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

any number is a number.Interpreted by formal implication, they offer no difficulty, for they assert merely that the propositional function

holds for all values ofxis a number impliesxis a number

any numberbe 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 numbercannot count as a number at all. The fact is that the concept

any numberdoes denote one number, but not a particular one. This is just the distinctive point about

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 `ϕ``x`, where `ϕ``x` is a propositional function, is an unrestricted variable; but the `ϕ``x` 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)

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.