The Principles of Mathematics (1903)

§ 127

There is still, however, a certain difficulty, which is this: a class seems to be not many terms, but to be itself a single term, even when many terms are members of the class. This difficulty would seem to indicate that the class cannot be identified with all its members, but is rather to be regarded as the whole which they compose. In order, however, to state the difficulty in an unobjectionable manner, we must exclude unity and plurality from the statement of it, since these notions were to be defined by means of the notion of class. And here it may be well to clear up a point which is likely to occur to the reader. Is the notion of one presupposed every time we speak of a term? A term, it may be said, means one term, and thus no statement can be made concerning a term without presupposing one. In some sense of one, this proposition seems indubitable. Whatever is, is one: being and one, as Leibniz remarks, are convertible terms[89]. It is difficult to be sure how far such statements are merely grammatical. For although whatever is, is one, yet it is equally true that whatever are, are many. But the truth seems to be that the kind of object which is a class, i.e. the kind of object denoted by all men, or by any concept of a class, is not one except where the class has only one term, and must not be made a single logical subject. There is, as we said in Part I, Chapter VI, in simple cases an associated single term which is the class as a whole; but this is sometimes absent, and is in any case not identical with the class as many. But in this view there is not a contradiction, as in the theory that verbs and adjectives cannot be made subjects; for assertions can be made about classes as many, but the subject of such assertions is many, not one only as in other assertions. Brown and Jones are two of Miss Smith's suitors is an assertion about the class Brown and Jones, but not about this class considered as a single term. Thus one-ness belongs, in this view, to a certain type of logical subjects, but classes which are not one may yet have assertions made about them. Hence we conclude that one-ness is implied, but not presupposed, in statements about a term, and a term is to be regarded as an indefinable.(§ 127 ¶ 1)

§ 127 n. 1. Ed. Gerhardt, II, p. 300.