# Definition:Bound Variable

(Redirected from Definition:Dummy 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.

### Predicate Logic

In the context of predicate logic, the concept has a precise definition:

In predicate logic, a bound variable is a variable which exists in a WFF only as bound occurrences.

## Examples

### Algebraic Example

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:

$\forall x: \map P x$

the symbol $x$ is a bound variable.

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

That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$.

And so on.

### Existential Statement

In the existential statement:

$\exists x: \map P x$

the symbol $x$ is a bound variable.

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

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

### Family of Sets

Let $I$ be an indexing set.

Consider the union of the indexed family of sets $\family {S_i}_{i \mathop \in I}$:

$\ds \bigcup_{i \mathop \in I} S_i$

The variable $i$ is a bound variable, or dummy variable, such that $\ds \bigcup_{i \mathop \in I} S_i$ could as well be written $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ or $\ds \bigcup_{\gamma \mathop \in I} S_\gamma$, for example.

## 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.

## Also see

• Results about bound variables can be found here.