The Principles of Mathematics (1903)

§ 490

The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms; this is the case with all propositions asserting numbers other than 0 and 1. But the predicates or class-concepts or relations which can occur in propositions having plural subjects are different (with some exceptions) from those that can occur in propositions having single terms as subjects. Although a class is many and not one, yet there is identity and diversity among classes, and thus classes can be counted as though each were a genuine unity; and in this sense we can speak of one class and of the classes which are members of a class of classes. One must be held, however, to be somewhat different when asserted of a class from what it is when asserted of a term; that is, there is a meaning of one which is applicable in speaking of one term, and another which is applicable in speaking of one class, but there is also a general meaning applicable in both cases. The fundamental doctrine upon which all rests is the doctrine that the subject of a proposition may be plural, and that such plural subjects are what is meant by classes which have more than one term[132].(§ 490 ¶ 1)

It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on ad infinitum; we shall have to hold that no member of one set is a member of any other set, and that xu requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus xx will become a meaningless proposition; and in this way the contradiction is avoided.(§ 490 ¶ 2)

§ 490 n. 1. Cf. §§ 128, 132 supra.