The existential quantifier is a quantifier that describes a predicate as true for at least one item in its domain. The existential quantifier is expressed by the symbol "" which is read as "there exists". Formally this can be defined as follows

Let be a predicate and the domain of . A universal statement is a statement of the form "". It is defined to be true if, and only if, is true for at least one in . It is defined to be false if, and only if, is false for every in

A value for which the predicate the existential quantifier is applied to which is true is called a witness of the existential statement.

Example

Given the predicate over the domain (the set of all real numbers). We can use the existential quantifier to describe the solutions to this equation in the following way:

This can be read as "there exists some number such that equals "

Since the quadratic equation above has two solutions, namely and , the statement is true and has two witnesses.

Negation

The negation of an existential quantifier is a universal quantifier with the condition negated.

Formally:

Example

What is the negation of the statement “Some snowflakes are the same”?

"All snowflakes are not the same"