Propositional Function

propositional function, in logic, a statement expressed in a form that would take on a value of true or false were it not for the appearance within it of a variable x (or of several variables), which leaves the statement undetermined as long as no definite values are specified for the variables. Denoted as a mathematical function, A(x) or A(x₁, x₂, · · ·, x_n), the propositional function is an abstraction from propositional forms (or predicates). For example, “x is small,” “x ist klein,” and “x is not large” are all propositional forms. The substitution of some noun (or value) for x yields a specific proposition that is either true or false, but “x is small” itself has no truth-value. Abstracting from these three propositional forms yields a propositional function A(x), which has, for example, the value true when x takes the value mouse and has the value false when x is elephant.

Propositional functions permit the treatment in symbolic logic of propositions the truth of which is contingent upon variable components.