# Definition:Bound Variable/Predicate Logic

## 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.

Some authors gloss over the difference between:

- a
**bound variable**: a variable which exists in a WFF only as**bound occurrences**

and:

- a
**bound occurrence**of a variable which*may*otherwise exist as a**free occurrence**