# Definition:Free Variable

## Contents

## Definition

Let $x$ be a variable in an expression $E$.

$x$ is a **free variable in $E$** if and only if it is not a bound variable.

In the context of predicate logic, $x$ is a **free variable in $E$** if and only if it has not been introduced by a quantifier, either:

- the universal quantifier $\forall$

or

- the existential quantifier $\exists$.

## Also known as

A **free variable** is often referred to as an **unknown**, particularly in mathematical contexts.

In the field of logic, a **free variable** can also be referred to as a **real variable**.

However, this can be confused with a variable whose domain is the set of real numbers, so its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is discouraged.

The name arises in apposition to the name **apparent variable**, which is another name for bound variable.

## Also see

- Definition:Free Occurrence: a somewhat more precise concept, recognising the fact that a variable may appear multiple times in an expression, and not necessarily always of the same category.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.4$: Universal and Existential Quantifiers - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation