In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined.
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.
For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106): 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.
[1] A Propositional Function, or a predicate, in a variable x is an open formula p(x) involving x that becomes a proposition when one gives x a definite value from the set of values it can take.
According to Clarence Lewis, "A proposition is any expression which is either true or false; a propositional function is an expression, containing one or more variables, which becomes a proposition when each of the variables is replaced by some one of its values from a discourse domain of individuals.