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