A System of Logistic [Reprint 2014 ed.] 9780674435155, 9780674431652

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A SYSTEM OF LOGISTIC

LONDON : HUMPHREY MILFORD· OXFORD UNIVERSITY PRESS

A SYSTEM OF LOGISTIC BY

WILLARD YANORMAN QUINE, PH.D. SOCIETY Ο Γ F E L L O W S , H A R V A R D U N I V E R S I T Y

CAMBRIDGE · MASSACHUSETTS

HARVARD U N I V E R S I T Y PRESS 1934

COPYRIGHT, 1984 BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE

PRINTED AT THE HARVARD UNIVERSITY PRESS CAMBRIDGE, MASS., U . S . A .

TO NAOMI

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. RUSSELL

ACKNOWLEDGMENT I FAM indebted to Professors Whitehead and Sheffer for valuable criticism during the preparation of this work; also to Dr. Tarski, who convinced me of the defects of a certain subsidiary theory which I had included in an earlier draft. Discussion with the Harvard logicians and with Professors Carnap, Lukasiewicz, and Lesniewski has been an important source of stimulation. I wish to express my gratitude to the Department of Philosophy and the Society of Fellows of Harvard University, who provided the funds for publishing the book. For help in the initial arrangements of publication, during my absence, I am grateful to Professor Lewis. W. V. Q. HARVARD UNIVERSITY

March,

1934

FOREWORD I N T H I S book Dr. Quine has effected an extension of the scope of Symbolic Logic. The advance is more than an improvement in symbols. It extends to fundamental notions. He has introduced a generality adequate to the complexity of the subject matter; and the symbolism embodies the generality of its meaning. I have no hesitation in stating my belief that Dr. Quine's book constitutes a landmark in the history of the subject. The basic idea from which the book starts is that of 'ordination.' This idea includes the traditional subject-predicate doctrine as one of its particular exemplifications. But it is immensely more general. Logic should be adequate to explore every type of the connexity of things. The concept of 'ordination' makes some advance towards this ideal. Naturally it is a notion difficult to grasp in its generality. The danger of such generality is that it may become unworkable: that it may refuse to march. But in Dr. Quine's treatment a remarkable facility of treatment is achieved. For example, theorems concerning propositional functions involving any number of variables are special instances of one general exposition. In Principia Mathematica — to take an example to which Dr. Quine often refers — propositional functions with one variable and those with two variables have to be treated separately. Dr. Quine's theorems apply indifferently to such functions with any number of variables. Naturally, the auxiliary ideas of the book are of a generality which requires some effort to grasp. But the advance thereby effected is fundamental. In the modern development of Logic, the traditional Aristotelian Logic takes its place as a simplification of the full problem presented by the subject. In this there is an analogy to arithmetic of primitive tribes compared to modern mathematics. Dr. Quine does not touch upon the relationship of Logic to Metaphysics. He keeps strictly within the boundaries of his subject. But — if in conclusion I may venture beyond these limits — the reforma-

χ

FOREWORD

tion of Logic has an essential reference to Metaphysics. For Logic prescribes the shapes of metaphysical thought. Finally it is interesting to note the influence on Dr. Quine's thought of the work of Professor Η. M. Sheffer, and of the great school of Polish mathematicians. There is continuity in the progress of ordered knowledge. ALFRED NORTH HARVARD UNIVERSITY

October 8th, 1984

WHITEHEAD

CONTENTS FOREWORD BY A . N . WHITEHEAD

IX

INTRODUCTION I.

3

ORDINATION

10

C L A S S E S AND T Y P E

18

III.

PROPOSITIONS

26

IV.

CONGENERATION AND ABSTRACTION

35

R U L E S OF I N F E R E N C E

42

POSTULATES

51

M A T E R I A L IMPLICATION

60

INCLUSION AND I D E N T I T Y

75

U N I V E R S A L I T Y AND I M P L E X I O N

85

E Q U I V A L E N C E , D E N I A L , AND CONJUNCTION

95

II.

Y. VI. VII. VIII. IX. X. XI. XII.

E X T E N S I O N OF THE G E N E R A L CALCULUS T H E PROPOSITIONAL CALCULUS AS A CALCULUS OF C L A S S E S

105 .

121

XIII.

INTRODUCTION TO R E L A T I O N S

128

XIV.

DESCRIPTIONS

139

UNITY

146

DOMAINS AND R E L A T I V E PRODUCTS

153

D E R I V A T I V E NOTIONS

161

D E D U C I B I L I T Y OF THE S Y S T E M OF ' P R I N C I P I A MATHEMATICA'

173

T A B L E OF D E F I N I T I O N S

194

INDEX

195

XV. XVI. XVII. XVIII.

A SYSTEM OF LOGISTIC

INTRODUCTION

THE system of logistic to be presented in these pages is in its broad outlines closely allied to that set forth by Whitehead and Russell in their Principia Mathematica·, it is to be contrasted with the latter, however, in its possession of an additional dimension of generality. This gain in comprehensiveness takes place along the direction of relational degree* or, what is equivalent, the direction of the multiplicity of arguments admitted by propositional functions. The treatment of relations in PM + is confined to the dyadic case. It might in exceptionable terminology be added that monadic relations are treated in that work, reference being had to the logic of classes; but no treatment is presented for relations of degree higher than 2. It is true that the methods used in treating classes and dyadic relations may be applied in analogous fashion to triadic, tetradic, and pentadic relations, and so on, through a succession of as many supplementary chapters. It remains impossible, however, on the basis of PM, to adduce theorems in general about n-adic relations without having first specified n. The present system on the other hand has been so framed as to admit of the demonstration of theorems concerning relations in general, without regard to degree. Not only is the scope of the system infinitely extended thereby, relatively to that of PM, but furthermore the volume of necessary symbolic machinery is reduced. The mutually analogous treatments of classes and relations in the latter work give way here to a single treatment of classes and relations in general, and pairs of parallel theorems proved respectively for classes and dyadic relations in that work have as their correspondents in the present system single theorems which apply at once not only to classes and dyadic relations but to relations of any degree whatever. The tech* The term degree, as applied to relations, is due to Professor Η. M. Sheffer. To say that the degree of a relation is η is to say that the relation is ra-adic, i.e. a relation among η terms. By a natural extension we may speak also of the degree of a propositional function, referring thus to the number of circumflexes which occur in the expression of that function according to the notation "φχ", "ψ(χ,Ρ)",.

" x(x,y,z)", etc., of Principia Mathematica. + "Principia Mathematica" will hereafter be thus

abbreviated.

4

A SYSTEM OF LOGISTIC

nique which makes this possible brings about a similar generalization and condensation in the material relative to quantification. In numbers * 9 - * l l of Ρ Μ the quantification of propositions with respect to one, two, and three variables is successively treated. For any given n, a continuation of the same process through supplementary chapters would lead to a treatment of quantification with respect to η variables; the general case, however, for unspecified η cannot be dealt with by the machinery of that system. Here on the other hand this becomes possible, and PM's separate discussions of the first three cases are thereby telescoped into a single treatment of the general case. All these results depend ultimately upon including within the system an operation by which terms can be combined into sequences of terms; in this wise sequences become themselves terms of the system, so that the general variables "x", "y", etc., come to admit as values not only separate terms in the ordinary sense, but also, in particular, whole sequences of terms. It will be shown that the gain in generality and brevity thus achieved is not attended with any loss in the direction of minuteness of analysis, but rather that the whole formal system of P M is demonstrable from the present system. Increase of generality, as an objective of the present system, has been coupled with the objective of increase in rigorousness. There are points in P M at which the reader's common sense is depended upon to bridge considerable lacunae in the formal structure of that system; at such points an indeterminate amount of supplementary material makes its entry unheralded. Let us consider briefly the difficulties which thus beset a structural analysis of those developments in P M ( * 9 - * l l ) which relate to the propositional function. PM works with three primitive ranges of subject matter: things in general (alternatively individuals),

propositions,

a n d propositional

func-

tions. The letters "x", "y", "z", . • . are used as general variables, the letters "p", "q", "r", . . . as proposition variables, and the letters " φ " , "ψ", " χ " , . . . as function variables.* In the verbal discussions the latter are elaborated as "φχ", "ψ(χ,ί/)", etc., to indicate the de* Acknowledgment is due Professor Η. M. Sheffer in the use, throughout the present work, of his diction "proposition variable," "function variable," "class variable," etc., meaning variables taking as their values propositions, propositional functions, classes, etc., respectively. The propriety of this usage is patent, as over against the usage "variable proposition," "variable function," etc.

INTRODUCTION

5

grees of the functions represented. On the other hand the expressions "φχ", "\p(x,y)", etc., without circumflexes do not stand for prepositional functions, some inaccurately phrased passages in P M to the contrary notwithstanding. These are rather propositional expressions, ambiguous to the extent of involving variables; "φχ" represents, for any given φ and x, the proposition which is the value of the propositional function φ (or φζ) for the argument x. "Thus," we read,* " ' x is hurt' is the propositional function, and 'x is hurt' is an ambiguous value of that function." As primitive operations we are given non-conjunction f and universal quantification* symbolized in context as "ρ | q" and " (χ) . φχ" respectively. I have not reckoned assertion as a primitive operation of PM, for, as Professor Carnap has pointed out, 5 the assertion sign expresses no primitive idea of logic, but serves rather merely as a device for the presentation of the system, after the manner of theorem numbers, parentheses, etc. Without any further officially recognized primitives we are called upon to understand that a propositional function φ, or φζ, when endowed with an argument x, yields a proposition φχ; similarly that a function ψ, or ψ(ζ,ύ)), when endowed with arguments χ and y in that order, yields a proposition \p(x,y); and so on. Now it seems clear that some new primitive ideas are brought in at this point unheralded. Some manner of operation must be assumed whereby a propositional function φ, or φζ, and a term χ are so combined as to yield a propo* PM, Vol. I, p. 15. t Reference is had to Professor Sheffer's stroke function, which, in the second edition of PM (Vol. I, pp. xiii, xvi-xix), is made to supplant the former primitive ideas of alternation ("disjunction") and denial. The name "non-conjunction" is Professor Sheffer's; cf. his review of PM in Isis, VIII (1926), p. 229. But v. infra, p. 102 n. * According to the first of the two methods set forth in of PM both universal quantification and particular quantification, "(tlx). φχ", are taken as primitive; by the second method, however, only the universal variety is taken as primitive and the particular is then defined. The authors of ΡΜ are persuaded that under this second method the primitive operation of the calculus of so-called elementary propositions, namely non-conjunction, must be reintroduced for application to quantified propositions as another primitive operation over and above the original non-conjunction. It is irrelevant to the present discussion, however, either to introduce this complication of primitives or to enter upon the considerations which are supposed to call for it. 5 Abriss der Logistik, p. 9.

A SYSTEM OF LOGISTIC

6

sition φχ; this would seem to be a primitive binary operation, expressed notationally by the juxtaposition of "φ" and "x" or by the substitution of " x " for " I " in " φ ζ " . For convenience let us refer to this primitive operation as predication. Furthermore, the corresponding operation for the dyadic case seems to be another primitive idea: the ternary operation, namely, whereby a function φ (or φ(ζ,ύή), a term χ and a term y are so combined as to yield a proposition \j/(x,y). The triadic case appears to involve an analogous primitive quaternary operation, and so on. This supplementary series of primitive operations does not put an end to the difficulties. Inferences in P M involving propositional functions depend upon a common-sense understanding that such expressions as "φχ", "φχ", etc., may represent any propositions about x. No attempt seems to be made to reduce such inferences to explicit rules of inference. As a random example consider the third step in the proof of * 1 0 ' 3 5 , where it is deduced from *10"23.

(χ) .φχ Dp.

= ·. (3x)

.φχ-D.p

that (1)

(χ) : ρ . φχ . D. ρ : D : (3χ)

. ρ . φχ . D.

p.

This inference depends upon putting "ρ.φχ" for "φχ" in * 1 0 ' 2 3 . Now "φχ" is not a variable, but a complex of two variables; if therefore this inference is to be regarded as a single step of substitution, the rule of inference involved cannot be the usual rule of inference by substitution upon variables, but must be a special rule making for substitution upon complexes such as "φχ ". Given any theorem containing expressions "φχ", "φν", etc., but no occurrences of " φ " outside such contexts, the rule would be such as to allow the inference of any result of replacing all occurrences of "φχ", "Φν", etc., respectively by propositional expressions involving "x", "y", etc., homologously and differing from one another in no other respects. An analogous rule would be needed for substitution upon such complexes as "x(x,y)", and so on.* To leave the matter at this stage is of course to leave the theory of propositional functions almost completely unformalized. Alternatively we may insist upon reduction of the given inference to terms of substitution upon separate variables. Granted that "x" remained "x" in passing from * 1 0 ' 2 3 to (1), what then was substituted for • Cf. pp. 187-188, infra.

INTRODUCTION "φ" ? The answer would be that the substitute for "ψ" was "p. φζ", this being the notation in PM for that case of ψ or ψζ corresponding to the determination of φχ as ρ . φχ. Here some primitive operation is presupposed whereby a propositional function is derived from a proposition; the notational expression of this operation consists in replacing a term of the proposition by a variable and placing a circumflex over the latter. Let us refer to this operation as abstraction, following Dr. Paul Weiss.* Given predication as primitive we still need abstraction, as is readily seen from the propositional function ρ , φζ. Being propositional expressions, "p" and "φχ" are meaningfully combined in "ρ . φχ" through the notion of conjunction as an operation upon propositions; the function ρ . φζ can thereupon be explained only as having been derived in turn from the proposition ρ. φχ (or ρ . φ-y, etc.) by some such operation as abstraction, for the occurrence of the symbol of propositional conjunction within the functional expression "ρ .φζ" would otherwise remain unexplained. Conversely, given abstraction as primitive we can no more dispense with the primitive idea of predication. Without the latter we have simple proposition variables "p", "q", etc., entering such combinations as "p . q", etc.; before abstraction can be used as it is used in PM, e.g. in deriving a propositional function ρ . φζ from a proposition ρ . φχ, we need the notion " a proposition involving x " as expressed by "φχ" itself, and this calls for a primitive idea of "involving," or predication of φ relatively to x, or some equivalent notion. We observed that parallel to ordinary or binary predication a further primitive operation of ternary predication is involved in P M in the generation of expressions of the form "(x,y,z)" of the literature may be interpreted as that case of 4>w where w is the triad x,y,z, i.e. the result of iterated ordination x,(y,z). Correspondingly for the cases involving four and more arguments. According to the method of generation above described, an n-ad Χι,Χί,. . .,Xn-i,xn is the result of the operation of ordination relatively to the term Χι and the (n — l)-ad x2,. . .,x n -i,x n . Since sequences are thus generated from the primitive operation of ordination, that primitive operation is itself explained conversely by the notion of sequence, which latter has the advantage of being already current in mathematics. By a sequence is meant nothing more than a complex en* Aside from detail, this device is due to Peano. Cf. his Formulaire de Mathematiques (1901), §1, *9'1. f But v. infra, p. 51, note t. * V. infra, pp. 47, 92, 95, 99-100.

12

A SYSTEM OF LOGISTIC

do wed with a discrete order; * as written in linear form it m a y involve any manner and amount of repetition. The primitive operation of ordination, yielding x,y, may now be described as the operation of so prefixing χ to the (n — l)-ad y as to obtain t h a t n-ad the first term of which is χ and the rest of which is y. The operands χ and y will be referred to hereafter as respectively the affix and the base of the sequence x,y. I t has been seen t h a t an n-ad is the result of the operation of ordination upon two terms the second of which is an (η — l)-ad. I t follows further t h a t an n-ad is itself an (n — l)-ad, namely an (n — l)-ad whose last term is a sequence: for it is seen from the given convention of parentheses t h a t the n-ad Χι,Χι,. . .,x n _i,x n _ x ,x n is the (n — l)-ad XljX2f · *jXn—2, (xn-i,xn)· T h u s h-ads come to be special kinds of fc-ads wherever h exceeds k. Triads, tetrads, etc., become thus not mutually exclusive categories but rather species each of its predecessor. The series is completed terminologically by introducing " d y a d " as synonymous with "sequence," and " m o n a d " as the completely general category, synonymous with " t e r m . " The words " d y a d " and " m o n a d " will be forthwith abandoned in view of their superfluity; thus " n - a d " is to be read " t e r m " and " s e q u e n c e " in the respective special cases where η is 1 and 2. I t is important to note that, whereas any (η + l)-ad is an n-ad, it does not follow t h a t any n-ad is an (n + l)-ad if one of its η places is occupied by a sequence; this is true only if its last place is occupied by a sequence. T h u s the sequence (x,y),z is no more a triad than is the sequence y,z; neither is a triad, namely, unless ζ happens to be a sequence. New and idle terminology could of course be introduced so as to designate the sequence (x,y),z as a "sinistral triad," as distinguished from a normal or " d e x t r a l " triad x,y,z, which is to say x,{y,z). More complex terminology would be needed in order to designate the triads (a:,y),z,w [i.e. (x,y),(z,w)] and x,{y,z),w [i.e. x,{{y,z),w)~] and the sequence ((x,y),z),w as this or t h a t abnormal kind of tetrad, and progressively more complex as one progressed beyond tetrads. Whether * Whereas it may be convenient for some mathematical purposes to include also under the notion of sequence the degenerate case of a "one-place sequence," not to be distinguished from the single element occupying that place, on the other hand it has proved more convenient in the present work to confine the notion of sequence to the case of two or more places, and thus to identify it with the result of the operation of ordination.

ORDINATION

13

sinistral and normal triads should ultimately turn out to be the same, and consequently all higher cases should telescope in analogous fashion, depends upon whether or not ordination be taken as associative. The operation has not been taken as associative in the present system, because of considerations which will now be explained. It is desirable so to fashion the system that every general principle demonstrable in the usual logistic (e.g. in the system of PM), as applying to all terms x, will remain demonstrable here with unrestricted generality despite the fact that the primitive operation of ordination increases the range of the variable " x " to the extent of allowing that variable to take entire sequences as values. In other words, it is desirable so to frame the system that whatever is formally demonstrable of everything exclusive of sequences will not break down when sequences are included. Other things being equal, the advantage of such a procedure in point of simplicity and convenience is obvious. As principles applying to terms in general, sequences and otherwise, there would be the general principles of the usual logistic. Then there would be further general principles obtaining for sequences in general but not applying to terms other than sequences. Such a principle, besides making mention of a sequence x,y, would variously involve one or both of the operands χ and y in isolation, wherefore no expression could be introduced in lieu of "x,y" except it have that same compound form; an unarticulated variable " z " could not even be fitted to the phrasing of such a principle, having as it does no expressed affix and base to play the respective individual roles of χ and y. Such principles would however hold at once of triads, tetrads, etc., indeed of sequences of any length, since y, in x,y, is itself free to be determined as a sequence of any specifications. Then, over and above such general principles of sequences, there would be various formal principles applying to all triads (and hence in particular to all tetrads, pentads, etc.) but having no application to shorter sequences. Just as the principles last considered involved explicit mention of x,y and various separate mentions of χ or of y, the principles now under consideration would mention x,y,z and refer also, severally, to y or z. We could go on similarly to adduce principles true of all tetrads but applying to no other triads; and, in general, for any given n, we should have formal principles true of all n-ads but having no more general application. Everything

14

A SYSTEM OF LOGISTIC

formal deduced regarding n-ads in general would serve ipso facto as a treatment of m-ads wherever m exceeds n; to proceed thence to adduce further properties of m-ads inapplicable to shorter sequences would· simply be to add somewhat more detail to the system, without affecting the generality of what had preceded. Such a regime of initially complete generality, supplemented by progressively less general detail, would not obtain in the alternative procedure; rather should we find ourselves in a maze of speciality at the outset. If namely there were formal principles failing of sequences but demonstrable of all else, we should have at the start so to limit their generality. A general calculus would hence have to be replaced by a calculus of non-sequential terms. Having once introduced machinery for thus separating off the non-sequential terms from terms in general, then in the case of any general principle continuing to hold of all terms including sequences we should be called upon conversely to provide expressly for that unlimited generality. The situation would be repeated at the level of triads, for to apply to χ,ζ,ιν what had been proved for x,y would be to assert of the sequence z,w what might be a property only of non-sequential terms y. The awkwardness would recur analogously at the emergence of tetrads, of pentads, and so on. In fashioning the present system, therefore, it is important so to proceed as to make for the first of these alternatives, namely that otherwise valid formal principles will continue to hold upon the introduction of sequences. Let us now consider the bearing of this ambition upon the question of the associativity of ordination. The converse of a relation R, expressed by the notation "R" in PM, is the relation which y has to χ if and only if χ has the relation R to y. The following therefore is a theorem of the usual logistic and of ΡΜ in particular: y has the relation R to χ if and only if χ has the relation R to y. As will subsequently * be seen, relations are handled in the present system as classes of sequences, wherefore the above principle would here assume the following form: (1) y,x is a member of the class of sequences R if and only if x,y is a member of the class of sequences R. If the desire expressed in the preceding three paragraphs is to be realized, the principle (1) should hold true even though χ or y be in turn a sequence. Let us then take y in (1) as a sequence y,z\ we obtain * Pp. 18-19, infra.

ORDINATION

15

(2) (y,z),x is a member of R if and only if x,(y,z) is a member of R. Again, let us take χ in (1) as z,x; this yields (3) y,(z,x) is a member of R if and only if (z,x),y is a member of R. Consider now the consequence of assuming the associativity of ordination, i.e. that (a,b),c is identical in every case with a,(b,c). According to this associative law, (3) yields (4) (y,z),x is a member of Ii if and only if z,{x,y) is a member of R. Combining now the equivalences (2) and (4), whose first parts are the same, we infer (5) x,(y,z) is a member of R if and only if z,(x,y) is a member of R. Suppressing the parentheses in (5) according to the convention, we have (6) x,y,z is a member of R if and only if z,x,y is a member of R. In this way, granted (1) as a general principle of the system and granted further the associativity of ordination, (6) would become a theorem of the system, " t r u e " for all values of the variables. Such a result would obviously be disastrous, excluding as it would all triadic relations save those of a very arbitrary pattern. Betweenness is a familiar triadic relation violating (6), since χ can be between y and ζ without z's being between χ and y. Among a multitude that could be devised, the above is one example of the irreconciliability of the associative law for ordination with the objective of conforming sequences to such formal laws as hold in complete generality for all other kinds of terms. From the example we readily detect the essential manner in which the associative law interferes with the conformity of sequences to otherwise general principles. Where χ and y are not sequences, they are the only terms relatively to which the operation of ordination can yield the sequence x,y. If on the other hand either χ or y is a sequence, such determinacy of the operands is swept away by the associative law; where e.g. a; is a sequence z,w, the sequence x,y or (z,w),y is according to the associative law indiscriminable from the sequence z,{w,y) and is hence indifferently the result of the operation of ordination both relatively to z,w and y and relatively to ζ and w,y. Principles depending upon the unique determinacy of the operands to ordination (e.g. (1) above) thus remain valid for all non-sequential values of χ and y, but break down, under the associative law, when χ or y is a sequence. The resolution to retain for sequences all otherwise valid formal principles

16

A SYSTEM OF LOGISTIC

leads therefore to the abandonment of the associative law for ordination. Hence sinistral and normal triads are not identified, nor consequently do the higher cases telescope to the "normal" forms; rather is the infinite variety of modes of iteration of ordination to be preserved. The terminology "sinistral triad" and its cumbersome sequel will of course not be used; given all conceivable variety of two-place sequences, either of whose places may in turn be occupied by a sequence, and so on, the cases of iteration in the right-hand direction are designated by convention as "triads," "tetrads," etc., whereas no such special terminology is appropriated to the other manners of iteration. Thus an n-ad acquires no more particular designation through exhibiting a sequence in one of its η places unless the place concerned is the last. Wiener * and Kuratowski + have invented methods of defining sequences on the basis of the logic of classes. The definitions are alike in principle, but Kuratowski's is slightly simpler in detail. He defines x,y as that class of classes whose only members are the two following: (1) the class whose only member is x, and (2) the class whose only members are χ and y. This definition fulfills the essential requirement, namely that x,y be distinct from y,x except where χ = y. Sequences of any given length could be defined by obvious extensions of the same device. Relations then emerge, as in the present system, 1 as classes of sequences. According to the theory of class types the above definition gives meaning to "x,y" only if χ and y are of the same type. Thus relations, as classes of sequences, are provided for on the basis of Kuratowski's device (and Wiener's likewise) only when they are relations between or among terms of the same type. Wiener deals with this difficulty by introducing a system of supplementary definitions to accommodate heterogeneous relations; the structure of relational theory ceases thereby to be uniform for all types, but aside from this inelegance the method is successful.

In relational theories based thus upon definition of sequences as classes there remains the necessity, as in PM, of developing separately the mutually analogous calculi of classes, dyadic relations, triadic relations, and so on; these do not become special cases of a single calculus of classes as in the present system,? despite the fact that relations become a special kind of classes (of classes of classes). It is true that the inclusion and the logical ad* Norbert Wiener, "A Simplification of the Logic of Relations," Proceedings of the Cambridge Philosophical Society, XVII (1914), pp. 387-390. t Kazimierz Kuratowski, "Sur la notion de l'ordre dans la Theorie des Ensembles," Fundamenta Mathematica, II (1920), p. 171. * Cf. pp. 18-19, infra. 5 Cf. p. 3, supra·, pp. 115, 191-193, infra.

ORDINATION

17

dition and multiplication of n-adic relations become merely special cases of the corresponding notions for classes, those cases namely where the classes involved are n-adic relations; it is true also that the null n-adic relation may be identified with the null class of classes of classes.* Beyond this point, however, the relationship breaks down; n-adic-relational negation and the universal n-adic relation do not telescope into their classial analogues, but require rather to be sharply distinguished from the latter and from their m-adic-relational analogues as well. Whereas — R, the classial negate of an n-adic relation or class of n-ads R, is the class of all those classes of classes, both n-ads and otherwise, which do not belong to R, on the other hand —nR, the n-adic-relational negate of the same relation R, comprises just those members of — R which are n-ads; —mR, finally, comprises just those members of — R which are m-ads, and is hence distinct from both — R and —nR if 1 m n. With the universal relation the situation is parallel. It should be noted finally that no economy of primitive ideas would result from introducing such a definition of sequences within the present system; whereas the operation of ordination would no longer stand as primitive, on the other hand the notion of predication or membership, provided for in the present system by ordination,1· would have to be adopted as a primitive idea. * Cf. Wiener, op. at., p. 390.

t Cf. pp. 26-28, infra

CHAPTER II CLASSES AND TYPE To EXPRESS the membership of a term ι in a class y I shall for the moment adopt Peano's notation "xey". We have now come upon propositions: "xey", namely, expresses the proposition that χ is a member of y. This notation will subsequently be discarded and the notion of membership, rather than being a primitive idea, will be found to be altogether superfluous. Let us accept the notion provisionally, however, as if it were primitive. Since a term χ may be in particular a sequence, or more particularly a triad, tetrad, etc., a class y of which χ is a member may be a class of sequences, a class of triads, of tetrads, etc. In the development of the theory of types which is to follow it will be seen that of any class of which a sequence can be said to be a member, only sequences can be said to be members; of any class of which more specifically a triad can be said to be a member, only triads can be said to be members; and, in general, of any class of which an n-ad can be said to be a member, only n-ads can be said to be members. A class of sequences is called a relation in extension, or briefly a relation. A class of triads, or in other words a relation the sequences belonging to which are triads, is called a triadic relation. A class of tetrads similarly is called a tetradic relation, and in general a class of n-ads an n-adic relation.* Wherever m exceeds n, since m-ads are then a special kind of n-ads, ra-adic relations are a special kind of n-adic relations. Thus pentadic relations are tetradic, tetradic relations are triadic, triadic relations are relations, and relations finally are classes. The notation "zRw" of PM, "z has the relation R (or y) to w," gives way here (pending the discard of the notion of membership) to " (z,w) ey", this being merely a case of the form "xey". Thus " ζ has the relation y to w" means "The sequence z,w is a member of the class (of sequences) y." Similarly "z, u, and ν stand in the triadic * Obviously we are to interpret "n-adic relation" simply as "class" and "relation" in the respective special cases where η is 1 and 2.

CLASSES AND TYPE

19

relation y" receives expression as " (z,u,v) ty", this being the case of the form " (z,w) ty" where w is u,v. Hence "z, u, and ν stand in the triadic relation y" means "z has the relation y to the sequence u,v," which means in turn " T h e sequence z,(u,v) (which is to say the triad z,u,v) is a member of the class (of triads) y." Similarly for the further cases. The copula of membership, adopted temporarily to facilitate present discussion, is the connective to govern which the theory of types is required. The theory of types fulfills its purpose of avoiding contradictions * by branding such and such combinations of symbols as meaningless. All canons of meaningfulness belong to the prosystematic stage where we discuss our use of notation. T h u s e.g. "p Da" could not, within the formal system of P M , be explicitly declared to be meaningless, since the system is concerned, not with the symbols " p " , " D ", and " a " , but with their denotations; the meaninglessness of "p Da" comes about rather through the convention, announced informally in the explanatory prolegomena to P M , t h a t the letters "a", "ß", "y", • . . are class variables, together with the fact t h a t "pDq" is defined only for "q" as a proposition variable. The canons of type are of the same nature as this; their peculiarity is merely t h a t the cases which they brand as meaningless do not flaunt their meaninglessness typographically in so simple a fashion as does the expression "pDa". The canon of type (e.g.) which says t h a t two classes cannot significantly be said to be members reciprocally of each other is none the less a prosystematic canon of symbolism — a convention, indeed, governing the use of the interponent sign " e " ; otherwise one would be at a loss to give a meaning to meaninglessness. In terms of our tentatively primitive copula of membership, therefore, the office of the theory of types m a y be summed u p thus. For the avoidance of contradictions, the use of this connective must be so restricted, given one term, t h a t only certain choices will be significantly admissible for the other. All the significantly admissible choices for the other term, given the one, are said to constitute a type. Since variables and even logical constants, as symbolized, always refer ambiguously insofar as concerns the types of the terms represented, the immediate description of notation is inadequate to enforcing the desired restrictions. Hence the need for an organon of significant use of * V. PM, Vol. I (2d. ed.), pp. 37-65.

A SYSTEM OF LOGISTIC

20

the copula of membership, supplementing the notation of that connective; this organon is the theory of types. Since all canons of significance must be confined to the status of informal rules of procedure, the manner of expounding them is immaterial so long as clarity is achieved; considerations of postulational economy and elegance do not enter as in the case of formal postulates. Whether one start with canons governing the significant use of the symbol " e " , and derive the hierarchical scheme of types thence b y verbal argument, or merely present the canons through the medium of a direct description of the type hierarchies, is wholly a question of expository, rather than logical, technique.

I shall adopt the latter

course and move directly to a description of the system of types which is generated b y the requirements of significance with regard to the copula. T h e scheme of types involved b y the present system is a direct adaptation and extension of the scheme of types of classes and relations in P M . W e are to suppose an infinite set of infinite hierarchies of types such that, where "xty"

is significant, the type of y must

belong to the same hierarchy as the type of x, and must be the type next higher in that hierarchy than the type of x; in other words, (i) the copula can meaningfully be applied only to terms belonging to consecutively ascending types of the same hierarchy. This consideration suffices, given the lowest type of some hierarchy, to generate that whole infinite hierarchy. For, let χ be of the lowest type of some hierarchy. Then the second type of that hierarchy will consist of all terms y such that "xty"

is significant.

T h e third type of the hierarchy will con-

sist of all terms ζ such that "ytz"

is significant, where y is of the

second type; similarly, in general, the n-th type of the hierarchy will consist of all terms ν such that "utv"

is significant where u is of

(n — l)-st type. Whatever is of second or higher type in any hierarchy is called a class; more particularly, whatever is of third or higher type in any hierarchy is called a class of classes; whatever is of fourth or higher type is still more particularly a class of classes of classes, and so on. Such is a general description of any hierarchy of types, relatively to the initial type of that hierarchy. Now these initial types themselves are related one to another, or generated one from another, according to the principle (ii) that two sequences are of the same type if and only

CLASSES AND TYPE

21

if the affix and base of the one are of the same types respectively as the affix and base of the other. T h u s x,y and z,w will be of the same type if and only if χ and y are of the same types respectively as ζ and w. Now let y be a non-sequence, class or otherwise. By the above principle the sequence x,y will be of the same type as y itself if and only if χ and y are of the same types respectively as the affix and base of y. But y is not a sequence and hence has neither affix nor base; therefore the sequence x,y must be of different type from y. Then the sequence v,x,y, which is to say v,(x,y), will be of different type from x,y, since the base x,y of v,x,y is of different type from the base y of x,y. Similarly u,v,x,y will be of different type from v,x,y, since the base v,x,y of the first has been seen to be of different type from the base x,y of the second; and so on. By such reasoning we see in general t h a t a sequence is always of different type from its base, i.e. a,b is always of different type from b (provided only t h a t the base b is not an infinite sequence); this can always be deduced from principle (ii) by the above form of argument, since b must be of some one of the forms "y", "x,y", "v,x,y", "u,v,x,y", etc., where y is not a sequence. A strictly analogous argument shows also t h a t a sequence a,b is always of different type from its affix a (provided t h a t a is finite in length). I t follows also (among finite sequences) t h a t x,y,z, which is to say x,(y,z), is of different type from (x,y),z, since neither their respective affixes χ and x,y nor their respective bases y,z, and ζ can be of the same type. T h u s not only is the operation of ordination taken as nonassociative, b u t diverse associations are consigned moreover to separate types. The operation of ordination thus combines terms of any given types into sequences belonging to new types; ordination relatively to these sequences generates further types of sequences, and so on. From each of these types springs a hierarchy of types. The resulting scheme of types may be conceived schematically as follows. Where a is any type, let us represent by " a ! " the next higher type in the same hierarchy. T h u s a! is the type of classes of terms of type a. Terms which are neither sequences nor classes are all of one type, which we m a y call λ; then classes of terms of type λ constitute the type λ!, classes of such classes constitute the type X!!, and so on. Here we have an infinite hierarchy of types, initial to which is the type λ. Now by the notation " a t b" let us represent the type of

22

A SYSTEM OF LOGISTIC

those sequences x,y such that χ is of type a and y is of type b.* Thus λ Τ X will be the type of those sequences x,y such that χ and y are both of type X, i.e. are neither sequences nor classes. The type (Χ t λ)! is the type of classes of such sequences, or in other words (since relations are classes of sequences) the type of relations between terms of type X. The type (Χ Τ X)!! will then be the type of classes of such relations, the type ( λ Τ λ ) ϋ ! will be the type of classes of such classes of relations, and so on. Here we have a second hierarchy of types, whose initial type is X t X. A third hierarchy is generated similarly from the type X i t X , i.e. the type of sequences x,y such that χ is of type X! and is hence a class of terms of type X, and y is of type X. The second type of this hierarchy is (λ! Τ X)!, i.e. the type of classes of sequences of type λ! Τ X, or in other words the.type of relations which classes of terms of type X bear to terms of type X. The third type of the hierarchy, (X! t Χ)ϋ, is the type of classes of such relations, and so on. The hierarchies λ Τ Χ, (λ Τ λ)!, (λ Τ Χ)ϋ, . . . and Χ! Τ Χ, (λ! Τ Χ)!, (λ! Τ λ)!!, . . . are but two of the double infinity of hierarchies of the form "alb, (alb)I, (at?))!!, . . ." which are to be had by variously selecting a and b from the primary hierarchy Χ, Χ!, X!!, . . . . That is to say, along with the hierarchy springing from X t X and the hierarchy springing from Χ! Τ X we have the analogous hierarchies springing from λ Τ λ!, from X! t X!, from Χ ϋ ΐ λ , from Χ ϋ ΐ λ ! , from Χ ϋ ί λ ϋ , from XTX!!, from X! t X!!, from X!!! t X, and so on. This double infinity of hierarchies is only a beginning: a beginning namely of that infinitely richer infinity of hierarchies of the form " a t 6, ( a t δ)!, ( a t 6)!!, . . ." where a and b, rather than being selected exclusively from the primary hierarchy Χ, λ!, X!!, . . . , are selected from any of the double infinity of hierarchies already generated. For example, taking α as X t X and b as X, we obtain the hierarchy whose initial type is (X t X) t X, i.e. the type of sequences (x,y),z where x, y, and ζ are all of type X. The second type of this * Like " a ! " , the notation "a t b" is introduced here merely by way of temporary abbreviation to facilitate the present strictly informal discussion. The notation " a f 6 " will be observed however to coincide with the notation of an operation subsequently (pp. 161-164, infra) to be introduced under the name of perordination. It is readily seen that in the special case where the classes a and b are types the perordinate a f b is the same as a f b in the present temporarily adopted sense.

CLASSES AND TYPE

23

hierarchy, namely ( ( λ Τ λ ) Τ λ ) ! , is then the type of the relations which sequences of type λ ΐ X bear to terms of type X; the third type of the hierarchy, ( ( x t x ) t x ) ! ! , is the type of classes of such relations, and so on. Again, taking a as X and b as Χ ΐ X, we have the hierarchy whose initial type is X t λ ΐ X,* i.e. the type of all triads of terms of type X. The second type of the hierarchy, namely (Χ t Χ ΐ λ)!, will then be the type of all classes of such sequences, or in other words the type of all triadic relations among terms of type X. The third type of the hierarchy, namely ( λ ί χ ί λ ) ϋ , will be the type of classes of such relations, and so on. Again, taking a say as ( x t x ) ! and b as (λ! Τ λ ) ϋ , we have the hierarchy whose initial type is (Χ ΐ X)! t (Χ! ΐ Χ)ϋ, i.e. the type of all sequences x,y where χ is a relation between terms of type X and y is a class of such relations as are borne to terms of type X by classes of terms of type X. The second type of this hierarchy, namely ((Χ ΐ Χ)! Τ (Χ! ΐ X)!!)!, will be the type of classes of such sequences, or in other words the type of relations borne by relations of type (Χ Τ X)! to classes of type (Χ! Τ X)!!; described in full, ((Χ Τ Χ)! ΐ (Χ! ΐ X)!!)! is the type of relations borne by relations between terms of type X to classes of relations borne to terms of type X by classes of terms of type X. The third type of the hierarchy, namely ((Χ ΐ λ)! t (λ! Τ λ)!!)!!, will be the type of classes of such relations, and so on. Any types already generated m a y be linked by the arrow operation, as in the above examples, to form the initial type of a new hierarchy; far from being confined in our linking material to the double infinity of hierarchies of types first generated, we m a y also draw upon any of the infinities of infinities of hierarchies generated from those. The further this generation proceeds, the more prolifically new hierarchies spring u p from which to select types to be linked by the arrow operation to form the initial types of further hierarchies. The progeny of all this is the infinite array of infinite hierarchies constitutive of the scheme of types. In particular, we find among these hierarchies those special hierarchies which spring from a type of tetrads, or of pentads, or of hexads, etc., whose four places, or five places, or six places, etc., are occupied respectively b y terms of a n y given types. T h u s where * The parentheses in " λ t (λ f λ)" are automatically suppressed by the general convention of parentheses enunciated earlier (p. 11).

24

A S Y S T E M OF LOGISTIC

a and b are respectively e.g. the types λ ϋ and λ ί (λ Τ λ!)! Τ λ!,* the hierarchy a t b, (α Η ) ! , (a t b)Ü, . . . has λ ϋ t λ ί (λ Τ λ ! ) ! Τ λ ! as its initial type, i.e. the type of tetrads x,y,z,w such that χ is a class of classes of terms of type λ, y is of type λ, ζ is a relation of terms of type λ to classes of terms of type λ, and w is a class of terms of type λ. T h e second type of the hierarchy, namely ( λ ϋ Τ λ Τ (λ Τ λ!)! Τ λ!)!, will be the type of classes of such tetrads, or in other words the type of tetradic relations holding among terms which are respectively of types λ ϋ , λ, (λ t λ!)! and λ!.

T h e third type of the hierarchy,

namely ( λ ϋ Τ λ ΐ (λ t λ!)! Τ λ ! ) ϋ , will be the type of classes of such tetradic relations, and so on. In sum,., every possible construct upon λ, in terms of the binary operation represented b y the arrow and the unary operation represented b y the exclamation point, is a type; the system of types is the system of all these constructs. T y p e s are said to belong to the same hierarchy when their representations differ from one another, if at all, only in the number of terminal exclamation p o i n t s ; f they are ranged in that hierarchy in the order of multiplicity of terminal exclamation points. Those types whose expressions in terms of λ lack terminal exclamation points are thus initial to their respective hierarchies. I t is to be noted that all sequence types are initial to their respective hierarchies, since such types are never of the form " a l " (e.g. ( λ ϊ Τ λ ϋ ) ϋ ) , but rather have always the form "alb" λ ! Τ λ!!).

(e.g.

Thus every sequence belongs to the lowest type of its

hierarchy. This is, indeed, to be expected, since anything belonging to the second or higher type of any hierarchy is a class, whereas a sequence, although it m a y in particular be a sequence of classes, can never itself be a class. From this minimal position of sequence types it follows moreover that every relation is of next lowest type of its hierarchy, since a relation is a class of sequences and is of next higher type than the sequences whereof it is a class. Taking all the hierarchies in columnar fashion and considering their constitutive types as a collective stratification of all the columns, we see from the above that sequences are confined to the lowest stratum * I.e. λ Τ ((λ f λ!)! t λ!); cf. preceding footnote. t " T e r m i n a l " is to be taken here as meaning "applied to the whole expression"; thus ( Χ | λ ) ! is of the same hierarchy as λ f λ, whereas λ f x ! (meaning X f (λ!)) is of a different hierarchy from X f X.

CLASSES AND TYPE

25

and that relations are confined to the next lowest. According to the foregoing sketch the lowest stratum is not, conversely, made up exclusively of sequence types, but contains in addition the type λ of terms which are neither sequences nor classes; similarly the second stratum is not made up exclusively of relation types, but contains in addition the type λ! of non-relational classes of non-classes. It must now be remarked, however, that the type λ and the hierarchy which it initiates are, so far as concerns the demands of the system, no more than expository fictions; no formal considerations presuppose the existence of the type λ of terms which are neither sequences nor classes. The foregoing description of the type scheme can be appropriately relativized by reconstruing λ as any type arbitrarily chosen as origin. The generation of hierarchies can then be run backward indeterminately just as it can be run forward infinitely: if namely λ is a sequence type a t b, we have antecedently to the hierarchy λ, λ!, λ!!, . . . the more fundamental hierarchies a, al, all, . . . and b, bl, bll, . . .; if on the other hand λ is a class type cl, a new type c becomes prefixed to the hierarchy Χ, λ!, λ!!, . . . . Then a or b in the first case, or c in the second case, may in turn be either a sequence type or a class type, and the extrapolation of new types in the backward or downward direction thus continues. This relativity of type does not disturb the principle that all sequence types are initial to their various hierarchies; the relativity tells us only, conversely, that there is not necessarily any initial type other than sequence types.

C H A P T E R

III

PROPOSITIONS I N EXPRESSIONS of the form "xey" we have encountered propositions; it is now to be observed that this is a completely general form for propositions, i.e. that the expression of any fact whatever, or its denial, can be translated into the form "xey". Let φ, namely, be any proposition involving any term or sequence of terms x; then ρ has or may be given the form "So-and-so is true of x." The state of affairs presented by ρ is therefore presented equally by xey where y is the class of all terms (or sequences of terms) of which so-and-so is true. Hence any proposition involving any terms whatever can be set over into the form "xey". The adoption of "xey" as the general form for propositions appears to represent a step in the direction of the traditional subjectpredicate doctrine: for χ may in a manner be construed as the subject of the proposition, and y as the predicate. I t must be observed, however, that this usage depends upon a numerical generalization of the ordinary notion of subject, χ may in particular be either a subject in the ordinary sense of the term, or a sequence of a plurality of such subjects. Thus the difficulties which beset the traditional subjectpredicate doctrine in the treatment of relations do not arise here; y is itself an n-adic relation in the general case, and can be construed as a predicate in the traditional sense only in the special case where χ is not a sequence. Since a proposition is construed here as a complex of the form "xey", we can determine any proposition uniquely by specifying y and x, or, what comes to the same thing, by specifying the corresponding sequence y,x. Conversely, since "xey" expresses a proposition provided only that y is of next higher type than χ (in the same hierarchy *), a proposition is determined by every sequence y,x whose affix y is of next higher type than the base x. There is no risk of ambiguity, therefore, in identifying every proposition xey with the cor-

* The condition of sameness of hierarchy is hereafter always to be understood in connection with the phrase "next higher type."

PROPOSITIONS

27

responding sequence y,x and thus completely eliminating the copula of membership symbolized by " « " , provided that the various propositional connectives (implication, conjunction, equivalence, et al.) be properly adapted to the change. The fact is that the superstructure in question admits readily and naturally of the required adaptation, as the reader will see upon examination of subsequent chapters. For the present it remains to consider the significance and immediate consequences of this reinterpretation of propositions as sequences. I t has been pointed out that according to the theory of types the propositional expression "xey" can have meaning only when y is of next higher type than χ; in other cases "xey" is merely a combination of signs running counter to the prescribed rules of combination, and represents nothing. When however "xey" is rewritten as "y,x" and the proposition is identified with the sequence thus represented, it cannot be demanded correspondingly that for the significance of "y,x" y must be of next higher type than χ; on the contrary, the operation of ordination whereof the sequence y,x is the result is legitimate irrespectively of the types of the respective operands. In terms of the interpretation of propositions as sequences the demands of type hitherto attaching to the expression "xey" assume rather the following form: whereas y,x is of course a sequence irrespectively of what y and χ may be, it is that special kind of sequence called a proposition if and only if y is a class next higher in type than x. Proposition, like e.g. triadic relation, thus constitutes a subdivision of the system of types: it embraces just those sequences, namely, which are of such type that their affixes are of next higher type than their bases. In terms of the notational devices of the preceding chapter, a proposition type is thus any sequence type of the form " a ! t o " . What was originally a canon of meaningfulness is therefore seen to evolve into a mere definition of a subdivision of the system of types. The essential economy of the device is already apparent. Instead of the inefficient membership copula, only a part of whose notationally possible interpositions are significant while the rest yield notational refuse, there now stands the versatile comma, all possible interpositions of which yield expressions significantly denotative of sequences, which latter may or may not be sequences of the kind called propositions. The device is economical further in obviating the need of explaining propositions as of some new category lying outside the given

28

A SYSTEM OF LOGISTIC

scheme of types. Rather do propositions find their place already provided in the basic ontology of the system, namely as a kind of sequences. A proposition, while always a sequence, may be more specifically a triad, a tetrad, or anything further. Thus, taking χ in the earlier explanation as the sequence z,w, we see that the proposition "z stands to w in the relation y," formerly written "(z,w) ey", becomes the sequence y,(z,w), which, by the convention for the suppression of parentheses, is y,z,w. The proposition in question is hence the triad y,z,w, where y is a class (more specifically, a relation) of next higher type than the sequence z,w. In precisely similar fashion a tetrad y,z,u,v may be a proposition, namely when y is a triadic relation of next higher type than the triad z,u,v. The tetrad y,z,u,v is then the proposition formerly expressed by "(z,u,v) ey". Similarly for all further cases; in general, an n-ad χι,χ·ι,. . ,,xn is a proposition when and only when xx is an (n — l)-adic relation whose type succeeds that of the (η — l)-ad Χΐ,Χζ,. . .,χη· All this is of course only a recapitulation of what has already been provided for in complete generality by characterizing y,x as a proposition when y is a class next higher in type than x. Thus far we have seen the office of the type theory as arbiter of meaningfulness give way to the office of classifier of sequences as propositions or otherwise. The latter office will be found, however, to carry with it the former. There are namely two kinds of contexts the significance of which demands that a certain term mentioned therein be a proposition. A variable or complex whose context makes this demand upon it will be said to occur in propositional position. The conditions for being a proposition which are imposed by the theory of types are thus conditions as well for the meaningfulness of a context, when they are applied to a term mentioned in propositional position in that context. One manner of propositional position is occurrence as an entire postulate or theorem of the system; it is set down as a demand of significance that to stand as a postulate or theorem an expression must represent a proposition as against any other kind of sequence. Thus e.g. the theorem 4 1 , which reads "V,x" ("x is a member of V"), demands for significance that for whatever value be given the variable " x " the typically ambiguous constant " V " (duly to be defined)

PROPOSITIONS

29

be determined as of next higher type. Otherwise " V , x " , though none the less denotative of a sequence, would not represent a proposition and hence could not meaningfully stand as a theorem. The other manner of propositional position which has been promised the reader must be withheld until the notion of abstraction has been introduced.* I t has been suggested f t h a t we need only one kind of variables, the completely general variables " x " , " y " , etc., and t h a t whenever the range of a variable must be confined to some subdivision of the system of types, say classes, or more specifically relations, or perhaps sequences, or more specifically propositions, such confinement is always necessitated b y and reflected in the nature of the context and hence demands no further advertisement through use of special class variables, relation variables, sequence variables, proposition variables, etc. In formal developments such confinement of the type-range of a variable usually arises through the occurrence of t h a t variable, or of a complex involving t h a t variable, in propositional position. 1 If the variable " x " enters a given context in propositional position, the range of that variable is of course confined for the meaningfulness of t h a t context to propositions. If on the other hand the expression "x,y" occurs in propositional position, so t h a t it must be taken as representing a proposition, then the range of " x " is confined to classes: for x,y is a proposition only if χ is of next higher type than y, and whatever is of non-minimal type is a class. If the expression "x,y,z" occurs in propositional position, then, analogously to the preceding case, oc must for significance of the context be of next higher type than the sequence y,z; the range of the variable "x" is therefore confined for t h a t context to relations. The next and last example involves D, whose definition i is such as to compel it to be of relational type. If the expression "D,x" occurs in propositional position, so as to be forced for the significance of its context to represent a proposition, the range of the variable " x " is confined for t h a t context to sequences: for where D,x is a proposition the relation or class of sequences D must be of next higher type t h a n x, which latter must therefore be a se* V. infra, pp. 35-36. t P. 10, supra. ' The only other way in which such confinement can arise is through the occurrence of the variable as sole occupant of the square brackets indicative of the primitive operation of congeneration. V. infra, p. 35. § V. infra, p. 153.

A S Y S T E M OF LOGISTIC

30

quence. I t is in such ways that an endless variety of restrictions of the type range of a variable m a y depend, in the present system, upon the occurrence of that variable either directly in propositional position or within some complex having propositional position. Although the general variables " x " , " y " , etc., suffice in the described manner for all purposes to which a variable may be put, it will nevertheless prove psychologically convenient to introduce distinctive proposition variables and class variables in lieu of the general variables " x " , " y " , etc., when the range of the general variable is already restricted b y the context to propositions or to classes. As proposition variables I shall use "p",

"q",

"r",

"s",

and "t",

reserving the rest

of the alphabet for the general variables. A s class variables I shall use lower-case Greek letters, reserving however " e " and " i " for certain constants subsequently to be defined. Hence e.g. the expressions " x " and "y,z"

in propositional position will be rendered rather as

and " a , z " respectively.

"p"

I t is important to note that the generality

of the variables " x " , " y " , etc., is not diminished b y the adoption of these special variables; the former continue of course to admit as values propositions and classes along with everything else. "p",

"q", etc., and "α",

"β",

Indeed,

etc., m a y themselves be regarded from

the strictly formal standpoint as just so many further general variables of exactly the same sort as " x " , "y",

etc.; then, from a less

formal and more expository standpoint, I aid the reader's intuition b y adopting marginally the unofficial agreement to conduct the ostensibly random selection of variables in such a way, in practice, as to draw upon the subclass of variables "p",

"q",

etc., only when the variable

occurs in propositional position, to draw upon the subclass of variables " α " , " β " , etc., only when the range of the variable is restricted by the context to classes, and to use the variables " x " , " y " , " z " , etc., whenever neither of these restrictions obtains. Before he reaches the formal presentation of the system the reader will have been apprised of those simple circumstances under which an expression m a y be said to occur in a context in propositional or otherwise restricted position. Through the device of singling out certain variables for exclusively propositional and classial purposes I therefore impart no information to the reader which he could not determine from the formal context independently of such typographical conventions; the device serves rather merely as a pony b y means of which he may be spared a portion of his pains.

PROPOSITIONS

31

I shall use the variables "p", "q", etc., and "α", " β " , etc., not only in formal developments, under the conditions described, but also whenever I have occasion in informal discussion to mention propositions or classes through the medium of variables. This especial distinction which the convention of special variables accords propositions and classes could of course be bestowed equally upon any other arbitrary subdivisions of the scheme of types. Variables "P", "Q", "R", etc., might e.g. be used in strictly analogous fashion for relations, " Χ " , " Υ " , " Z " , etc., might be used as sequence variables, the Hebrew alphabet might be used for special triadic-relation variables and the Cyrillic for triad variables, and so on through all the resources of typography. It has seemed to serve best the interests of clarity and expediency in the present work, however, to confine the use of special variables to the signalizing of typical restriction relatively only to two subdivisions of the type scheme, class and proposition. The variables "φ", "φ", etc., indicative of propositional functions in ΡΜ have no place in the present system; their offices are taken over by class variables " a " , " β " , etc. Propositional functions, as elements of the system of P M , may be regarded as replaced here by their extensions, i.e. the classes which they determine. Theorems of ΡΜ involving a function variable " φ " may be translated into the language of the present system by rendering "φ" as "a". The expression "φχ", indicative in P M of the predication of a propositional function or property φ relatively to a term x* gives way here to the expression " a , x " indicative of the proposition that ζ is a member of the class a; s i m i l a r l y t h e "φ(χ,υ)"

of PM gives w a y h e r e t o "a,x,y"

(or " a , ζ " ) ,

which denotes the proposition that the sequence x,y (or z) is a member of the class of sequences a, or in other words that χ has the relation a to y. This assimilation of propositional functions to classes and of predication to membership represents no actual impoverishment of logistic, but only the elimination of useless lumber: for, as will subsequently * be shown, all theorems of P M involving function variables or predication can be proved in the present system under the indicated manner of translation. It is true that certain theorems can be proved of classes in the present system, and also in P M , which cannot be proved of propositional functions in P M . The fundamental theorem * Cf. pp. 5-7, supra. t Chapter XVIII.

32

A SYSTEM OF LOGISTIC

of this kind is the principle of extensionality. This principle is proved for classes in P M as the theorem *20"43 and in this system as 6'22, namely that a and β are identical if every member of α is a member of β and vice versa. The corresponding proposition for propositional functions is not demonstrable in PM, namely that φ and φ are identical if, for every χ, φχ = φχ. On the other hand the latter is equally incapable of disproof in PM, so that the disparity is a purely negative one residing in the incompleteness of the system of propositional functions in PM. All that can be proved of propositional functions in P M depends upon the extensions of the functions, and hence can be proved of classes. The exclusion of the propositional function from the present system does not presuppose the non-existence of propositional functions, nor does it involve the doctrine that all that can be said of propositional functions within or outside logic depends upon their extensions; rather does the exclusion relate only to the bounds of the subject matter of the system. Nor does this suppression of propositional functions in favor of classes weaken the present system relatively to that of PM, since P M itself can prove nothing concerning propositional functions beyond what relates to their extensions. The adoption of the doctrine of propositions as sequences was explained on strictly technological grounds, and from that point forward technical matters of type and manipulation were considered in view of that doctrine. Mention must now be made of the bearing of the doctrine upon the more philosophical side of logic. Wittgenstein * identifies the proposition with the symbol which, as we should ordinarily say, expresses the proposition. I have only to suggest, without wishing to contend over terminology, that the word "sentence" is already fairly well suited by general usage to that purpose; it is with the nature of propositions, not in the sense of sentences, but in the sense of what sentences may be taken to symbolize, that I am here concerned. In my doctrine of propositions as sequences it is not the sentence but the denotation of the sentence that is construed as a sequence. Professor Whitehead f emphasizes the non-assertiveness of propositions as contrasted with judgments, which latter may be said to involve the evaluation of propositions as true or false. A proposition * Tractatus Logico-Philosophicus, 3 Ί 2 , 3'31. t Process and Reality, pp. 281-284, 293.

PROPOSITIONS

33

itself, e.g. the proposition predicating redness of this book, is regarded by Whitehead as the noncommittal notion of redness as, or as if, attaching to this book, or of this book as, or as if, red. The doctrine of propositions as sequences stands in striking agreement with Whitehead's point of view; it presents a definite technical entity fulfilling just the demands which he makes of a proposition. The proposition predicating redness of this book is for me the sequence redness, this book or class of all red things, this book. Nothing could be more noncommittal, less assertive, than the connexity constitutive of association in a sequence. As is well known, two propositions which are materially equivalent, i.e. both true or both false, are indistinguishable so far as concerns their properties within the ordinary calculus of propositions. From this it does not follow, of course, that materially equivalent propositions are identical. Such a conclusion would make for the existence of but two propositions, the true and the false; propositions would become truth values, and sentences would become, as with Frege,* symbols of truth values. At least two of every three sentences would thus be synonymous — an appalling ratio of verbiage to content, even for the hardiest extensionalist. If on the other hand Frege's conclusions be rejected, what is propositional identity to involve over and above material equivalence? In seeking more exacting conditions of identity than material equivalence, danger is to be apprehended of having to leave the terra firma of algorithmic logic and tread more metaphysical ground; the usual formal procedure, therefore, is to avoid the question by handling material equivalence as such and never mentioning the identity of propositions. It is a singularity of the doctrine of propositions as sequences that it gives a definite meaning to the identity of propositions which goes far beyond the trivial condition of material equivalence, and that in so doing the doctrine continues to confine itself to strictly logistical territory, and, indeed, to import nothing exceptionable to the most ardent extensionalist. When propositions are construed as sequences, identity between propositions becomes merely a case of identity between sequences. From the definitions and postulates of the present system it will subsequently be proved (7"3) that sequences are identi* Grundgesetze der Arithmetik, Vol. I, p. 50.

34

A SYSTEM OF LOGISTIC

cal when and only when their respective affixes are identical and their respective bases are identical, i.e. that x,y = z,w . = . χ — ζ . y = w.

Taking the sequences x,y and z,w then in particular as propositions, so that χ and ζ are classes a and β next higher in type than y and w, we see that the propositions a,y and ß,w are identical when and only when« = /Sand?/ = w. This is patently a far stricter connection than is demanded in mere material equivalence; it is a different connection moreover from that which obtains when propositions may loosely be said to "convey the same message." Let a be for example the class of all cities now smaller than Chicago, let β be the class of all cities now larger than Paris, let y be Paris and let w be Chicago. Thus a,y is the proposition that Paris is now smaller than Chicago, and ß,w is the proposition that Chicago is now larger than Paris. Not only are these two propositions both true and hence materially equivalent, but furthermore both "tell us the same thing," however this convenient phrase may be analyzed; yet the two propositions are not identical, since a is not identical with β, nor, indeed, is y with w. If on the other hand we take 7 as the class of all cities now having fewer than three million inhabitants, and ζ as the capital of France, we find that the propositions a,y and 7,z are identical: for a = 7 and y = z. Thus, whereas only difference in respect of truth value is relevant to the so-called propositional calculus, propositions of the same truthvalue may still fail of identity by differing in respect of the terms whereof they are sequences. This result depends upon no resort to anything which the extensionalist could object to as intensional (in the technical sense of involving discrimination between predicates having the same extension). Identity is defined in the present system * in a strictly extensional manner, such namely that χ and y are said to be identical if y belongs to every class to which χ belongs. The a and β upon whose identity, along with that of y with w, the identity of the propositions a,y and ß,w has been seen to depend, are themselves in turn conceived in pure extension, being taken as classes. The doctrine of propositions as sequences thus not only endows propositional identity with a thoroughly definite and satisfactorily discriminative meaning, but does so quite without commerce beyond the strictly extensional medium of mathematical logic. * P. 78, infra.

CHAPTER CONGENERATION AND

IV ABSTRACTION

far only one primitive operation has been introduced, the binary operation of ordination. Let us turn now to the second primitive operation of the system, a unary operation which I shall call congeneration. The result of the operation is represented notationally by placing the sign of the operand χ in square brackets, thus: "[x\". The operation is coordinated with the scheme of types according to the following conditions: (i) the operand must be of non-minimal type (i.e. a class), and (ii) the result [x] of the operation is of next higher type than the operand x* The first of the two conditions tells us that for significance of "[x]" "x" must represent a class, i.e. that "x" has classial position in any context involving "[#]"; therefore, according to the conventions for the use of special variables, in any context exhibiting "[x]" the general variable "x" will give way to a class variable " a " . Thus " [ « ] " is the general expression of the result of congeneration. From the conditions (i) and (ii) together it follows furthermore that [a] is in every case a class of classes, a class namely whose members are classes of the type of a. THUS

The result [a] of the operation of congeneration upon a class a is called the congenerate of a, and is to be construed as the class of all genera, or superclasses, of a. Thus [a] is the class of all those classes 7 such that a is included in 7, i.e. such that every member of α is a member of 7. If β is of the same type as a and hence of next lower type than [a], then according to the developments of the preceding chapter the sequence [α],β will be a proposition, the proposition namely that β is a member of [a], i.e. that β is a superclass of a, i.e. that β is a class in which a is included. The sequential expression "[a],ß" in prepositional position may therefore be read " a is included in β " ; it answers to the "a C β " or (where the classes a and β are relations) to the "Q OR" of PM. The remaining primitive idea of the system is abstraction, an operation involving the bound variable.* The result of abstraction * V. supra, p. 26 n.

t Cf. p. 38, infra.

36

A SYSTEM OF LOGISTIC

is represented notationally in the form "xy", subject to the following conditions: (i) the sign under the circumflex must be a variable, and (ii) the expression following the circumflex must be a propositional expression. The latter condition embodies the second variety of propositional position, anticipated earlier.* In view of that condition a general variable "y" occurring in the context "xy" will always give way to a proposition variable "p", according to the conventions of special variables. Let " " b e any propositional expression. The result of abstraction x( ), called the abstract of " " with respect to "x", is a class, and may be described as the class of all terms χ such that . Thus, taking " " a s the expression " 0 < χ < 1 " of the arithmetic of rational numbers, the abstract χ (0 < χ < 1) is the class of all terms χ such that 0 < χ < 1, or in other words the class of all proper fractions. Similarly, by way of an example in terms exclusively of the primitive ideas of the present system, the abstract x{a,x,y) is the class of all terms χ such that a,x,y, i.e. the class of the affixes of those members of the relation a which have y as base, i.e. the class of all terms bearing the relation a to y. Correspondingly x(a,y,x) is the class of the bases of those members of the relation a which have y as affix, or in other words the class of all terms to which y bears the relation a. Again, £(a,x,x) is the class of all terms bearing the relation a to themselves. Since "a,x,y" occurs in propositional position in " x(a,x,y)", the class a must for the significance of that context be a relation and more specifically a relation whose members are sequences with bases of the type of y. Similarly the significance of "x(a,y,x)" demands that a be a relation whose members are sequences with affixes of the type of y. Significance of " £(a,x,x)" demands only that α be a relation whose members are sequences whose affixes are of the same type as their bases. A trivial example of abstraction is x(a,x); this is the class of all terms χ such that a,x, i.e. the class of all members of a, and hence is identical with a itself.1· As a further example consider the abstract x(x,y). Since "x,y" occurs here in propositional position, the conventions governing the * P. 29. t Cf. 6-38, p. Ill, infra.

CONGENERATION AND ABSTRACTION

37

use of special variables direct that " x " be written here as a class variable "a". The abstract under consideration is therefore a(a,y), i.e. the class of all classes to which y belongs. Again, consider the extremely simple abstract xx. Here the variable " x " itself, occurring as it does in propositional position, should by the conventions be rendered instead as a proposition variable " p " . The abstract thus becomes pp, i.e. the class of all propositions ρ such that p, or in other words the class of all true propositions. Finally, consider the abstract χ([χ],β). Here "x" occurs in classial position (namely in "[x]"), and should therefore be rendered instead as a class variable " a " ; the abstract is therefore α([α],β), i.e. the class of all terms (classes) a such that α is included in β, or in other words the class of all subclasses of β. The class x( ), made up as it is of all terms χ such that , will have y as a member if and only if the result of substituting the sign of y for "x" throughout " " expresses a true proposition. It is now to be observed that there is no need of demanding that the variable " x " occur anywhere in the propositional expression " the degenerate cases of "x( ) " where " " is free from occurrences of "x" need not be excluded. The above description of x( ), namely as the class of all terms y the substitutions of whose signs for "x" throughout " " express true propositions, can be retained also in the case where " " contains no occurrence of "x". In this degenerate case the substitution of any sign for " x " in " " is of course vacuous, yielding as its result merely the original " in this case therefore either every or no "substitution" for " x " in " " yields the expression of a true proposition, according as " " itself expresses a true or a false proposition. If therefore " " is free from occurrences of " x " and expresses a true proposition, the class χ( ) comprises all terms y and is hence the universal class; if on the other hand " " is free from occurrences of " x " and expresses a false proposition, the class x{ ) comprises no terms and is hence the null class. Thus, whereas we have seen that £(0 < χ < 1) is the class of all proper fractions, on the other hand £(0 < ζ < ]) is the universal class or the null class according as ζ is or is not a proper fraction. In particular x(0 < \ < 1) is the universal class, and x(0 < 3 < 1) is the null class. From the earlier discussion it is clear that the "x( ) " of the present system corresponds exactly to the similarly expressed device

38

A SYSTEM OF LOGISTIC

in P M , with the sole qualification that I do not require "x" to occur in " ", whereas in P M , in practice at least, that demand is made. Even this divergence admits of immediate reconciliation: if namely " " contains no occurrence of " x " , the expression may be elaborated into an equivalent form exhibiting " x " immaterially. Thus if " x " does not occur in " " the expression " x ( ) " of the present system may, through recourse to the notions of identity and conjunction, be regarded as corresponding to the "x(x = χ . ) " of P M . Since, regardless of what y may be, the proposition expressed by " y = y • " will be true or false according only as the proposition expressed by " " i s true or false, it follows that either every or no substitution for " x " in " χ = χ . " will express a true proposition according as " " expresses a true or a false proposition. Thus, where " " i s free from occurrences of " χ " , "x(x = χ . )" agrees with the "x( ) " of the present system in representing the universal class or the null class according as " " expresses a true or a false proposition. We see therefore that the failure to insist upon the occurrence of "x" in " ", for significance of "x( ) " , does not result in any real difference of behavior on the part of the general notion " x{ ) " of this system relatively to the corresponding notion of P M . In "x( )", "x" is a bound variable.* This means, not that "£( ) " like " y + χ — χ " retains the same denotation irrespectively of what constants or complex expressions be substituted for " x " , but rather that in " x ( ) " " x " simply does not admit of such substitutions. To rewrite "x" throughout "£( ) " as some new variable is of course merely to impose a trivial change of notation; introduction of a constant or other construct for "x" in such an expression, on the other hand, yields nonsense. The bound variable exists in Ρ Μ in a variety of contexts: the " x " in class expressions of the form " x ( ) " , the " x " and "y" in relational expressions of the form " x y ( ) " , the " x " in quantified expressions of the forms " (x) . " and "(3.x) . ", and the "x" in descriptive expressions "(ix)( ) " , are all bound variables. In the present system on the other hand the bound variable occurs only * The adjectives "apparent" and "real," as applied to variables in PM, have been abandoned here in favor of the more appropriate terminology "bound" and "free" current on the Continent.

CONGENERATION AND ABSTRACTION

39

as "x" in expressions of the form "x( )". Here therefore a bound variable is always marked at its initial occurrence by a circumflex. The propositional expression following the circumflex, i.e. the expression " " in "£( )", is called the scope of the bound variable "x". It remains to consider the type of the class £( ). Since this class is to embrace as members terms the substitutions of whose signs for " x " in " " express true propositions, the class x( ) must be interpreted as of next higher type than a term the substitution of whose sign for " x " in " " yields a propositional expression. The type of £ ( ) is hence determined insofar as the type of any term χ is determined by propositionality of the context " "; x( ), namely, is of next higher type than a term χ such that " " expresses a proposition (rather than some other manner of sequence). Thus e.g. the class ά([α],/3) must be of next higher type than a term a such that [α],β is a proposition. But in order that [α],/3 be a proposition the class [a] must be of next higher type than β and hence a must be of the same type as β; the class, α ([a],/3) is therefore determined as of next higher type than β. Similarly, the class pp must be of next higher type than some term ρ such that ρ is a proposition; hence pp is determined as to type only to the extent that it must be a class of propositions, i.e. that it must be of some type of the form "(α! ία)!".* As in all strictly formal practice, so in particular in dealing with the abstract, we are concerned not with unique determination of type but rather with typical ambiguity variously restricted. There are cases in which £ ( ) is completely ambiguous as to type except for the fact that it must be a class; such a case is £(x = x), since "χ = χ " is (in view of the subsequent definition of identity) a propositional expression irrespectively of the type of x. The type of x( ) is also of course completely indeterminate if " x " does not occur in " ". In all these cases, however, it should be noted that in practice further determinants enter beyond those having to do with intrinsic features of the expression "£( )"; given namely any degree of internal determinacy of type, or none whatever, the wider context in which "£( ) " occurs will in general impose upon that class supplementary restrictions of type. An example from the formal postulates or theorems will illustrate the manner in which the various conditions of type and significance * Cf. pp. 21-22, 27,

supra.

A SYSTEM OF LOGISTIC

40

interact throughout a symbolic context to restrict or determine, relatively to one another, the types of the various terms. Let us consider the longest of the postulates, namely 1"6: I M & ( [ r ] , ί([β(β , Y(A,X,y))}

, β{β, Y([Z(ß,X)]

, I(Y,V)))))],

VP)] , WP

According to the above discussion, the complex abstract (1)

χ([θ(θ,

y(a,x,y))],

θ(θ,

0(0*03,χ)],

z(y,y))))

will be of next higher type than a term χ such that (2)

[ 0 ( 0 , y(a,x,y))]

, 0 ( 0 , y([Hß,x)]

,

Kw)))

is a proposition. Since however " β , χ " occurs in propositional position in (2), namely as the scope of a bound variable " ζ ", χ must be of next lower type than β in order that the expression (2) be significant at all, whether as denotative of a proposition or otherwise. Hence χ must be of next lower type than β if the expression (2) is to denote a proposition. Therefore the abstract (1) is of next higher type than a term χ of next lower type than β; (1) is thus of the same type as β. Considering now a slightly wider context in 1'6, we see that " Μ

, χ(ίθ(θ,

y(a,x,y))],

0 ( 0 , y([z(ß,x)],

Κί,y)))Y'

occurs in propositional position, standing as it does as the scope of the bound variable " a " . Therefore [ f ] must be of next higher type than (1). But the congenerate [ f ] is of next higher type than its operand f ; therefore f is of the same type as (1). But (1) was seen to be of the same type as β. Hence ζ is determined as of the same type as β. The postulate Γ 6 contains four free (i.e. not bound) variables, namely " β " , " ζ " , " τ " , and " ρ " ; all other variables are marked by a circumflex at their initial occurrences, and are therefore bound. I t has been seen that of these four free variables the first two, " β " and " ξ " , must for significance of Γ 6 be determined as representing terms of the same type. Their relative determination of type is thus complete. They are also partially determined as to absolute type, to the extent namely that they must be classes: for " ζ " has classial position by occurring in the context " [ f ] " , * and " β " has classial position in that "β,χ"

stands in propositional position.f I t is for this reason, in

fact, that the conventions of special variables prompt the use here of the class variables " ξ " and " β " instead of general variables " u " and "v".

On the other hand the significance of Γ 6 makes no demands

upon the types of γ and ρ relatively to β or ξ or to each other. Like β * Cf. p. 35, supra. t Cf. p. 29, supra.

CONGENERATION AND ABSTRACTION

41

and f, however, y and ρ receive from their contexts certain minimal restrictions of type in a non-relative way. Thus y must be a class, since "y,y" has propositional position; hence the use of the class variable "y" instead of a general variable. Furthermore "p" has propositional position, standing as it does as the scope of the bound variable " η " (and also, later, as the scope of a bound variable "w"). Hence the use of the propositional variable " p " instead of a general variable. The variable " a " has relational position in the context "y(a,x,y)", since "a,x,y" has propositional position in that context.* Therefore the abstract (3)

fi([f]

, *([0(0 , y(a,x,y))]

, 0(0 , y([z(ß,x)]

,

t(y,y)))))

must be a class of relations, since the bound variable " a " occurs here in the context "y(a,x,y)". Now if "[μ],ν" occurs in propositional position, ν must be of next lower type than [μ] and hence of the same type as μ. But " [ ά ( Μ , x([0(0, £ ( « ) ) ] , 0 ( 0 , M(ß,x)],

Ky,y))M

, VP"

occurs in propositional position in Γ 6, namely as the scope of a bound variable "w". Therefore rjp must be of the same type as the class of relations (3). Hence ήρ must also be a class of relations, or, to be less specific, a class of classes. Since the bound variable of an abstract represents so to speak a specimen member of that abstract, we may regard the bound variable " η " as having classial position when, as in this case, "rjp" is determined for significance of its context as representing a class of classes. Hence the use in 1'6 of a class variable "η" instead of a general variable "z" as bound variable in "ήρ". Any system, whether of logic or of dynamics, lays claim to the truth of the propositions which stand as its theorems and postulates; logic, however, is in part a discussion of propositions, and hence, unlike dynamics, may have to do with truth as a part of its subject matter as well. The sole channel through which truth becomes thus involved in the subject matter of the present system, Professor Whitehead has pointed out, is the operation of abstraction. Abstraction yields classes which depend for their contents upon the truth of propositions of the form involved in the abstraction; upon such classes, in turn, I base the definitions of the "truth functions" of the ordinary propositional calculus. * Cf. pp. 29, 36, supra.

CHAPTER

V

RULES OF I N F E R E N C E LIKE any system presupposing no deductive basis beyond itself, this system requires in addition to its formal postulates certain informal rules in accordance with which to infer consequences from those postulates. The four rules here adopted correspond fairly closely to those used in P M . The statement of the first of these rules of inference presupposes an explanation of what is meant technically in the present work by substitution. The substitution of an expression Ε for a variable "x" in an expression E' consists of the following operations: If there is any bound variable whose circumflexed occurrence lies in E' and whose scope contains an occurrence of "x", and if the letter used for that bound variable occurs also as a variable in E, let the bound variable be rewritten in E' so as to be alphabetically distinct from all variables in E. These adjustments having been made, let Ε be written in lieu of "x" throughout the thus adjusted form of E'. The first informal rule of inference is the rule of substitution: The result of the substitution of any expression for any free variable in any theorem or postulate may, if significant, be set down as a theorem. Although not mentioned as an informal rule in the first edition of P M , the rule of substitution is of course employed at every turn in that work, as has been recognized by Russell on subsequent occasions.* The adjustment of bound variables called for in the above definition of substitution is required for the prevention of vicious confusion of free and bound variables. Given e.g. the theorem (8-2)

ι'χ = y(x = y)

to the effect that the unit class of χ is the class of all terms identical with x, we might, but for that stipulation regarding bound variables, simply replace the free variable "x" by "y" in 8 2 and infer, by au* Introduction to Mathematical Philosophy, p. 151; PM, Vol. I (2d ed.), p. xix.

RULES OF INFERENCE

43

thority of the rule of substitution, the undesirably monistic consequence (1)

t'y

=

y{y

=

y)

which tells us that the unit class of y consists of all self-identical terms, or in other words (since everything is self-identical) that the unit class of y is coextensive with the universe. According to the above explanation of what is meant by a substitution, on the other hand, the rule of substitution does not countenance the inference of (1) from 8" 2. Since the variable " y " occurs under a circumflex in 8"2, and its scope (viz. "x = y") contains an occurrence of the free variable "x" upon which the substitution is to be made, and since the expression "y" to be substituted contains (in fact consists of) an occurrence of the variable "y", the substitution of "y" for "x" in 8"2 proceeds, according to the definition of substitution, by our first rewriting the bound variable " y " in 8'2 as some other letter " z " and then replacing "x" by "y". The result i'y

=

&(y

=

z),

unlike (1), is a trivial but unobjectionable consequence of 8'2, differing from the latter only in the letters chosen for the variables. The above argument shows also that the process of substitution must be similarly circumscribed for the system of PM, since the theorem 8"2 is valid also in that system. The second informal rule of inference is that of subsunvption: Given

any

where

" a "

theorem and

or " x "

postulate are

any

"

" , we

may

infer " [ α ]

, x(

)"

variables.

This rule answers in effect to the rule of inference stated in P M as the informal primitive proposition * 9 ' 1 3 : the rule namely according to which, given a theorem or postulate " " containing a free variable "x", one may infer the theorem " (x). ", i.e. " F o r every x, ." The universal quantification " (x). " will appear in the present system as "U , £( ) " where U is defined as [F], the universal class V having been defined in turn.* In terms of this system PM's rule * 9 ' 1 3 would therefore assume the following form: (2) Given a theorem or postulate " " containing a free variable "x", one may infer the theorem " [ F j , £( ) " , i.e. "V is included in £(- - -)." Clearly however the condition that "x" occur in " " i s superfluous, * Cf. pp. 88-89, infra.

44

A SYSTEM OF LOGISTIC

since, if "x" does not occur in " ", truth of makes x( ) the universal class * and hence assures us of the inclusion of V in x( ). Thus (2) gives way to (3) Given a theorem or postulate " - - - " , infer "[V], £(---)" where " x " is any variable. The reference of (3) explicitly to the constant " V " is, in contrast to the reference of the rule of subsumption to an unrestricted class variable " a " , an idle complexity: idle in that any class« must be included in x{ ) if the universal class V is included in x( ), and a complexity inasmuch as V, not yet defined, has the relatively complex form "£([7(7,#)] , y(y,x))" in terms of primitive ideas.* Instead of (3), therefore, I adopt the rule of subsumption as originally enunciated; the offices of (3) can always be filled by first applying the rule of subsumption and then substituting " V " for the free variable " a " in the result on the authority of the rule of substitution. I t might be thought desirable, in order to avoid conflict of scopes on the part of similarly written bound variables, to incorporate into the rule of subsumption the stipulation that "x" be a variable not already occurring as a bound variable in the given theorem or postulate " ". For, let the theorem or postulate " " contain an occurrence of " CC äS 81 bound variable; then let us write " " as " - - £ ( · · · ) - - " , where " · · • " is the scope of "x". Now without the restriction just now suggested the rule of subsumption would permit us to infer, from the theorem or postulate " - - £ ( • · • ) - - " , the theorem "[α] , - £ ( · · · ) - - ) " · The latter theorem is dangerously ambiguous in failing to determine all occurrences of " x " within " · · · " as occurrences specifically of that bound variable " x " which is marked by the inner circumflex rather than occurrences of the bound variable " x " which is marked by the outer circumflex. The suggested restriction upon the rule of subsumption is unnecessary, however, if relatively to such conflicts of scope the convention be a d o p t e d t h a t if the same letter is used for two bound variables of one of which lies within

the scope

the scope of the other, then all occurrences

that letter within the smaller scope are to be construed

of

as cases of the bound

variable whose scope is the smaller. In practice this convention need * C f . p . 37,

supra.

t Cf. definitions D2, D3, and D4, p. 194, infra.

RULES OF INFERENCE

45

never be applied, for one can always so conduct the choice of variables as to obviate expressions of the form " £ ( - - £ ( · · • ) - - ) " . Enunciation of the third informal rule of inference had best be preceded by definition of material implication: Dl.

ρ D q for [xp] , yq

I.e., "pDq" is to be used as an abbreviation for "[xp] , yq". The expression "xp" is that simple case of "x( ) " where the prepositional expression " " is the single variable " p " ; similarly for "yq". D l thus utilizes the degenerate cases of abstraction where the bound variable does not occur anywhere in its scope; cases of "x{ ) " , that is to say, where "x" does not occur in " ". I t was observed in the informal discussion of abstraction that in the degenerate case in which " x " does not occur in " " the expression "£( ) " denotes the universal or the null class according as is true or false; hence xp will be the universal class V or the null class A according as ρ is true or false, and correspondingly for yq. If therefore ρ and q are both true, the proposition \xp] , yq becomes [V],V, i.e. "V is included in itself"; if ρ is false and q true, [ϊρ] , yq is [A],V, i.e. "A is included in V"; if ρ is true and q false, [xp] , yq is [V],A, i.e. " V is included in A"; if finally ρ and q are both false, [xp] , yq is [A],A, i.e. "A is included in itself." All these propositions of inclusion on the part of V and A are true except the third, namely [V],A. Hence [xp] , yq is false only in case ρ is true and q. false; as definiens for material implication "[xp] , yq" therefore gives the customary meaning to "p D q", namely " N o t both ρ and not-g." The above discussion serves solely to indicate informally to the reader's intuition the agreement of D l with the usual definition of material implication; it is therefore irrelevant that the notions "V", " Λ " , " n o t " , and " b o t h " , used in the discussion, have thus far not been formally defined for the present system, and that the truth or falsity of the four simple propositions of inclusion referred to has not been formally derived. From the strictly formal standpoint D l is a mere convention of abbreviation and demands no justification. Since definitions are conventions of notational abbreviation and thus lie outside the scheme of postulates and theorems, the rules of inference whereby we transform one theorem into another are enunci-

46

A SYSTEM OF LOGISTIC

ated with no thought of their being applied to definitions. For generality definitions are nevertheless expressed for the most part with help of free variables, as in the case of D l , and the application of such a definition to a given case proceeds by means of substitutions for those variables analogous to the substitutions performed upon postulates and theorems. Whereas a definition involving no free variables is a specific statement of abbreviation, a definition involving free variables is a general rule of abbreviation the special applications whereof are had by making various substitutions for those variables in definiens and definiendum. Thus D l is meant to tell us not only that the symbol "p D q " is an abbreviation for the symbol " [xp] , yq ", but also that the symbol "pD p" is an abbreviation for the symbol "[xp] , yp", that the symbol "[α],β .D.qDp" is an abbreviation for the symbol "[£([ V is equivalent to the proposition expressed by the substitution of "y" for "x" in " ". The manner in which this equivalence may in any given case be proved, with the aid of the rule of concretion, is seen from the following example. In 3'4 the principle of self-identity (x = x) will be proved. Substituting "z(x(a,x,w) , y)" for " x " in 3'4 on the authority of the rule of substitution, we have S(£(a,x,w),

y) = z(x(a,x,w)

, y).

Applying the rule of concretion to this theorem so as to transform the right-hand side, we have z(£(a,x,w),

y) =

z(a,y,w),

which, according to the definition of material equivalence,1 is x(a,x,w) • P. 42.

, y . = . a,y,w.

t p. 37, supra.

» Cf. D7, p. 95, infra.

RULES OF INFERENCE

49

Such is the proof that x( ) , y is equivalent to the proposition expressed by substituting "y" for "x" throughout " in the special case where " " is "a,x,w". The corresponding equivalence for any other case of abstraction can be proved analogously. The further propositional operations of material equivalence, denial and conjunction will be introduced into the present system as abbreviations of contexts in which the operands stand as scopes of bound variables,* just as was done in the case of material implication in D l . On the basis of D l and these subsequent analogous definitions, therefore, the rule of concretion as stated above can be applied relatively to expressions of the form "£( ) , · · · " whether they occur in notational contexts of the form "z(x( ) , · · · ) " or in any of the contexts (4) «*(- - - ) , . . · . 3 " (5) " D.f(---) , ..." (6) --),.·· •» (7) " • = .*(---),·•·" (8) " H - - - ) , - • • • " (9) " — . * ( - - - ) , · · · " (10) . £(- - - ) , . · · » For example, given a theorem having the form (4) or containing an expression of that form, "x( ) , - · · " can be replaced therein by the result " - - · · . of writing " · • · " for "x" throughout - -", by the following process. Application of D l to the given theorem enables us to replace (4) by

αϊ)

"im---), ···)],( )". Application of D l to the resulting theorem then replaces (12) therein by " - - · · · - - D so that " £ ( ) , · · · " in the original theorem has given way to " - ". Theorems involving the forms (5) to (10) can be subjected to the rule of concretion in analogous fashion. Thus far it has been seen that an expression "x( ) , - - · ", occurring in a theorem in any of the contexts (4) to (10) or as the scope of a bound variable, can by means of the rule of concretion be replaced in that theorem by the result " - - · · · - - " of putting " · · · " for "x" * Cf. pp. 95-97, 99, 100, infra.

50

A SYSTEM OF LOGISTIC

in " ". The replacement is also possible when "x( ) , ···" occurs, not as a part of a theorem in the described fashion, but as the entirety of a theorem: that is to say, it is possible given a theorem "x( ) , · · · " to deduce " - " as a theorem. This manner of inference is never required in practice, but is readily performed with the aid of the theorem 2"5 of self-implication. Substitution upon the latter yields the theorem *(---),···.:>.*(---),···, which, having as it does the form (5), enables us according to the preceding discussion to deduce

* ( - - - ) , · · · •=>--

·

But the antecedent of this theorem has been supposed given as a theorem; the consequent can therefore be inferred by the rule of inference of the consequent. The rule of concretion corresponds in P M to informal understandings, mainly tacit, which govern the treatment and use of the notion of propositional function.* A minimum of those presuppositions, adapted to the devices of the present system and so fashioned as to take on the status of an informal rule of inference, constitutes the rule of concretion. In a complete inventory of the informal rules of inference, account should perhaps be taken of our right to replace definiens by definiendum and vice versa. Professor Sheffer f maintains that this involves an informal rule of inference, which he calls the rule of replacement. Definitions have exactly the same status for the present system as for P M : that namely of notational conventions serving to introduce abbreviations. It is sufficient here merely to mention the fact that such recourse to abbreviations may be construed as depending upon an informal rule of inference.* f * Cf. pp. 6-9, supra, and pp. 187-190, infra. In his lectures. * For a novel and thoroughly rigorous procedure treating definitions not as notational conventions but as assertions of equivalence integral to the system, cf. Stanislaw Lesniewski, "Ueber Definitionen in der sogenannten Theorie der Deduktion," Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie, XXIY (1931), Classe III, pp. 289-309.

CHAPTER

VI

POSTULATES EXPRESSED in terms exclusively of the primitive ideas, the formal postulates of the system are as follows: 1-1.

[zflMxp],«)] , xp)] , Ap

12.

[ ά ( [ χ ρ ] , x([yp],

1-3.

[γ], *([[«]] J ( [ f ( M , a ) ] , i ( 7 Ä ) )

ι·4.

[w([a],ß)],

m m , y ) ] ,

15.

[w([mp]

, ί ( Μ , *ρ))] , 2([ί(α,«)] , 4(0,2)))] , ([y(p

D.

[ 7 ] , xp)]

, ζ (a,ζ

. D.

β,ζ))]

,

t»([a],|S)

[y(pD. [7] , xp)] , ζ (a, ζ . D. β,ζ) . D. [α],β * Cf. pp. 46-47,

supra.

POSTULATES

53

and the following successive reductions of 1'6: I M ä ( [ r ] , Ä([3(ö , yia,x,y))3 , 0(0 , y(ß,x . D . 7,2/))))] , νρ)~\ , wp [Ä([f] , χ([θ(θ , y(a,x,y))] , 0(0 , y(ß,x • D. 7,2/))))] , ν Ρ - ^ Ρ By the definition of identity subsequently to be introduced * this last is reduced in turn to [ ä ( C r ] , *{yW,x,y)

= y(ß,x .D.y,y)))],

vv • 3 V

Expressed with help of the defined notions of material implication and identity, rather than exclusively in terms of primitives, the set of postulates therefore assumes the following somewhat less forbidding aspect: (l-i) (1-2)* (1-3) (1-4) (1-5) (1-6)

[xp],a . Dp : D ρ [a([xp] , x([yp] , y(a,x,y)))]

, [üp]

[y],a([[a]],ß([ß],a.D.y,ß))

[α],)3 . D :. [β],γ . D : α,χ . D. y,x [y(pD. [ά(Μ

[7] , ί ρ ) ] , Ha,ζ .D.ß,z).D. , i(y{a,x,y)

= y(ß,x . D . y,y)))]

[ά],β ,τΐρ.Ορ

Let us now consider these postulates severally, in an informal manner, allowing ourselves access to the whole of logic to the end of determining the intuitive meaning and the validity of the postulates relatively to the usual notions and principles of logistic. Throughout the ensuing discussion the reader should consult the postulates in the form in which they have just now been rendered, rather than turning back to their official enunciation in terms of the primitive ideas. The antecedent of the antecedent of Γ1, namely [xp],a, affirms the inclusion of xp in a. Since xp is the null class if ρ is false, and since the null class is included in any class a, we see that if ρ is false the proposition [£p],a will be true. The antecedent of 1"1, namely [xp\,a . D p, is therefore false if ρ is false, since it then affirms the material implication of a false proposition by a true one. If on the other hand ρ is true, the consequent of 1"1 will be true, for that consequent is ρ itself. Since a material implication holds if either its antecedent is false or * Cf. D2 and D3, p. 78, infra. t Note that the antecedent in this implication is not reduced further by D1 to Μ , x(y(a,x,y) = y(ß,x. D . y,y)). D p, since the scope of the bound variable "a" contains an occurrence of "a". Cf. p. 46, supra. * Unchanged.

54

A SYSTEM OF LOGISTIC

its consequent true, l ' l is verified irrespectively of the truth-value of p. We see therefore that l ' l is an acceptable postulate, since, interpreted into the terms of the familiar logistic and appraised by the principles thereof, it is true no matter what class and proposition a and ρ may be. The purpose of l ' l is to lead to Peirce's law (2*2) and thus to the theorem (5-35) that a proposition materially implied by its own denial is true. Let us turn next to 1"4. This postulate is immediately recognized as the syllogistic conclusion from the principle of the transitivity of inclusion, (3-95) and the principle that any member of a subclass is a member of the superclass, (3'25) [a],7 . D : a,χ . D. y,x. The purpose of 1'4, on the other hand, is the deduction of those two principles. For the understanding of 1'5 it is to be noted that "p D. [7] , xp", i.e. "If p, then 7 is included in xp" expresses a true proposition no matter what proposition and class "p" and " 7 " may represent; this is seen from the fact that 3tp is the universal class if ρ is true, and that any class 7 is included in the universal class. Since "p 3 . [7] , £p" expresses a true proposition, "y(pD. [7] , xp)" (a case of "y( )" where " y " does not occur in " ") must represent the universal class. Hence the antecedent of 1*5 says in effect that the universal class is included in z(a,z . D. β,ζ), i.e. that ζ (a,ζ . D. β,ζ) is universal, i.e. that a,ζ . D. β,ζ for every term z, i.e. that every member of α is a member of β. Since the consequent of 1 "5 is to the effect that a is included in β, 1*5 in its entirety postulates that if every member of a is a member of β then a is included in β. Whereas in P M this is a vacuous consequence of the definition of inclusion, in the present system it expresses a relationship between implication and inclusion which is not contained in the definitions. Γ5 serves also the additional purpose of making possible the derivation of "p D. [7] , £p" itself as a theorem (3* 3); this is of value because of its important special cases 2-3 ( p D . q D p ) a n d 4"4.

1*3 tells us that any class (of classes) 7 is included in the class of those classes α such that [a] is included in the class of those classes β

POSTULATES

55

such t h a t [β],α . D . y,ß. Since the inclusion of f in η is equivalent to the membership in η of every member of ξ, 1"3 may be interpreted as telling us t h a t every member a of the class (of classes) 7 is such t h a t every superclass β of a (i.e. every member β of [a]) is such t h a t [β],α . D . y,ß. Hence, more briefly, 1"3 tells us t h a t if α is a member of 7 and is included in β, then [β],α . 3 . -γ,β; i.e., t h a t if α is a member of 7 and a and β are mutually inclusive, then β is a member of 7. Since 7 may be a n y class of classes, Γ 3 is now recognizable as the extensionality principle t h a t if a class a and a class β include each other, β belongs to a n y class of classes to which a belongs. The purpose of 1"3 is to lead to the theorem (3'8) t h a t mutually inclusive classes are identical, or in other words t h a t a class is uniquely determined by its members. The two remaining postulates, 1"2 and Γ 6, are concerned with relations. The bound variable " a " has relational position in each of these postulates, for "a,x,y" occurs in each in prepositional position.* Since the inclusion of f in η is equivalent to the membership in η of every member of f , we may interpret Γ 2 as saying t h a t every member of ά([£ρ] , £([yp] , y{a,x,y))) is a member of [w>p], i.e. t h a t any relation α is a member of [wp~\ if (i) [ ί ρ ] , %([yp], y(pi,x,y)), i.e. t h a t ibp is included in every relation a satisfying (1), i.e. that wp is included in every relation a such t h a t xp is included in the class of those terms χ such that yp is included in y(a,x,y). Now since zp is the universal or null class according as ρ is true or false, the inclusion of zp in β demands t h a t β be universal if ρ is true, b u t makes no demand upon β if ρ is false. The inclusion of zp in β is hence, in general, equivalent to the material implication by ρ of the universality of β. T h e last paraphrase of 1 2 m a y hence be f u r t h e r paraphrased as follows: If p, then any relation a is universal provided t h a t ρ implies t h a t t h a t class is universal which consists of those terms χ such t h a t ρ implies t h a t the class y(a,x,y) is universal. Here ρ m a y be any proposition; if in particular it is a false one, the above becomes obviously true b u t uninteresting: for a false proposition materially implies a n y proposition. If on the other hand ρ is true, it can be dropped as a * Cf. p. 41, supra.

56

A SYSTEM OF LOGISTIC

premiss in the above; we are then left with the following: Any relation a is universal provided that that class is universal which consists of those terms χ such that the class y(a,x,y) is universal. In other words, if for every term χ (of proper type) it is true for every term y (of proper type) that a,x,y, then it is true for every term ζ (of proper type and thus necessarily a sequence) that a,z. Whereas the four postulates Γ1, 1*3, 1'4, and Γ5 must be provided for also in such a system as that of P M , either by directly analogous postulates or (as is the case in P M ) by an alternative set of postulates and definitions which together yield theorems corresponding to those yielded by the postulates 1 Ί , 1'3, Γ4, and Γ5 of the present system together with the definitions of the latter, on the other hand the system of P M has no place for an analogue of 1'2 either as a postulate or as a theorem. 1'2 has been seen to mean in effect that if for every χ it is true for every y that a,x,y, then for every ζ it is true that a,z; since however a single term ζ cannot be a sequence in P M , the consequent of this implication could not be rendered in P M except as a direct repetition of the antecedent. In the present system on the other hand sequences are terms and relations are classes of sequences, so that the universality of a relation a is merely the universality of α as a class. In order to deduce the universality of this class a of sequences from the universality of the class of those terms χ such that for every y the sequence x,y is a member of a, we need special provision from the postulates; it is provided by 1'2. Let us now consider the last postulate, 1'6.* It was observed f that the bound variable " a " occurs here in relational position, and that (2) a ( [ f ] , t(i)(*,x,y) = 003,®. D. τ , y ) ) ) is consequently a class of relations. 1*6 tells us that if the class of relations (2) is included in rjp, then p. By negation and transposition of antecedent and consequent, this becomes: If not p, then (2) is not included in ηρ. Here ρ is any proposition; furthermore, falsity of ρ is equivalent to nullity of the class ηρ. Hence this last paraphrase of Γ6 can be interpreted, without use of " p " , as denying the inclusion of the class of relations (2) in the null class of relations, i.e. as denying * For further discussion of Γ6 v. supra, pp. 40-41. t P. 41.

POSTULATES

57

the nullity of (2), i.e. as affirming that there is a relation belonging to (2), i.e. that there is a relation a such that f is included in (3)

£(y(a,x,y)

= y(ß,x

. D.

y,y)).

Since there are no restrictions upon choice of ζ, the whole amounts to the following: There is a relation a such that any class whatever, no matter how large, is included in (3). In other words, there is a relation a such that (3) is universal, i.e. there is a relation a such that, for every term x, y{a,x,y)

= y(ß,x

. D.

y,y),

i.e. there is a relation a such that, for every term x, the' terms y such that α,χ,y are the same as those such that β,χ . D. y,y. Hence 1'6 tells us in effect that, given any classes β and y, there is a relation a such that a,x,y is equivalent for every χ and y to β,χ . D. y,y. In expressing relations in P M use is made of the form "xy( )", which is analogous to the form " x( ) " used in the expression of classes in that work and this. The relation a whose existence 1'6 has been seen to postulate could, by means of the device "xy{ )", be expressed as "xy(ß,x . D. y,y)"; there would then be no need of the postulate Γ 6, since the existence of a relation a satisfying the condition specified in 1"6 would be demonstrable simply by indication of xy(ß,x . 3 . y,y). It is for this reason that no postulate analogous to 1'6 is required in PM. In the present system, however, the device "xy( ) " is not at hand; the primitive idea of abstraction provides only for the case of a single circumflexed variable. Since variables can stand as sequences in this system, it is true that various expressions of the form "xy( )" can be rendered here immediately in the form "z( ) " by letting the single bound variable " z " play the role of sequence which, in the form of notation "xy( )", is played by the two bound variables "x" and "y" in combination. Thus the form "z( ) " is adequate to the expression of such cases of the form "xy( ) " as ay(a,y),

yThese operations of the abstraction of propositional functions are not recognized as primitive in PM, but figure in the unsystematized presuppositions in which the treatment of propositional functions in that work abounds. In the present system the use of propositional functions as forerunners of classes and relations is altogether eliminated, and PM's primitive operations of the abstraction of propositional functions with respect to one, two, and more variables give way to my primitive operation of the abstraction of classes. One operation of class abstraction, namely that with respect to a single variable, has * Cf. pp. 180-182, 188-190, also pp. 146-147, 153, 157, 161, 164, 167, infra. t Cf. p. 7, supra, and PM, Vol. I (2d. ed.), pp. 71-84, 187-188, 200-201.

POSTULATES

59

been made to suffice; but in making it suffice I have needed the additional postulate 1"6, as has just now been seen. The postulate 1"6 thus represents in effect the strict formalization of a part of the unsystematized presuppositions of PM's theory of propositional functions.* The only direct use made of 1'6 is in proving the theorem 7Ί5. The only use of the latter, in turn, is in proving the theorems 7" 2 and 7'21, which assert that sequences are identical only if their respective affixes are identical and their respective bases are identical; i.e. if x,y = z,w then χ = ζ and y = w. Such further theorems as depend upon Γ6 do so through 7'2 and 7'21. * Cf. p. 9, supra.

CHAPTER VII MATERIAL

IMPLICATION

THE foundations of the system have been presented in their entirety; we may now proceed to the derivation of theorems. Of the theorems which are to be derived in the course of the book, the present chapter contains all and only those which are expressed exclusively in terms of variables and the sign " D " of material implication. The chapter is thus occupied with the derivation of that extremely simple subdivision of the calculus of propositions which may be called the calculus of material implication. No new definitions need be introduced as yet, as the only one relevant to this chapter is D l , already enunciated.* The notation of proofs which will be used throughout the book is an extension and compaction of the scheme used by Lukasiewicz in connection with the calculus of propositions.* The method has the theoretical advantage over those used in P M and elsewhere, that it is completely rigorous, involving no dependence upon common sense for the negotiation of ellipses. It has moreover a practical advantage over the usual methods, in that it is actually easier to follow; whereas a step of ordinary proof may call for some reflection, ingenuity or trial and error on the reader's part preparatory to his comprehending the precise manner in which the multiplicity of signalized theorems and postulates combine to yield the enunciated consequence, on the other hand the notation of proof here adopted depicts graphically in each case the manner and order in which definitions or rules of inference are applied to antecedent theorems or postulates in achieving the step of proof under consideration. In effect, proofs are written in such a way that they could be checked by a robot. The reader's insight is none the less to his advantage, for it enables him to move more rapidly than the robot and to avoid many of the laborious substitution proc* P. 45, supra. t Cf. Jan Lukasiewicz, "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls," Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie, X X I I I (1930), Classe III, pp. 55-56.

MATERIAL IMPLICATION

61

esses which are the robot's lot; under this notation of proofs it remains possible, however, for the reader to check his insight at will by returning to the robot's methods. Finally, it is interesting to note that the present scheme, despite its greater rigor and greater facility, does not occupy perceptibly more space than the usual method of logistical proof. Let us proceed to the deduction of theorems; the notation of proofs will be explained progressively as it is used. 1-4 (a / vp; β / vq; 7 / vr) + c2 + DS1 — 21.

pDq.DzqDr.D.pDr

The first line here is the proof and the second line is the theorem proved, recognizable immediately as the principle of the syllogism or of t h e transitivity

of material

implication.

The expression "1-4 (a /vp; β / vq; y / vr)" in the proof indicates that "vp", "vq", and "vr" are to be substituted respectively for the free variables "α", "β", and " y " in the postulate 1'4,* on the authority of the rule of substitution.* This yields the theorem (1) [w([vpl , vq)] , » w·)] , K[y(vp, x)] , y(vr, a;))). The next entry in the proof, "c 2 ", means that the rule of concretion is to be applied twice to the result (1) of the substitutions. Application to (1) of the rule of concretion i consists in finding within the theorem (1) some expression of the form "z(w( ) , •••)", and replacing it by "&(- - · • · - - ) " where " - " i s the expression resulting from substitution of " · · · " for "w" in " ". There are only two expressions of the form "z(w( ) , • · · ) " in (1), namely "y(vp , χ)" and "yivr, x)". The two applications of the rule of concretion to (1) which are demanded must therefore be applications relatively to these two expressions. Since "υ" does not occur in the expression "p", the substitution of "x" for "v" in "p" is vacuous; the result is simply "p". Thus, where "z(w( ) , • · · ) " is "y(vp, x)", * All reference to the postulates, in formal proofs, relates of course to the official enunciation of the postulates in terms of primitive ideas (p. SI, supra) rather than to the forms which the postulates assumed through application of definitions in the course of the subsequent informal discussion. t P. 42, supra. ' Cf. p. 48, supra.

62

A SYSTEM OF LOGISTIC

the expression "z(- - · · · - - ) " by which the former is replaced according to the rule of concretion is simply "yp". Similarly the expression with which the rule of concretion replaces "y(vr, χ ) " is simply The entry "c 2 " in the proof of 2 Ί serves therefore to derive from (1) the theorem (2)

, £?)] ,

, vr)~\ , Z([yp]

,

yr)).

5

The next entry in the proof of 2 Ί , namely "D 1", indicates fivefold abbreviative application of Dl to (2). By abbreviative application of a definition is meant that in which definiens is replaced by definiendum, rather than vice versa. Thus "D 5 1" means that five expressions of the form of the definiens in Dl are to be found in (2) and replaced by the corresponding definienda according to Dl. More specifically, five expressions of the form "[ά( ) ] , & ( · · · ) " must be found in (2), such that "a" does not occur in " " nor "b" in " · · · * these are then to be replaced by expressions of the corresponding form " D • • • ". Three simple expressions of the required form are apparent in (2), namely "[?p] , vq", "[vq] , vr", and "[yp] , yr"; replacement of these according to Dl turns (2) into (3)

[?)))]

,

,%

Dp)).

The next entry in (16), namely "D 6 1", indicates six abbreviative applications of D1 to (19). The three simplest of those applications reduce (19) to IM0([4p] » Kq D p))] ,z(pD.qD p))] , w(pD. q D p), which is reduced successively by the three remaining applications of D1 to: [w([{/{p

D.qDp)]

,Z(pD.qD

,w(pD.qDp)

p))]

[ib(p D . q D ρ -.D ·. ρ D . q D ρ)] , ύ)(ρ D. (20)

pD.qDp

:D:pD.qDp.:

qDp)

D:pD.qDp

(Since these were the only six possible abbreviative applications of Dl to (19), the instruction "D 6 1" was unambiguous.) Thus far we have seen 1-5 (a / vp; β / v(q Dp); y / xq) + c2 + D61 to yield (20). Therefore, since 2-3 is "pD. q Dp", (16) in its entirety coincides with (17). Thus the sign " — " in the proof of 2"3 maintains the required balance. The manner in which the proof of 2*3 was written may thus be read as meaning that the substitution of 2"3 for " p " and " q " throughout 2'2 yields the material implication of 2'3 by the result (20) of the operations 2 6 (21) (a / vp; β / v(q Dp); y / ig) + c + D 1 upon 1'5. Now an implication, when thus adduced in a proof, represents an application of the rule of inference of the consequent; * having deduced (20) by the operations (21) upon Γ 5, and having deduced (17) by the operation (p / 2*3; q / 2'3) upon 2*2, we may apply the rule of inference of the consequent to these two results and thus infer the theorem 2*3 which was to be proved. The proof of 2'3 is thus to be interpreted as follows. By complex use of the rule of substitution, as described, the theorems (17) and (18) * P. 47, supra.

MATERIAL IMPLICATION

67

are derived respectively from the theorem 2 2 and the postulate 1'5. By two applications of the rule of concretion the theorem (19) is derived from (18). By six abbreviative applications of the definition D l , the theorem (20) is derived from (19). By means of the rule of inference of the consequent the theorem 2'3 is then inferred from (17) and (20). The notation in which the proof of 2'3 is set down may be read directly, i.e. without explicit reference to the lemmas (17), (18), (19), and (20), as follows: When the theorem arising from 2"2 by the substitutions (p / 2"3; q / 2'3), and the theorem arising from 1*5 by the substitutions .(a / υρ; β / v(g Dp); y / &q)

followed by two applications of the rule of concretion and six abbreviative applications of Dl, are given the respective roles of " D · · ·" and " " in the original enunciation of the rule of inference of the consequent, the theorem 2'3 is inferred. The method of checking a proof, e.g. that of 2"3, falls into two simple processes. The first consists in observing that the reference numbers mentioned outside parentheses in the line of proof are smaller than the reference number of the theorem to be proved; i.e., in the present case, that the numbers 2"2 and 1'5 are smaller than 2'3. This is clearly a check against circularity of proof. The occurrence of "2*3 " in the context " (p / 2'3; q / 2"3)" is ignored in this check, since it stands in parentheses; in such context it is purely abbreviative, and imports no circularity. The second and more important part of the checking process consists in determining whether " — " marks the balance which it is supposed to mark. This process consists in carrying out, mechanically or intuitively, the operations indicated on either side of " — ", and comparing the results to see if they are the same. In the case of the proof of 2'3, performance of the operations indicated to the left of " — " has been seen to give (17), and the totality following " — " has been seen to constitute (17) again, namely the implication of 2-3 by (20). A proof whose expression, as in the case of 2 1 and 2" 2, exhibits no entries after the sign " — " save the enunciation of the demonstrandum itself, may be called an immediate inference from the theorem or postulate cited to the left of " — "; such a proof involves only the application of definitions and informal rules of inference to the given

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theorem or postulate. Thus the proofs of 2 Ί and 2 2 are immediate inferences from the respective postulates Γ4 and 1Ί. The proof of 2 3, on the other hand, is not an immediate inference. Tarski and Bernays have proved that the three theorems thus far derived, namely 2*1, 2'2, and 2 3, are adequate to the entire calculus of implication.* All theorems of the calculus can be derived from those three by means of the rule of substitution and the rule of inference of the consequent. In the remainder of this chapter the seven most important of these derivative theorems will be derived, together with nine others which, though of less intrinsic interest, are needed here or in the sequel. In the following proof two new features demand explanation. 21 - 2 1 2'4.

( q / p D q ; r / q ) D . 2'2 ( p / p D q ) D pD.pDq:D.pDq

This theorem makes for the suppression of a repeated premiss. It is to be noted that tacit substitutions are involved to the left of " — " in the above proof. Written in full in the earlier manner, the proof would appear as follows: (22) 2Ί (p / pD . pD q; q / pD q .D q . pD q; r / pD q) ~ 2 - 1 ( q / p D q ; r / q ) D . 2"2 ( p / p D q )

D2'4

Let us check the proof in this form. First, 2 Ί and 2'2 are smaller numbers than 2 '4; therefore there is no circularity. Second comes the question of balance. The substitutions upon 2 Ί indicated to the left of " - " yield (23) pD. pDq : D :. p D q . D q pDq :: D : : . pDq

.D q

pDq

D. pD q ::D:.

pD . pDq

.

pDq.

The substitutions upon 2 Ί and 2'2 indicated after " yield respectively (24) pD. pDq : D :. pDq .Dq : D . pDq, (25) pDq .Dq : D . pD q .: D . pD q. Now (23) is recognizable as "(24) D. (25) D 2'4", so the proof (22) balances in the required fashion. Since in this case the right-hand side of the proof contains two implication signs, two applications of the rule of inference of the consequent * Cf. J. Lukasiewicz and A. Tarski, "Untersuchungen über den Aussagenkalkül," Comptes Rendus des seances de la Societe des Sciences et des heitres de Varsovie, X X I I I (1930), Glasse III, p. 43.

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-

are involved in the derivation of 2 4. The results (23) and (24) of substitution upon 2 Ί yield, b y the rule of inference of the consequent, the theorem "(25) D 2 * 4 i . e . pDq.'Dq:D.pDq.iD.pDq::D:.

pD.pDq:D.pDq.

This result and the result (25) of substitution upon 2 - 2 then yield 2'4 by a second application of the rule of inference of the consequent. The right-hand side of a proof may exhibit in this way any number of implication signs, provided only t h a t the scope of each is smaller t h a n t h a t of the preceding. B y way of example let us anticipate the proof of 5"86 below,* namely 2-1 - 5-85 (q / . . .) D :. 2"7 (p / . . .) D : 2"5 Z>. 2*3 D 5"86. Here there are four applications of the rule of inference of the consequent. Applied to 2 Ί (with tacit substitutions) and 5*85 (with indicated substitutions), the rule yields the theorem to the effect t h a t (26) 2-7 (p / . . .) D : 2 5 D. 2 3 D 5'86. Applied to (26) and 2"7 (with the indicated substitutions), the rule yields the theorem (27) 2-5 D . 2 3 D 5'86. Applied to (27) and 2'5, the rule yields (28) 2-3 D 5-86. Applied finally to (28) and 2"3, the rule yields the theorem 5'86. I t was observed t h a t the substitutions upon 2 Ί were left tacit in the left side of the original statement of the proof of 2 "4, and t h a t the proof assumes the form (22) when those substitutions are recorded explicitly. Actually however the recording of those substitutions as in (22) is redundant; the required substitutions are already uniquely determined in the original form of the proof. The substitutions upon 2"1 to the left of " — " must be such, namely, as to provide the required balance between opposite sides of " — ". Since the right-hand side, namely (29) 2-1 ( q / p D q ; r / g ) D . 2 - 2 ( p / p D q ) D 2 - 4 , comes out as the proposition designated earlier as (23), the substitutions upon 2 Ί to the left of " are determined uniquely by the condition t h a t they also must yield (23); the required substitute for a variable in 2 1 will be the homologously situated segment of (23). Thus, since " p " is the antecedent of the antecedent of 2 Ί , and the * P. 101.

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antecedent of the antecedent of (23) is "p D . ρ D q", the substitution upon "p" for the derivation of (23) from 2Ί must be "pD. ρ Dq". Similarly, since " q " is the consequent of the antecedent of 2 Ί , and the consequent of the antecedent of (23) is "p Dq . D q : D. ρ Dq", this last must be the substitution upon " q " in 2 Ί . Similarly, since "r" is the consequent of the consequent of the consequent of 2*1, and the consequent of the consequent of the consequent of (23) is " p D q " , the substitution upon " r " in 2 Ί must be " ρ Dq". Considerations of balance thus determine the substitutions upon 2 Ί to the left of " — " uniquely as those indicated in the expanded form (22) of the proof; that expanded form is therefore superfluous, determining as it does nothing not already determinate. It is in view of such considerations that substitutional instructions to the left of " — " in proofs will in general be suppressed. This does not mean however that the reader is called upon to determine and supply the required set of substitutions as was done in the above example. A proof in which the substitutions to the left of " •• " are tacit can be checked rather in a different and easier way. The first part of the checking process in these cases is as before, namely an inspection of reference numbers to determine freedom from circularity. The second part of the process, however, namely the testing of balance, assumes a different form when the substitutions to the left of " — " are tacit. The question in these cases is no longer, "Does the expression to the left of ' — ' come to the same thing as that to the right?" but rather, "Does the expression to the right of ' — ' have the form of the theorem mentioned to the left of ' —' ?" or in other words, "Can what is expressed to the right of ' —' be derived by the rule of substitution from the theorem mentioned to the left of ' — ' ? " In the simpler cases at least this question does not call for the labor of discovering and making the necessary substitutions upon the theorem mentioned to the left of " — ", but can be answered rather by mere attention to patterns. Thus in the case of the proof of 2'4 the question is whether (29) has the form of 2 Ί , i.e. is derivable from 2Ί by substitution upon the free variables "p", "q", and "r" therein. The affirmative answer can be reached by writing out (29) in full, i.e. as (23), and then observing that (23) has that form "

. - ) » p^

.. ••

I ·

b/

•••

•·

a/

*

•·

. —' • •

. - )

"

Β _•/ •

whereof 2 "1 is the general expression. The proof of 2'4 is thus checked

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71

without taking explicit cognizance of the substitutions upon 2*1 as was done in (22). Indeed, with practice the reader will occasionally find himself dispensing even with the explicit expansion of the right-hand side of a proof. Thus, in order to determine that (29) is of the form of 2 Ί we do not need even to expand (29) into the form (23) as was done in the above reasoning. I t is sufficient merely to observe that the following conditions are fulfilled: (a) the consequent of 2 Ί , when subjected to the set of substitutions (q/pDq; r / q) in accordance with (29), must be the same as the antecedent of 2*2 subjected to the indicated substitution (ρ / ρ Dq); (b) the consequent of 2*2, when subjected to the substitution (ρ / ρ Dq), must be the same as the consequent of 2'4; (c) the antecedent of 2"1, when subjected to the substitutions (q / ρ Dq; r / q), must be the same as the antecedent of 2"4. These three conditions are, together, obviously equivalent to the exhibition by (29) of the form of 2*1. The separate conditions (a), (b), and (c) are so simple as to admit of verification at a glance, rather than demanding laborious substitutions. The conditions corresponding to (a), (b), and (c) would of course be completely different if the theorem cited to the left of " •» " were other than 2 Ί ; in each case however the appropriate conditions would arise with equal obviousness from a consideration of the form of the theorem involved. The reader need not try to gain at once a working command of the technique sketched in the past two paragraphs. The description is intended rather as a prediction of the manner in which with practice the reader will tend of his own accord to clip the corners of the explicit robot procedure. What is of prime importance is to observe that at most the foregoing discussion is occupied only with practical shortcuts of procedure, and in no way calls upon intuition to fill in lacunae in the expression of proofs. Rather is the proof e.g. of 2'4 fully expressed, despite the omission to the left of " — " of the list of substitutions upon 2 Ί . Like the proofs of 2 Ί , 2'2, and 2 - 3, that of 2 - 4 can be run through in strictly mechanical fashion. The method consists in first determining the required list of substitutions, by the routine method of homologies described earlier, and then proceeding as was done above at the beginning of the discussion of 2 "4. The list of substitutions to the left of " — " will be suppressed as in the proof of 2 4 unless, as in the proofs of 2 Ί and 2'2, those substi-

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tutions are to be overlaid by applications of definitions or of the rule of concretion; in such cases the substitutions will be indicated explicitly. Let us now continue with the theorems. The next is the so-called principle of identity, or of the reflexivity of material implication. 2-4 - 2-3 (q / p) D 25. pDp According to the foregoing explanations, the proof of 2"5 is checked by observing that " 2 3 (q / p) D 2"5", which is to say pD. pDp :D. pDp, is of the form of 2*4. (The tacit substitution upon 2'4 which is involved in this proof is thus the simple substitution of " p " for "