The Principles of Mathematics (1903)

§ 84

It only remains to say a few words concerning the derivation of classes from propositional functions. When we consider the x's sicj tjat ϕx, where ϕx is a propositional function, we are introducing a notion of which, in the calculus of propositions, only a very shadowy use is made--I mean the notion of truth. We are considering, among all the propositions of the type ϕx, those that are true: the corresponding values of x give the class defined by the function ϕx. It must be held, I think, that every propositional function which is not null defines a class, which is denoted by x's such that ϕx. There is thus always a concept of the class, and the class-concept corresponding will be the singular, x such that ϕx. But it may be doubted--indeed the contradiction with which I ended the preceding chapter gives reason for doubting--whether there is always a defining predicate of such classes. Apart from the contradiction in question, this point might appear to be merely verbal: being an x such that ϕx, it might be said, may always be taken to be a predicate. But in view of our contradiction, all remarks on this subject must be viewed with caution. This subject, however, will be resumed in Chapter X.(§ 84 ¶ 1)