Existential quantifier in the context of "Quantifier (logic)"

Existence is the state of having being or reality in contrast to nonexistence and nonbeing. Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does not know whether the entity exists.

Ontology is the philosophical discipline studying the nature and types of existence. Singular existence is the existence of individual entities while general existence refers to the existence of concepts or universals. Entities present in space and time have concrete existence in contrast to abstract entities, like numbers and sets. Other distinctions are between possible, contingent, and necessary existence and between physical and mental existence. The common view is that an entity either exists or not with nothing in between, but some philosophers say that there are degrees of existence, meaning that some entities exist to a higher degree than others.

In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (${\displaystyle \exists }$) in formal proofs.

Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."

In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ${\displaystyle \forall }$ in the first-order formula ${\displaystyle \forall xP(x)}$ expresses that everything in the domain satisfies the property denoted by ${\displaystyle P}$. On the other hand, the existential quantifier ${\displaystyle \exists }$ in the formula ${\displaystyle \exists xP(x)}$ expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

The most commonly used quantifiers are ${\displaystyle \forall }$ and ${\displaystyle \exists }$. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula ${\displaystyle \neg \exists xP(x)}$ which expresses that nothing has the property ${\displaystyle P}$. Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström.