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 ...

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 $\bigvee x$ 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 $\exists x: P \left({x}\right)$.

Note, however, that it is also used to symbolize the concept most.

Beware
Now, you have to be careful with most. It has to be interpreted in the same way as some.

Compare these sequents:

... which can be epitomised by:

... which one has to admit seems plausible.

On the other hand, check this out:

... an example of which reasoning may be:

Well I don't know about you, but I've never been beaten at chess by an amoeba.

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