Peirce's Logic of Relations and Other Studies 9783110856019, 9783110133257

Table of contents :
Preface
I. Individuality and Quantification
II. Of Lovers, Servants, and Benefactors
III. De Morgan and the Logic of Relations
IV. The Relational Formulae of 1883
V. Some Icons of Second Intention
VI. The Relation of Representation
VII. Frege’s Pragmatic Concerns
VIII. A Dialogue with Velian on Truth
IX. Bradley and Continuous Relations
X. The Logic of Idealism and the Neglected Argument
XI. The Semiotics of Common Names
XII. Common Natures and Mathematical Scotism
XIII. Set Theory and Royce’s Modes of Action
Index

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PEIRCE'S LOGIC OF RELATIONS AND OTHER STUDIES

PEIRCE'S LOGIC OF RELATIONS A N D OTHER STUDIES R. M. Martin

Ψ

FORIS PUBLICATIONS Dordrecht - Holland Cinnaminson, N.J. - U.S.A.

© 1980 R.M. Martin All rights reserved ISBN 90 70176 173 Printed in Holland by Intercontinental Graphics Dordrecht

For Morton White with gratitude

CONTENTS

Preface I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII.

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Individuality and Quantification Of Lovers, Servants, and Benefactors De Morgan and the Logic of Relations. The Relational Formulae of 1883 Some Icons of Second Intention The Relation of Representation Frege's Pragmatic Concerns A Dialogue with Velian on Truth Bradley and Continuous Relations The Logic of Idealism and the Neglected Argument The Semiotics of Common Names Common Natures and Mathematical Scotism Set Theory and Royce's Modes of Action

Index

11 25 46 54 62 67 80 87 98 . 110 121 136 148 155

7

PREFACE

C. S. Pierce was surely one of the distinguished logicians of the nineteenth century, as Morton White has aptly observed in his Science and Sentiment in America. More than this, he was surely one of the two greatest, being equalled only by Frege, who is usually given the palm. Even up to the very moment, however, Peirce's contributions to formal logic, especially to the logic of relations, have not been studied with any systematic thoroughness. This neglect is rather astonishing, especially in view of the widespread interest in the work of Frege. Both Peirce and Frege were deeply concerned not only with exact logic but with natural language and semiotics as well. To be sure, Peirce's semiotics has had a proper airing in recent years among linguists, due in part to the influence of Roman Jakobson. Peirce's semiotics is probably superior to that of Frege in its usefulness for linguistics, as now seems well recognized. The supposition is that, if Peirce's contributions to formal logic were more widely known and understood, they too would be seen to be of major importance. The various papers of this volume are concerned with themes, and variations upon them, to be found in Peirce's most important logical writings, as contained in the Collected Papers (Volume III, Exact Logic). Some of the papers here are expository and help to make clear and readily available precisely what Peirce's contributions are. Others are more analytic and critical and concerned rather with what Peirce might, and perhaps should, have written than with what he actually did. Still others are concerned with variations on Peircean themes, to be found in the work of Duns Scotus, Ockham, Hobbes, De Morgan, Bradley, Frege, and Royce. All in all, these papers are intended either to help make known Peirce's contributions to formal logic, especially those concerned with the logic of relations, or to illustrate their depth and fecundity in showing how they relate to the work of other authors. More specifically, the various papers of this volume are as follows. In I Peirce's notorious difficulties with the notion of individuality are considered, especially in connection with the theory of quantification. In 9

10

Preface

II his first major paper, that of 1870, explicitly concerned with the logic of relations, is discussed rather thoroughly, and brought up to date. Ill is devoted to the work on relations of Augustus De Morgan, Peirce's most important predecessor in the field. In IV Peirce's paper of 1883, the main content of "Note B" from the Studies in Logic by Members of the Johns Hopkins University, is studied in detail. Attention is called, in V, to the most important of the so-called icons of second intention. In VI the Peirce representation relation is studied in the light of modern semantics. VII is devoted to Frege's contributions to pragmatics, which are surprisingly close to Peirce in spirit if not in letter. Peirce's anticipation ofthe semantic truth-concept is discussed, in VIII, in the form of a dialogue with Velian, the Athenian stranger. In IX his doctrine of continuous relations is compared with that of Bradley. The attempt is made, in X, to formulate a logic of objective idealism, in connection with a discussion of Peirce's "Neglected Argument for the Reality of God." XI is concerned not directly with Peirce but with the semiotics of common names, a variation of a Peircean theme. In XII some notions due essentially to Duns Scotus are shown to harmonize with modern set theory. Finally, in XIII, Royce's theory concerning modes of action is shown to harmonize with modern set theory, by considering the theory of classification and generalizing it "to the very tip-top." These various papers, although closely interrelated, may be read independently of each other. The authorwishestothankthelnstituteforAdvancedStudy, Princeton (through Grant FC 10503 of theNational Endowment for the Humanities) as well as Northwestern University for support of the research reported in this volume. Also thanks are due, and herewith expressed, to the editors of the Transactions of the Charles S. Peirce Society, Idealistic Studies, Ratio, the Journal of Philosophical Logic, and Philosophia, and to Fromann Verlag, for permission to borrow or reproduce material originally published by them.

I

INDIVIDUALITY AND

QUANTIFICATION

It is not until 1885, in his "On the Algebra of Logic, A Contribution to the Philosophy of Notation," 1 that Peirce first, in his published papers, achieves full clarity concerning the quantificational idioms involving 'some' and 'all'. "We now come to the distinction of some and all, a distinction which is precisely on a par with that between truth and falsehood, . . . " he writes (3.393). "All attempts to introduce this distinction into the Boolian algebra were more or less complete failures until Mr. Mitchell2 showed how it was to be effected. His method really consists in making the whole expression of the proposition consist of two parts, a pure Boolian expression referring to an individual and a Quantifying part saying what individual that is." By a 'pure Boolian expression' Peirce means here approximately what we would now call 'a sentential function' and by 'the Quantifying part' what we would now call 'the quantifier'. It is interesting that it took Peirce so long to become clear concerning quantification, and one can only speculate as to the reasons. But before doing this, it will be helpful to note what the variables (if any) are allowed to range over and the terms to designate, in his earlier logic papers. In the first published one3 in 1867, he lets "the letters of the alphabet... stand for classes," and then comments parenthetically that these are classes "of events or things" (3.1). Peirce, following Boole here, is interested only in relations between and operations upon these classes. When he wishes to speak of a member of one of these classes he refers to it as an individual. And in speaking of identity he states that "a =p ¿is to

1

Collected Papers (Harvard University Press, Cambridge: 1931-1958), 3.559-3.403A. References throughout are to the Collected Papers and not to the unpublished material, which is not yet available in definitive form. 2 0 . H. Mitchell, "On a New Algebra of Logic," in Studies in Logic by Members of the Johns Hopkins University (Little, Brown, and Company, Boston: 1883), pp. 72-106. 3 "On an Improvement in Boole's Calculus of Logic," Collected Papers, 3.1-3.19. Cf. Emily Michael, "Peirce's Early Study of the Logic of Relations, 1865-1867," Transactions of the Charles S. Peirce Sotiety X (1974): 63-75.

11

12

Individuality and Quantification

mean that a and b denote the same class — the same collection of individuals." It is clear that the individuals here are either "events or things." The same letters are sometimes used for both individuals and classes, and thus it is often difficult to know which are being talked about. Even more puzzling, the letters are allowed to take on a second meaning (3.13), namely, "the proportion of individuals ofthat class to be found among all the individuals examined in the long run." Now a proportion is presumably best expressed as a ratio of cardinal numbers. For a clear symbolism, Peirce also needs here expressions for classes as well as expressions for the cardinal numbers of those classes. In addition, expressions for the members of those classes are needed. And since "things" and "events" are very different, two styles of expressions for the members are needed. Part of the inadequacy of this first paper is that a cogent symbolism for the various kinds of entities dealt with is not developed. In "Upon the Logic of Mathematics" (3.20-3.44), of 1867, in addition to the foregoing types of entities there seem also to be compound pairs of the events admitted. Thus (3.21) 'a¿>' is supposed to represent "an event when a and b are events only if these events are independent of each other, in which case ab =p a,b." A definition of 'independent' is then given, but the notion remains elusive. The meaning of 'ab =f= a,tf here likewise is obscure. Whatever the pair-event ab is, where a and b are "independent" events, it cannot be the intersection of classes a and b, 'a,b' being the notation here for such. In the final sections of this paper, Peirce again allows the letters to stand for cardinal numbers. But his explanation is not very clear. What he wishes to say is roughly as follows. Let η be the class of New England States, and m be the class of sides of a cube. Then we may construe n i to mean, not that the two classes are the same, but that their cardinal numbers are. Peirce is struggling here of course with the notion of cardinal number of a class, of which Russell will make much later. Peirce said later (4.333) of this paper that "it is now [c.1905] utterly unintelligible to me, and is, I trust, by far the worst I ever published. Nevertheless, it is founded upon an interesting idea, worthy of a better development." Whatever the truth-value of the first sentence may be, there can be no doubt but that the second one is an understatement indeed. In his 1870 paper, explicitly on notation, 4 there is a very considerable improvement. Here (3.63) "the letters of the alphabet... denote logical 4

"Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic," 3.45-3.149.

Individuality and Quantification

13

signs. Now logical terms are of three grand classes." These are: absolute terms, which involve "only the conception of quality... for example,... 'horse', 'tree', 'man' [quotes added]." The second class consists of terms which involve "the conception of relation, and which require the addition of another term to complete the denotation." Examples are 'father o f , 'lover o f , 'servant o f . These are the simple relative terms. In addition there are the conjugative terms which involve "the conception of bringing things into relation, and which require the addition oí more than one [italics added] term to complete the denotation . . . . They regard an object as median or third between two others, that is as conjugative; as giver of—to —, or buyer of—for—from—[italics added]." Note that Peirce here is still in the grips of class-logic, all of these terms standing for classes. "No fourth class of terms exists involving the conception of fourth," it is remarked. However, it is interesting that Peirce adds immediately that "I shall commonly denote individuals by capitals, and generals by small letters," thereby recognizing (for the first time in print, it seems) that individuals are not to be treated on a par with classes. Signs for individuals are in effect a fourth class, although they do not therewith involve tetradic relations in the way in which absolute terms, simple relative terms, and conjugative terms involve monadic, dyadic, and triadic relations respectively. Peirce comments that no fourth class of terms exists involving the conception of fourth, "because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship." But Peirce is not quite happy with this statement, for he adds that "whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives." Unfortunately Peirce nowhere established this fact by a cogent proof, and in fact the matter is much more complicated than he realized.5 Following De Morgan, Peirce speaks (3.65) of a "universe" oruniverse of discourse as consisting of "individuals about which alone the whole discourse is understood to run. The universe, therefore,..., is different on different occasions." He adds, however, that the "discourse may run . . . [also] upon the qualities or collections of the individuals it contains." There is thus recognized implicitly a narrower and a wider sense of 'universe of discourse' which it will be good to bear in mind. 5 Cf. W. V. Quine, Selected Logic Papers (Random House, New York: 1966), VIII and XXII.

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Individuality and Quantification

Numbers are needed here, just as in the preceding papers. "I propose to assign to [the designata of(?)] all logical terms, numbers; to an absolute term, the number of individuals it denotes..." (Note the use here, for the first time apparently in the published corpus, of 'denotes' in the sense of multiple denotation. 6 ) Numbers are also assigned to the designata of relative and conjugative terms in special ways. "I propose to denote the number of a logical term by enclosing the term in square brackets, thus [t]." [t] is then the cardinal number of the class t, where 't' is an absolute term. It is interesting that, with the exception of differentiating notationally between events and things, Peirce explicitly here provides separate notations for the various kinds of entities under discussion. There is a marked improvement in this respect over his earlier papers. When it comes to the assignment of number to the designata of simple relative and conjugative terms, Peirce is less successful. To a relative term he proposes to assign "the average number of things so related to one individual. Thus in a universe of perfect men . . . , the number of 'tooth of' would be 32." But what is the average number of 'enemy of', 'benefactor of', 'servant o f , to take just a few of the relative terms Peirce introduces? Here some reference to the time would seem essential. But even so, what average number can be assigned to 'enemy of Jones at times V ? A good deal of spelling out is needed here, and it is doubtful that a clear or useful notion would emerge. And similarly for the average numbers Peirce assigns to the designata of conjugative terms. Several notational confusions persist. And all of these seem traceable to the second of Peirce's supererogatory requirements concerning notation in general (3.61). This is that "it is desirable that, in certain general circumstances, determinate numbers should be capable of being substituted for the letters operated upon, and that when so substituted the equations should hold good when interpreted in accordance with the ordinary definitions of the signs, so that arithmetical algebra should be included under the notation employed as a special case of it." Thus the equal sign has the characteristic that (1)

χ = y if and only if [x] = [y],

if 'x' and 'y' are absolute terms. "Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are predicable of the same collections . . . . So, to write 5 < 7 is to say 5 is apart of 7, just as to write f < m [where f is the class of Frenchmen and m that of men] is to say that Frenchmen are part of men . . . " Having made desirable nota'Cf. the author's Truth and Denotation (The University of Chicago Press, Chicago: 1958).

Individuality and Quantification

15

tional distinctions, Peirce blurs them by thinking that this second supererogatory requirement is in any way desirable. That ( 1 ) holds as a principle is by no means tantamount to saying that the relation of equality between classes is "nothing but" the relation of identity between the numbers of those classes, the relata being different kinds of objects altogether. It is remarkable, however, for Peirce to have surmised that numbers that are equal are predicable of the same collections, but he does not carry out the promise of this remark. Let a and β be numbers, and let 'a(x)' express that a is predicable of x. Then (2)

' a = /3 = (a(x) - (/$(x))\

where 'x' is an absolute term, expresses what is desired. Also it is a pity Peirce confutes the ' < ' of '5 < 7' with that of 'f < m' two very different relations being involved. And clearly '5 < 7' does not mean that 5 is a part of 7 in any recognizable sense of 'part'. Of course we do have that (2)

f < m - [ f ] < [m],

where the right-hand occurrence of ' < ' is for the less-than relation. Having introduced the square brackets to differentiate a cardinal number from that of which it is a cardinal number, Peirce should have distinguished clearly the symbol for class-inclusion from that for being less-than between numbers. And similarly for equality and identity, and the various signs for the Boolian operations. Peirce "defines," as he says, Ό' for the null class by the equation (3)

'x + 0 = x'.

"This interpretation is given by Boole," he writes, "and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have (4)

[0] = 0."

But this is not very neat after all, for (4) does not unambiguously say what it is supposed to, that the cardinal number of the null class is zero. The Ό' is being used ambiguously for the null class and for the number zero. Peirce uses the "letters of the alphabet" as constants throughout, subdivided into the "three grand classes" of terms. He never mentions variables explicitly, but uses italic V and ' / as such in effect, and as having either absolute, simple relative, or conjugative terms as substituends. The remaining comments concerning notation, in this paper on notation, are for the most part concerned with logical constants, either for

16

Individuality and Quantification

various Boolian operations or relations, or for various notions used to try to fit the logic of relative and conjugative terms into a Procrustean Boolian framework. With so clumsy a notation and so unyielding a framework, it is remarkable that Peirce succeeds as well as he does. The details here are of very minor interest, for it is only in later papers that the logic of relatives really takes wing, but with the help of a much improved notation. There are still a few comments in the 1870 paper on notation, however, which are relevant to the topic of individuality. Peirce notes (3.92), almost as an aside, that "individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can." And in 3.93 he speaks of a "logical atom, or term not capable of logical division" and the context makes clear that the logical atom is a unit class. "The logical atom . . . must be one of which every predicate may be universally affirmed or denied . . . . Such a term can be realized neither in thought nor in sense [italics added]." Not in sense, because predicates of sight are not applicable to objects of taste. Similarly "in thought, an absolutely determinate term [with respect to the affirmation or denial of every predicate] cannot be realized . . . . A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate." In this way Peirce distinguishes between the individual, or absolutely indivisible logical atom, and the singular, or "that which is one in number from a particular point of view." In other words, the singular is not absolutely indivisible, "but indivisible as long as we neglect differences of time [and place] which accompany them. Such differences we habitually disregard in the logical division of substances . . . . There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse [italics added], Note that the singular is something construed "from a particular point of view" and is the result of conventionally neglecting some "difference" or other in some discourse. The individual, on the other hand, if we are to be strict in our terminology, is absolutely so in the sense that all possible "differences" are brought into account and that there is no relativization to a discourse. A discourse, to be sure, carries its "universe," its ontology, along with it. Different ontologies, different discourses — although not necessarily conversely. The entities of the discourse are always singulars, but not individuals. The notion of being a singular is internal to the discourse, that of individual, external to it, to use Carnap's well-known but somewhat hackneyed distinction. Discourses, it well be recalled, are "different on different occasions."

Individuality and Quantification

17

Also any discourse D may presumably be broadened into a more inclusive one D ' taking account of "differences" that D neglects. In this way any singular of D may be made more determinate in D'. "Such a term as 'the second Philip of Macedón' is still capable of logical division—into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them," Peirce writes. We are not to construe this passage as stating that individuality is to be determined by occupying "only one place at one time," for even there further determination is possible —for example, Philip kind, Philip brave. Some further discourse might be available in which Philip kind is one singular, Philip brave another, and both at the same place and time. Peirce nowhere, however, suggests how a new singular term such as 'Philip brave' is to be introduced or defined in terms of 'Philip' and 'brave', where these latter are presumed already available. This is a ticklish problem and needs a very considerable discussion. A related problem is concerned with the individuation of events, which has been under considerable discussion recently. There seem to be at least two approaches to the problem. We may regard Philip at a place-time as a singular and then distinguish Philip kind from Philip brave intentionally by bringing in Frege's Arten des Gegebenseins.1 Philip-at-the-placetime taken under the predicate description 'kind' is then one intentional entity, but, taken under the predicate 'brave', quite another. A second method is to admit Philip's states of being as also singulars, and to recognize that Philip-at-a-place-time kind is a very different state from Philip-at-a-place-time brave, provided 'kind' and 'brave' are not synonymous predicates or paraphrastic of each other. 8 Somehow Peirce thinks (3.92) that the "logical laws of individual terms are simpler than those which relate to general terms." Note that the definition of 'individual', however, is not especially simple. Peirce's comment would seem to rule out the possibility of a calculus of singulars (Lésniewski's mereology).9 And in view of that calculus, developed to be

'See G. Frege, Begriffsschrift, §8, and "Über Sinn und Bedeutung," second paragraph. See also the remarks at the end of VI below. Most of the references to Frege may be found in Translations from the Philosophical Writings of Gottlob Frege, ed. by P. Geach and M. Black (Basil Blackwell, Oxford: 1952). "See the author's Events, Reference, and Logical Form (The Catholic University of America Press, Washington: to appear) and "Some Comments on Events and Actions," in Action Theory, ed. by M. Brand and D. Walton (Reidel, Dordrecht: 1976), pp. 179-192. 'See especially H. S. Leonard and N. Goodman, "The Calculus of Individuals and Its Uses," The Journal of Symbolic Logic 5 (1940): 45-55, and Truth and Denotation, Chapters XI and XII.

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Individuality and Quantification

sure only years later, his statement is dubious indeed. The very definition of 'individual' can be given only in terms of a calculus of singulars (as in 3.216), which are allowed to intersect each other, to be subordinated to each other, and the like, and the laws governing them are not, in any obvious sense anyhow, any simpler than those concerning classes. The logic of individuals is thus wholly contained in the logic of singulars. And although it is true that any individuals χ and y are either identical or mutually exclusive, the proof of this circumstance requires laws as complex as those of classes. It is interesting that Peirce does not mention here the summation of two individuals, a notion he has in effect already employed a few paragraphs back, perhaps unwittingly. Thus (3.69), where h is a class of horses, Peirce writes, and where H, H', H", etc., are individual [singular] horses, (5)

'h = Η + , H' + , Η" + , etc.'.

But the ' + , ' of course is significant only as between class-expressions. To make sense of the right-hand side of this equation, we must let Ή ' designate rather a unit class, a class whose only member is a single horse, or we must allow ' +,' to take on a new designation, namely, the operation of summation as between individuals. If we do this latter, 'h' must be a term for a sum individual, not for a class, if (5) is to mean what it is supposed to. Peirce apparently nowhere distinguishes between an individual (or singular) and the class whose only member is that individual. The footnote to 3.93 is one of Peirce's clearest statements about the absolutely individual. There he says that it "cannot exist, properly speaking . . . . All . . . that we perceive or think, or that exists, is general . . . . That which exists is the object of a true conception. This conception may be made more determinate than any assignable conception; and therefore it is never so determinate that it is capable of no further determination." Here the absolutely individual is correlated with something that is "true," at the limit of the long run of inquiry, presumably. And such entities may be discussed only in the most inclusive kind of discourse possible, in which all determinations can be made, this discourse itself perhaps to be formulated only at the limit of the long run of inquiry. The paragraph 3.94 seems to have little to do with the distinction between the individual and the singular, but rather with the (Scotistic?) distinction between the Individuum signatum and the Individuum vagum.

Peirce's example of the former is 'Julius Caesar', and of the latter 'a certain man' and perhaps 'any individual man'. But of course, if we are to keep to the strict terminology of 3.93, 'Julius Caesar' is a term fora

Individuality and Quantification

19

singular, not for an individual. The phrase 'a certain man' hinges much upon 'certain' and how it is to be construed in the context; some uses of this phrase are no doubt to be handled as Russellian descriptions. Peirce's comments about 'any individual man' are remarkably perspicacious, and may be construed as foreshadowing the so-called selector operator. A term such as 'any individual man', he notes, "is in one sense not an individual [sic] term; for it represents every man. But it represents each man as capable of being denoted by a term which is individual; and so, though it is not itself an individual term, it stands for any one of a class of individual terms." Let 'M' be the predicate 'man'. Then '(eM)', where e is the selector operator, comes rather close to being what Peirce has in mind here. It is not an individual term but still stands (in some sense) for any member of the class of men. And nothing can be predicated of it "which cannot be predicated of the whole class." Thus if 'is mortal' is predicable of (eM), it is predicable of every man. No ontological matters not already considered are involved in Peirce's discussion of "infinitesimal relatives" (3.100-130), suchas 'Bishop of the see of —' where in place of '—' an individual (singular) term is inserted. When we come to "elementary relatives," however, in (3.121-134), something genuinely new is needed. Elementary relatives are "relations which exist [hold] only between mutually exclusive pairs ( . . . or triplets, or quartettes, etc.) of individuals, or else between pairs of classes in such a way that every individual of one class of the pair is in that relation to every individual of the other." Here ordered pairs, triples, and so on, of individuals constitute new ontological categories, as well as pairs of classes where there are pairs of individuals between every member of the one with every member of the other. Peirce simply introduces a new notation 'A:B' for pairs and lets the matter go at that. He seems insensitive to the enormous ontological extension this introduction involves. One might reasonably have expected that in this paper Peirce would have introduced terms for relations themselves, dyadic, triadic, and so on, as one of the "grand" classes of signs. But this is not done and all discourse concerning relations up to this point has taken place wholly by means of the simple relative and conjugative terms. But now, for the first time in Peirce's published logic papers, ordered pairs, triples, and so on, are explicitly admitted, and he remarks immediately that "every relation may be conceived of as a logical sum of elementary relatives." Not only are pairs of individuals needed, but logical sums of them as well, these sums not being of any of the kinds introduced up to this point. Thus, although relations are introduced here explicitly it is in a roundabout, devious way; relations reduce to sums of elementary relatives,

20

Individuality and Quantification

where neither these sums nor elementary relatives had been previously provided for. No doubt it would have been better to have introduced relations as an additional "grand" class ab initio, as indeed Peirce will do some years later. In the 1880 paper, "On the Algebra of Logic" (3.154-3.251), there is no notational improvement over the preceding ones. Nor is there any further clarification concerning individuality, and there is still no explicit introduction of the quantifiers. However, symbols 'Σ' and 'Π' for sums and products of simple relative terms — now called 'dual relatives' — are introduced, and this is a step in the direction of the quantifiers. However, there is some confusion as to how these function. Peirce still does not explicitly differentiate relatives from relations. When he writes (3.223) 7=

Σ (A : B)'

he speaks of / as a dual relative but not as a relation. Yet this formulais presumably to be understood as saying that the relation I is identical with a certain sum of ordered pairs. The dual relative is identical, to be sure, with some other sum, a class-sum, but it cannot be identical with a sum of ordered pairs. The most revealing paragraph in this 1880 paper, so far as concerns individuality, is the definition of'individual' (in 3.216), already referred to above. "Just as in mathematics we speak of infinitesimals and infinites, which are fictitious limits of continuous quantity..., so in logic we may define an individual, A, . . . " The requirements Peirce lays down for A are in effect that it cannot be a subclass of the null class and that any χ which is properly included in A is not included in the null class. This last clause is tantamount to saying that A is included in all non-null classes that are included in it. This definition is a remarkable anticipation of that of a Boolean atom in Tarski's paper on Boolean algebra.10 It may also be given as a definition of'atomicindividual' or 'unitindividuaP in the calculus of individuals." Of course, Peirce fails to distinguish between classes and their elements, so that by an individual here he can mean only a unit class, a class with just one member. Peirce also defines the notion of a simple individual, which turns out to be merely the negative of an individual. "The individual and the simple, as here defined," he writes, "are ideal limits, and every statement about either is to be interpreted by the doctrine of limits." Also (3.217) "every term may ,0 A. Tarski, "On the Foundations of Boolean Algebra," in his Logic, Semantics, Metamathematics (Clarendon Press, Oxford: 1956), XI, esp. p. 334. "See the author's Semiotics and Linguistic Structure (The State University of New York Press, Albany: to appear).

21

Individuality and Quantification be conceived as a limitless logical sum of individuals . . . , thus, a = A! + A2 + Aj + A4 + A5 + etc

"

Peirce's discussion is marred, here and elsewhere, by an unanalyzed use of 'etc.', a little word that Frege (and Russell) worked hard and long to characterize in the theory of mathematical induction. It is not the inductive use of 'etc.' that is relevant here, but the summational one. Thus α here is actually what later is written as '(xlx Ρ α)', for the sum of all unit parts of a, where 'P' is the sign for the part-whole relation and Ί ' stands for an operation of unit summation.12 Also in place of'term' here Peirce should have written 'singular' (3.93), terms being expressions, singulars and individuals being (in the first instance anyhow) nonlinguistic. But in any case, Peirce's definition of'individual' is a remarkable anticipation of a modern notion. Much of the notational inadequacy of these early papers is due no doubt to Peirce's following "too much in the footsteps of ordinary numerical algebra" (3.154, footnote 1), in accord with the second supererogatory notational requirement of 3.61, already remarked upon and lamented above. This is not the occasion to explore at all deeply just what it is that Peirce achieves in his "On the Logic of Number" of 1881 (3.252-3.288). He apparently liked this paper, for c. 1905 he commented upon it as follows (4.331): "As for Dedekind, his little book Was sind und was sollen die Zahlen? is most ingenious and excellent. But it proves no difficult theorem that I had not proved or published years before, and my paper had been sent to him." A detailed comparison of this paper with Dedekind's book would be of interest, but it is doubtful that the result would be to Peirce's advantage. (But cf. Collected Papers, 4.341 ff.) However, let us keep now to the present topic, Peirce's early ontology and theory of individuals. In 3.53, Peirce comes close to recognizing relations as such. "Let r be a relative term, so that one thing may be said to be r of another, and the latter by the former." Here r must clearly be adyadic relation, not a "simple" nor a "dual relative." Peirce's interest in this paper is in alternative systems of "quantities" that stand in certain relations to each other. The notion of quantity is taken as unanalyzed. In the discussion of "semi-infinite" quantity there is a "minimum number . . . called one." And in the discussion of "discrete simple quantity infinite in both directions," there is a number one that is to be understood "in a new sense." But even with the admission of alternative kinds of 12

As in Semiotics and Linguistic Structure. See also XI and XII below.

22

Individuality and Quantification

quantities and alternative numbers one, the real ontology of this paper lies hidden, like the greater portion of an iceberg. In 3.262 and 3.263 Peirce purports to give recursive definitions of ' + ' and ' X ' in terms of the notion 'next greater than'. But no recursion equations are given, and no analysis of this kind of inductive definition in terms of the ancestral of a relation.13 The definitions thus fail to define 'x + / and 'χ χ γ for variable V and variable ' / . For adequate definitions avast hidden ontology must be brought to the surface with which to analyze the ancestral or to justify the recursion equations. On this matter both Dedekind and Frege had a much deeper mathematical intuition than Peirce had and went to great pains to show how logically sound definitions of ' + ' and ' X ' may be given. In the 1882 paper, "Brief Description of the Algebra of Relatives" (3.306-3.322), Peirce lets "A, B, C, etc., denote objects of any kind," and "sums" of them are "absolute terms." In addition certain expressions are said to denote dual relatives (relations), and ordered pairs of "objects" are formed as in earlier papers. There are some additional numerical functions and coefficients at work, in effect taking the place of quantifiers.14 Also, for the first time explicitly, Peirce introduces here triple relatives, and therewith a whole new ontological category unawares. In the "Note B" paper of 1883 (3.328-3.358), we find a fascinating first paragraph: "A dual relative term, such as 'lover', 'benefactor', 'servant', is a common name signifying a pair of objects [italics added]. Of the two members of the pair, a determinate one is generally the first, and the other the second; so that if the order is reversed, the pair is not considered as remaining the same." Note the Hobbesian talk of "common names," not heretofore appearing in Peirce's logical writing.15 Pairs of objects reappear here as earlier, and "a general relative may be conceived as a logical aggregate of a number of . . . individual relatives" or "pairs of objects." Here again we have logical sums of pairs handled in terms of the numerical functions and coefficients used previously. Towards the end of this paper Peirce comes remarkably close, for the first time, to a satisfactory notation for quantifiers. Here he in effect defines 'Σ,Σ,·/,·,·' to express, where 7' stands for the dual relation of "See especially *90, Prindpia Mathematica (Cambridge University Press, Cambridge: 1910-1913). '••The transition from the use of these functions and coefficients to the full-fledged quantifiers is well described by W. V. Quine in his review of Vol. Ill of the Collected Papers, Isis XXII (1934): 285-297. 15 α. IX below.

Individuality and Quantification

23

being a lover of, that "something is a lover of something," and 'Π,-Σ^/,·,·', that "everything is a lover of something." Several other examples are given, showing at last that a notation for the quantifiers is satisfactorily available in full array. It is not until we arrive at the 1885 paper, however, that the quantifiers over singulars are introduced on their own, and not in terms of the numerical functions and coefficients, which was more or less a complete failure, as Peirce himself commented. Peirce credits his student, Ο. H. Mitchell, with showing how the distinction between 'some' and 'all' is to be effected within the "Boolian" framework. However, Mitchell's method is limited to monadic quantification and his notation is still hampered by following "too much in the footsteps of ordinary numerical algebra." Peirce's notation is surely superior and is capable of handling iterated quantifiers in all possible ways. The quantification theory of this paper has been well described by George Berry, and need not be repeated here." However, a few comments concerning ontology and notation are in order. At the very beginning of this 1885 paper, Peirce speaks of tokens, indices, and icons. Especially interesting is the use here made of'token'. Tokens (3.360) are "signs [that] are always abstract and general, . . . They include all general words, the main body of speech . . . . " Tokens thus include signs for classes and relations. "Without tokens there would be no generality in the statements [3.363], for they are the only general signs; and generality is essential to reasoning." Peirce does not quite say here how general signs are related to the objects that fall under them, or what their designata are (if they have any). He is more successful concerning indices. The index is a sign that stands "in a direct dual relation . . . to its object independent of the mind using the sign [3.361] . . . . Demonstrative and relative pronouns are nearly pure indices . . . ; so a r e . . .the subscript numbers which in algebra distinguish one value from another without saying what those values are." It is by means of indices that Peirce forms his notation for quantification. Their purpose is precisely to "distinguish one value from another withoui saying what those values are." Note that indices are neither names nor constants of any kind. They are more like variables, and Peirce uses them essentially as such. They become the carriers of what is being talked about in a given discourse.

"See his "Peirce's Contributions to the Logic of Statements and Quantifiers," in Studies in the Philosophy of Charles Sanders Peirce (Harvard University Press, Cambridge: 1952): 153-165. See also Atwell R. Turquette, "Peirce's Icons for Deductive Logic," ibid., Second Series (The University of Massachusetts Press, Amherst: 1964), pp. 95-108.

24

Individuality and Quantification

One additional comment. It is most interesting that in some of the icons of second intention (3.398-3.401) quantifiers over classes are admitted. Here too a whole new ontological category creeps in almost unawares. 17 Let us conjecture now as to why it took Peirce so long to gain a clear understanding of the quantifiers. His earliest work, on "Boolian" algebra, is of course concerned with general or class terms, whereas quantifiers are, in the first instance anyhow, over singulars. Singulars are relative to a discourse, and the choice of them varies from discourse to discourse. Only "individuals" remain as discourse-invariant, but these are like infinitesimals or "ideal limits" and "cannot exist, properly speaking." It would not do to allow quantifiers over that which does not exist, nor ab initio over second-intentional entities which are in their very nature of a general character. A third reason is the very algebraic character of these early papers. In algebra par excellence quantifiers are not needed. Also quantifiers really become indispensable only when the full logic of relations is arrived at. Monadic quantification theory is merely a variant of Boolean algebra, but the theory of iterated quantification goes hand in hand with relations of higher degree. Another possible reason is that Peirce did not approach logic merely as providing the foundations of mathematics, as Frege did. Hence he was not concerned with the exact logic of arithmetical functions (one-many relations) right from the start, as Frege was. Nor does functionality permeate his logical work through and through as it does that of Frege. It might be thought that some of the critical comments above are too severe, that we are expecting too much of Peirce, that we are confronting him with higher standards of rigor than those current in his day, that we are condemning him for not knowing what is now known, and so on. A remarkable feature of Peirce's logical writing is, however, that it invites these criticisms. One does not approach his work as though it were a century old ; one reads him as though he had written yesterday. His ideas are for the most part so astonishingly modern that one is shocked to encounter antiquated methods, inadequate formulations, and the like. It is a tribute to his greatness that his mode of writing solicits criticisms in accord with contemporary standards of rigor and in the light of contemporary knowledge. Peirce and Frege are unique in this respect among the logicians of the nineteenth century.

17

For further comments, see V below.

II

OF LOVERS, SERVANTS, A N D BENEFACTORS

Peirce wrote no paper under the title "On the Algebra of Relatives." However, such a title would be apt for all of his work on the logic of relations up to 1883 when he fully grasps the significance of the quantifiers for the first time. In his very first paper on the subject, in 1870,1 he treats of "relative terms," standing for certain kinds of classes, and terms for relations are considered primarily as components of such terms. Relations are not allowed to stand on their own feet, so to speak, except in very restricted identity contexts, until 1883. All of the early work on the logic of relations is thus embedded within a kind of algebra of classes. Peirce dismisses at the outset the notation of De Morgan,2 in which terms for individuals, classes, and relations are all admitted and recognized as of distinct kinds, by noting that "Boole's logical algebra has such singular beauty, so far as it goes, that it is interesting to inquire whether it cannot be extended over the whole realm of formal logic, instead of being restricted to that simplest and least useful part of the subject, the logic of absolute terms . . . " Peirce thinks that it can, but only at the price of embedding relations within compound terms for relatives. Peirce comments (3.68) that he will "adopt for the conception of multiplication the application of a relation, in such a way that, for example, / w [7' standing for the relation of lover of and 'w' for the class of woman] shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in mind." But this notation is not strictly that of De Morgan, it would seem, who considers rather the "composition" of two relations, not of a relation with a class. Thus Peirce's notion of the application of relation to a class here is something new. The 7w' we may 1 "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic," Collected Papers, 3.45-3.149. 2 "On the Syllogism, No. IV, and on the Logic of Relations," Transactions of the Cembridge Philosophical Society, 1860, pp. 331-*358. Recall I above and cf. III below.

25

26

Of Lovers, Servants, and Benefactors

think of as standing for the class of all lovers of women, i.e., for /"w or

x{Ey)(xly·yew)

in essentially the notation of Principia Mathematica (*37.01). In the paper under review Peirce puts forward a number of principles concerning the application of a relation to a class, many of which very likely are given here for the first time. Although the systematic context in which they are embedded is not satisfactory from the modern point of view, it will be useful to enumerate the most important of these principles. They are interesting on their own account and antedate by thirty years or so the corresponding principles of Principia Mathematica itself. By so doing we will gain a clearer picture than otherwise of precisely what it is that Peirce contributed to the logic of relations in this significant paper of 1870. First it is noted, where Y stands for the class of servants and'm' for that of men, that (1)

s(m + , w) = im + , í w

and (2)

(/ + , j)w = /w + , íw,

which express of course, in current notation, that (1')

s"(m υ w) = (j"m υ 5"w)

and (2')

(/ Ù j)"w = (/"w υ s"w).

The ' Ù ', we recall, is the PM symbol for the logical sum of two dyadic relations (*23.03), and ' u ' for the logical sum of two classes (*22.03). In addition to the notion of the application of a relation, Peirce also introduces here the relative product of two relations, but only within the context of applying it to a class. Thus '(i/)' stands for the relative product of the relations s and I, i.e., for the relation (siI) or xy(Ez)(x: s ζ· ζ Iy) in the terminology of PM (* 34.01). This is used by Peirce in a context such as \sl) w' or '(s/1) "w', for servants of lovers of women. Peirce then enunciates the law that (3)

(si) w = í(/w).

This would appear to be an associative law, and indeed Peirce mistakes it as such by commenting that with (2) and (3) "all the absolute conditions

Of Lovers, Servants, and Benefactors

27

of multiplication are satisfied." But just as (2) when spelled out as (2') is seen not to be a distributive law, so (3) when spelled out as (3')

(s//)"w = j"(/"w),

is seen not to be an associative law after all. Peirce lets Τ stand for the relation of being identical with, so that rl is the relative produce of relation r with 1. He then states that (4)

r\ = r,

and here the ' = ' is really the sign for relational identity. In this formula the V , note, is allowed to occur other than in the context of the application of a relation. Let 'g' stand for the triadic relation of giving. Then Peirce lets 'gxy' stand for the class of givers of y to x, i.e., for z(z g xy). But note here that χ and y are no longer classes but individuals. Thus gxy is not the result of the application of the relation g but rather of some quite new primitive operation. And if in place of 'x' here we put in 'o' for the relation of being owner of, Peirce seems to think we have a term of the same kind. But '(go)h', or 'g(oh)', where h is the class of horses, is strictly meaningless. g here is a triadic relation and no meaning has been given to 'go' where g is triadic and o dyadic. Peirce thus has considerable difficulty eking out ways of expressing giver of a horse to an owner of a horse and distinguishing this from giver of a horse to the owner of that horse. But the former is the class x(pe)(pz)(y

e h · / gzy· (E u){u e h · ζ o u)),

and the latter the class x(E>>)(Ez)()> eh-χ

gzy-ζ

oy).3

Similarly Peirce tries to distinguish between 'g/wh' and 'g(/w)h' and comments that "the associative principle of multiplication must in some form or other be abandoned at this point." He introduces a rather clumsy notation, however, to purport to enable him to express giver of a horse to a lover of a woman and to distinguish this from giver of a woman to a lover of her. 'g/wh' he takes to stand for the former, i.e., for x(Ej>)(Ez)(Ez)(j> e h · ζ e w · χ g uy · u l z), but 's(/w)\ for x(Ey)(Ez)(y

e w · xgzy-

zl y).

'This class is null if a meaning postulate is available to the effect that if χ gives ζ to y then y is not the owner of τ (at the time), for all χ, y, and ζ.

Of Lovers, Servants, and Benefactors

28

Just how he accomplishes this remains a bit obscure, and his notation for differentiating these classes cannot be regarded as systematically satisfactory. The notation 7,íw' is introduced to denote lover of a woman that isa servant

of that woman,

i.e., f o r

;c(Ey)(j; e w χ Iy· χ sy) or (/

Concerning

ή s)"w.

Peirce states a commutative law, that

(5)

s,l =

l,s

or (50

(ί ή /) = (/ ή

s).

Clearly /, s is the logical or Boolean product of the relations s and /, just as I +, s (in (2)) is the Boolean sum of them. These are of course very significant notions, extensions of the corresponding notions for classes, and apparently appear here for the first time anywhere. Peirce takes 7 W ' to express involution, i.e., to stand for the class x(j>)(je w => χ Iy), the class of lovers of every woman. Here of course w is a class. "Then," Peirce adds, i\s')v/ will denote whatever stands to every woman in the relation of servant of every lover of hers; and s (/w) will denote whatever is a servant of everything that is [a] lover of a woman." In other words we are to understand the expression 's1', where /is a relation, to stand for the relation xy(z)(z

l y => χ s ζ).

This kind of involution is due to De Morgan.4 Involution is thus ambiguous, and we must bear in mind always which kind is being used. It then holds that 0 ' ) w = i('w>,

(6)

this formula connecting the two kinds of involution. In other words (6')

x(y)(y

e w z> (z)(z

Iy

χ s

ζ)) = Jc(z)(z e /"w =>

χ s ζ).

It should be remarked that, although Peirce cites these principles only 'Op. cit., pp. 341 ff. Cf. W. V. Quine, A System of Logistic (Harvard University Press, Cambridge: 1934), p. 91.

Of Lovers, Servants, and Benefactors

29

for constant 7', V , 'w', and so on, they may readily be generalized to apply to all relations and classes (of appropriate type). Note also that involution is a kind of exponentiation, related somewhat to that numerical operation. The class of servants of every man and woman will be j m + , w or x(y)(y e (m υ w) => χ s y). m

And s , î or (i π j ) is the class of persons who are both servants of men and servants of women. Clearly then m

(7)

w

w

j . m + ,w

=

jm^w

or (7')

5(muw) = (i m η í w ).

However, Peirce speaks of '¿ m ,j w ' as denoting the class of servants "of every man that is a servant of every woman," i.e., *ΟΉ(.ν € m ' (z)(z e w 3 y s ζ))

χ s y).

muw

But this class is by no means identical with j ( l The reading given to the right-hand side of (7) is thus incorrect. Peirce goes on to stretch the analogy with ordinary mathematics to the utmost in citing a "binomial theorem" for involution. The notation, however, is so obscure that it is difficult to decide whether he has enunciated a correct formula or not. And in any case, the meaning of the formula is very complex, ordinary arithmetic being presupposed by it. The formula is thus not a formula in the algebra of relatives but at best only in a much more extended framework not clearly delineated. The principle (8)

(i,/) w =

or (8')

w

(s ή /) = (j

w

η /w)

is analogous to (7) or (7 '), and connects the involutes of a logical product of relations with the class product of the respective involutes of those relations. In a similar way, of course, (7) or (7') connect the involute of a given relation with respect to a class sum with the sum of the involutes of that relation with respect to the two summands. Note of course that more general forms of (7') and (8') also obtain. Conjugative terms are those involving triadic relations such as 'givers of—to —' or 'buyer of—from—for —'. "The application of involution to conjugative terms presents little difficulty after the explanations

30

Of Lovers, Servants, and Benefactors

which have been given under the head of multiplication," Peirce notes. "It is obvious that betrayer to every enemy [italics added] should be written ba, just as lover of every woman is written */ w» "

he adds. He then proceeds to symbolize such expressions as 'betrayer of a man to the same enemy of him', 'betrayer of every man to some enemy of him', and so on. "These interpretations [of the symbolic expressions introduced] are by no means obvious," he says, and later (3.144) comments that "the treatment of conjugative terms presents considerable difficulty, and would no doubt be greatly facilitated by algebraic devices. I have, however, studied this part of my notation but little." It is remarkable that Peirce thinks he has introduced any satisfactory notation here at all, lacking, as he does until 1883, the analytical power of the quantificational notation. Peirce lists 85 "general formulae" summarizing what has thus far been obtained. These include the various principles already listed, the transitivity of the relation for class inclusion as well as for that of relational inclusion, the various Boolean laws concerning logical sums and products, concerning the null and universal classes, concerning relative products, concerning involution, and a few more. Peirce states these formulae i.e., the ones which are completely general, in terms of variable V , y , and 'z'. He never quite tells us, but it is clear that these letters are allowed to range over both classes (including simple relatives and conjugatives) as well as over relations themselves. The various Boolean principles for relations are thus perhaps stated here for the first time. In many of the formulae, however, the letters cannot be construed in this ambiguous way, and it is often difficult to tell whether the ambiguity is allowed or not. Also some of the formulae, e.g., the binomial formula, presuppose the availability of cardinal numbers and are thus not strictly formulae within the algebra of relatives at all. Peirce cites 87 more formulae, bringing the total to 172. Let us look at those of this list which are of interest from the point of view of the historical development of the logic of relations.5 Concerning relative products the following laws are of interest. (We 5 The formulae which presuppose a numerical measure as well as certain other material, being only of minor interest, will not be considered here.

Of Lovers, Servants, and Benefactors

31

will use hereafter Peirce's numeration). First (90)

lía— ζ s y), this latter notion having been already introduced by De Morgan. "All the laws of . . . [backward involution] but one are the same as for ordinary involution," Peirce notes, "and the one exception is of that kind which is said to prove the rule." It is that analogous to (6) above we have '('w) = (/J>w,

(126)

that "the things which are lovers to nothing but things that are servants to nothing but women are the things which are lovers of servants to nothing but women." In other words (126')

;t(j)(z)((;t ly • y s ζ)

ζ s w) = x(z)(x(l/s)

ζ => ζ e w).

Note that the formula: (127)

C+'^w = 'w, 'w

is analogous to (7) and '(w,m) = ; w, 'm to (8). Note that these principles may readily be generalized. But why

35

Of Lovers, Servants, and Benefactors (126) is an exception that proves the rule is by no means clear. Two principles that interrelate involution with negation are that Is = 1 - ((1 - l)s)

(124)

(where 1 - χ is the negative of χ, χ being either a class or relation) and (125)

>s = 1 - (/(1 - s)).

Although, we recall, involution is significant where the term 'x' in 'x>'' or ' x y ' may stand for either a class or a relation, (124) and (125) are significant only where 7' is a relation. Note that these principles may be generalized (where we put 'R' for 7' and 'S" for V ) to express that (124')

xy{z)(z Sy=> χ Rz) =

-(-Λ/5)

and (125')

xy(z)(x Rz=> ζ Sy) = - ( Ä / - 5 ) .

The key principle connecting backward and forward involution is that '(Í w ) = ('¿) w

(130) or (130') y s ζ)),

/ y => (z)(z e w => y s ζ)) = Jc(z)(z e w

(j)(x ly =>

that "things which are lovers of nothing but what are servants of all women are the same as the things which are related to all women as lovers of nothing but their servants." Here too a more general form obtains. Peirce introduces the converse of a relative χ in terms of a special operator K, so that if χ is servants of so and so Kx is the class of masters or mistresses of those servants. Thus in effect Κ S"a =

x(Ey)(yea-x^Sy),

u

where S is the converse of the relation S. Actually Peirce uses 'K' ambiguously as applying either to a relative term or to a relational one, although in a way he has no right to do this latter. In any case, the following principles Peirce indicates (in a table) without giving them special numbers. Let us write •Ra\ 'Rs\

iR

a\

and'ÄS',

36

Of Lovers, Servants, and Benefactors

using the PM variables in place of Peirce's ambiguous ones, and assuming these defined essentially in Peirce's way. Then clearly the following principles obtain. ( - R of course is the negative of R, and - a of a.) U

(R/S) = ( U 5 / U Ä ) ,

(125A)

Ra = ~R — a

(125B) or

xy(z)(z e α Γ3 χ Rz) = xy(z)(x - Rz s

R =

(125C)

ζ e - a),

R

~ -S

or χy(z)(z S y => χ Rz) = xy{z)(x -Rz R

(125D)

D Ζ -S y),

-Ra

a =

or Ry

y e a) = x(y)(y e - a => χ -R y), R

(125E)

-R's

S =

or xy(z)(x Rz rs ζ S y) = χy(z)(z -S y => χ -R ζ), (125 F) -R" = R-a

or

and

x(y)(y e a

χ -R y) =

R y => y e - α),

(125G) -Rs=

R

—S

xy(z)(z S y => χ -R z) = xy(z)(x R ζ => ζ -S y). All of these principles are acceptable. (Two more principles are hinted at in Peirce's table, however, whose meaning is not clear. And in any case they cannot be or "(R/-S)

=

(-R/S),

Of Lovers, Servants, and Benefactors

37

or \-R/S)

=

(-R/S),

or = (υΛ/ —u5').)

"(R/-S)

Concerning the universal and null classes and relations we have the following principles concerning involution. (Again, Peirce cites some of these only in an ambiguous form.) (133A)

α =V

or x(y)(x A y

y e a) = V,

and k

(133 B)

R =V

or xy{z)(x

Az=>zRy)=A.

Similarly (133C)

RA=V

or xp(z)(z Ay=>xRz)

= V.

Also (134 A)

UR

A = A if R is an "unlimited relative[?]"

or (x) ~ (Ey)yRx=> x(y)(x ^Ry => y e Α) =Λ, and (134B)

UR

À =A

or (χ) ~ (Eζ)ζ Ry 3 xy(z)(x

U

R ζ => ζ Λ y) = Α

and (134C)

Α α = Λ if a is not null

38

Of Lovers, Servants, and Benefactors

or ~ α = A =3 •xCyH.V e a = > x A ^ ) = A , and (134D)

Λ Λ = A if J? is not null

or

~ R = Λ 3 xy(z)(z Ry=>xh.z)=A. Likewise, where 1 is the relation of identity,

(135A)

= R

or xy(z){x 1 z^zRy)

= R

and l

(135B)

a = a

or

*(.)>)(* 1 y 3 y e a) = a. Also

Rl = R

(135C) or

xy(z)(z 1

xRz) = R.

If y is an "infinitesimal," then (136A)

y

l = y, for such y.

It is not easy to decide what an infinitesimal here is supposed to be. However if a relation R is many-one, or (Cls -> 1) as in PM, *71.02, and is such that (x)(Ey)x R y, we have what we need, namely, (136A')

(R e ( C l s ^ l ) · (x)(Ey);t Ry) => Rl = R,

or

Hyps => xy(z)(x Rz => ζ 1 y) = R, as may readily be shown. Likewise, Peirce notes, if ζ is individual, then (136B)

V = z.

Or, better, 1 = {Z}

Of Lovers, Servants, and Benefactors or *(y)(ye{Z}=>xlZ)

= {Z}.

Also, if R "is less than unity", V

(137A)

R = λ

or ~ R = V = xy(z)(x V ζ => zRy)

= xy(z)z R

And if R is a "limited relative", then Rw = A , for such R

(137B) or

(Ex) ~ (y)xRy

=> x W ^ f V n

xRy)=A

and Λ ν = Λ , for such R

(137C) or

(Ex) ~ (y)xRy

=> x p ( z ) ( z \ y ^ xRz)

Finally we have (138A)

*V = V

or

and (138B)

*V = V

or i j ! ( z ) ( x Ä z 3 ζ Vj>) = V. And also Va = V

(138C) or x(y)(y

e α => χ V j ) = V,

and (138D)

VÄ = V

=À

40

Of Lovers, Servants, and Benefactors

or xy(z)(zRy

D X V Z ) = V.

Let us return now to lovers, servants, and benefactors, withfivemore interesting principles (3.118). First we have that If α —< b then bc —< "c

(143) and

If a - < b then ca -< cb.

(144)

In (143), for significance, a and b must be relations, but c may be either a relation or a class. And in (144) c must be a relation, but a and ¿may be either both classes or both relations. We thus have the four principles that (143 A)

R ε S z> sa c. Ra,

(143B)

R

(144A)

a c β Z5 Ra c Rß,

S => ST ε RT,

and (144B)

SD TR ε TS.

R ε

The reader may easily verify that these are valid formulae. As a consequence of (143 A) Peirce says that we have that "whatever is lover to nothing but what is servant to nothing but women stands to nothing but a woman in the relation of lover of every servant of hers." Thus we should have that ls

(145)

vf —< ' w .

However, (145) does not hold, but needs the hypothesis that everybody has a servant — an assumption perhaps taken for granted in 1870 but not in 1976! Thus we have only that (145')

(j>)(Ez)z s y => x{y)(x{l/s) y => y e w) er x(y)((z)(z

s y =>

χ l z) => y e w).

Further, Peirce's reading is incorrect. It should be to the effect rather that whatever is lover of a servant of nothing but women stands to nothing but women . . . etc. (The Editors call attention to this second reading but do not observe that Peirce's reading is incorrect.) Clearly (145') may be suitably generalized. Peirce notes next that "the following formulae can be proved without

Of Lovers, Servants, and Benefactors

41

difficulty," that (146)

/( J w)

< \ or / ' l i w c

(y

w,

that "every lover of somebody who is servant to nothing but a woman stands to nothing but women in the relation of lover of nothing but a servant of them," and that (147)

('j)w

'(jw) or ('j)"w c '(j"w),

that "whatever stands to a woman in the relation of lover of nothing but a servant of hers is a lover of nothing but servants of women." Clearly (146) becomes: (146') yew),

x(Ey){z)(y s ζ z> ζ e w)· χ / y) e x(j>)((z)(x / ζ => ζ s y) =>

and (147) becomes: (147') x(Ey)((z)(x y e w)),

l ζ z> ζ s y)· ye w) e x(z)(x l ζ => (Ey)(z s y-

both of which hold quantificationally, as well of course as more general forms. A few additional properties of negation are as follows. (Ill)

(α π — a) = À and (R π —R) = À ,

(167)

(α υ - α ) = V and (R ù -R) = V, -(a Πβ) = ( - α υ -β) and -(β ή S) = (-R Ù -S),

(168)

-(R°) = -Ra and ~(RS) = ~RS.

Incidentally, nothing has been said thus far concerning involution where both terms are for classes. Peirce seems to suggest such a notion early in the paper (3.56), before speaking of either relatives or relations. No clue whatsoever is given as to how involution in this sense is to be understood. (Nor is there any mention of involution in Peirce's two earlier published papers.) We are told merely that involution should obey the formulae: (9)

{xyy = x W

(10)

xy+'z = xy, xz,

and the obscure "binomial" theorem. Leaving this last aside for the moment, we should have then that (9')

(αβ)ν = a(M

42

Of Lovers, Servants, and Benefactors

and (10')

(aß

=

αν),

Π

for any classes α, β, and γ. These clearly will obtain if we define W

e β

as

χ e

a)',

as may readily be verified. Perhaps it is this interpretation of involution for classes that Peirce had in mind. Almost everything of permanent interest in Peirce's paper has been covered in the foregoing, except the rather obscure material involving a numerical measure and the "binomial" principle in its non-numerical form. This principle is stated first (3.77), where e is the relation of emperor of, c the relation of conqueror of, and f is the class of Frenchmen, as (e + , c){

— ef + , Ipe{~P,

cp + ,

cf.

Peirce tells us that the right-hand of this equation "will denote whatever is emperor of every Frenchman or emperor of some Frenchmen and conqueror of all the rest, or conqueror of every Frenchman." In other words, and more generally, we have the principle that (R

ύ

S)a

=

(.Ra

υ

x(Eß)(ß

c

( y ) { ( y e a - ~ y e ß ) ^ x S y ) ) u S

α· a

~α

c

ß-(y)(y Ε β ^

xRy)·

) .

Clearly also there is another binomial principle (involution being ambiguous) concerning just relations. Thus also we have that (R Ù S ) (z)((z

T

=

( R

T

0

xy(EU)(UxRz)·

ST).

And concerning the inrolution of just classes, we have also that (α ~

Υ β)Υ

= ( a r U JC(ES)(5

x e S ) => x e ß ) ) u

< = y ~ y < = S - ( x e S = > x e a ) - ( ( x e y

ßr).

The reader may readily verify that these principles hold in view of their quantificational and abstractional structure. Peirce presumably lumps all three of these together. Note also, of course, that the principles hold only within a "second-intentional" or second-order logic. Three binomial principles of backward involution also hold, that is, in their modern, general form, (129Α')

Λ

( Α Υ β)

y e a ) - ( y ) « x R y Q29B')

R(S

O

χ S ζ ) · ( ζ ) ( ( ζ Ry-

=

( R a Υ JC(ES)(5 Ε R - ~

~ χ S y ) => y e β ) ) U ) =

(RS

~ zuy)=>

Ú xy(EU)(U χ Tz)) 0

Ä χ e γ)) U αγ). Of these, again, Peirce states only (129A') in a very restricted form. As an instance of (129A') Peirce notes that "those persons who are lovers of nothing but Frenchmen and violinists consists first of those who are lovers of nothing but Frenchmen; second, of those who in some ways are lovers of nothing but Frenchmen and in all other ways of nothing but violinists, and finally of those who are lovers only of violinists." Peirce construes here 'lovers in some way' as 'bears asub-relationof-the-relation-of-loving to'. This is not perhaps a very happy reading. Presumably everyone loves his spouse, so that the relation of being married to is a sub-relation of that of loving. But one would not then wish to say that the relation of being married to is a "way" of loving. Related to these binomial principles are two principles involving products rather than sums. Thus, Peirce notes, "lovers of French violinists are those persons who, in reference to every mode of loving whatever, either in that way love some violinists or in some other way love some Frenchmen." Peirce states this as an equivalence, so that we should presumably have, in more general form, that (131) R"(a Π β) = (R"a Π x(S)(S ^Rz>xe η R"ß).

(S"a U (Ä ή -S)"

β))

"This logical proposition is certainly not self-evident," Peirce comments, "and its practical importance is considerable." In fact, (131') cannot hold as it stands, however, for the left-hand side could stand for the nullclass without the right-hand side's doing so. But we do clearly have that (131") Ä > n / i ) c ( r a n i ( S ) ( S E j i D x e ( S " » u ( Ä n - s ) " ß ) ) η R"ß), as may easily be seen quantificationally with the abbreviations eliminated. Note also that Peirce's reading of it is not quite correct, no account being taken of the ' i ? " a ' and '/?"/5' terms on the right. Similarly "to say that a person is both emperor and conqueror of the same Frenchman [Frenchmen] is the same as to say that, taking any class of Frenchmen whatsoever, this person is either an emperor of some one of this class, or conqueror of some one among the remaining Frenchmen." Peirce's symbolization of this is somewhat obscure, but apparently it is supposed to state, more generally, that (132)

(R ή S)" a = χ{β)(β

c a id χ e (R"(a η -β) U

S"β))).

44

Of Lovers, Servants, and Benefactors

His reading thus does not quite accord with what he states. Here also, however, the left-hand side might stand for Λ without the right-hand side's doing so, so that we have only that (132') (R ή S)"a c x(ß)(ß c a 3 χ e

Γ) -β) U S"β))).

Peirce's attempt to embed triadic relations within the Procrustean Boolean framework causes difficulties. He would surely have made progress here if only he had developed a notation in which individuals, classes, and relations were clearly separated from each other. De Morgan, on the other hand, it is ironic to note, had such a notation but apparently never went on to reflect upon triadic relations, or, in general, relations of higher degree. Peirce occasionally observes that one principle is a special case of another, or that "it is easy to show" such and such. But no acceptable proof in the modern sense is ever given in this paper. But note that all but a few of his formulae, and more general forms of them, are readily provable in the modern theory of (virtual) classes and relations as based on quantification theory with identity and abstraction. That Peirce was able to put forward acceptable and important principles independent of that theory, on the basis of his clumsy notation and inadequate deductive framework, is remarkable indeed and well attests to his extraordinary logical insight. It is interesting to note that some of the foregoing principles are already contained in De Morgan's paper of 1860, in particular some of those concerned with the notions of relational negation, the converse of a relation, relative products, and involution. De Morgan, however, makes little of logical sums and products of relations, and in general of the relational analogues of the Boolean notions. He does not mention the universal and null relations and makes little of the identity (of individuals). Nor does he interrelate these notions at all deeply. Accordingly, his list of principles is less rich and extensive than that of Peirce. There is one respect in which the foregoing account has perhaps been over-generous to Peirce. In accord with his own requirements, relational terms are allowed to occur syncategorematically only within the context of a relative, that is, only as a component in a compound context designating a class. Peirce's three "grand classes" of terms in this paper, it will be recalled, are absolute terms (for classes), simple relative terms (for classes such as lovers of women, lovers of betrayers of women, and so on), and conjugative terms (involving a triadic relation, such as givers of such and such to so and so). Strictly, then, relations are not allowed to occur in any other contexts, not even identity ones. However, Peirce does allow them to do so occasionally. Above, wherever possible, we

Of Lovers, Servants, and Benefactors

45

have done so also, even perhaps in some cases where Peirce himself did not explicitly intend it. In these cases we have perhaps unduly extended the letter of his writing, but hopefully not its intent. In fact, it is no easy task to eke out of Peirce's logical writing just what he is trying to say. As C.I. Lewis aptly noted in 1918, Peirce's logic papers "are brief to the point of obscurity: results are given summarily with little or no explanation and only infrequent demonstrations. As a consequence, the most valuable of them make tremendously tough reading, and they have never received one-tenth the attention their importance deserves"6 — nor have they in the intervening years. And Lewis adds that "any who find our report of Peirce's work unduly difficult or obscure are earnestly requested to consult the original papers." Ego quoque. The foregoing calls attention to the main content of the paper of 1870 and helps to make it readily accessible. Peirce himself wrote (in 1903) of this paper that in it "I made a contribution to this subject which nobody who masters the subject can deny was the most important excepting Boole's original work that ever has been made." This estimate is acceptable within the proper contraints. Aristotle and Leibniz should have been mentioned, as well as De Morgan. And it is quite clear that, when he made this judgment, Peirce was not aware of the work of Frege.

6

A Survey of Symbolic Logic (University of California Press, Berkeley: 1918), p. 106.

III

DE M O R G A N A N D THE LOGIC OF RELATIONS

Augustus De Morgan's "On the Syllogism, No. IV, and on the Logic of Relations" of 18591 is surely a landmark in the history of logic. In this paper a theory of dyadic relations may be said to have been formulated for the first time. Peirce was aware of the contents of this paper apparently in 1866 or 1867 and expressed again and again his indebtedness to it.2 Even so, it is remarkable that Peirce failed to follow De Morgan until 1883 in one important respect. It was in his paper of that year that Peirce for the first time gains the full effect of variables over relations and over individuals and admits relations as entities fully on a par with classes.3 But De Morgan did so right from the beginning. In this respect his work is superior to anything Peirce did on relations until 1883. By that date Peirce at last has full possession of the quantifiers, in terms of which of course the logic of relations takes on its fully modern form. 4 Let us look at De Morgan's paper in some detail. We will then be in a position to see precisely just what Peirce could have gained from it and what he added to it. "Any two objects of thought brought together by the mind, and thought together in one act of thought, are in relation [p. 339] . . . . All our prepositions express relation . . . but the preposition of is the only word of which we can say that it is, or may be made, a part of the expression of every relation . . . though the same thing may nearly be said of the preposition to. When relation creates a noun substantive, of is unavoidable: if A by its relation to Β be C, it is a C of B. A volume might be written about the idiom of relation: but it would be of the matter, not of the form, of the subject." This subjectivistic account of what a relation is is all that is provided, but none of the formal work in any way 1

Loc, cit. See especially Emily Michael, op. cit. 3 See IV below. 4 Recall I above. 2

46

De Morgan and the Logic of Relations

47

depends upon it. And note the interesting comment about prepositions as expressing relations, which has an extraordinarily modern ring.5 However, De Morgan does nothing more with prepositions and sets immediately to work on the relations they express. De Morgan concerns himself only with the "formal laws of relation" so far as is necessary for the treatment of the syllogism." Just as Peirce has his Procrustean "Boolian" equations, so De Morgan has Procrustean syllogisms. "Let the names Χ, Y, Z, be singular.... I do not use the mathematical symbols of functional relation φ, φ, etc. : there are more reasons than one why mathematical examples are not well suited for illustration. The most apposite illustrations are taken from the relations of human beings: among which the relations which have almost monopolized the name, those of consanguinity and affinity, are conspicuously convenient, as being in daily use." De Morgan, like Peirce, is interested in the use of relations outside of merely mathematical contexts, and in his examples, the beginnings of the modern anthropological theory of kinship may be seen. De Morgan lets (1)

'X .. LY'

express "that X is some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y." And he lets (2)

'X . LY'

express that X is not one of the Ls of Y. Although it is not explicitly stated, (2) is to be taken as the denial of (1). Thus (2) we may also write now as (2')

'~X..LY\

Notice how close we are here in (1) and (2) to the modern notation and to that of Peirce in 1883 and after. De Morgan introduces immediately the notion of the composition of two (dyadic) relations. The formula (3)

'X .. (LM)Y'

expresses that "X is one of the Ls of one of the Ms of Y." Actually this is also written as ' X . . L(MY)'

or

'X..LMY',

but De Morgan does worry as to whether the compounding in the left s

Cf. Semiotics and Linguistic Structure.

48

De Morgan and the Logic of Relations

formula here is the same as that of (3). But clearly (3) expresses what, essentially as in Principia Mathematica, we would write as (3')

'X (L/M) Y'.

The other formulae have no strict meaning unless '(MY)' is suitably defined. The formulae would then presumably be ambiguous as between ' X L (M'Y)'

and

'X L M'Y',

expressing that X bears L to the one object bearing M to Y, or that X bears L to the class of objects bearing M to Y. There are still other possibilities here, but it is clear that De Morgan intends all of these formulae in the sense of (3'). Of course (3') is logically equivalent to '(EW)(X e L'W· W ε M'Y)', which is closer to De Morgan's literal reading that X is one of the (or is a member of the) Ls of one of the Ms of Y. The 'e' of class membership is read 'is one of'. ('(M'Y)' may be defined as in Principia Mathematica, *30.01, and M'Y' is characterized by *32.13). There is no doubt but that De Morgan has discovered here the notion of the relative product of two relations in essentially its modern form. He also has the notion of the ordinary logical sum of two relations but does "not at present find it necessary to use" it. Thus (4)

'X .. (L,M)Y'

expresses that "X is either one of the Ls of Y or one of the Ms [of Y], or both." Two further interesting notions are symbolized by 'LM" and 'L-M'. The use of the little superior and inferior accents by no means accords with the importance of the notions symbolized by means of them. The first is to "signify an L of every M [of Y]" and the second "an L of none but Ms [of Y]." Clearly these notions may be written, in terms of relational abstraction, as 'XY(Z)(Z M Y o X L Z ) ' and '£Ϋ(Ζ)(Χ L Z a Z M Y ) ' respectively. De Morgan gives these notions no names, but—but recall II above — Peirce christened them in 1870 (and 1902) as the ordinary (or forward or relative progressive) involution and backward (or relative regressive) involution of L and M, respectively.6 6

Collected Papers, 3.77, 3.113, and 3.640.

De Morgan and the Logic of Relations

49

De Morgan introduces next the important notion of converse. "The converse relation of L, L - 1 , is defined as usual: if Χ .. LY, Y .. L - 1 Χ: if Χ be one of the Ls of Y, Y is one of the L - 1 s of X." And conversely, of course. The contrary [or negation of a] relation is introduced as follows. "If X be not any L of Y, X is to Y in some not-L relation." It would be better if 'not-L' and 'L' here were interchanged. Thus X . . not-L Y if and only if X.LY. In true iterative style, De Morgan considers contraries of contraries, contraries of converses, and converses of contraries, enunciating the following principles, with mere whisps of proof. (5) not-L is the converse of (not-L) - 1 . (6) L - 1 and (not-L) - ' are contraries of each other. (7) The contrary of a converse is the converse of the contrary. (8) If a first relation be contained in a second, then the converse of the first is contained in the converse of the second; but the contrary of the second in the contrary of the first. (9) The conversion of a compound relation converts both components, and inverts their order. (10) Where there is a sign of inherent quantity [ " ' o r ' ' ' ] , if each component be changed into its contrary, and the sign of quantity be shifted from one component to the other, there is no change in the meaning of the symbol. In essentially the familiar notation of Principia Mathematica, where L is the converse of L, - L its contrary, and ' Ζ U L X)).

"A relation is transitive when a relative of a relative is a relative of the same kind," that is, L is transitive just where (15')

(X)(Y)(Z)((X L Y · Y L Ζ) = Χ L Ζ).

Dè Morgan notes that (16) "a transitive relation has a transitive converse." (16') (X)(Y)(Z)((X y

U

LZ)nX

U

L Y ' Y L Z ) D X L Z ) 3

(X)(Y)(Z)((X u L Y.

LZ).

He notes also the following theorems for transitive L. (17')

L ε £Y(Z)(Z U L Y => X L Z),

(18')

L ε £Ϋ(Ζ)(Χ u - L Ζ => Ζ - L Y),

(19')

L ε &Y(Z)(Z - L Y => Χ u - L Ζ),

(20')

L ε £Ϋ(Ζ)(Χ U L Ζ r> Ζ L Y),

De Morgan and the Logic of Relations (21')

υU L Ε £ Ϋ ( Ζ ) ( Χ L Ζ => Ζ U L Υ),

(22')

υJ L ε Χ Ϋ ( Ζ ) ( Ζ - L Y d X

(230

J U' L Ε £Ϋ(Ζ)(Χ

U

U

- L Ζ),

- L Ζ => Ζ - L Υ),

υ'L Ε £ Ϋ ( Ζ ) ( Ζ L Υ Ο Χ U L Ζ),

(240

U

(250

— L Ε £ Ϋ ( Ζ ) ( Ζ L Υ => Χ - L Ζ),

(260 (270 (280

51

— L Ε ΧΫ(Ζ)(Χ L Ζ => Ζ - L Υ),

υ - L ε ΧΫ(Ζ)(Χ U L Ζ a Ζ υ - L ε Χ Ϋ ( Ζ ) ( Ζ U L Υ => Χ

U

- L Υ),

Υ

- L Ζ),

(290

(Χ)(Υ)((Χ ( U L/-L) Υ ν Χ (-L/ U L) Υ) => Χ - L Υ),

(30')

(Χ) (Y) ((Χ ( L / u - L ) Υ ν Χ ( U -L/L) Y) ζ> Χ U - L Υ).

The proofs are all obvious in view of their quantificational structure and the transitivity of L. De Morgan gives no proofs of these theorems but subjoins instances in words where L is the relative ancestor of and U L descendant of. The paragraph giving these instances is so delicious that it must be cited in full: "An ancestor is always an ancestor of all descendants [17'], a nondescendant of none but non-descendants [18'], a non-descendant of all non-ancestors [19'], and a descendant of none but ancestors [20']. A descendant is always an ancestor of none but descendants [21 '], a nonancestor of all descendants [22'], a non-descendant of none but nonancestors [23'], and a descendant of all ancestors [24']. A non-ancestor is always a non-ancestor of all ancestors [25'], and an ancestor of none but non-ancestors [26']. A non-descendant is a descendant of none but non-descendants [27'], and a non-descendant of all descendants [28']. Among non-ancestors are contained all descendants of nonancestors, and all non-ancestors of descendants [29']. Among nondescendants are contained all ancestors of non-descendants, and all non-descendants of ancestors [30']." After this peroration of principia et exempla, De Morgan returns to his old stamping ground, the syllogism, enriched now with the admission of relational formulae. His work here is of considerable interest in its own right, but lies beyond the confines of the present discussion. However, the whole of his theory of relational syllogisms seems reducible to just one form. "The universal and all containing form of syllogism is seen in the statement of X .. LMZ is [as?] the necessary consequence of X .. LY and Y . . MZ." In terms of the truth functions and quantifiers

52

De Morgan and the Logic of Relations

we clearly have that (31')

(X)(Y)(Z)((X L Y · Y M Ζ) => X (L/M) Ζ).

It is from this that all other valid relational, syllogistic forms are derivable. It is of interest to observe that, although De Morgan has full possession of the notions of relative product, relative progressive involution, and relative regressive involution, he makes no mention of the relative sum of two relations, that is, of ΧΫ(Ζ) (X L Ζ ν Ζ M Y). This notion remains to be introduced by Peirce in his 1870 paper and the "Note B" paper of 1883.1 And neither Peirce nor De Morgan mention the relative simultaneous involution of L and M, ΧΫ(Ζ)(Χ L Ζ = Ζ M Y). X would be said to bear the simultaneous involute of L and M to Y, just where X bears L to all and only the things thatbearMto Y—lover of all and only the benefactors of, or servant of all and only the lovers of and so on. It is interesting also that De Morgan does not introduce explicitly the logical product of two relations, nor the universal or null relation. The reason is no doubt that his Procrustean bed is that of the syllogism rather than that of Boolean algebra. These notions also must await the 1870 paper of Peirce. Also De Morgan lacks the notion of a definite (Russellian) description, and thus cannot handle such important notions as (L'X), the one object or individual bearing L to X, as implicitly noted above. Nordoes he introduce the class of referents of a given relation, L'X, the class of all objects bearing L to X, nor that of the class of all relata of a given referent, L'Y. Nor does he introduce the notions of the domain and converse domain of a given relation. Nor that of the plural descriptive function, L"K, the class of all objects that bear L to some member of K. Peirce likewise fails to introduce these notions anywhere. The difference, however, is that they were all within De Morgan's grasp. He was clear that objects, classes, and relations are different kinds of entities, and thus tobe handled appropriately. Peirce never satisfactorily distinguishes individuals (or singulars) from classes, as we have noted in I, and it took him many years clearly to separate out relations from ' Collected Papers, 3.45 ff., 3.154 ff., and 3.323 ff. Recall II above and see IV below.

De Morgan and the Logic of Relations

53

relatives or classes of objects standing in given relations to such and such, as noted in II. Both logicians had available the notion of identity, but it is inconceivable that Peirce could ever have hit upon the notions mentioned in the preceding paragraph, especially that of a definite description, prior to 1883, whereas De Morgan had all the necessary tools implicitly at his command. Of course, after 1883, with quantifiers explicitly available to Peirce, the whole of relational logic takes on new wings, including that of triadic relations. The pity is that, in the years after 1885, Peirce became a Daedalus preoccupied with the existential graphs, and lost his wings accordingly. But that is another story. In conclusion, some criticism of C. I. Lewis's comments on De Morgan is in order." Lewis says (p. 45) that for De Morgan "L or M, written by itself, will represent that which has the relation L, or M . . . , and LY stands for any X which has the relation L to Y . . . , " and (p. 50) that the "introduction of quantifications and the systematic ambiguity of L, M, etc., which are used to indicate both the relation and that which has the relation, hurry . . . [De Morgan] into complications before the simple analysis of relations, and types of relations, is ready for them." But it is not true that De Morgan exploys his relational symbols in this ambiguous way. On the contrary, he is explicit (p. 341) that 'Χ', Ύ', and 'Z' are singular terms and he always uses 'L' and 'M' as terms for relations. Expressions of the kind 'LY' do not occur except in the context 'X .. LY' or 'X . LY'. Nor is it true that complications result from this alleged ambiguity. De Morgan's treatment of atomic formulae containing relational terms is surprisingly modern, more so than that of Peirce up to 1883, as we have noted. Further, it is a positive merit of De Morgan's work that the "quantifications" are introduced, namely, the relative progressive and regressive involutions of two relations. It is remarkable that he handles these so skillfully, lacking the quantifiers, as he does, and given the clumsiness of his notation. Lewis concedes that "it should always be remembered that it was De Morgan who laid the foundation [of the logic of relations]." Indeed, towards the end of his paper, De Morgan himself writes that "here the general idea of relation emerges, and for the first time in the history of knowledge, the notions of relation and relation of relation are symbolized." By 'relation of relation' is meant, of course, not relations of higher type between relations, but relative products. One cannot dissent here from De Morgan's own estimate of what he had achieved. "In A Survey of Symbolic Logic, pp. 45-51.

IV

T H E R E L A T I O N A L F O R M U L A E OF 1883

Peirce is one of the first logicians explicitly to have introduced special letters for relations, as noted in II above. He treats some of them almost as though they were variables and in terms of them enunciates certain general laws or "formulae." Relations as such, as opposed to the "simple relative" and "conjugative" terms that were the concern of earlier papers, constitute the main interest of the "Note B" paper of 1883.' Let us look at this important paper in some detail in order to determine just what principles in the modern logic of relations Peirce was aware of. Most of these he and De Morgan were the first to have put forward in any exact form. We will let this paper speak for itself, rather than compare it in detail with that of 1870. "A general relative [relation] may be conceived as a logical aggregate of a number of . . . individual relatives [ordered pairs of individuals]. Let / denote "lover"; then we may write (1)

I = Σ , Σ ^ Μ ) . . ( 3.29).

The clearest way to interpret this formula, for present purposes, is that it identifies / with î j ( l ) i j , the relation between any i and j where (J)ij. (Actually Peirce thinks of (l)ij as a numerical coefficient, but this is cumbrous and not very helpful.) In later formulae Peirce uses the 'Σ' as an existential quantifier, so that his two uses should be carefully distinguished. In any case we will not go far astray if we think of (1) as expressing essentially that (10

/ =

ijlip

And the context makes it clear that any other term for a dyadic "general relative" may be put in in place of 7' and the result will be a formula. 'CollectedPapers, 3.328-3.358. We will part from Peirce's notation slightly, here and throughout, in order to keep the notation uniformly linear (except for circumflexes, which are not used by Peirce anyhow). The notation used will be an adaptation, as in the papers above, of the familiar one of Principia Mathematica. 2

54

The Relational Formulae of 1883

55

"Every relative term has a negative . . . The negative of a relative includes every pair that the latter excludes, and vice versa" (3.330). Peirce does not state it, but clearly the following formula expresses what is intended here. (2')

-/ = y ~

m

Likewise "every relative has also a converse, produced by reversing the order of the pair. Thus the converse of "lover [ o f ] " is "loved [ b y ] " . " This Peirce says, "is defined b y " the equation (3)

lij - I j i ,

or, better, (3')

u/y

- Iji.

He then notes that the "following formulae are obvious, but important," and lists five principles that we would now write as (4')

—1=1,

(5')

uu/ = /,

(6') (7')

_u/

=

u^

/ ε b = -b ε -/,

and (8')

/ ε b =

/

u

ε u¿.

Peirce merely lists these formulae without proof, and hence we shall do likewise for the moment. Also he takes ' = ' as given and does not distinguish identity here from material equivalence. Nor does he distinguish (dyadic) relational inclusion notationally from either class inclusion nor from material implication, nor these latter from each other. Nor does he distinguish dyadic relational identity from either class or individual identity, nor these latter from each other. It is interesting to note also that most of these formulae are to be found in the paper by De Morgan, discussed in I I above. "Relative terms can be aggregated and compounded like others" (3.331), and thus we have logical sums and products of two dyadic relations. (9) (10)

(/ + b)ij = l i j + b i j . (l,b)ij =

lij χ bij.

The ' + ' in (9) is of course ambiguous, and, strictly, (9) may be expressed,

56

The Relational Formulae of 1883

(/ ύ b)ij = (lij ν bij).

And similarly (9') becomes (10')

(/ ή b)ij = {lij- bij).

Peirce then lists the "main formulae of aggregation and composition" as, in modern notation,

(110

(/ ε s-b

ε s) r> (/ Ú b) ε j,

(12')

( j e | . j E j ) 3 ( j e ( / ñ b)),

(13')

(/ Ú b) ε s => (/ ε s-b

(14')

s ε (/ ñ b) 3 (ί ε I · s ε b),

(15') (16')

ε s),

((/ Ú i ) ñ i ) e ((/ ns)U(b

(is)),

((/ ù S) ή (b u S)) C (/ ή (é 0 s)).

"The subsidiary formulae need not be given, being the same as in nonrelative logic." "We now come to the combination of relatives" (3.332-3.337), namely, relative products and relative sums. "The two combinations are defined by the equations (17) (18)

(lb)ij = Σχ(1)ίΧ ( / t b)ij = n{(l)ix

ib)xj, t

(b)xj}.

It is interesting to note that here 'Σ' and ΎΓ are being used correctly for quantifiers, and this for the first time in Peirce's published logic papers. And the formulae Peirce gives concerning these products and sums are perhaps the most important and original in this paper. But first let us put (17) and (18) into a more familiar notation. (17') (18')

(l/b)ij

= (Ex)(lix·

{I t b)ij = (x)(lix

bxj), ν

bxj).

It is interesting to note, incidentally, that relative sums are not introduced in Principia Mathematica, so that the use o f ' t ' here is Peirce's own. First there are associative 1 ws for both relative sums and products. (19')

(20')

(/t(étJ)) = ((/tè)Ti), ψ ψ s)) =

((l/b)/s).

The Relational Formulae of 1883

57

Then there are "two formulae so constantly used that hardly anything can be done without them." (210

(l/(bfs))^((l/b)U),

(220

( ( / 1 b)/s) ^ ( / 1 (b/s)).

In addition there are the "obvious and trivial" formulae, (23')

((l/s) Ú (b/s)) t ¿)) = ( - / t - ¿ > ) ,

(31')

u

(/ ύ b) = (u/ O u¿)t

58

The Relational Formulae of 1883

(32')

u

(/ ñ b ) = ( υ / ή υΖ>),

(33')

u

( / 1 b) = (ub t u

(34')

u

/),

(//è) = ( u ¿>/ u /)·

Here again, the formulae concerned with relative sums and products are of especial interest. Some of them were already known to De Morgan. In addition to introducing letters for relations, a special notation is introduced for "individual dual relatives" or ordered pairs of individuals. These are of two types, Peirce notes, A:A

and

A:B.

But we may let (35')

(A:B) = Íj(i = A· j = B),

therewith gaining a general formula, for of course then (A: A) = ij{i = A-j = A). Peirce speaks of pairs such as (A:B) where ~ A = Β as alio-relatives, pairs such as (A:A) then being self-relatives. More specifically, then, an alio-relative is any pair such as ij{i = A-j = Β · ~ A = B), and a selfrelative a pair ij(i = A-j= Β· A = B). Peirce states another principle verbally when he notes that "the negatives of alio-relatives pair every object with itself." Thus (36')

—y(i = χ·j = y· ~ χ = y)zz.

Peirce distinguishes also concurrent from opponent relatives. "Relatives containing no pair of an object with anything but itself are called concurrents as opposed to opponents." Note that it is a relative or relation here that is concurrent or opponent, not just a pair. Note also the use of 'containing' in the sense of relational inclusion. To frame these definitions properly, then, the sign for relational inclusion will be needed. Thus (37')

Concurrent / = (*)(j>) ((*:>>) ^ I ^ χ = y),

(38')

Opponent / = (*)(>>) ((*:>>) ^ I

~ x = y).

Concurrent relations are those contained in identity and opponent ones are contained in diversity (or totally irreflexive). Peirce comments that "the negatives of concurrents pair every object with every other." Thus (39')

Concurrent /

(x)(y)(~ χ = y => (χ:y) ε I).

Another formula implicit in the notation and use of ordered pairs here

The Relational Formulae of 1883

59

is of course that (40')

(x:>>) e I = Ixy.

Peirce never states this principle, thinking it perhaps too trivial. We now turn to the universal and null dyadic relations V and Λ respectively, characterized by the following formulae. (41')

Λ ε /,

(42')

/ ε V,

(43')

(/ ύ λ ) = /,

(44')

(/ ñ V) = /,

(45')

(/ ù V) = V,

(46')

(/ ή λ ) = λ .

And concerning combinations we have that (47')

( / 1 V) = V,

(48')

(//Λ)=Λ,

(49')

(V ΐ /) = V,

(50')

(Λ//) = Λ.

Let I be the relation of identity and J its negative. Then also (51')

( / 1 J) = /,

(52')

(//I) = /,

(53')

(J t /) = /,

(54')

m

= /,

and of course (55')

(/ Ù - / ) = Y,

(56')

(/ ή - / ) = A.

Finally there are two "highly important formulae:" (57')

1 Judg A,q'

express that author judges A for reader q. No doubt then we are to think of (2) as a fuller explanation of (1), leaving open who the persons ρ and q actually are. Perhaps in (2) we merely quantify them away, in which case (1) would be equivalent to '(£p)(Eq)p Judg A,q\ In §3, Frege observes that "in language the place occupied by the [grammatical] subject in the word-order has the significance of a specially important place; it is where we put what we want the hearer to attend to specially." Note the inessential transition here from "author" in §2 to "we" and from the "reader" to the "hearer." "All such aspects of language are merely results of the reciprocal action of speaker and hearer; . . . " But such aspects are not taken account of, Frege tells us, in his formalized language, where "only that part of judgments which affects the possible inferences is taken into consideration." Nonetheless it is eminently in the spirit of Frege's formalized languages to do so, as the development of systematic pragmatics well attests. It is interesting that Frege introduces here (§5) the material conditional in terms of affirmation and denial rather than in terms of truth and falsity (as he does later (1893) in the Grundgesetze, §12). Again, this is a step towards pragmatics. Thus

is said to stand for the judgment that A and Β are both affirmed or denied, or A is affirmed and Β denied, but A is not denied where Β is affirmed. And similarly for negation, conjunction and the two disjunctions, as well as Modus Ponens and other rules of truth-functional inference, in § § 6-7. In the oft-discussed §8 of the Begriffsschrift the Bestimmungsweise (later to be called the Art des Gegebenseins) is introduced as a way of determining the content of a name, the "same content" being determinable in different ways. The judgment 'I—A = Β' can expresses that "the symbol Ά ' and the symbol 'B ' have the same conceptual content, so that 'Λ' can always be replaced by ' 5 ' and con-

82

Frege's Pragmatic Concerns

versely." The pragmatic element is here at a minimum. Even so, it is the person or user of language who gives or produces two modes of determination by means presumably of differing linguistic expressions, which are not necessarily synonymous and are not mere paraphrases of each other. These modes of determining are in effect modes of reference — the speaker merely refers to one and the same entity in different ways. And reference is of course par excellence a notion of pragmatics. Note that in §8 Frege does not introduce material equivalence, but rather this notion of "equality of content." He has not confused the two, but one would have expected a discussion of material equivalence to follow §§5-7. In speaking of functions of two or more arguments (§9), Frege comments that the "speaker usually intends the subject to be taken as the principal argument; the next in importance often appears as the object." In some such way as this then does linguistic structure reflect the speaker's intentions, about which more will be said below. In the discussion of generality (§11) nothing is said with a pragmatic twist other than in the use of the notions of affirmation and denial. In "Funktion and Begriff" (1891) Frege explicitly states that "not merely numbers, but objects in general, are now admissible" as arguments for a function, "and here persons must assuredly be counted as objects" (Geach-Black, p. 31). This is perhaps not a profound comment, but it is good to have it stated once and for all. In "Uber Begriff und Gegenstand" (1892) there is the interesting footnote [ibid., p. 46) to the effect that perhaps "the same word is never taken in quite the same way even by men who share a language." Frege comments also that the same "thought" may be variously expressed, and that the differences do not concern the sense "but only the apprehension, shading, or coloring of the thought, and is irrelevant for logic." But not of course for systematic pragmatics, logic in its modern extended form. In "Über Sinn und Bedeutung" (1892) the Bestimmungsweise reappears as the Art des Gegebenseins, but with no essential difference of doctrine. New, however, here are the splendid pragmatical statements concerning "ideas" and "internal images." "The referent and sense of a sign (ibid., p. 59) are to be distinguished from the associated idea. If the referent is an object perceivable by the senses, my idea of it is an internal image Such an idea is often saturated with feeling; the clarity of its separate parts varies and oscillates. The same sense is not always connected, even in the same man, with the same idea. The ideáis subjective: one man's idea is not that of another . . . . " Frege goes onto note also that hence "one need have no scruples in speaking simply of the sense, whereas in the case of an idea one must, strictly speaking,

Frege's Pragmatic Concerns

83

add to whom it belongs and at what time." (The distinction between a sense and an idea will be sharpened later, in "Der Gedanke" (1918-19), and the pragmatical character of the latter made more explicit.) A second pragmatical level is mentioned (p. 61) when Frege speaks of "the coloring and shading that poetic eloquence seeks to give to the sense. Such coloring and shading are not objective, and must be evoked by each hearer or reader according to the hints of the poet or speaker." Frege nowhere develops a theory concerning such "coloring and shading," however, and his comments on this subject remain merely suggestions. In Frege's papers preceding "Der Gedanke" (1918—19' ), we see then that there are a few pragmatical whisps of doctrine, even if mere "trifles light as air." In this late paper, however, and to a lesser extent in its sequel "Die Verneinung" (1919, Geach-Black, pp. 117 ff.), pragmatical notions take on a new significance. In speaking of interrogatory sentences ("sentence-questions"), for example, a distinction is made between the thought or sense contained in them and the "something else" that makes them interrogatory, namely, the request. Similarly "two things must be distinguished in an indicative sentence: the content, which it has in common with the corresponding sentence-question, and the assertion" ("The Thought," p. 294). Immediately Frege gives us the famous triumvirate "(1) the apprehension of a thought — thinking, (2) the recognition of the truth of a thought —judgment, (3) the manifestation of this judgment — assertion," followed by the comments that "we perform the first act when we form a sentence-question. An advance in science usually takes place in this way, first a thought is apprehended, and, after appropriate investigations, this thought is finally recognized to be true. We declare the recognition of truth in the form of an indicative sentence. We do not have to use the word 'true' for this. And even when we do use it, the real assertive force lies, not in it, but in the form of an indicative sentence and where this loses its assertive force the word 'true' cannot put it back again . . . . " Frege is not saying here that the apprehension of a thought is identical with a questioning of it, but rather that we do not question a thought prior to apprehending it. Someone else of course might question of us a thought that we do not apprehend, but we would not do so of him 1

See "The Thought: A Logicai Inquiry" (tr. by A. M. and Marcelle Quinton), Mind LXV

(1956): 289-311.

84

Frege's Pragmatic Conceras

without some apprehension however dim. Frege does not introduce the notion of degree or extent of apprehension, but in a sophisticated theory we would no doubt wish to do so — or if not, at least a comparative notion in accord with which we might be said to apprehend one thought more fully than we do another, or that one person apprehends a thought more fully than another does. Also some reference to the time or act of apprehending is no doubt needed, so that we could be said to apprehend a thought more fully at one time than another. After a thought is apprehended, and perhaps questioned, it may be recognized or judged as true, perhaps only "after appropriate investigations." It may then be declared or made manifest in the form of an indicative sentence or assertion. Here also degrees of recognition-astrue and even of assertion could be considered, as well as the times at which specific recognition-as-true acts and assertion-acts take place. Note that Frege is very explicit in distinguishing 'true' from 'recognized as true'.2 In "Der Gedanke" 'true' is used as a predicate in essentially the sense of modern semantics. Clearly a thought might be true without its being recognized as such, as Frege here merely suggests but explicitly states in the very last sentence of the paper. If we disregard numerical degrees but bring in the persons and times, (1), (2), and (3) may be symbolized as 'ρ Apprh α,ί' '/> Judg a,t,q' and l

p Assrt a,t,q'

respectively. Apprh here is inner-directed, so to speak, whereas Judg and Assrt are outer- or other-directed, from person ρ to person q? Frege does not explicitly recognize the other person here, although he did do so, it will be recalled, in §2 of the Begriffsschrift. It is most interesting that Frege, in speaking of this triumvirate, speaks of performing an act. For some purposes, it may be useful to have variables available for such acts, as in event logic.4 Thus we may let ',Apprh,a,i>e' 2

On this distinction see the author's "Truth and Its Illicit Surrogates," Neue Hefte für Philosophie 2/3. 'Cf. the author's Logic, Language, and Metaphysics (New York University Press, New York and University of London Press, London: 1971), Chapter I. 4 Cf. Belief, Existence, and Meaning, esp. Chapter IX; Logic, Language, and Metaphysics, Chapters VII and VIII; and Events, Reference, and Logical Form.

Frege's Pragmatic Concerns

85

express that e is the act of apprehending a at t, if in fact there is one and only one such act. Once one apprehends a for the first time, one no doubt continues in a state of apprehension of it unless something intervenes to make it otherwise. And similarly with judgment. But in the case of assertion, if taken in the sense of explicit manifestation of a by means of spoken sounds of some kind, the act of assertion presumably ceases when those spoken sounds die away. It is sometimes said that a fundamental task of pragmatics is to account for deixis, for the egocentric particulars Ύ, 'this', 'here', 'now', and so on. Frege makes some valuable comments concerning these words. For example, he notes that "the present tense is used in two ways: first, in order to give a date; second, in order to eliminate any temporal restriction where timelessness or eternity is part of the thought" ("The Thought," p. 296). This ambiguity of the present tense is reflected in pragmatics in the doctrine of taking the logical or source forms of linguistics as tenseless (which Frege later refers to as "a tense of timelessness") and then tensing them by the addition of a clause concerning the present time now.5 The past tense is then formed from the tenseless present by a clause concerning a time or event before or earlier than the now. Concerning T , Frege remarks that "the same utterance containing the word Ί ' will express different thoughts ,in the mouths of different men, of which some may be true, others false" ("The Thought," same page). By 'utterance' here Frege means an utterancetype or -shape, not an utterance-event. One such utterance-type may be true as spoken by person ρ at t, but false perhaps as spoken by q at t, or by q at t', or even by ρ at t', for some other t'. Frege is effect recognizes here that truth for eternal sentences (in essentially Quine's sense) must be handled somewhat differently from truth for occasion sentences. Frege is brilliant in bringing to light the differences between thoughts and "ideas" and between ideas and objects of the "outer" world. Ideas cannot be seen or touched or smelled or tasted or heard. They are "had," whereas thoughts are apprehended or judged or asserted. Ideas need a "bearer" or an agent, unlike both thoughts and outer objects. Further, any one idea has only one bearer — "no two men can have the same idea." Frege could have noted here also the need for a time reference. Ideas occur in some sense in time, thoughts as such do not. A superb footnote ("The Thought," p. 302) summarizes the matter thus: "One sees a thing, one has an idea, one apprehends or thinks [or asserts] a 5

Cf. Zellig Harris, "The Two Systems of Grammar: Report and Paraphrase," in his Papers in Structural and Transformational Linguistics (D. Reidel, Dordrecht: 1972), and "On Harris's Systems of Report and Paraphrase," in Events, Reference, and Logical Form.

86

Frege's Pragmatic Concerns

thought. When one apprehends or thinks a thought, one does not create it but only comes to stand in a certain relation... to what already existed beforehand," namely, the relation Apprh or Judg. Of course under the rubric 'ideas' Frege assembles a multitude of disparate entities, "sense-impressions,... creations o f . . . imagination, of sensation, of feelings and moods, . . . of inclinations, wishes and decisions." In a detailed discussion one would wish to separate these rather sharply from each other, and much of what one would wish to say of one one would not wish to say of another. It thus remains to be determined whether what Frege claims for "ideas" may be correctly asserted of all these types of entities. "Die Verneinung" (1918-19) and "Gedankengefüge" (19236) contain no new pragmatic thoughts not already apprehended or judged or asserted in the foregoing. In calling attention to pragmatical ingredients in Frege's linguistic writings, we must be careful not to force upon him anything not clearly contained in his work — hence the many quotations above. The form, for example, Apprh a,t\ of course is not suggested by Frege, and perhaps he would not even approve of it or of its use in the present context. The explicit introduction of such a form, however, seems a natural next step in the development of the logic of apprehension. And similarly for the other forms suggested. Without such explicit forms, these or some others like them, it is difficult to see how the philosophy of language can develop beyond its incipient stages. Indeed, the kind of clarification Frege introduced in the foundations of mathematics could scarcely have been achieved without introducing a suitable notation for the notions of logic. The same no doubt is true of the study of language. The various forms suggested are thus not intended to distort Frege's views but rather to pave the way towards the more systematic development that is quickly coming to the fore in contemporary logico-linguistics.

'See "Compound Thoughts" (tr. by R. H. Stoothoff), MindLXXII (1963): 1-17.

Vili

A D I A L O G U E WITH V E L I A N ON T R U T H

Velian. There seems to be some similarity between what my old friend Peirce once told me1 and what I have been hearing recently concerning the new semantics. R.M.M. Undoubtedly so, for your friend Peirce had remarkable insights and anticipated many recent developments. But what are you thinking of in particular? Velian. Well, Peirce told me once some fascinating things about truth, and I understand that modern semantics is concerned very largely with that notion. R.M.M. Quite so, but do you remember just what it was that Peirce said? Velian. Yes, I remember it quite well, for it struck me at the time as being very perspicacious and a considerable improvement over most of the philosophical talk I had heard concerning truth and falsity. He commented that "before anything can be true or false, it is necessary... that something should be said, whether by writing, by speech, or in thought." It seemed to me splendid to regard truth and falsity as primarily applicable to things said, that is, to items of speech or of thought. Peirce was insistent, you remember, that in order to reason at all our thoughts must be embodied in speech. Speech has a certain concreteness about it that we can get at and subject to exact study, whereas thoughts tend to be rather elusive and hence more difficult to get at. R.M.M. Undoubtedly, on both the points you have made. Velian. Peirce went on to say that what is true or false "must be said concerning something." R.M.M. Of course. Velian. ... and that "something definite must be said of that subject, some predicate." R.M.M. Yes, indeed. Velian. He went on to state that "the subject must be designated by 1

Collected Papers, 6.350 ff.

87

88

A Dialogue with Velian on Truth

a word or other sign" and that "the predicate must be signified by some word or other sign." R.M.M. This is very interesting, but I am wondering whether Peirce intended any significant difference here between 'designated' and 'signified'? Or was he just varying the words for literary effect? Velian. I am not sure of that. He did not tell me just how he was using these words. He went on immediately, however, to use 'applicable' in what seemed at first to be essentially thesame way as he used 'signified' or 'signifies' —the tense is immaterial here. He spoke of the predicatesign as being applicable or inapplicable to something, and I suppose it is inapplicable to just those things to which it is not applicable. R.M.M. Do you think he was using 'signifies' and 'applicable' (or 'applicable to') as synonyms? Velian. No, on second thought, it would seem rather that he had three notions here. Peirce was a master of language, you know, and he usually said just what he meant, although not always as carefully as he should have. R.M.M. Perhaps we should say here that a word signifies a predicate and that that word (or phrase) is the predicate-sign. The predicate-sign may then be said to be applicable or inapplicable to things. Velian. Yes, this seems in accord with Peirce's usage. R.M.M. We still have the problem of understanding his use of 'designates' and 'signifies', however. Velian. We should not worry too much about this, I think. Peirce used them only in his preliminary explanations and the real nitty-gritty of what he had to say was done in terms of 'applicable'. R.M.M. Well, if that is the case, we have something here close to my denotational semantics.2 When I say that a given sign denotes something, this is merely to say that, if it is a predicate-sign, it is applicable to that thing, and, if it is an individual-sign, that it designates the individual of which it is the sign. (This is not fully accurate, but is the real gist of the matter anyhow.) And similarly the predicate-sign may be said to designate or signify the virtual class of just the objects to which it is applicable. Thus denotation or applicability are the basic ideas, and 'designates' and 'signifies' in essentially Peirce's uses are definable in terms of it. Velian. Possibly, but I am not sure whether this accords with what Peirce said or not. He was not concerned with systematic details as to what is definable in terms of what. In any case let me go on to tell you how he handled 'true' and 'false'. 2

See especially Truth and Denotation.

A Dialogue with Velian on Truth

89

R.M.M. By all means. Velian. He noted that "if it is said that the predicate-sign is inapplicable to something to which the subject-sign is applicable, that must be true or false." R.M.M. I assume that Peirce was speaking here only of what we now call "atomic" sentences, which arise from atomic sentential forms by replacing the variable by a constant. Thus if 'Px' is an atomic sentential form, 'Pa' is an atomic sentence where 'a' is an individual-constant designating some object which is a value for the variable 'x'. Velian. No, I think you are getting ahead of the story. Peirce was speaking here, it would seem, of subject-predicate sentences in the sense of Aristotelian logic. For he went on to comment that if it is false that the predicate-sign is inapplicable to something to which the subjectsign is applicable, "then whatever there may be to which the subject-sign is applicable the predicate-sign is also applicable." We might try to symbolize what he was saying as follows. Let "S' Appi χ' express that the subject-sign'S' is applicable to the thing x. And similarly then " P ' Appi χ ' will express that the predicate-sign 'P' is applicable to χ. Peirce then was perhaps saying that 'All S are P' is true or false according as to whether (Ex)CS' Appi χ - ~ 'P' Appi χ) or not. And if not, then of course (x)('S' Appi χ => 'Ρ' Appi χ). Thus 'All S are Ρ' may be said to be true just where 'P' is applicable to everything that'S' is applicable to, but false otherwise. R.M.M. Very good, but this does not explain the truth or falsity of the atomic sentences of which I was speaking a moment earlier. Velian. Well, you have raised a difficult matter. Peirce never quite distinguished, you know, atomic sentences from ones of the form 'All S are P' where 'S' is applicable to just one object. He perhaps thought there was no need to do so, in accord with his objective idealism no doubt. This is a difficult matter which Peirce never quite became clear about. R.M.M. Did he go on to consider truth as pertaining to sentences of other forms? Velian. Yes, he did. I remember asking him about the truth of the sentence 'It rains'. He replied that in order for an assertion of this kind "to be true or false, this assertion must refer to some time and place,

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A Dialogue with Velian on Truth

and the circumstances under which the assertation [sic !] was made must have indicated a time and a place. That indicating circumstance, of which speaker and auditor had experience, was the subject-sign; and we may presume that the assertion was in meaning equivalent to these two: first, there is some time and place indicated by these circumstances to which the description 'it rains' is applicable; and secondly, whatever time and place these circumstances indicate is an occasion to which the description 'it rains' is applicable." This seemed to me a quite cogent account of the matter. R.M.M. But was not Peirce using 'applicable' in this account in a different sense from that above? Here it is an "assertion" or whole sentence that was said to be applicable, not just a subject- or predicatesign, as before. Further, it is applicable not to a thing but to something called 'circumstances', whatever these are. It is difficult to say just what a circumstance is. I think we can do Peirce one better on this matter if we use event logic3 to help analyze this kind of sentence. Let us accept Peirce's suggestion that a place and a time be brought in. To say that it rains is then to say that there is a raining going on at some place and at some time. Note the two distinct uses of the preposition 'at' here, one to stipulate the place and one the time.4 Where 'e · e At Place ρ · β At Time t)'. Of course we have given this sentence no tense, only the logical tense of timelessness. If we wish to give it the present tense we can add 'e During (the present moment)' as an additional conjunct, expressing that e takes place during the present moment. Velian. You are right that Peirce does use 'applicable' in his account in a more extended sense, and perhaps he is not entitled to this. You have carried the analysis a bit further and made it more explicit, and this seems to me all to the good. R.M.M. Did Peirce say anything about the truth and falsity of conjunctions and disjunctions as being determined in suitable ways by the truth or falsity of their components? Velian. Yes he did, but before doing so, if I remember correctly, I 3 4

See especially Events, Reference, and Logical Form. On the logic of prepositions in general, see Semiotics and Linguistic Structure.

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asked him how he would handle the truth or falsity of 'If I had upset my inkstand I should have spoiled my manuscript'. R.M.M. Ah, this is a very difficult kind of sentence to handle. What did he reply? Velian. Well, he gave me a rather obscure account in which he distinguished between "positive" and "virtual" assertions, but he never told me quite what the distinction is. Also he used 'applicable' again in the extended sense in which whole sentences "apply" to circumstances, which you seem to disapprove of. R.M.M. Yes I do disapprove of that kind of usage, because it is so obscure. Whatever we explain by means of it is then at least equally obscure. Further it seems to me that subjunctive conditionals—for that is what this sentence is — should be handled somewhat as questions, commands, and the like. And even assertions, for that matter. Frege seemed to me on the right track here in suggesting that such sentences be handled pragmatically by bringing in explicitly the speaker and his mode of speech.5 Velian. Well, you may be right on this. I will accept it provisionally. After all, I was not fully convinced by Peirce's account either. R.M.M. Incidentally, it occurs to me that we have said almost nothing about 'not'. Did Peirce say anything explicit about this little word? Velian. Yes he did, especially as prefixed to a predicate-sign. He had borrowed from De Morgan, you know, the idea of a universe ofdiscourse. He then noted that "given any sign whatever, which we may call P, we can always frame a sign which shall be applicable to every object of the universe of discourse to which Ρ is inapplicable and which shall be inapplicable to every object to which Ρ is applicable." This new sign he called 'not-P'. It took me a while to see that Peirce was here stipulating a special use of 'not', and that there are many other uses of it that he was not concerned with. R.M.M. This explanation of 'not' strikes me as rather peculiar, to say the least. No wonder you hesitated to accept it right off. Peirce's use of 'P' here was not a happy one. He should have used " P " instead, should he not? Then it would have been clear that he was combining 'not' with " P " , not with 'P'. Velian. He did say that "if Ρ is a word, this [new] sign may be formed by simply prefixing not, μή; as man, not-man; rightous, not-rightous; I, not-ir R.M.M. s

Well, this helps to bring out my point. He should have said

Recall VII above.

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'not-'man' ', 'not-'righteous' and 'not-'Γ ' here. If we apply his explanation to 'man', the new sign is to be applicable to all the things to which 'man' is inapplicable. If 'man' is P, then the new sign has to be 'not-P' or 'not-'man". Something here is surely amiss. Probably what Peirce intended to say is that where 'P' is the old sign, we frame the new sign 'not-P' to be applicable to just those things to which 'P' is inapplicable. You might think that my criticism here is a mere quibble, but actually it involves the distinction between the use and mention of expressions, a distinction that we now realize is very basic to all clear discussion in the philosophy of language. Velian. Yes, what you say makes good sense. R.M.M. There is still a difficulty, though, with 'not-I', a double difficulty in fact. I, you, he, she, and it are not the same kind of thing as whatever the subject- and predicate-signs are supposed to refer to. To handle 'not-I' something like Lesniewski's calculus of individuals is needed, so that I together with not-I make up the whole cosmos. Further, when you use Ί ' you refer to someone quite different from the person referred to when I use T . The same word on different occasions of use may refer to very different things. Not so with'man', which presumably is applicable to just the same objects, no matter who uses it or when. Did Peirce take this difference into account, do you think? Velian. Not in his conversation with me, but he must have been aware of it. As to the calculus of individuals, I don't think Peirce had any conception of this or of its need. After all, he was never very clear, you remember, with what he thought an individual is anyhow.6 R.M.M. Quite. But you have not yet told me how Peirce handled 'true' and 'false' as applied to conjunctions and disjunctions. Velian. Oh yes, I must tell you this because it is quite in accord with what we would wish to say today. Peirce explicitly spoke of assertions here. He said that "if A and Β are two assertions, and if there is a third assertion C equivalent to asserting both A and B, then I say that the copulative assertion, C, is true in case both A is true and Β is true, but is false in every other case, whether A is false or Β is false." R.M.M. Yes, this is splended. Velian. Peirce went on to say that "if D is equivalent to asserting that C is false, then I say that the disjunctive assertion, D, is false in case both not-Λ and not-5 are false, but is true in every other case whether not-A or not-5 be true." R.M.M. Yes. This explanation is a bit roundabout, but it is quite correct. Note, though, that Peirce has no right to use 'not-Λ' and 'not-5' 6

Recall I above.

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here, according to me. You recall that he has not really explained 'not' in this kind of context. Even if we allow him 'not-P'rather than'not-'P", where 'P' is a predicate-sign, we still do not have 'not-Λ' explained where A is an assertion. Velian. Well, yes, but I think Peirce intended his explanation of 'not' to include these cases. R.M.M. Perhaps, but if so, would it not have been clearer for Peirce to have given it in the same manner as for conjunction and disjunction? In other words, would it not have been better for him to have said something like this: 'not-Λ' the negation of A, is to be regarded as true just where A is false? Velian. Yes, no doubt. R.M.M. Here is another point we have not mentioned. Peirce does not seem to have distinguished an assertion from the sentence asserted, does he? But surely we should do so, should we not? He himself often wished to refer to the speaker, and sometimes also to the person spoken to. In assertion there are then at least three factors, if we include the sentence asserted. Perhaps there is a fourth also, the time or the occasion of asserting. In any case, would it not have been better for Peirce to have spoken here of 'true' and 'false' as pertaining only to declarative sentences rather than to assertions? Velian. Yes, no doubt. Especially if assertion, along with commands, questions, and subjunctives, are to be handled in the Fregean, pragmatical way you suggest. R.M.M. Did Peirce say anything more about 'true' and 'false' as applied to sentences containing quantifiers? Velian. Yes he did, but here again I am afraid he was a bit obscure. He distinguished somehow between being true and being particularly true, but I did not understand just what he was saying. He went on to talk about a universe of discourse in which there are no black tulips. He said that in such a universe "green, blue, white, reality, non-existence, and anything else you please are universally true of black tulip, while not even being black or being a tulip is particularly true of black tulip. Everything is universally true of it but universally nothing is false of it. Nothing is particularly true of it, but particularly everything is false of it." I assented to all this, at Peirce's insistence, but perhaps should not have done so. R.M.M. Well, I grant you this is quite a bit to swallow, but perhaps we can make sense of it. Let us try anyhow as follows. Let us define " P ' UnivTr ' S " to express that 'P' is universally true of 'S', that is, that 'All S are P' is

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true in the sense of Peirce's explanation above. Then of course 'blue' UnivTr 'black tulip'. And similarly for other predicate-signs. And if "P' PrtlrTr 'S" expresses that 'All S are not-P' is false, then "P' PrtlrFls 'S" may express that it is not the case that 'P' PrtlrTr'S'. Then 'blue', and so on, will all be particularly false of 'black tulip'. How's that? Velian. Not too bad. But among the predicate-signs here do you include 'reality' and 'non-existence' as Peirce does? R.M.M. Oh dear, must we go into that?7 Velian. Well, as I recall, Peirce went on to say some rather strange things about nothing. He said that, in the universe of existence we have been talking about, "a black tulip is nothing. Therefore, instead of nothing being unutterable, and all that, . . . universally it is real and non-real, utterable and unutterable, etc. But particularly it is none of these." R.M.M. But all this makes good sense if we equate being nothing somehow with the null class Λ. For then the class of black tulips is Λ in the sense of identity. If 'real' is then a predicate-sign, it is universally true of Ά', and so is 'non-real'. But particularly not, because Λ has no members. Velian. Peirce also said that "since of nothing, everything is true, it follows that of everything, being is true universally, but not at all that everything is universally true of being. Moreover being is particularly true of everything of which anything is particularly true; but not everything." Can you make sense of this too? R.M.M. Well, we must assume that 'being' or 'exists' can be handled as predicates. Then I suppose we would say that 'exists' UnivTr Ύ', where V is the universal class. This says that everything exists or that being or existence is true universally. To say that "not everything is universally true of being" is abit more difficult. Let 'a' be an expressional variable. Then ~ (a)a UnivTr 'exists', that is, not all expressions (in particular, predicate-signs) are universally true of 'exists'. This seems to express pretty closely what Peirce said. 7

Cf. "On Existence, Tense, and Aspect," American Philosophical Quarterly, to appear.

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And to say that being is particularly true of everything of which anything is particularly true; but not everything" is to say that (a)((Eb)b PtclrTr a => 'exists' PtclrTr a) and ~ (a)((b)b PtclrTr a => 'exists' PtclrTr a). The first "principle," if such it be, holds if we assume that (a)((Eb)b Appi χ => 'exists' Appi χ), a rather reasonable assumption. And the second clearly holds in view of the presumed law that (x)(b)(b Appi χ => ~ (not-ò) Appi χ). I think we may assume that Peirce accepted both of these laws concerning applicability without deigning to tell us. Velian. Very good. But Peirce really was confused, it seemed to me, when he insisted that "whatever is nothing is, of course." I protested that this is contradictory, but Peirce did not seem to think it was. R.M.M. There is still more we should say, no doubt, about 'nothing' and what it refers to. We have taken it thus far to stand only for the null class. But there is also Lesniewski's null individual, a rather strange but interesting kind of object. The null individual is the one object that does not exist, you remember, and it has a useful role to play if you care to admit it at all.8 If we do we can make good sense of the contention that "whatever is nothing is," for the nothing is then merely the one and only null individual. Velian. I suspect that Peirce might have welcomed this suggestion. He could then have distinguished three ways of handling 'nothing', regarding it as designating now the null class, now the null thing, and the third in terms of quantification when we say that there is no such and such, or ~ (Ex) .. .χ Of course this third amounts essentially to the second. R.M.M. Very good, Velian, I agree with you entirely. Perhaps also there is a dyadic-relational sense, even a triadic-relational sense of 'nothing', but Peirce obviously did not include these here. In fact, he did not include relational sentences in any of the comments he made to you on truth, did he? «Cf. "Of Time and the Null Individual," The Journal ofPhilosophy LXII (1965): 723-736.

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Velian. No, apparently not. To handle them, I suppose a wider notion of applicability would be needed, in accord with which we would say that a dyadic-relational word would apply to objects x and y, χ and y taken in that order. But then would we not need a notion of triadicrelation applicability, and so on? R.M.M. You raise an interesting question. It is astonishing that Peirce, with his love of relations, triadic ones in particular, did not face up to it. Actually one can do without these more complicated kinds of applicability, although at a price and only if one has suitable technical devices available. Thus we may say that a dyadic-relation predicate 'R' applies to individuals a and b, just where the monadic-relation (or virtual-class) predicate Ίχ 3 Rxb}' applies to a, or equally well, where '{y 3 Raj} ' applies to b. But note that 'a' and 'b' here must be individual constants, not variables. Velian. Peirce and I squabbled a bit at the end of our conversation, and he came close to calling me names. I am afraid both of us said heatedly some things that would have been better said cooly if we had taken into account the distinctions you and I have been making. R.M.M. No doubt. There should be progress in these matters, after all. Before we break off, there are two or three more things I should like to ask you about what Peirce said about truth. It is often contended that truth should be defined recursively, in terms of a higher-order logic. The people who say this are usually Tarskians and they mean that the satisfaction relation, in terms of which Tarski defines 'true', is itself to be introduced recursively. The idea is first to define 'satisfies' for atomic sentences or sentential functions, then for negations, disjunctions, and quantifications. Was there any suggestion in Peirce's comments to you that this was the way to go about it? Velian. Well, I suppose there is a hint to that effect, in that Peirce gave separate clauses for disjunction (and conjunction), hinted at a separate clause for negation, and started out with a separate clause for what he took the simplest subject-predicate sentences to be. There was no separate clause for quantifications, however. That the clauses he did give are to be taken together simultaneously to provide for negations of disjunctions, disjunctions of these, etc., etc., was not clear from the way in which Peirce worded the phrases. I think it would be saying too much to say that he explicitly anticipated a recursive definition here, but he surely came close to it. R.M.M. What about the satisfaction-relation? Do you think Peirce would have welcomed Tarski's development of semantics in terms of this? Velian. I doubt it. He would have welcomed the technical details,

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but he would have thought satisfaction too artificial and remote a relation, too remote from the field of our toil and struggle. Peirce was interested in actual language, you know, and he therefore sought for the most immediate ways in which language relates to the world. The relation of applicability, what you have called 'denotation', provides such a way. After all, Peirce's whole semantics is based upon relations such as this, you know, and is complicated enough as is. R.M.M. Yes, these are good points. The relation of applicability or denotation, it matters not how we call it, has a long and robust history too, you know. Ockham had something like it; Scotus too, although less explicitly so. Hobbes made much of it in that magnificent chapter O f Names' in the Leviathan. Mill too, in his Logic, but less clearly. And Woodger, that prince of logistic biologists, has recently revamped the theory of it, at least to some extent.9 I am not suggesting that we bow down before history, but we should not disregard it either. Velian. Quite. Logic and history are of course two of the great pillars upon which philosophic study should rest Well, I must be off now. R.M.M. It has been splendid talking with you, Velian, and I am grateful for what you have told me. It is good to see that, in spite of your good fortune in having conversed with so great a thinker as C.S.P., you do not take respite under the shadow of a great name, but allow the subject to grow and develop in the light of what we now know. We should not block the road to inquiry, let us not forget, but must get on with the painstaking job of advancing knowledge. Arrividerci.

'See XII below.

IX

BRADLEY A N D CONTINUOUS RELATIONS

We may paraphrase Charles Péguy1 by noting that the philosophical classics are new every morning and nothing is as old as today's latest philosophical fad. Even fads are not without their value, however, if some new approach or method is introduced and shown to be really contributory to "progress in clarification" in either historical understanding or in the pursuit of new knowledge. In any case, the great enduring philosophical views present a continual challenge to be updated in the light of new knowledge. We know much more now about individuals, classes, relations, predication, identity, and the like than was known in the days of Peirce and Bradley. It may be helpful to examine the views of these writers on these topics, especially as concern continuous relations, in the light of what we now know. The result will be not only an updating but to some extent a defence and rational reconstruction of some views which have long been in disrepute among writers who pride themselves upon logical clarity. Let us begin with Peirce, one of the first philosophers who may be said to have studied relations with real seriousness, as noted in II above. And the primary question, for present purposes, is whether Peirce's work on the logic of relations may be reconciled with his objective idealism, with his "law of the mind," and with his synechism. To attempt to reconcile them in detail would be a long, hard task. The main difficulty to be overcome, it would seem, is that as yet we have no successful formulation of the kind of applied logic needed for a "rational reconstruction" of objective idealism. Once this is supplied — and it is no easy matter to do so — the law of the mind, synechism, and the like, should follow suit. We must be clear straightaway that some entities must be available as values for our variables —or, if the logic is formulated combinatory-wise without variables, as denoted by the primitive one-place predicates. 2 'Basic Verities (Pantheon Books, New York: 1943), pp. 126-7. As noted by Donald Davidson, in a paper read at the New York Philosophy Club, Fall, 1972. 2

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This surely is a point Peirce would not wish to deny, even though the exact status of variables and of individuals in his logic papers is dubious. 3 That something like this is the case was well pointed out by A. B. Kempe in his Memoir of 1886,4 for which Peirce had nothing but unbounded admiration as he says in effect again and again. Kempe notes at the very beginning that "whatever may be the true nature of things and of the conceptions which we have of them . . . , in the operations of reasoning they are dealt with as a number of separate entities or units. These units come under consideration in a variety of garbs — as material objects, intervals or periods of time, processes of thought, points, lines, statements, relationships, arrangements, algebraical expressions, operators, operations, etc., etc.; occupy various positions, and are otherwise variously circumstanced." What a splendid statement this is! Note how catholic the list of entities is, including physical objects, times, mathematical objects, mental ones, syntactical ones, abstract ones, and even "arrangements" which presumably would include states, acts, processes, events, and the like. Note that "processes of thought" are included, the very stuff of idealism. Did Peirce note this item in Kempe's list, do you suppose, and store it away for future reference? Whatever one's ontology is, Kempe's point is that if we are to reason, the items of that ontology must be "dealt with as a number of separate entities." And even if one is an absolute monist, if reasoning is to take place, some subdivision of the fundamental unity into separate entities would seem a necessary presupposition. An especially interesting statement of essentially this same point is to be found, incidentally, in Royce's article in Ruge's Encyclopedia of the Philosophical Sciences (1914).5 "Without objects conceived as unique individuals," he notes, "we can have no Classes. Without classes we can . . . define no Relations, without relations we can have no Order. But to be reasonable is to conceive of order-systems, real or ideal. Therefore, we have an absolute logical need to conceive of individual objects as the elements of our ideal order systems [italics in original]; This postulate is the condition of defining clearly any theoretical conception whatever . . . . To conceive of individual objects is a necessary presupposition of all orderly [thought and] activity." This is an admirable statement, 3

Recall I above. Cf. also Gresham Riley, "Peirce's Theory of Individuals," Transactions of the Charles S. Peirce Society X (1974): 135-165 and Manley Thompson, "Individuals and Singular Terms in Peirce's Philosophy," presented at the meeting of the Peirce Society, December 28, 1975. 4 "A Memoir on the Theory of Mathematical Form," Philosophical Transactions of the Royal Society, 177 (1886): 1-70. s See Royce's Logical Essays, ed. by D. Robinson (Wm. C. Brown Co., Dubuque, Iowa: 1951), p. 350.

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by an idealist, note. Would that Peirce had been as explicit! Of course in practice, when Peirce turned to the concrete task of formulating principles in the logic of relations, he is forced ultimately to admit individuals very much in the modern manner, which is essentially that of Kempe, De Morgan, and Royce. What should Peirce's logic have looked like, then, if it is to accord with the objective idealism of his later cosmology? He should have turned his back on Boolean algebra altogether, it would seem —it was useful to him only as a heuristic anyhow — and should have developed both it and the logic of relations only virtually. His variables should then range over the fundamental soul-stuff and its various manifestations. The soul-stuff forms a genuine continuum and has what Peirce called 'Kanticity', namely, all its parts or manifestations are alike. These manifestations are the only entities Peirce can condone here as individuals, it would seem. Note that a part-whole relation must be available in accord with which one can say that one bit or manifestation is apart of another. One must characterize this part-whole relation carefully, of course, but presumably this may be done within a suitable framework. Identical entities are then those that are mutually parts of each other. This part of relation is one of the fundamental illative relations. (Peirce has several, it will be remembered, but we should perhaps regard them all as specializations of just one.) Further, this part-whole relation is an intentional relation relating things of the mind, so to speak, and must be handled accordingly.6 How do we pass on, from the theory of the part-whole relation concerning the fundamental soul-stuff and its manifestations and subdivisions, to the actual objects Peirce thought his logic applicable to — lovers, servants, benefactors, French violinists, owners of horses, betrayers of men to their enemies, and the like? Peirce, it must always be remembered, is a great advocate of logica uteris and we must not forget that logical principles guide us not only in formulating theories but in carrying out all manner of practical reasoning as well. Well, to individuals or certain bits or manifestations of soul-stuff suitable general predicates must be applicable. Some of these are monadic, some dyadic, and so on. Just what does 'applicable' here mean? For the characterization of this, we need, it would seem, something like the Scotistic notion of haecceity. A manifestation of soul-stuff is an haecceity of a class a if the predicate 'is an a ' applies to it. And for two manifestations of soul-stuff to be haecceities of a relation R, is for the predicate 'R' to apply to them in a suitable order. And so on. ' O n intentional relations, see "Events" in Events, Reference, and Logical Form. Cf. also St. Thomas, Summa Theologiae, I 13, 12, c.

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It is interesting that Peirce has no explicit theory of predication, as already remarked above, predication being accommodated in terms of one of the illative relations. Thus, for him to say that Sortes is mortal is merely to say that Sortes (or, better, the class whose only member is Sortes) bears the illative relation to being mortal. But we may say here just as well that Sortes is an haecceity of mortality. If the notion of haecceity is available, it would seem that a theory of predication may readily be provided. It is interesting also to note that Peirce has a very clear conception of the logic of second intentions, of what we now call 'the logic of second order'. In his 1885 paper {CollectedPapers, 3.359 ff.) there is a remarkable section devoted to "icons of second intention," as we have noted in V above. In all of these there occurs a quantifier over classes or relations. Here and in Chapter 14 of the Grand Logic of 1893, Peirce is explicitly aware of the importance of second intentions for the analysis of number. He remarks (Collected Papers, 4.97) that "although I cannot, in this work, carry the student deeply into second intentional logic, yet it will be indispensable to look upon quantity somewhat in that way; for quantitative thought, like the traditional 'chimaera bombitans in vacuo,' feeds upon second intentions." Does Peirce anywhere envisage or suggest or state an icon of third intention? Apparently not, at least explicitly. And although he had read a good deal of Cantor's work on set theory, it is by no means clear just how much of it he fully understood. Nor, apparently, did Peirce know of Zermelo's axiomatization of set theory of 1908, nor did he envisage any himself. If he had, he would perhaps have recognized that a much wider notion of haecceity is needed than that which he considers, one in which haecceities of haecceities are admitted and handled very much as Zermelo handles sets of sets. Entities which have haecceities are a peculiar kind of object, for they may be regarded as both unum in multis and unum de multis, in the well-known terms of Duns Scotus. In XII below there will be an exploration in some detail of what the Zermelo set theory would look like if formulated on a Scotistic basis, presupposing essentially the calculus of soul-stuff discussed above (albeit differently interpreted). It should be in some such way as this, it would seem, that we can harmonize Peirce's work in logic with his later cosmology, in which there is still ample room for tychism, fallibilism, probability, pragmaticism, and the like. (See also X below.) Peirce's law of the mind is "that ideas tend to spread continuously and to affect certain others which stand to them in a peculiar relation of affectibility. In this spreading they lose intensity and especially the power of affecting others, but gain generality and become welded with

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others ideas" (6.104). Affectibility here is to be handled in terms of some suitable intentional relation between manifestations of soul-stuff, the full definition of which poses problems. And similarly for the notions of intensity and generality. In one key passage on continuity (6.168), Peirce remarked that "it seems necessary to say that a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition breaks the continuity." But if there is to be reasoning concerning the continuum, manifestations of it must be recognized in some fashion, as we have noted above following Kempe and Royce. We may recognize manifestations and parts of them without "defining" them in such a way as to break the continuity. Further, all manifestations are of soul-stuff and are alike in being relata of the fundamental illative relation. Not only does continuity for Peirce pervade the fundamental soulstuff and its parts, it must characterize even the classes and relations admitted as well. For the moment let us consider only the latter. (Here we are very close to Bradley, as we shall see in a moment.) Peirce's clearest passage on the continuity of relations is perhaps that (of 1904) in one of his Letters to Lady Welby.1 Consider 'Cain kills Abel'. "Here the predicate appears as '— kills —'," Peirce writes.8 "But we can remove killing from the predicate and make the latter '—stands in the relation — to —'. Suppose we attempt to remove more from the predicate and put the last into the form '—exercises the function of relate of the relation — to —', and then [,] putting 'the function of relate to the relation' into another subject [,] leave as predicate '—exercises — in respect to — to —'. But this 'exercises' expresses 'exercise the function'. Nay more, it expresses 'exercises the function of relate', so that we find that though we may put this into a separate subject, it continues in the predicate just the same. Stating this in another form, to say that Ά is in the relation R to B' is to say [in part] that A is in a certain relation to R. Let us separate this out thus: Ά is in the relation R1 (where R1 is the relation of a relate to the relation of which it is the relate), to R to B'. But A is here said to be in a certain relation to the relation R1. So that we can express the same fact by saying Ά is in the relation R1 to the relation R1 to the relation R to B', and so on ad infinitum. A predicate [or relation] which can thus be analyzed into parts all homogeneous with the whole I call a continuous predicate [or 7

Ed. by I. Lieb (Whitlock's, New Haven: 1953), pp. 24-25. 'Peirce's use of repeated dashes here is not a happy one, the relation of killing not being totally reflexive and similarly throughout.

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relation]. It is very important in logical analysis, because a continuous predicate obviously cannot be a compound except of continuous predicates, and thus when we have carried analysis so far as to leave only a continous predicate, we have carried it to its ultimate elements." Note that Peirce allows the new relation R1 here to have itself as one of its relata. But clearly this is undesirable. If (1)

Ά R 1 R,B'

expresses that A is a relate of the relation R with Β as relatum, R1 is here a special triadic relation. In (2)

Ά R1 R'.R.B',

however, 'R 1 ' is ambiguous. It would be better to let R2 be a quadratic relation, so that we would write here (2')

Ά R 2 R^R.B'

instead. This would seem to accord with Peirce's intent. And so on. Peirce, when he wrote this letter, did not take into account the theory of types and allows in (1) and (2) or (2') relations to have as a relatum a relation of the same type. This of course will never do. In (1) and (2), if they are to be significant in accord with type theory, the 'R 1 ' must be of type one higher than 'R', and 'R 2 ' must be of type one higher than 'R 1 '. And so on, of course. And clearly R 1 , R 2 , and so on, are inhomogeneous relations, having entities of different types among the relata. Bradley nowhere speaks of "continuous" relations yet all relations for him are continuous in just Peirce's sense. The reader will already have noticed that Bradley's celebrated comments on relations in Chapters II and III of Appearance and Reality9 are in essential agreement with that of Peirce discussed above. Consider the quality-words 'white', 'hard', and 'sweet', and suppose they are "now the subjects about which we are saying something." Suppose we wish to say that one of these qualities A is in certain relation to another fi. "But what are we to understand here by ii?" Bradley asks (p. 17). "We do not mean that 'in relation with fi' is A and yet we assert that A is 'in relation with B\ In the same way C is called 'before £)', and E is spoken of as being 'to the right of F . We say all this, but from the interpretation, then 'before D' is C and 'to the right of F is E, we recoil in horror. No, we should reply, the relation is not identical with the thing. It is only a sort of attribute which inheres or belongs. The word to use, when we are pressed, should not be is, but only has. But this reply comes to very little. The whole question is '(Clarendon Press, Oxford: 1893).

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evidently as to the meaning of has; and, apart from metaphors not taken seriously, there appears really to be no answer. And we seem unable to clear ourselves from the old dilemma. If you predicate what is different, you ascribe to the subject what it is not ; and if you predicate what is not different you say nothing at all." Bradley clearly confuses use and mention here and fails to distinguish the is of identity from the is of predication. But even so, his view in this passage may be reconstructed as follows. Let 'a.n-ob]cct-in-relation-Rto-B' be taken as

(3)

'(ex· χ R B)'.

Then Ά = (ex · χ R B)' states that A wan-object-x-in-relation-R-to-B.10 Here we can express essentially what Bradley intends, assuming that (3) is available as significant either primitively or by previous definition, in terms of identity. But we may also express what he intends by means of 'has' or 'is a member of' or mere predication. To say that A has the attribute of being-in-relation-R-to-B is to say merely

(4)

Ά e x(x R B)',

that A is a member of the class of entities that bear R to B. Here likewise the class abstract and the 'e' of membership must be available either primitively or by previous definition. (For the present discussion the differences between attributes and classes may be disregarded.) Of course Bradley would say no doubt that this comment "comes to very little," for relation-expressions occur in the definientia and the availability of the word 'has' is taken for granted. But of course some predicates must be allowed if any rational discourse is to take place at all, as urged above by Royce. Bradley is presumably putting forth an argument, in Chapters II and III, that is intended to be "rational." Thus he must presuppose some individuals with suitable predicates upon them for any argument to take place in logically acceptable terms. Bradley's "old dilemma" is indeed old but not really a dilemma, resting upon a confusion of identity with predication. We should say rather: If you ascribe a predicate to A, the predicate itself, being a linguistic entity, must be distinct from A. And according to the theory of types, it is not even significant to say that what the predicate stands for is or is not distinct from A. In a set theory, however, or rather in a 10 The 'e' in (3) stands for the selector operator whereas in (4) it stands for classmembership. This ambiguous usage is customary, '(ex· Gx)' in general may be read 'any selected object having G'. Such expressions are analogous to Russellian descriptions without a uniqueness condition.

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semantics for such, such statements can be made. In such a framework, "if you predicate what is not different," as in ' x e x \ it is not the case that you are saying "nothing at all." Whether you are saying something true or false in that theory depends upon what is substituted for 'x' and upon the way in which formulae of this form are handled within that theory. 11 Bradley goes on to "attempt another exit from this bewildering circle," that of the relations R 1 , R 2 , and so on, above. But this gets us no further, he thinks, and "we are forced to see, when we reflect, that a relation standing alongside its terms is a delusion." And a similar "argument," of course, holds for attributes and classes. This is not the occasion to reflect upon Bradley's views —nor Peirce's either — concerning what individuals are. Ultimately, of course, for Bradley (p. 217) "there is nothing which, to speak properly, is individual or perfect, except only the Absolute." Still, insofar as Bradley's metaphysics can be made systematic, and any "rational" discourse concerning substance, qualities, relations, space, time, motion, change, perception, causation, activities, things, selves, phenomena—whether we call them "real" or merely "apparent" — is allowed to take place, suitable logical means must be provided. Details aside, this framework will presumably not be so very different from that needed for the later cosmology of Peirce, mutatis mutandis. In many respects Bradley's Absolute is akin to Peirce's one vast primordial continuum of soul-stuff or feeling. What are we to make of this doctrine of continuous predicates or relations? Is it of any real interest? Is it logically sound? Is Bradley's use of it defensible? Well, it can be made to be logically sound if brought into accord with type theory (or, alternatively, set theory). But it is also merely a matter of definition and thus rather trivial. Clearly (1) could be defined as Ά R B', (2') as (1), and so on, if 'R' is either a primitive or previously defined relational constant, and 'R 1 ', 'R 2 ', and so on, are construed as of proper type. But such a doctrine would not establish anything at all significant, for all relations would then'be "continuous." For from any relation R, whether dyadic, tetradic, or whatever, a sequence of definitions such as (1), (2'), and so on, can be generated. It is thus difficult to see why this doctrine "is very important in logical analysis," as Peirce thought it was, consisting of merely aseries of trivial definitions and failing as it does to differentiate "continuous" from any other kind of relation. " I n the system of Gödel's The Consistency of the Continuum Hypothesis (Princeton University Press, Princeton: 1940), for example, we have as a theorem that (x) ~ χ e χ, whereas in the system of Quine's Mathematical Logic (W. W. Norton, New York: 1940) 'V e V holds, where V is the class of all elements.

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Note, by the way, that a similar doctrine applies to monadic relations or classes, as Bradley in effect observed. If we say that A is an element of a class a, then here too A may be said to stand in the relation of being a relate of the relation of being an element of with respect to a. And so on. It might be argued that we have put the cart before the horse, that Ά R B' should be defined in terms o f ' R 1 ' , 'R 2 ', and so on, rather than the other way around. But then we should wish to know which of the infinity of definientia would be satisfactory, Ά R 1 R,B\ ' A R ' R 1 , R,B', or etc. All of them would be perhaps, but even so each o f ' R " , 'R 2 ', and so on, would presumably b e a primitive. It would be rather extravagant to have an infinity of primitive relations, the main use of which would be to define simple atomic sentences of the form ' A R B ' , to say the least. It should be observed that neither Peirce nor Bradley carry the dissection of relations far enough. To analyze Ά R B' as (1) or (2) is, it might be thought, not sufficient. We might contend that two additional relations are needed, one relating A to R, and another R to B. And then for each of these, two more. And so on. But this proliferation would be as harmless and otiose as that already given. Another comment is perhaps worth making. One can, if one wishes, handle (dyadic) relations as classes of ordered couples, as is now well known. Let the ordered couple be regarded as {{A},{A,B}}, the class of classes consisting of the class whose only member is A and of the class whose only members are A and B. Where A and Β are individuals of type 1, the class is of type 3. Dyadic relations between individuals are then classes of type 4, and to say that A R Β is merely to say that is a member of a certain class of couples. If for some reason, then, one thinks that relations are objectionable or delusory, one can harmlessly regard them as classes of suitable higher type instead. Bradley seems to suppose that because a relation is continuous, it is therefore "a delusion," and because the relation of predication is continuous, attributes likewise are delusory. But these do not logically follow and the missing steps in the argument are nowhere supplied — nor indeed can they be supplied. We may well take attributes, relations, and individuals as being in some sense "unreal," but still carry out rational discourse about them. The very words 'real' and 'unreal' will themselves occur in that discourse and be governed by appropriate logical, linguistic, and metaphysical principles. Much of what has been said above concerning the rational reconstruction of objective idealism is, or can be seen to be, in essential accord

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with much of what Bradley says about relations. Consider, for example, his contention (p. 125) that relations "are a development of and from the felt totality. They inadequately express, and they still imply in the background that unity apart from which the diversity is nothing. Relations are unmeaning except within and on the basis of a substantial whole, and related terms, if made absolute, are forthwith destroyed. Plurality and relatedness are but features and aspects of a unity." Apart from the "felt totality" no meaning is ascribed to statements concerning the diverse items that are "parts" of it. And no single statement ever can express adequately the basic unity. For this nothing less than the postulation of all truth would suffice. In particular, relational statements are meaningless except where the many items, including the "felt unity," are admitted as relata. Still more particularly, of course, this holds of the basic illative relation. If we humans had total understanding concerning any item we would presumably be in a position to postulate all truth about it. " L e t us consider . . . the subject that is presented," Bradley comments (p. 157). " I t is a confused whole that, so far as we make it an object, passes into a congeries of qualities and relations. And thought desires to transform this congeries into a system. But, to understand the subject, we have at once to pass outside it in time, and again also in space. On the other hand these external relations do not end, and from their own nature they cannot end. Exhaustion is not merely impracticable, it is essentially impossible. And this obstacle would be enough; but this is not all W e can neither take the terms with their relations as a whole that is selfevident, that stands by itself, and that calls for no further account; nor, on the other side, when we distinguish, can we avoid the endless search for the relation [or relations] between the relation and its terms." N o w to experience an "object" is one kind of activity; to talk about it, quite another. T o do this latter is to ascribe certain predicates to it, including relational ones, and therewith to transform a congeries of singular statements into a vast holistic network. And of course we "pass outside" the subject in time and space, and also as regards temperature, velocity, mass, force, acceleration, and so on. And we never exhaust the subject. Think how dull this would ordinarily be ! W e can, however, avoid the endless regress in the search for the relations between a relation and its terms, as we have seen above. But we do not "take the terms with their relations as a whole that is self-evident... and that calls for no further account." W e take them rather, as above, as the only linguistic means we have for saying what we wish, not as in any way involving statements of self-evidence, nor as in any sense requiring no further account. The further account has been supplied to some extent by modern logic,

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and a still better account may be at hand some years hence when the new logico-linguistics will perhaps have come of age. "The relational form," Bradley notes a couple of pages later (p. 159), "is a compromise on which thought stands, and which it develops. It is an attempt to unite differences which have broken out of the felt totality. Differences forced together by an underlying identity, and a compromise between the plurality and the unity — this is the essence of relation." For Bradley, yes, the felt totality having Peirce's property of Kanticity, that all the "differences" are alike each other and alike to the felt unity. The relational form of sentence is a "compromise" only in the sense in which all language is. Language is not to be equated with reality, any more than the word is to be equated with its object. Nor is language to be equated with thought; it is merely one of the best articulated means we have for expressing the latter. We have only begun in recent years to understand and characterize the many complex relations between word and object, relations such as denotation, designation, satisfaction, determination, and the like. Note that the virtual treatment of classes and relations is eminently Bradleyan. "The relation . . . does not exist beyond the terms [p. 159]; for, in that case, itself would be a new term which would aggravate the distraction." If terms are admitted as designating elements of the diversity, relations may be thought of as virtual, not "real" in the sense of being values for variables. Virtual classes and relations, remember, are never, never values for variables; expressions for them are introduced rather merely as façons de parler, with certain ones taken as primitives. Bradley would surely wish to have no commerce with higher-order logic or with set theory, which (as ordinarily formulated) take sets, classes, relations, and the like, as values for variables. The harmless triviality of the doctrine of continuous classes and relations by no means vitiates the underlying logic for objective idealism as sketched above.12 The fundamental illative relation for part-to-whole is taken as primitive, and there is then no need for defining sentential contexts containing it via circuitous definitions in terms of ' R \ and so on. And similarly for any other class or relational constants needed as primitives. Could it be that Bradley, and Peirce here also, failed to grasp the need for primitive or undefined expressions in the construction of an exact 12 In commenting upon Bradley's use of continuous relations in The Principles of Mathematics (Cambridge University Press, Cambridge: 1903), p. 99, Russell commented that "the endless regress is undeniable, if relational propositions are taken to be ultimate, but it is very doubtful if it forms any logical difficulty," and, a bit later, that "the endless regress . . . is logically quite harmless."

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theory? If such a need is not recognized, the doctrine of continuous classes and relations might conceivably appear as more cogent than otherwise. And although Bradley had little interest in modern logic — idealists for the most part, Royce excepted, have been hostile to it — his metaphysics can be brought into harmony with it by suitable logicolinguistic devices. A detailed formulation of these devices is one of the crying needs of contemporary idealism. The tentative suggestions above may perhaps be useful as first, informal steps towards such a formulation.

χ

THE LOGIC OF IDEALISM A N D THE NEGLECTED ARGUMENT

The N.A., as Peirce somewhat affectionately called it, consists of a "nest of three arguments for the Reality of God." 1 The first arises from "Musement" and is perhaps best described in terms of the psychology of discovery. Yet musement "inevitably" leads to "the hypothesis of God's Reality." Thus this, the "Humble Argument," then gives way to the N.A. proper, which is in part reminiscent of the traditional argument from design. Also every human heart "will be ravished by the beauty and adorability of the Idea" of God's reality, by the notion of an Ens necessarium. Indeed, "a latent tendency toward belief in God is a fundamental ingredient of the soul." The third argument of the nest "consists in a study of logical methodeutic, illuminated by the light of a first-hand acquaintance with genuine scientific thought — the sort of thought whose tools literally comprise not merely Ideas of mathematical exactitude, but also the apparatus of the skilled manipulator, actually in use. The student, applying to his own trained habits of research the art of logical analysis — an art as elaborate and methodical as that of the chemical analyst, compares the process of thought of the Muser upon the . . . [universe] with certain parts of the work of scientific discovery.. .until.. .there is "evolved"... an explanatory hypothesis." He seeks "by logical analysis to Explicate the hypothesis, i.e. to render it as perfectly [clear and] distinct as possible." He will find that "the Plausibility of the hypothesis reaches an almost unparallelled height among deliberately formed hypotheses. So hard is it to doubt God's reality, when the Idea has sprung from Musements, that there is a great danger that the investigation will stop at this first stage, owing to the indifference of the Muser to any further proof of it " Within this third argument deductive argumentations 2 may take place 1

Collected Papers, 6.452-6.492. "An 'Argument' is any process of thought reasonably tending to produce a definite belief. An 'Argumentation' is an Argument proceeding upon definitely formulated premisses" (6.456). 2

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from the explicated hypothesis as premiss. In addition the hypothesis "will exert a commanding influence over the life of its believers." The Muser will come to desire "above all things to shape the whole conduct of [his] life and all the springs of action into conformity with that hypothesis." In short the third argument seems to consist of a "rational reconstruction" of the first two as well as a pointing beyond to the realm of human values as life ideals. A full analysis of the N.A. would require going rather deeply into a discussion of what Peirce understands by Abduction, Deduction, and Induction, for these are supposed to correspond, in some rough way anyhow, with the three arguments. But it is only after the explication, or explicit formulation, of the "hypothesis" has taken place that any logical inferences involving it, of no matter what kind, can be carried out. It would therefore seem best to try to formulate the hypothesis first as a basis for worrying about the validity of inferences concerning it. This lack of formulation is perhaps the most neglected aspect of the N.A. We must never forget that Peirce holds to an objective idealism, a metaphysical view that haunts in one way or another almost everything he wrote of a philosophical nature. And in the N.A., he does not disappoint us in this regard. "... God, in His essential character as an Ens necessarium, is a disembodied spirit" who "probably has no consciousness." Peirce nowhere gives us a definition of 'Ens necessarium', but does comment at the very beginning of his paper that'God' is "the definable proper name, signifying Ens necessarium ; in my belief Really creator of all three Universes of Experience." 3 Later (6.490) a hint is given of items that should be taken account of in the definiens. "A disembodied spirit, or pure mind, has its being out of time, since all that it is destined to think is fully in its being at any and every previous time." Peirce goes on to speak of "order" and "Super-order" but this adds precious little towards a clear-cut definition oí'Ens necessarium'. For the remainder of this discussion, then, let us .take Έ Ν ' as an 3

The Three Universes, it will be recalled (6.455), are as follows. The first comprises "all mere Ideas, those airy nothings . . . [whose] Being consists in mere capability of getting though . . . . " The second "is that of the Brute Actuality of things and facts," and the third "comprises everything whose being consists in active power to establish connections between different objects, especially between objects in different Universes . . . . " On the definability of 'God', see Bowman Clarke, Language and Natural Theology (Mouton and Co., The Hague: 1966), p. 98, where he says that "a clear and precise definite description of 'God' is basic to any discussion of theological language and the first task of natural theology." Cf. also William L. Power, "Linguistic Structure and Theology," in Philosophy of Religion and Theology: 1975 Proceedings, ed. by James Wm. McClendon, Jr. (American Academy of Religion, Tallahassee, Fla.: 1975).

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abbreviation for this phrase and regard it as a primitive or undefined term. As we go on it will be necessary to part more and more from the letter of what Peirce wrote, although keeping in essentials to its spirit. For details concerning the "real internal constitution" of God's nature, let us use the material in "On God and Primordiality."4 The discussion there, it will be recalled, was on a somewhat Whiteheadian basis. Let us now reflect upon that theory along more Peircean, idealist lines. But first a few additional introductory comments. Metaphysical idealism, the great philosophia perennis, seems to have attracted little recent attention from workers in modern logic. It is thought perhaps that the two are hostile to one another, and never the twain shall meet. Peirce, as well as Royce, were idealists deeply concerned with the exact logic of their time, Peirce of course contributing notably to it. Neither, however, could truly be said to have integrated explicitly his idealist metaphysics with his interest in logic, although the possibility of such for both surely lurks in the background.5 In this present paper, concerned only with Peirce, we attempt to show not only that the twain for him should meet, but that they must meet if the N.A. is to be given a suitable foundation and his over-all philosophical view properly understood. One of the most explicit of Peirce's comments concerning objective idealism is no doubt that of 1891 in "The Architecture of Theories," that the "one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws. But before this can be accepted it must show itself capable of explaining the tridimensionality of space, the laws of motion, and the general characteristics of the universe, with mathematical clearness and precision; for no less should be demanded of every philosophy" (Collected Papers, 6.25). Peirce's use of'explaining' is perhaps too strong. A metaphysics may well accord with or accommodate principles without "explaining" them "with mathematical clearness and precision." And of course Peirce himself nowhere came anywhere near doing this latter. He never states a metaphysical principle, in particular an idealist one, with mathematical precision, nor does he ever even suggest how this might be done. The ideal of doing so set forward here is admirable, however, and we should make every effort to live up to it if possible. Peirce claims that he derives his view from Kant, in three respects. We must (6.95) "recognize that all our knowledge is, and forever must be, 4

The Review of Metaphysics 29 (1976): 497-522. On Royce, see Bruce Kuklick, Josiah Royce: An Intellectual Biography (Bobbs-Merrill, Indianapolis: 1972), p. 26, pp. 205 ff., and passim. Cf. also XIII below. 5

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relative to human experience and to the nature of the human mind." Also "as soon as it has been shown concerning any conception that it is essentially involved in the very forms of logic or of other forms of knowing, from that moment there can no longer by any rational hesitation about fully accepting that conception as valid for the universe of our possible experience." And thirdly, there is "the flat denial that the metaphysical conceptions do not apply to things in themselves. Kant never said that. What he said is that these conceptions do not apply beyond the limits of possible experience. But we have direct experience of things in themselves," these being ultimately "psychical" in accord with idealism. Peirce considers 'That is red', not tarrying over the logic of the demonstrative 'that'. Nor is he here, or indeed anywhere, explicit as to just what a "form of logic" is. What he seems to suggest is not only that all of our knowledge of redness is relative to human experience and to the nature of the human mind and that the conception of redness is "valid for the universe of our possible experience," not just actual, but also that we have direct experience of redness regarded as a psychical thing in itself. From the point of view of objective idealism, which regards (6.24) "the physical law as derived and special, the psychical law alone as primordial," these three Kantian claims are acceptable with suitable constraints. All our knowledge, and indeed all being whatsoever, is relative to and consists of experiences, actual or possible, akin to the nature of the mind. That those experiences must be "human" and that the mind involved is the human mind, is a Peircean addendum. 6 Also if conceptions are always of something, as presumably they are (if only the null thing),7 our conception of redness is always of some actual or possible experience. Just how we go, precisely, from such experiences to the general conception of redness as a thing in itself is a bit mysterious, but in any case all the latter can be is in turn some psychical entity or other. Perhaps it is to be regarded as a logical sum in some fashion of all actual and possible redness-experiences. In any case, the thing in itself redness, should be distinguished sharply from the multifarious, actual rednessexperiences, if we are to make progress towards a clear formulation of the theory. Let us return now to the N.A., and attempt to provide a logicometaphysical foundation for it. A host of problems awaits us, which we must consider one by one. How, precisely, is the£Wrelated to a "pure" or "disembodied" spirit? 'Recall 4.551, where Peirce writes that "thought is not necessarily connected with a brain." But since "there cannot be thought without Signs, . . . signs require at least two Quasi-minds, a Quasi-utterer and a Quasi-interpreter " 7 Cf. "Of Time and the Null Individual."

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This latter notion is no doubt the fundamental notion of metaphysical idealism. It would seem appropriate therefore to take a predicate for it as a primitive. Let this be 'PS', regarded as applicable to the fundamental individuals of the system. We may presuppose the usual first-order logic-cum-semantics with identity, virtual classes and relations, mereology or the calculus of individuals, syntax, denotation, and event theory. 8 Let 'P' stand for the relation of part-to-whole, so that 4x Ρ y' expresses that χ is a part of y in the sense of the calculus of individuals. Some PS's have no effect unless suitably embodied. If the Three Universes are to be accommodated in all their multifariousness, a notion of embodying must be brought in, another notion quite fundamental to idealism. Let 'x Emb γ express that χ embodies y or has y as an embodiment. And let U,, U 2 , and U3 be the Three Universes. Immediately we have the Principle of Existence for PS, Axl.

I—3!PS,

and the Principle of Idealism, Ax2.

1— (x) (PS χ ν (Ey) (PS y · (χ Ρ y ν y Emb χ)) ).

Thus PS exists (in the sense of having at least one member) and everything is either a PS or a part of one or an embodiment of one. What now is the Ens necessariuml Surely it is a disembodied PS, and exists in the sense of not being the null entity. (And of course it is unique, the statement to this effect being an instance of a law of identity, that HWOO«*

= EN·y

= EN) 3 χ = y).)

Thus there are the Principle of Existence for EN, Ax3.

I— E\EN

(or I— ~ EN = N, where Ν is the null individual), the Principle of Disembodiment, Αχ4.

I

(Ey)y Emb EN,

and the Principle of Spirituality, Ax5.

I— PS EN.

Further, we have Principles of Embodiment, Ax6. 8

I—(x)((U,x ν U 2 x ν U 3 x) => £WEmb x)

See especially Events, Reference, and Logical Form.

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and Ax7.

|—(x) (>>)(* Emb y r> χ = EN).

These tell us that the entities of the Three Universes are all embodiments of EN, and that anything that embodies anything is identical with EN. It follows that Emb is a one-many relation. And in view also of Ax4, it follows that nothing embodies itself (that Emb is totally irreflexive), and also that it is asymmetrical. Let 'Before' and 'During' stand for suitable temporal prepositional relations, of taking place before or during respectively. Then we have the Principle of Atemporality for EN. Ax8. I— 0 ) ( ~ EN Before χ·~ ~ χ During EN).

EN During χ · ~ χ Before EN·

In this way the EN "has its being out of time." And for the entities of the three universes we have a Principle of Temporality, that Ax9.

I— (x)((U,x ν U 2 x ν U 3 x) == (C 'Before χ ν C 'During x)),

where C'R is the campus or field of the dyadic relation R. Peirce asks us, in a crucial passage (6.490), "to imagine, in such vague way as such a thing can be imagined, a perfect cosmology of the three universes. It would prove all in relation to that subject that reason could desiderate; and of course all that it would prove must, in actual fact, now be true. But reason would desiderate that that should be proved from which would follow all that is in fact true of the three universes; and the postulate from which this would follow must not state any matter of fact, since such fact would thereby be left unexplained. That perfect cosmology must therefore show that the whole history of the three universes, as it has been and is to be, would follow from a premiss which would not suppose them to exist at all. Moreover, such premiss must in actual fact be true. But that premiss must represent a state of things in which the three universes were completely nil. Consequently, whether in time or not, the three universes must be absolutely necessary results of a state of utter nothingness. We cannot ourselves conceive of such a state of nility; but we can easily conceive that there should be a mind that could conceive it, since, after all, no contradiction can be involved in mere non-existence. A state in which there should be absolutely no super-order whatsoever would be such a state of nility. For all Being involves some kind of super-order." Peirce tells us a little more about the EN, but this difficult passage gives us the very heart of what he conceives it to be. How are we to untangle it? Let us recall the somewhat Whiteheadian account of the "real internal

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constitution" of God's nature as givenin'OnGodandPrimordiality." 9 There the primordial valuations play a fundamental role, and it will be useful to bring them in here in order to spell out something close to Peirce's notion of "a perfect cosmology of the three universes." He tells us nothing more about this "cosmology" so that we have precious little to go on. Hence we may be allowed some liberty in developing the notion in a somewhat novel way. In so doing we will in effect be suggesting a way of characterizing the structure of the divine will as construed on an idealist basis. The primordial valuations may be subdivided into primordial determinations, desires, tolerations, and detestations. And of course it is only the EN that entertains these. Accordingly we may let ΈΝPrDtm a,xu .. .,*„', ΈΝ PrDsr a,xl,... ,x„', 'EN PrTol and 'EN PrDtst a,x„..

.,*„'

respectively express that the EN primordially determines, desires, tolerates, or detests the circumstance described by having the «-place predicate a apply to the objects xt,..., xn in this order. Roughly a primordial determination cannot be violated but must obtain if the cosmos is to be the way it is. A primordial desire maybe violated or fail to obtain. The primordial determinations and desires together constitute the primordial obligations. The tolerations are of what is allowed, neither necessarily praised nor condemned. They are primordially neutral, as it were. The detestations are the primordial prohibitions. Even though prohibited, they may be permitted. In fact all of the primordial valuations are of what may be said to be permitted. Hence the divine will may be said to be permissive and may be subdivided into the determinative, the desiderative, the tolerative, and the prohibitive wills.10 In terms of these various kinds of valuations, we will be able to "imagine a perfect cosmology" with essentially the properties Peirce wishes it to have. Principles concerning the primordial valuations similar to those in "On God and Primordiality" may now be laid down mutatis mutandis. Note, however, the crucial difference that it is now the fiVwho does the 9 Cf. also the author's Whitehead's Categoreal Scheme and Other Papers (Martinus Nijhoff, The Hague: 1974). 10 Cf. St. Thomas, Summa Theologica, I, q. 19, a. 12.

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primordial valuating; but the EN here is much else beside. On the Whiteheadian account the real internal constitution of the primordial nature of God is wholly described in terms of the primordial valuations. On the idealist account they are performed by the EN, so to speak, but by no means exhaust its nature. Let all of the axioms concerning EN be grouped together now into one grand class of formulae, presumably infinitely large. The primordial obligations are so complex, and there are so many of them, that it is doubtful that a finite enumeration could ever exhaust them. This class is to provide for the "postulate" Peirce speaks of in the passage cited. We must distinguish the various kinds of primordial stipulations as above in order to provide for the various properties he wishes the "postulate" to have. The primordial desires constitute a "perfect" cosmology in the most lofty possible sense. All "that reason could desiderate" is stipulated by them. Further, all that we may prove from the primordial determinations "must, in actual fact, now be true of the three universes," everything happening in accord with the divine determinative will. Further, the postulates "from which this would follow must not state any matter of fact," they themselves being merely stipulative primordially. And these postulates do not suppose the three universes to exist at all. Yet these postulates or premisses must clearly be true — they are meaning postulates, if you like, stipulative of the behavior of the EN. They "represent a state of things in which the three universes were completely nil." The three universes are thus "absolutely necessary results of a state of utter nothingness." Although it is difficult for us to conceive of such a state, it is no problem for the EN, where of course we speak of his "conceiving" only analogically. A full description of the situation is much more complex than Peirce seems to realize. He seems not to have provided for chance in the primordial vision. It is curious that here he makes no mention of tychism. Also there is the tragic difference between what is primordially desired and what is merely tolerated that should be taken into account. And then there are the prohibitions that are more often violated than not. In short, where a kingdom of perfection is envisaged, in the actual stream of history "mere wreckage," in Whitehead's telling phrase, is the usual result. The EN, we recall, is the "creator" of the three universes, not just their inventor or compiler or rearranger. Genuine creation for Peirce is ex nihilo. On the Whiteheadian account there is no creation strictly but merely a rearrangement. The EN, however, is genuinely the creator by being embodied in items in one or the other of the three universes. Thus 'embody' becomes in effect merely a synonym for 'create'.

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The possibility is left open for the EN to be a person in some suitable analogical sense of that word. Peirce tells us that "in endless time it [the EN~\ is destined to think all that it is capable of thinking," and "all that it is destined to think is fully in its being at any and every previous time." Its "thinking" is presumably to be identified with the primordial envisagement and is concerned in part at least, with the entities of the three universes. TheisiV is of course "out of time," but the entities about which it is "destined" to "think" are "fully in its being" at all times, in accord with the Principles of Idealism and of Temporality, each entity χ of one of the three universes being in time, or at least in what Peirce might have called 'quasi-time'. The EN strictly does all its thinking out of time, but "in endless time" all that it "thinks" determinatively comes to obtain. "Order," Peirce tells us, "is simply thought embodied in arrangement," the actual course of history. "Super-order" is "that of which order and uniformity are particular varieties." The super-order may perhaps be thought of merely as primordial, the actual order then being merely the one that actually comes to obtain. "Pure mind," we are told, "as creative of thought [i.e., of anything at all], must, so far as it is manifested [embodied] in time, appear as having a character related to the habit-taking capacity, just as superorder is related to uniformity." The use of 'just as' here is unwise. We may accept the contents of both parts of this sentence, but the one is by no means "just as" the other. The uniformity of the actual world is merely one of the possibilities in the divine envisagement. Embodied pure mind, on the other hand, is such as to have a habit-forming capacity. Another is a "tendency to grow." These requirements must be built into the primordial valuations in some suitable fashion. One way is as follows.11 Let 'x Cpbl a' express that χ is capable of having the one-place predicate a apply to it. Let 'e Becomes e" express that e becomes e' in the course of time, 'ft Des χ' expresses that the individual constant ft designates x, and ' xY)\ And (1), ' x / , both names unshared, could perhaps be taken in a weaker sense than identity. We could let ' x / express that χ is a part of y, again in the sense of mereology. And then 'x = y' would be immediately definable as \xyyx)\ Woodger remarks (p. 20) that "this very simple language would . . . be able to cover a great deal of the ground covered by traditional set theory without using the notion of set." His subsequent work has to some extent born this out for certain areas of biology. In the intervening years, however, a good deal of expansion in other directions has turned out to be needed, if the program of getting along without sets is to be made workable. Woodger's comments are of especial interest if the doctrine of common names as to be taken seriously. They are among the most sophisticated up to their time. Note that the key phrase in the explanation of 'juxtaposition', namely, 'everything named by the first (or left hand) is also named by the second (or right hand) name' 10 They were not so taken Journal of Symbolic Logic "No wonder then that in predicates and terms as a

in our joint paper, "Towards an Inscriptional Semantics," The 16 (1951): 191-203. our joint paper we retained the syntactical distinction between fundamental one.

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joins hands across the centuries with Ockham's requirement that "subject and predicate [the common term] should stand for the same thing." And both of these are remarkably close to Hobbes' observation (p. 17) that "when two Names are joyned together into a Consequence, or Affirmation; as thus, A man is a living creature·, ... If the latter name Living creature, signifie all that the former name Man signifieth, then the affirmation or consequence is true·, otherwise false. For True and False are attributes of Speech, not of Things. And, where Speech is not, there is neither Truth nor Falsehood " Let us go on now to try to make sense of the doctrine of common names of the basis of the suitable theory of denotation and designation. As a part of the object-language, with which we must begin, let us take an appropriate first-order formulation of the calculus of individuals. Let 'P', for the part-whole relation, be taken as primitive. Thus 'χ Ρ / expresses that individual χ is a part of individual y. The axioms characterizing this relation are such as to admit both units and a null individual. 12 A unit, roughly speaking, is an entity not further subdividable in the system by means of the part-whole relation, and the null individual is the one entity that is taken to be a part of all entities whatsoever. To handle sums of units an operation of unit summation is taken as primitive. Thus '(xl—χ—)' is to stand for the logical sum of all units χ suchthat —χ—, where '—χ—' is a formula containing 'x' as its only free variable if any. A definitional abbreviation or two are needed. We may let 'Null x'

abbreviate

'(>>)x Ρ

so then χ is null just where it is a part of everything. Then 'Unit x' may abbreviate '(~Null χ · (y)((~Null y-y 3 χ Ρ y))'.

Ρ χ)

An individual χ is thus a unit provided it is itself not null but is apart of all its non-null parts. Also let 'χ PP y

abbreviate

'(χ Ρ y • ~ y Ρ χ)'.

The definiendum here expresses that χ is a proper part of y. Note that units are then describable as entities that are non-null and have no non12

On atoms or units, see A. Tarski, "On the Foundations of Boolean Algebra," and on the null individual, see "Of Time and the Null Individual." Cf. also C. S. Peirce, Collected Papers, 3.216, for a remarkable anticipation of this definition, as noted in I above.

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The Semiotics of Common Names

null proper parts. Identity is merely mutual part-to-whole, so that 'x = y

may abbreviate

'(χ Ρ y · y Ρ χ)'.

As logical axioms, in addition of course to those required for the quantifiers and truth-functional connectives, we have the following characterizing the part-whole relation P. Axl. Ax2. '—χ—' Ax3. '—y—' Ax4. Ax5.

h (*)(?)(* Pj = (z)(z Ρ * => ζ Ρ y)). I— ( M x l — χ — ) Ρ y = (λ) ((Unit χ · —χ—) s i P >>)), where is a formula containing 'x' as a free variable. I— (x)(x Ρ (>>1—y—) = (7)((Unit y · y Ρ χ) => —y—)), where is a formula containing y as a free variable. I— (x)(~ Null χ 3 (Ey)(Unit y-y Ρ χ)). I—(Ex) ~ Null χ.

Axl is the Principle of Part to Whole, and Ax2 and Ax3 art Principles of Unit Summation. Ax4 is the Principle of Composition by Units, to the effect that all non-null entities have unit parts. It is a consequence of this principle that all non-null things are composed of unit parts. Ax5 is an Existence Principle. So much now for the mereological part of the object language. When we turn to the specific common and proper non-logical names, no means are thus far available for expressing what is ordinarily achieved by predication. For this purpose a new primitive is needed, for the relation of subsumption. Let 'x Subs y" express that individual χ is subsumed in (complex) individual y. Thus suppose χ is Jones and y is the sum individual designated by the common noun 'man'. Then Jones is said to be subsumed in man. All the sentences otherwise stateable in terms of one-place predicates, may now be stated in terms of'Subs'. For example, 'All men are mortal', becomes here merely '(x)(x Subs man

χ Subs mortal)'.

The relation of subsumption will be seen to play the role here ordinarily played by predication. There is a choice as to how widely 'Subs' is to be construed. There are the three possibilities suggested by Woodger. We may allow sentences of the form a Subs b to hold where a and b are both proper names, or where α is a proper and b a common one, or where both are common ones. If we choose to allow only the second case here, no limitation need result, for the other two cases may then easily be handled, as already noted. Thus all true sentences of the form ra Subs b will be such that α is a proper name and b a common one. A list of primitive names is presumed available. These need not be

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given, however, as subdivided into two kinds, proper and common. The distinction between proper and common names will emerge rather in the axioms given concerning them and in their role in the formal semantics. Thus if we have an axiom or theorem of the form a Subs b where a and b are both primitive names, a must be proper and b common. In the formal semantics a and b will both be said to designate their objects, and b will be said to denote the object designated by α as well (presumably) as others. The various non-logical axioms required to characterize the primitive names may be stated in terms of 'Subs'. The use of 'Subs' is essential for this, of course, wherever we have a formula that would ordinarily be expressed by means of one-place predicates. Thus where a and b are any two primitive names, I—

a = b · ~ Null a)1.

And there might also be axioms to the effect that certain entities are subsumed in certain entities. Thus I— Γα Subs b1, for suitable primitive a and b. Also there might be some interrelational or "meaning" postulates. Thus, for example, if 'man', 'animal', and 'rational' are all primitive, it might be postulated that I—(x)(x Subs man = (x Subs rational · χ Subs animal)). We also must have some specifically logical axioms governing'Subs', interrelating this suitably with 'P'. There are no doubt many ways of doing this, We will sketch what appears to be the simplest way, modelling closely the behavior of 'Subs' with that of 'P\ Thus we might assume the following. Ax6. \— (Ex)(Ez)(~ χ = ζ · — χ z—) => (x)(x Subs (y 1(Ez(y Ρ ζ —ζ—)) = —χ—), where '—χ—' is a formula containing 'x' as its only free variable if any, and '—ζ—' differs from '—χ—' only in containing free occurrences of 'z' wherever and only where there are free occurrences of 'x' in '—χ—'. Ax7. I— (Ex)(Ez)(~ χ = ζ — χ y—) => (x)(w)((—χ—w = ( y 1(Εζ)(_μ Ρ ζ — ζ — ) ) · χ ΡΡ w) => χ Subs w), where (etc., as in Αχό). Ax8. I— (x)(^)((~ (z)z Ρ χ · χ Subs y) => χ PP y). Ax9. I— (x)(^)(x Subs y => (Ez)(~ ζ = χ · ζ Subs j>)), AxlO. I— (x) ( y) ((Unit y-y Ρ χ · (Ez)z Subs x) => (Ez) (yPz-z Subs χ)). Ax6 is a kind of Aussonderungs-schema. It enables us to handle com-

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The Semiotics of Common Names

plex names on the right side of 'Subs'. Ax7 and Ax8 are Interrelational Principles for PP and Subs. Ax9 is the Principle ofMultiplicity to the effect that there is always a second entity subsumed in y if anything at all is. And AxlO is the Principle of Purity, stipulating that any unit part of χ is a part of something subsumed in x, provided there is anything subsumed in χ at all. The effect of this axiom is to exclude irrelevant material from being parts of entities in which something is subsumed. The set of logical axioms suggested is by no means definitive. No doubt it can be simplified and improved in various respects, and there are alternatives to be considered. Perhaps also, in a fuller development, it would be desirable to introduce a second kind of identity, subsumptional identity (as in XII below), to be contrasted with mereological identity, each having its separate properties. Let us now, very briefly, make the semantics of the foregoing system precise, within a metal-language based on denotation and designation. Let 'Den' and 'Des' both be primitives and let 'a Den x' express that sign a denotes individual x as ex pluribus unum, as one perhaps out of many. This use of 'denotes', as noted above, is closely akin to usages of Ockham, Hobbes, and Mill, 'a Des χ' expresses that α designates χ uniquely. The semantical metal-language must of course contain a syntax as a part. Little need be said about this latter here, however, it being merely a standard classical one based on concatenation and shape-descriptions. In the informal definition of 'formula' and 'term' above a list of primitive names of the system was presupposed. Thus in the formal syntax we may let 'PrimNm o' be defined by enumeration to express that α is an expression to be found in that list. We also need the notion of a common unit-sum name as one of the form \yl(ßz)(y Ρζ·—ζ—))'. We may let 'ComUnitSumNm a' express that a is an expression of this form, '—ζ—' itself of course being a suitable object-language formula. Also there are proper unit-sum names of the form '(yl—y—)'. Let 'PropUnitSumNm d express that a is of this form. Then

The Semiotics of Common Names 'Nm a' may abbreviate PropUnitSumNm a)\

129

'(PrimNm a ν ComUnitSumNm a ν

A name simpliciter is then either a primitive one or a common or proper unit-sum name. Note that, just as a matter of syntax, every common unit-sum name is a proper one, but not conversely, proper ones not necessarily having the additional clause. We are now in a position to state the Rules of Designation and Denotation. Those for designation are as follows. DesRl. DesR2. sum name DesR3.

J— (a) (x)(j>)((a Des χ · a Des y) => χ = y). I— a Des χ, where in place of 'x' a primitive or proper unitis inserted and in place of 'a' its shape-description. I— (a)(x)(a Des χ z> (PrimNm a ν PropUnitSumNm a)).

DesRl is the Principle of Uniqueness. DesRl governs the designation of primitive and proper unit-sum names. DesR3 is the Principle of Limitation, to the effect that only primitive or proper unit-sum names designate. The Rules of Denotation are as follows. DenRl. |— (a)(x)(a Den * ^ (Ey)(~ y = χ· a Den y)). DenR2. |— (Ez)(z Subs y => (χ) (a Den χ = χ Subs y), where in place of 'y' a primitive or common unit-sum name is inserted and in place of 'a' its shape-description. DenR3. |— (a) (x) (a Den χ => ((PrimNm a · (Ey) (Ez) (a Des y · ζ Subs _y)) ν ComUnitSumNm a)). DenRl is the Principle of Multiplicity, that whatever denotes must denote at least two objects. DenR2 governs the denotation of primitive common names, and of common unit-sum names. DenR3 likewise is a Limitation Principle, to the effect that only primitive or common unitsum names denote. Finally, a rule is needed stipulating that the names admitted are just the expressions that either designate or denote. DenR4. |— (a)(Nm a => (Ex)(a Des χ ν α Den χ)). And still another, to the effect roughly that any common unit-sum name designates an χ just where it denotes just the entities subsumed in x. DenR5. |— (a) (ComUnitSumNm a => (x)(a Des χ = (j)(a Den y = y Subs χ))). The use throughout of common unit-sum names should be commented on. It is by means of such expressions that defined common names are introduced. Thus if 'rational' and 'animal' are primitive names but

130

The Semiotics of Common Names

'man' is not, we might define 'man' as \y 1(Εζ)0> Ρ ζ · ζ Subs rational · ζ Subs animal)'. In this way 'man' is defined once and for all and may occur now in all possible contexts as a substituend for a variable. Note that the notions of being a proper or common name are of only minor importance here in the formal semantics. The reason is that one and the same name can function as a proper one in some contexts and as a common one in others. Thus where r a Subs ft1 holds a is proper and b common. But in another context rb Subs c , b may function as a proper name. The door is left open thus for analyzing individuals as common sums of their parts, as well as of forming higher-order compounds of which common sums are parts. But no more need be said of these possibilities here. The semantical truth-predicate may now be defined on the basis of the foregoing. Thus far there has been no need to introduce for the operation of concatenation. It is desirable to do so now, however, in order to give a precise form to the truth-definition. Thus where a and b are any object-language expressions, (α ^ b) is the result of concatenating a with b. And rather than to introduce specific shape-descriptive names within formulae, single quotes will be used instead. Let 'Sent a' express within the syntax that α is a sentence, i.e., a formula containing no free variables. We may then immediately, within the semantical meta language, let 'Tr a' abbreviate '(Sent α · (*)('(>> 1(Ez)(y Ρ ζ.' ~ α ~ '))') Den x)\ Thus a is said to be true — for variable 'a' — if and only if a certain commun unit-sum name containing a denotes all objects. A good deal more could be said concerning this definition. However, it is sufficiently close to other well-known definitions in the literature so that this seems hardly necessary. Concerning truth we have a few theorems as follows: I-(o)(Sent a => (Tr α = Tr ('(*)'~a))). I— (û)(Sent a => (Tr a ν Tr ( ' - ' ^ a))). I—(a)(¿>)((Sent a · Senté) 3 (Tr('(' - a ^ 'ν' ^ b ^ ' ) ' ) = (TravTrb))). Especially important is the Adequacy Theorem. I— T r a = , where in place of ' ' a sentence is inserted and in place of 'a' its shape-descriptive name. We then have the particular case of the Adequacy Theorem for the

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131

relation of subsumption. f—(a)(6) (Tr (α o 'Subs' η b) = ((Ex)(E>>)(a Des χ · (ζ) (A Den z = z Subs y) - χ Subs y) = (Ex)(a Des χ · b Den χ) = (Ex)(a Des χ · (y)(a Des y ο b Den >>))). This principle incorporates many of the informal ideas above. It is the Ockham-Hobbes-Woodger Theorem in its present semantical garb.13 A second method of handling common names and their semantics is that of treating predication by disquotation in the meta-language rather than by means of a special object-linguistic primitive 'Subs'. We may thus proceed essentially as above but with the deletion of all the material concerning 'Subs'. The object-language then reduces merely to the part concerned with the part-whole relation. In place of saying that χ Subs man, for example, we say now merely that 'man' Den x. In fact we could now define 'x Subs a' as 'a Den x', where in place of 'a' we put in a common name and in place of 'a' its shape-description. This definition-scheme would not give a definition of 'x Subs γ for variable 'x' and variable ' / , however, but only for variable 'x' and constant 'a'. To provide for a more general definition of 'Subs', therefore, some other method is needed. But this presents no difficulty. We may let 'x Subs γ abbreviate '(Ea)(a Den x · (z)(Unit ζ => (ζ Ρ y = (Ew)(z Ρ w a Den w))))'. Note that this gives us a completely general definition, for variable 'x' and variable y . Using this definition, we may transfer the theory of subsumption now to the semantical meta-language, and reconstrue all the non-logical axioms accordingly. The logical axioms Ax6-Ax8 are dropped altogether. There is a genuine attractiveness to this second method. We merely pay the metal-language overtime and have it do the work of the subsumptional theory of the object-language. The kind of duplication between 'x Subs / and 'a Den x' seems unnecessary, provided we are careful to handle the quotes correctly. There is nothing really sacrosanct about the object-language anyhow. Of course we must have one to get the hierarchy of languages started, but it need not be especially powerful if the power needed can be achieved at a higher linguistic level. Note of course that the theory of denotation for the object-language "Cf. Peirce's Collected Papers, 6.350, and recall X above.

132

The Semiotics of Common Names

now is much more restricted than that above, since it concerns only its part-whole part, so to speak. DenRl is dropped altogether with some other minor changes. To achieve the full effect of the theory of denotation above we must now move into the meta-meta-language. There are several other alternatives that might be worth developing. A three-place part-whole relation might be taken as primitive, 'JC Pyz' would express that Λ: is a y-pañ of z. In all true instances of this form y would be replaced by a name, e.g., 'Jones is a man-part of'man' and 'Jones is a Jones-part of man'. Then 'χ Ρ y could be defined as 'χ P x γ and 'x Subs y as

l

xPyy\

An axiom would be to the effect that I— (*)(.y)(z)(x ρyz

(y = x\y=

ζ)).

The other axioms concerning Ρ and Subs could then be given essentially as above. Another possibility is that the null individual be dropped. This may readily be done with suitable reformulation. It is, however, convenient to retain it for various reasons, although its role, as with Carnap, could be played equally well by some other special entity. 14 The foregoing theories have made fundamental use of unit individuals, and of unit sums. It is well known that mereology may be formulated without these, however, and no doubt they could be dispensed with here also, if it is thought desirable to do so, with suitable minor changes. The syntax presupposed in the foregoing is the classical one based on shapes or sign-designs rather than upon inscriptions or sign-events. But here too an inscriptional basis could be assumed instead, with appropriate changes in the semantics. And throughout 'Subs' has been taken only in the sense essentially of Woodger's (2). It could of course be construed to include (1) also, in which case the theory of identity takes on a somewhat different form. And still another way of construing 'Subs' is to include Woodger's (3). It is time now to reflect very briefly and summarily upon how relations may be handled within the very narrow kind of framework above. This is a rather long story, narrated elsewhere, so that only a brief summary need be given here. 14

See Meaning and Necessity (University of Chicago Press, Chicago: 1947), p. 36.

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133

Although the logic of relations is a well-developed discipline, it presupposes (as usually formulated) a higher-order logic or a set theory. The exception is of course the theory of virtual relations (including classes), but even this is not used here. The theory above is so restricted —so extreme a nominalism, if one wishes to call it such—as even to exclude the theories of virtual classes and relations. Let us hark back for a moment or so to the origins of modern relation theory to see if any neglected items can be found that might be serviceable. "A dual relative term, such as 'lover', 'benefactor', 'servant', Peirce noted in 1883, "is a common name [!] signifying a pair of objects. Of the two members of the pair, a determinate one is generally the first, and the other the second; so that if the order is reversed, the pair is not considered as remaining the same.'" 5 According to this, typical of many similar passages in Peirce's writings, the most important feature of a dual relative term, or dyadic relation, is the order of its terms, distinguishing the first from the second. And Russell somewhat later, in The Principles of Mathematics (p. 95), noted that it is characteristic "of a relation of two terms that it proceeds, so to speak, from one to the other. This is what may be called the sense of the relation." Note that Peirce explicitly calls relative terms 'common names' but does not provide a theory of the latter, on a par with proper names, in which the theory of relatives could be developed. Let us now attempt to see how this could be done if 'From Russ ' and ' T o r ^ ' are added to the foregoing as new dyadic relational primitives. The subscript 'Russ' indicates that these words are taken to express the Russellian sense of a relation.16 Before doing this, let us note that a theory of events, acts, processes, states, and the like, may be added to the foregoing by allowing another kind of common name, namely, gerundives, expressions such as 'lovings', 'benefittings', 'servings', 'sittings', 'talkings', 'refusings', 'lamentings', and so on and on.17 And let us allow also a new style of variable, V, 'e¡, and so on, perhaps V , V", and so on, also, to'range over events and the like. To say that e is a loving act or state is then to say merely that e Subs lovings. But of course e is a loving from someone as agent to something as object or patient. Suppose it is person χ who loves person y. We could then stipulate that the e here bears the relation From to χ and To to y. Thus 15 Collected Papers, 2.328. " F o r more on 'From' and 'To' see Semiotics and Linguistic " S e e especially Events, Reference, and Logical Form.

Structure.

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The Semiotics of Common Names

'x Love γ becomes short for '(Ee)(e Subs lovings · e From Russ x.

e

Toruss y).

Incidentally, note how this definition, and others like it, are in accord with the remark of De Morgan, quoted at the beginning of III, that "the preposition of is the only word of which we can say that it is, or may be made, a part of the expression of every relation . . . though the same thing may nearly be said of the preposition to." De Morgan's use of'of ' here is handled by Trom R u s s ' and his use of'to' by TORuss\ These two notions are the only ones needed for "the expression of every [dyadic] relation." To the list of primitive names, a list of gerundives may now be added, not all of which need be dyadic. Some of them, however, are labelled as such, and only these are subject to characterization by means of 'From Russ ' and 'To Russ '. Concerning these we have inter alia the following rules. I— (e)(e Subs g => (Ex)(E>>)(e From Russ χ · e To Russ j)), where g is a dyadic gerundive. I {e)(x){e From Russ χ = e To Russ x) · ~ (e)(x)(e From Russ χ = e Ρ x)· ~ (e)(x)(e From Russ χ = e Subs χ)· ~ (e)(χ)(e To Russ χ = e Ρ χ· ~ (e){x){e Toruss χ = e Subs χ). This second principle assures that the Russellian From- and Torelations do not collapse into either Ρ or Subs. And the first principle assures that whatever is subsumed in a sum designated by a dyadic gerundive is in fact genuinely dyadic, i.e., bears From Russ to something and To Russ to something. Thus dyadic-relational terms may be defined in terms of the dyadic gerundives. And triadic relations may be introduced by bringing in other relations than From and To. Thus, to take an example of Peirce, 'x sells y to z' utilizes the preposition 'to' to indicate the actual recipient of the action.18 Thus among the primitive (or perhaps defined) dyadic gerundives we must have 'to-ings' in this sense. Then 'eToActuaiRecipience e · e' To Russ x)'.

may abbreviate '(Ee')O' Subs To-ings. *>' From Russ

To say that e bears the prepositional To-relation of actual recipience to χ is to say that there is a state e' subsumed in the sum of all to-ings whatsoever and which bears the Russellian From-relation to e and the Russellian To-relation to x. 18

Cf. Randolph Quirk et al., A Grammar of Contemporary English (Seminar Press, New York and London: 1972) p. 322.

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135

In some such way as this all prepositional relations other than From Russ and To Russ may presumably be introduced. And now 'sells' may be introduced as a triadic relation as follows. l

x Sell y,z' abbreviates XEe")(e" Subs sellings· e" FromRUSS x· e"

TORUSS y · e" TOActualRecipience

z

)'·

To say that χ sells y to ζ (in Frege's "time of tenselessness") is to say that there is an action subsumed in sellings which is from χ to y (in the Russellian senses) and which bears the To-relation of actual recipience to z. In this way the triadic relation for selling is traced back to the ultimate 'From Russ ' and 'To Russ ' and suitable dyadic gerundives. With a sufficient supply of prepositional relations available by definition from primitive gerundives, there would seem to be no limit on the possibility of definitions of this kind, for triadic relations as well as relations of higher degree. Note that although most of this paper has been concerned with the denotational or extensional semantics of proper names, a pragmatics and an intensional semantics are also included. Not only are human persons allowed as variables for variables and their uses of language or speech-acts explicitly accommodated, a method is also forthcoming, in the various papers referred to, for handling deixis and the egocentric particulars, as well as intensions using the Fregean ylri des Gegebenseins. In this way not only is a semantics provided but a full semiotics including a pragmatics. Note also that on this kind of a basis, mathematics need not be sacrificed, for a suitable way of providing for a very powerful theory of real numbers is forthcoming by a suitable construction.19 It would seem then that the foregoing theories, with the extensions suggested, provide the most powerful yet developed on so narrow a nominalistic basis, for logico-linguistics and the methodology of science. A final technical point. With the addition of the Russellian relations and the gerundives, and with the new style of variables, many minor extensions must of course be made in the material above. But this may all be accomplished without difficulty.

"See "On Mathematics and the Good" in Whitehead's Categoreal Scheme and Other Papers.

XII

COMMON N A T U R E S A N D MATHEMATICAL SCOTISM

Scotus' doctrine of the "common nature," the natura communis, has been the subject of much comment for several centuries, and indeed this subtle doctrine is worthy of the doctor subtilis. Very roughly the view may, for present purposes, be construed as follows. There are individual things or supposita, each of which has an intelligible structure or "nature" regarded as something distinct from it. The intelligible structure may be common to more than one thing. What precisely, ontologically speaking, are these structures? The answer is that they are in effect universale, a universal being that which is both unum in multis et de multis, a unity which is in many as well as predicable of many.1 Regarded as in multis the universal is a common nature; regarded as de multis it is a predicate. The notion of a predicate, however, is a notion of second intention and need not have an ens reale correlated with it. A common nature, on the other hand, is an ens reale an entity really existing in the world along with the supposita. The supposita and the common natures are thus of the same ontological category or logical type. The problem now becomes that of interrelating the two. In particular, supposita and the common natures being both entia realia, how are they distinguished? The answer is: by haecceity, which characterize the individual suppositum but not the common nature. This latter is "contracted" by haecceity into the individual. We cannot say that the common nature is predicable of the individual, for this would be to turn it into a predicate which it is not. Clearly now, in order to make logical sense of this doctrine, we have two tasks before us: to be very clear concerning the nature of the common natures and as to how they are related to individual existing things by haecceity. It might be thought that the Scotistic doctrine is of merely historical interest, and that we need not be concerned with it if our task is to get ahead with the advancement of late twentieth century knowledge. This 'See especially In Metaph., VII, q. 13 and q. 18, and Oxon., II, d.3, q.l.

136

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is not the case, however, in view of the recent discussion of nominalism versus platonism in the philosophy of mathematics and elsewhere. The question arises as to whether Scotus' doctrine provides perhaps a third alternative, being clearly neither one nor the other of the opposing views. Let us examine this question in the light of the logical reconstruction of the Hobbes-Woodger theory of common names, put forward in the preceding paper. Indeed, it will turn out that common natures and common names have more in common than being called 'common'. "Of Names," as Hobbes wrote in the celebrated passage already quoted in the preceding paper, "some proper and singular to one onely thing; as Peter, John, This man, this Tree: and some are common to many things; as Man, Horse, Tree; every one of which though but one Name, is nevertheless the name of divers particular things; in respect of which it is called an Universall·, there being nothing Universali but Names; for the things named, are every one of them Individual and Singular." Note that universale here as with Scotus are names, not entia realia and hence are of second intention. Hobbes of course has no truck with common natures, with the result that there is no unum in multis for a "Universali" to name. And if there is nothing for them to name, it is difficult to see how they can be regarded as "names" at all. Of course they may denote, but cannot name. Woodger also, quite independently of Hobbes, has been friendly to the doctrine of common (or "shared") names, as we have noted. But he likewise does not face up to the problem as to what the shared names stand for or designate. The theory of the preceding paper may be summarized roughly as follows. By way of logical background the usual classical first-order quantification theory is presupposed, augmented with a suitable formulation of the calculus of individuals or theory of the part-whole relation (essentially Lesniewski's mereology). In terms of this theory sums, products, and negatives of individuals may be built up, so that as values for the variables we have actually now not only individuals but all mereological compounds of them as well. The ontology or universe of discourse of the theory is in this way considerably more extended than that of mere quantification theory. The sums of individuals are especially useful in connection with common names. The semantics of this theory may be handled in terms of a relation of denotation together with one of designation. The names may then be said to denote or designate their objects, some designating just one object, some denoting more than one. Those that designate one and only one object are, following Hobbes, said to be proper; those denoting two or more, are said to be common. But a common name may also be said to designate the logical sum of all the individuals it denotes. Designation differs from denota-

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tion, it will be recalled, in being a many-one relation, so that a name designates only one thing but may denote many. In addition to the part-whole relation, another logical primitive is needed for a relation of subsumption between individuals, so that an individual may be said to be subsumed in an individual regarded as a logical sum of individuals. Of course every individual is a sum, and the interesting cases are where these logical sums are regarded as common names. For example, Socrates is subsumed in the common entity man. Here also Socrates is a proper part of the sum of individuals man, and he is also denoted by 'man' and of course designated by 'Socrates'. A suitable supply of primitive non-logical common names is presumed available in the system, but no non-logical predicates are admitted at all. Likewise no relation of predication is admitted, no copula or whatever relating subject with predicate, there being no predicates. In place of the relation of predication we have the relation Subs. What in the usual logic is expressed by means of predicates and predication is expressed now by means of common names and the relation Subs. Note, incidentally, that the supply of primitive names is given in the usual form as 'man', 'tree', 'stone', and so on, not in the form of a logical sum. They are given intensionally, so to speak, and not by enumeration. Thus among the primitive common names we would never find '(Adam υ Eve υ Cain υ Abel υ . . . ) ' , where ' u ' is the symbol for the addition of individuals. It should be evident that there is an interesting connection among these three theories, that of Scotus, that of Hobbes if his "Universalis" are given designata, and the logical reconstruction suggested. (Hobbes we need not consider further here.) The parallel is sufficiently striking to suggest that we identify Scotus' common natures with the designata of common names. Much of what is said of the former can be said in terms of the latter. Similarly we might construe haecceity relationally, and identify it with the relation Subs itself. Thus just as Socrates is subsumed in mortality so he may be said to be "contracted" from mortality by haecceity, or, in short, to bear the relation of haecceity to mortality. The common nature is now the universal regarded as unum in multis, being the unique designatum of a common name. Regarded as unum de multis, the universal is apparently only post rem or "in the mind." It is easy enough to make a predicate out of a common name, however, if we wish. We may merely define 'Mortal x' as 'x Subs mortal', and similarly for other cases. We might then regard the predicate as existing only in the mind or second-intentionally. Predicates therewith

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become predicable de multis in the usual sense. In this way the unum in multis is clearly an ens reale, and the unum de multis is an ens rationis. A second way of construing the unum de multis is to regard it as what is designated by the common name, bearing in mind that that name also denotes. Thus 'man' designates a certain logical sum of entities but it also denotes each and every individual man. In other words, regarded as being designated as a unique object, the entity man is an unum in multis·, but regarded as being an entity designated by a name that denotes each and every individual man, it is an unum de multis. The difference between designation and denotation is of course crucial here. It has been pointed out that the use of 'common' in 'common nature' is misleading, for the common nature is " . . . a certain nature which is ordinarily referred to as common or universal nature but which, in reality, is not common, not universal, nor particular." 2 The common name, on the other hand, is genuinely common in the sense of denoting at least two objects. It is not clear that there is a real conflict here, however. If Socrates is the only man existing (or that ever has existed or ever will exist) he still may have a common nature, but he could not properly be said to have the common nature man. He would still have the common nature rational if there is some other rational being, and the common nature animal if there are other animals. Thus here common natures are regarded as genuinely common in essentially the same way that common names are. Picking up now the remainder of Grajewski's comment, we note that common natures are clearly not universale, for then they would be predicates or the designata of such. Nor are they particulars in the sense of being designata for names that do not also denote. On the other hand, the common natures are universal in the sense of having more than one object subsumed in them and they are particulars in the sense of being designata of names that also denote. Suppose there is no suppositum at all to have a given common nature. Scotus maintains that if every member of a species were to be destroyed, the common nature would be destroyed also. In this case the appropriate "common" name denotes nothing at all, and strictly is no longer either common or proper. It is clear that the suppositum and the common nature are never the same (with the possible exception of the one cosmic or world entity). One of the principles of the theory of subsumption is that whenever χ 2

M. Grajewski, The Formal Distinction of Duns Scotus (The Catholic University of America Press, Washington: 1944), p. 143. See also A. Wolter, The Transcendental and Their Foundation in the Metaphysics of Duns Scotus (The Catholic University of America Press, Washington: 1946).

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Subs y, χ is then a proper part of y, and hence of course cannot be identical with it. It was suggested above that haecceity be construed relationally in terms of Subs. A suppositum is then an entity that is the haecceity of (or is subsumed in) some common nature or other. And similarly a common nature is an entity of which something is an haecceity. The so-called "formal distinction" between the common nature and haecceity then becomes the distinction between the occupants of the domain and those of the converse domain of the relation Subs, in the terminology of Principia Mathematica. Joseph Owens has emphasized that Scotus, unlike St. Thomas, recognizes a "metaphysical" mode of being, in addition to the "real" and "logical" ones. The real or physical mode is made up of individuals or supposita, and the objects of the logical mode are somehow in the mind but not real. The objects of the metaphysical mode are not supposita and in this respect are like entities in the logical mode. Yet they are real and in this respect like objects in the physical mode. The inhabitants of the metaphysical mode are precisely the common natures. Owens points out that we cannot gain a proper understanding of Scotus with recognizing explicitly the metaphysical mode.3 Scotus tends to refer to common natures by words such as 'humanity', 'whiteness', and the like. The reason is no doubt that'humanity' is nota predicate. Socrates may be said to have humanity, to be subsumed in it, or to be an haecceity of it, but it would never be appropriate to say that Socrates is a humanity or that he is humanity. Perhaps a good way of arranging the matter would be to write all common names in their abstract form, as here, and then to define such words as 'Man', 'White', and so on, as predicates. Thus, in place of the kind of definition of 'Mortal x' suggested above, we would now let 'Man x' abbreviate 'χ Subs humanity', 'White x' abbreviate 'x Subs whiteness', and so on.4 The Scotistic view is often referred to as a moderate realism, the point of contrast with nominalism being that the common natures are entia realia. For nominalism, only the individuals or supposita are —this is essentially the view of some contemporary nominalists also. In the 3

See his "Common Nature: a Point of Comparison between Thomistic and Scotistic Metaphysics," Medieval Studies (Pontificial Institute of Medieval Studies, Toronto), XIX (1957): 4 For scalar adjectives this handling is of course much too simple. See "On How Some Adverbs Work" in Events, Reference, and Logical Form.

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rendition here, in addition to the individuals, mereological compounds of individuals are also admitted, as well as the relation of subsumption. The view is thus closer to a moderate realism than to a nominalism. But a mereology (as formulated on the basis of first-order logic as here) may be regarded as a nominalism, and if a theory of subsumption is added there is no increase in the ontology. The mereological compounds might be thought less "real" than the individuals comprising them, however, and in any case they are surely more artificial. The compound sum of all human beings is not "given" in nature in the simple way that human beings individually are. It takes a considerable cognitive leap even to comprehend what this sum is. It is in some such sense as this that the common natures may be thought to inhabit a metaphysical mode of being distinct from that of the individual supposita. The Scotistic theory of common natures is intimately linked with epistemological matters and it is surely a mistake to view it in abstraction from them. We have seen in fact that it is essential to bring in semantical matters concerned with the reference of general names, more particularly, with their denotation and designation. The theory of reference is in effect epistemology (or at least part of it) in modern dress. As Peirce noted: "The notion that the controversy between realism and nominalism had anything to do with Platonic ideas is a mere product of the imagination which the slightest examination of the books would suffice to disprove." 5 This statement no doubt exaggerates the matter somewhat. But it cannot be emphasized enough that the most significant issue here is over how the semantics of general terms is best to be handled. To summarize concerning universale. For Scotus there are actually three kinds: the physical, the metaphysical, and the logical. The physical here is identified with the designatum of a common name, that is, a certain compound individual ; and the metaphysical, with that individual regarded as the sum of the various entities denoted by the common name. The logical universal is then given by definition as above and is a res secundae intentionis. It is only the universal in this third sense that may be said to be predicable of anything. Much of importance has been omitted in these rough introductory remarks. Their intent has been only to point out that there is a striking similarity — to within minor technical details — between the Scotistic theory of the common nature and haecceity and the mereological system with the relation Subs. This latter may perhaps be regarded as providing a suitable logical foundation for the former. Whatever the merits of this contention, enough surely has been shown to suggest that 5

North American Review 113 (1871), p. 454.

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a logistic reformulation or rational reconstruction of the Scotistic theory on the basis of the mereological system might be possible and, if so, would be well worth undertaking. Let us turn now to some reflections of a somewhat more technical character. Peirce was surely an admirer of Scotus. "The works of Duns Scotus have strongly influenced me," he commented at one point. "If his logic and metaphysics, not slavishly worshipped, but torn away from its medievalism, be adapted to modern culture, under continual wholesome remainders of nominalistic criticism, I am convinced that it will go far toward supplying the philosophy which is best to harmonize with physical science." 6 Elsewhere Peirce spoke of the "vast logical genius of the British Duns Scotus" (2.166). Peirce often thought of himself as a "Scotistic realist" (4.50, 5.77, n.l, 5.470) and much of his work no doubt is illumined if the influence of Scotus is recognized.7 However, nowhere does Peirce attempt a sustained account of just what this Scotistic doctrine is. To attempt to show precisely how it could harmonize with physical science would be an immense undertaking indeed. It is not clear even that this has ever been shown for any physical view, in full detail, even if we confine ourselves to the physics of Peirce's day. An important part of physics, systematically conceived, is the full classical mathematics of real and complex numbers, the theory of functions of them, and such further modern developments as depend upon these in one way or another. Let us attempt to show now, in the remainder of this paper, that what is essentially the Scotistic doctrine, not slavishly worshipped but sympathetically reformulated in modern terms, stripped of such medievalism as is not now scientifically acceptable, formulated in accord with first-order logic and semantics and in terms of the positive achievements of contemporary nominalism, can be harmonized with mathematics in this strong, classical sense. In so doing, of course, we shall have to part considerably from both Scotus and Peirce. 6

Collected Papers, 1.6. 'See especially Charles McKeon, "Peirce's Scotistic Realism," Studies in the Philosophy of Charles Sanders Peirce (Harvard University Press, Cambridge: 1952), pp. 238-250; Edward C. Moore, "The Influence of Duns Scotus on Peirce," Studies in the Philosophy of Charles Sanders Peirce, Second Series (The University of Massachusetts Press, Amherst: 1964), pp. 401-413; John F. Boler, Charles Peirce and Scholastic Realism (The University of Washington Press, Seattle: 1964); Robert Almeder, "Peirce's Pragmatism and Scotistic Realism," Transactions of the Charles S. Peirce Society IX (1973): 3-23; Ralph Bastian, "The Scholastic Realism of C. S. Peirce," Philosophy and Phenomenological Research 14 (1953): 246-249; and Robert Goodwin, "Charles Sanders Peirce: A Modern Scotist?" The New Scholasticism 35 (1961): 478-509.

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The theory to be put forward will differ somewhat from that sketched above, and must be more oriented to the foundations of mathematics. To provide the necessary changes, let us start from scratch. As basic logic let us presuppose, as above, the usual classical quantification theory of first order augmented by what is essentially a suitable formulation of Lesniewski's mereology or calculus of individuals. (The universal quantifier '(x)', V for disjunction, and for negation are taken as primitive. Let 'P' be the primitive for the part-whole relation between individuals. Thus 'χ Ρ / expresses that individual χ is a part of individual y. And also an operation of unit summation is taken as primitive. Thus any expression of the form '(xl—χ—)' is to be a term, more strictly, a quasi-atomic one, where '—χ—' is a formula containing 'x' as a free variable.) The theory of subsumption may be handled as above in terms of 'Subs' as a primitive, which, however, will be construed in a new way. And also of course there is a list of primitive, non-logical names. Identity may be defined familiarly as follows. ' χ = γ abbreviates '(z)(x Subs ζ o y Subs ζ)'. A few mereological definitions may now be given. ' ( x u y ) ' abbreviates '(zl(z Ρ χ ν ζ Ρ >>))', '(χ Π y)' abbreviates '(zl(z Ρ χ · ζ Ρ >>))', ' - χ ' abbreviates '(zi ~ ζ Ρ χ)', 'Ν' abbreviates '(xl ~ χ Ρ χ)', 'W' abbreviates '(xlx Ρ χ)'. These will be recognized as definitions of notations for the logical sum of two individuals, the logical product of two individuals, the negative of an individual, the null individual, and the world individual respectively. An entity is said to be a unit or an atomic individual if and only if it is not part of Ν and is a part of all its parts that are not parts of N. Thus 'Unit x' abbreviates ' ( - χ Ρ Ν · (>>)((- y Ρ Ν · y Ρ χ) => χ Ρ >>))'. The mereological axioms comprise Axl-Ax4 of the preceding paper, the Principle of Part to Whole, Principles of Unit Summation, and the Principle of Composition, interpreted appropriately. In addition we have the following. Ax5. I Ax6. I

χ = y, where χ and y are distinct primitive names. χ = N, where χ is a primitive name.

Ax5 is the Principle of Distinctness and Ax6 the Principle of Non-Nullity, for the primitive names.

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We now need some specifically Scotistic axioms governing haecceity as construed in terms of subsumption. Ax 7. |— (x)(.y)((x Subs y ν (z)(z Subs χ ζ Subs >>))=> χ Ρ y). Αχ 8. 1— (x)(y)((Unit y-y Ρ χ·(Εζ)ζ Subs χ) => ( E z ) ( y Ρ ζ · ζ Subs χ)). Αχ 9. I— (*)(>>)((ζ)(ζ Subs χ = ζ Subs y) ^ χ = y). Αχ 10. I— ΟΟΟΉ,ν Subs (zl(Ew)(z Ρ w — ν ν — w Subs χ)) = (— y—-y Subs χ)), where '—w—' is a formula differing from the formula '—y—' only in containing free occurrences of 'w' wherever and only where there are free occurrences of y in '—y—', 'χ' and 'ζ' not occuring freely in '—w—'. Ax 7 is the Principle of Inclusion, that haecceity or subsumption is included in the part-whole relation. Ax 8 is the Principle of Purity, stipulating that any unit part of * is a part of some haecceity of x, provided χ has any haecceity at all. In other words, irrelevant material is excluded from being parts of common natures. Ax 9 is a Principle of Extensionality. Ax 10 is the Principle of Common-Nature Formation. It enables us to build up a common nature whose haecceities are just the haecceities y of a given common nature χ where—y—. This is, of course, a very powerful assumption. We now turn to the specifically mathematical axioms formulated on the Scotistic basis. Many of these are existence assumptions, or at least assumptions of relative existence, for common natures of a complex kind. Axil. I— (Ex)x Subs (y\(Ez)(y Ρ ζ · ~ (Ew) w Subs z)) · (x)(x Subs (yl ( E z ) ( y Ρ ζ · ~ (Ew) w Subs z)) = ~ (Ew) w Subs x). Axl2. \— (x)(y)(z)(z Subs (wl (E«)(w Ρ U · (« = χ ν u = 7))) s (ζ = χ νζ = Αχ13. \— (x)(y)(y Subs (zl(Ew)(z Ρ w · (E«)(w Subs u · u Subs χ))) s E«)(j> Subs u · u Subs χ)). Axl4. ( X ) ( J ) ( J Subs (zi (Ew)(z Ρ W · (u)(u Subs w => u Subs χ))) Ξ (M)(M Subs y => u Subs χ)). Axl5. I— (Ex)x Subs {yl ( E z ) ( y Ρ ζ · ((w)(~ (Ε«)« Subs w => w Subs Z ) · ( W ) ( H ) ( ( W Subs ζ · (V)(V Subs u = v = w))=> u Subs ζ))))· (x)(x Subs (_yl ( E z ) ( y Ρ ζ · ((w)(~ ( Ε φ Subs w => w Subs z) · (w)(w)((w Subs ζ · (ν) (ν Subs u = ν = w)) is u Subs ζ)))) = ((w)(~ (Ε«)« Subs w => w Subs χ) · (w)(«)((w Subs χ · (ν)(ν Subs a = ν = w)) d « Subs χ))). Axl6. 1— (x)(((Ey)_v Subs χ · (>>)(>> Subs χ => (Ez)z Subs y) · ~ (E>>) ( E z ) ( y Subs χ · ζ Subs χ · ~ y = ζ) · (Ew)(w Subs y · w Subs z))) ((E>> Subs (zl (Ew)(z Ρ W · («)(« Subs χ => (Εν)(ί)((ί Subs u · t Subs w) = t = v)))) · ( y ) (y Subs (zl (Ew)(z Ρ w · («)(« Subs χ => (Εν)(ί)((/ Subs u · t

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Subs w) = t = v)))) = («)(« Subs χ 3 (Εν)(ί)((ί Subs u-t Subs y) = t = ν)))). Axl7. I — (Ex)—JC—ID ((Ey)_Y Subs (zl(Ew)(z Ρ W — w — ~ ( E M ) (H Subs w—«—)))·(>00> Subs (zl(Ew)(z Ρ w — w — ~ (Ea)(«Subs w—u—))) = (—y— ~ (E«)(w Subs y—u—)))), where (as needed about '—JC—'—a—', and ' — y — ) . Axl8. \ - (xXÍj'Xj' Subs χ r> (Ez)(w)(—u—w— = w = z)) = ((Ej)^ Subs (zl(Ew)(z Ρ H> ·(«)(« Subs W = (Ev)(v Subs χ · —V—«—)))) · (>0 (y Subs (zl(Ew)(z Ρ w· («)(« Subs w = (Ev)(v Subs χ·—ν—«—)))) = (m)(« Subs y = (Ev)(v Subs χ —ν—u—))))), where (as needed, about '—u—w—' and '—ν—u—'). Although some of these axioms look somewhat long and cumbrous, actually they are merely adaptations of well-known axioms for a certain formulation of set theory, essentially that of Zermelo with some additions.8 Also they are stated in such a way as actually to exhibit the relevant unit-sum name, and to incorporate a suitable instance of a logical principle of abstraction. In general, of course, the principle of abstraction, that (x)(x Subs (>>1 (Ez)(_y Ρ ζ — ζ — ) ) = —χ—). does not hold in this system, but suitable instances of it do. The additional clauses in some of the axioms stipulated such instances. Axil is the Principle of the Existence of the Null Common Nature, together with the relevant principle of abstraction. Axl2 is the Principle of the Pair, that there is a common nature covering any two entities. (Note that it is formulated in such a way as to incorporate its own principle of abstraction.) Axl3 is the Principle of the Sum Common Nature, that given any common nature, there is a common nature whose haecceities are just the haecceities of its haecceities. Ax 14 is the Power Principle, that given any common nature, there is a common nature consisting of all common natures included in it. Axl5 is the Axiom of Infinity, with an appropriate principle of abstraction. Axl6 is the Axiom of Choice, with its app opriate principle of abstraction. Axl 7 is the Axiom of Foundation and Axl8 is the Replacement Axiom. In addition to Axl—Axl8, of a logico-mathematical character, some purely empirical principles are needed together with some meaning postulates governing the primitive names. Thus we would have also that I—χ Subs y, for suitable primitive χ and y. Principles of this kind would stipulate that such and such objects are the 'Almost any alternative first-order axiomatization of set theory could have been used here equally well. It should be noted that Axil and Axl2 are provable from the others.

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haecceities of such and such common natures. Also we might have a meaning postulate as follows, where 'man', 'animal', and 'rational' are all primitives. I— (jc)(x Subs man = (x Subs rational · χ Subs animal)). A word or two concerning identity is in order. Note that the sign for it is defined in terms of Subs. This relation we might speak of as subsumptional identity. But there is also mereological identity, being mutually part of. Clearly now but not conversely.9 We might feel a little uncomfortable that the converse does not obtain, but this is part of the price we must pay for a moderate realism. We pay the unit sums overtime and make them do some of the work of abstract objects, with the result that alitile abstractness must creep into the notion of identity. So much now for the axiomatic framework of the object-language. The mathematical part of it, as observed, is essentially the Zermelo system grafted upon a Scotistic foundation. But note of course that no such things as sets occur as values for variables, the theory here being a moderate realism. It is thus a kind of set theory without sets. In place of sets there are the common natures, and in place of the relation of membership we have here the relation Subs. It is well known that on the basis of axioms such as Ax9-Axl8, huge portions of classical mathematics can be built up — not all, of course, the phrase 'all mathematics' having been a myth long before Gödel's proof of incompleteness. To provide the formal semantics for this system we may proceed essentially as in the preceding paper. A syntax of shapes is presupposed, with concatenation and shape-descriptive names. Let 'a Des χ' and 'α Den χ' express primitively that a designates χ and a denotes χ respectively. And let 'PrimNm a' be defined by enumeration to express that a is one of the primitive names. Let 'PropUnitSumNm a' be defined to express that a is a proper unit-sum name, i.e., an expression of the form '(xl—χ—)'. Similarly 'ComUnitSumNm a' expresses that a is a common unit-sum name, of the form '(jcl (Ey)(x Ρ y—y—))'. Every common unit-sum name is also a proper one, but not conversely. We then let 'Thus, for example, where a and b are given non-unit individuals, the entity ( x l (Ey) ( i P y · Unit.y·,yP(a υ b)))is mereologically identical w i t h ( x l ( E y ) ( x P y • (y=avy = b))), but not subsumptionally so.

Common Natures and Mathematical Scotism 'Nm a' abbreviate ComUnitSumNm a)'.

'(PrimNm

a

ν

PropUnitSumNam

147 a

ν

The rules of Designation and Denotation are now as follows. DesRl. I— (x)(j>)((a Des χ · a Des y) 3 χ = y). DesR2. I— a Des χ, where χ is a primitive or proper unit-sum name and a is its shape-description. DesR3. I—(a)(x)(a Des χ 3 (PrimNm a ν PropUnitSumNm a)). DenRl. \— (Ez)z Subs y => (x)(a Den χ = χ Subs y), where y is a primitive or common unit-sum name and a is its shape-description. DenRl. |—(a) (x) (a Den χ => ((PrimNm a · (Ey)(Ez)(a Des y· ζ Subs j ) ) ν ComUnitSumNm a)). DenR3. |— (a)(Nm a 3 (Ex) (a Des χ ν a Den χ)). DenR4. |— (α)(ComUnitSumNm a => (x)(a Des χ = {y)(a Den y = y Subs χ))). DesRl is the Principle of Uniqueness, and DesR2 specifies precisely what the primitive and proper unit-sum names designate. It is the Principle of Designation. DesR3 is the Limitation Principlefor Designation, that only primitive or proper unit-sum names designate. DenRl is the Principle of Denotation, specifying precisely what denotes what. DenR2 is the Limitation Principle for Denotation, that only primitive names (that designate something having an haecceity) and common unitsum names denote. DenR3 is the Limitation Principle for Names, that only expressions that designate or denote are names. And DenR4, the Principle of Common Unit-Sum Names, interrelates Des and Den by specifying that a common unit-sum name designates an χ just where it denotes just the haecceities of x. A full semantics for the theory is provided by these rules, including a theory of truth. If the foregoing considerations are acceptable, we have a genuine third alternative, in the philosophy of mathematics, between the platonistic and nominalistic extremes.10 Moderate, Scotistic realism provides this, with no sacrifice of any of classical mathematics. The full power of set theory is achieved in terms of the theory of common natures and subsumption. With a suitable choice of primitive names, a linguistic framework can be provided also for empirical science as well, at least for any area of empirical science that may be regarded as an applied set theory. In this way Peirce's conviction that Scotistic realism supplies "the philosophy which is best to harmonize with physical science" is shown at least to be a cogent one. 10

Cf. Whitehead's Categoreal Scheme and Other Papers, VI and VII.

XIII

SET THEORY AND ROYCE'S MODES OF ACTION

In his valuable but neglected contribution of Ruge's Encyclopedia of the Philosophical Sciences, already referred to in IX above, Josiah Royce called attention to "modes of action" as "logical entities." "Such objects . . . have never been regarded heretofore," he wrote,1 "as logical entities in the sense in which classes and propositions have been so regarded. But in fact our modes of action are subject to the same general laws to which propositions and classes are subject." Although he does not explicitly say so, these "laws" are taken to be the usual Boolean laws. For, given any mode of action such as singing, Royce notes, there is the contradictory mode of action of not singing. Similarly there are logical sums and products of modes of action, singing or dancing and singing and dancing. Likewise "between any two modes of action a certain dyadic, transitive and not totally non-symmetrical relation may obtain or not obtain. This relation may be expressed by the verb 'implies'." Thus singing and dancing implies singing. Finally there are the two special modes of action, which are contradictories of each other, of doing nothing and doing something. These of course correspond with the Boolean null and universal elements, implication with Boolean inclusion, and contradictories, logical sums, and products with the analogous Boolean operations. It is very likely true that Royce's comments here about the logic of modes of action are the most thorough and systematic up to his time. The point of the present paper is to carry forward some of his observations in the direction of modern set theory in terms of the notions of event logic. Straightaway we should make a distinction that Royce seems not to have made, namely, that between a mode or type of action and an action-event or -occurrence. To sing or to be singing is a mode, but only a particular singing, spatio-temporally localized with a particular

1

Royce's Logical Essays, p. 374.

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person as singer, etc., is a singing-event. Let ' Imp · Imp e · e ByAgent p)}\ Thus to say that e is to say that e is an action-event of person p's F-ing, so to speak, or performing an action of the mode . Royce considers a "calculus of modes of action" in which, in addition

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to the Boolean laws, the following principle of density holds: If and are distinct modes of action such that Imp , there is always a mode of action , distinct from and , such that Imp and Imp . In other words, between any two distinct modes of action there is another distinct from both. Royce does not illustrate this principle, however, and it seems to play a more theoretical role for him than seems warranted. What mode of action is there between singing and dancing and just singing? It is not clear just what Royce's answer would be. He states that "the question is not whether there actually lives any body who actually does all these things [or modes of action considered]. That, from the nature of the case, is impossible. The question is as to the definition of a precisely definable set of modes of action." Royce seems to wish to retain the principle for systematic reasons that appear somewhat dubious. He wishes the calculus of modes of action to exhibit some at least of the properties considered in his system Σ.2 Royce is interested in modes of action as a means of characterizing the mathematical notion of set. A set is the result somehow of a "voluntary act, since all classification involves a more or less arbitrary norm or principle of classification."3 And again, "every classification of real or of ideal objects is determined in any special instance, by a norm or principle of classification which we voluntarily choose. And in so far classifications are arbitrary, and may be said to be "creations" or "constructions"" 4 — "freie Schöpfungen des menschlichen Geistes," in the famous phrase of Dedekind. The notion of a set or class should, in accord with this conception, be generated out of acts of classification, regarded as our perhaps most fundamental intellectual activity. To think or speak about objects at all is to classify them, or to put them in relation with other objects, or to note that they are already classified or stand in relation with other objects, and so on. "An individual [p. 350] . . . may be classed with other individuals," and to do so is an "act of will." "If the various individuals in question are viewed as if they were already given, the act of classing them thus, that is[,] of asserting that these individuals belong in the same class, is again an act of the will. Its value is so far pragmatic. We accomplish in this way some purpose of our own, some purpose of treating things as for some special reason distinguished or, on the other hand, undistinguished." To be sure, we 2 Ibid., 3 Ibid., i

pp. 379-441.

p. 354. Ibid., p. 363.

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should distinguish an act of assertion from an act of classifying, even though both are to be handled in terms of pragmatics. Still, Royce is calling attention here to something both novel and important. "[_Á]ll classes," he continues, in persuasive italics, "are subjectively distinguished from other classes by the voluntarily selected Norms, or principles of classification which we use. Apart from some classifying will, our world contains no classes. Yet without classifications we can carry on no process of rational activity, can define no orderly realm whatever, real or ideal." Royce is not claiming here that classes are distinguished from other classes only "subjectively" by voluntary acts, for, once performed, these acts are as objective as one could wish for. And note also that the resulting classes are wholly determined, in an appropriate sense, by the corresponding acts. The "Norm" is something else again, and should embody the "reason" or "purpose," or something of the kind, for classifying in just that way. And the "classifying will" is wholly determined by the person who does the classifying. Apart from people performing acts of classifying, there are no classes at all, neither in a Platonic heaven nor in a subterranean limbo. This pragmatic way of viewing classes seems never to have been brought into accord with modern set theory. An interesting and very natural way of interpreting set theory would emerge, it would seem, if this could be done satisfactorily. Let us attempt now to see how such a harmonization can be brought about. The fundamental sentential form of pragmatics needed may be taken as (1)

'p Classify x\

that ρ classifies χ tenselessly, so to speak. (But of course tensed forms may readily be given.) More specifically, for ρ to bear the relation Classify (or Clsfy for short) to χ is for/? to classify x as one usually among others, ρ merely associates χ with the other objects (if any) he is simultaneously classifying. Then ', Clsfy,xW expresses that e is an action-event of p's classifying χ in this sense. Of course e may be a complex act of classifying y, z, and so on, also. In fact if the full complexity of e is spelled out, that is, if the various objects that are classified in the act e are enumerated or suitably described, the resulting class is wholly determined. Thus it is eminently natural to identify the resulting class with e itself. Classes are nothing but actionevents of classifying, on this view, nothing more, nothing less.

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An action-event e of classifying may itself in turn be suitably classified in another action-event e'. Thus we would have that, for the given person p, < p,C\úy,x>e-(p,

Clsfy,e>e'.

In this way by a kind of iteration sets of sets may be generated, of all manner of complexity. The question arises now as to the axioms characterizing 'Classify'. What principles, in other words, are to govern classificatory actions? Well, classifying nowadays is not the simple dichotomous division of antiquity, nor the mere distinction of genus and species, nor yet the elementary Boolean affair of a century back. Nothing less than the full sophisticated techniques of modern set theory, one might contend, will suffice if one wishes a theory of sufficient breadth and power to handle classificatory procedures in the well-developed sciences. For this purpose, then, let us give now a pragmatized formulation of the famous Zermelo-Frankel set theory, used in the preceding paper, in terms of acts of classification. The Zermelo-Frankel axioms, it will be recalled, comprise the Axiom of Extensionality, Axioms of Subset Formation, the Axiom of the Power Set, the Axiom of Infinity, the Axiom of Choice, the Axioms of Replacement, and the Axiom of Foundation. These seven axioms and axiom-schemata constitute one standard formulation of that system.5 All of them may be regarded as axioms concerning classificatory procedures on the part of the persons who formulate the system. Hence we may let person ρ now, in all seriousness, be merely (Zermelo υ Frankel), the compound individual consisting of those two distinguished mathematicians. 6 The relation of identity for the system is given by the following definition. 'e, = e 2 ' abbreviates '(e')(e' =3 ,Clsfy,i?2>*>')'· Thus e¡ and e2 are the same provided any act e' ofp' classifying e, is also an act of p's classifying e2. The Axiom of Extensionality is then that ( e i)(^2)((, e )(Clsfy,e)e, = e2)

= e2).

The Axioms of Subset Formation are to the effect that (e)(Ee')'*)«/>,Clsfy,*>e' == (< A Clsfy,jc>c· —x—)), 'See, for example, H. Wang, From Mathematics to Philosophy (Humanities Press, New York: 1974), p. 184, and recall Axll-Axl8 from the preceding paper. 'The ' u \ it will be recalled, is the sign from the calculus of individuals for the summation of persons. It is used ambiguously also for the summation of virtual classes.

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where '—χ—' is a suitable sentential form built up from atomic ones of the form 'e"' but not containing V or V as free variables. And so on for the other axioms. The Zermelo-Frankel system is merely one among many alternative systems of set theory, and the others could no doubt be formulated here equally well, including the Russell type-theory. One's preference for one or the other will rest for the most part on internal technical considerations. Once chosen, however, one presumably works only with the one system, at least within a given context. Royce calls his view 'Absolute Pragmatism'. The laws governing one's classificatory procedures, once chosen, are to be held fast and not tampered with. And even if we were to compare one system with another, the laws of each reign supreme within their context. In comparative set theory, in other words, each theory as such must be given a kind of absolute status. It was commented above that tensed forms of '/> Clsfy JC' may also be introduced, so that we can say that ρ classified χ at some time in the past, or even will classify χ in the future. Person ρ might have classified χ in one way in the past but do so now in another. In allowing tensed forms for Clsfy JC' we gain also a notation for constructivistic, or even intuitionistic, types of set theory. Person ρ might be said to have done just such and such acts of classification up to a certain point, just those needed, say, for a given proof of a theorem. In some renditions of intuitionism, it will be recalled, the creative subject is explicitly brought in as well as his acts of proof and the like.7 But even acts of proof are meta-acts of classification. A sentence may be classified as provable just where the creative subject has himself constructed a proof of it on the basis of sets he has already constructed. Thus, the tensed forms for lp Clsfy JC' may not be altogether devoid of interest for metamathematics and intuitionism. In allowing the set theorist the privilege of the axioms needed for a given theory, we are attributing to him vast intellectual powers of classification. But this is just what must be allowed him if set theory is to do its work providing a foundation for mathematics and the exact sciences. No horse-and-buggy kind of procedures here will suffice in this day of sophisticated methodologies. Royce observed that "every classification of real or of ideal objects is determined in any special instance, by a norm or principle of classification which we voluntarily choose." 8 And according to the O.E.D. to classify is "to arrange or distribute in classes according to a method or system." If we wish to specify the norm or method in a special case, 7

As in the work of Myhill, Kreisel, Troelstra, van Dalen, and others. 'Ibid., p. 363.

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we may do so in various ways, by bringing in additional clauses. Thus where ,Clsfy,x)e, we might wish to say that e is performed for such and such a purpose. Let the purpose be stated in a sentence a. Then 'e For Arpóse a' expresses that e is done for the purpose of bringing it about that a obtain. Or the norm might be, and usually is, because the object classified satisfies such and such a sentential form or function. Let 'e BecauseOfReason a' express that e has whatever a expresses as its reason. Let a express now, in a particular instance, that the object χ satisfies some sentential form b of just one variable. Here a is a metameta-linguistic sentence. Thus ',Clsfy,x>e· e BecauseOfReason a' expresses that χ is classified in e for the reason that it satisfies the given sentential form. And similarly for all manner of other norms. Thus in identifying classes here with complex acts of classifying, there is no neglect of the norms or purposes of such acts. (To be a little more precise, we should write here ',Clsfy,x}e · (Ec) (c Des χ · a = rc Sat è1 ) · e BecauseOfReaSon

.

where 'Des' and 'Sat' stand for relations of designation and satisfaction.) Acts of classifying are mental acts and set theory, on this present account, becomes a branch of the wider theory of such acts. The view is thus compatible with metaphysical idealism, if the individuals or Urelemente are also ultimately taken as manifestations of mind. This would not seem to hold of other renditions of set theory, for which an objective realm of sets as "abstract objects" sui generis is needed. No wonder this kind of a view arises naturally out of reflections upon Royce, combining as it does an exact logic with certain features of pragmatism within a context compatible with metaphysical idealism.

INDEX

Almeder, R. 142 Aristotle|45, 121 Art des Gegebenseins 17, 76ff, 81ff, 135

Harris, Ζ. 85 Hobbes, T. 22, 97,121, 131,137 Hungerland, I. C. 121

Bastian, R. 142 Berry, G. 23 binomial principie 29,42ff Boler, J. F. 142 Boole, G. l l f f Bradley, F. H. 98ff

identity 14ff, 27, 59,63, 124, 126, 143 interprétant 67 ff involution 28f,34ff,48ff

cauculus of individuals 125ff, 143ff Cantor, G. 101 Carnap, R. 72,74,122,132 Church, A. 122 Clarke, Β. I l l classification 150ff common names 22, 121 ff, 136ff connotation 69f converses 49ff, 55ff

Kant, 1.79, 112f Kempe, A. B.61;99f, 102 Kuklick, B. 112

Jakobson, R. 9 Jevons, W. S. 121f

Leibnitz, G. 45 Leonard, H. 17, 123 Lesniewski, St. 17, 92, 95,123, 143 Lewis, C. 1.45, 53

Davidson, D. 98 Dedekind, R. 21 f, 150 De Morgan, A. 13,25,44f, 46ff, 54, 58, 91, 134 denotation 73, 88, 122, 128f, 137f, 147 designation 68,70, 73, 88, 122, 128f, 137f, 147 Duns Scotus 18,101,136ff

McKeon, Ch. 142 Michael, Ε. 11 Mill, J. S. 97,121f Mitchell, Ο. H. 11,23,60 Moore, E. C. 142 negations 35ff, 47,49f, 55ff, 91 ff null class 15, 37ff null relation 37ff, 52, 59 numbers 12ff, 21f, 65f, 101

Ens necessarium llOff Frankel, A. 145,152f Frege, G. 17,22,24,45, 68, 76ff, 80ff, 91, 93, 122, 135 Gödel, Κ. 105,146 Goodman, Ν. 17,123 Goodwin, R. 142 Grajewski, M. 139

Ockham 97, 121, 125, 131 ordered pairs 58f Owens, J. 140 Péguy, Ch. 98 Principia Mathematica 26ff, 38,48, 54, 56, 60, 65 Power, W . L . I l i products 25ff, 30f, 52, 55ff

155

156

Index

Quine, W. V. 13, 22, 69, 85, 105, 122, 123 Quirk, R., and associates 135 relations 13ff, 25ff, 46ff, 54ff, 98ff, 134f relative products 26ff, 3If, 47ff, 56ff relative sums 56ff Riley, G. 99 Royce, J. 99f, 102,109, 112, 148ff Russell, B. 52, 60,70ff, 104, 108, 122,123, 153 Saint Thomas 100,116 Schröder, E. 61 signs 67ff Strawson, P. 123 subsumption 126ff, 138ff sums 15, 18f, 21, 25ff, 30f, 55ff, 137f

Tarski, A. 20,96,122,125 Thompson, M. 99 truth 87ff, 130f Turquette, A. 23 universal class 30, 39 universal relation 30, 37ff, 52, 59 Wang, H. 152 White, M. 5 , 9 Whitehead, Α. Ν. 112,115ff Wolter, Α. 139 Woodger, J. H. 123f, 125f, 131f, 137 Zermelo, E. 101, 145, 152f