Existential Quantifier

first-order-logic

Definition

Existential Quantifier

The existential quantifier is true under an interpretation if holds for some re-assignment of :

Semantics

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

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

Relation to Universal Quantifier

The existential quantifier is the dual of the universal quantifier (). In classical logic, they are related by De Morgan’s laws for quantifiers:

Multiplicity

The statement does not specify how many elements satisfy , only that the set of such elements is non-empty. For “exactly one”, the unique existential quantifier is used.