The Principles of Mathematics (1903)

§ 22

The next fundamental notion is that of a propositional function. Although propositional functions occur in the calculus of propositions, they are there each defined as it occurs, so that the general notion is not required. But in the class-calculus it is necessary to introduce the general notion explicitly. Peano does not require it, owing to his assumption that x is an a is general for one variable, and that extensions of the same form are available for any number of variables. But we must avoid this assumption, and must therefore introduce the notion of a propositional function. We may explain (but not define) this notion as follows: ϕx is a propositional function if, for every value of x, ϕx is a proposition, determinate when x is given. Thus x is a man is a propositional function. In any proposition, however complicated, which contains no real variables, we may imagine one of the terms, not a verb or adjective, to be replaced by other terms: instead of Socrates is a man we may put Plato is a man, the number 2 is a man, and so on[18]. Thus we get successive propositions all agreeing except as to the one variable term. Putting x for the variable term, x is a man expresses the type of all such propositions. A propositional function in general will be true for some values of the variable and false for others. The instances where it is true for all values of the variable, so far as they are known to me, all express implications, such as x is a man implies x is mortal; but I know of no à priori for asserting that no other propositional functions are true for all values of the variable.(§ 22 ¶ 1)

§ 22 n. 1. Verbs and adjectives occurring as such are distinguished by the fact that, if they be taken as variable, the resulting function is only a proposition for some values of the variable, i.e. for such as are verbs or adjectives respectively. See Chap. IV.