What is "propositional function"

n. (context logic English) An expression containing algebraic symbols that serve to represent words or other elements of a sentence or proposition

A propositional function in logic, is a sentence expressed in a way that would assume the value of true or false, except that within the sentence is a variable (x) that is not defined or specified, which leaves the statement undetermined. Of course, the sentence can consist of several such variables (e.g. n variables, in which case the function takes n arguments). As a mathematical function, A(x) or A(x, x, · · ·, x), the propositional function is abstracted from predicates or propositional forms. As an example, let's imagine the predicate, "x is hot". The substitution of any entity for x will produce a specific proposition that can be described as either true or false, even though "x is hot" on its own has no value as either a true or false statement. However, when you assign x a value, such as lava, the function then has the value true; while if you assign x a value like ice, the function then has the value false.

Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):

"...it has become necessary to take propositional function as a primitive notion.

Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types.

A Propositional Function, or a predicate, in a variable x is a sentence p(x) involving x that becomes a proposition when we give x a definite value from the set of values it can take.