Existential Quantifier

The existential quantifier is a symbol of symbolic logic which expresses that the statements within its scope are true for at least one instance of something.

The symbol ∃, which appears as a backwards “E”, is used as the existential quantifier.

Existential quantifiers are normally used in logic in conjunction with predicate symbols, which say something about a variable or constant, in this case the variable being quantified.

The existential quantifier ∃ (which means “there exists”), differs from the universal quantifier ∀ (which means “for all”).

Examples

For example, if the predicate symbol Bx is taken to mean “x is a ball”, then we may formalize an expression using a existential quantifier:

∃xBx

Translated back into English, this reads as “there is an x such that x is a ball”, or more simply, “there is a ball”.

We may formalize the expression “Some P are Q” using an existential quantifier and a conjunction:

∃x(Px ∧ Qx)

This reads as “There exists an x such that x is a P and x is a Q”, which may be literally as “at least one P is a Q”, or more generally, “some P are Q”. (In logic, the word “some” is almost always taken to mean “at least one”.

Limit ∃ to one instance

The existential quantifier always means “at least one”, which means that there may be one or more of the specified thing in existence. Sometimes, it may be useful to say that there is only one. In these cases, an existential quantifier is written as ∃!, which means “there exists exactly one”. For example, we may say ∃!xBx, using the meaning presented above, to say “there is only one ball”.