# Propositional function

In propositional calculus, a **propositional function** 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 sentence may contain several such variables (e.g. *n* variables, in which case the function takes *n* arguments).

## Overview

As a mathematical function, *A*(*x*) or *A*(*x*_{1}, *x*_{2}, ..., *x*_{n}), the propositional function is abstracted from predicates or propositional forms. As an example, consider the predicate scheme, "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 a value is assigned to *x* , such as lava, the function then has the value *true*; while one assigns to *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.[1]

A Propositional Function, or a predicate, in a variable *x* is a 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."[2] Lewis used the notion of propositional functions to introduce relations, for example, a propositional function of *n* variables is a relation of arity *n*. The case of *n* = 2 corresponds to binary relations, of which there are homogeneous relations (both variables from the same set) and heterogeneous relations.

## References

- 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. - Clarence Lewis (1918)
*A Survey of Symbolic Logic*, page 232, University of California Press, second edition 1932, Dover edition 1960