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A propositional function in logic, is a statement expressed in a way that would assume the value of true or false, except that within the statement is a variable (x) that is not defined or specified, which leaves the statement undetermined. Of course, x could also consist of several variables.
As a mathematical function, A(x) or A(x1, x2, · · ·, xn), 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.
- "...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.
- Tiles, Mary (2004). The philosophy of set theory an historical introduction to Cantor's paradise (Dover ed.). Mineola, N.Y.: Dover Publications. p. 159. ISBN 978-0-486-43520-6. Retrieved 1 February 2013.