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 ...".

Some authors call this the particular quantifier.

There are variants of this symbol:
 * $$\exists !$$ 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 $$\exists !$$.

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

See the definition of the propositional expansion of $\exists x$.

Variants
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 haven't been beaten at chess by any amoebae.

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