Peirce & Triads

Peirce &TriadsAuthor(s): Arthur SkidmoreSource: Transactions of the Charles S. Peirce Society, Vol. 7, No. 1 (Winter, 1971), pp. 3-23Published by: Indiana University PressStable URL: http://www.jstor.org/stable/40319601 .

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Arthur Skidmore

Peirce & Triads

As every student of Peirce knows, he was very fond of making trichotomous analyses and classifications. Awareness of the odd effect this practice must have on his readers led him to write the following amusing apology:1

The author's response to the anticipated suspicion that he attaches a

superstitious or fanciful importance to the number three, and forces divisions to a Procrustean bed of trichotomy. I fully admit that there is a not uncommon craze for trichotomies. I do not know but that psychiatrists have provided a name for it. If not, they should. ... it might be called triadomany. I am not so afflicted; but I find myself obliged, for truth's sake, to make such a large number of trichotomies that I could not [but] wonder if my readers, especially those of them who are in the way of knowing how common the malady is, should suspect, or even opine, that I am a victim of it. (1.568)

By his doctrine of the categories I mean a family of assertions by Peirce, all to the effect that a given field of discourse can be divided by trichotomy into exactly three exhaustive and mutually exclusive classes. This doctrine, in its various formulations, is employed throughout Peirce's writings on virtually every philosophical subject discussed by him.

In one formulation, the doctrine of the categories asserts that there are exactly three distinct kinds of appearances, whatever may be said to be in any way present to the mind. In this case, Peirce calls the three kinds First ness, Secondness, and Thirdness, and he has referred to the doctrine of the categories in this formulation as his 'one contribution to philosophy' (5.469). He employs the doctrine in many



4 ARTHUR SKIDMORE

other fields as well, asserting for example that there are but three distinct kinds of characters, relations, and supposable objects, to name just three.

Sometimes Peirce seems to employ the doctrine of the categories merely as a suggestive rule of thumb, or in what I would like to call a regulative way. That is, sometimes Peirce merely tries to find a divi- sion by trichotomy with the expectation that one can be found.

There is no good reason to object to this regulative employment of the doctrine of the categories. In fact, the doctrine has in my opinion more than proved its suggestive worth. The evidence for the existence of the three kinds predicted by the doctrine is in such cases independ- ent of the doctrine, amounting to a separate body of evidence for each case.

However, it is indisputable that Peirce sometimes employs the doctrine in an unrestricted way. Sometimes Peirce asserts the trichotomy of a field antecedent to any other evidence.

It is this employment of the doctrine that I wish to criticize. That is, I do not believe that Peirce is justified in claiming a priori that every field of discourse can be divided by trichotomy into exactly three exhaustive and mutually exclusive classes.

Peirce himself thought that there was ample justification for his unrestricted employment of the doctrine of the categories. This con- fidence was afforded him by a claim of his which I shall call Peirce' s Thesis: that all indecomposable predicates are divisible into exactly three kinds.

It is extremely difficult to say precisely how Peirce's Thesis is connected with what we have been calling his doctrine of the categories. It might be supposed, for example, that Peirce's Thesis is itself an instance of the doctrine of the categories. That is, the three exhaustive and mutually exclusive classes in this case are of predicates.

But this would be I think a misleading way of looking at the matter. For Peirce's Thesis has been singled out by me for special treatment precisely because Peirce offers an argument for his claim that there are exactly three kinds of indecomposable predicates. He does not treat this claim regulatively. That is, he does not propose an empirical examination of the class of predicates to ascertain whether they fall into three kinds as predicted. Nor does he treat it as an unrestricted employment of his doctrine. On the contrary, he gives reasons why there must be just three kinds.

At least partly because he had such an argument for his Thesis,



Pe ir ce & Triads 5

Peirce subsequently took the Thesis as evidence for his doctrine of the categories in its various formulations. And when he employs the doctrine of the categories in an unrestricted way, his argument for the Thesis constitutes the sole evidence for the doctrine.

I am claiming here that Peirce always has in mind his Thesis whenever he employs the doctrine of the categories, in any of its formulations, and that he considers the Thesis to provide conclusive evidence for the doctrine. Peirce unmistakably asserts such a connection between his Thesis and one formulation of the doctrine of the categories, namely, that there are exactly three elementary meta- physical conceptions:

Kant taught that our fundamental conceptions are merely the ineluctable ideas of a system of logical forms; nor is any occult transcendentalism requisite to show that this is so, and must be so. Nature only appears intelligible so far as it appears rational, that is, so far as its processes are seen to be like processes of thought. ... It follows that if we find three distinct and irreducible forms of rhemata, the ideas of these should be the three elementary conceptions of metaphysics. (3.442)

Peirce of course does discover 'three distinct and irreducible forms of rhemata'; this is precisely Peirce's Thesis, since, as we shall later see, predicates are the same things as rhemata.

In another place Peirce, using an early version of the argument for the Thesis, uses 'concept' almost interchangeably with 'predicate':

[M]y researches into the logic of relatives have shown beyond all sane doubt that in one respect combinations of concepts ex- hibit a remarkable analogy with chemical combinations; every concept having a strict valency. . . . Thus the predicate "is blue" is univalent, the predicate "kills" is bivalent ... the predicate "gives" is trivalent, since A gives B to C, etc. Just as the valency of chemistry is an atomic character, so indecomposable concepts may be bivalent or trivalent. Indeed, definitions being scrupu- lously observed, it will be seen to be a truism to assert that no compound of univalent and bivalent concepts alone can be trivalent, although a compound of any concept with a trivalent concept can have at pleasure, a valency higher or lower by one than that of the former concept. Less obvious, yet demon- strable, is the fact that no indecomposable concept has a higher valency. (5.469)



6 ARTHUR SKIDMORE

Evidently, then, there is a close connection perceived by Peirce between the Thesis and the assertion that there are but three sorts of concepts.

It is easier to see how Peirce would apply the Thesis to show that there are but three kinds of characters and but three kinds of propositions, as is revealed by the following:

Any character or proposition either concerns one subject, two subjects, or a plurality of subjects. (3.359)

Propositions can clearly be divided according to the nature of their predicates. 'Character' is often used by Peirce in the sense of the signification of predicates. Consequently these too can be divided according to the nature of predicates, namely, those which express them.

The remainder of this paper aims to develop Peirce's argument for his Thesis and to show that this argument is invalid. That is, I wish to show that Peirce does not prove that there must be exactly three kinds of indecomposable predicates.

Several conceptions must first be developed, that of a subject, of hypostatic abstraction, of a predicate, and of the formal representation of propositions. We return to Peirce's conception of the composition of predicates and to his argument for his Thesis in V.

I

The subject of a proposition is that part of it which indicates the object of the proposition:

The interpretant of a proposition is its predicate; its object is the things denoted by its subject or subjects (including its gram- matical objects, direct and indirect, etc.). (5.473) The function of subjects is indexical. That is, in the last analysis they

indicate what they do by bringing the interpreter into a brute connection with their objects:

[A] symbol, in itself, is a mere dream; it does not show what it is talking about. It needs to be connected with its object. For that purpose, an index is indispensable. No other kind of sign will answer the purpose. (4.56)

An indexical word, such as a proper noun or demonstrative or selective pronoun, has force to draw the attention of the listener



Peirce & Triads 7

to some hecceity common to the experience of speaker and listener. (3.460)

Paradigm cases of subjects of propositions whose mode of meaning Peirce takes to be clearly indexical are proper names. Peirce even offers as a criterion for identifying the subjects of a proposition that of re- placing the symbol in question by a proper name:

Each part of a proposition which might be replaced by a proper name, and still leave the proposition a proposition is a subject of the proposition. (4.438)

The function of proper names is merely to indicate an object: The expressed subject of an ordinary proposition approaches most nearly to the nature of an index when it is a proper name which, although its connection with its object is purely intentional, yet has no reason (or, at least, none is thought of in using it) except the mere desirability of giving the familiar object a designation. (2.357)

Peirce notes that on further analysis the kinds of subjects of propositions in addition to subjects which are singular can be described simply as universal and particular2 quantifiers:

Every subject, when it is directly indicated, as humanity and mortality are, is singular. Otherwise, a precept, which may be called its quantifier, prescribes how it is to be chosen out of a collection, called its universe. . . . [I]n necessary logic . . . only two quantifiers are required; the universal quantifier, which allows any object, no matter what, to be chosen from the universe, and the particular quantifier, which prescribes that a suitable object must be chosen. (2.339)

II

The operation of hypostatic abstraction transforms a proposition into another proposition which has the following characteristics: (a) it is logically equivalent to the first, and (b) it has one more subject than the first. This additional subject produced by hypostatic abstraction will be abstract, that is, its object will be an abstraction, or an ens rationis.

Perhaps the easiest way to understand the process is to consider an example given by Peirce:



8 ARTHUR SKIDMORE

[H ]ypostatic abstraction, the abstraction which transforms 'it is light' into 'there is light here/ which is the sense which I shall commonly attach to the word abstraction ... is a very special mode of thought. It consists in taking a feature of a percept or percepts ... so as to take propositional form in a judgment (indeed, it may operate upon any judgment whatsoever), and in conceiving this fact to consist in the relation between the subject of that judgment and another subject, which has a mode of being that merely consists in the truth of propositions of which the corresponding concrete term is the predicate. Thus, we transform the proposition, 'honey is sweet,' into 'honey possesses sweetness.' Sweetness might be called a fictitious thing, in one sense. But since the mode of being attributed to it consists in no more than the fact that some things are sweet, and it is not pretended, or imagined, that it has any other mode of being, there is, after all, no fiction. (4.235) There are basically two different modes of hypostatic abstraction,

and consequently two different sorts of abstractions which result as objects of the new subjects produced. One mode proceeds typically by transforming the predicate 'is an X' into the predicate 'is a member of the collection of Xls.' So that 'George is a man' would be transformed into 'George is a member of the collection of men' or 'George is a member of mankind.' This last proposition has two subjects, 'George' and 'mankind.'

The other mode proceeds by transforming 'is an X' into 'possesses the character of being an X.' Thus 'George is a man' would become 'George possesses the character of being a man' or 'George possesses humanity,' with subjects 'George' and 'humanity.'

Although he is sparing with examples, Peirce holds that the operation can also be used to transform relative propositions. An example would presumably be the transformation of 'George loves Martha' into 'The relation of loving holds between George and Martha,' where the subjects of the second are 'George,' 'Martha,' and 'loving.'

Finally, we note that Peirce thinks that the operation can be per- formed upon all ordinary propositions, and not merely those for which there exists in the language an appropriate abstract noun. Conse- quently we should expect to find such abstract subjects upon the analysis of any ordinary proposition.

Summarizing, hypostatic abstraction is an operation upon any proposition which produces a logically equivalent proposition. This



Pe ir ce & Triads 9

new proposition contains an additional subject which is abstract, that is, which indicates an object whose being consists in the manner of being of something else. There are two broad classes of such abstract subjects produced by the operation: collections, as when 'John is happy' is transformed into 'John is a member of the class of happy things/ and attributes, as when 'John is happy' is transformed into 'John possesses happiness/

III

Peirce isolates the predicate of a proposition by striking out one or more subjects of the proposition:

In any proposition, i.e., any statement which must be true or false, let some parts be struck out so that the remnant is not a proposition, but is such that it becomes a proposition when each blank is filled by a proper name. The erasures are not to be made in a mechanical way, but with such modifications as may be necessary to preserve the partial sense of the fragment. Such a residue is predicate. (2.358)

That which remains of a Proposition after removal of its Subject is a Term (a Rhema) called its Predicate. (2.95) As may be seen from the passage last quoted, Peirce sometimes calls

predicates rhemes or rhemata. He states that there is only a subtle difference between a rheme and a predicate. 'Predicate' is a relative term, whereas 'rheme' is not:

Let a heavy dot or dash be used in place of a noun which has been erased from a proposition. A blank form of proposition produced by such erasures as can be filled, each with a proper name, to make a proposition again, is called a rhema, or, rela- tively to the proposition of which it is conceived to be a part, the predicate of that proposition. (4.438)

We may expect, then, to find the terms used practically interchangeably, though I will in the sequel only use 'predicate.'

Peirce remarks at several places (for example at 4.438 and 2.358) that what the predicate of a proposition is taken to be depends upon how we choose to analyze the proposition. He seems, though, to settle upon the usage that the predicate of a proposition results from striking out all those expressions regarded as subjects under a given analysis:



IO ARTHUR SKIDMORE

Taking any proposition whatever, as 'Every priest marries some woman to some man/

we notice that certain parts may be struck out so as to leave a blank form, in which, if the blanks are filled by proper names (of individuals known to exist), there will be a complete proposition (however silly and false). Such blank forms are, for example:

Every priest marries some woman to , marries to some man, marries to

The last of the above blank forms is distinguished by contain- ing no selective word such as some, every, any, or any expression equivalent in force to such a word. It may be called a Predicate. . . . (2.379) The predicate of a proposition is thus the result of erasing the subjects,

which are either singular, possibly abstract, a universal quantifier, or a particular quantifier.

IV The final preliminary to our discussion of Peirce's Thesis is a presentation of two of the methods devised by Peirce for the purpose of representing ordinary propositions. The first of these is adapted from his 1885 paper, "On the algebra of logic; a contribution to the philosophy of notation" (3.359-403), in which he presents a formal system re- markably similar to those modern systems roughly known as 'predicate logic' or the 'logic of quantifiers.' Peirce calls his system "general algebra of logic" (3.447). I shall refer to it in the sequel as his 'quantification theory.'

The second method for the representation of propositions is based upon the system called by Peirce his "existential graphs." Peirce's writings on existential graphs contained in the Collected Papers appear mainly in Book 11 of Volume iv. Roberts3 has provided an account of much more material by Peirce on the system of existential graphs not contained in the Collected Papers.

Since my aim is to emphasize the dissimilarities between the two



Pcirce & Triads n

ways of expressing propositions provided by the two systems, I shall take considerable liberties in presenting Peirce's quantification theory and his system of existential graphs. In particular, those features which are not essentially distinct I shall present as identical, though I shall indicate the places where such alterations have been made. The point of presenting the two systems is to show that Peirce had two distinct methods of representing the structure of what he called "relative propositions," that is, propositions whose predicates have more than one blank place.

Both the quantification theory and the existential graphs are pre- sented by Peirce in three segments, each of the last two being a further complication of the preceding segment. Peirce's reason for presenting his systems in this way is to enable him to show how much logical machinery is required for the analysis of reasoning of various degrees of complexity. This is indeed his reason for the subtitle of the 1885 paper (see 3-363-4)-

The first segment of both systems is a presentation of sentential, or truth-functional, logic, called 'Non-relative logic' in the quantification theory and 'Alpha part' in the existential graphs. Since these are not essentially different, I shall present them as the same, a variant of the Alpha part of the existential graphs as the nonrelative logic part of the quantification theory.

The second segment of both systems is a presentation of symbolism for expressing all propositions expressible in modern predicate logic. In this sense, the second segments may be viewed as equivalent. It is here, however, that the two systems become essentially dissimilar, and consequently where we shall pay closest attention.

The third segment of both systems is Peirce's ventures into the domain of higher-order logic, identity theory, set theory, and modal logic. We will not be concerned here with the third segments, since the second segment of each system is adequate for the representation of all propositions, though not, Peirce would say, for the representation of all valid inferences.

The Alpha part of the existential graphs is a two dimensional system, but can be adequately represented as linear. Propositions are ex-

pressed by writing out instances of them. We shall use letters where convenient as abbreviations of instances of propositions, as Peirce himself does in the nonrelative portion of the quantification theory.

The two logical operations on propositions are conjunction and



12 ARTHUR SKIDMORE

negation. Negation is expressed by enclosing the expression to be negated in a self-returning line. We shall employ pairs of brackets for this purpose. Conjunction is expressed by juxtaposition.

We give a few illustrative examples. If 'P' is taken as an abbreviation for 'Paul is a pauper' and 'R ' is taken as an abbreviation for 'Robert is a rascal/ then 'Paul is a pauper, but Robert is not a rascal' would be represented as

P[Q] and 'If Paul is a pauper, then Robert is a rascal' as

[P[QM The nonrelative logic of the quantification theory differs from the

above by using small letters instead of capital Roman letters, and in taking other truth-functional connectives as basic.

The second segment of the quantification theory is called by Peirce "First-intentional logic of relatives." It provides for a representation of propositions which reveals their subject-predicate structure, analyzed according to the procedures mentioned earlier. The additional symbolism required for this purpose contains letters for representing predicates, quantifiers, and relative pronouns. Peirce ordinarily uses letters of different types for representing predicates of different kinds. I shall employ numerical superscripts to distinguish the number of blank places had by the predicate. Peirce employs a capital pi for the universal quantifier, a capital sigma for the particular quantifier, and small letters written as subscripts to perform the function of the relative pronouns. I shall use 'U' for the universal and 'E' for the particular quantifier, and '*,' 'y,' and 'z for the pronouns.

A few examples: Suppose 'F ' represents ' is a woman' and

'G2' represents ' is a lover of ' Then 'Someone loves

someone' would be represented as

Ex Ey G 2xy and 'Some woman loves herself as

Ex Fx G2xx The second segment of the existential graphs is called the 'Beta

part.' It provides similarly for the separate expression of predicates, although these are frequently written out in full by Peirce, together with a device for indicating the number of empty places in the predicate, which we shall continue to indicate by superscripts. However,



Peine & Triads 13

the Beta part of the existential graphs does not contain quantifiers or letters for relative pronouns. Instead, and in order to perform the same function, it contains a symbol called by Peirce the line of identity.

The line of identity functions as if it were a particular quantifier together with its relative pronouns. No separate symbol is used or required by Peirce for the expression of the universal quantifier, since its function is accomplished by the use of the line of identity together with the operation of negation. (Peirce took both quantifiers as basic in his quantification theory because it enabled him to express all propositions in normal form, that is, with all the quantifiers occurring at the left.)

To illustrate, we shall give examples of propositions to be represented, then represent them in Peirce's quantification theory, then represent them in quantification theory without using the universal quantifier, and then represent them in the existential graphs:

'Something is a woman':

Ex Fx F

'Something is not a woman':

Ex[Fx] \-F] 'Someone loves everyone':

ExUy G 2xy Ex [Ey[G2xy] ] [ [ G2-\ ] An essentially more complicated case, according to Peirce, is pro-

vided by the following:

Let 'F' represent ' is a negro,' 'G' represent

' is an albino,' and 'H' represent

' is handsome.' Then 'Some negro albino is handsome':

Ex Fx GxHx F G H

A branching line of identity, such as the one in this example, is called by Peirce a line of ter-identity . This symbol will figure prominently in Peirce's argument for his Thesis.

V We are now in a position to consider in detail the assertion which I have called Peirce's Thesis, that there are exactly three distinct kinds



14 ARTHUR SKIDMORE

of indecomposable predicates. So far, predicates are distinguished merely according to the number of blank places they contain- a predicate with one blank place, as

' is handsome,' is monadic; a predicate with two blank places, as

' despises ,' is dyadic; and a predicate with three blank places is triadic, such as ' gives to '

By introducing the notion of the composition of predicates to form other predicates, Peirce thereby becomes capable of speaking of predicates which are indecomposable, that is, which cannot be analyzed as the result of the composition of other predicates. His claim is that there are just three kinds of these, those with one, two, or three blank places.

We begin by examining briefly Peirce' s conception of the composition of predicates. The composition of predicates takes place, according to Peirce, by identifying a subject of one with a subject of another (see 3.422). To explain what he means by this, it is easiest to give a few examples.

Peirce shows how the predicate ' is a grandparent of

' can be construed as the composition of the predicate '

is a parent of ' with itself. (1.294) To say 'A is the grandparent

of B ' is just to say 'A is a parent of something which is a parent of B.' A particular quantifier, according to the quantification theory, or a line of identity, according to the system of existential graphs, has been used to bind together two blank places.

Let 'F2' represent ' is a parent of

' Then ' is a parent of something which is a parent of

' would be represented in the two formal systems as follows:

ExF2xF2x F2 F2

Similarly the predicate ' is an aunt of ' could be

construed as the result of combining ' is a sister of '

with ' is a parent of ' Letting 'G2' represent

' is a sister of ,' ' is an aunt of ' would be represented as:

Ex G2x F2x G2 F2

A slightly different way of putting Peirce's Thesis is this- take any predicate you like, either it is indecomposable, in which case it contains either one, two, or three blank places, or it can be analysed as composed of such indecomposable predicates.



Peirce & Triads 15

Some parts of the Thesis seem unobjectionable. It does seem clear, for example, that no dyadic predicate can be construed as composed of monadic predicates alone. For all the blank places would be used up by the particular quantifier or by the line of identity.

Other parts of the Thesis which are harder to accept and which are of special importance are the following: (a) There are instances of indecomposable triadic predicates, and (b) Every predicate having four or more blank places can be analyzed as composed of predicates having at most three blank places.

We shall first consider claim (a), that not all triadic predicates can be analyzed as composed of dyadic or monadic predicates. Peirce argues for claim (a) in the following:

[W]e cannot build up the fact that A presents C to B by any aggregate of dual relations between A and B, B and C, and C and A. A may enrich B, B may receive C, and A may part with C, and yet A need not necessarily give C to B. For that, it would be necessary that these three dual relations should not only co- exist, but be welded into one fact. Thus we see that a triad cannot be analyzed into dyads. But now I will show by example that a four can be analyzed into threes. Take the quadruple fact that A sells C to B for the price D. This is a compound of two facts: first, that A makes with C a certain transaction, which we may name E; and second, that this transaction E is a sale of B for the price D. Each of these two facts is a triple fact, and their combination makes up [as] genuine [a] quadruple fact as can be found. The explanation of this striking difference is not far to seek. A dual relative term, such as "lover" or "servant," is a sort of blank form, where there are two places left blank. I mean that in building a sentence around "lover," as the principal word of the predicate, we are at liberty to make anything we please the subject, and then, besides that, anything we please the object of the action of loving. But a triple relative term such as "giver" has two correlates, and is thus a blank form with three places left blank. Consequently, we can take two of these triple relatives and fill up one blank place in each with the same letter, X, which has only the force of a pronoun or identifying index, and then the two taken together will form a whole having four blank places; and from that we can go on in a similar way to any higher number. But when we attempt to imitate this proceeding with dual relatives, and combine two of them by



1 6 ARTHUR SKIDMORE

means of an X, we find we have only two blank places in the combination, just as we had in either of the relatives taken by itself. (1.363)

Dyadic predicates thus can only combine to form dyadic predicates, since the use of the particular quantifier in linking two blank places of two dyadic predicates results in an expression with only two blank places, as in the examples of aunt and grandparent above.

However, [{three dyadic predicates were to be combined, one would arrive at an expression with three blank places. For example, suppose 'F2' represents the predicate

' is two feet from .' Then the triadic predicate

' lies on the same circle of radius two feet as and ' could be representd as:

Ex F2x F2x F2x F2 F2 F2

Thus there is no reason why plural predicates could not be composed out of dyadic predicates alone.

Further, Peirce's example of an indecomposable triadic predicate can be analyzed as composed of dual predicates by taking advantage of the operation of hypostatic abstraction, as follows. Let 'F2' represent ' is a giving by ,' 'G 2' represent

' is a giving of ,' and 'H2' represent

' is a giving to '

Then ' gives to ' could be expressed by: Ex F2x G2x H2x ^F^r-G^-H2

A gives C to B if and only if, for some D, D is a giving by A and D is a giving of C and D is a giving to B.

Peirce became aware by 1892 that triadic predicates were so analyzable as compounds of dyadic predicates. Yet he rejects such a refutation of his Thesis in the following way:

My position has been modified by the study of Mr. Kempe's analysis. For, having a perfect algebra for dual relations, by which, for instance, I could express that "A is at once lover of B and servant of C," I declared that this was inadequate for the expression of plural relations; since to say that A gives B to C is to say more than that A gives something to C, and gives to somebody B, which is given to C by somebody. But Mr. Kempe (par. 330) virtually shows that my algebra is perfectly adequate



Peirce & Triads 17

to expressing that A gives B to C; since I can express each of the following relations:

In a certain act, D, something is given by A; In the act, D, something is given to C; In the act, D, to somebody is given B.

This is accomplished by adding to the universe of concrete things the abstraction "this action." But I remark that the di- agram fails to afford any formal representation of the manner in which this abstract idea is derived from the concrete ideas. (3424)

Before considering this reply by Peirce, it will be useful to consider briefly part (b) of Peirce's Thesis, that all plural predicates can be composed out of triadic predicates. Peirce notes that triadic predicates can be combined in such a way that the number of blank places increases without limit. But there might well be plural predicates which do not occur anywhere in such a series of predicates. Conse- quently Peirce is concerned to show that they are in fact so composed:

[E]very tetradic relation, or fact about four objects can be analyzed into a compound of triadic relations. This can be shown by an example. Suppose a seller, S, sells a thing, T, to a buyer, B, for a sum of money, M. This sale is a tetradic relation. But if we define precisely what it consists in, we shall find it to be a compound of six triadic relations, as follows:

1) S is the subject of a certain receipt of money, R, in return for the performance of a certain act As;

2) This performance of the act As effects a certain delivery, D, according to a certain contract, or agreement, C;

3) B is the subject of a certain acquisition of good, G, in return for the performance of a certain act, At,;

4) This performance of the act Ab effects a certain payment, P, according to the aforesaid contract C;

5) The delivery, D, renders T the object of the acquisition of good G;

6) The payment, P, renders M the object of the receipt of money, R. (7-537)



1 8 ARTHUR SKIDMORE

The method here is similar to that described above in 1.363. The important thing is to see that in both of Peirce's 'reductions' abstraction is employed.

We now return to Peirce's reply to the proposed reduction of giving. Notice that Peirce objects on the grounds that no representation has been given in the reduction of the operation of hypostatic abstraction. But if Peirce objects to the employment of hypostatic abstraction in reducing triadic predicates to dyadic ones, then his objection applies with equal force to his own reduction of tetradic predicates to triadic ones. We thus see that Peirce has given no reason to prevent the combination of plural predicates out of dyadic predicates which would not equally prevent the combination of plural predicates out of triadic ones.

Recapitulating the argument so far, Peirce has argued for part (a) of his Thesis by insisting that triadic predicates cannot be composed of dyadic ones, since only two blank places remain after two subjects have been identified. We have seen that this argument is invalid, since he had restricted his attention to combinations of just two dyadic predicates. If three dyadic predicates are combined, then three blank places may remain. Further, we indicated a general method for employing Peirce's operation of hypostatic abstraction to show how any triadic predicate can be composed of three dyadic ones. We then saw that Peirce's objection to this rejection of part (a) of his Thesis requires that he give up part (b), the claim that all plural predicates are analyzable as compounds of triadic predicates. He must therefore give up either part (a) or part (b).

Peirce attempts to avoid this dilemma by offering another criticism of the view that triadic predicates can be composed of dyadic ones. This criticism in effect maintains that quantification theory itself is misleading with regard to the combination of predicates, and that the system of existential graphs rectifies this essential error of quantification theory. Consider again the representation of our proposed reduction of the predicate

' gives to ' Notice that the representation in the system of existential graphs requires a branching line of identity. A branching line of identity is itself taken by Peirce to represent a triadic relation. Thus a triadic relation cannot be reduced in the way we supposed without the use of a relation itself triadic. Perhaps Peirce's clearest statement of this point is as follows:

The other premiss of the argument that genuine triadic relations



Peirce & Triads 19

can never be built of dyadic relations and of qualities is easily shown. In existential graphs, a spot with one tail X represents a quality, a spot with two tails R a dyadic relation. Joining the ends of two tails is also a dyadic relation. But you can never by such joining make a graph with three tails. You may think that a node connecting three lines of identity is not a triadic idea. But analysis will show that it is so. (1-346) Peirce 's defense of his Thesis here amounts to showing that the

system of existential graphs is fundamentally superior to the system of quantification theory; otherwise its only advantage might be construed as appearing to preserve Peirce's Thesis.

We have remarked in Part IV that the chief difference between the quantification theory and the existential graphs, as they have so far been exposed, lies in the use of lines of identity versus quantifiers and small letters. As a matter of fact, however, a system of quantifiers and letters is introduced into the system of existential graphs, although Peirce takes pains to insist that they are to be construed as mere abbreviations:

In such a case [when lines of identity cross], and indeed in any case in which the lines of identity become too intricate to be perspicuous, it is advantageous to replace some of them by signs of a sort that in this system are called selectives. (4.460) Peirce argues at some length why the selectives are undesirable.

These arguments are in effect demonstrations of the superiority of un- abbreviated graphs over quantification theory, since this is essentially the difference between the two theories:

The respect in which selectives violate the general idea of the system is this: the outermost occurrence of each selective has a different significative force from every other occurrence- a grave fault, if it be avoidable, in any system of regular and exact representation. . . . When lines of identity are used to the exclu- sion of selectives, no such inconvenience can occur. . . . (4-473)

Peirce sums up his objections to the use of selectives (quantifiers) in a highly complex footnote (4.561^1), in which he argues for the superiority of lines of identity over selectives on grounds of simplicity, iconicity, and analyticity.

But whether the combination of predicates be represented in a



20 ARTHUR SKIDMORE

system which represents the combination by means of a triadic or a dyadic symbolism is, after all, irrelevant. For this can have nothing to do with whether the predicates being combined or resulting from a combination are triadic or not. The criterion for the type of predicate one has is solely concerned with the number of blank places the predicate contains. If the manner of composition of the predicate also be taken into account, then another criterion for the classification of predicates is being surreptitiously employed. We shall shortly consider the desirability of such a new criterion.

Peirce remarks in various places that the composition of predicates is itself irreducibly triadic (see 1.363, 6.32if). But this point, even if true, would have nothing to do with whether the resulting predicate were triadic or not, since this is a matter to be decided by counting the number of its blank places.

We may, however, revise the conception of how predicates are to be classified, as suggested in the last discussion, and take into consid- eration the nature of the subjects which are to fill the blank places: If the blank place of a predicate is to be filled by the name of an abstraction, then the character of the predicate in which that abstraction appears as a 'transitive element' should be taken as affecting the character of the given predicate. Accordingly,

' is a giving by ' would be construed as at least triadic in nature, since the first blank place is filled by the name of an abstraction derived from the predicate

* gives to ' Consequently, we will

not be said to have reduced the triadic predicate ' gives

to ' to a combination of dyadic predicates, since each of the dyadic predicates is seen to be triadic in nature.

However, now Peirce's proposed reduction of higher predicates to triadic predicates, insofar as it depends upon hypostatic abstraction, would similarly fail. Notice, for example his introduction of the abstractions As and A& in the example given above of the reduction of ' sells to for the sum ' The abstraction As results from hypostatic abstraction on the predicate 'S is the subject of a certain receipt of money, R, in return for a delivery, D, according to a contract, C The abstraction A^ results from hypostatic abstraction on the predicate 'B is the subject of a certain acquisition of good, G, in return for a payment, P, according to a contract, C Consequently, each of predicates (1), (2), (3), and (4) of the example would be at least tetradic in nature.



Peirce & Triads 21

There may well be innumerable examples of tetradic predicates which are mere complications of triadic predicates, just as there may be innumerable examples of triadic predicates which are mere complications of dyadic ones, as in our case of * lies on the same circle of radius two feet as and .' However, those plural predicates which can only be construed as complications of lesser predicates by aid of hypostatic abstraction will not now be construed as reduced. And the presumption is -that there are many such predicates, since Peirce only analyzes plural predicates with the aid of hypostatic abstraction.

Thus if we modify the principle of the classification of predicates in order to take account of the nature of their subjects, part (a) of Peirce's Thesis may be preserved, but at the cost of giving up part (b).

As a final criticism of Peirce's Thesis, I would like to point out that there is a sense in which predicates cannot be composed of others at all. For in order to find the type of a predicate, we should, strictly speaking, remove all the subjects which can be removed, as we saw in Part 111. The predicate involved, for example, in ' is a parent of something which is a parent of

' is thus not a dyadic predicate, but a tetradic one: ' is a parent of and is a parent of

' This is so since 'something which' is a subject which can be removed. In this strict sense, then, the result of combining two predicates will always have as many blank places as the sum of the blank places of the predicates combined. Consequently none of the examples of reductions of predicates are what they claim to be.

Three possibilities for Peirce's Thesis emerge from the preceding discussion, depending upon (a) whether we ignore Peirce's stated criterion for calculating the type of a predicate, and (b) whether we take the nature of subjects into account in classifying predicates, and they are all bad:

(a) If we take seriously Peirce's stated criterion for calculating the

type of a predicate strictly according to its blank places, then we find that Peirce has failed to offer a meaningful account of composition in terms of which indecomposability can even be defined. For then none of the examples of the composition of predicates which we con- sidered have fewer blank places than the total number of blank places of the predicates combined. Hence, for example, no tetradic predicate could possibly be composed of two triadic predicates, since two



22 ARTHUR SKIDMORE

triadic predicates combined would always yield a predicate with six blank places.

(b, i) If we do not include the nature of the subjects in classifying predicates, then Peirce has given no convincing reason for rejecting the analysis of

' gives to ' in terms of dyadic predicates. Indeed, all triadic predicates can be reduced to dyadic predicates by the general application of hypostatic abstraction given by Peirce. In this case, Peirce' s Thesis fails since it would appear that there are only two kinds of indecomposable predicates, monadic and dyadic ones.

(b, ii) If the nature of the subjects is taken into account, then Peirce has not shown that tetradic predicates can in general be reduced to triadic predicates. In this case nothing prevents the number of kinds of predicates from being indefinitely large.

VI

Peirce originally erred, perhaps because of a false analogy with chemical valencies (see, for example, 3.470), in thinking that he could prove that triads were irreducible to dyads. He also felt that all higher relations could be built up out of triads because of their fecundity in pro- ducing polyads. These two views together, he thought, guaranteed his Thesis.

Here is an instance of another suggestive analogy employed by Peirce:

Now, no combination of roads without forks can have more than two termini; but any number of termini can be connected by roads which nowhere have a knot of more than three ways. (1-371) The mistake here lies in ignoring the possibility that three termini

of roads without forks could coincide, as may be indicated by connecting the three roads in the following at X:

X

None of these roads has a fork, but their combination has three termini.

Peirce was never able to overcome this error. For as we have seen, his attempts to prevent the construction of triads out of dyads either



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fail to do so or else require him to give up the construction of polyads out of triads.

The University of Kansas

Notes

1. All references are to the Collected Papers of Charles Sanders Peirce. Citations are made in the customary way, by volume and paragraph. Thus, '1.568' refers to Volume 1, paragraph 568.

2. 'Particular' is used here instead of the more usual 'existential' not merely because it is more faithful to Peirce's usage, but also because it is a less misleading term.

3. Don Davis Roberts, The Existential Graphs of Charles S. Peirce (unpublished doctoral dissertation, Urbana, Illinois).



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