It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"[1]).

The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related formula editors.

Consider the formal sentence This is a single statement using existential quantification.

The domain of discourse, which specifies the values the variable n is allowed to take, is therefore critical to a statement's trueness or falseness.

Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate.

For example, the sentence is logically equivalent to the sentence The mathematical proof of an existential statement about "some" object may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by a nonconstructive proof, which shows that there must be such an object without concretely exhibiting one.

In symbolic logic, "∃" (a turned letter "E" in a sans-serif font, Unicode U+2203) is used to indicate existential quantification.

represents the (true) statement The symbol's first usage is thought to be by Giuseppe Peano in Formulario mathematico (1896).

Afterwards, Bertrand Russell popularised its use as the existential quantifier.

Through his research in set theory, Peano also introduced the symbols

For example, if P(x) is the predicate "x is greater than 0 and less than 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is greater than 0 and less than 1" can be symbolically stated as: This can be demonstrated to be false.

Truthfully, it must be said, "It is not the case that there is a natural number x that is greater than 0 and less than 1", or, symbolically: If there is no element of the domain of discourse for which the statement is true, then it must be false for all of those elements.

That is, the negation of is logically equivalent to "For any natural number x, x is not greater than 0 and less than 1", or: Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically, (This is a generalization of De Morgan's laws to predicate logic.)

A common error is stating "all persons are not married" (i.e., "there exists no person who is married"), when "not all persons are married" (i.e., "there exists a person who is not married") is intended: Negation is also expressible through a statement of "for no", as opposed to "for some": Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:

There are several rules of inference which utilize the existential quantifier.

Symbolically, Existential instantiation, when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation while substituting an existentially quantified variable for a subject—which does not appear within any active sub-derivation.

The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name.

must be true for all values of c over the same domain X; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P(c) might unjustifiably give more information about that object.

In category theory and the theory of elementary topoi, the existential quantifier can be understood as the left adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the universal quantifier is the right adjoint.