Definition:Bound Variable

Definition

A bound variable is a variable which, when it occurs in an expression, can be replaced with another variable without changing the meaning of the statement.

Examples

In algebra:

$x^2 + 2 x y + y^2 = \paren {x + y}^2$

both $x$ and $y$ are bound variables.

Universal Statement

In the universal statement:

$\forall x: P \paren x$

the symbol $x$ is a bound variable.

Thus, the meaning of $\forall x: P \paren x$ does not change if $x$ is replaced by another symbol.

That is, $\forall x: P \paren x$ means the same thing as $\forall y: P \paren y$ or $\forall \alpha: P \paren \alpha$. And so on.

Existential Statement

In the existential statement:

$\exists x: P \paren x$

the symbol $x$ is a bound variable.

Thus, the meaning of $\exists x: P \paren x$ does not change if $x$ is replaced by another symbol.

That is, $\exists x: P \paren x$ means the same thing as $\exists y: P \paren y$ or $\exists \alpha: P \paren \alpha$. And so on.

Also known as

A bound variable is also popularly seen with the name dummy variable. Both terms can be seen on $\mathsf{Pr} \infty \mathsf{fWiki}$.

In treatments of pure logic, this is sometimes known as an individual variable.

Some sources call it an apparent variable, reflecting the fact that it only "appears" to be a variable.