ABSTRACT
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 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 member of any class of propositions of constant form will contain that term. Thus x, the variable, is what is denoted by any term, and x, the propositional function, is what is denoted by the proposition of the form in which x occurs. We may say that x is the x is any x, where x denotes the class of propositions resulting from different values 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 more in detail.
