Are Predicates and Relational Expressions Incomplete?

Philosophical Review

Are Predicates and Relational Expressions Incomplete?Author(s): Peter LongSource: The Philosophical Review, Vol. 78, No. 1 (Jan., 1969), pp. 90-98Published by: Duke University Press on behalf of Philosophical ReviewStable URL: http://www.jstor.org/stable/2183814 .

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ARE PREDICATES AND RELATIONAL EXPRESSIONS INCOMPLETE?

WE MAY represent how things are in the world by constructing a model in which relations between its elements represent

how the corresponding things are related. The elements of the model stand for the things whose relations to one another are being repre- sented, and that these elements "are related to one another in a determinate way represents that things are so related to one another" (Tractatus, 2. 15). This, however, as Wittgenstein recognizes, does not seem to hold of a proposition, if only because propositions, even when they do contain proper names, contain besides expressions with a different role expressions which contribute in diverse ways to the sense of a proposition. So, having asserted (at 4.01) that a proposition is a picture of reality, Wittgenstein says: "At first sight a proposition-as it is set out on the printed page, for example-does not seem to be a picture of the reality of which it treats" (4.01 I). But then he seems to make an exception in the case of relational propositions, for in the next section we read: "It is obvious that we perceive a proposition of the form 'aRb' as a picture. In this case the sign is obviously a likeness of what is signified" (4.012). We may be puzzled by this remark, for a proposition of the form "aRb" is not like a proposition written in hieroglyphic script: it does not "depict the fact it describes" (4.016). An example of such a "picture-proposition" would be one representing the fact that one thing is next to another by putting a sign for the first thing next to a sign for the second. What, then, is Wittgenstein's point? His point is, I think, that we perceive a proposition of the form "aRb" in the same kind of way as we should the corresponding picture-proposition. Both are likenesses of what is signified. (But there are degrees of likeness.)

Here we may be inclined to object that such a proposition cannot be a likeness of what is signified because it contains a relational expression. And to the question "but why is this a reason for denying that such a proposition is a likeness of what is signified ?" we may think the reply is obviously "because you are expressing a relation by means of a sign for a relation and not by means of a relation between signs." But now the question arises: is it possible to contrast propositions expressing a relation by means of a sign for a relation with propositions expressing a relation by means of a relation between signs?

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ARE PREDICATES INCOMPLETE?

Miss Anscombe, in her book on the Tractatus, draws a contrast between our ordinary relational propositions and propositions that express a relation by means of a relation between signs. She concedes that we could express relations with what she calls a "material content" by means of a relation between signs, but she goes on: "but if you were quite generally to express relations between things by relations between their signs, then you would need to have as many different relations between signs as we in practice have words to express relations."' Her point, of course, is that we could express that a man A is next to a man B by writing, say, the sign "A" next to the sign "B" or, again, that we could express that A is bigger than B by leaving out "bigger than" and making the sign "A" bigger than the sign "B," and so on; but that the number of signs for relations is greater than the number of discriminable relations between signs. In practice it would not be possible to discriminate, and operate with, as many relations between signs as we have signs for relations.

In effecting this contrast, Miss Anscombe does not see herself as in disagreement with Wittgenstein. She allows that in the elementary propositions the way the names are combined is what expresses. After all, they consist only of names, so how these are combined must be expressive. But why should not the way the names are combined or related be what expresses in our ordinary relational propositions? Indeed, is not this the very point Wittgenstein is making at 3.1432? But he must have seen the difference between "AB" and "A is next to B," so, however 3.1432 is to be interpreted, he cannot have wished to assimilate propositions that differ in the way these do! But how do these two propositions differ? The first does not contain a sign for a relation, whereas the second does, but is this difference a difference between a proposition that does, and a proposition that does not, express a relation by means of a relation between signs?

That Wittgenstein sees that it is not is one of the insights of the Tractatus. We have as many relations between signs as we have signs for relations just because each relational sign sets up a relation between signs, which is what expresses in the proposition. This is surely the point of the much-quoted passage at 3.1432: "We should not say: 'The complex sign "aRb" says that a stands to b in the relation R' but: 'That "a" stands in a certain relation to "b" says that aRb.' " If we wish to understand this passage aright, we should look carefully at the preceding remark on the essence of a propositional sign. There

1 An Introduction to Wittgenstein's Tractatus (London, 1959), p. 101.

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we read: "The essence of a propositional sign is very clearly seen if we imagine one composed of three-dimensional objects (such as tables, chairs, and books) instead of written signs" (3.143I). Now the essence of a propositional sign is very clearly seen in such a case, precisely because none of the constituent signs are signs for relations. Each chair, book, and table stands for what is a possible term of a relation, and that these objects stand in a certain relation to one another is what expresses the sense of the proposition. And this is how it is with "A is next to B": that "A" stands in a certain relation to "B" is what says that A is next to B. We have to remember that "What is essential in a proposition is what is common to all propositions that can express the same sense" (3.341). And what is common to "A is next to B," "AB," and a model in which a table (standing for A) is next to a chair (standing for B) is that a relation is expressed by means of a relation. Of course, the expressive relation in "A is next to B" is not one for which we have a name: it can only be described. Confining ourselves to the written script, it is the relation which "A" has to "B" through "A" being written to the left, and "B" to the right, of the sign "next to." It is this indirect, asymmetrical relation in which "A" stands to "B" which says that A is next to B. So the expressive relation in "A is next to B" is different from the expressive relation in "A is bigger than B." The two relations differ in that "A" is related to "B" through being related to different written characters: the two sign-relations are characterized by different signs.

We misunderstand 3.1432 badly if we take its point to be one about our way of expressing the direction of a relation: namely, that it is the fact that "a" is written to the left of "b" in "aRb" which expresses the direction of the relation, so that if we were to reverse the order of "a" and "b," putting "b" to the left of "a," we should, if the relation were not symmetrical, express a different sense. As the preceding remark should indicate, Wittgenstein is not here concerned with the "accidental features" of our notation, but with its "essential features." Now it is not essential that the direction of a relation should be expressed by a relation between signs, but it is essential that a relation should be expressed by a relation between signs. Of course, in describing, in a given case, the particular relation that is expressive, we cannot but mention accidental features; for these features just "are those which result from the particular way in which the propositional sign is produced" (3.34). And the particular way in which we produce propositional signs that express relations is by setting up sign-relations that are indirect or sign-mediated, in which the direction of the

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relation is expressed by a relation between the signs for the terms. If, therefore, "A is next to B" does not depict the fact it describes, the reason is merely that the symbolizing relation is not the relation you set up by writing "A" next to "B," but such an indirect relation as we have described. The sign-what symbolizes-is a likeness of what is signified.

In fact, the point made at 3.I432 is not to be confined to relational propositions. Perhaps it can be brought out more clearly with such propositions, but it applies no less to subject-predicate ones. We can say the same thing, mutatis mutandis, about, say, "Socrates is bald" as we have said about "A is next to B." Thus we should not say that the complex sign "Socrates is bald" says that Socrates has the property of being bald. We should say instead "that the name 'Socrates' has a certain property says that Socrates is bald." The expressive property here is the relational property that "Socrates" has through being written to the left of the sign "bald." Here likewise it would be an illusion to think that one could contrast expressing a property by means of a predicate, a sign for a property, with expressing a property by means of a property of a sign.

Could one not imagine that a people should discover our way of expressing relations? Suppose a language is spoken in which there are no signs for relations, but in which certain spatial relations are expressed by relating signs directly. For example, one writes "AB" for "A is next to B," "A B" for "A is near to B," "AB '" for "A is to the left of B," and so on. As we may imagine, misunderstandings between those who use this notation are not uncommon, since some of the sign-relations they use are not readily distinguished from one another. Thus "A B" is sometimes confused with "AB" and "AB" is sometimes confused with their sign for "A is above B," which they produce by writing "A" over the top of "B." It now suddenly occurs to one of them that all their difficulties can be removed if they simply introduce into their script a number of arbitrary shapes which are easy to reproduce and distinguish, and use these to set up indirect relations between their signs. In this way, they can avoid the ambi- guities to which the old notation gave rise. For the expressive sign- relations in the new notation will be as easily distinguished as the shapes which distinguish them! Such written shapes we shall call signs for relations: their role in this language is no different from the role of relational signs in ours.

I shall call relational signs the indexes of sign-relations, since it is they that indicate which relation is expressed, symbolized, by the

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PETER LONG

sign-relations they characterize. And I shall call predicates the indexes of sign-properties, since it is they that indicate which property is expressed, symbolized, by the sign-properties they characterize. Thus, "next to" indicates that the sign-relation it characterizes is the symbol for the relation next to, and "bald" indicates that the sign-property it characterizes is the symbol for the property bald. Here we may compare Wittgenstein's use of "index" at 5.02, where he is contrasting the arguments of functions with the indexes of names. He says there that in the name "Julius Caesar," "Julius" is an index: it "indicates that the sign as a whole" is the name of a certain man, the Caesar of the Julian gens. But the use of this sign is "the result of arbitrary convention" and "we could choose a simple sign" to do the same work. In the same way, the use of the sign-relation which "next to" characterizes is the result of arbitrary convention, and we could choose a direct sign-relation to do the same work. We could write "AB" instead of "A is next to B." Of course, there is this difference between "Julius" and "next to": whereas the first is an index of a proper name, an element of language, the second is not. The symbol of which "next to" is an index is not a sign that can be uttered: it is a relation between such signs.

It is an illusion, then, to think that one can contrast propositions expressing a relation by means of a relational sign with propositions expressing a relation by means of a sign-relation. This illusion finds expression in a wrong understanding of what it is for an expression to stand for, be a sign for, a relation. We readily think of the difference between the roles of "A" and "next to" in "A is next to B" as if it were a difference, not in how they symbolize, or how they contribute to what symbolizes, but in what they stand for, and this makes it difficult to see how the proposition is a complex sign and not merely a complex of signs. We contrast what "next to" stands for with what "A" stands for and say such things as "Whereas 'A' stands for a particular, 'next to' stands for a relation, which is something of a radically different kind." But there can be no question of a sense for "what 'next to' stands for" in which we can draw a contrast between what "next to" and "A" stands for. To draw such a contrast is to repu- diate the indexical role of "next to" and treat this sign, though only an expression within the proposition, as if it had of itself the role of the sign-relation of which it is the index; as if relations could, so to speak, have proper names, and "A is next to B," as containing such a name, be of a form logically incommensurable with that of "AB"! But if we call "next to" a proper name, it is a proper name with a difference.

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For it is only when combined with other signs of suitable type that it stands for a relation. We cannot name a relation by writing down "next to" alone, but must write down such a proposition as "A is next to B" or "There are men x andy such that x is next toy." But if the expression can have the role of the sign-relation in "AB" only when thus combined with other signs, how can it have the role of the sign-relation ?

Here we are attacking only a "false" conception of what it is for an expression, an element of language, to stand for, designate, a relation. We are not saying that it is wrong to speak of an expression as standing for, designating, a relation. To say that "next to" stands for, is a sign for, a relation is to say no more than that it is a relational sign, and we are not denying that the sign or expression "next to" is a relational one. But it is only as an index of a sign-relation that it is a sign for a relation. If we call "symbols" only those signs that are not indexes, we can say: No expression can be the symbol for a relation. What expresses, symbolizes, a relation is not an expression.

Frege, as is well known, held that the expressions for properties and relations, predicates and relational expressions, are functional expressions and, as such, are incomplete. In this they are to be contrasted with proper names, which are complete. Now Ramsey found it difficult to understand how any one kind of expression could be called incomplete. If we call predicates and relational expressions incomplete, there is no reason, it seemed to him, why we should not call proper names incomplete also, since expressions of either type have to be combined with other expressions of suitable type in order to form a proposition. As we complete "(is) bald" by combining it with "Socrates," so we complete "Socrates" by combining it with "(is) bald." Ramsey, however, overlooks the fact that predicates and relational expressions are indexes, and so fails to see that there is a sense, fundamental to logic, in which such expressions, as against proper names, may be termed incomplete. For as indexes of sign- properties and sign-relations they have a signifying role within a proposition or clause only through characterizing such non-verbal symbols. And, since these symbols are set up only through expressing the appropriate form of proposition or clause, we have a reason for writing "the expression 'bald' " as "the expression 'X (is) bald' " and "the expression 'next to' " as "the expression 'X (is) next to r.' " This does seem to explain why such expressions should be thought to involve the form of a proposition, in a sense in which proper names do not.

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We have, then, a reason for saying that in "Socrates is bald" the name "Socrates" completes the expression "(is) bald." That is, we have a reason for saying not merely that the name attaches to, is combined with, the predicate, but that it completes it. Now if we think, with Ramsey, that there is no logical reason why we should not also say that the predicate completes the name, we should ask ourselves what we should say of a proposition containing only names. What should we say of "Socrates Plato," where this is written in a notation in which putting one name next to another means that what the first name stands for is next to what the second name stands for? Is "Socrates" here completed by "Plato"? Either we say that it is, in which case "completes" is merely being used in the sense of "is combined with," or we deny that it here makes sense to speak of the constituents completing one another. But why, then, does it make sense in the case of "Socrates is bald" but not here? To this question, the only answer one could give is that whereas it would be absurd to suggest that a notation is possible in which, although there are names and predicates, a proposition does not result from attaching a predicate to a name, there is no absurdity in the idea of a notation in which, although there are names, a proposition does not result from attaching a name to a name. Indeed, our notation is one such! But what is the source of this asymmetry? It is this: if in the formation "aP," in which "a" is a name, "P" has the role of a predicate, then "P" must serve as an index of a property of "a," where the fact that "a" has the property says that aP. So the asymmetry we marked between attaching a predicate to a name and a name to a name derives from the indexical role of a predicate, and so from the sense in which a predicate, as against a name, is incomplete.

So the expression for a property or relation is incomplete because, to put it paradoxically, it does not itself express a property or relation but only serves as an index of what does. It does not itself symbolize but only contributes to what does symbolize. To say this, however, is not to express agreement with Frege. For although he contrasts names with predicates, calling the first complete and the second incomplete, he does not see the difference between their roles as rooted in the alien roles of names and indexes. He sees it rather as deriving from a difference in the type of entity they stand for: whereas the Bedeutung of a proper name is an object, which is something complete, the Bedeutung of a predicate is a concept (property), which is something incomplete. But if predicates are indexes, such a contrast must be meaningless. The role of a name can be contrasted with that

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of an index, but the very nature of this contrast undermines the possibility of the Fregean contrast.

It is at this point, it seems to me, that we find an inconsistency in Frege's thought. He cannot hold both that predicates "stand for something," "have Bedeutung," in the sense in which (genuine) proper names do, and hold that predicates, as against names, are incomplete. For the incompleteness of predicates, which he sees as somehow deriving from an incompleteness in what they stand for, is merely a special case of the incompleteness of indexes, and no expression with the role of an index can "stand for something" in the sense in which proper names do.

This inconsistency finds expression in the following difficulty, which is essentially the difficulty Frege epitomized by the paradoxical statement that the concept horse is not a concept. If "next to" in "Socrates is next to Plato" stands for the relation next to, then we must be able to replace "next to" by the expression "the relation next to" and still have a proposition. But this we cannot do: since it is not an incomplete expression, "the relation next to" does not combine with the two names to form a proposition. It seems therefore that we must deny that "next to" does stand for the relation next to, so that in using the expression "the relation next to" we are not mentioning a relation. For we are mentioning a relation when we use the expression "next to"-only that is something we cannot really say!

Here we go astray from the first step. Of course "next to" stands for the relation next to, but it is a mistake to conceive of this "standing for" on the model of a proper name's standing for a man, or a horse, or whatever it may be. That "next to" stands for the relation next to means only that it is an index of a (nonverbal) symbol for the relation next to, and the truth of this does not require that we should be able to replace "next to" by "the relation next to" in "Socrates is next to Plato" and still have a proposition.

Nevertheless (it may be argued), there remains a difficulty over the nounlike phrase "the relation next to." It does not stand for a relation because "Socrates the relation next to Plato" is no proposition. But if it does not stand for a relation then even when we say that "next to" is an index of a symbol for the relation next to we shall not really be saying what we mean.

There is a confusion in this argument. It begins by denying that "the relation next to" stands for a relation on (what is in effect) the ground that this expression is not a relational one. And so far we can go along with it: that this expression is not an index of a symbol for

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a relation is a reason for denying that it stands for a relation. But this sense of "stands for a relation" will not fit what follows. The implication of what follows is that we shall not be saying what we mean when we say that "next to" is an index of a symbol for the relation next to-shall not be saying that "next to" is a symbol for a relation !-unless the expression "the relation next to" is an index of a symbol, which it is not. This expression does not contain the index "next to," which is why these words, as they occur within it, are italicized. But then no expression can contain such an index. This, however, does not mean that we fail to say what we mean when we say that "next to" is an index of a symbol for the relation next to. We have to recognize the peculiarity of expressions which, although indexes, are not the indexes of further expressions. We say what ''next to'' is an index of by using the noun-like phrase "the expression next to," but this peculiarity means that such a phrase is a makeshift and can be got round. So, instead of saying that "next to" is an index of a symbol for the relation next to, we can say: it is an index of a symbol-namely, a sign-relation-which is set up by expressing a proposition or clause of the form "X (is) next to r.52

PETER LONG

University of Leeds

2 In his book, Wittgenstein's Tractatus, p. I38, Stenius says that the word "round" in "The earth is round" is employed as a characteristic (his italics) of a linguistic predicate which names roundness. In a footnote to the same page he adds: "Following Frege's terminology we might call the word 'round' in 'The earth is round' 'incomplete' or 'unsaturated' as a symbol, since it does not name roundness but refers to it only as a 'part' of a linguistic quality naming roundness." The sense in which I have called predicates and relational expressions "incomplete" seems very close to this. What Stenius here calls a "part" of a linguistic quality is what I have called an index of a sign- property.

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Are Predicates and Relational Expressions Incomplete?