Variable Assignment (First-Order Logic)

first-order-logic

Definition

Variable Assignment (First-Order Logic)

Let be a structure. A variable assignment is a function

that maps each variable to an element of the domain.

Variants

For a variable assignment , a variable , and an element :

is the assignment identical to except it maps to .

Example

Let and let assign:

Now take . This means: copy , but overwrite with :

Only changed. stays untouched.

When evaluating quantifiers, we often need an assignment that differs from only at :

This denotes that for all , . The value can be any element in .

Example

Let and let assign:

Consider two candidates for :

			?
one			yes — only differs
another			no — differs too

The relation only cares about variables other than staying the same.

vs.

builds one specific variant: you pick the new value for . relates any two assignments that agree everywhere except possibly . The notation is used in quantifier semantics (“does there exist some -variant such that…?”) because is not fixed to one value — it ranges over all of .

Dynamic Nature

Unlike a structure, which is fixed, a variable assignment changes as we traverse a formula — specifically when evaluating quantifiers.