The Principles of Mathematics (1903)

§ 86

The discussion of the preceding chapter elicited the fundamental nature of the variable; no apparatus of assertions enables us to dispense with the consideration of the varying of one or more elements in a proposition while the other elements remain unchanged. The variable is perhaps the most distinctively mathematical of all notions; it is certainly also one of the most difficult to understand. The attempt, if not the deed, belongs to the present chapter.(§ 86 ¶ 1)

The theory as to the nature of the variable, which results from our previous discussions, is in outline the following. When a given term occurs as a term in a proposition, that term may be replaced by any other while the remaining terms are unchanged. The class of propositions so obtained have what may be called constancy of form, and this constancy of form must be taken as a primitive idea. The notion of a class of propositions of constant form is more fundamental than the general notion of class, for the latter can be defined in terms of the former, but not the former in terms of the latter. Taking any term, a certain number of any class of propositions of constant form will contain that term. Thus x, the variable, is what is denoted by any term, and ϕx, the propositional function, is what is denoted by the proposition of the form ϕ in which x occurs. We may say that x is the x in any ϕx, where ϕx denotes the class of propositions resulting from different value of x. Thus in addition to propositional functions, the notions of any and of denoting are presupposed in the notion of the variable. This theory, which, I admit, is full of difficulties, is the least objectionable that I have been able to imagine. I shall now set it forth in more detail.(§ 86 ¶ 2)