The Principles of Mathematics (1903)

§ 88

We may distinguish what may be called the true or formal variable from the restricted variable. Any term is a concept denoting the true variable; if u be a class not containing all terms, any u denotes a restricted variable. The terms included in the object denoted by the defining concept of a variable are called the values of the variable: thus every value of a variable is a constant. There is a certain difficulty about such propositions as any number is a number. Interpreted by formal implication, they offer no difficulty, for they assert merely that the propositional function x is a number implies x is a number holds for all values of x. But if any number be taken to be a definite object, it is plain that it is not identical with 1 or 2 or 3 or any number that may be mentioned. Yet these are all the numbers there are, so that any number cannot count as a number at all. The fact is that the concept any number does denote one number, but not a particular one. This is just the distinctive point about any, that it denotes a term of a class, but in an impartial distributive manner, with no preference for one term over another. Thus although x is a number, and no one number is x, yet there is here no contradiction, so soon as it is recognized that x is not one definite term.(§ 88 ¶ 1)

The notion of the restricted variable can be avoided, except in regard to propositional functions, by the introduction of a suitable hypothesis, namely the hypothesis expressed by the restriction itself. But in respect of propositional functions this is not possible. The x in ϕx, where ϕx is a propositional function, is an unrestricted variable; but the ϕx itself is restricted to the class which we may call ϕ. (It is to be remembered that the class is here fundamental, for we found it impossible, without a vicious circle, to discover any common characteristic by which the class would be defined, since the statement of any common characteristic is itself a propositional function.) By making our x always an unrestricted variable, we can speak of the variable, which is conceptually identical in Logic, Arithmetic, Geometry, and all other formal subjects. The terms dealt with are always all terms; only the complex concepts that occur distinguish the various branches of Mathematics.(§ 88 ¶ 2)