The Principles of Mathematics (1903)

§ 64

The connection of denoting with the nature of identity is important, and helps, I think, to solve some rather serious problems. The question whether identity is or is not a relation, and even whether there is such a concept at all, is not easy to answer. For, it may be said, identity cannot be a relation, since, where it is truly asserted, we have only one term, whereas two terms are required for a relation. And indeed identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with? Nevertheless identity must be something. We might attempt to remove identity from terms to relations, and say that two terms are identical in some respect when they have a given relation to a given term. But then we shall have to hold either that there is strict identity between the two cases of the given relation, or that the two cases have identity in the sense of having a given relation to a given term; but the latter view leads to an endless process of the illegitimate kind. Thus identity must be admitted, and the difficulty as to the two terms of a relation must be met by a sheer denial that two different terms are necessary. There must always be a referent and a relatum, but these need not be distinct; and where identity is affirmed, they are not so[52].(§ 64 ¶ 1)

But the question arises: Why is it ever worth while to affirm identity? The question is answered by the theory of denoting. If we say Edward VII is the King, we assert an identity; the reason why this assertion is worth making is, that in the one case the actual term occurs, while in the other a denoting concept takes its place. (For purposes of discussion, I ignore the fact that Edwards form a class, and that seventh Edwards form a class having only one term. Edward VII is practically, though not formally, a proper name.) Often two denoting concepts occur, and the term itself is not mentioned, as in the proposition the present Pope is the last survivor of his generation. When a term is given, the assertion of an identity with itself, though true, is perfectly futile, and is never made outside the logic-books; but where denoting concepts are introduced, identity is at once seen to be significant. In this case, of course, there is involved, though not asserted, a relation of the denoting concept to the terms, or of the two denoting concepts to each other. But the is which occurs in such propositions does not itself state this further relation, but states pure identity[53].(§ 64 ¶ 2)

§ 64 n. 1. On relations of terms to themselves, v. inf. Chap. IX, § 95.

§ 64 n. 2. The word is is terribly ambiguous, and great care is necessary in order not to confound its various meanings. We have (1) the sense in which it asserts Being, as in A is; (2) the sense of identity; (3) the sense of predication, in A is human; (4) the sense of A is a-man (cf. § 57, note), which is very like identity. In addition to these there are less common uses, as to be good is to be happy, where a relation of assertions is meant, that relation, in fact, which, where it exists, gives rise to formal implication. Doubtless there are further meanings which have not occurred to me. On the meanings of is, cf. De Morgan, Formal Logic, pp. 49, 50.