Definition:Bound Variable

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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.


Sources

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