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.

Variants
The following variants of this notation exist:
 * $\exists !$ means there exists uniquely, or, there is one and only one.
 * $\exists_n$ means there exist exactly $n$.

Thus $\exists_1$ means the same thing as $\exists !$.

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

Some authors use $\lor x$ or $\bigvee x$ to mean $\exists x$, which is appropriate when considering the propositional expansion.

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 Giuseppe Peano in volume II, number 1, of Formulario Mathematico (2nd edition) 1896.

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

Also see

 * Existential Statement


 * Universal Quantifier
 * Universal Statement


 * Fallacy of Generalisation