The Principles of Mathematics (1903)

§ 103

The first method which suggests itself is to seek an ambiguity in the notion of ∈. But in Chapter VI we distinguished the various meanings as far as any distinction seemed possible, and we have just seen that with each meaning the same contradiction emerges. Let us, however, attempt to state the contradiction throughout in terms of propositional functions. Every propositional function which is not null, we supposed, defines a class, and every class can certainly be defined by a propositional function. Thus to say that a class as one is not a member of itself as many is to say that the class as one does not satisfy the function by which itself as many is defined. Since all propositional functions except such as are null define classes, all will be used in, in considering all classes having the above property, except such as do not have the above property. If any propositional function were satisfied by every class having the above property, it would therefore necessarily be one satisfied also by the class w of all such classes considered as a single term. Hence the class w does not itself belong to the class w, and therefore there must be some propositional function satisfied by the terms of w but not by w itself. Thus the contradiction re-emerges, and we must suppose, either that there is no such entity as w, or that there is no propositional function satisfied by its terms and by no others.(§ 103 ¶ 1)

It might be thought that a solution could be found by denying the legitimacy of variable propositional functions. If we denote by kϕ, for the moment, the class of values satisfying ϕ, our propositional function is the denial of ϕ(kϕ), where ϕ is the variable. The doctrine of Chapter VII, that ϕ is not a separable entity, might make such a variable seem illegitimate; but this objection can be overcome by substituting for ϕ the class of propositions ϕx, or the relation of ϕx to x. Moreover it is impossible to exclude variable propositional functions altogether. Wherever a variable class or a variable relation occurs, we have admitted a variable propositional function, which is thus essential to assertions about every class or about every relation. The definition of the domain of a relation, for example, and all the general propositions which constitute the calculus of relations, would be swept away by the refusal to allow this type of variation. Thus we require some further characteristic by which to distinguish two kinds of variation. This characteristic is to be found, I think, in the independent variability of the function and the argument. In general, ϕx is itself a function of two variables, ϕ and x; of these, either may be given a constant value, and either may be varied without reference to the other. But in the type of propositional functions we are considering in this Chapter, the argument is itself a function of the propositional function: instead of ϕx, we have ϕ{f(ϕ)}, where f(ϕ) is defined as a function of ϕ. Thus when ϕ is varied, the argument of which ϕ is asserted is varied too. Thus x is an x is equivalent to: ϕ can be asserted of the class of terms satisfying ϕ, this class of terms being x. If here ϕ is varied, the argument is varied at the same time in a manner dependent upon the variation of ϕ. For this reason, ϕ{f(ϕ)}, though it is a definite proposition when x is assigned, is not a propositional function, in the ordinary sense, when x is variable. Propositional functions of this doubtful type may be called quadratic forms, because the variable enters into them in a way somewhat analogous to that in which, in Algebra, a variable appears in an expression of the second degree.(§ 103 ¶ 2)