The Principles of Mathematics (1903)

§ 104

Perhaps the best way to state the suggested solution is to say that, if a collection of terms can only be defined by a variable propositional function, then, though a class as many may be admitted, a class as one must be denied. When so stated, it appears that propositional functions may be varied, provided the resulting collection is never itself made into the subject in the original propositional function. In such cases there is only a class as many, not a class as one. We took it as axiomatic that the class as one is to be found wherever there is a class as many; but this axiom need not be universally admitted, and appears to have been the source of the contradiction. By denying it, therefore, the whole difficulty will be overcome.(§ 104 ¶ 1)

A class as one, we shall say, is an object of the same type as its terms; i.e. any propositional function ϕ(x) which is significant when one of the terms is substituted for x is also significant when the class as one is substituted. But the class as one does not always exist, and the class as many is of a different type from the terms of the class, even when the class has only one term, i.e. there are propositional functions ϕ(u) in which u may be the class as many, which are meaningless if, for u, we substitute one of the terms of the class. And so x is one among x's is not a proposition at all if the relation involved is that of a term to its class as many; and this is the only relation of whose presence a propositional function always assures us. In this view, a class as many may be a logical subject, but in propositions of a different kind from those in which its terms are subjects; of any object other than a single term, the question whether it is one or many will have different answers according to the proposition in which it occurs. Thus we have Socrates is one among men, in which men are plural; but men are one among species of animals, in which men are singular. It is the distinction of logical types that is the key to the whole mystery[69].(§ 104 ¶ 2)

§ 104 n. 1. On this subject, see Appendix.