Variable Assignment (Logic)

logic

Definition

Variable Assignment

A variable assignment (or valuation) $\sigma :\mathcal{V}\to \mathcal{U}$ is a function that maps logical variables from the set $\mathcal{V}$ to elements of the universe $\mathcal{U}$.

Variant Notation

When evaluating quantifiers, we often need an assignment that differs from $\sigma$ only at a specific variable $x$:

$${\sigma }^{\prime }\stackrel{x}{\sim }\sigma$$

This denotes that for all $y\in \mathcal{V}\setminus \{x\}$, ${\sigma }^{\prime }(y)=\sigma (y)$. The value ${\sigma }^{\prime }(x)$ can be any element in $\mathcal{U}$.

Dynamic Nature

Unlike an interpretation, which is fixed for a structure, variable assignments are “dynamic” and change as we traverse the structure of a formula (specifically when encountering quantifiers).