nim/> < CI‘P = P“(1*T-P“P“Q.‘T
[*37*32] = d‘(T\P) - P“C1‘(T\ P)
[*93111 .*34-36] = min/. < a < (7 l J P) : D h . Prop
H P e 1 Cls . D : a « gen -P. = . (3 T) . T c Potid ‘P. a = ?'“7pp
098-2-3]
*9332.
*93-33. h:Pfl —>Cls.ae gen‘P . 0 . P“a t gen‘/ J
[*93-2-31 .*91-28 281]
*93 34. (-:P < l-.Cls.3.P“5 , ‘Pegen‘/ > [*93-22-33]
*93 36. b : P e 1 Cls. a t gen'P. T . Potid'P .O.T“a e gen ‘P
Dem.
V • *91-341 . *37*33 - *34 2 . D
*-:S,T e Potid ‘P. a = $*‘S‘P . Z> . e Potid‘P . P“a = {Cnv‘(S| 7*)] “if'P (1)
*" • (1) - *93-32 . D \-: Hp (1) . P e 1 -► Cls. 3 . 7*“a e gen'P (2)
h • (2). *10-11-23-35 . *93-32 .31-. Prop
aW PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
*93 36. h : P € I -> CIs. D. s‘ge n‘P = P*“1i'P
I)em.
h . *93-32 . D h :: Hp.D:.
'/ « ^geii‘7*. = : (3 T ). Te PoticPP. y e T“H*P :
[*37105] = : (3 7». Pe Potid'P . x c 7?P. xfy :
[*11*55] = U\[.t)ix( li'P u^T). PePotid^.xPy:
[*4111) = : (gx). x C Z?P . x(*‘Potid‘P) y :
[*1)1*55] = : <gx)./f 7?P . xP+y :
[*37*105] s :;/ c P^ f 7?P ::Dh. Prop
*9337. >-: /> ff 1 -» Cl*. D . C‘P = P*“7?7' u />‘(I“Pot‘/> [*93*274*3(5]
*93 38. h Pc 1 -> CIs . D : x <;>‘<I“Pot‘P . = .!*+** C (VP . .r e C‘7 J
Item.
h . *93*271*3(5. D
H :: Hp. D i. x * p‘(V t ¥ol i P . 2 :x« C‘P .x^e 7y*Z?P:
(*37 • 103.* 10*51 ] sue C‘7': yP*x • • //~ e 7?P :
[*!)3 101.*22*848) = s xeCPPiyP^x. D v . y e (4*7' u - WP :
| *90*13.*33*16] = ix<L H P : yP+x . D„ .
y t ((VP v - IVP) n ((VP u l)‘P):
[*22*69.*24*21] = : * « C‘P: yP*x • D y . y « <I‘P:: Z> H . Prop
*93 381. h 7 > «Cls—► 1 . D : x€p , d tt Pot t J J . = . P*‘x C 1>‘P . x € C l P
*93 382. I *P c 1 1 . D : x e 7 >‘CI“Pot‘P n ;,‘<I“Pot‘7~. = .
P^xsj 7V-rC D‘PnG‘P.xcC‘7^ [*93*38*381*261 . *90-31*311]
*93 4. I-: Pc 1 -* CIs. CPP C D‘P . 3 ! 7?P. Te Potid^ . D . 3 ! liun/a'P
Vein.
I-. *93*13 . D h : Hp . Z> . 3 ! ndli/d'a f* C‘P) (1)
h . *93*113. *33*181 . D h :. Hp . D : min/U'PC D‘P :
[*37*431] 3 : 3 ! D . g ! P‘W r ‘(I‘2\
[*93*31] D.aIm?n / . < a ^ (7 , |7 , ) ( 2 )
h. (1). (2). *91*17. DK Prop
*93 41. h : P € 1 -> CIs. d‘P C D‘P . 3 ! JVP . D . gen‘P e CIs ex 3 excl
[*93*2*4*25 . *84*13 . *24 63]
SECTION E] INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION
587
*93 412. h . P‘‘p‘d“Pot‘P C p‘d“Pot‘P
Deni.
I-. *93-261.31-. P“p‘d“Pot‘P = P“p‘(I“Potid‘P
[*40 37] Cp‘P‘“d“Potid‘P
[*43 411] Cp‘d“|P‘‘Potid‘P
[* 9 l-24] Cp‘d“Pot‘P .31-. Prop
*93 42. h : P « 1 _► CIs . 3 . P“p‘d“Pot‘P =p‘d“Pot‘ P
Dem.
I-. *93-261.3h .P"p‘d“Pot‘P = P“p‘d“Potid‘P ( 1 )
- (1) . *72-34 . *91-35 . *10 24.3
I-: Hp . 3 . P“p‘a“Pot‘/ > = p , P“ , a“Potid‘7 >
[*43-411] = p‘d‘‘|P“Potid‘P
[*91-24] -p‘d“Pot‘P: 3 \-. Prop
*93 431. I- .p‘d“Pot‘P = p‘d“|P"Pot‘P
Dem.
h . *91-264-304 . 3 I-. Pot‘P -i‘Pu| P“Pot‘P .
[*53-14] 3 I- ,p‘d“Pot‘P = Q‘P n p‘d“| P“Pot‘P .
[*91-271-283.*40 151-23] 3 I-. p‘d“PofP = p‘d“ | P“I>ot‘P . 3 . Prop
The following propositions, not being needed in subsequent propositions,
are here inserted without proof, merely for the sake of their intrinsic interest.
*93 6 . 1 -:Tt Potid'P. 3 .~PJT-~P«‘T= T\ “Potid'P = | 7'“Potid‘P
*93-61. 1-: T « Pot‘P. 3 . Pot' TC~P„‘TC Pot‘P
*93 62. I-: Te Pot‘P. 3 .p‘d“Pot‘T=p‘d“'p u , r~p‘d“ Pot‘P
*93 63. h:S.Te Pot‘P .xSx. 3 . (ay) .y(S\T)x
*93 64. I -:Se Pot ‘P .xSx. 3 . x e p‘d“Pot‘P
*93 66 . 1-. C‘(P„, A /) C p‘d“Pot‘P
*93 66 . I-: a ! (P,*, A I) . 3 . a ! p‘d“Pot‘P
*94 OX POWERS OF RELATIVE PRODUCTS
Sit in out n/ of *94.
In this number wo shall Ik> chit Hvconcerned with propositions connecting
p..\v.Ts of R S with powers of X It. If p is a power of R S, S\P<R will be
a power of X It If r is a power of R S, it is a product of the form
lR X) (li S) ...Ml X).
If we transfer the initial R to the end. we get a power of S It Thus the
is a power of X R. say T. such that
/' R = R T.
•re
If /{«!-» CIs .<!•(/? X) C I)‘ R. we find
R (X /OKX R) ... (X It) R = ( R X,j (R S)...(R [ S)
by rearranging and observing that R | It m / f D*R. Thus
R ( ' -* c,s • <i ‘<it X)c wit.Pt pot• it\s .D.( a V).r* \ws r.p=r\t\ It
Expressions of the form R T R are constantly needed. They will be
specially dealt with in *150. and will occur constantly in the sequel.
The above connections of Pot‘< R j S) and Pot‘(X| R) are embodied in the
following projjosilions:
*9414. b. /?“Pot‘(X X)-/f “Pot‘(X|/0
*94 21. V . Pot*(X /0-(X||/?)“|Pot‘(7?|5)ui‘/|
*94 31. b : R € I -> CIs. ( V(R X) C D‘X.D. Pot‘(X X) - (R\\ R)“?ot‘(S\R)
From *94 4 to *94-54. the propositions are all concerned with m‘CI“(/?|X)
am I pH I“(Xj R ). We prove
*945. »-.//< I “Pot‘(X R) = r ‘il“R “Pot‘(X|/e)
*94 51. b : Ii€ I —►CIs. D . ;>‘(I“Pot , (5!/?) = X«p*a < ‘Pot < (/J|X)
Finally we prove (*94-5354) that if either R is one-one and Cl ‘(R |X) C D‘R,
or X is one-one and <I‘(X /OCD‘5, then 7 >‘CI‘‘Pot‘(tf|X) is similar to
y ,‘U“Pot‘(XjX).
The only proposition of this number which is ever subsequently referred
to is the last, *94 64. which, owing to the fact that the Schroder-Bernstein
theorem has been already proved (*73-88), is only used in *95-23. But *1)5-23
itself is never referred to again. The reader may therefore omit the reading
of the propositions of this number (as also of *1)5) without detriment to the
understanding of what follows; he should, however, read the summaries.
SECTION E j
ON POWERS OF RELATIVE PRODUCTS
589
The chief importance of the propositions in the present number is when
R and 5 fulfil the hypothesis of the Schroder-Bernstein theorem, i.e.
R, S e 1 -> 1 . d‘R C D‘S . a ‘S C D*R.
In this case, 72 J jS gives what we may call a “reflexion” of into part
of itself; this part may be again reflected by R J 5 into a part of itself, and so
on. The terms in D‘R which are eliminated sooner or later by this process of
reflexion constitute s t gen t (R | S), since any one reflexion eliminates terms which
constitute one generation of R S. The terms not eliminated by any number
of reflexions constitute p'G^Pot^/S | S). These two sets of terms together
constitute D'(/i | S), i.e. D‘R. In this number and *95 we shall prove that,
with the Schroder-Bernstein hypothesis,
9 ‘gen‘(A | S) sm «‘gen‘(&| R) . p‘(J“Pot*(/21 S) 8tnp t d ii Pot i (S\ R).
These two propositions together yield a proof of the Schroder-Bernstein
theorem, in virtue of *93274-'275. This proof is essentially the same as
Bernstein’s published originally by Borel*.
The nature of the two proofs of the Schroder-Bernstein theorem, namely
Zcrmelo’s (that given in *73) and Bernstein’s (that t jo be given in this number
and *95) will be best apprehended by means of figures.
In Zermclo’s proof, we first prove that if 72 is one-one, and 0 is a class
contained in D‘R and containing Cl *R, then 0 is similar both to D‘R and to
d*R. In the figure, the points of the outer rectangle form D‘R, those of the
inner rectangle form CPJ2, and those of the outer oval form 0. Thus the shaded
portion of the figure is 0 — d‘R. We now define a class of classes k by the
following characteristics: a is a member of k if (1) a is contained in D‘R,
V/
(2) a contains the whole of the shaded area, (3) R“a C a, t.e. if x is a member
of a, so is any term to which x has the relation R. Our proposition is obtained
by considering p**, i.e. the area common to all the members of k. We prove
• Legont tur la tMorie ddet functions (Paris, 1898), Not* I (pp. 102—7).
V
PROLEGOMENA TO CARDINAL ARITHMETIC
[part II
590
(*7381) that p t K€K. and (*73*811) that /?“/>** docs not contain any of the
shaded area. In the figure, R“p *k is the smaller oval. We then prove (#73 83)
that p t ie consists entirely of the shaded portion and the smaller oval. Hence
/i (the larger oval) consists of two mutually exclusive parts, namely p*K and
<1 , It — R“p*K, the latter being that part of the inner rectangle which lies
outside the inner oval. Assuming now that R is one-one, p*/c is similar to
v v/
R“p'«\ hence, adding (\*R - R'^/k, it follows that is similar to and
therefore to I)‘/(.
In order to obtain hence tin* Sellr.kler-Bernstein theorem, it is only
necessary to replace R by R S and by CI‘5. and to assume further that
S is a one-one whose domain contains (I 1 /?. Then D*R = S), and wc
obtain (*73*87) U'.Vmii WR. and therefore D'tfsin D*/?, which was to be
proved.
In Bernstein's proof, we have the two relations R and .S' from the beginning.
In the left-hand part of the figure, the outer rectangle is D*R, which- D*(/i|5),
the oval is (1*5, and the second rectangle is Q‘(R j S). Thus the points of the
outer but not the second rectangle form the first generation of RfS. Within
(1*(/?|5) wc can form a third rectangle, which will be | S),
i.e. (I*(/f | »S')\ The points belonging to the second rectangle but not to the
third form the second generation of R\S. We can proceed in this way to
continually smaller rectangles. The points which sooner or later are left outside
some rectangle form s‘gen‘( R \ S ); those which arc common to all the rectangles
form />‘G“Pot*(7?| S). A similar analysis, exhibited in the right-hand part of
the figure, may be applied to D‘S. which is thus divided into 6-‘gen‘(S , | R)
and p , Q it Pot t {S\R}. We prove in this number (*1)453) that, with a
hypothesis which is part of the hypothesis of the Schroder-Bernstein
theorem, />‘d“Pot‘(tf| S)sm;>‘d“Pot‘(S| R)\ in the next number (*9571)
wc prove that with the hyjiothesis of the Schroder-Bernstein theorem,
s‘ S cn‘(R | <S)sm s‘gen‘(S| R). Hence by addition. D'Rsm D‘S.
SECTION E]
ON POWERS OF RELATIVE PRODUCTS
591
*9412. h : P e Pot‘(P | S) . 3 . ( a P) . Te Pot‘(S| P) . P | P = R | T
Dem.
h.*34-21. 3K(PjS)|P = P|(S|P) ( 1)
t-. *91-36 . *34-27 .3h:Te Pot‘(S I R) . P | R = R | T . 3 .
r|S|« e Pot‘(S|-R)-P|i£|S'|Ji=ii|r|s,ie.
[*10-24] 3.( a r).7’',rPot‘( 1 S|P).P|P|,S|P = .R|Z'' (2)
h • (2) • *1011-23 .31-: ( a P) . Te Pot‘(S | R). P \ R = R\T.>.
(^T-).T' ePot\S\R).P\R\S\R~R\T (3)
H. (1). (3). *91-171.3 K Prop
*9413. I- : Te Pot ‘(S | P>. 3 . ( a P) . P e Pot ‘(R \S).P\R=R lT
[Proof as in *94-12]
*94 14. h . | P‘‘Pot‘(P | S) - R | “Pot‘(S | R)
Dem.
V . *9412 . *43111 1 . *37 1 . 3 I- : Pe Pof(P | S) . 3 . | R‘Pe R |“Pot‘(S| R) ■
[*37-61] 3 h . | P“Pot‘(P | S)C P|“Pot‘(S| P) (1)
I-.*9413. *4311 101 .*371.3
\--.Te Pot ‘(S | P) . 3 . R\‘Te\ P“Pot‘(P \ S) :
[*37-61] 3h.P|“Pot‘(S| fi)CIP“Pot‘(«|S) (2 )
*" • (1). (2) .3K Prop
*94-2. h : P e Pot‘(P | S) v t‘/. 3 . £>’| P | P e Pot‘(S | R)
Dem.
. *34-21 . 3 h . S | (P | g) | P = (S | Rf.
[*91-352] 3KS|(P|S)|P e Pot‘<.S|P)
h -*34-21 .*91-282.3
l-:S|P|P e Pot‘(S|P).D.S|(P|P|S)|/j = (S|P|P)|5|P.
(S|P|P)|S|PePot'(S|P)
K(l). (2). *91-171 l-fi e Pot , (-S'|g) D
<pb
'-■■PePot‘(R\S).O.S\P\RePot‘{S\R)
h • *504 . *91-351 . 3 I-. s 1 1 1 P e Pot‘(S | P)
•-•(3). (4). DP. Prop
( 1 )
( 2 )
(3)
(4)
*94-201. I-: Te Pot*(S| P) . 3 . ( a P) .Pe Pof(P | S) w PI. T = S\P | P
Dem.
h • * 50 '4 - *51-16 . 3h.P|P = S|/|P. Ie Pot‘(P | S) w PI.
[*1024] 3h .( a P).P«Pot‘(P|S)wt‘/ ,S\R = S\P\R ( 1 )
H.*91-282.*34-21.3(-:PePot‘(P|P).r = S|P|P.3.
r P|P|S«Pot‘(P|S).r|S|P = P|(P| P|S)|«.
L * 10 ' 24 ] 3-(aC)-0«Pot‘(P|S).r|p|p = s|(?|p (2 )
PUOLKCOMEXA TO CARDINAL ARITHMETIC
[PART II
K *304. *3421 . D
b : P = 7 . I = S P | /.'. D . T | S ! R = N | (It S) \ R .
[*0I35I] :>• (a<?>• Q € P-*‘<#15). 2*151 7?«.S' \Q\R (3)
V . (2) .13). *10 11 23 . D h : <aP>. P < Pot *(R \ S) o PI . 7* = £» j P | /?. D .
WJ). Q< Pol ‘(R | .S'). T\S\R = S\Q\R.
[*22-58] D . (aV). <?c Pot‘< P | .S') u ,‘7.7*1 .S | 7? = S\Q I R (4)
h. (I). (4). *01171 ' * n ' 1 1 . D h. Prop
l . <pi
*9421. b. Pot‘(S| /0 = (.S'H rt>“;Pot‘</?|.S->ui‘/;
l)em.
H. *04 2. *43 112. *37 61 . D h . (N fl)";Pot‘(7e | tf) u i‘/| C Pot‘(S | R) (1)
K *04 201 . *43*102 . *37 1 . D I-. Pot‘(.V| R)C{S 7f)“|Pot‘(77 | S) v <‘/| (*2)
H . (I). (2). D H . Prop
*94 22 h :. (I'/t C D*5. v . D‘S C (t‘R : D .
Pot‘(5| ff)-{S i 70“P»tirl'<ft|£)
Dan.
H . *1)4-21 . *43" 112. *50 4. *5331 . D
I-. Pot‘(»s*| R)m{S\ 70“P»t‘( R\8)v # < (,s'| /o ( 1 )
H . *37-321 . D b : l\‘R C l)‘£’. D . l)‘7f - l)-(/71S).
[*33101] D.D'R CC‘{R\S).
[*50-63] D. 7 |* C‘< R | 6') | 77 = 77.
[*34-23] D . .S' | 7 r C U (R | 8)\R-S\R.
[♦43112] D.<3|| 70‘/ r^(^|5)-5|7? (2)
Similarly H : DSSC (W7 .0.(8 R) 1 1 |* C\R\S) = S\R (3)
K.(!).(2).(3).D
b : Hp. D . Vol\S\ R) = <S j! 77>“Pot‘(77 | S)«P(S|| 7?)*/ |* C‘(77 1 8)
[*01*23] =(S|| 77)“Potid‘(/21 8) : 0 I-. Prop
*94 3. b R e 1 -> CIs. (T‘(77 | 8) C D‘77. D :
P € Pot‘(7? I S ). = . (a 7'). 7' € Pot‘(S1 70. P = R I 2' | 77
Dcm.
b .*04*12. Dh:7 , £pot‘(72|S).D.(a7 T ).7 , £Pot-(5|7?).P|7r|^=7?|7’|^ (D
I-. *01-271 . D h Hp. D : P £ ?ot<(R | S ). D. d‘7> C D‘77.
[*72G] D.7 j |7?|tJ = P (2)
H.(l).(2). DH:. Hp. D: P £ Pot‘(721 S'). D.
(g 7'). Te Pot*(S \R).P = R\T\R (3)
h.*0413. DI-: T’fPot'O&l!*).:>.
( 3 P). P £ Pot‘(P | S). P | R | R = 771 T | R (4)
593
SECTION E]
ON POWERS OF RELATIVE PRODUCTS
I-. (2) . (4) . D h Hp. D : 3T<= Pot‘(<S| R) . D .
(a^) • P « Pot‘(R |S).P=«| T\ii.
[*13195] D.«|r|ieePot‘(«|5):
[*13'12] D : Tt Pot‘(S\R). P-R\T\R . D . Pt Pot‘(if | S) (5)
H.(5).*1011-21-23.D
I- Hp . D : (^T) . Tt Pot ‘(S \ R) . P - «j T\ R .3 . P t Pot ‘(R \ S) <(j)
K(3).(6).3K Prop
*94-31. I- : R e 1 -* CIs . C I\R \ S) C D‘ft . 3 . Pot‘(/e | S) - (R || ,R)“Pot‘(S| R)
[*94-3]
The following series of propositions lead up to the proof that when
« e 1 -+ 1 . a‘<« | S) C D‘R, or St 1 -» 1. d‘(S\R) C D‘S, we have
p‘d“Pot‘(R | S)sm p‘d“Pot‘(S | fl).
*94 4. I- .p‘d“Pof(R | S) -p‘(F“| S“| Jl“Pot‘(« | S)
~p‘S‘“ CI“| «“Pot‘(/f | S)
= p , S‘“R“ , d“Po\.\R | S)
Dem.
K *93-431. Dh.p'd*
‘Pot‘(/i 1 8) - P‘d“ |(fi| S)“Pot‘(R I S)
[*43 201.*37-33]
-p‘d“\S“\R“Pot\R\S)
(1)
[*43-411]
-p‘*'“‘a“|/e“Pot‘(«iS)
(2)
[*43-411J
- p‘S“‘R‘“(l“Pot‘(R | S)
(3)
H . (1) . (2) . (3) . D h . Prop
*94*401. h ./>«a«Pot*(/e 16') = p‘ci“/f |«s|“pot«(/2 | sy
Dem.
K *93431 .*91*304.3
\-.p‘<l“Pot‘(R I S) = I S)\“Pot‘(R I S)
[*43*2.*37*33] = p‘(I“/e |“6'| “Pot‘(7i I .S’) . D h . Prop
*94-402. h . p*(l“R | C
Dem.
I- . *4311 . *34 36 .DK (i>). (I‘/e | *P C d‘i J
h.(l). *40-451. D h . Prop
*94-41. \-:Se 1 —> CIs . d‘(S| 72) C D‘S . D .
S“p‘d“Pot‘(7* | = />‘d“| R“Pot‘(R | £)
Dem.
h .*4012 . *91-351 . D 1- .p‘d" \ R“Po\.‘(R | S) C d‘| | S)
[*43-111] Ca<(*|S|^)
[*34-36] C d‘(S | R)
R«CW 1
38
a)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
K(l).DP:Hp.D. p 1 <I“ | 7?“Pot‘< 77 | S) C IPS.
[*72 502] 3 ./>‘< 1“| 77“Pot 4 < R\ S) - S“S‘y<l“[ 77 44 Pot 4 (77 | S)
[*72-34] = S 4 yS“ 4 (l“| 77 4 ‘P..t 4 (77I S)
I* 94 ' 4 ! = S 4 y<T“Pot 4 (77 I S) : 3 h . Prop
*94 42 K : 77 * 1 —►CIs. 3 . 77 4 yCl“P..t‘< R | .s') = ya“| 77 44 Pot 4 (77 j .S')
I Jem.
H . *72*34.31-: Hp. 3 . 77 4 y<l“P«>t 4 </7 | R)-j)‘R‘“(l“Pot‘(R \ S)
I* 4 '*' 4 ' J J -//U“|77“Pot 4 (/7|S):3 h . Prop
<94 43. b : 77. N* 1 ->Cls.<|‘(S| R) CD'S.
S'yCP'Pot 4 !77 I ,S') = 77 4 yU 44 Pot‘(77 |S) [*!)4-41 42]
*94 441 K:S« I -*C|s.(I‘<S| 77)C1PS.3.
.s ,4 y(|“Poi‘(/r^ = //(I“^|“Po t 4 (.s'| R) [*9414 41]
*94442. b : Rt 1 ->Cls. 3 . 77 4 y<l“Pot 4 <77 | S) = />‘< 1 44 77 |‘ 4 Pot 4 (S| R)
[ *94 14-42]
*94 5 K/>‘U“Pot 4 <S| 77) -/i 4 <I“77|“Pot 4 (S| 77)
Dan.
h.*94402.3h .;/(l“77|“Pot‘(6*|77)C/> 4 a“Pol 4 (S| 77) ( 1 )
1-. *94-402.31-. />*< I“S|“771 44 Pot 4 (S | 77) C/> 4 ([“7? |“Pot 4 (S| 77).
[*94 401) 3 b ./> 4 (J“Pol*(S| 77) C;>*<!“77 |“Pot*(5| 77) (2)
b .(1).(2). 3 1-. Prop
*94 51 1-: 77e 1 ->Cb. 3./> 4 <I“Pot 4 (S| 77)= 77 4 y(I“Pot 4 (7?1 S)
[♦94.V442]
*94 52. b:S*\-* CIs. U 4 (S | 77) C IPS. 3 .
7 >‘<I"Pot‘(S| 77) = S 4 ya“Pot 4 (7? | S) [*94-5-441]
*94 53. H : 77 « 1 —* 1. <I 4 (77 | S) C IP77.3 .
/>‘<I“Pot‘(77 | S) sm yj 4 U“Pot 4 (S | 77)
Dent.
h . *93 2G1.3 b . p 4 (l l4 Pot*(7? | S) C U‘(77 | S) (1)
b . (1). 3 b : Hp. 3 . y> 4 Cl 44 Pot 4 (771S) C IP 77 (2)
b . (2). *94-51 . *73-21.3 b . Prop
*94 64. b:Se 1 -»1 . CI 4 (S| 7?) C IPS. 3.//a 44 Pot 4 (77|S) smy>‘<J“Pot‘(S | 77)
*94-53
...S/71
77. Si
[Or. *94-52 . *93 2(>1 . *73 22]
SECTION E]
ON POWERS OF RELATIVE PRODUCTS
595
r s.^i
' R,S,M\
*946. I -R | S = S \ R . 3 M e Pot ‘R . V e Pot‘5 .^.M\N = N\M
Dem.
(-. *34-27-28 (1)
H • (1) • *91171 J,/| S = fM. 3
<pj\I
h Hp . M e Pot‘R .0 : AI\S = S \ AI: (2 )
D : N f Pot‘S. 3 . AI\N= N \Af:. 3 I-. Prop
*94 61. t-:.iJ|S = S|7f.D: J »/ ( Pot‘^.3. J l/|S po = S I „M/:
Dem XePot‘S.O.N\R„-lt po \ir
f-. *43 42 . 3 (-. At | S,„ = i‘M | “Pot‘S ( i)
K . (1) .*94-6.3 I-: Hp. Me Pot‘12.3 . M\ S |K , = i‘| M“Pot‘S
t* 43 ’ 421 ] ~S W \M (2)
h ‘( 2 >sri - ^ • Hp . Nc Pot‘S. 3 . A r 112,„ = R llo | iV (3)
h . <2). (3) . 3h. Prop
*9462. l-:ie|S = S|ie.3.72 po |S I>0 = S po |^
Dem.
I-. *43-42. *94-61.3 h : Hp. 3 . R„ | S„ -i‘| R^-PotfS
[*43-421] — Spo | R ,„: 3 I-. Prop
*9463. h:ii|S = S|/e.3.(«|S) w) Gfi 1 K> |S I10
Dem.
f-.*91-502. 3K.ft|,SG.R MO |.S po (i)
K*94-61. 3 h : Hp . Af G R^ | Sp. .O.Af\R\SC | R | S,, 0 | SJ
t* 91 ' 6 ”] G R„IS,„ ( 2 )
h - (1) • (2) . *91171.3 h Hp. 3 : Afe Pot ‘(R | S) . 3 . Af G R,„ | S,„ :
[* 41161 ] 3 : (R | S)„ G Jt„ | S M 3 h . Prop
*9464. K.3.(/J|S)»Gif»|S»
Dem.
*" • *34-36 . 3 I-. D\R \ S) C D‘R . d‘(S | R) C (l‘R.
[*33-16] 3 h : Hp. 3 . C‘( R\S)C C‘R ( 1 )
Similarly I-: Hp . 3. C‘(R | S) C C‘S ( 2 )
h - (1) - (2). *50-6 . *35-31.3 h : Hp . 3 .1 f- C\R \ S) G /f C‘R | /(• C‘S (3)
4 ■ ( 3 ) • *94-63 . *91 -54. 3 1-. Prop
38—2
*95. ON THE EQUI-FACTOR RELATION
Siininwry nj
The purpose of this iiiiinhi-r may be explained as follows. Consider the
scries of relation*.
it, p\n\Q, i n \n v. r\i<\Q\...\
it is required to find a means of defining this scries without the use of numbers.
If we u*ed numbers, and had f lu* definition given later (*301 > of P*. where v is
any finite integer, the general term of the series would be P ¥ , I{ | (f. But we
have not yet defined numbers, and we therefore desire some means, not
involving numbers, ot expicssitig what is intended when we say that, in
a given term of the series, the same power of P and of (} is to be involved.
This we do as follows. I’sing the definition of /'||y in *43, we have
/ ,; i R\Q*-ii"'-W*R.!»\K\Q--U’iQrR ....
Tims the general term ol our series is got by taking any power 6’ ol
(/')! V). and forming *N‘ It. The whole of the terms of the series are therefore
constituted by the terms which have to It the relation (/' (})+•, i.e. they are
|sg‘( /'(|For eonvenience of notation we put*
Thus the class of relations we wish to consider is (P+Qflt.
To illustrate the nature of (/**V) , /f, suppose If is the relation • first
cousin," while P is the relation of child to parent and (J is the relation ol
parent to child. Then P\li\(J is the relation “second cousin," P : \li\Q :
is the relation “third cousin, and so on. Thus ( P+Qyli is the class of all
relations of consulship which do not involve a difference of generation; and
".#• \v‘( P*Q)‘li\ y will mean "x is a cousin of y in the same generation."
Most of the propositions in this number are inserted because they are
required in the proof of *95*52, which states that, under suitable circum¬
stances. s\P*QYR « l —> 1. This proposition itself is proved mainly because
it is required in the proof of *!).V63, which states that, if P, Q arc one-one s
each of which has its converse domain contained in its domain, and if the
first generation of P is similar to the first generation of Q. then the sum
of the generations of P is similar to the sum of the generations of Q. This
leads immediately to a proposition (*!)5'71) which is half of the Sehroder-
Bcrnstein theorem (the other half being *94*53 or *1)4*54), namely: “II
• Thin notation is used in the present number only. In •257, we tfhall introduce a different
and wholly unconnected meaning for (/'• V)- A temporary definition is indicated by the letters
••Dft'* followed by a reference in square brackets to tho number or numbers in which the
definition is used.
ON THE EQUI-FACTOR RELATION
597
SECTION E]
R and S are one-one’s each of which has its converse domain contained in the
domain of the other, then the sum of the generations of 72 | .S is similar to
the sum of the generations of R .”
*9501. (i-**(3) = sg‘((P|ie)*! Dft[*!)5]
*951. h :: M e ( P*Q)‘R . = :.«< M : A r c p . O k . P| zV| Q « /t: . M e ^
Dem.
.*32 18 .(*95 01). D
h :: Me(P*Q)‘R . = :. M {P\\ Q)*R :.
L*9011] =:.^ e C‘(P||Q):. N * ,x. T(P\\Q)N. 0„ T . Tcp:
R e ft i • M e ft :.
[*43-302102] a z.iVeft. T-P\N[Q . D rA .. Te ft: Re ft: D M . Me ft:.
1*13191] ■ :. N e ft . D. v . -P| ^ : R e ft: . M e ft:: Dh. Prop
*95 11. h : .cf>R: <f>N . D v • 0 (P|*V| Q): D : 31 e ( P*Q)*R . D. v . </>,!/
Dem.
K.951^>.D
h :: il/€ (7>*Q)‘72 .0 :.<fiR: <fiiV . D v . <f> (P\N \(?) .O . <*>,1/ (I)
1-. ( 1 ). Comm . *10*11*21 . D 1- . Prop
*95 12. h :. # e (p*g)'/i . D 3/ . <*> (P|i»/| (?) : D : 2\ r * (P*Q)‘R - P72 . D. v . </,A r
Devi.
H. *43112. D
h :. Hp . = : A/«(7 J *(?)‘/2 . D.„ . <f> |( P\\QY3f ) :
[*37*03] s : Ar€(7'||(?)“(P*(?)‘/e . O x .<f>N (1)
h .*90311 7 3
it
h : N€(P*Q)*R-i‘R.D.Ne(P I! Q)“{P*QyR (2)
h . (1) . (2) .DK Prop
*9513. h . ft € (P*Q)*R [*951J
*95131. b.P\R\Qe(P*Q)‘R
Dem.
H.*90151^P.DI-:S(P||0)ft.D.S(P|IQ) ll ,ft (1)
It
h . (1) . *43*102 . (*95 01) . D h . Prop
*95132. Me ( P*Q)‘R .D.P\M\Qe ( P+Q)*R
[*90-172 ® . *431021
'»0S PROLEGOMENA TO CARIUNAL ARITHMETIC [PART II
*95 14 H 077: .V, < P*Q)‘li . <*>.V. D v . 0< P\X\tj): D: .l/c<P*<?)‘/7. D,,. <f>M
Dew.
h.*0.V|3132.DI-:. Hp.D:
0/7. /; f ( : .V < i P+<p*R . 0.V. D.v • 7»| iV| (JtilWH .<f>(P\N\Q):
| *05 11) D : .1/ c (7'*<7>‘7?. D v . M t \ P*Q)*R . 0.1/ D h . Prop
Tin* »>•• of *05*11 in the Iasi line «»f the above proof proceeds by sub¬
stituting .1/ € (P*(p*R • 0-17 for 0.1/.
*95 21. h:.l/«</'*<?»•//.D-igS 7>.N« \\>t*Psj'*I. T* WtUJv i*f ,M=S\R\T
Dem.
K*50 4.Dh. 77 = /|77|7.
|*5l*Ili] D H . (;.|N. T ). Nt Pot *P » i*1 . Te Pot 4 V ^ f*/. 77 - N|72| T < 1)
H . *n I 36*351 . *50 4. *34 27 2s. D
H :.Sc Pot 4 7* vi*/ . 7’c Pm 4 y * i 4 /. J/- S| 7?| 7\ D .
7 , |N € Pot 4 /'*# 4 /. r|V# Pot 4 (fvi 4 /.7^.»/|y-(7 , |5)|7J|(7 , |Vi.
|*ll■:wp.<a.S\7 , ‘ , ).N , tPoi 4 7 , ui 4 7.rcPoi 4 Vui 4 /. /'|J/|V-S‘|7i|T (2)
h.<2).*ll IP35.D
h : ( 5 |.S. 7’). S« Pot 4 7' ui‘/.7’< Pot 4 V w‘/..l/ = N| 771 7’. D.
(gS. '7’). .V c Pot 4 Pm Pi. Tt Pot 4 y v i 4 / . P\ .1/1 Q - N| 7717’ < 3 >
l-.<l).(3).*05*ll . D h . Prop
*95 211. h : < 1*77 C ("V • .V t <7'*V) 4 77 . D .
<gS. '7b. St Pot 4 P sj i*I m 7’* Poti.l 4 ^. ,1/ - S| 77| 7’
Dem.
H . *50 02 4 . D h lip . D : »S' : R / [C*Q = S| 77| / :
[*51*230.*01*23] D : u[S. T ). Sc Pot 4 7'w 4 7. T’cPotkPQ. il/=S|77| 7\ =.
(gS. 7'). .V c Pot 4 Pm PI. Te Pot*y v / * /. ;l/=S177, T :
[*05*21] D : <gS. Tt . g«Pot 4 />vf 4 /. 5T«Potid‘Q. A/ = S | 7717’
D h . Prop
*95 212. h : I > 4 77 C C*P . 5/ € (7 , *<7> l 77. D .
<gS. y). S€ potid‘7*. T c Pot 4 V. /*/ . M = S\R\T
[Proof as in *05*211]
*95 22. h : 1 >*/7 C C*P . <1*77 C Oy. M c (7^*Q)‘77. Z>.
(gS. 7*).Sc Potid 4 7 > . Te Potid 4 Q . M = S\R\T
[Proof as in *05 211]
*96 221. h-.Tt Pot *Q . D . (gS). S c Pot 4 P. S17712* e(P*Q)*R
Dem.
h . *05131 . *01*351 . D 1. <gS)-SePot'P. S| 77| Qe(P*Q)‘R (1)
h . *05*132 . D
h : S c Pot 4 P . rc Pot 4 Q. S1771 Fc (P*(^) 4 77 . D.P\S\R\T\Qe (P*Q)‘R •
[*01*30] 3 • (3-$*) - S' ^ Pot 4 P. S' | R | T | Q e (P*Q)‘R (2)
h . (1). (2). *01*373. D 1. Prop
SECTION E]
ON THE EQUI-FACTOR RELATION
599
*95*222. h :ScPot*P. D .(gP). TePot'Q .S\R \ Tc(P*Q) ( R
[Proof as in *95-221]
*95 23. h : M € (P*Q)‘R . D . Af (P rt j Q,,) R
Dem.
f- . *3218 . (*95 01) . D h : Hp . D . M |(P|| Q)*} R .
[*43-202] D.j»/{(P|)|(|C))*fi.
[*43-202.*9464] D . ,1/ |(P j)* | (| Q)*] R .
[(*910102)] D . J1/(P, t | <?„) U.Dh. Prop
*95 24. H : J/ c (7>*Q)‘P . D . jl/ (Q ts P. t ) P [Proof as in *95-23]
*95 3. K a ! R . d‘Q C D‘Q . G‘P C . D: P« Potid'Q . D . 3 ! P | 7
Dem.
K *50-62. D I- : Hp . D . 1 /(" = R .
[*1312] D.alieK/rO-Q) ( 1 )
I-. *91-27 . *33 181.3h:.Hp.r ( Potid'Q . O : Q‘TC D‘Q :
[*34--35] D:a!r.D.a!(r|Q) (2)
1- • (1) . (2) . *91-371 . D I-. Prop
*95 301. I-:. a ! R . D‘P CO'P.D'fiC <J‘P .D:Se Potid'P . D . a ! 51 R
[Proof as in *95-3]
*95 302. I- :. CI‘Q C D ‘Q . (I‘R C D ‘Q . D : 7’e Potid‘Q . D . d‘(/e | T) C I )‘Q
Dem.
H . *91-271 . *34-36 . D b : T c Potid € Q . D . G‘(P | T) C G‘Q (1)
f-. (1). *22-44. Dh. Prop
*96-303. f- D‘P C Cl ‘P . D‘P C CI‘P .D:3« Potid‘7 J . 3 . D‘(S| P) C G‘P
[Proof as in *95-302]
*96 304. h G‘Q C D‘£ . CI‘P C D‘Q . D‘P C d‘P. D‘P C d‘P. D :
Sc Potid‘P . Pe Potid‘Q . D . D‘(S| P| T) C G‘P . G‘(P| P| P) C D‘Q
[*95-302-303 . *34 36]
*96 305. I-Hp *95 304 . D : M c (P*Q)‘R . 3 - T)‘M C G‘P. (I*M C D'Q
[*95-304-22]
*96 31. h Hp *95-304 . g ! P . D : P € Potid'P . P c Potid'Q. D . g ! S\R | P
Dem.
y . *92142 143 . D h Hp .DzSe Potid *P . Tc Potid'Q . D .
D‘P C a‘S. a*P C D*P.
[*34-361] D.g!P|P|P:.DK Prop
*95 32. h s . Hp *95*31 . D : Mc(P*Q)‘R . D . g ! 3/ [*95 31 22]
*96 33. h : a*/e C . D . d‘(S |ii|r)C T“~B‘Q
Dem.
I-. *34-36.3 h : Hp. D . d‘(S | R) C
[*37-32-2] D . d‘(S I R | r> c T“~B‘Q :DK Prop
PROLEGOMENA TO CARDINAL ARITHMETIC
( PART II
*9534 H : (I 'R . McUWjyR . D . i^T). T, YoluYQ Al*MCT“B‘Q
[*95-33 2II)
*95 35 H: <J ( I -*r|s.<l‘/fC/i‘<j. D.(Ha).a*gcn‘(?.Cl‘iVCa
[*95 3+. *93-32)
*95 351. I ->CU. O - // C 7/‘y. 3 :
y. r «YoiWQ .auvts\K\ n a < i‘(N' | /? | n. d . r = r
I bin.
h . *95*33. D hHp.D:
7\ 7’ * Poli«Ptf. 3 ! /,' I AMI'IS* I /7 I D. D. 3 ! r"/?Qn .
|*93-3) D.fl!milled*To min v *CI‘3T' •
|*!»3 24.Tmns 1 >) D. 7’= T D I-. Prop
*95 352 h :./*•('Is—* 1 . L>*/f C /**/'. D:
.S', .S '1 PoticP/'. ! I >*(N| /?I /•) A 1 >‘(5' I /f I 7"). D . S■ *s"
[ 'Proof :i> iii *95 351 ]
-9536 I-:. V* I •"♦Cls.U‘/^c7?V.<|! /M>‘/fC<l‘/\
1)*/^ C < l‘/ > . < C ]YQ :
S. S' < INitiipy*. T. T * Poti«PQ. .S'l K\ T-S | K \ T . D . T - T'
I bin.
h .*‘13-31 . *93‘101 . D 1-Hp. D :
n. .s'f i»»ti.i‘ 7 ' . r. r * Poti«i«v . s| /*I '/•- n # | /?| r . d .
UlS\H\r.S\R\T-S'\K\r.
1*22 24] D. g ! <I‘(S| /f | T >« U«<S'| /f | 7”).
I] D.r-r :.DKl«rop
*96-361. 4 1.l>‘/f C« i /\ >|! «. 1>‘/'CCI«/'.
a*«CD‘Q.a‘QCD‘Q. 3 :
.s-, .s- 1 Potw/'. 7'. r ( i>oi i.py. si«i r- .s- 1 7f | r. d . s - 5'
[Proof ns in *95*36]
*95-37. }■:./*€ CIs —* 1. « l —> CIs. D‘7? C 7?7~. CI‘7? C/?‘Q. 3 ! R
D'/'Ca'i’.d^CD^.^:
,s. .s-«PotiH •/*. r, r ( p..tid‘<?. s\r\ •/'= .s- 1 «i r .o.s-s’.t-t-
(*!I5-36'361 ]
*95-38. 4 :■ 3 «<1‘« • 3 : 7 1 * Poftf. 3 . K | 7’+ H
I-. *91-271.34: T ( Pot ‘Q .O.a\RT)C (I ‘Q.
[*93101] D.a i («|7)nS : Q = A (1)
4 . *24 54. 34: Hp. D . ~ JCI‘71 n~B‘Q = Aj < 2 >
4. (1). (2). *1314. D 4. Prop
SECTION E]
ON THE EQUI-FACTOR RELATION
601
*95*381. h a ! P‘P n D‘P . D:5e Pot‘P . D . 6* | P * P
[Proof as in **>5*38]
*95*382. h a !i?‘P ^ D‘P . v . a ! 2**Q n d‘P : D :
5c Pot/P . Pc Pot ‘Q . D . 5 \ R | P* R
Dem.
V . *91*271 . *93*101 . D h : Pc Pot‘Q. D . Cl‘<51 R | T) * ~B‘Q = A (1)
H . *24*54 . D h : a ! ~B*Q * d‘P . D . ~ |d‘P cT5‘Q = A) (2)
I- . (1) . (2) . *13*14 . D h a ! P*Q " d‘P . D : 7’c Pot‘Q. D.5| R\ T+R (3)
K . *01-271 . *9312 . D h : 5c Pot‘P . D . D‘(5| P | P) n P‘P- A (4)
h . *24*54 . Dhrg! P‘P r. D‘P . D . ~ {D‘P P‘P - A j (5)
H . (4). (5) . *13*14. D h :. a D ‘R . D : 5c Pot‘P. p .5| P| P* P (6)
K(3).(0).Dh. Prop
*95 383. h a 8 R : D‘P C ~B‘P . v . d‘ R CB*Q : D :
5 c Pot‘P . Pc Pot ‘Q . I> . 51 P | P+ P [*95*382 . *33*24 . *22*621]
*95 4. h : 3/ c (P+Q)‘R . 5c Pot‘P. 7’c Pot‘£ . 51 R \ Te(P*Q)‘R . D .
5|3/| Tc(P*Q)‘R
Dem.
h.Simp. Dh: Hp.D.5| P| Pe(/'*Q)‘P (1)
I- .*91*34.*95*132. D
h : Hp . 5| 71/1 7’c <7'*g>‘P . D . 5| P| A/\Q\T-P\S\M \T\Q.
P\S\3f\T\Qe(P*Q)*R.
[*13*13] D.S\(P\M\Q)\ Pc(P*Q)‘P (2)
K ( 1 ). (2). *95*14. DH. Prop
*95*41. h.Pf Cls —* 1 . (£ c 1 —* CIs . D‘P C d‘P. C VQ CD'Q.D:
5,5" c Potid'P . T, T c Potid'Q . D . 51515' | A r | T'\T\T *= S' \ N \T
[*92*15*151]
*95*411. Hp *95*41 . D‘P C C‘P . d‘P CC'Q.D:
5e Potid'P . Pc Potid'Q . 3f c (P*Q)‘R . D . 71/ -5| 5| M \ T\T
[*95*41*22]
*96*42. h :. Hp *95*41 1 . D : 3/ c (P*Q)‘R - i‘R . O . P\M\Q e (P*Q)‘P
Dem.
h . *95*411 . *91*351*281 . D
h s. Hp. D s M c(P*Q)‘P-D-P |(P I M\Q) | Qe(P*Q)‘R
h. (1 ). *95*12.3 h. Prop
(1)
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
G02
*95 43 I- Hp*93411 . Hp*9.V3s2 . D : .Se Potid 4 A . Te Potid 4 // .
A S\n T;Qc{P*Q) t K.O.S\R\T€(P»Q) t R
Den*.
K . *!I.V4*> :m-2 . *91*2*3. D b lip. D : St Potid 4 A. Tt Potid 4 /?.
ASIA T\ f?c(AU?)'A.D.A|A|S,Jt T\U\Q€KP^R (D
h . *93-41 . D h Up. D Potid 4 A. 7'e Potid 4 /? . D .
A | A| £ | A| 7’| <?I ^ = A | A | 7* (2)
I- Dh . Prop
♦95 431. H : Hp *95-43..S\ Potid 4 A. T< Potid 4 (J. M c(P*(J)‘Ii.
P\S\M\T\Qc( A*/?) 4 A . D . S |-1/| 7'e/ A*<?)‘ A’
Dew.
V . *95*22 . D K : Hp. D . t;.|£\ D. N < Pot id 4 A. T't Potid 4 /?. .1/ = S' | A | T .
/*1 8 1.1/1 T\Qt(P*<}yli.
| ♦91*341 ] D . (y.V. D. 6% Potid 4 A. r# Potid 4 /? - J/ - 5' | A | T .
.s’ | S' t Potid 4 A. 7 M 7'e Pot id* V. A l A | S' | A | T | T\Qt{P*QYR .
|*93 43) D . C 4 .S-. 7*'). .S"« Potid 4 A. 7’’« Potid 4 /?. M - .S’ | A | 7” .
.s’ | N A | 7” | 7*c( A*/?)‘A.
|*13I93) D.S| J/| 7 , t(A*V) 4 A:DK Prop
*95 44. 1- lip♦95*43. *s’« Potid 4 /*. Tt Potid 4 /?. D:
.1/ € (A*<?) 4 A. N| . 1/1 Tt (A*/?)• A. D. .s’ | A | 7’«( A*/?) 4 A
Item.
h . Id . D H :: 4>M . s. w :.s'| .1/ |A<t A*/?) 4 A. D. S| A| 7 , < (A*/?) 4 A:. D. 0 A (1)
H .*95*431 .*91*3.3
I-Hp. D ::S| P\M\Q\ Tt(P*Q)‘ll . D .s’| M \ 7’e/A*/?) 4 A
|*2 27) D .s’| 4/1 7’«<A*/?) 4 A. D. 6'| A| T’e/A*/?) 4 /* : D .
6 ’| A 7\(A*/?) 4 A (2)
h . (2). Comm . D
h :: Hp . D A | M I 7’e (A*/?) 4 A. D . S\ A 17'c(A*/?) 4 A : => :
Si (A J/ Q)|7 , €(A*/?) 4 A.D.£ A | 7’e (A*/?) 4 A (3)
H .(3). D I- s. Hp. Hp(l). D s 4>M . D. <f> (A |.l/1 /?) (4)
h . (1). (4). *9514 . D b : Hp . Hp( 1). M €(A*/?) 4 A . D . 0-1/ : D H . Prop
*95 45. h Hp *95 43. S. S’ e Potid 4 A. T. Tt Potid 4 /?.
S | S' | A | T | T c (A*/?) 4 A. D: S | A | 7’e (A*/?) 4 A. = . S' | A | Tt (A*/?) 4 A
Dem.
h . *95 44.D h: Hp.tf| A| T't(P*<2yR . D . S ; A j Act A*/?) 4 A (1)
K . ♦91*34. D h Hp. D: S' |S| A; T\ T t (A*/?) 4 A:
[*95 44] 3 : S| A | Te(P*QyR . 3 . S' \ A) (A*Q) 4 A (2)
I- .(1).(2). 3 t- ■
SECTION E]
ON THE EQU1-FACTOR RELATION
603
*9546.
Devi.
I-Hp *95-41 . g ! R . D ‘R C B‘P . Q‘R CB‘Q . 3 :
Te Pot ‘Q . 3 . R | T~ e (P*Q)‘R
I-. *95 38.31:. Hp . Te PofQ . 3 : R | T * R :
t* 95 ' 42 ] 3 : R I T e (P*Q)‘ R . 3 . P | iJ| T\ Qe(P*Q)‘R .
[*95-32] D.a!^|iJ|r|0.
t* 34 ' 31 ]
[* 34 ' 3 ] D.gJD'PoD'ie (1)
1 . *9312 .31: Hp . 3 . D'P n D'R = A (2)
■ (2). (1). Transp . 3 h : Hp . Te Pot ‘Q . 3 . R \ T~ e ( P*Q)‘R : 3 1 . Prop
*95 47. (- : Hp *9546 . S * Potid'P .T.T'e Potid'Q.
S\R | T, S |/e | T'e(P»Q)‘R .O.T= T'
Devi.
V . *91-46.31:. Hp. 3 : (gif) : Ue Potid'Q : T ■= U\T‘ .v . T = U\T (1)
1. *50-62 . *91-35.3 1: Hp . 3 . S = S | / f C‘P. /fC-P e Potid ‘P (2)
1 . *95-45 . *33-24 . *22 621 .(2). 3
h:H P- tfrPotid'Q.T- U\T' .O.I[C‘P\R\ Ue(P*Q)<R. UePotid‘Q.
[*50 63] 3 . R Ue(P*Q)‘R . Ue Potid ‘Q .
[*95-46.Transp] . 3 . U ~ « Pot‘Q. U e Potid.
[*91-23] 3.17=/[ <?'<?.
[*91-27.*50-63] 3 . U\ 7” = T'.
[*1312] O.T-T' (3)
Similarly 1: Hp . Ue Potid'Q . T = U\ T. 3 . T = T' (4)
h • (1) • (3) . (4) . 3 1. Prop
*96 471. I: Hp *95 46 . S, S' e Potid ‘P . Te Potid‘Q .
S | R | T, S' | R | T e ( P»QY R . 3 . S = S'
[Proof as in *95 47]
*95-61. f- ; Hp *95-46 . M, M'e (P*Q)‘R . g ! a *At r> CP At '. Z> . At = At'
Dem.
h • *95-22 . D I-: Hp . D . (gtf, S', T, T') . S, S' c Potid'P . T, T' e Potid'Q .
AI = S\R\T.At' = S'\R\T'.
S\R\T,S'\R\T' c(P*Q)*R.
g ! d‘(S\ ft\T)n a ‘(S’ | R | T') .
[*96-351] 3 . (a S,S', T) . S, S'* Potid Te VoUd‘Q. M=S\R\T.M‘ = S'\R\T.
S\R | T, S' | S| Te (PmQ)‘R.
[*96-471] 3 . ( a s, T) . Se Potid'P. Te Potid'Q . M = S\R\T. M'= S\R\T.
[*13172] 3 . M = M ': 3 1 . Prop
PROI.KIJOMKXA TO CARDINAL ARITHMETIC
(PART If
*95 511. K : H|> *93 40. M. M' < < !>*(})• K . 3 ! I>M /* PM/' ,D.M= M'
| Proof as in *115*3 I ]
*95 52. h : /'. V K «■ I ->1 . Ii‘/'C<l‘/\(I‘QCI)‘Q.D‘/ec"?7».(I < //C/i‘Q.D.
x‘(/V^)‘//c 1 -► 1
Item.
h . *95*21 . *3432. 3 l-: // = A . 3 .(/ , */^) < // C /‘A .
I*33*n4| 3.* 4 < I'+fJ)* /f = A .
1*72-11 3 . a‘i /**())*]{t 1 —> I (1)
H . *1)210- . #95*21 . *71*252 . 3 H : H,». ,l/e( /'*(,)rt - 3 . A/ c l -► 1 (2)
I-. *41 11 . 3 h is A IWjHi y.x\#\ n+ifVK: z . 3 .
<a .1/. .1/). .1/. .1/' < (/'*v ) f /* • .v*. x.i/ 'r.
|*33*14] D .(M.1/ .1/ ). .1/. A/V</'*V)‘//.^%.^m.a!DM/n DM/' (3)
I-.|3>. *95*511 . 3
h : 111>. a ! //. II|> t3). 3. <a A/). M «(/V^)‘ //. .M/y. .#0/.*.
| (2) ] 3.//-: (4)
Similarly
h : 11 |I. a ! //. .#• \i't /'*V »• H :. .y ;*< /V^) 4 /e* *. 3..» « // (5)
I- .(4). (•’»). *7II72. DH : lip. a ! H .0 .AlWJVR < 1 -* I (0)
h.(l).(0).3H. Prop
>956 H: D 4 //C<l 4 /M» 4 /*CU 4 / , .C|‘//-/?Q.^€l ->Cls.3.
(I 44 ( l**QYR™gen*Q
Item.
h .*02*143.31*: Hp..S'< Poii«l 4 /'. 3.(I«$-U 4 7\
(II,,| 3 . I) 4 // C (I*.S.
[*.‘{7*322] 3 . <1*(6’
[#37*32] 3 . (1 4 (.S| // D = '?“<I 4 // (1)
h . (I). 3 K : H|». .s« P«»li«l 4 /* . 7*c P<»ti«l'Q. 3 . < l‘(S j 7f, T) = T“li l Q . (2)
[*93*32] 3.a 4 (S|//;7 , )€ K cn 4 Q (3)
H .(3). *95 22 . 3 h : Up . 3 . (l 44 (/ > *^) 4 // Cgen 4 g (4)
I-. (2). *95-221 .*93-32.31-: Hp. 3. geiPQCCI^i^Q) 4 // ( f >)
I- .(4) .(5) .31-. Prop
*96-601. h:ci‘/ec m) .a 4 vc d*q.D 4 // = .p£cis-> 1.3.
D 44 ( P+QYR = gen 4 /'
[Proof as in *95*0)
*95*61. AYPCWP.ll'QCD^.D^i^B'P AVR- B‘Q-3 •
AP*QYRt 1 -* 1 • \YAP*Q)‘R -s 4 gen 4 /\(I 4 * 4 </>*Q) 4 J* = s‘gen‘<2
[#95*52*6*601 .*4143*44]
SECTION E]
ON THE EQUI-FACTOR RELATION
*9562.
*9563.
Dem.
h : Hp *95 61 . D . s'gen'P sm s*gcn*Q [*95 61 .*73 2]
H : P, Q € 1 1 . d‘P C D‘P .d*QC D‘Q . ~B*P sm ~B‘Q . D .
s*gen‘P sm s*gen*Q
v
I-. *95*02 — . D f- : P, (J, /£ € 1 —> 1 . d‘P C D *P . d *Q C D‘<2 -
*9565.
D*R = B*P.tt*R = B*Q.D. s < gen t ] i sms* gen* Q ( 1 )
b . (1) . *10 11-23-35 . *731 .DK Prop
*95 64. h : P,Q e 1 -> 1 . d‘P C D'P. (S‘Q C D‘<J . R*Psm B*Q .
//d“Pot‘P = A .p*Q**Yot*Q = A . D . P‘7 J sm D‘<2
[*95-63 . *93-274 . *33181]
*95 65. b : P t Q e 1 ->1 . d‘P C I VP . d‘<? C I >*<? . 2?P sm ~B*Q .
C‘P = P+“!pP . <7‘Q = $*“/?<? . ^ . G'Psm
[*95*63 . *93-36]
The following example may illustrate the scope of *95-65. Let R, S be
the generating relations of two well-ordered series, neither of which has a last
term. Put P = R — R *. Q -* S — S 1 . Then P is the relation of immediately
preceding in the 72-series, and Q is the relation of immediately preceding in
the iS-series. We shall have
P, Q e 1 -> 1 . d‘P C D*P . a *Q C \VQ.
Also, except in certain exceptional cases, B*P, B*Q are the first derivatives
of the two series (including the first terms of the two series).
"C‘P-/V‘2PP”
states that, starting from any term of the series and going backwards, a finite
number of steps will bring us to a member of the first derivative, which is
true. Hence, by *95*65, neglecting certain exceptional cases, we arrive at the
result that if the first derivatives of two well-ordered series have the same
cardinal number of terms, then the series themselves have the same cardinal
number of terms. This proposition can of course be proved otherwise; the
above is merely mentioned as an illustration of the results of *95 65.
*96 7. h : R, S e 1 -> 1 . d‘72 C I VS . (I‘S CD *R.D. ~B*(R | S) sm ~B*(S | R)
Dem.
b . *93101 . *24-412 . *37 16 321 . D
h : Hp. D . /?(«| S) - (D‘.R - d‘S) « (d‘S-S“d‘R) .
R) = (D‘S- <3‘.R) « (d‘R - R‘‘d‘S) (1)
I-. *71-38 . *37 32 . D 1-: Hp. D . R“(.D‘R —J1‘8) = d‘R — W'd'S (2)
I-. *71-381. *37 32.31-: Hp . D . S‘\d‘S - S“d‘R) = D‘S - S“S“d‘R
[*72-602] =D ‘S-d‘R (3)
PROLEGOMENA TO CARDINAL ARITHMETIC
| PART II
<>m>
K.(2>.(3>.*7*21-22.Dh: H ( ». D . \)*1{ - fp.S'sm (|‘7? - jf“<T‘S.
WX-iVIi (4)
b . *24*21 . DH : Hp . D . (I VR — (l*,S') r> (<I‘.S' — N“(I*/?) = A .
<<!*/{ — 5 m CI‘jS) n( DSS* - < I ‘/O = A (5)
K . (1).(4).(">). *7*71 . D b . Prop
*9571. b:R.S€l-*\M'RC\) t S.(l‘SC\VR.'}.s*f'en t (R\S)sni$*gon l (S\R)
Di-in.
K . *34-3t». *37321 . D b : 11p. D. < |‘< R \ S) C1 >‘< R \ S ). (1‘(.S'| R )CI )‘(51 R) (1)
b . *71 252 . D h : Hp. D . /f | N.£| J? e 1 -»1 (2)
>- . < 11. ( 2 ) . *!i->*7 93. D b . Prop
Phis proposition anil *!)+'.”>3 or *!»4-.">4 together reconstitute the Schrtidcr-
Bemslcin theorem (*73*88). For. in virtue of *f»3-274*275 and *7371, they
logrther give
/f. c 1 -»1 . < I'/f C I >‘.s* .iVSCWH.O. («(R | S) sin C‘(S | R),
and with this hyj>othi*si.s
C‘( R | S) = WR. C‘(.S*| R) = 1 >‘.S\
*96. ON THE POSTERITY OF A TERM
Summary of *96.
By the “posterity" of a term with respect to a relation R we mean the
class R^x. In the present number, we shall be chiefly concerned with the
relation (R*‘x)' | R, i.e. the relation R confined to the posterity of x. We shall
also be concerned with (R^x)] R+ and (R+‘a:)' j R lto , which, as is proved in
*9613, are respectively
{(**'*) 1*i* and ((**'*)
The most interesting case is when /£eCls—»1. In this case, R#‘x is in
general shaped like a Q, with x at the tip of the tail; that is, R^fx may be
divided into two parts, the first an open series, the second a closed series. If
y is the junction of the two. we shall have
xR^ . zR^y {zR lHf z),
yR#z . D . zR lMi z;
in fact, (gP) : P * Pot ‘R : yR+z . 3,. zPz.
• We have also, when jReCls—»1,
y.ze R m ‘x . D : yR+z • v . zR*y.
It thus appears that R*x is divided into two parts, the first consisting of
those terms z for which ^(zR^z), the second of those for which zR vo z. The
first wholly precedes the second; the first exists if ~(x/i po a:), the second if
a! I^po A/). Every term in R^x has one and only one immediate
predecessor, except the term (if it exists) at the junction of the tail and circle
of the Q; this term has just two immediate predecessors, one in the tail and
one in the circle. But if either the tail or the circle is null, then every term
ln Pvo*x has only one immediate predecessor, and therefore
Put • Dft
J R ‘x = R**x f\ 2 (~ (zR^)} Dft
(these definitions being only to apply within *96). Then J R x is the open
Part of the series and I R ‘x is the circular part. The open part wholly
precedes the circular part, provided PeCls— *1; i.e.
R e Cls—»1 - 3 - J*‘* C TfR^IJx.
l'ROLF.COMKXA TO CARDINAL ARITHMETIC
(PART II
(JOS
If •/,..*.*• and 1,'j' both exist. •/,.*.»• has a hist term, say y. The successor of
this term. R*y, is th«* only term in R#*x which has two immediate predecessors
in R#*x, namely // and l*{/ lt € xr\ R*R*y).
Tin- most important applications of the propositions of the present number
are in the theory of tinitc and infinite, both cardinal and ordinal. When R
is many-one, then if ] ,Sx exists, or, more generally, if J t *x has a last term.
R*-*’ is a finite class, i.t. what we shall call a "CIs induct" (cf. *120). That is.
we have
I" S R «(’Is—► I . K! max,..*./ ,.*x . D . //**.#• c CIs induct.
If exists, but has no last term, R+x is a progression (cf. *122) when
its terms are arranged in the order generated by R. That is, giving to N*.
and o> the meanings given by Cantor (cf. *123 and *203). and using “ Prog"
for t he class of one-one relations which generate progressions, we have
h : R « (’Is—* I . ~ K ! max #; * J ,*x . $| ! ,/ w V. D .
R+'x «• bt „., Rj. r) 1 R € p r „g. Of^x) 1 R t „ f *>.
Another very important proposition in tin* proof of which the present
number is useful is *12147. which proves that if R is either one-many or
many-one. and a and ; are any two terms whatever, then R+*u n R^z (which
we call the "interval from a to z) is always a finite class. The proof that
progressions are well-ordered series depends ti|>on the propositions of this
number, since it uses *122 *23. which depends upon *!K»*52.
'I'he present number begins with a series of propositions (ending with
*!M> I(>) on o'] R ln , an<l o'] /»'*. both in general and when a= R# f x. We then
proceed 1 to a few propositions-(*!»G 2—25) on ( R+'s) *] R when /f«?l-*Cls;
with the exception of *!MJ *24. these propositions are all used in the cardinal
theory of finite and infinite. They are, however, less important than the
subsequent pro|K)sitions, which arc* concerned with R # ‘x when ReC Is-* I.
If R is a many-one relation, and .#• is a member of 1) 'R, the relation R in
general arranges R*x (i.e. the posterity of x) in a
figure such as is here given. The relation R holds
between each dot and the next, starting from x, and
travelling round the circle jn the sense indicated by
the arrow. The dots from / toy constitute J^x, and
the dots in the circle constitute /„‘x. y is the last
term of Jk*, t.e. maxj,‘/ K ‘j; w is R'y, and z is
i‘(R‘iu n I u ‘x), or, what comes to the same thing,
|(/ K *x)1 R\‘w. w is the only term which has more
than one immediate predecessor in R*x\ w always
f v
R
R \.
TV
y
SECTION E]
ON THE POSTERITY OF A TERM
009
exists if neither J R ‘x nor I R x is null, and conversely, if iu exists, neither
J R x nor I R x is null. The proof of these propositions is long; the following
are useful stages in the proof.
If xlix, the whole posterity of x is x itself (#96*33); if xRy and yRx,
x and y constitute the whole posterity of x (#96 331), and so on. The
successors of members of I R *x belong to I R ‘x (#96*341), and the predecessors
of members of J R *x, if they belong to R+'x, belong to J R ‘x (#96*351). (It
should be observed that, since R is only assumed to be many-one, not one-
one, every member of R*‘x may have any number of predecessors which do
not belong to R**x.) We have a series of propositions, beginning with
#96*4, which deal with the hypothesis yRw.zRw. We prove (#96*42) that
if yRw . zRw and yR lt0 z, then zR lKt z, i.e. z belongs to I R x. We prove
(#96*431) that J R ( x wholly precedes I M ‘x; that (J R *x) ] R and (I n *x)'\R are
both one-one (#96*45), so that if yRw . zRw . y z, one of y and z must belong
to J R l x and the other to I R l x (#96*441). Hence it follows (#96 453) that if
either xR po x (in which case J H *m- A) or ( R+*x) 1 R^ G J (in which case
I R x= A), then (/£**#)*] R is a one-one relation. (This proposition is used
twice in the cardinal theory of finite and infinite, namely in #121*43 and
#122*17.) Hence we arrive at the proposition (#96*47) that if two different
members y and z of R+ l x both immediately precede a term w, then one of
y and z (say y) is the last term of J R x, w is its immediate successor and z is
the immediate predecessor of w in I R x, i.e. we have
y = m&.x R t J R t x . iv - . z = (( I R ‘x ) 1 R\‘R‘max R ‘J R ‘x.
Thus y, z, w are unique if they exist. We prove next (#96*475) that y, z, w
exist when, and only when, neither I R *x nor J R *x is null.
It follows from the above propositions that if R is one-one, either I n *x or
J R x must be null (#96*491), i.e. the posterity of a term is either an open
series or a cycle, and cannot have the f^-shape.
#96 01. /= %< m r. 2 (zR lHt z) Dlt [#96]
#96 02. J R x =*R*‘x - I R ‘x Dft [#96]
#96*1. hzzeljx.s. xR+z . zR^ [#20*3 . #32*181 . (#96*01)]
#96*101. h s zeJ R *x. = . xR ** —( zR ^) [#96*1 . #22*93 . (#96*02)]
#96*102. h • R+x = ./ R *x \j I R *x . J R *x r\ I H *x = A [#24 41*21 .(#96 01*02)]
#96*103. .(J R ‘x)'\R po <iJ
Dem.
V . *96101 . D V y 1 z . = : xR+y . ~(yR„y) • yR„* :
[•1314] 3:y + «r.Dh.Prop
R&W I
39
610
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*96104. b : I, : 'x = A . = . ( 1 R l<t CJ. = . J n * x = if*'*
Dem.
h • * ! »61 • 3 H /».*..•■- A . = : xft,./. D„ .~(yR„y) :
(*13196] = : -rft*i/. i/ft,.,* . 3„,. /, * . :
s:(ft»‘r)1ft w> G./
K(l).*9G102.3l-. Prop
*9611. (-.(a-|ft),.,Ga-|ft,.,
Dem.
b . *01-502 . *35-46 . D b . a 1 R G a 1 R (H1
h . *351 . D
H :. /* G a *| R ^,. D : . y (* 1 /f)s. D . x«a. xR x<t y. yRz .
[*01-511.*351) D.*<«1 ‘
[*341] D:P|(«17?)Gflt1/? lw
h .(1). (2). *01-171 . D h : /*€ Pot‘(a *]/?). D . / J G a *] /?,„:
1*41151) D b . (a 1 /?V, G a 1 /f |(0 . D K . Prop
*96 111. b : R“a C a . D . (a 1 R) x „ = a 1 R„
Dem.
h . *01-502 .Dh.a‘]/£G(a'| /{),„
b . *00*22 . *01-54 .Db:. Hp. D : Pe Pot ‘R ,xt a. xPy .0 .yea:
[*:)-.-1 .Fact] 3 : ft« Pot‘ft. .* (a 1 ft) <j . //ft.-. 3 . y (a 1 ft) z :
[*91511 ] 0: Pe Pot ‘ft . a 1 ft C (a 1 ft),... 3 . (a ] ft) | ft G (a ] ft),„
1-.<1). (2). *91-378.3 I-:. Hp . 3 : ft * Pofft. 3 . a] ft C(a1 ft)^ :
r*+l 52] ^ : 0 1 C (o 1 ft) w :
[*»C I1] 3 : a"| ft|« = («1 ft)p„ :. 3 I-. Prop
*96 112. (• : a C D‘ft . ft“a C a . 3 . (a 1 ft), = a 1 ft.
Dem.
( 1 )
(»)
(2)
(D
( 2 )
*96121.
*96122.
*9613.
h . *35 02 . *37 4 . D I-: Hp. D . C*(a 1 ft) - a w ft“a
[*22 02]
f*-.0-5] 3 . / f C‘‘(a 1 ft) = a 17
K *50-53. 3 H . a"] / f- C‘ft = (a n C'R) ] /
h. (2). *22021 . 3h:Hp.D.a1/fC‘ft = a-]7
(-.*91-54. 3l-:(a1ft)* = (a‘1ft) po ia/(‘ C‘(a ] ft)
h . *91-54. *35-42.3 I-: a 1 ft* = a 1 ft M c; a 1 /1-C‘ft
l-.(l). (3). (4). (5). *96111.31-. Prop
( 1 )
( 2 )
( 3 )
(4)
( 5 )
(- :ft“aCa.3.(ft[-a) po =ft M Ca [Proof as in *96111]
(-: a C ([‘ft . ft“a C a . 3 . (ftf- 8)*= ft„f a [Proof ns in *96"112]
(-. (ft»‘x) 1 = |(/e„‘. I )1.ft| po [*96111 . *90 163]
SECTION E]
ON THE POSTERITY OF A TERM
61 J
*96*131. I- : * e D‘R . D ,{R**x) 1 R* = \(R*‘x) ] /*]* [*96112 . *90163]
*96-14. :x € C‘R . D . R*‘ x = l*x u*R ih> *. x [*91 *54 . *32 33]
*96141. I- . C\a 1 1%) = R*“ a
Dem.
I-. *35-61 . *37-4 . *9014.31. C‘(a 1 ft*) = (a n C‘R) u ft*“ a
[*90-331] = ft*“a .31. Prop
*96142. I-. C‘(a'|ii |10 ) = (a r. D‘ft) v R,„“a [*3561 . *37 4 . *91-504]
*96 143. I-. C‘(a -] R 1H> ) = ft*“( a n D‘ft)
Rem.
h . *37-261 . *91-504.31. ft 11( ,“a = ft llo “( a r> D‘R) (1)
1.(1). *91-546 . *96 142.31. Prop
*96 144. 1 : a « CI‘ ft C ft*“(« n D‘ft) . D . C‘(a 1 ft,.,) - ft*‘‘a
Dem.
1. *22 62 . Z> 1: Hp . D . ft*‘«(a « D‘ft) = (a n (3‘ft ) u ft*“( a « D‘ft)
[*91-546] = (a « Cl'ft) u (a « D‘R) u ft 1K) “(a n D ‘R)
[*37-261 .*91 -504] = (a n C‘ft) u R„“<x
[*91-544] - R„“o
1.(1). *96143.31. Prop
*96 16. I-. D‘((ft*‘x) 1 ft) . ft*‘x « D‘ft . d‘((ft*‘x) ] R) =%„‘x
Dem.
1. *35-61.31. D'^ft - *'*) 1 ft) = R+'x n D‘R
1. *37-4 .31. CI*((ft*‘x) 1ft) -
[*91-74] -*R^‘x
1. (1). (2) . D 1-. Prop
*96 161. 1: x e D‘R . D . C‘((ft*‘x) ] .ft) = ft*‘x
Dem.
I-. *9614 . D h : Hp . D .%*<* « D‘/* = i*x w « D‘R) •
[*22*63] D . fef m n D‘tf) u^o** =
[*9614] ~*R* € x
H. (1). *9615. DK Prop
*96-152. . R^^R^x = [*9017]
*96153. h . = R v **R 9 ‘b = [*91574]
*96154. H . i2 # ) [*96141 152]
( 1 )
( 1 )
( 2 )
( 1 )
39—2
012
* 96155 .
Pew
PROLEGOMENA TO CARDINAL ARITHMETIC [PART I
H . D ‘|<^*‘*>1 7 ? J =*Ti*‘* « I>‘ It • U «|cX ^)1 R vo\
*96 156 .
Pern.
h . *35-61 . *!>1 104 . D I-. 1>‘!<K»‘*> 1 R,„l = l>‘/f
h . *37 4 . 3 h . 1
(*!IG'153)
h . (I). (2). D H . Prop
t-. 1{ ,-.\ = <i' cr ' U*/?) ««!•**
( 1 )
( 2 )
K •!>(>• 155. D
h . c* :<i- (/?»‘x n d* «>«
[*!ll 54J .HVnfKn I >*/<)« (#,„'*" D‘A)u R,„‘x
[*22 02.*331(U] = <|V a 1>‘«) « H,J* ■ 3 I" • I’rop
*96167. h:/ ( I)‘/iO.C|l«,V)1'U=V l*!)615014]
h : ,r~« ])‘/f . D . (/f*‘x) 1 = A
* 96158 .
Pan,
h . *9 1 504 .DHsHp.D.u^cD*/?^.
[*33-4]
h.(l).*9(»155.DK Prop
D . 7* 4 * - A
( 1 )
*96 169 . h : 51 ! </f*‘x)1 D. C«|(«*‘-r)1 /f,„| = li+'x [*!»G 157 158]
H . (rt* 4 *) 1 ^ ^ C T?* 4 **
* 9616 .
Pern.
* 962 .
Dew.
* 9621 .
Deni.
h . *351 . D H : y |(7f*‘.r) 1 7?) * . a . y « 7f* 4 * • •
[*90 16.*4 71 ] a . y « 7f* 4 .r. yRz . * € 7?* 4 x .
[*36-13] s . y (R [ R*x) z : 3 H . Prop
h : /? € 1 —» Cls . D . (7?* 4 .r)1/? =
V . *72-55 . D Y : Hp. D . <rt*‘.r) 1 7< « R \ R^R+'x
[*91-74] = R r A*/* :Dh - Pr °P
h . ^ e i _> Cls . xBR . D . (7?* 4 *) 1* = R r**‘ 1
h . *9614 . D h : H p. D . 7* f/F*‘* = i? T ® ^ f* TV* (1)
1-. *35 64 . *93 1.3 h : Hp . D . (l‘(7* [ l 4 *) = A .
[*33241] D.Rt£x=A •- ( 2 >
H . (1). (2). Dh:Hp.D.7*r^*‘* = ^*Po‘*
[*96-2]
SECTION E]
ON THE POSTERITY OF A TERM
613
C\s.~(xRx).D.{R *)1 RGJ
*96-22. H : R e 1
Dem.
H . *31*11 . D I- : a:Qy . yity . D . o;Qy . yity . yQx .
[*1024.*34*1] D.a:Q|iJ|Qa; (1)
I- . (1) . *92132 .DH:i*el-»Cls.D:<3e Potid *R . xQy . ytfy . D . ar/fc: :
[*10*11'21'23*35.*91*55]
[Transp]
[*13196]
[*32-181 .*35-1]
D
D
D
xR^y . y/iy ■ D . :
(xllx) . xR m y . D . ~ (yRy) :
~ (ar/ta;) . xR#y . yRz , D . y ^ 2 :
~ (xRx) . D . (R#‘x) *] RG JD h . Prop
*96 23. H : 7i e 1 —* Cls . x&K . D . 7„‘x = A . («»‘x) 1G J
Dem.
I- . *3111 . D I- : xQy . yTy . D . xQy . yTy . yQx .
[*.341] D . xQ | T | Qx ( 1 )
l-.(l). *92132.3
I- R < 1 -» Cls . D : Q, T( Potid ‘ft . xQy . yTy . D . xTx :
[*91-271] D : Q « Potid'ft . Te Pot ‘R.xQy . yTy .3.xeCPft
[*1111-3-35 r,4.*91-55.(*9105)] D : y ( ft*‘.r . yft^y . D . x « d ‘R :
[Transp.*93-1] D : xftft . D . ~ (y « ft*‘.r . y/f.^y) :
[*961. *1011-21] D : xftft . D . 7„‘x -A (2)
K (2). *90104.3 1-. Prop
Cls. C*‘ft = ft,“ft‘ft . O . ft,*, G J
*96-24. I- :K. 1
h . *37105 . D I-:. Hp . D :y«C‘ft . 3 . (gx) .xcli-R . xft*y :
[*91 -504] D : yft^ . D . (g x ) . x.7?ft . xR m y :
[*4'7 .*3218-181] D : yR t „z . 0 . (gx) . xBR . y « *R*‘x. yR,„t.
[*96-23] D . yjz :. D 1-. Prop
*96 25. I-:. R e 1 —♦ Cls . xBR . xR+y : yR % z . v . zR+y : D . xR„z
Dem.
H. *90-17. D I-: xR m y . yR+z . D . xR+z
K *92-31 .*91-76. D
h Hp . D : xR+y . zR+y .3: xR+z . v . zR^x
h . *91-604 . *931 . Z> H : . D . ~ (*i? |(0 .r)
H . (2) . (3). D I- s. Hp . D : xR % y . zR+y . D . xR+z
h . (1) . (4) . D 1-. Prop
( 1 )
( 2 )
(3)
(4)
PROLEGOMENA TO CARDINAL ARITHMETIC
[TART II
The following propositions lead up to *00*32, i.e.
f-: ft € 1 —¥ 1 . .rft*y. 3 . ft* 4 .r v R+‘x = ft* 4 y v ft* 4 y.
which is a proposition used in the following number (#07).
*96*3*301*302*303 are also frequently used elsewhere.
*96 3 h : xR # y . D .V* 4 y C R+*x f*9017]
*96301. H :.#ft*y . 3. 7? # ‘.rC7?*‘y [*90*17]
*96302 h:.ft«Cls-> 1 . xR^y . xR«z . 3 : yft*r . v . zR m y 1*92*311]
*96 303. H R e CIs —► I . xR+y . xR*~ . y + z . 3 : yR x *z . v . zR l<t y
(*96*302 . *91*542]
*96 31 h : ft e CIs -► 1 . xR # y . 3 . ft* 4 .r C ft* 4 y v ft* 4 y [*96*302]
*96 311. H : ft e I -+ CIs. xR+y . 3 ."ft* 4 // Cft*V v, ft* 4 ./* [*92*31 ]
*96 32. I*: ft a 1 —► 1 . .rft*y . 3 . ft#*/ ft* 4 .r = ft*'// v ft* 4 y
Dew.
h . *96*301 *31 .Dhftf CIs -» 1 . xR+y . 3 . ft, 4 .** sj*R m *x C ft* 4 y w ft* 4 // (1)
I- . *96*3*311 . 3 h : ft e I —♦ CIs . xR+y . 3 . ft, 4 // u *R+‘y C ft, 4 .** u ft* 4 .* (2)
K(l).(2). 3 K Prop
*96 33. h : ft c CIs -* 1 . xRr . 3 . ft* 4 x = i*x
De.w.
K*71*171.31*Hp.3sr-x.«ftw.3 x .^.w-a; (1)
I- .(1).*13*15. *90 112 3h:*ft*y.3.ye.r (2)
h . *9012. 3hHp.3.xV (3)
h . (2). (3). 3 h Hp. 3 : xft*y .a .year:. 3 I*. Prop.
<—
*96 331. h : ft eCIs -* 1 . xRy . y/fr*. 3 . ft* 4 * = i 4 j* v t 4 //
Dew.
4 —
1- .*90151*162.
3h:Hp.3.f 4 xui 4 yCft* 4 a*
a)
h.*71*171 .
3 I-:. Hp. 3 : z — x . zRw . 3 r , tf . w * y.
[*51*232]
3,, IP . we l‘x v t*y
( 2 )
h.*71*171 .
3 h :. Hp . 3 :z = y.zRw. 3 rir . w = x.
[*51*232]
3 ''".luc^xv l‘y
(3)
h.(2).(3).
3 1-:. Hp. 3 :zei‘xu i*y . zRiv. 3, (IC . wei‘x w i 4 y
(4)
h . *5116 .
3 H . .r e i‘x v l 4 y
<5)
h . (4) . (5) . *90 112.3 h :. Hp . 3 : xR m z .3 . z e l‘x v i*y
( 6 )
h.(l).(6).3h.
Prop
This process of proof can obviously be extended to any finite cycle of
terms.
SECTION E]
ON THE POSTERITY OF A TERM
615
*96 34. h : R e Cls —» 1 . 3 . R I>0 "S {zR,„z) C t (zR^z)
Dent.
I-. *31'11 . *341 . 3 h : zR po z . zRw . 3 . wR \ /2 IJO | Riv (1)
I- • (1) . *92113 . Dh:. Hp . 3 : zR„z . zRw . 3 . ,vR,„w :
O 20 ' 3 ] 3 : zj z (zR„z). zRw .D.wei (zR,„z ):
[*37171] 3 : R“S (zR„ e ) C J (zR,„z) :
[*91-71-53] 3 : R„“2 (zR^ z) C J (z R IHt z ):. 3 I-. Prop
*96 341. h : R e Cls -> 1. 3 . R„“I„‘x C I H ‘x
Dem.
I- . *37-21 . (*96 01) .31-. R,„“I k ‘x C R^".R m ‘x r> R^'i (zR„z)
[*90163.*91-002] C*R„‘x n R, n “2 (zR^z) (1)
. (1) . *96 34 . 3l-:Hp.D. «„“/*•* C «.<* « 2 (zR^z)
[(*96 01)] C I K ‘x : 3 1-. Prop
*96 342. h : iZ e Cls —>1.3. R„“I n ‘x C /„‘x [*96 341 . *91-71]
*96-36. I- :. R e Cls —► 1.3: ~{wR,^,w ). zR,„w . 3 . ^(zR^z)
[*96-34. Transp]
*96 351. h : R e Cl« -» 1.3 . R,„“J l: ‘x « *R m ‘x C J„‘x
Dem.
h . *96*35 . Fact. *96*101 . D
h Hp . D : w e J H *x . zR XKt w . z c R m ‘x .D.z c J K *x :. D h . Prop
*96-352. H : R e Cls -* 1 . D . R m **J R *x r> 5v* C J R ‘x [*91*543 . *96 351]
The following propositions are lemmas for *96 45*47.
*96 4. 1- : R c Cls -* 1 . S, Tc Pot l R . ySy . yTz . D . zSz
Dem.
h .*31-11 . D H : Hp . D . zT\S\Tz .
[*92*133] D.sSssDh. Prop
*96*401. h : R e Cls —► 1 . S, T e Pot * R . ySy . yTz . y Rw . zRw . D . wSw . wTw
Dem.
h . *31*11 . D h : Hp . D . wRz . zTy . ySy . yTz . zRw .
[*34*1*2] D . {Cnv‘<7*| R) | 51(7*1 R)} w (1)
H. *91*282. D hs Hp . D . T\R c Pot‘R (2)
h . (1) . (2). *92*133 . D h s Hp . D . u;5u/ (3)
h • *31*11 . D H : Hp . D. wRy • yTz . zRw .
[*341] D.w/.RIT’l.ftttf.
[*91*351.*92*133] D. k/7W (4)
I-. (3) . (4) . 3 h . Prop
016
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*96 402. h : ft e CIs —► 1 . T e Pot* ft . y fty . yTz . yRw . zRw .O.y — w.y = z
Dent.
h. *71171. D I-: Hp. 3 .y = (1)
h . *96*4 . *91-351 .DhiHp.D. zRz .
[*71171] 0.z = w (2)
H.(l).(2).3h.Prop
*96 403. bsRe CIs —♦ 1 . S. Te Pot* ft. yS \fty . yTz . yRw . zRw . 3 .
wSy . . y = z
Dent.
3I-: Hp. D . wit | <S | fty.
O.wSy (1)
3h: Hp . z8\Rz .
0.wR\S\Rz.
3.
h. *31*11 .
[*92133)
h. *96 4. *91 3+3.
[•3M1]
[*92 133]
C2)
<•*<>
P .(1>.(2). *92 101 .*71171 .Dh: Hp.D.»/ = :
h . (1). (2). (3) . 3 H . Prop
*9641. b : ft c Pis —♦ 1 . S, T c Pot*ft . ySy .yTz . yRw. zRw . 3 . y — r
Deni.
b . *91-264 304 . 3 h . Pot*ft - f‘ft u| ft**Pot*ft .
|*51 -236) D h :. £« Pot* ft. = : S « ft. v. ($|S'). S' e Pot ‘ft. S - S' | ft (1)
b . *96-402 . 3
I- S = ft . 3 : ft « CIs —» I . 7’« Pot ‘ft. //«S// . yTV . yftit*. *ftw (2)
I- . *96-403 . 3
I- <a«'). S' € Pot*ft . S - S' | ft. 3 s
ft e CIs —► 1.7’« Pot*ft . //fty. yTz . yftw . sftw ,D.y = z (3)
h . ( 1). (2) . (3). 3 I -Se Pot ‘ft. 3 :
ft e CIs—» 1 . Tc Pot*ft .ySy.yTz,yRw. zRtu.D .y = z :. 31- .Prop
*96*42. V i Re CIs —» I . yRw. zRw. yR x *>z • 3 - zR x ^z
Deni.
. H . *31 11 . 3 b : Hp . 3 . wRy. yR x ^z .
[*92111] 3 .wR+z.
[Hp.*34 l ] 3 . zR | ft <*2 .
[*91-52] 3 . zR XHi z : 3 H . Prop
*96 421. h :. ft t CIs 1 . y. r c ft*‘x. yftw. zRxo . y + z . 3 : yft llo y. v . rft,^
Dem.
I-. *96-303 .Dh. Hp . 3 : yftpo* • v . zR x *y (1)
b . *96-42 . 3 I-: Hp . yR x «,z . 3 . ^ft,^ (2)
I-. *96*42 . 3 H : Hp. zR x<t y . 3 . yftp©y (3)
1*. (1). (2) • (3) .31*. Prop
SECTION E]
ON THE POSTERITY OF A TERM
017
*96 431. h : R <r Cis -» 1 . y e J,,‘x . 2 e I„‘x . 3 . yR^z
Dem.
h . *96102 . D h Hp . D : y * * :
096 303] D : yR XH) z . v . zR lh0 y ( 1 )
V . *96-341 .Dhr.Hp.D: *7*^ . D . y e :
[Transp.*96 l02] D : y € . D . ~(zR llo y) ( 2 )
H.(2). Dh:Hp.D—(*/*,*?) (3)
^ • (1) - (3) . DK Prop
*96*432. h : R e Cis —> 1 . y.z e I R x. yRtv . zRw ,D.y=z
Dem.
h . *961 . D : Hp . 3 . ( 3 S, T) . S,T e Pot‘« . ySy. zTz ( 1 )
K *96 303.3 1-:. Hp. D:y = z: v:( 3 tf): Ue Vot‘R tyUt. v . zUy (2)
I" • (1) • (2) . 3 1-:. Hp. 3 :
V “ * : * : (aS. T, U)iS.T,Ut Pot ‘R . ySy. zTz -.yUz.v.zUy (3)
h.*96-41. 3 1-:. Hp . 3 : ( a S, U) . S, UePot'R.ySy. yUz. 3 .y-z ( 4 )
h. *96-41. 3h:.Hp.D:( a r,ir).r.H«Pot*^.2r*.2t^.3.y-2 (5)
h . (4) . (5) .3 1-:: Hp . 3 :.
( 3 -Sf, Z\ U) t S,T, U t Pot ‘R .ySy.zTz:yUz.v.zU,,:0.y = z ( 6 )
1-. (3) . ( 6 ) . 3 1-. Prop
*96'44. h It e Cis —* 1 . y, 2 r R*‘x . yRw.zRw.yk z .3 : i/ e I ,,‘x. v . z e I"‘x
[*96 421 1]
*96-441. I- :. ft « Cis -♦ 1 . y, z ,*R m ‘x. yRw .zRw.y + z. 3 :
Dem. W * 7 "‘ X ' V * • / "‘ X • * f 7 *‘* • v ’ ^ £ / *‘ a: • * f • / "‘ r
h . *96-432 . Transp . (*96*02) . D
I- Hp . D : ze I H *x .D.ye J R *x ly € I R *x . D . * €*/*'*: (1)
I- . (1) . *96*44 . Dh. Hp . D : y e J R x . z c . v . y c I R x . z e J R x (2)
V . *91*502. *96*341.3
h Hp . D : z « 1 ,*x .3 .tv c I H *x : y « I R *x .!>.«/* I n ‘x :
[*96*44] Dzwel^x (3)
h . (2) . (3). 3 h . Prop
*96 442. h : 7d e Cis —* 1 • y, z c J R x . yRw . . 3 . y = z
[*96*44. Transp]
The following proposition (*96*45) is important.
*96 46. h : R e Cis -> 1 . 3 . (J R *x) ] R, (I R *x) ] R e 1 -> 1
[*96*442-432]
*96-461. V R € CU-* 1 : J u ‘x = A . v - I R 'x = A : 3 . (/£*'*) 1 R e 1 -♦ 1
[*96-45-102]
PROLEGOMENA TO CARDINAL ARITHMETIC
| PART II
*96 452. H A t Cls —» 1 . 3 : g ! J K *x . = . x c J l: *x
Dem.
h . * 1024 . Dhjf J h x • 3 • 3 ! J n x (* >
h . *90342 . 3 h : Hp . x c 7 /; ‘x. 3 . A # ‘x C / i; ‘x.
[*96102] D.J l: ‘x= A (2)
h . *96101 . 3 h : a ! J /; ‘x. 3 . a ! /T*‘x.
[*9013] 3.x/f*x (3)
h . (3). (2). Transp. 3 1*: Hp. 3 !«//,‘x. 3 . x e R m *x — I l: *x .
[(*!>6 02)] 3 . x e .7 /; ‘.r (4)
I-. (1). (4). 3 H . Prop
*96 453. H A e Cls —> 1 : x A.^r. v . ( A*‘x ) ] A,*, G .7 : 3 . <A*‘x) 1A e 1 -» 1
Deni.
H . *96 452 . Transp . 3 H : R < Cls 1 . xA 1 K ,x. 3 . J H *x «= A 0 )
h . *96 104 . 3 I-: R * Cls-> 1 . (R+'x )] A |IO G J. 3 . 7,.‘x- A (2)
H.(l). (2). *96451.31*. Prop
*9646. h:/if Cls —► 1 . y, 'J c J R x . A 1 //. Ay c 7,/x . 3 . y = y'
Dem.
H . #92111.3
I- : A € Cls —► 1 . i/ € J, { ‘x . Ii 1 !/ « 7,,‘x. y A |H ,y'. 3 . A‘y c 7,/x . 7f‘yA*y'.
[*90-342] 3 .y'du'x (l)
h . (1). Transp . 3 h : 7i e Cls -* 1 . y,y' c «7«‘x. A‘y c 7 / .*x . 3 .~(yA |K> y') (2)
I- .(2) y * 3 h : cCIs-* 1 .y,y' tJ/i'x. lt'yel^x. 3 .~(y'A 1 ( „y) (3)
h . (2). (3). 3 h : Hp . 3 — (yR l ^y) (y'A |H> y).
[*96*303.Transp] 3 . y = y : 3 h . Prop
*96 461. h : 7f c Cls -► 1 . y c «7 /{ ‘x. 7*‘y € 7 /; ‘x .3 . y = max/J/.T
Dem.
h. *14-21.3h Hp.3:E!A‘y:
[*3013] 3 : A‘y~c J*‘x. = .~{R‘y cJr'x) .
[*71-371.Transp] = .y~e R“J H ‘x (1)
h . (1). *93115 . *96102.3 I-: Hp . 3 . y max i: (J*‘x) (2)
I- . *96-431.3 h Hp . y e . 3 : yR^lVy :
[*91-504] 3:y / €D‘A:
[*71164] 3:E!A‘y':
[*3013] 3 : R*tf*»€j R ‘x. = .~(R‘y e J n ‘x).
[*71-371.Transp] =.y'~cR“J R ‘x (3)
SECTION E]
ON THE POSTERITY OF A TERM
619
H . (3) . *93115 . Dh. Hp -Dry max* (J,<‘x) . D . y e J R ‘x . R'y'^eJ^x .
[*96‘ 10 2] D.y'e •/*** . £‘y' e I R ‘.v .
[*9646] D.y = y' (4)
H - (2) . (4) . *30-31 .DK Prop
*96-462. h : « e Cis -* 1 . y c J n ‘x . 2 e I R ‘x . y/i?<;. 2 ^* 4 , . D .
V = maX|,V " <a? • w = • z « [(/„'*) 1 Ryii'nmK,.'./,.**
h .*96-441102.*71-361. D
h Hp.D.wt . w» = R*y .
[*96-461] I> . y =, maxjjV***. w; = .ft'max*',/,/* (1)
h . *96-45 OhrHp.D.^ {(/*'*) 1 /*)'«; (2)
h.(l).(2).DI-.Prop
The above proposition, since it exhibits y, *, «/ as functions of « and /e,
shows that there is at most one w in R m <x having more than one immediate
predecessor, and that this one has exactly one immediate predecessor in J R x
and one in I R *x. (These results require *96 441, in addition to *96 462.)
Thus we arrive at the following proposition:
*96 47. (-R « CIs -» 1 . y, t , R„‘ x . yRw . xRw . y* z . D : w = lfme.x u ‘J„‘x :
V = >na x„‘J„‘x . * - \{I„‘x) 1 R\‘R , ina.x B , J n ‘x. v .
a - ma x„‘J„‘x . y = {(/„•*) 1 R]‘R‘mnx„‘J„‘x
[*96-441-462]
We still have to prove
R e Cis -* 1 . a ! J, t *x . a ! I R *x . D . ( a y, z,w).y,ze li m ‘x . yRw . zRw . y * z,
or, what comes to the same thing because of *96 441.
Re Cis -* 1 . a ! J R ‘x . a ! I R *x . D . ( a y, z,w). ye J R *x . z e I R *x . yRw . zRio.
This is effected in the following propositions.
*96 472. hzRe Cis -> 1 . a / t «*. a ! I R ‘x . D . ( a y) . y «• . R‘y e I R ‘x
Dem.
V . *901 .Dht.xe J R *x . R“J r *x C . D : «/**y .D.ye J„‘x :
[*96104] D :/*'*= A (1)
h . (1) . Transp . *96452 .Dh:Hp.D. a ! R“J n ‘x - J R *x .
[*71-401] D . ( ay , z).ye J R *x . * = 72‘y . z~eJ R *x .
[*13195] D . ( a y). y € J B ‘x . R*y~eJ B ‘x .
[*96102] D . ( 3 y). y e ,/^‘a;. R‘y e I R *x s D h . Prop
<>•20
PROLEGOMENA TO CARDINAL ARITHMETIC
| PART II
*96 473. b : RcC ls-» 1 .g ! 7,/x.g ! J,/x.3. El max^J^x.ElRUxuix^J^x
[*06 40147 2]
*96 474. h : /^ € CIs -*1.*= rt‘max / /./ l /x . 3 .
E ! !(/ / /x)'| /f|‘w. E ! max,/J,/x . R\*w = max,, 4 ./,/#
/Jr
( 1 )
(2)
(3)
W
H . *71 361 . Dh: Hp.D.( max,/./,/x) Riu .
[*14 21] 3 . E! niaxp*./ ‘x.
[*03’11] 3 . max,//,// e ./,/x.
[(1 ).*!Mj*45] 3 . !(./,/x) *| /f j‘w = max,/-/,/x
K(2). *0311 .D H:Hp.3. max ,/./,/./-<>-* R“J n *x .
(*7 I 371.*30” 13] 3 . /^inax,/./,/.r'vf .//,‘x •
[H|).*00*102] 3 . ••• € /,/.»•.
[*061 .*01 •32] 3 . fr/t^w. w/f* 17tw .
[*341] 3 . (g*). irR lm ,w . wR+z . zR>u .
[*06-342] 3 . (g*). i c /,/x. s/fw .
[*06 43] 3 . E! |(/,/x) *1 R\*w
b . (2). (3) .(4).3 h . Prop
*96 475 I- :. Rc Cls-» I . 3 : E ! //‘max,/ J ,S.r . s . g ! ./„*/. g S I M ‘x
|*06 473-474]
This proposition ami *06 43'47 embody the main results of this number.
*96 48. I -R t CIs-* 1 . S - (/^‘x) ] R . w € R^J.r. D :
W —> « -I
/#• = /f 4 max,/./,/.#•). = . SS a u >« 1 : m =» //‘iniix,//,/.r . = . S*i<; f 2
Deni.
h . *06 13 . *33 41 . 3 H : Hp. 3. g ! S‘w (1)
v
h . *06-47 . 3 H Hp. 3 : (gy, z ). . // + z . 3 . = //‘max,//,// :
—► v
I (I ).*52'41J 3 : 8*w ~~ t 1.3 . u; = //‘max,/,/,/x (2)
w —►
h . *96-474-102 . 3Hs. Hp.3:w = R*iim x R € J R a x . 3 . S‘w~c 1 (3)
v —►
b . (2) . (3). Transp . 3 I-Hp . 3 i~(w = //‘max,/./,//).^.^€ 1 (4)
h . (2). *52 4 . *34101.3 b :. Hp. 3 : S‘we 2.3 . w— //‘max,, 4 ./,, 4 # (o)
h . *06-474102 . 3 b :. Hp. 3 : u> = //‘max,, 4 ,/,, 4 # . 3 .
E ! {(■/,/*) 1 R\‘w . E ! !</,/x) 1 . i € {iJ K ‘x) R\‘w » l l \{I n*x) *\ R}‘w = S‘w •
[*96102.*54 101] 3 . SHoe 2 (6)
h . (5). (6). 3 b :. Hp. 3 : w = R‘nvix R € J n *x . = . S'w € 2 (7)
h. (4). (7). 3 h. Prop
In the above proposition we write u ~(w = R i m*x R ‘J n i x)’' rather than
v
“ w ^ R l m&x u l J R x’' because the latter implies the existence of fi'ma x R g J R x.
SECTION E]
ON THE POSTERITY OF A TERM
621
*96 49. h :: R e Cls — 1 . D (/?»«,) 1 R e 1 -» 1 . = s I,‘ x _ A . v . J„‘x = A
Dem.
h - *9648 . Transp .Dh:. Hp . S = (R**x) *] R . D :
w c R^x . w = R‘m&x R ‘J R ‘x . = . w e R lto ‘x . S‘w ^el :
[*9615.*9152] D : w -» R € mn.x R € J R € x . = :
[*14-204] D : E ! R‘max R ‘J R ‘x . = . (gu») . weG‘£ r .~S‘t 0 ~« 1 :
[*96 - 475.*71-1] D : g ! . g ! /*«* .s.S~c 1 ->Cls .
[*71-261103] (1)
I- . (1) . Transp .Dh. Prop
*96 491. h R c 1 —* 1 . D : /,,‘x = A . v . J n *x = A
Dem.
H. *96-49. Dh.Hp.xt D*R . D : I n ‘x = A . v . .///a: = A (1)
h . *91-54-504 . D h : Hp .x^e T)‘R . D . R+‘x = i‘x rs C‘R . — ( xR po x ) .
[*96-1] D . I R *x — A (2)
H • (1) • (2) . D h . Prop
*96 492. h c 1 —> 1 .xe I)‘R . D :
~ (xR 0 x) . = . / n ‘x *= A : xR lto x . s . J*** = A
Dem.
H .*961101 . D
h : I R *x - A . D . ~(xR po x) : a: * D‘ii . ~(ari* |(0 a:) . D . g !
h . (1) . *96-491 .Dh. Hp . D : ~( x R iioX ) . = . A
Similarly h Hp. D : x/2 |ic a:. = . = A
h . (2) . (3) . D 1- . Prop
The above proposition is used in *122 52.
The following propositions, not being needed in the sequel, are merely
stated :
( 1 )
( 2 )
( 3 )
h : R e Cls-* 1 . g ! J R ‘x . g ! I R ‘x . D . I R *x rs R“J r *x e 1 . J R <x * R“I r <x * 1
h : R € Cls-* 1 . J R *x = A . D . (R*‘x) 1 / € Pot'I^A*'*) 1 /2)
*96 6. \-:Rel->l.xe D‘R . D . ~R„‘R‘x «= 7?*'* -~R 99 ‘x ^ t‘*
Dem.
h . *717 . D I-Hp. D : y « ’R vo , R‘x . = . yR„ j .
[*92'11] = . yR*x. x tD'R .
[Hp.*4-71] = . t/R+x :
[*3218.*96-14 D : R„‘R‘x = ~R m ‘x=~R vo ‘x u t'*:. D . Prop
*96-601. I-: A e 1 -»1 . <r « d‘.R. D ,*R vo ‘R‘x =.%*x=*R po ‘x « i‘ar
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*96 502. b z R € l —»Cls. xRy. 3 . R*y = R+x v t*y
Dem.
h . *9031 . 31-:: Hp. 3 zR+y • = : zR+(R*y) .v .2 = y:s Dr . Prop
*96 51. I-: R e 1 -¥ 1 . a C R+“H‘R . a C R,J‘a . 3 . a = A
Dem.
h . *37 1 05. 3 V :. Hp. 3 : y € a. 3 y . (gx). a* « a . .
[*3218] D y . g ! a a R x Jy
[*1418-21] D : 5‘a c a. 3 . g ! a R‘x .
(*90-5] D . g ! a a /V* :
| Transp] D : a a /V r = A . x/f»/ . 3 . ~ e a .
[*5121 1] ^ • a u «‘jf) = A •
[ *90*502] D.aA/Vif-A (l)
H . *9l\50+ .Dh:.aC /?,*“« .D:aC <I*/f:
[#93104] D : . 3 . a aA (2)
h . ( 1 ).(2). *90112. 3 H :. Hp . 3 : .r * H‘R . . 3 . a a /?*«'/ - A .
[*9013] ^ 3.y~«a:
[*37*105] 3!V^/«A fl -A (3)
h. *22-621. 3h : Hp.3.a= 7{+ tl R*Rr\a (4)
h. (3). (4). 3 K Prop
*96 52. : Re\-+ \ . a C RJ'B'R . g *•«• 3. g * min
Dem.
s/
I-. *96 51. Transp. 3 h : Hp . 3 . g ! a - R 0 “a (1)
K(l). *93111 . Ob. Prop
This proposition is used in *122*23.
*97. ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES
Summary of* 97.
In this number, we consider not only the posterity of a term, but the
ancestry and posterity together. i.e. ~R*‘x v R#*x. We put
R l x = R* x u (i‘x n C‘R) kj*R*x Df.
^Thus the whole family of a term. i.e. its ancestry and posterity together,
is R*'x. The most important case here is when Re 1 1; in this case families
are mutually exclusive, i.e. we have
I- s R € 1 i . D * Cls ex 9 excl.
In case^c 1-^ and y belongs to a family which has a beginning, i.e.
in case g ! R+*y B‘R, the whole family of y consists of the posterity of the
beginning, i.e. we have
whence I": 12 e 1 —»1 . xBR. x%. D . *R.‘y ~
^^
*97-21. h : R el-» 1 . D . R#“s*gcn‘R = R+“B*R
When Rel->1, the relation of gen -R to *R+“~B‘R may be pictured as
the relation of rows to columns. E.g. let the field of R consist of the dots
• . /T\. . B 1 " .
A I-
>1
1
t
i"
in the accompanying rectangle, and let each dot have the relation R to the
dot below iL Then the top row is &R, the second row is (I‘R-<1‘R*, the
third is Q R t -(1‘R*, and so on; thus the rows are the generations of R.
Again, if x is any dot in the top row, the column beginning with a: is *R#*x,
and if y is any member of this column, the column is *R*‘y. Thus the columns
are the families of R. It will be seen that in the case represented by the
above figure, every family consists of a selection from the generations, and
every generation consists of a selection from the families, i.e.
R**R*R C D“e A 'gen ‘R . gen‘R C D“es*R^ g ‘B‘R.
021
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
The circumstances under which this occurs will be considered in the
present number (*973—*47). Tin* results are summed up in *97*47.
The remaining propositions (*97 5—58) are concerned with circular
families of one-one relations. If Re 1 —► 1, /f*‘.r is a circular family if
In that case, we have xR x ^/. D.///?,«*»•; moreover there-is a definite
power of R. say P. such that every member of the family of a-has the relation
/' lo itself (*97*54). (The same will hold, of course, of all powers of P.) The
families of a 1 —> 1 are all either circular or open, t.e. we have (#97*55) either
<—i . ,
// € R+x • • ///e |0 y. or y € R+*x. D y .~(yR t „y). The ^-shaped families con¬
sidered in *9G are not. possible for a 1 —* 1, since in such families the term at
iIn- junction of the tail and the circle has two predecessors. The family of
anv member of .v‘gcn 4 /? must be open (*97*57). The family of a member of
/>*<!“Pot‘/f nerd not be closed, but cannot have a beginning; if open, it
forms a series of type *<o or + t*>, according as it has or has not an
V/
• nil*. Finite open families are contained in *‘gen‘i? a «‘gen‘ii; families
of type to are contained in s'gerfR n/ZCI^Pot'/i; those of type *to, in
#*gt?n‘/f r\ y>‘( 1 11 Pot*; those of ty|»e*a> + c«> and circular families arc contained
in />‘<I“Pot‘/f ny/tF'Pot'ii. Those of t\*|»e *to + to are distinguished from
circular families by the fact that in the former we do not have xR tHl x, while
in the latter we do have this.
In addition to the propositions already mentioned, the most useful pro¬
positions of the present number are the following:
* 97 13. h .tv* - V* w ^*‘* r
*97 17. h . /V-r = R** u ^ " ^ik,^ w R^x
*97 5. b : R e CIs -* 1 . xR^x . xR l>0 y. D . yR,^
*97 501. b : « 1 —> CIs . xR llo x . yR %to x . D . xR v ^y
*97 01. R*x = R*x v(i*xn C‘R) yj R*x Df
Observe that “ i*x r\ C‘R” means that x is to be included if it is a member
of C‘R, but not otherwise; for i‘xr\C‘R = i‘x if xeC‘R , and otherwise
t‘x r\ C*R — A.
• Horc the typo is the type of converses of relations of typo u, i.e. tho typo of the
negative integers in order of magnitude, ending with -1, u being the type of the positive
integers in order of magnitude, and therefore *« + w being the type of negative and positive
integers in order of magnitude.
A RELATION INTO FAMILIES
625
SECTION E] ANALYSIS OF THE FIELD OF
*971. 1- :.yeR‘x.= :yRx. v ., J = X , xtC ‘R. v. xRy
Dem.
V . *3218181 . *51*15 . (*97 01) . D
I" :• y eR ‘ x • = • yft* . v . y = X . y C C‘R . v . xRy :
[*13193] =iyRx. v .y = x.xeC‘R. v .xRy:. D f-. Prop
*97101. I - z y eR‘x . = . x e*R‘y
Dem.
t-. *3218181 . *51-15 . (*97 01). D
I-a e R‘y . e : xRy. v . * = y . a: e C‘R . v . yRx :
[*971] =:ye*R‘ x D h . Prop
*9711. h . s‘R“C‘R = C‘R
Dem.
h . *07*1 .*4011 . D
[*33 13 131.*13195] = : y * V‘R . v . y , C‘R . v . y e Q‘R : * *
[*3316] aiytC‘Ri. 3 1-. Prop
*97 111. = .xe*R‘x.= , 3 1*R‘ X
Dem.
h . *971 .Dh.xe . = : ar/te. v . x e C*R :
[*3317] ^ s: xeC'R ^
«! '*' ■’ I ij&r** -
h . ( 1 ) . (2) . Dh. Prop (2)
*97 12. H . A~e R+“C‘R
Dem.
y . *97 111 .*37-63 . 3 t- :ae*R+"C‘R . 3. . 3 > a (1)
H.(l). *24-63.3 1-. Prop ‘ ' ’
*9713. 1-. R„‘x = R % ‘x u R+'x
Note R m is to mean («*). not (fl)*. The latter is unmeaning, since is
never a homogeneous relation, and therefore its square and higher powers
are unmeaning. a 1
Dem.
h • *9012 .DHs y = x.yc C‘R . D . yR^x :
[*5115] C *R^x .
[*9014] D h . i‘x r\ C‘R+ C i£^c ( 1 )
^ • (1)‘* (*9701) .Dh. Prop
a«c w i
40
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
^ y ^^
#9714 I-: R e 1 -► 1 . x/?*y . D . /V* = R *'J [*«6*32 . #9713]
*9715. I-: ft € 1 -> l . r c ftyy . D . 7?*'* = /Vy
Dew.
I-. *9713 . D K Hp . D :xft*y . v . yft*x (1)
H . (1). *9714 .DK Prop
*9716. hsftcl -»1 .D.^'C'/eeCIsex’excl
Dem.
. *9715 .Dh. Hp.D:xe ft*‘y .xr RJs . D, . R m *x — ft*‘y • R** = R * :
(*13 171] Or.R^ymR^z:
[* 10-23] Z> s a ! ft*‘y A H+'z . D. ft*‘y - ft*‘* (1)
H . (1). #1111*3 .*37*53 .0
h:. Hp.Dsa.tfeTv'C^ft.a! a -/9 (2)
h . (2). *9712 . *34 132 . D I-. Prop
*97*17. I-. /V* - /V* - ft*‘x v7? l>0 ‘x - ft 1>0 ‘x u 7F*‘*
Zfe»i.
h . *97*13 . *91*54. D h .7?*‘x- u <i‘x a C‘ft) u ft,*'* (1)
[*91 *504.(*97 01)] - /f |K /* (2)
h . (1). *91 *54 . D H . TV* “ /?,-/* u Vx - ft*‘x w ft^x (3)
h . (2). (3) .DK Prop
^ ^^
*97 18 KC‘(ft[^‘*)=^‘**
Dem.
V . *37*41. D K C‘(ft £ T?x) C/£* (1)
h.*971.*3613. D
h :.x€C‘R . y e ft‘x u ft‘x. D :x(ft £ ft‘*)y. v . y (ft £ ft'*)* :
[*33* 17] 3 : J*. y c C ^‘*) ^ (2)
K (2). *971 . Dhxc C*R . 3 .^C C‘(ft [*£) (3)
I- .*97111 .Trausp. D H: *~€ OR . D . 7*‘x C C‘(ft £ ft‘x) (4)
h. (1). (3). (4). D H. Prop
*97 2. h : xftft . 3 ./V* = ft***
Dem
h . *93104. *9713 . D h : Hp. D . ft*‘* = i‘x u ft*‘x (1)
I-. *93101 . *9012 . D h : Hp. D . * € R* l x (2)
h . (1) . (2) . D h . Prop
SECTION Ej ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES 627
*97 21 . t-:Re 1
Dem,
1 . D . R+“s‘gen‘R = R*“B‘R
* 9722 .
*97 23 .
Dem.
h . *9714-2 . D h Hp . D : xBR . xR m y . D . *R*‘ X = .
[ * 37 ' 62 ] I.JISyt'RS'B'R :
[*93 .36] => : y e s ‘gen‘R . D . %‘y € ^‘WR :
[*37-61] D : 7?*“s‘gen‘7* C ^“WR
H . *97-2 . *93-22 . D H . R C*R*“s‘gen‘R
I" . (1). (2) . D h . Prop
F- : Re 1 —> l . D . R^'B'R v /£*“/>'(! “Pot‘7£ = ^R^'C'R
[*97-21 . *93 37]
I- :: R“C‘R e 0 w 1 . a x,y c C‘7i . D,. y sx -y. v . xifcy . v . y R x
( 1 )
( 2 )
h. *52-4. (*5401). D
f-:s.i?«C^ e 0ul. a ••••a,/3e 4 R“C‘R.D mtll .a-/3..
[*37-63]
[*97-1]
[*4*71]
[* 101 ]
[*1315]
= ::x,ye C*R . D x<y . 7*‘x -= 7i‘y ::
= ::x,ye C*R . D x . y g Rx. v . xe C‘R . * -x.y. x R t .,
• v . y e C‘72 . * - y. v . y7^
- ::x »y< . 3 x . 8» - v.^ai.v. x7£s : =, :
_ • *7?y. v.z-y.v.yRz (1)
D :: x, y € G u 7i. D x y x72x. v . x — x. v . x7*x : = s
x7fy . v . X = y . V . y7?x
, , - ^z.v'-xRy. V ,x = y .v .yRx (2)
F-. *101 . D (- ili.yiOfiij, x . ye C . R . _ r : xRy.v ,x=y.v .yRx:. ~>
X ^ 2 • v -X — * . v . zRx : y7?z . v .y = z . v . zRy :.
3:.xl?* . v.x = *. v.xRx: = ivRz . v.y — *. v.zRu (3)
h .*33132 .Transp. *1314. D J K '
rJj 911 * CR ! Z ~ tC ‘ H '■ ° - V • zltx) .x±z-.~(yRz . v . zRy) . y^z-..
yRz .v.yi.v. ztfy ( 4 )
h . (3) .(4).Dh ::. x.ye C l R . D x y : x7*y . v . x = y . v . yTdx :. D ::
< 5 >
*97 231. F-:. <■ 0 u 1 . = : x e C‘fi . Z>, . C‘R -7?‘ x « i‘ x yj*R‘ x
Dem.
H . *97*23 . *3218 181 . *51 15 . D
hs.WlJeOwl. 5 :a ?e ^.D.^cJxwt‘xu^ (1>
. *33152 . *51 2 . D h : x e C‘72 . D . 7£‘x w i*x \j *R‘x C C‘R (2)
K(l).(2).DKProp
: •
40—2
628 PROLEGOMENA TO CARDINAL ARITHMETIC [PART II
*97-24. b :. /?***C‘/? « Oul.sut OR . 3 X . OR - 7?**x w /F*‘*
Deni.
b. *07*231 .*0014.3
h :. fiV'C* /? € 0 u 1 . = : a- 6 OR . 3,. C*/? = TV* u i *.r v /?*** (1)
b . *00 12 . 3 b : x* OR . 3 . i«r C 7?*‘x (2)
b . (1). (2). *22-62 . DK Prop
^^
*97 241. b :: /?*“(."/? «0 ul.= :. x.yeOR . 3 XiV : x/?*y. v . yR+x
/)em.
b. *07-24. *3218 181 . 3
^^
b /?#“C*/? fOul . == nxeOR . 3, yeOR . =„ : x/?*y . v . yR*x (1)
b . *00' 13.3b:. x/?*y . v . yR+x : 3 . y e C*/? (2)
b . (1) . (2) • *4*73.3b. Prop
*97 242. b :: V^OR e0u 1 . =:. x. ytC‘R . 3 x . y : x = y . v . */? |lo y . v . yR„ 0 x :.
= :.V('‘/^0u 1
[*01-542 . *07 23 . *01*504]
The remaining propositions of this number (except *07*5 ft*.) are concerned
with proving that, under certain hypotheses,
C D*‘c A *gen*/?, i.e. /?* M * f gen‘/? C D M « A *gen*/?,
and gen*/? - # *A C 1 3“« A */?*“^*-
These propositions have the merit of proving the existence of selections
in the eases to which they apply.
*97 3. K7?*|*/?•/?« 1 -> *
Deni.
b. *0012.3
b :. x, y € OR . R+x - R+y • 3 : y c R^x :
[*91-54] 3: y = x. v ,xR l<t y (1)
b . *01504 . 3b: xR x ^y . 3 . y c Cl*/? :
[Transp.*93*101 ] 3 b : y < ~B‘R . 3 . ~(x/? llo y) (2)
b . (1). (2) • 3b:x,ye^*i?.^*‘x = J? # *y.3.x = y (3)
b . (3). *71-55 . *72 12.3b. Prop
*97-301. b . / r « (R*h‘~B‘R
Dem.
b . *7217 . 3 b . / [* OR € 1 -» Cls
b . *0015 . 3 b . I [* ~B‘R G R *
b . *50-5-52.3 b . a*/ I* 2?i? = ZPi?
h . (1). (2) . (3). *8014.3b. Prop
( 1 )
( 2 )
( 3 )
SECTION E] ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES
629
*97-31. h . (B‘R) 1 Cn v‘R# e e*‘R***B‘R . D‘|( B‘R) 1 Cnv'jR*) =~B‘R
Dem.
K *97 3. *85-13 ^*.3
• ---S* (R*)JB‘R . 3 . S | Cnv'B* e *Jr+‘&R
I-. (1) . *97-301 . Dh./ C ~B‘RI Cn v?R m e t*?R 0 “1}‘R .
[*50-61 ] 3 I- . (B'R) 1 Cnv'-tf* r ts‘*R 0 ‘‘B‘R
I-. *35-62 . *33-431.3 h . D‘[(B‘R) -| Cnv'^.l = ~R‘R
h . (2) . (3) . 3 h . Prop
*97 32. I -.B‘Re D ••e A ‘*R 0 ‘ r B‘R [*9731]
*9733. h:.Rel ->1 . a C s‘*R m “0 . 0 C s‘*R m “a . 3 . *R 0 “a - *R m “0
Dem.
h . *9715 . Fact . 3 h Hp . 3 : y ,0 . x t*R m ‘y . 3 .*R 0 ‘y - *R 0 ‘x . y e 0 .
[*3762] 3.5v*«**‘7S
h • (1) • *10 11-21-23 . *40 4. 3 f-Hp . 3 :xe«‘rt»“/9.3 . .*R m ‘xe*R m <‘0 :
[Hp.Syll] 3 : x« a . 3.. *R 0 ‘x e*R 0 “0 :
[*37-61] 3:fl,“.C«,‘‘/3
h . *404.3 I-Hp . 3 : y e 0 .3 . (gx) .xea.ye R 0 ‘x.
[*97-15] 3 • (3*) • x e a . *R m ‘x — *R 0 ‘y.
[*37-62] 3 .*R 0 ‘y e*R m “a
y . (3) . *37-61.3 y : Hp. 3 .*R 0 "0 CR 0 “a
I-. (2) . (4) .31-. Prop
*97 34. H : R e 1 -» 1 .3eDVfi,“« • 3 -*R»“a = *R m “0
Dem.
h . *83 6-62 . 3 h Hp . 3: x « a. 3. . g ! jS n *R 0 ‘x : 0 C s‘*R 0 ‘H
( 1 )
( 2 )
(3>
O)
( 2 )
(3)
(4)
( 1 )
( 2 )
I- . *40 4 . *97101 . 3 y x e a . 3, . g ! R n R 0 ‘x . = . a C s‘R 0 "0
t-. (1). (2) . *97-33.31-. Prop
*97 341. y-.Rtl-tl.0e D “eeCR 0 ,7 B , R . 3 ,*R 0 “0 =*R 0 "’b‘R
[*97-34 B ‘ a R . *97-2]
*97 36. l-:ie«CU-»1.7’e Potid'Ji . ~B‘R C D'T. 3 .
Cnv'ft.ft, [- B‘R) | T\ e eS*R 0 “~B‘R. d‘((«* f- ~B‘R) | T\ = T“l3‘R
Dem.
I-. *97-3 . *92 101.3 y -. Hp. 3 . Cnv*{(fl. f~B‘R) | T\ e 1 —* Cls
< 1 )
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
f- . *35101 .*30-4.3
T)'/. = .{&x).xi?i t R.a=*R 1f! t x.xTij (2)
h .*91 • 58 . 3 b s. Hp. 3 : .# 7// . 3 .»/ e 7f*'.r:
*1312] 3 : a = . xTy . 3 . y € a (3)
H.(2).(3). 3 h Hp . 3 :a [(T^fTf'T?) 7*) y. 3«.„. y e a :
[*23 1.*31131] 3:Cnv‘j(7f*rT?7?) 7’j G c (4)
h . *37 321 . *35-65.31-: Hp. 3. D*[<7?*f 7**7?) 7*1 = 7?*“7?*7? (5)
I- . (1). (4>. (.-»). *80 14. 3 V : Hp. 3 . Cnv‘((tf* [* Tf'T?) | 7*J * < A t4 R»“~B‘R (0)
H . *35-65 . 3 h - 1?‘<R •
[*37-32] 3 A\‘\(XrH‘R)\r] = r**7?7? (7)
K((i).(7).DK Prop
*97 36. h : /? € CIh-» 1.7< Potid‘7?.7?7? C D‘7\ 3 . f“J?R e
[*97-35]
*97 37. 1-: 7? « 1 -> 1 . CI‘7? C 1>*7?. 3. gcn‘7? C D “'ARS'D 1 R
Dem.
h . *9214. 3 I-Hp. 3: Tt Potid ‘R . 3 .7?* 7? C I>‘7’ ( 1 )
h . *93 32 . 3 H :. Hp. 3 : a * gen *7?. s . (gD. 7*< Potid* A . a = fi'li'R (2)
K (1). (2). *97-36. 3h. Prop
*97 38. »-:/?«! —* 1 • <1*7? C D*7? . 3 . JV‘7?7? C l>*‘c A ‘gcii*7?
Dem.
h . *93-36. *40-52 .3 h:Hp.3. s*7?*“77*7? = *‘gcn‘7? (1)
h . (1). *84-43 . *97-37 . *93 25 . *97 16-21 . 3 h . Prop
*974. h:.S'« Pot*7? . 3 .S“Z?£ = A
Dem.
h . *91-31. 3 H : Hp. 3 . < 3 7*). Tc Potid‘7?. 5= 7?| 7*.
[*37-341] 3 • ( 37 *) • 7*€ Potid‘7 ?, 5 **/?*/? — T**./?**/?/?
[*37 2G1-29.*93 101] = A .
[*10-35] D.S“B‘R = A : 3 f-. Prop
*97 401. h :.*€ D‘7? : 5 c Pot* 72 .xSy. 3$. „. ye D‘R : 3: Se Pot* 72. 3 s .;reD‘&
Dem.
h . *3313 . 3 H Hp. 3 : £« Pot* 7? . xSy . 3*, y . (gr). i/Rz . arSy.
[*341.*3313] D s>v .x € D*(S| R ):
[*10-28.*33-13] 3 : *$ c Pot *R . or e D‘S. 3 S . .r € D‘( 6 ’ | R)
1-. (1). *91*373 .DP. Prop
(1)
SECTION E] ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES 631
*97*402. h :. R € Cls —» 1 . * e T>‘R : ( a S) . S e Pot *R . x ~ e D‘S : D .
(a S) . s e Pot‘/e. s‘x €~b (S r
Dem.
H . *97-401 . Transp . D h : Hp. D . (a«S. y) . Se Pot‘i£ . xSy . y ~ e J)‘R .
[*91-271.*3314.*93101] D . ( 3 S.y). Sc Pot‘7* . xSy . y e ~B‘R .
[*71-321] D . (gS). Se Pot‘72 . S‘x€~B‘R .Oh. Prop
*97 403. h : R € Cls— ► 1 . x e B‘R . Te Pot ‘R . ~B ( R «= T“li‘R . D .
(3^) . 5 € Pot'ft . x ~ e D ‘S
Dem. h . *92131 .Dh.Hp.D: xTy . xTz . zRw . D . yflw, .
[* 3314 ] D.y~€~B‘R (1)
h . (1). *11*11 , 3 , 35 . D
h :: Hp O :. xTy : (g*. w) . xT* . zftw O . y ~ c~B‘R :.
[*341 .*33 13] D :. xTy. x e D‘(T\ R) O . y ~ €
[Transp] D s. xTy . y « B‘R O . x~eD‘(T\R) (2)
h . *10-24 . D h : Hp.x~« D*T . D . (gS). 6 Pot'/Z .x~ c D‘S (3)
h . *37 105 O h : Hp . xTy O . y c 7?i* .
[< 2 >] :>.x~el>‘0r|/e) (4 )
h . (4). *10 11-23-35 . *33 13 . D
h : Hp.x«D‘7’0.x~«D < (7 , |/2).
[*91-282] D . (gS). Sc Pot‘7* .x- f D‘S (5)
h . (3) . (5) O h . Prop
*97 41. b : R e Cls—► 1 . x e~B‘R . T e Pot ‘R . IvR = T“lvR . D .
*97 42.
Dem.
*97 43.
[*97-402-403]
(gS) . 5 e Pot‘7e . S‘x e IVR
b:Rel-+l.x€B*R.S,TePot‘R.-B*R = r“'B‘R.S*xe'B‘R.D.S-T
h . *37 6 . D h : Hp . D . (gy) . y e~B l R . S‘x = T l y (1)
h . *37-62 .(l).Dh:Hp.Di‘xf S‘*B‘R n T**B*R .
O 93 3 ] D . S‘x c tmn /t ‘<l‘S * tmn J{ ‘<1‘T.
[*93 24.Transp] D . 5 = T O h . Prop
h:i£*l-*l . r € PotVft.2?ie=2^‘^/eO.Z?‘72CD‘T
jDem. h . *97-42 . D
h Hp .xeB‘R O : Sc Pot *R . S*xc~B‘R . D . T‘xe~B‘R :
[*1011-21-23] D : (gS) . .Sc Pot'i* . S‘x€~B‘R . D . T‘xell‘R :
[*97-41] D : 2*x e "~B‘R s
[*14-21] D : E ! T*x :
[*33-44] Ds*6 D‘T sO h . Prop
032
PROLEGOMENA TO CARDINAL ARITHMETIC
[PART II
*97 44. P : 7? e 1 —* 1 . «S\ 7’ c Pot*/? . B'R = T“B l R . 3 ! S“/P/?.D . B‘R C D *S
Dem.
P. *91-45. D I-:. Hp.D:(atf): Ue Potid*/? :S = 04 T.v.T = U\S (1)
P . *97 4 . DP:. Hp. D : 0 € Pot‘7? . S = 0|7*. D. 5**3*/? = A :
|*91*23] D : 0cPotid*/?.S«0 7’. 3 !.?“/?7? . D . U = I[C‘R.
[*50 03.*91-271] D . .S = 7*.
[*97-43] D.MCD'S (2)
P . *91-34. D P Hp. D : 0e Potid*/? ,T=U S. D . 7* =* £ 1 0.
[*34 30] D . D‘7CD‘S.
[*97-43] D./P/<C1>‘S (3)
P . (1). (2) . (3). Df- . Prop
*97 45. P: R < 1 -> 1 ./P7? < gen*72 . D . gen*/? - PA C 1>‘*< A */?*“7P7?
Dem.
P . *97-44 . *1011-23-35 . *93 32 . D
P : R c 1 -> 1 TP /?1 gen*/? . .S’ € Pol*/? . 3 ! S“&R . D . B‘R C1 >*5.
[*97-30] D. 5**/P/? € D**€ A ‘/5y*U*/? (1)
P .(1).*13-12 . D
P : /?c 1 ->1 ."3*7? egen*/?. St Pot*/? . a = S“B f R . 3 ! a .
D.atD **c A */?***/**/? (2)
P . (2). *1011-23-35 . *93-32 . D
P: 7?e l-> 1 .If 1 lit gen*/?. a € gen‘7? . 3 ! a. D. a € D*‘« A */?***/?*/? (3)
P. (3). *53-52. DP. Prop
*9746. P : 7? € 1 -»1 .TP 7? € gen*/?. D . /?*“/?/? C D‘* fA ‘(gen‘7? - PA)
Z>em.
P . *93-30 . *40-52 .DP: Hp . D . «*/?*“ JP 7? = s‘gen‘7?
[*53 18] = P(gen*7?-f‘A) (1)
P . ( 1 ) . *84-43 . *97-4516-21 . *93 25 .DP. Prop
*97 47. P : 7? e 1 -> 1 .3*7? * gen*7? upA . D .
gen*7? -t‘AC I)‘*e A ‘7?**‘/P/? .7?*“/?*/? C D“ eA ‘(gen‘7? - PA)
Dem.
P . *93-32 .DP: TP7? = A . D . gen‘7? = PA (1)
P . (1) . *37-29 . D P i~B‘R = A . D . gen‘7? - PA = A . 7?*“7P7? = A .
[*2412] 3 • gen*/? — i*A C D* *€**/?**‘TP/? .
%r&R C D“eA*(gen*7? - PA) (2)
633
SECTION E] ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES
I-. *24*3 . Fact. D
I- : R e 1 -> 1 . a IB‘R .~B‘R = A.D..R e l—►l.g! ~B‘R. a‘R C D ‘R . (3 )
[*9341] D.A~egen ‘R.
[*51-222] O . gea'R — t‘A = gen‘R .
[(3).*97-37-38] D . gen ‘R - t‘A C .
*R m ,r B‘R C D“ ei ‘(gen ‘R - t‘A) (4)
(•. (2). (4) . D I- : R e 1 -»1 ,~B‘R = A . D . gen'R — i‘A C D“eS*R+“li‘R .
K * 97 - 45-450 C D^(gen A) (5)
—> « ^ ^
h : R e 1 -* 1 . B*R € gen*R . D . gen 4 7* - 1 *A C D 44 ^ 4 #*"#'^ .
h . (5) . (6) . D K Prop «*“** C " ‘‘ A > (6 >
*975. t- : Re Cls—> 1 . xR IK> x . x R^y . D . y72 1>0 a:
Dem.
I- . *92-111 . D P Re Cls—► 1 . xR^x . xRy . D : yR^x :
O 91 ' 54 ] Ozy-x.v .yR^xz
f H P] 3 : yR lto *
I- . *101 .*341 .Dh:.72eCls-*l .xR xto x.PeVot*Rz
xPy . D y . yR lHt x : xP i Rz : D : (gy) . yi^a;. y7*s :
[*92-111] DzzR+xz
09154] 2 : z = x. v . zR llo x:
t H P] 3 •* (2)
I-. (1) . (2) . *91171 *Df*:/2e Cls—> 1 . xR lHJ x . P e Pot *R . xPy . D . yR lto x :
[091 05)] 3 P : R * Cls—► 1 . xR lto x . xR^y . D . y72 1(0 a:: D h . Prop
*97 601. h : R e 1 -*Cls . ar/? 1KJ x. y7* I>0 a:. D . xR lto y [Proof as in *97o]
d)
*97 61. h/2el-»l. . D . 7i* 4 * = R+*x - 7** 4 a: = ft* 4 a: ^ R**x
O97-5-501-17]
*9762. h : Re 1->1 .xR^x .xR lH) y .D .*R m ‘x ='*R m *xf\ll m *y [*97-5-501-51 14]
*9763. biRel-+l.P € Pot *R . xPx . y e 7?* 4 a:. D . yPy [*92132133]
*97 64. P : A < 1 -»1. xR^x . D . (gP) . /> € Pot 4 R . 7^ f = 7 P TO‘*
[*97-53]
*97 66.
Dem.
D h s. Hp . arA^a; - D : y e A* 4 ar. D v . yR^y
h . *97-53.
P • jjT^ • Transp .Dh Hp . ~ (xR^x). x e R*y . 3 . ~ (yR^y)
( 1 )
( 2 )
h . (2) . *97101. D h Hp . ~ (xR^x) .D:ye R+*x . D v . ~ (yTipoy) (3)
h . (1). (3). D h . Prop
634
PROLEGOMENA TO CARDINAL ARITHMETIC
—» <->
[PART II
*97 56.
1-:. /?€ 1—>1 ,j-e B*J{. 0 z yc 7?*‘.r . D,. ~ < y/? 1(0< y)
[*!M>*23*I . *5)7-55]
*97 57.
y /?< 1 —> 1 . -re s‘gen‘7? . D : »/ e /?**.»• . D v . ~ (y/? l>0 y)
[*5)7*21-56]
*97 58.
h :. 7? c 1 —►CIs . D : .r € s‘gen‘7? . D . 7?*‘.r C s‘gen‘7? :
<-»
.*• € y/G“Pot‘7? . D . /?**.# C ;/<[“
Pot‘7?
hem.
y . *5)3*412 . DK /?“//< I “l\»t‘/? C j>‘CI“Pot‘7? (1)
[*OO l01.*93-273.*37 265] D y . /t'Vgvn'ACj'gi'n'/? (2)
h . *5)3*33. #4013-38-43 . D f*: // « 1 —>Cls. D . }?“«<gcn‘J? C s‘gen‘7?. (3)
[#5)0*101.*03*271.*37-265] D . /?“/>•< I “Pot‘7? C />‘<l“Pot‘/? (4)
h.(1).(2).(3).(4). *00 22 . *40-5-52 . D
h :/?« 1 —►< ’!*. D . .v*/? # ‘Vgen‘7? C s‘gen‘7? .
s‘/?*“;>'<!“Pot‘7? C />‘(T“Pot‘/? : D h . Prop
It. follows from this proposition that every family is either wholly contained
in the generations of It or wholly contained in />‘G“Pot‘7?, which may be
called the residue of t he field of /?.
APPENDIX A
* 8 . THE THEORY OF DEDUCTION FOR PROPOSITIONS
CONTAINING APPARENT VARIABLES*
All propositions, of whatever order, are derived from a matrix composed of
elementary propositions combined by means of the stroke. Given such a
matrix, any constituent may be left constant or turned into an apparent
variable; the latter may be done in two ways, by taking “all values” or
“ some values.” Thus, if p and q are elementary propositions, giving rise to
p\q t we may replace p by <f>x or q by >/ry or both, where <f>x, yfry are pro-
positional functions whose values are elementary propositions. We thus
arrive, to begin with, at four new propositions:
(*).(<t>x\q), (g*).(^|?), <yM/>|*y>. (ay)-(pl^.v).
By means of definitions, we can separate out the constant and the variable
part in these expressions; we put
*8 01. }(#). <f>x\\q . ■= .(a*). ($* 1 9 ) Df
*8 011. \(^x).<t>x\\q.~.(x).(<f>x\q) Df
*8 012. p\ ((y) . yfry\ . - . (ay) . (p\yfry) Df
*8 013. p|((ay)-*y|---(y).(p|*y) Df
These definitions define the meaning of the stroke when it occurs between
two propositions of. which one is elementary while the other is of the first
order.
When the stroke occurs between two propositions which are both of the
first order, we shall adopt the rule that the above definitions are to be applied
first to the one on the left, treating the one on the right as if it were ele¬
mentary, and are then to be applied to the oue on the right. Thus
[(x) . 4>x) | {(y) . yfry) . = : (g*) : <£x| {(y) . yj,y] :
- 2 <a*) s (ay) • (<M ^y)-
The same rule can be applied to n propositions; they are to be eliminated
from left to right. If a proposition occurs more than once, its occurrences
must be eliminated successively as if they were different propositions. These
rules are only required for the sake of definiteness, as different orders of
elimination give equivalent results. This is only true because we are dealing
with various functions each containing one variable, and no variable occurs on
both sides of the stroke; it would not be true if we were dealing with func¬
tions of several variables. We have e.g.
(a*) 2 (y) • = = = (y) = (a*) • (4> x \ ^y)-
• This chapter is to replace »9 of the text.
APPENDIX A
030
Hut we dn not have in general
(a*) :(//)• x( x > //> : 5 : (y) : <3 X > • x (*• y) •
here the right-hand side is more likely to be true than the left-hand side.
For the present, however, we are not concerned with variable functions of two
variables.
It should be observed that this possibility of changing the order of the
variables is a merit of the stroke. We have
(gx): <y> • <t>‘ I: = : (y): (gx). ‘MV'7/ : = : (H* r ) $x. v . (y) . ~ yfry.
That is, these equivalent proj>ositions are true when, and only when, either <f>
is sometimes false or is always false. But if we take e.ff.
4 >r v >\ry . ^ <f>j v ^ yfry
wo shall not get the same result. For
(gx) : (!/) • 4> r v 't'H . ~ 4>r v ^ yfry : D : (y) . yfry . v . (y) . ^ ^ry,
w herons (//>: (gx) . </>.# v yfry . — <f>.r v ~ >\ry does not imply this.
Written in stroke notation, after some reduction, the above matrix is
1 4 >' I (1^)111 I (‘ I > I •
Here both ./• and // occur on both sides of the principal matrix. Thus in order
to be able to change the order of " (gx)" and "(•/).’ it is sufficient (though
not always necessary) that the matrix should contain some part of the form
<f>r | \fty. and that x and y should not occur in any other part of the matrix.
(This part may of course be the whole matrix.) We assume the legitimacy ot
this interchange by a primitive proposition, and in practice arrange to have
all the g-prefixes ns far to the right as possible, because this facilitates proofs.
Our primitive propositions are the following:
* 81 . h. (a r » y > • <*»« I ( 1 4>y) Pp
On applying the definitions, this is seen to be
h : <f>a . D . (gx). 4>x.
*811. h .<f).r\(<t>a\<f,b) Pp
On applying the definitions, this becomes
h : (x ). <f>x . D . <pa . <f>b.
We have <fxt\(<f>a\4>b) . v. <t>b j ( tf>a j </>&)
and by *81 h : 4>« (4> (l 4>*>) • 3 • (3*) • 4 **! (<*>“ I 4>b) •
<f>b i (<f>a | 4>b). D . (gx) . <f>x | ( 4>a | <f>b),
but we cannot deduce (gx). <f>x\(<f>a { (f>b) without *811 or an equivalent.
*812. From “ (x) . <f>x ” and “ (x). tf>x D y\rx " we can infer “ (x) . 4rx ," even
when <f> and yfr are not elementary. Pp
*813. If all occurrences of x are separated from all occurrences of y by a
certain stroke, we can change the order of x and y in the prefix, i.e. we can
replace “ (y) : (gx) . <f>x I yfry " by “ (gx) : (y) . <f>x | yfry ” and vice versa. Pp
PROPOSITIONS CONTAINING APPARENT VARIABLES
637
The above primitive propositions are to be assumed, not only for one or
two variables, but for any number. Thus e.g. * 8*1 allows us to assert
b : <f> (a,, a t , ... a n ) . D . (g.r,, x 3 , ... x n ) . <f>(x lt x t , ... x n ).
*8*2. I-: (x) . <f>x . D . <f>a ^*8ll^J
In what follows, the method of proof is invariably the same. YY T e first
apply the definitions until the whole asserted proposition is brought into the
form of a matrix with a prefix. If necessary, we apply *813 to change the
order of the variables in the prefix. When the proposition to be proved has
been brought into this form, we deduce it by means of *8111, using *812 in
the deduction if necessary. It will be observed that *8T is l~z<f>a . I>.(gar).<£./•.
Hence, by *812, whenever we know <f>a, we can assert *81 is often
used in this way.
*8 21. I- (ar) . <f>x D \Jrx . D : (gx) . <f>.c . D . ( r >\x) . yfr.c
Dem.
Applying the definitions, and using *813, the proposition to be proved
becomes
(i/» y) : (a*. *> \4>* O'* | yfrx)\ | [\<py\('l '2 \ ^w)} I [<f>y '| (^' I
Putting z = w = z = xu = x, the above becomes
(y. y) • (a*)• \4 >*Ii +*)} W* I +*)] i i 4>y ', | ^r))].
By *81, the proposition to be proved is true if this is true. But this is true
by *811, putting y, y' for a, b and <f>y | (yfrx 1 \frx) for <f>a. Hence the proposition
is true.
*8*22. h : <f>a v <f>b • D • (a#) . <f>x
b . *811.3 b . (a*) • <f>z) | (~ <f>a , ~ 4>b) (1)
Transp . D h : <f>z) I <f>a ~ tf>b) . D . (<f>a v <f>b) | (<f>z <f>z) (2)
b - (1) • (2). *8-21 . D h . < a *) . (<f>a v <fib) | (<fiz \ <f>z) (3)
b . (3) . *8T'21 . DK (g*, «/) . (<f>u v <f>b) i (tpz j <f>tv) .
[(*8 012 013)] D h : <f>a v <fib . D . (g*) . <f>x :D\-. Prop
These propositions, as well as all the others in *8, apply to any number of
variables, since the primitive propositions do so.
*8 23. h : (a* 1 ) • <f> x v <f>c . D . (a®) • <t> x
Dem.
Applying the definitions, this proposition is
(*) s (3 y. 2 ).(fcv <f>c) | (<f>y | 4>z),
i.e. • (x) : <f>x v <f>c . D . (a^) •
which follows from *8*22.
APPENDIX A
038
I'he following propositions are concerned with forms of the syllogism.
*8 24. I - yO 7 . D 7 . D . (gx). <£x : D :p . D . (gx).
Dem.
Applying the definitions, we obtain a matrix
( /°7> l (7 «</»'• 0//)) (/> (fa <t»») p (<*>// (fir))]
. , „ !the same with accented letters! 1
with a prefix
(• r . //. **■'. $0 s (a*» w > 11 ■ **. «»'. «\ *»')•
Mv * 8 * 1 , this will be true if it is true for chosen values of z % w, u, v, z\ w, u\ v.
Put : = a =j. w —v= I/, z' = u = x . w' = v = >j . Then what has to be proved
becomes
p Dr/O v . D . <*>/.«/>// : D :p . D . <*>r . $,j s.r/. D . $x'. : D : y>. D . 0 /.
w hich is true by Syll. Hence the proposition follows.
*8 241. H :: (x). 0, . D . y> : D yO 7 . D : (x) . <£.r . D . 7
/<**>•-• I/' <7 7>! l\4>!/ (7 7)i !+* (7 7)1].
the matrix of the proposition to be proved is
and the prefix is <x) : (g//. /, r'). Putting y - s - y' - t ' =
reduces to </>./Oy>. D : yO 7 . D . <£. 1 0 7 . which is true by Syll.
proposition is true by * 8 ’I.
the matrix
Hence the
*8 25. 3. <3*). <f >'s D <gx> . <*» . 3 . (gx). O : ;>. 3 . (gx).
Deni.
Put /(.c, y. X. a).-. |£, <>fry ^*‘)||[|p (+>• >fre) j jy,
Then the projiosition to be proved, on applying the definitions, is found to
have a matrix
\P ( 4>a \/{T,i/,z t u, v,m. n) u'.v\ m\n)\
with the prefix
(<i, 0, 1 /, s, •/', z) : (gx, u. r, m. a. x\ v, m, n).
Put x — u . x = b . u = v = i/ . m = n = z . u = v = y . m = n = z.
Then the matrix reduces to
j>. D . <f>n . <t>0 : D <fja . D . yjri/ . yfrz : 0 : p . D . yjrp . yfrz
yfnj. yfrz D :p. D . yfri/' . yfrz,
which is true by Syll. Hence our proposition results by repeated applications
of *8*1*18.
Analogous proofs apply to other forms of the syllogism.
*8 26. h : <f)ii v <f>b v <f>c . D . (gx) .<f>xv<pc
Deni.
H : tf>a v <£& v <f>c . D . (<£</ v <£c) v (</>£» v <£c)
f- . *82*2 .Dh:(^v^c)v (</>& v <f>c) . D . (gx) ,<f>xv<f>c
h .(!).(2) - *8*24. D H . Prop
( 1 )
( 2 )
PROPOSITIONS CONTAINING APPARENT VARIABLES
*8-261. h : <f>a V <f>b v <f>c . D . (gar) . <£.*•
[*8-25-26-23]
It is obvious that we can prove in like manner
and so on. <t>a v <pb v <f>c v <f>d . D . (ga?) . <fxc
639
*8-27. K :: y . D . (gar) . *x : D /O y . D . D . (gx) . «£ar
Dem.
Put /(*. y, «. v).=. Ipi (iftx, <t>i/)) I \ p I (<(,„ I </> U )}.
Then the matrix is
[q | (<t>a | <f>b )j | [j(/> D q) | /(x, y, u, t/» | {(p D y) \f{x\ y\ u, v')j]
and the prefix is (a. b) : (gar, y, u, v, x', y\ u\ v').
Putting x= u = x'=u=a.y=v = y’ = v'=b, the matrix becomes
q . D . <f>a . <f>b : D p D q . D : p . 2
which is true. Hence the proposition.
*8 271. V :: y . D . (gar, y) . <f> (x, y) : D p D q . D : p . D . (gar, y ) . ^ (iC> y)
[Proof as in *8 27]
It is obvious that we can prove similarly the analogous proposition with
</> (a;,, ar a , ... ar„) in place of <f> (ar, y).
*8 272. h ::.p . D : y . D . (gar) . <f>x D :: r D /> . D r . D : y . D . (g*> . 0 *
Dem.
7 . D . (gar) . <£ar is (gar, y). q | (<f>x j <f>y). Hence the proposition results from
*8 271 by the substitution of p for q, r for p, and q\(<f>x | <f>y) for <f> (x, y).
*8 28. I ~ z:p.D . (gar) . <f>x z D q . D . (gar) . <f*x z D z pv q . D . (gar) . <j>x
Dem.
Put /(ar, y,z,w). = . {(/> v y) | (£ar j <*>y)) | {(/> v y) | (<f>z |
Then the matrix is
I P I ( 0 a I <f> b )\ I [(('/ I (<t> c I <t>d)) |/(ar, y, *, « 0 ) | |(y | («£c' | ')) |/(x', y', «,'))]
and the prefix is
. (a, 6 , c, rf, c', d') : (gx, y, *, «/, x', y, z\ w').
Ihe matrix is
p . D . <£a . : D y . D . <t>c . : D -/(x, y, z, u/)
y . D . <f>c'. <f>d'. D ./(x', y', s', w/),
while ' f(*,y,z, w) ,=zpvq.D. <f>x . <f>y . ^>2 . <j>w.
Call the matrix /*(ar, y, z, w, x\ y, z\ w).
Then hsp.D. F(a, b, a, b, a, b, a, b),
: ~p . D . F(c, d, c, d, c, d', c, d').
h : F(a, b, a. b, a, b, a, b) . v . F (c, d, c, d, d, d> c, d').
Hence
APPENDIX A
040
Hence, by the extension of *8-261 to eight variables,
*" • n\ x, y, z, w) . F(x, y, z, w, x, y, z , w ),
which was to be proved.
*8 29. V <f>xO yfrx . D : (x) . tfrx . D . (a:) . \jrx .
Item.
Applying the definitions, our proposition is found to have a matrix
(tfixDfx) [l<fiyl(y/r/t i>Jrp)l !<t>y'l(yjsu' |>K>1]
with a prefix (after using *818)
<". ">'):<g.n y. y).
The matrix is equivalent to
<t>.r D y\tx . D : 4>y . D . yfni . yfrv : <f>y '. D . yfru' . \Jrv\
Calling this M {x.y.y), we have to prove
<:**• y. y') • M (•«% y. y>.
If yjru .yf/v. yffu . yfrt?, M (x. y, y) is always true. ( 1 )
If put ur = // = y = a. Then if <f>u is true. <f>u D \fni is false and
M <". ", ") is true. Hut if <f>u is false. (f»i . Z> . and <f>u . D . \jru\frv are
true, so that M (u, ", u) is true. Hence
^ M ('0 (gr. y, y) . J/ (a;, y, y). ( 2 )
Similarly if '^ylrvv'^yfni'v^yjrv. ( 3 )
( 1 ), ( 2 ), and (’I) exhaust possible cases. Hence the result by *8 28.
We are now in a position to prove that all the propositions of * 1 —*5
remain true when one or more of the propositions p. 7 . arc first-order
propositions instead of being elementary propositions. For this purpose, we
take, not the one primitive proposition which Nicod has shown to be sufficient,
but the two which he has shown to be equivalent to it. namely:
/O p and p D 7 . D . 7 D p | g.
We show that these are true when one, or two. or three, of the propositions
p. 7 , a* are first-order propositions. From this, the rest follows. The first
of these primitive propositions, /O p, gives rise to two cases, according ns we
substitute (x). <£.c or (%\x). <f>x tor p \ the second primitive proposition gives
rise to 26 cases. These have to be considered one by one.
*8 3. h z(x) ,<f>x.D. (x) . <f>x
Applying the definitions, this is (gar) : (y, z ). <f>x | (<f>y | $z), which follows
from *8T1 by *8*13.
*831. H : (gar) ,<f>x.D. (gx) . <f>x
Applying the definitions, this is (x) : (gy, z ). tf>x | (<f>y 1 </>*). This is *81.
This completes the proof of p D p.
641
( 1 )
( 2 )
PROPOSITIONS CONTAINING APPARENT VARIABLES
*8 32. 1-:. («). <^r. D . 7 : D : 51 g . D . {(*) .
Putting />. = .(*). <f>x, the proposition to be proved is
By the definitions, (Pi ~ ?)! ~ l<S I ?> I~0> I *»•
P l~9 • = • (3“) • 4>a I (919),
p|*- = - (a*) -<t>x |«,
~<P I *)• = •(*. y) . (£* 1 s) j (<£y | s),
(* I 9) l~(P I *) ■ = • (a*. y) ■ (S | 9) | |(<#>x j s) | (<f>y | s)j.
,, • = • (s I 9) I (( 4>* I s) | (*y | s)J.
Then ~ [(»| 9) I ~(P | S)1 . - . (x, y, x'. ,f) ./(*, y) |/< x ', y').
By (1) and (2), the proposition to be proved is
<“) : (ax.y.x.y-) . |^,a|(9|9)| | |/(x,y)|/(x',y')].
I Utting x-y-x -y'-a, the matrix of this proposition reduces to
<f>a D . D . s | D <£a J s,
which is our primitive proposition with substituted for p, and is therefore
true. Hence the proposition follows by *8 1.
In what follows, the reduction of the proposition to be proved to a matrix
and prefix by means of the definitions, proceeds always by the same method,
and the steps will usually be omitted.
*8 321. I-( a *) .«x.D. 9 :D:,| 9 .d. (< a x) . ^ j s
prefu'is 0bt ' lin thC S “ me matriX “ in * 8:J2, b,,t th ° °PP° sit « prefix, ».*. the
rn. ... . , ( x > y> at, y ): (ga).
1 he matrix is equivalent to
<f>aDq.5zq . D . <f> x D~ s . <f>,j . <f>x' . <f>y' D~s.
Calling this/a, we have to prove ( 3 <i)./a , for any x,y,x',y'. We have
<f>a .D ,f a .
Also 4>a . 9 . D -..fa . = : . D . <£x D . <f>y 0~s . <f>x D~«. ^.y' D~s:.
D :./a.
Hence <f>a . D ./a.
Hence by *8-1-24 *x . D . (ga) ./o,
and similarly for <f.y, fx’, <f>y. Hence by *8 261
^v^yv <£x'v 4>t/ .O. (get) .fa.
AIbo • ~4>y —*x' — 4 >y". D .fa .
Hence by *8-28 [ * M ‘* 4] 3 • (3<*) •/«•
T — 4 >y — <^r' — <£y': 3 - (a a) -/a.
Hence, by *812, (ga) .fa, which was to be proved.
HtW I
41
APPENDIX A
(542
*8 322. I- p . D . (x). >/fx : D : s I(x) . \px J . D . /> s
l)em.
Pllt /y.=.(« >/ry) |(/> S) (/> S».
Then the proposition to be proved is
(.'/■ y) ■ (a*. 0 • \p' W t c )l l (/y /y')-
The matrix here is equivalent to
p . D . >/r A . >/rc : D : s >^y . D . /> *: s >Jr#/ . D . p s.
Putting b = y . c = //, this follows at once from the primitive proposition, which
gives
p D y/r y . D : .v ifry . D . y> «,
/> D '/ry' . D : .v >/ry'. D . p S.
Hence the proposition.
*8 323. h D . (gx). D : .v j(g.r). . D . /> «
We have the same matrix ns in *8-322, but the opposite prefix, i.e.
(^.c)s(ay.y).
Putting y = b . y' = c. the matrix is satisfied, jus in *8 .322.
*8 324. h./O'/.D: |(x). x x l 7 • ^ K- 0 . x-'*)
J)eni.
Put f(x,y t z). — .{x x V)ll(Plxy) (P X*)l- T,lcn the matrix is
I /> I (71 7)! 11/<*. //• -) /(*'. /. *')|
and the prefix is (.r, x') : (gy, s,y, s'). Putting
y=-z=*x.y’ = z'~x,
the matrix is equivalent to
/; I) 7 . D : *.r 17 . D .p X ar: x*‘\q . 3 ./>| X-r'.
which follows from our primitive proposition by Comp.
*8 326. I-;0 7 . D : l(gx) • X*I 17 • ^ I (<3*) • X- 1 ’!
I) cm.
The matrix is the same ns in *8.32+, but the prefix is the opposite, i.e.
(y> z, y,
('ailing the matrix M(x,x), we have, if 0 w. e^.^xw'.
M(x t x ). = :: p D 7 . 3 7 0 6* • 3 : p . D . 0 y .0 z:.qO 0x .D :p ,0y'. 0z.
H ence 0.7 . 0* • . 0z '. D . Jl/ (x, x) . D . (ga:, x '). <1/ (ar, x) (1)
But ~Ox .~0x'. D . A( (x,af)- Hence
~ Ox. D . M (x, x). D . (gar, x'). J/ ( x , x') ( 2 )
Similarly with By, 6x, Oy . Hence the result follows as in *8 321.
This ends the cases in which only one of p, q, r in
pDq.3i9\q.0.p\s
is of the first order instead of being elementary. We have now to deal with
the cjises in which two, but not three, are of the first order.
643
PROPOSITIONS CONTAINING APPARENT VARIABLES
*8 33. h :.(x). <f>x . D . (x) . yfr X: 3 . ,| ((*) .**).;>. ((x) . ^.j )s
Putting /(*, y, *). = .(«( *x) I ((<*.y \s)\( 4 >z \,)] , the matrix is
(•#>« I (*& 1 yfrc)} I [f(x, y, z)\f(x\ y, *')J
and the prefix is (a, x, x') : (g&, c, y, *, y', The matrix is satisfied by
6 = *.c = x'.y = 2 = y' = 2 ' = «,
in which case it is equivalent to
Hence Prop. ^
We have the same matrix in the three following propositions, only with
different prefixes. J
*8 331. I- (x) . 4>x . D . (gx) . yfrx : D : * | ((gx). \frx] . D . ((x) . «£x) | s
Here the prefix to the matrix is (a. b, c): (gx, y, z, x\ y’, z'). The matrix
is satisfied by x = b . x — c. y •= z = y' = z' = a. Hence Prop.
*8 332. I-(gx) . <#>x . D . (x) . yfrx : D : * | {(x) . *x| . D . |(g. r ) .^ x \\s
becomes Pr0fiX " ( *' V ' *' y ‘‘ *"> 1 <S°* b ‘ c >- Writing r for ~s, matrix
<t>a . D . yfrb . yfrc : 0 :. yfrx D r . D . <f,y y o r : yfrx D r. D . <*y' v <£*' D r .
(Here only a, 6. c can be chosen arbitrarily.) This is true if £y, 0 2 , Ay', d,z'
are all false. Suppose 4>y is true. Put a - y. Then if yfrb or is Mae,
f “ ■ J • r* • yc is false, and the matrix is true. Therefore if ^x is false, put
- c =*x ; if ^ 18 false, put b — c - x\ If yfrx and yfrx' are both true, putting
a — y.O*"C=x, the matrix becomes equivalent to
r.D.^v^Dr:r.D.«/,/v^'Dr,
which is true. Hence if <f>y is true, the matrix can be made true. Similarly
,or y, * - Ihis exhausts possible cases. Hence Prop, by *8 28.
*8-333. I-(gx) . tpx. O . (gx) . yfrx : D : s | |(gx) . yfrx) . D . ((gx) . 0x) | s
Dem.
The matrix is as before, and the prefix (after using *813) is
(6, c, y, z, y\ z ) : (ga, x t x).
Call the matrix if (a, x, x'). Then
b : >\rb . D . M (a, b, b) . D . (ga, x, x') . (a, a:, x') (1)
b : yfre . D . Af (a, c, c) . D . (ga, a:, af). if (a, x, x) (2)
b : ~ yjrb . ~ yjrc . <f>y . D . if (y, b, c) . D . (ga, x, af) . if (a, a:, o') (3)
(1).(2).(3).D b :<£y. D. (ga, x, x).Af (a, x, x) [using *828] (4)
Similarly for <j>y', <f>z, <f>z Hence by *8 28
b s <f>y v tf>y' v <pz v <{>z '. D . (ga, a:, a-'), if (a, a:, x) (5)
But b ~ <f>y . ^ ^> 2 ! .D z 4>y v tpz D r. (fry'v 4>zf D r :
D : if (a, ar, V)
C* 8 ’ 1 ] 3: (3°. x,af).Af (a, x, x') (6)
b . (5). (6) . *8 28 - D b . (ga, a:, a :'). Af (a, x, x') . D h . Prop
41—2
APPENDIX A
644
This ends the cases in which p and 7 but not s contain apparent variables.
We take next the lour cases in which p ami s, but not 7 , contain apparent
variables.
*8 34. h :.(./ ).<£./ . D . 7 : D : I(j-) . x-t-; 7 . D . -(a:) . 0.r| |(x) . x**'!
Putting/(.r, //. 2 . u. v). = .(x-r 7 >j !<$.'/ X r > X 1 ’)!- the matrix is
<4>" ^//) |/(x, y. 2 , m. c) /(x\ y\ z, u, i» )].
(This is also the matrix of the three following propositions.)
The prefix is (a, x, x') : (g.y, z. u, v, #/. s'. u\ v).
The matrix is e«|iiivalent to
<t>« 0//.D ./(x. y, z, ii. e) ./(*, //'. c'. /»', v)
and /(.**. //, '•). = : x-r 7 • ^ • 4>!/ X- • 4>" X v :
= : 7 D ~ x* r • ^ • 0 // ^ X* • 4 >l1 ^ ~ X r *
Put ting »/ = // = //' = »#'=»/ . - = r = x. *' = v = x', the matrix is satisfied. Hence
Prop.
*8 341. I-(x) . 0x . D . 7 : D : |(gx). x» ’ 17 • 3 • • 0' I |<3*) • X r !
Matrix as in *8 34. Prefix (a, z. v, z. v ):('.{ x. y, u. x\ y, n ).
Matrix is equivalent to
<f>n D 7 . D 7 D ~ x ,c • ^ ^ ^ X* • 0 " ^ ^ x»’:
7 D ^ x*' / • ^ • $!/’ ~ X’* • 0 '* ^ ~ X , ’ , ‘
U <t»i is false, this becomes true by putting //= " *»'/=* u' — u. If <f>a is true,
the matrix is true if 7 is false. Suppose 7 true. Then the matrix is
equivalent to
~ X»* • 3 • 0.y ^ ~ X 2 • 0" ^ ^ X w : ^ X r • ^ • 0y ^ ~ X-* • 0' 1 ’ ^ ^ X v •
This is true if x-. X 1 '* X- • X" nre false. If one of them, say \z. is true, put
./• = x «= z, and the matrix is true. This exhausts possible cases. Hence Prop,
by *8*28.
*8 342. y (gj). 0x . D . 7 : D : |(x) . xr| 1 7 . D. \(& x ) • 0' I1 !(- r ) • X* l
Matrix as before. Prefix (after using *8*13) (x. y. 11 , x\ y. u) : (go, 2 , v. z'. v).
Call the matrix M (a. z. v. z'. v ). Then
h : ~ X‘ r • ^ • M (a. a*, •»*. x) (1)
t*: ^ x*'*’ • D . M (a.x'.x'.x.x) ( 2 )
h 2 7 • X x • X*' • ^ -M7 -^(7 •
D . 3/(«, 2,0, «»') (3)
0 ,~{(f>yD q).
D . i)/ (y, r, e, 2', e) ( 4 )
Similarly if ^7 . <J>u or ~7 . ^»y' or ^7 . </>«'. Hence by *8*1*28
H : ~q . </>y v </»» v <f>y v <#>«'. D . (ga, 2, i», 2', i»'). Jl/ (a, 2, u, 2',»') ( 5 )
\-:~<py.~<pu.~<py'-^<t>"'' ^ - «/»y ^~x 2 • <t> u ^~x v ‘ W ^~X Z ' • f u ' 1~X V t
O.M(a,z,v,z',v) (6)
(5). (6 ). D y : ~7 • 3 • (3 a * z '» O • M ( a > z > v > z > v ') < 7 )
h . (1) . (2) . (3) . (7) . D h . Prop
PROPOSITIONS CONTAINING APPARENT VARIABLES
645
*8 343. b (ga;) . 4>x . D . q : D : |(gx) . x *\ I q • => • «3*) • <M I ((3*) - x x \
Prefix to matrix is (y, z, u, v , y', y, te\ v') : (go.or,a;').
Call the matrix
/(a, ar, x).
It is true if
~X Z -~X V -^X 2 ' -~x v '
(U
Also xz . q
• ^ •/(«, z, z) . D . (ga, ar, x') .f(a,x,x)
(2)
Similarly if we have
X» • q or x 2 -qorxv'.q
(3)
From (1) . (2). (3), by *8 28, q . D . (3a.ar.aO .f(a,x,x')
"No\v <f>a .^q.O. f (a, x, x'). Hence
(4>
<*>y —<? . D ./(y, ar,#'). D. (ga, a:, x') . f (a. a:, x)
Similarly for <f>z .~q, <f>y' .~q, <f>z' .^q. Hence
0y v^v^y'v^' ,~q . 5 .(•aa.x.x) .f(a,x,x) (5)
liut ~<t>y .~<f>y'.~~<f>z'. D .f(a,x,x') (6)
By (5) and (6), ~<7 . D . (ga, x, x') .f(a, x, x') (7)
1- . (4) . (7). *828 .Dh. Prop
In the next four propositions, q and r are replaced by propositions con¬
taining apparent variables, while p remains elementary.
*8*36. b i.p . D . (a?) . yfrx : D : ((a:) . *a:) | ((ar) . yfrx\ . D . p | {(a:) . x x \
Putting 7 . =* . (x) . yfrx, s . =*. (a:) . yfrx, the proposition is
(p ^7)|~|(s|7)|^(p|s)|.
We have by the definitions
• (3&. c).^|^c,
P I c) • V I I ^c),
5 l?- = •(a*.y)-xylV r *.
7>l*- = .<3*)-/>lx*.
~<Pl*). = .(*,w).(p| X *)|(p| X u;),
(«I *7> I ~(P I«) - = : (x, y) : (g z, w) . (xy | | j(p | *r) I (p I X w )i-
f(x, y, z, w). = . ( X y | yfrx) H(plx 2 )l(pl X w )l
~ W*1 ?) I ~(P I s >l • = : (3 : (*. w.z', w').f(x,y,z,w)\f(x',y,z',w),
(p I ~ <l) I ~ {(* \q) | ~ (p |«)) . = : (x, y, X, y ) : (g b, c, z, w, w') .
\p!(yfrb\ yfrc )j | [f{x, y, z, w) \f(x\ y , z, w')\.
Writing 0£ for '—^a?, the matrix is equivalent to
Put
Then
p . D . yfrb . yfrc : D yfrx D 0y . D z p . “D . 0z . 0w :. yfrx D 0y . D : p . D . 0z '. 0 m/.
This is satisfied by putting b = x.c=x.z = w = y.z=w=y. Hence Prop.
The same matrix appears in the next three propositions; only the prefix
changes.
*8*361. b :.p.D.(x).yfrx:D: {(ga:) . X A I {(*) - ^1 - ^ -p\ K3*) • X x i
Same matrix as in *835, but prefix (x, z, w, x',z\ w) : (g6, c,y, y f ).
Matrix is true if 0s . 0w . 0z' - 0w\
Assume ~ 0z, and put y = y' = z.b*~x.c = af.
APPENDIX A
046
We now have yjrx D By . = ,~y}rx and /> . D . 02 . : = ,^p. Hence matrix
is equivalent, to
p .D . yfrx . yfrx : D :.^>/rx. D ,^y> . D : p . D . Bz . £he',
which is true. Similarly if <vfl«'v^fe'v'v^w'. Hence Prop, by *8*1‘2-S.
^8 352. h :•/>. D . (gx). >/r.c : D : |(x). *x; | {(g*). | . D . /> | {(*) . \ x \
Same matrix, hut prefix (b t c,y,y') : (g.c,*, w,x' t z, to').
Satisfied by x = b . x = c . z = tv — y . *' = a*' = y'. Hence Prop.
*8353. h :./). D . (gar), yfrx : D : |(g.r). *rj | |(3- r ). yfrx j . D . p | ((g.r) . *.r)
Same matrix, with prefix (b,c,z, w, z\ tv ): (yx, y, x\ y).
If ^b is true and Bz false, matrix is satisfied by x = x = b . y — tj = z, be¬
cause these values make yjrx D By and yfrx D By' false. Smilarly if yjrb is true
and Btu or Bz or Bw is false, and if yfre is true and Bz, Bio, Bz' or Bio is false.
It remains to consider ~\frb . ^ yfre : v z Bz . Bio . Bz '. Bw.
The second alternative makes the matrix true, because it gives
p . D . Bz . Bw: p . D . Bz '. Bw'.
The first alternative gives
p . D . yfrb .yfrez D : ^p Z
D: p • D . $z • Bw z p . D . Bz '. 0m/,
so that again the matrix is true. Hence Prop.
This finishes the cases in which one or two of the three constituents of
/O 7 . D . s 1y D /> | .v remain elementary. It remains to consider the eight cases
in which none remains elementary. These all have the same matrix.
*8 36. h (x) . <f>x . D . (.c) . yfrx z D : [(x). x x \ I ((•»)• ^ | ((•*•). **i
Putting p .=> .(x). 4>x, 7 . = . (x). yjrx, s . = . (x) . *.r, we have
~7-"-<3&. c). yfrb\yf/c,
P\~<J • * : (3") : (b. c ). ^a|(^r6|^c),
«l7
/>!*• = • (3*.
—<y^l«) • = •(*. w. *>).
(«l*y)l — (y>l«) - == : <^y> : (3*- «*.»>-(xyl^)ll(^lx ,<, )|(<#*Mlx*’)l-
Put/(.r,y, = .(xy|^)||(^lx M, )l(^ M lx l, )I- Then
— K^l v>l —(/^l ^>1 - = : (3*» y. y') : (*. w, »«'.*0.
f( x , v> z, W, II, o)\f(x,y, z\ w, u, o),
(p I ~ 7 ) I ~ ((* 17) I ~ ( P I s )) • = z («. •»-. y. y) • (3^. c, Z, W, u, V. z', u/, u\ v ).
|^»« I I ^c)111/(^. y. z, w, u, v) I f(x, y\ z', to, n\ v')J-
Writing Bx for t ^ ie niatr * x * s equivalent to
<f)a .0 . yfrb . yfre z D yfrx D By . D . <f>z D Bw . <^>w D Bo z
yfrx DBy'.D. <f>z' D Bw '. <f>u' D Bo.
This is satisfied by b = x . c = x . z = u = z = u' = a . w = v = y . xo = v = y .
Hence Prop.
PROPOSITIONS CONTAINING APPARENT VARIABLES
647
*8 361. h :.(«). 0* . D . («).yjrxzDz {(gar). X x\ | {(ar). yfrx \. D. \(x). <f>x) | fax ). *arj
Same matrix, hut “all ” and “some ” are interchanged in arguments to x ,
i.e. in y, w, v, y', w', v. The g-variables are therefore b, c, y, y, z, z ', u, u'.
If put z = u = z' = u = a, and matrix is satisfied.
If <f>a is true, matrix is true if ~>/r& v ~yfrc, i.e. if ^yfrx v ~ yfrx', since b, c
are arbitrary. Assume yfrx . yfrx'. Then matrix reduces to^
0y.D.<f>zD0w.<f>uD 0v : 0y . D . <f>z' D dw . <f>u D 0u.
If 0w, 0v, 0w\ 0v' are all true, this is true.
If ~0w, put y — y' = w, and matrix is satisfied.
Similarly if ~0v, ~0w' or ~0v'. Hence Prop.
*8-362. b :.(x).<f>x.D. (gar). yfrx : D :{<*). X x\ | fax ). yfrx }. D. {(#). 0ar) | {(ar). X x\
Matrix as in *836. Prefix results from *836 by interchanging "all ” and
“some” among ^-arguments, i.e. b, c, x, ar'. Hence Prop results from same
substitutions as in *8 36.
*8 363. b (ar) . <f>x . D . (gar) . yjrx : D : ((gar) . X x] | {(gar) . yfrx J .
D . {(x) . <f>x] | {(gar) . X x\
Results from interchanging "air' and "some,” in *8361, in the yfr-
arguments, viz. b, c, x, ar'. The g-variables are therefore ar, ar', y, y\ z, z' t u, u\
and the proof proceeds exactly as in *8 361, interchanging ar, ar' and b, c.
*8*364. b (gar) . <f>x . O . (ar). yfrx: D: {(ar).^arj | {(ar).>/rar). D. {(gar).<£ar| | [(x). X x\
The proposition is what results from *8 36 by interchanging " all ” and
" some ” in the </>-arguments, viz. a, z, u, z, u'. Hence the g-arguments arc
a, b, c, w, v, w, v. If 0y is true, put w = v = w = v' *= y, and the matrix is
satisfied. If 0y is true, put w «= v >= w' = v = y', and the matrix is satisfied.
Assume ~0y . ~0y'. The matrix is true if yfrxD0y and yf/x' D 0y‘ are false,
i.e., since 0y, 0y ' are false, if yfrx and yfrx' are true. If yfrx is false, put b = c = x
and a = y \ then <f>a . D . yfrb . yfrc is false, and the matrix is true. If yfrx is
false, similarly. Hence Prop.
*8 366. b (gar) . tf>x . D . (ar) . yfrx : D : {(gar) . *ar) | {(ar) . yfrx ) .
D . {(gar) . <f>x] | {(gar) . X x)
Prop is what results from *8*364 by interchanging "all ” and “some” in
the ^-arguments, viz. y, w, v, y\ w', v'. Hence the g-arguments are a, b, c, y, y.
Matrix is true if 0w . 0v . 0w . 0v. Assume ~0w, and put y = y = w. Matrix
is true if yfrxO0y and yfrx'D 0y' are false, i.e., in the present case, if yfrx and
yfra / are true. Suppose one of them false, and put b = ar. c = ar'. Then yfrb . yfrc
is false. Therefore <f*a . D . yfrb . yfrc is false if <f>a is true ; therefore the matrix
is true if <f>a is true. Therefore if <f>e is true, the matrix is true for a = z.
Similarly if <fm, <f>z' or <f>u' is true. But if all are false, matrix is also true.
Hence matrix is true when we have ~0w and ~ yfrx v ~ yfrx'. Similarly for
~0v, 0v/ or r-^Ov with ^yfrxv ~yfra?. We saw that matrix can be satisfied
APPENDIX A
018
for '■'w 0n\ ~6v, ~*0,V or ^*$o with yfrx.-tyx. Hence it can be satisfied for
*>*$w v ^0o. And we saw that it is true for 6w. dv . $w ‘. Ov.
This completes the cases. Hence Prop.
* 8 366. V (gx). <f>x . D . (gx). : D : j ( . r ). x . c ] | |(g. r ) . yjr.v\ .
D. |( a x).^j||(x).x.c|
Prop is what results from *8 804 by interchanging “all” and “some”
among ^-arguments. viz. b. c, a. x. Hence ^-arguments are a. x, x, w, v, w, v.
I h*- proof proceeds as in *8*364, interchanging b, c and x, .v.
■•8 367. h (HOT) . <*>x . D . (gx). yfr.r : D : '(gx). x .r\ | |(gx) . yjrx j .
D. Kg-*-) • <M 11<3*> • X*\
1 r«»p is what results from *8*365 by interchanging “all” and “some"
among ^-arguments, viz. b, c. x. x. Hence the g-arguments are a, x, x\ y, y.
The proof proceeds as in *8*365, interchanging b, c and x, x.
'I his completes the 20 cases of /Or/.D. .s*]y D/j|a\ Hence in all the pro¬
positions ol *1—*.) we can substitute pro|>ositions containing one variable.
The proofs for propositions containing 2 or 3 or 4 or ... variables are step-by-step
l he same. Hence the propositions ol *1—*.» hold of all first-order propositions.
The extension to second-order propositions, ami thence to third-order
propositions, and so on. is made by exactly analogous steps. Hence all stroke-
functions which can be demonstrated for elementary propositions can be
demonstrated for propositions of any order.
It remains to prove ~ |<x) . <f>.r\ . = . (gx) . ~£.»- and similar propositions.
*8*4. ~ |(x). <f>x\ . = . (gar) . ~~<f>.r
Dent.
h .*8-1 .
:> h: 4* 1 4> r • ^ • <ay) • <f >>-1 <t>!/
(i>
y .(i).*8-2i.
D y : (gx) . <f>x | tft.r . D . (gar, y) . <f>x\<f>y
•
[(*801 012)]
D y : (g.r).'v^, D .~J(x). <£.r|
(2)
We have
y s p\*i • 5 • pIp
(3)
M3).
D 1-: tf>x\<f>y . = . tf>.r\<f>.r v <f>y\4>y
(■*)
K (4). *8*22*24.
. D h : <f*x\4>y . D . (gx). <£.r|£x
(5)
[(*8*011)]
y (3*-!/)-A#*!/) • 3 -p : a : (a\.y).
/(*.*>:>/» (0)
h.(5).(6).
3 I- : (a*, y) • . 3 - (gx) . «£a*|</>.r
•
•
[(*801*012)]
D h : ^|(x). 0x|. D . (ga).^fr
0)
h.(2).(7).
D h . Prop
h : ~ ((gar) . £x| .
= . (x). ~<£x
[Similar proof]
*8 42. f- z.p. D . (g.r) . <f>x : = : (gar) . p D <f >.r
Dein.
\-z.p.D. (gar) . <f>x : = : p\ (~(gx) . 0x j :
[*8 41 ] S : P | f(x) . <f>x j :
[(*8*011)] = : (gx) . :
[*8*21] = : (gar) . p D <f>x D I-. Prop
PROPOSITIONS CONTAINING APPARENT VARIABLES
G49
*8 43. I- p . D . (x) . <f>x : = : (a:) . p 3 4> x
[Similar proof]
Other propositions of this type may be taken for granted.
*8 44. h (a:) . <f>x . D : (a;) . yfrx . D . (a:) . <f>x . \Jrx
Bern.
h <f>z . D : yfrz . D . <f>z . yjrz ( 1 )
l-.(l).*81 .DI-::.(ax):s.(ay)::W:.^. Z> : . D . <f>z . yjrz ( 2 )
V . (2) . *8-42-43 . D h . Prop
*8 5. If -F(y>, q, r, ...) is a stroke-function of elementary propositions, and
P. <?. r > ••• are replaced by first-order propositions p x , q lt r Xt .... we shall have
p = p x .q = q x .r=r lt ... D : F (p, q, r, ...) . » . F{p Xt q lt r„ ...).
This.,follows from
*>i . = . (a) . <f>x : D i p s p, . D . />, = y>|</ . tflp, = </|p,
Pi • = • (3*) - s D :p up t . D .y>, . ?|p, 3 qjp,
both of which are very easily proved.
APPENDIX B
*89. MATHEMATICAL INDUCTION
The difficulties which arise in connection with mathematical induction when
the axiom of reducibility is rejected have- been explained in the Introduction
to the present edition. Retaining the definition of R # (*90 01), we have
H xR+y . = : x e C‘R : R u n C/i. jcm-
The " /x’’ which occurs here jus apparent variable must be of some definite
order. If k is a class of classes, and the members of k are of the order con¬
templated in the definition of R+. we cannot infer
j7V/ • ^ : R^p't C p‘k .xcp'tc .0 . ye p l K
nor yet */?*'/. D : R u s*k C s*k . x e s*k . D . y e s* k.
It is necessary, primd facie, to have
a € k . . R**a C a
in order to be able to argue from xe p‘* to ye p*K or from xes*K to yes*K.
In the following pages, we shall show how to avoid the resulting complications.
Let us denote by “ n„, " a variable class of the with order, and put
*89 01. rR*,„U .«: ./• e < H R : R'•p m C p, H . j- t p,„ . . y € Df
Since every class of a lower order is equal to some class of any given higher
order, R+ m G R+„ if m > n. We shall show that
m > 5 . D . /f*,„ « R^.
Hence we take R jis R+, and the complications disappear.
« In *90, substituting R+ m for /?* and p„, for p and <f>,„z for <f>c, the first
proposition involving an invalid induction is *9017. where we use the fact
that R+*x is hereditary. It is obvious that is a class of order w + 1,
and therefore, although
we cannot infer
y € R#„,‘x. l/R+m *. D . 2 6 R*,,, 1 : r.
In this case, however, as in many others, there is no difficulty in substituting
a valid induction. Put
* = %n | R“p,n Cp, n .X€ /*,„].
d ■ V/
Then R#,,,'* = 2 )f *- ^ ow we have not “©rely R^pU C p*K but also
w
Pn 6 K . D . R“ft, n C p m .
Hence the induction is valid.
MATHEMATICAL INDUCTION
651
The proofs of R £ G and analogous propositions are easily re-written
so as to be valid.
The next difficulty—and this one is more serious—arises in connection
with *90 31. The present proof uses the fact that
x(If C‘Rs* R#\R)z
is a hereditary property of z. But it is a property of a higher order than those
by which R * is defined; i.e. if R m is R+,„, then x(I [ C‘R vy R#, n | R) z is of
order m + 1. Let us prove first
R 0 svR*\R<lR^
where
*89 02. R 0 = If C*R Df
The proof is as follows:
*891. b .R 0 v R+\RGR #
Dem.
V/
H :: xe ft • R“ft uRz : D • z« ft (1)
H .(l).Comm .Db:.:r = a.v.u€/A. uRz : D : xe ft • R li ft C ft • O . se ft
Z) \- zz x = z z. v z. x e ft. R“ft Q ft . D . u e ft z uRz D :
xt ft . C ft. D. z e ft zz
= R**ft C ft. . u e ft z uRz
D : xe ft. R“ftC ft. D^.zeftzz
D h a:/f 0 * . v . . u/?* : D . D b . Prop
*89101. . R 0 v R\R#(iR m [Proof as in *90 311 ]
*89 102. H : R e Cls 1 . D . R+ = R 0 c/ R | R+
Dem.
h Hp . R<x e0.R lf 0C0.Z)zxe i‘x \j 0. R“(i‘x v 0) C0z
D : xR+y . D . y e i‘x v 0 (1>
b • (1) - Com m.Dh:.Hp.y + x. x/**y . D : ^‘a: c 0 . R“0 C 0 . D . y e 0 (2)
b • (2) • D b : Hp . xR+y . a: 4= y • ^ • a: {R | R#) y (3)
b . (3) . *89101 . D b . Prop
*89 103. h : ifc « 1 —» Cls .D . R+ = R 0 \j R#\R ^*89 102 ~ j
*89104. b * = a(are a . C a) . D : a: (i2 |/?*) z . ^ . zc P ‘R ‘“k
Dem.
b : Hp . xRy . D . y c p*R ttt K
b : Hp .aex.ye R“a . yR+z .D.ie /2“a
b . (2) . Comm . D b : Hp . y e p , R tit K . yR*z ."5 . z ep t R“*K
b • (1) • (3) . Db. Prop
( 1 )
( 2 )
(3)
APPENDIX B
652
*89*105. b Hp *89104 . R e Cls —► 1 . D : .r (ft | ft*) z . = . sxp'Rfi"*
Dem.
K = ft“ M .
D : // € R“0 v - }VR . E! ft\y . D . R<y € R“Q v - D‘ft:
D : ft“<ft“£ v - D‘ft) C R“(3 ^ - D‘ft
b : H|>( 1). D ..r € R**{3
K (1). (2). D b :. Hp (I>. - « yi«ft“‘* .D.:( ft‘‘(ft“tf u - D‘ft)
D.rt/3
b: Hp(1).D.£C/t
K(»).(4).Dh. Hp. z(p'R'“K . D : ft 4 ** M . ft‘> C /x . D . j € M :
D:.r<ft|ft*)*
b .(’»). *89’104 . D b . Prop
(1)
( 2 )
( 3 )
(4)
(5)
*89106. b : ft«Cls-*l . D . ft+| ft <• ft| ft*
Dem.
: .r (R+\ li):. = . x € R“/>*k ( 1 )
b . (I >. *89'105 . *40*37 .DK Prop
It is now necessary to take up the subject of intervals (cf. *121). Our
further progress depends upon the fact that in suitable circumstances the ft-
intcrval between x and y, i.c. R+*x c\ R m *y, is an inductive class.
*89*11. b : R € (, Is —> 1 . xRz . - R+y • 3 • R (j'ny) = i‘.r \j R (z >—* y)
Dem.
I-. *89*102 . D b :: Hp. D :.xR+u .= :.#• = u . v . zR*u (1)
I-: Hp. j 1 ■ « . D . n * R (x*-*y) (2)
h Hp. sR+u . D : »R*y. D.uc ft (.*•*-< y) (3)
K(2).(3).Dh: Hp.D.(*rv R( z n-iy) C R (x*-<y) (4)
l-.(l). D b : Hp. D . R(x>-*y)C i*x\j ft (*>-«,/) (5)
I-. (4). (5) .OK Prop
*89*111. b . I> . ft(s ►—<//) = A
*89*112. I :R* Cls -> 1 . xRz . .rft*y — (zR+y) . D . .r = y. ft (.d-h y) = R (xt-*x\
[*89102]
*89*113. I-: R e Cls—* 1 .if C‘R . ~(*ft \ ft*r) . D . ft (* -, <T ) =, «. r
Dem.
b :. Hp . D : yft*o*. 3 .~(.rft | ft # y)
D : xft*.y . yft*a-. D . a-R^y . ~(.rft 1ft*y).
[*89 102] ^ 0.x = y:.Dh. Prop
*89*114. b : ft e Cls —* 1 . R“a Ca.xea- ft“a . D —(xft | ft**) [*89*105]
*89*115. b :. R c Cls —» 1 . R tl a Ca.xea- R“a . D . ft (.rnnar) = i‘x
[*89*113*114]
MATHEMATICAL INDUCTION
653
We now take as the definition of an inductive class the property proved in
★121-24, i.e. we put
Cls induct = p [rj e p . D, >y . 7) SJ i*y € p : A e p : D M . p e p) Df.
That is to say, if il/» $£((gy) • £- v ” i*!/}.
put Cls induct = Df.
There will be different orders of inductive classes according to the order of p.
p must be at least of the second order, since i*y is of the second order; at
least, not much could be proved if we took p to be of the first order. We put
Cls induct,* = A/+ in ‘A Df.
We have (3/*,) . A - p 3 z (^p 3 ). v = p ,. D . (g^) . v v i
Now (a/^).77 — Pi is a third-order property. Hence
‘y = ms -
*89 12. h : p e Cls induct,. D . . p = p 3
This proposition is fundamental.
*8913.
Put
h e Cls —► 1 z rj e p. D,, >y
[★8911111-112-113]
. rj v i‘y e p z A € p z ~(xR 17?* x) . xRz : D :
li (z^y) e p . D . It (xt-iy) e p
*89 131. R m (x>-iy) - R+ m ‘ x rs R^y Df
Then
* = a rn (R“a, n C a m .xea m ).\ = J3 m (R“/3 m C (3 m . y e 0 m ) . D .
Rm («»-«y) - p‘x * p‘*~
Thus R m (x>-*y) is a class of order in + 1. Moreover we have
*89 132. h R e Cls-* 1 . xRy . D (yR | R*y) . D —(xR \R*x)
Dem.
h :~(y72|72*y) . xRy . D . (ga) . R“a Qa .yea — R“a .xRy (1)
h : Hp . R“a C a . y e a — R**a . xRy . y = i*x \j i*y \j R**a .
D . R“y = l*y yj R‘y sj R“R“a . (2)
D . R“y C y (3)
h :.~(y/2| 72* y) . D :~(y72y) :
Ds xRy.D.x^y (4)
h: y e a —R“a . xRy . D . x~e a (5)
h :.~(y72|72*y). Dz~(yRhy)z
D : x72y . D . x~*R‘y (6)
- (2) - (4) . (5) - (6) • D h : Hp (2) .'*-*{yR | R*y) ."D .xey — R**y (7)
h . (3) . (7). D h : Hp (2). D .~(xR | R*x) (8)
h . (1) . (8) .Df. Prop
654
APPENDIX B
*89133. h R e Cls —► 1 : rj e p. . „ . »; v Cy e p : A e /z : x7?e : D :
^{zR | R*z) . (^HHy) € p . D | 7?*.r). It(x*-*y) e p
[*8!)'13'132]
*89 14. h c Cls -* 1 .~(y/?| R+ m y ). 3 : xR*,m+»y • 3 .
R,„ (-r»-Hy) e Cls induct**,
Pem.
By *89-133, ~(zR | R*,»z ). R„, (*>-*y) € p , u *, is a hereditary property of z if
*/ € /*»«.♦! .Di.p.yv i*y € : A € /x,„*,.
Moreover this property is of order in + 1. And by *89-113, y has this property
if ~(y/{|/f* M y). Hence .r has this property if xR*, m + lt y. Hence with this
hypothesis we have
. /*,„ (xi-«y) «,***,,
Rm ( r*-*y) € Cls induct,,,*,,
which was to Ik* proved.
*89 15. h : R € Cls -► 1 . R“a m C a m . ye a, n - R“a m . 3 :
*R*<•»+»y • 3. /?*(xi-Hy)« Cls induct**, [*8911414]
We have R m + X (xtny) C R,„ (x>-*y),
Cls induct,,,*, C Cls induct*.
The next point is to prove
p e Cls induct* . 7 C p . D . 7 C Cls induct*.
This can be proved for Cls induct,, and extended to any other order of inductive
classes. The proof is as follows.
*89 16 . h : a^e Cls induct,. y e Cls induct, .3.310-7
Deni.
Hp.3:(a^): A***,:. 3*,,.£ v €/ 4 , s 7 C/£> .(1)
A e p 3 : fi e p*. 0 f , ¥ . i'ye p 3 :y e n,. a ~ e A . A c ft*:
3 :3 ! a — A . A e/x, (2)
3 !a-/9.aC/3w‘y.3.a = /9 v e*y ( 3 )
(3) . 3 Hp (2). 3 : >9 e /x,. a ~ e /x, . 3 ! a — £ . 3 .
£w‘ye/x,.a + £vt‘y.3!a-(/9v«‘y) • (4)
(4) . 3 :. Hp (2). 3 : /9 € /x, . 3 ! a - ^. D . /3 w £‘y e ^ . g ! a _ (^ w t ‘y) (5)
(2). (5). 3 h :. Hp(2). 3 :/3e Cls induct,. 3.#e/x,. a !a-/9 ( 6 )
( 1 ). ( 6 ). 3 V . Prop
*8917. H : 7 < Cls induct,. a C 7 . D . a e Cls induct, [*8916 . Transp]
It follows that, with the hypothesis of *8915, R m (xnny), R m+l (xt-iy), etc.
are all of them inductive classes of the (m + l)th or any lower order.
MATHEMATICAL INDUCTION
655
*89 18. I- :. R e Cls —> 1 . y, « ( . ~(yR | R*#) . D : y R^ z . „ . zR ^ y
Deni.
Put f = R 3 (x hh y) r\ R 3 (x *-* z).
h . *8912 1417 . D h : Hp . D . f c Cls,, i.e. f is a class of the second order
h Hp .~(yR*zz) . D : u c £ . D . u/e*,y . a/?*,* . w + y . w + z .
[*89102] 3 . R‘ U R^ . .
C H P] D.rt'uef
b.(l).(2).Db:Hp(2>.D.ye£
b:Hp(2).D.y~e£
h . (3) . (4) -3b:. Hp . D : yR^z . v . zR^y 3 b . Prop
( 1 )
( 2 )
(3)
(4)
*8919. h:Re Cls-» 1
Dem.
h:Hp +
[*89121517]
1- : Hp(l).D.y,^/^cf
b • (1) • (2) • 3 b . Prop
R il y~t C^.\-R^ 3 x n ^ — R tl fj 7 .D.XeOul
-ft*(**-«y) n iZ,(**-<*) . 3 . R“£C g .xe ( [as above]
3-y,2€£ ( 1 )
( 2 )
*89 2. b : i* c Cls —> 1. xR^y. R s (y >-<y) c Cls induct,. 3. /i,(a;HHy) c Cls induct,
Dem.
As in *8911111112,
b i? e Cls —► 1 . a:/iz . D :
^-ft (*»-Hy) . v . R(xv-iy) = i*x. v . R(x*-iy)-> A (1)
b • (1) • 3 b Hp(l) : A^:af / i.D, iU .ayt < ut / i.O:
^ . D ./2(x»-Hy) 6 ^4 (2)
b • (2) . 3 b : e CJs —* 1 . xR+,y . 72, (y *-«y) e Cls induct,. D .
R t (arnny) e Cls induct,: Db. Prop
To deal further with the case in which y(/2| R^)y, proceed as follows:
Having proved
R e Cls —► 1 . xR^y . R, (yn-«y) c Cls induct, . 3 . R, (xt-iy) e Cls induct,,
we have to prove Rt(y*-*y) c Cls induct,; for this purpose, put
S = (- i‘y) 1 R.
Then 5 e Cls —* l . S G R.
Observe that yRy .D.R (y nny) = i*y,
yR*y - 3.22 (ynny) = t‘y ^ i<R* y .
Assume, therefore, ~ {yRy) •~(yR x y)-
We have S**y. * R“(jt — t‘y) . S“/t = R“y. — i*y. Hence
S**fjL C fx ,. R*y «/*• = • C /4 - i£'y e /*,
<S‘V C/t.y«/*. = . /2‘V C ft .ye y..
APPENDIX B
<>56
Hence ‘y = **‘^7 =R*'J-
1 lence 8 , ( R t u+-*y) = 77, ( Ft*!/*-*!/) - (•/>-*!/)
because y
Moreover xve have ~ ( 1/8 | S*y) because //~ e IPS.
Hence by *8914. R.(y*-*y)e CIs induct.. Hence generally:
*89-201. I-: 77 c CIs —* 1 . x77*,y . D . 7?,(x>-<y) € CIs induct,
We have 77,(x>-«y) C 77.<x»-«y).
11,.nee by *89*17, 77_. (xHHy) «< Ms induct.. D. 77. (.#**—•//) e CIs induct,. Hence
*89 21. h : 77 c CIs —¥ 1 . D . 77. (rwy) e CIs induct,
because ''•'(•r77*,y) • ^ • /7,(.ri—«y) — A.
*89*22. b 77 « CIs -> 1 .y % Z€ 77*/.#-. D : y77*.; . v . zR*<y
[Proof as in *89*18. using *89*21 instead of *89*14]
*89221.
*89 23.
Potid M ‘J<-(/?„)*/ R.. l>f
I- ,S\ T€ Potid/77 . D : 8R t JT. v . TR,^S [*89*22 (
*89*24. b : 77 « CIs —> 1 . 77“X C X . .r < X . D . 77*/a* C X
Here X is assumed to be of more than the third order.
Dem.
f-Hp . y « 77*/. r - X.):;tXft 77, (xhHy). D. * + y .
D . 77‘r ( X a 77, (xn-iy) (1)
h . *89*2117*12 . D h : Hp. D . (g/O. X n 77,(xi-«y) = (2)
H • (I). (2) • D b : Hp (1). D . (g/i-). X ^ 77, (.r^ny) = ^. 77“***C^•
D . 77*a‘x C\r\ R t (x>-ny) (3)
4 —
h . (3) . D b Hp. D : y 6 77*/.r - X . I) . y * X :
D : 77*/x CX:. DK Prop
Hence if X is an inductive class, it can be used in an induction no matter
what its order may be, if 77 € CIs—> 1.
*8925.
*8926.
h : 77 € 1 —* CIs. D . 77^ (xt-ny) e CIs induct,
h 77 e 1 —» CIs . y, z c 77*,‘x. D : y77*j2 . v
MATHEMATICAL INDUCTION
657
*89 28. h-.R e (1 -» CIs) u (Cls —» 1). D . R^ = i‘Potid,‘«
Dein.
y-.Tt Potid/ii . xT<j . yRz .0 . T\ Re Potid JR ,x(’T\R)z
Hence h : s'Pot id/ft = ft . D . R“*S‘x C*S‘x ( 1 )
I- .(1). *89-24.3h/(t Cls —»l.Hp(l).D (2)
h :• Hp(l). ft“/* C y.. 0 : iF'x C M . D . ft“7\, C M :
Jnr.^.S'C,. (3)
K(3).Dh:Hp(l).D.S' s C V* (4)
K . (2) . (4) . D h : Hp(2) . Z> .S'* = ft*/® (5)
H . (5) . 3 I- : R e Cls -» 1 . D . ft*, - i«Potid/ft ( 6 )
Similarly 1- : R e 1 -» Cls . 3 . ft*, = i‘Potid/ft ( 7 )
I-. ( 6 ). (7) . D I-. Prop
*89 29. h:R e (l~. Cls) w(Cls —►1).D. ft* otml = ft*, [*89 24-27]
We have now to obtain an analogous result when R is not one-many or
many-one. For this purpose, we use R,, which is one-many.
We P rove «*,»«'* = »'(/£.
whence, since = (A)*,,
it follows that R*u+m = ft* s ,
so that for a relation which is not one-many or many-one we obtain the ad¬
vantages of unlimited induction by proceeding to ft*,. The proof is us follows.
*89 3. 1 - : ft, = ft. D . *‘S+„‘i‘x C~R mm ‘x
Devi.
h Hp . D ::aS+i‘x.= D f . R“£en: . aep
D l*x € CVy : £ e C\‘y . D, . R“% « 01*7 : =>y • * * Cl ‘7
3 :• x e y • R“y Q y. D y . a C 7 :.
D a C D h . Prop
*89 31. h : R< = D . R+^l^'x C 8 7 3+ m *i t x
Dem.
h. *89101 .Dh.S|S # GS # .
D h . S“7?+‘i‘x C .
D h . C (1)
h . (1) . *40-38 .DhtHp.D. R^s^S^^x C (2)
= J (i‘* e /j . S'V C . 3.«SVt‘ar = s i p t \ (3)
h . (2) . D h s *‘^‘*‘ 0 : £ Clan . D . ll^'x C 8^S+‘l‘x (4)
h . (3) . D h . a'S'U/i'x « CIs^, (5)
J-. (4) . (5) . D h . Prop
r & w 1 42
A PI’KXDIX It
■•89 32. H = .s*‘( /<. >* *.r
I h’ ni.
H . *89 3 29 . D h . .v‘( /;.•)*‘.r C 7^‘r
h . ( I ). *89'31 . D h . Prop
*89 33. h . /f*.,.- /.’*
prill.
As ill *K9-:p>. h . /(+«.„,‘x = .v‘(
= .v'(/M#V
[*89-321 = . D h . Prop
*89 34. : ylt*,r . .» € X . /f“X C X . D . .y c X [*89 33]
11 ere X is supposed to he of any order, however high. Hence, so far ns
inatheniatical induction is concerned, all proofs remain valid without the
asiom of redueihilily provided " lt.+ is underst«>od to mean "
APPENDIX C
TRUTH-FUNCTIONS AND OTHERS
•
In the Introduction to the present edition we have assumed that a function
can only enter into a proposition through its values. We have in fact
assumed that a matrix f\ ( <f> ! 2) always arises through some stroke-function
F{p, q, r, ...)
by substituting </> ! a, <f> ! 6, <f> ! c, ... for some or all of p, q, r, .... and that all
other functions of functions are derivable from such matrices by generalization
— i.e. by replacing some or all of a, b, c, ... by variables, and taking “all
values” or "some value.”
The uses which we have made of this assumption can be validated by
definition, even if the assumption is not universally true. That is to say, we
can decide that mathematics is to confine itself to functions of functions which
obey the above assumption. This amounts to saying that mathematics is
essentially extensional rather than intensional. We might, on this ground,
abstain from the inquiry whether our assumption is universally true or not.
The inquiry, however, is important on its own account, and we shall, in what
follows, suggest certain considerations without arriving at a dogmatic con¬
clusion.
There is a prior question, which is simpler, and that is the question
whether all functions of propositions are truth-functions. Or, more precisely,
can all propositions which do not contain apparent variables be built up from
atomic propositions by means of the stroke? If this were the case, we should
have, if fp is any function of propositions,
psq.O.fp =/q.
Consequently, according to the definition *13 01,
psq.O .p-q.
There will thus be only two propositions, one true and one false. This was
Frege’s point of view, but it is one which cannot easily be accepted. Frege
maintained that every proposition is a proper name, either for the true or for
the false. On grounds not connected with our present question, we cannot
regard propositions as names; but that does not decide the question whether
equivalent propositions are identical. It is this latter question that concerns
us. • That is to say, we have to consider whether, or in what sense, there are
functions fp which are true for 6ome true values of p and false for other true
values of p.
, Two obvious prxmd facie instances are “ A believes p ” and “ p is about A.”
We may take these instances as crucial. If A believes p and p is true, it does
42—2
APPENDIX C
660
not follow that A believes every other true proposition q : nor, if A believes p,
and p is false, does it follow that A believes every other false proposition q.
Again, the proposition "A is mortal ” is about A ; but the proposition “B is
mortal,” which is equally true, is not about A. Thus the function “ p is about
A " is not a truth-function of p. This instance is important, because the
notation “<£./ ” is used to denote a proposition about x t and thus the conception
involved seems to be presupposed in the whole procedure of propositional
functions.
We must, to begin with, distinguish between a proposition as a fact and
a proposition as a vehicle of truth or falsehood. The following series of black
marks: "Socrates is mortal, is a fact of geography. The noise which I should
make if I were to say "Socrates is mortal " would be a fact of acoustics. The
mental occurrence when I entertain the belief “ Socrates is mortal ” is a fact
of psychology. None of these introduces the notion of truth or falsehood,
which is, for logic, tin* essential characteristic of propositions. We shall return
in a moment to tin' consideration of propositions as facts.
When we say that truth or falsehood is, for logic, the essent ial characteristic
of propositions, we must not be misunderstood. It does not matter, for mathe¬
matical logic, what constitutes truth or falsehood; all that matters is that
they divide propositions into two classes according to certain rules. Let us
take a set of marks
■*j. • r j. ••• •'*»»•
Let us put, as unexplained assertions,
F^) <m<n).
Lot us further introduce the symbol x r \ x t , and assume
T(x r | x.) if F(x r ) or F(x a )\
F(x r \x,) if T(x r ) and T(x a ).
Assume further that, if p, q. s are any one of the x's or any combination of
them by means of the stroke, the above rules are to apply to p\q, etc., and
further we are to have:
T\p\(p\p)l
T\p0q.D .s\qDp\s],
where " p D q ” means " p \ (q | q ).” Further: given T (/> J ( q | r)J and T(p), we
are to have T (r).
Taking the above as mere conventional rules, all the logic of molecular
propositions follows, replacing “h ,p“ by "T(p).”
Thus from the formal point of view it is irrelevant what constitutes truth
or falsehood : all that matters is that propositions are divided into two classes
according to certain rules. It does not matter what propositions are, so long
as we are content to regard our primitive propositions as defining hypotheses,
TRUTH-FUNCTIONS AND OTHERS
66 I
not as truths. (From a philosophical point of view, this formal procedure may
be shown to presuppose the non-formal interpretation of our primitive pro¬
positions; but that does not matter for our present purpose.)
Throughout the logic of molecular propositions, we do not want to know
anything about propositions except whether they are true or false. Further,
we are concerned only with those combinations of propositions which are true
in virtue of the rules, whether their constituent propositions are true or false.
That is—to take the simplest illustration—we assert p\(p\p), but we never
assert any proposition j) that has not some suitable molecular structure,
although we believe that half of such propositions are true. Our assertions
depend always upon structure, never upon the mere fact that some proposition
is true.
A new situation arises, however, when we replace p by <f>lx. For example,
we have ^ -p\(p\p)
and we infer h . 0 ! x\(<f> l x\<f> l x).
We cannot explain the notation <f >! x without introducing characteristics of
propositions other than their truth or falsehood. Take for example the
primitive proposition (*811)
h . ( 3 *) . 4 >! x | (<f> l a | <f> ! b).
1 he truth of this proposition depends upon the form of the constituent pro¬
positions <f>lx, (pia, <p\b, not simply upon their truth or falsehood. It cannot
be replaced by
“»-.< a j>).p|(g|r)/'
which is true but does not have the desired consequences. We are therefore
compelled to consider what is meant by saying that a proposition is of the
form <f>la (where a is some constant). This brings us back to “ A occurs in p,"
which we gave above as an example of a function which is not a truth-function.
And this, we shall find, brings us back to the proposition as fact, in opposition
to the proposition as true or false.
Let us revert to our two instances: “A believes p" and “p is about A.”
We shall avoid certain psychological difficulties if we take, to begin with,
“A asserts p ” instead of "A believes />.” Suppose “/>” is “Socrates is Greek.”
A word is a class of similar noises. Thus a person who asserts " Socrates is
Greek ” is a person who makes, in rapid succession, three noises, of which the
first is a member of the class “ Socrates,” the second a member of the class
“ and the third a member of the class “ Greek.” This series of events is
part of the series of events which constitutes the person. If A is the series of
events constituting the person, a is the class of noises “ Socrates,” 0 the class
“is,” and y the class “Greek,” then “A asserts that Socrates is Greek” is
(omitting the rapidity of the succession)
( 3 *. y,*).a;ea.ye&.zey.xlywxlzKjylzCiA.
It is obvioas that this is not a function of p as p occurs in a truth-function.
42—3
GG2
APPENDIX C
1 1 we now take up ".l believes />.” we find the matter rather more com¬
plicated, owing to doubt as to what constitutes belief. Some people maintain
that a proposition must be expressed in words before we can believe it: if
that were so, there would not. from our point of view, be any vital difference
between believing and asserting. But if we adopt a less unorthodox stand¬
point, we shall say that when a man believes “Socrates is Greek" he has
simultaneously two thoughts, one of which “ means ’ Socrates while the other
“ ttH-ans Greek, and these two thoughts are related in the way we call
“predication.” It is not necessary for our purposes to define “meaning,"
beyond noticing that two different thoughts may “have the same meaning."
The relation having the same meaning" is symmetrical and transitive;
moreover, if two thoughts “have the same meaning,” either can replace the
oth. r in any belief without altering its truth-value. Thus we have one class
of thoughts, called “ Socrates." which all “have the same meaning"; call this
class a. We have another class of thoughts, called " Greek," which all “ have
the same meaning ; call this class Call the relation of predication between
two thoughts P. (This is the relation which holds between our thought of
the subject and our thought of the predicate when we believe that the subject
has the predicate. It is wholly different from the relation which holds between
the subject find the predicate when our belief is true.) Then "A believes
that Socrates is Greek " is
<:*K. &)•***•!/*&. jP'J . X, j/‘t C‘A.
Here, again, the proposition .as it occurs in truth-functions has disappeared.
It is not necessary to lay any stress upon the above analysis of belief,
which may bo completely mistaken. All that is intended is to show that
"A believes p may very well not be a function of />. in the sense in which
p occurs in truth-functions.
Wo have now to consider "p is about A.” e.g. “ * Socrates is Greek ’ is about
Socrates." Here wc have to distinguish (1) the fact, (2) the belief, (3) the
verbal proposition. The fact and the belief, however, do not raise separate
problems, since it is fairly clear that Socrates is a constituent of the fact in
the same sense in which the thought of Socrates is a constituent of the belief.
And tlie verbal proposition raises no difficulty, since each instance of the
verbal proposition is a series containing a part which is an instance of
" Socrates." That is to say, “ Socrates ” (the word) is a class of scries of noises,
say \; and “ Socrates is Greek ” is another class of series, say p ; and the fact
that “ Socrates " occurs in “ Socrates is Greek " is
Pep.D .(aQKQcX.QGP.
Thus we are left with the question : What do we mean by saying that Socrates
is a constituent of the fact that Socrates is Greek ? This raises the whole
problem of analysis. But we do not need an ultimate answer; we only need
TRUTH-FUNCTIONS AND OTHERS
663
an answer sufficient to throw light on the question whether there are functions
of propositions which are not truth-functions.
There are those who den}' the legitimacy of analysis. Without admitting
that they are in the right, we can frame a theory which they need not reject.
Let us assume that facts are capable of various kinds of resemblances and
differences. Two facts may have particular-resemblance; then we shall say
that they are about the same particular. Again they may have predicate-
resemblance, or dyadic-relation-resemblance, or etc. We shall say that a fact
is about only one particular if any two facts which have particular-resemblance
to the given fact have particular-resemblance to each other. Given such a
fact, we may define its one particular as the class of all facts having particular-
resemblance to the given fact. In that case, to say that Socrates is a con¬
stituent of the fact that Socrates is Greek (assuming conventionally that
Socrates is a particular) is to say that the fact is a member of the class of
facts which is Socrates. In the case of a belief about .Socrates, which is itself
a fact composed of thoughts, we shall say that a belief is about Socrates if it
is one of the class of facts constituting a certain idea which “ means ” Socrates
in whatever sense we may give to “ meaning.” Here an “ idea ” is taken to be
a class of psychical facts, say all the beliefs which “refer to” Socrates.
We can define predicates by a similar procedure. Take a fact which is
only capable of two kinds of resemblance such as we are considering, namely
particular-resemblance and predicate-resemblance; such a fact will be a
subject-predicate fact. The predicate involved in it is the class of facts to
which it has predicate-resemblance.
We shall assume also various kinds of difference: particular-difference,
predicate-difference, etc. These are not necessarily incompatible with the
corresponding kind of resemblance; e.g. It (x, x) and R (x, y ) have both
particular-resemblance in respect of x and particular-difference in respect of
y. This enables us to define what is meant by saying that a particular occurs
twice in a fact, as x occurs twice in R(x,x). First: R{x,x) is a dyadic-
relation-fact because it is capable of dyadic-relation-resemblance to other facts;
second: any two facts having particular-resemblance to R (x, x) have particular-
resemblance to each other. This is what we mean by saying that R(x, x) is a
dyadic-relation-fact in which x occurs twice, not a subject-predicate fact. Take
next a triadic-relation-fact R(x,x,z). This is, by definition, a triadic-relation-
fact because it is capable of triadic-relation-resemblance. The facts having
particular-resemblance to R(x, x, z) can be divided into two groups (not three)
such that any two members of one group have particular-resemblance to each
other. This shows that there is repetition, but not whether it is x or z that
is repeated. The facts of the one group are R (x, x, c ) for varying c; the facts
of the other are R (a, b, z) for varying a and b. Each fact of the group R ( x , x, c)
belongs to only two groups constituted by particular-resemblance, whereas
APPENDIX C
tin* tacts of the* group R{n, b, z) % except when it happens that a = b, belong
t«» throe groups constituted by particular-resemblance. This defines what is
un-ant by saying that x occurs twice and z once in the fact R (x, .r, z). It is
obvious that we can deal with tctradic etc. relations in the same way.
According to the above, when we say that Socrates is a constituent of the
lad that Socrates is Greek, we mean that this fact is a member of the class
of facts which is Socrates.
W hen we use the notation “falx" to denote a proposition in which “ x"
occurs, it is a fact that '* .r " occurs in “falx" but we do not need to assert
the- tact; the fact does its work without having to be asserted. It is also a
fad that, if " x ” occurs in a proposition p, ami p asserts a fact, then x is a
constituent of that fact. This is not a law of logic, but a law of language. It
might be false in some languages. For instance, in former days, when a crime
was committed in India, the indictment stated that it was committed “in the
manor of Hast (Jreemvich. These words did not denote any constituents of
the fact. Hut a logical language avoids fictions of this kind.
Tim notation for functions is an illustration of Wittgenstein’s principle,
that a logical symbol must, in certain formal respects, resemble what it sym¬
bolizes. All the facts of which x is a constituent, according to the above,
constitute a certain class defined by particular-resemblance. The various
symbols far, fax, %x, ... also all resemble each other in a certain respect, namely
that their right-hand halves are very similar (not exactly similar, because no
two ./'s are exactly alike). The symbols R (x.x). R(x,x,z), etc. are appropriate
to their meanings for similar reasons. The symbols are used before their
suitability can be explained. To explain why “far" is a suitable symbol for a
proposition about x is. as we have seen, a complicated matter. But to use the
symbol is not a complicated matter. Our symbolism, as a set of facts, resembles,
in certain logical respects, the facts which it is to symbolize. This makes it
a good symbolism. But in using it we do not presuppose the explanation of
why it is good, which belongs to a later stage. Ami so the notation “far" can
be used without first explaining what we mean by “a proposition about x"
Wc are now in a position to deal with the difference between propositions
considered factually and propositions as vehicles of truth and falsehood. When
we say "'Socrates’ occurs in the proposition ‘Socrates is Greek,”’ we are
taking the proposition factually. Taken in this way, it is a class of series, and
' Socrates ’ is another class of series. Our statement is only true when we take
the proposition and the name as classes. The particular * Socrates ’ that occurs
at the beginning of our sentence does not occur in the proposition ‘Socrates
is Greek’; what is true is that another particular closely resembling it occurs
in the proposition. It is therefore absolutely essential to all such statements
to take words and propositions as classes of similar occurrences, not as single
occurrences. But when we assert a proposition, the single occurrence is all
TRUTH-FUNCTIONS AND OTHERS
665
that is relevant. When I assert “ Socrates is Greek,” the particular occurrences
of the words have meaning, and the assertion is made by the particular oc¬
currence of that sentence. And to say of that sentence Socrates’ occurs in
it” is simply false, if I mean the ‘Socrates’ that I have just written down,
since it was a different ‘ Socrates’ that occurred in it. Thus we conclude:
A proposition as the vehicle of truth or falsehood is a particular occurrence,
while a proposition considered factually is a class of similar occurrences. It is
the proposition considered factually that occurs in such statements as "A
believes p ” and “p is about A."
Of course it is possible to make statements about the particular fact
“ Socrates is Greek.” We may say how many centimetres long it is; we may
say it is black ; and so on. But these are not the statements that a philosopher
or logician is tempted to make.
When an assertion occurs, it is made by means of a particular fact, which
is an instance of the proposition asserted. But this particular fact- is, so to
speak, “transparent”; nothing is said about it, but by means of it something
is said about something else. It is this “ transparent ” quality which belongs
to propositions as they occur in truth-functions. This belongs to p when p is
asserted, but not when we say “ p is true.” Thus suppose we say: “All that
Xenophon said about Socrates is true.” Put
X (/>).*= • Xenophon asserted p,
S (/>). = . p is about Socrates.
Then our statement is
X (p ). S ( p ) . D p . p is true.
Here the occurrence of p is not “ transparent.” But if we say
x e a . D x . <f >! x
we are asserting <i >! x for a whole class of values of x, and yet “ <f>lx'' still has
a “transparent” occurrence. The essential difference is that in the former
case we speak about the symbol or belief, whereas in the latter we merely use
it to speak about something else. This is the point which distinguishes the
occurrences of propositions in mathematical logic from their occurrences in
non-truth-functions.
Let us endeavour to give greater definiteness to this point. Take the
statement “Socrates had all the predicates that Xenophon said he had." Let
the series of events which was Xenophon be called X. Then if Xenophon
attributed the predicate a to Socrates, we might appear to have (writing
x -i y X z -i w f° r series x, y, z, w)
Socrates X had X predicate |aGX
Thus our assertion would be
Socrates ^ had ^ predicate ^ a G X . . Socrates had predicate a.
Here, however, there is an ambiguity. On the left, “ Socrates,” “ had,” “ pre¬
dicate” and “a” occur as noises; on the right they occur as symbols. This
GGG
APPENDIX C
ambiguity amounts to a fallacy. For, in fact, what I write on paper is not the
noise that Xenophon made, but a symbol for that noise. Thus I am using
one symbol " Socrates " in two senses: (a) to mean the noise that Xenophon
made on a certain occasion. (M to mean a certain man. We must say :
If Xenophon made a series of noises which mean what is meant by “Socrates
had the predicate a," then what this means is true.
For example: If Xenophon said “Socrates was wise,” then what is meant
by “Socrates was wise” is true.
Hut this does not assert that Socrates was wise. When I actually assert
that Socrates was wise, I say something which cannot be said by talking about
the words I use in saying it; and when I assert that Socrates was wise, although
an instance of t he proposition occurs, yet I do not say anything whatever about
the proposition — in particular I do not say that it is true. This is an inference,
not logical, but linguistic.
If the above considerations in any way approximate to the truth, we see
that there is an absolute gulf between the assertion of a proposition and an
assertion about, the proposition. The p that occurs when we assert p and the
p that occurs in "A asserts p " are by no means identical. The occurrence of
propositions as asserted i*» simpler than their occurrence as something spoken
about. In the assertion of a proposition, and in the assertion of any molecular
function of a proposition, the proposition does not occur, if we mean by the
proposition the p that occurs in such propositions as " A asserts p " or "p is
about A." When these latter are analysed, they are found not to conflict with
the view that propositions, in the sense in which they occur when they are
asserted, only occur in truth-functions.
When p is asserted, p does not really occur, but the constituents of p occur,
or an instance of p occurs. The same is true when a molecular proposition
containing p is asserted. Thus we cannot infer p = tj, because here p and 7
occur in a sense in which they do not occur when molecular propositions con¬
taining them are asserted.
Similar considerations apply to propositional functions. Suppose there are
two predicates a and f3 which are always found together: we may still say
that they are two, on the ground that a(.r) and £ {x) are fact* which do not
have predicate-resemblance. Hut the propositional function a (.'!') is solely to
be used in building up matrices by means of the stroke. The predicate a is
a class of facts, whereas the propositional function a (?) is merely a symbolic
convenience in speaking about certain propositions. Thus we may have
a (?) = &(?) without having a = &. In this way we escape the primd facie
paradoxes of the theory that propositions only occur in truth-functions, and
propositional functions only occur through their values. The paradoxes rest
on the confusion between factual and assertive propositions.
LIST OF DEFINITIONS
101.
pO<j
13 03. x ~y = z
233.
pvqvr
14 01. [(?*) (<f>x)] . + (ix) ( <f>x)
301.
p.q
14 02. E l(ix)(<t>x)
302.
p D q D r
14 03. [(»x)(^),0.r)(^)] ./[(ia:)(<px),
( ix)(>jrx ))
14 04. [OxX'l'*)].f{Ox)($x)Xlx)('lrx)\
401.
P = 9
402.
p = t/ = r
20 01. f\$(>\rz ))
4-34.
p.q .r
20 02. xe(<t>l$)
901.
~ {(«). <M
2003. CIs
9011.
~(x). <f>x
20 04. x,y e a
902.
—{(a :«)•♦*)
20 05. x,y,zea
9021.
~(Rx) . <f>x
20 06. x~ea
903.
( x).<f>x.v.p
20 07 (<*)./«
904.
p • V . (x) . <t>x
20 071. (a«)./«
905.
(•7{x) . <px . V ,p
20 072. [(?a) (<*>a)] . /(?«) (<*>a)
906.
p • v. (g*). <t>x
20 08. /|5(Vra)|
907.
(x). <f>x . v . (a y) • iry
20 081. a e yfr l a
908.
(a y) . yfrr/ . v . (x) . <f>x
2101. /f£p*(*.y)l
1001.
<3*> . 4>x
2102. a
1002.
<f>x D x \frx
2103. Rel
1003.
(f>x = x yjrx
21 07. (72) .fit
1101.
(x,y).<f>(x,y)
21*071. (a«)-/«
1102.
(x, y,z) . <f> (x, y, z)
21072. [(»/*) (4>R)] .f{iR) (<f>R)
1103.
<a*> y ) • <t> (*, y)
2108.
1104.
(a 37 . y>z)-4> ( x > y> z )
21 081. P {<f> ! <£, S) ) Q
1106.
<f>(x,y).D x . v .yfr(x,y)
21082. /(^(^i2)|
1106.
<f>(x,y).= miy .yfr(x,y)
21083. Re<f>lR
1301.
*=y
2201. flC/3
1302.
x^y
2202. arx&
008
LIST OF DEFINITIONS
2203.
34 03.
R-
2204.
— a
3501.
<*1 R
2205.
3502
ji r/3
2253.
a r\ ft r\ y
35 03.
2271.
35 04.
23 01.
It G .S'
3505.
R‘* T 0
2302.
li*S
35 24
a-\R\S
2303.
It u .S’
3525.
S\Rf 0
2304.
^ It
3601.
n«
2305.
It - .S’
3701.
R"0
2353.
It* S* T
37 02.
R.
2371.
It u .S’ o T
37 03.
li.
2401.
V
37 04.
R"'k
2402.
A
37 05.
E !! R“0
2403.
a
3801.
•'?
2501.
V
3802.
2502.
A
38 03.
«1>J
2503.
a**
4001.
p‘«
3001.
It*!/
4002.
S*K
3002.
&&•/
4101.
j/\
3101.
Cnv
4102.
s*\
3102.
I*
4301.
R\\S
3201.
It
5001.
I
3202.
tT
5002.
J
3203.
s g
5101.
i
3204.
g s
5201.
1
3301.
i)
5401.
0
3302.
a
54 02.
2
3303.
c
5501.
x lu
3304.
F
6502.
3401.
R\S
5601.
2
34 02.
R i
56*02.
2 r
LIST OF DEFINITIONS
669
5603.
0,
65 03.
R*
6001.
Cl
6504.
R(x)
6002.
Cl ex
651.
R(x .y)
6003.
Cls 3
6511.
6004.
Cls 3
65 12.
R(x,y)
6101.
R1
7001.
6102.
R1 ex
7301.
a sm >9
6103.
Rel 3
7302.
sm
6104.
Rel 3
8001.
r*
6201.
€
8401.
Cls 3 excl
6301.
t‘x
84 02.
Cl excl'?
63011.
t u x
84 03.
Cls ex 3 excl
6302.
855.
p lv
63 03.
ti .
8801.
Rel Mult
6304.
t u x
88 02.
Cls 3 Mult
63041.
t u x
8803.
Mult ax
6305.
t*K
9001.
R*
63051.
U'k
9002
R»
6401.
Ua
9101.
R.t
64 011.
t"‘x
9102.
Ru
64 012.
t”‘x
9103.
Pot -R
64013.
t*‘x
9104.
Potid -R
64 014.
t w x
9105.
iJpo
64 02.
t n *a
. 93 01.
B
64 021.
tio'a
9302.
rain/.
64 022.
t u ‘«
93021.
max/.
64 03.
U‘a
9303.
gen'P
64 031.
U u «
9601.
<P*Q) Dft [*95]
6404.
V«
9601.
J R *x Dft [*96]
64-041.
V«
96 02.
J R ‘x Dft [*96]
65*01.
9701.
R‘x
6502.
a(x)
10001.
Nc
670 LIST OF DEFINITIONS
10002.
XC
11201.
10201.
NC d (a)
11202.
2Nc‘*
10301.
X 0 c *a
11302.
£ X Q
10302.
N„C
11303.
M Xc"
10401.
X'c'a
11304.
Nc‘£ x cf t
104011.
XVa
11305.
ft x c Nc‘a
104 02.
X'C
113511.
a x x 7
104 021.
N*C
113541.
/x x 0 i/ x c cr
104 03.
11401.
II NV*
104 031.
n*
11501.
Prod‘*
10501.
X,c-«
11502.
C Is 5 atri thm
105011.
X,c‘«
11601.
a exp/3
10502.
N,C
11602.
105021.
x,c
116 03.
(Nc‘a)*
10503.
M(l»
11604.
105031.
11701.
10601.
N,oC*a
11702.
ft > Nc‘a
106011.
N»'c‘a
11703.
Nc‘a > »»
106012.
N ol c‘a
11704.
ft<v
106 02.
N 0 'c‘a
11706.
ft^v
106021.
•N oC ‘a
11706.
ft^v
106 03.
N«C
11901.
y-c*
106 04.
/*«»»
11902.
Xc‘a — 0 */
106041.
11903.
7-cNc‘/3
11001.
a 4- /3
12001.
NC induct
11002.
120011.
N<C induct
11003.
Nc‘a +o /a
12002.
Cls induct
11004.
M + c Nc‘a
120021.
Cls f induct
110561.
+0 ^ +0
12003.
Infin ax
11101.
* sui 8m X
12004.
Infin ax (*)
11102.
Crp
12043.
spec‘/3
11103.
Sill Sill
12101.
P(,*-y)
LIST OF DEFINITIONS
121011. P (x —t y)
161213.
x*±y 4+ P
121 012. P (x y)
16201.
2‘P
121013. P(xhi/)
16301.
ReP excl
12102.
P*
16401.
Psmor smor
12103.
finid'P
16402.
smor smor
121031.
fin‘P
16601.
QxP
12104.
Vp
166 421.
PxQxP
12201.
Prog
17001.
Pc
12301.
No
17002.
P .c
12302.
N Dft [*123—4]
17101.
Pdf
12401.
Cls refl
17102.
Pm
12402.
NCrefl
17201.
Tl‘P
124 021.
Nc *p e NC refl
17301.
Prod‘P
124 03.
NC mult
17401.
RcParithm
12601.
NCind
17601.
P exp Q
16001.
S'>Q
17602.
po
16002.
SfQ
18001.
P + Q
16003.
Q%y
18002.
fit + v
16004.
R'S'yQ
18003.
Nr ‘P + v
16006.
R'’S'’Q
18004.
fit- + Nr ‘Q
16101.
P emor Q
180 661.
fit -i- v + w
16102.
smor
18101.
P
16201.
Nr
181011.
x+¥ P
16202.
NR
18102.
M + l
16301.
1.
181021.
i + b
16401.
NRr(2T)
18103.
Nr‘/> + i
16601.
Nor‘P
181-031.
i + Nr‘.P
16602.
N 0 R
18104.
i + i
160 , 01.
P*Q
181661.
m + 1 + 1
16101.
P -\+x
181-671.
14-14-/*
16102.
xM-P
18201.
S
161-212. P-frx-fry
18301.
2Nr‘P
672
LIST OF DEFINITIONS
18401.
A* X v
23101.
PR~Q
184 02.
Nr ‘P x *
23102.
PK..Q
184 03.
m X Nr‘«
232 01.
(PRQ)^a
18432.
n X v X ST
23202.
(P RQ) 0 ,‘a
18501.
nNr‘7*
23301.
(.PPQbmx
18601.
A* exp r v
23302.
R(PQ)
186 02.
(Nr*7 J )exp r v
23401.
sc (P.Qyii
186 03.
Aicxp, (Nr*Q)
234 02.
os (P.QYli
20101.
trails
234 03.
chPQ)‘P
20201.
cnnncx
234 04.
con tin (7*<2)*7?
204 01.
Ser
234 05.
/* con tin Q
20601.
se«|/.
25001.
Bord
206 02.
prcci*
25002.
il
20701.
It/.
25101.
NO
207 02.
tl /.
25401.
less
207 03.
limax/.
254 02.
A*.
207 04.
limin/-
25501.
<
208 01.
cror‘7*
25502.
>
21101.
sect* 1*
25503.
N 0 O
21201.
9*7*
26504.
21202.
syni‘7*
25505.
21301.
A
25506.
/X< Nr‘7*
21401.
Dccl
25607.
Nr‘7* <
21402.
semi I)ed
25601.
M Dft [*256]
21601.
str'P
25602.
TV Da [*256]
21601.
Si-
25701.
(7 i*Q)*x
216 02.
dense* P
25702.
Q,u
21603.
closed*/*
25901.
A Da [*256]
21604.
perPP
25902.
A Ir Da [*256]
21606.
V*7 J
25903.
w A
23001.
72<2c„«
26001.
An
23002.
Qcn
26101.
Ser infin
LIST OF DEFINITIONS
673
26102.
fl infin
27604.
T P Dft [*276]
26103.
Ser fin
27605.
TV* Dft [*276]
26104.
n fin
30001.
U
26105.
H induct
30002.
Rel num
262 01.
NO fin
30003.
Rel num id
26202.
NO infin
30101.
R p Dft [*301]
26203.
Hr
30102.
num (R) Dft [*301]
26301.
CO
30103.
R*
263 02.
N
Dft [*263]
30201.
Prm
26401.
Ppr
Dft [*263]
30202.
(p, <x) Prm r (p, v)
264 429.
1 X a
30203.
(p, a) Prm (p, v)
26501.
a>,
302 04.
hcf (p,v)
26502.
N,
302 05.
1cm (p, u)
26503.
0>i
30301.
p/v
26504.
303 02.
0,
26505.
M
Dft [*265]
30303.
00,
26606.
N
Dft [*265]
30304
Rat
27001.
Comp
30305.
Rat def
27101.
med
30401.
X< r Y
27201.
T pq
304 02.
H
27301.
V
304 03.
H'
27302.
Rspq'T
Dft [*273]
30501.
X x, Y
27303.
(RS)pq
Dft [*273]
30601.
X+.Y
27304.
T RSPQ
Dft [*273]
30701.
Rat „
27401.
Rn
307011.
Rat,
27402.
Pn>‘«
Dft [*274]
307 02.
<n
27403.
Tp‘«
Dft [*274]
307 021.
>n
27404.
Mp<K
Dft [*274]
307 03.
<*
27601.
e
307 031.
>*
27601.
Ps
307 04.
Hn
27602.
A
Dft [*276]
307*05.
u.
27603.
Pm‘\
Dft [*276]
30801.
X-.Y
C74
LIST OF DEFINITIONS
30802.
-Y +, Y
33401.
30901.
X x, r
334 02.
FM trs
31001.
(-)
334 03.
FM connex
310011.
c-r
334 04.
FM sr
31002.
©n
334 05
FM sisyin
310021.
33501.
init'*
31003
<->*
33502
FM init
31101.
concord (/*,«». ...)
33601.
K
31102.
M +/* v
336011.
u.
31201.
H-,.v
336 02.
A n
31202.
H +„ v
35101.
FM subin
31301.
H- x«. v
35201.
T.
31401.
X + r Y
35202.
T m
31402.
X x ( Y
35301.
FM rt
31403.
35302.
FM ex
31404.
M +„ N
35303.
FM rt ex
31405.
M x. N
35401.
*9
33001.
ci ‘a
36402.
cx n *\
33002.
Abel
354 03.
FMgrp
33003.
fin'a
35601.
x m
33004.
FM
37001.
FM cycl
33005.
K*
37002.
K.
33101.
etmx**
37003.
/.
33102.
FM conx
37101.
w m
33201.
rep,‘P
37201.
V .
33301.
*5
37301.
M„ Dft [#373—5]
333011.
K *b
37302.
Prime
33302.
FM up
37303.
Dft [#373—5]
33303.
FM sip conx
37601.
(/*/*)«
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