Variable (Logic)

logic

Definition

Variable (Logic)

In logic, a variable is a symbol used to represent an unspecified element of the universe. Variables are the placeholders that quantifiers ($\forall ,\exists$) bind.

Usage

Variables differ from constants in that their meaning is not fixed by an interpretation $I$. Instead, their value is determined by a variable assignment $\sigma$.

Free vs. Bound

• Bound Variable: A variable that occurs within the scope of a quantifier (e.g., $x$ in $\forall xP(x)$).
• Free Variable: A variable that is not bound by any quantifier. A formula with no free variables is called a sentence.

Domain

All variables in a first-order language take values from the same domain of discourse $\mathcal{U}$.