# Propositional function

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Expression in propositional calculus

In [propositional calculus](/source/Propositional_calculus), a **propositional function** or a **predicate** is a sentence expressed in a way that would assume the value of [true](/source/Logical_truth) or [false](/source/False_(logic)), except that within the sentence there is a [variable](/source/Variable_(mathematics)) (*x*) that is not defined or specified (thus being a [free variable](/source/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](/source/Function_(mathematics)), *A*(*x*) or *A*(*x*1, *x*2, ..., *x**n*), the propositional function is abstracted from [predicates](/source/Predicate_(mathematical_logic)) 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](/source/Lava), the function then has the value *true*; while one assigns to *x* a value like [ice](/source/Ice), the function then has the value *false*.

Propositional functions are useful in [set theory](/source/Set_theory) for the formation of [sets](/source/Set_(mathematics)). For example, in 1903 [Bertrand Russell](/source/Bertrand_Russell) wrote in *[The Principles of Mathematics](/source/The_Principles_of_Mathematics)* (page 106):

- "...it has become necessary to take *propositional function* as a [primitive notion](/source/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 an [open formula](/source/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](/source/Clarence_Lewis), "A [proposition](/source/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](/source/Domain_of_discourse) of individuals."[2] Lewis used the notion of propositional functions to introduce [relations](/source/Relation_(mathematics)), for example, a propositional function of *n* variables is a relation of [arity](/source/Arity) *n*. The case of *n* = 2 corresponds to [binary relations](/source/Binary_relation), of which there are [homogeneous relations](/source/Homogeneous_relation) (both variables from the same set) and [heterogeneous relations](/source/Heterogeneous_relation).

## See also

- [Propositional formula](/source/Propositional_formula)

- [Boolean-valued function](/source/Boolean-valued_function)

- [Formula (logic)](/source/Formula_(logic))

- [Sentence (logic)](/source/Sentence_(logic))

- [Truth function](/source/Truth_function)

- [Open sentence](/source/Open_sentence)

## References

1. **[^](#cite_ref-Tiles_1-0)** [Tiles, Mary](/source/Mary_Tiles) (2004). [*The philosophy of set theory an historical introduction to Cantor's paradise*](http://store.doverpublications.com/0486435202.html) (Dover ed.). Mineola, N.Y.: Dover Publications. p. 159. [ISBN](/source/ISBN_(identifier)) [978-0-486-43520-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-43520-6). Retrieved 1 February 2013.

1. **[^](#cite_ref-2)** [Clarence Lewis](/source/Clarence_Lewis) (1918) *A Survey of Symbolic Logic*, page 232, [University of California Press](/source/University_of_California_Press), second edition 1932, Dover edition 1960

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