"Existential quantifier" redirects here. For the symbol conventionally used for this quantifier, see Turned E.
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists," "there is at least one," or "for some." It expresses that a propositional function can be satisfied by at least one member of a domain of discourse. In other terms, it is the predication of a property or relation to at least one member of the domain. It asserts that a predicate within the scope of an existential quantifier is true of at least one value of a predicate variable.
It is usually denoted by the turned E (∃) logical operator symbol, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.
Symbols are encoded U+2203 ∃ there exists (HTML: ∃
∃
as a mathematical symbol) and U+2204 ∄ there does not exist (HTML: ∄
).
Basics
Consider a formula that states that some natural number multiplied by itself is 25.

0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on.
This would seem to be a logical disjunction because of the repeated use of "or". However, the "and so on" makes this impossible to integrate and to interpret as a disjunction in formal logic.
Instead, the statement could be rephrased more formally as
For some natural number n, n·n = 25.
This is a single statement using existential quantification.
This statement is more precise than the original one, as the phrase "and so on" does not necessarily include all natural numbers, and nothing more. Since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, on the other hand, the natural numbers are mentioned explicitly.
This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce "5·5 = 25", which is true.
It does not matter that "n·n = 25" is only true for a single natural number, 5; even the existence of a single solution is enough to prove the existential quantification true.
In contrast, "For some even number n, n·n = 25" is false, because there are no even solutions.
The domain of discourse, which specifies which values the variable n is allowed to take, is therefore of critical importance in a statement's trueness or falseness. Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate. For example:
For some positive odd number n, n·n = 25
is logically equivalent to
For some natural number n, n is odd and n·n = 25.
Here, "and" is the logical conjunction.
In symbolic logic, "∃" (a backwards letter "E" in a sansserif font) is used to indicate existential quantification.^{[1]} Thus, if P(a, b, c) is the predicate "a·b = c" and $\backslash mathbb\{N\}$ is the set of natural numbers, then
 $\backslash exists\{n\}\{\backslash in\}\backslash mathbb\{N\}\backslash ,\; P(n,n,25)$
is the (true) statement

For some natural number n, n·n = 25.
Similarly, if Q(n) is the predicate "n is even", then
 $\backslash exists\{n\}\{\backslash in\}\backslash mathbb\{N\}\backslash ,\; \backslash big(Q(n)\backslash ;\backslash !\backslash ;\backslash !\; \{\backslash wedge\}\backslash ;\backslash !\backslash ;\backslash !\; P(n,n,25)\backslash big)$
is the (false) statement

For some natural number n, n is even and n·n = 25.
In mathematics, the proof of a "some" statement 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 but without exhibiting one.
Properties
Negation
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The $\backslash lnot\backslash $ symbol is used to denote negation.
For example, if P(x) is the propositional function "x is between 0 and 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is between 0 and 1" is symbolically stated:
 $\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$
This can be demonstrated to be irrevocably false. Truthfully, it must be said, "It is not the case that there is a natural number x that is between 0 and 1", or, symbolically:
 $\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$.
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
 $\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$
is logically equivalent to "For any natural number x, x is not between 0 and 1", or:
 $\backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$
Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,
 $\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$
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:
 $\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)\; \backslash not\backslash equiv\backslash \; \backslash lnot\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$
Negation is also expressible through a statement of "for no", as opposed to "for some":
 $\backslash nexists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\; \backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$
Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:
$\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash or\; Q(x)\; \backslash to\backslash \; (\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash or\; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; Q(x))$
Rules of Inference
Template:Transformation rules
A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.
Existential introduction (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,
 $P(a)\; \backslash to\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$
Existential elimination, when conducted in a Fitch style deduction, proceeds by entering a new subderivation while substituting an existentially quantified variable for a subject which does not appear within any active subderivation. If a conclusion can be reached within this subderivation in which the substituted subject does not appear, then one can exit that subderivation with that conclusion. 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. Symbolically, for an arbitrary c and for a proposition Q in which c does not appear:
 $\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash to\backslash \; ((P(c)\; \backslash to\backslash \; Q)\; \backslash to\backslash \; Q)$
$P(c)\; \backslash to\backslash \; Q$ 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.
The empty set
The formula $\backslash exists\; \{x\}\{\backslash in\}\backslash emptyset\; \backslash ,\; P(x)$ is always false, regardless of P(x). This is because $\backslash emptyset$ denotes the empty set, and no x of any description – let alone an x fulfilling a given predicate P(x) – exist in the empty set. See also vacuous truth.
As adjoint
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.^{[2]}
See also
Notes
References
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