The Principles of Mathematics (1903)

§ 92

From propositional functions all other classes can be derived by definition, with the help of the notion of such that. Given a propositional function ϕx, the terms such that, when x is identified with any one of them, ϕx is true, are the class defined by ϕx. This is the class as many, the class in extension. It is not to be assumed that every class so obtained has a defining predicate: this subject will be discussed afresh in Chapter X. But it must be assumed, I think, that a class in extension is defined by any propositional function, and in particular that all terms form a class, since many propositional functions whose truth defines the class should be kept intact, and not, even where this is possible for every value of x, divided into separate propositional functions. For example, if a and b be two classes, defined by ϕx and ψx respectively, their common part is defined by the product ϕx.ψx, where the product has to be made for every value of x, and then x varied afterwards. If this is not done, we do not necessarily have the same x in ϕx and ψx. Thus we do not multiply propositional functions, but propositions: the new propositional function is the class of products of corresponding propositions belonging to the previous functions, and is by no means the product of ϕx and ψx. It is only in virtue of a definition that the logical product of the classes defined by ϕx and ψx is the class defined by ϕx.ψx. And wherever a proposition containing an apparent variable is asserted, what is asserted is the truth, for all values of the variable or variables, of the propositional function corresponding to the whole proposition, and is never a relation of propositional functions.(§ 92 ¶ 1)