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 `ϕ``x`, 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

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