Universal Quantifier

first-order-logic

Definition

Universal Quantifier

The universal quantifier is true under an interpretation if holds for every re-assignment of :

Semantics

Formally, the semantics are defined using an interpretation and a variable assignment . The valuation function for a universally quantified formula is:

where denotes a variable assignment that is identical to except possibly for the value assigned to .

Relation to Existential Quantifier

The universal quantifier is the dual of the existential quantifier (). They are related by De Morgan’s laws for quantifiers:

Empty Domains

In a universe that is empty, any universally quantified statement is vacuously true, while any existentially quantified statement is false. However, in standard first-order logic, the domain of discourse is usually assumed to be non-empty.