Propositional function
Propositional function
is a function (q.v.) for which the range of the dependent variable is composed of propositions (q.v.) A monadic propositional function is thus in substance a property (of things belonging to the range of the independent variable), and a dyadic propositional function a relation. If F denotes a propositional function and X1, X2, . . . , Xn denote arguments, the notation F(X1, X2, . . . , Xn) — or [F](X1, X2, . . . , Xn) — is used for the resulting proposition, which is said to be the value of the propositional function for the given arguments, and to be obtained from the propositional function by applying it to, or predicating it of the given arguments.
Often, however, the assumption is made that two propositional functions are identical if corresponding values are materially equivalent, and in this case we speak of propositional functions in extension (the definition in the preceding paragraph applying rather to propositional functions in intension). The values of a propositional function in extension are truth-values (q.v.) rather than propositions. A monadic propositional function in extension is not essentially different from a class (q. v.)
Whitehead and Russell use the term propositional function in approximately the sense above described, but qualify it by holding, as a corollary of Russell’s doctrine of descriptions (q.v.), that propositional functions are the fundamental kind from which other kinds of functions are derived — in fact that non-propositional (“descriptive”) functions do not exist except as incomplete symbols. For details of their view, which underwent some changes between publication of the first and the second edition of Principia Mathematica, the reader is referred to that work.
Historically, the notion of a function was of gradual growth in mathematics. The word function is used in approximately its modern sense by John Bernoulli (1698, 1718). The divorce of the notion of a function from that of a particular kind of mathematical expression (analytic or quasi-algebraic) is due to Dirichlet (1837). The general logical notion of a function, and in particular the notion of a propositional function, were introduced by Frege (1879). — Alonzo Church