Definition:Existential Quantifier

Definition
The symbol $\exists$ is called the existential quantifier.

It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property.

That is:
 * $\exists x:$

means:
 * There exists at least one object $x$ such that ...

In the language of set theory, this can be formally defined:
 * $\exists x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} \ne \varnothing$

where $S$ is some set and $P \left({x}\right)$ is a propositional function on $S$.

Propositional Expansion
The existential quantifier can be considered as a repeated disjunction:

Suppose our universe of discourse consists of the objects $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ and so on.

Also known as
Some sources refer to this as the particular quantifier.

Semantics
The existential quantifier can, and often is, used to symbolize the concept some.

That is, Some $x$ have $P$ is also symbolized as $\exists x: P \left({x}\right)$.

It is also used to symbolize the concept most.

Historical Note
The symbol $\exists$ was first used by in volume II, number 1, of Formulario Mathematico (2nd edition) 1896.

However, was the first to use $\exists$ as a variable binding operator.

Also see

 * Definition:Existential Statement


 * Definition:Universal Quantifier
 * Definition:Universal Statement


 * Fallacy of Generalisation