Existential quantification

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Existential quantification

Type	Quantifier
Field	Mathematical logic
Statement	$\exists xP(x)$is true when$P(x)$is true for at least one value of$x$.
Symbolic statement	$\exists xP(x)$

Inpredicate logic,anexistential quantificationis a type ofquantifier,alogical constantwhich isinterpretedas "there exists", "there is at least one", or "for some". It is usually denoted by thelogical operatorsymbol∃, which, when used together with a predicate variable, is called anexistential quantifier( "∃x"or"∃(x)"or"(∃x) "). Existential quantification is distinct fromuniversal quantification( "for all" ), which asserts that the property or relation holds forallmembers of the domain.Some sources use the termexistentializationto refer to existential quantification.

Quantification in general is covered in the article onquantification (logic).The existential quantifier is encoded asU+2203∃THERE EXISTSinUnicode,and as\existsinLaTeXand related formula editors.

Basics

Consider theformalsentence

For some natural number$n$,$n\times n=25$.

This is a single statement using existential quantification. It is roughly analogous to the informal sentence "Either$0\times 0=25$,or$1\times 1=25$,or$2\times 2=25$,or... and so on, "but more precise, because it doesn't need us to infer the meaning of the phrase" and so on. "(In particular, the sentence explicitly specifies itsdomain of discourseto be the natural numbers, not, for example, thereal numbers.)

This particular example is true, because 5 is a natural number, and when we substitute 5 forn,we produce the true statement$5\times 5=25$.It does not matter that "$n\times n=25$"is true only for that single natural number, 5; the existence of a singlesolutionis enough to prove this existential quantification to be true.

In contrast, "For someeven number$n$,$n\times n=25$"is false, because there are no even solutions. Thedomain of discourse,which specifies the values the variablenis allowed to take, is therefore critical to a statement's trueness or falseness.Logical conjunctionsare used to restrict the domain of discourse to fulfill a given predicate. For example, the sentence

For some positive odd number$n$,$n\times n=25$

islogically equivalentto the sentence

For some natural number$n$,$n$is odd and$n\times n=25$.

Themathematical proofof an existential statement about "some" object may be achieved either by aconstructive proof,which exhibits an object satisfying the "some" statement, or by anonconstructive proof,which shows that there must be such an object without concretely exhibiting one.

Notation

Insymbolic logic,"∃" (a turned letter "E"in asans-seriffont, Unicode U+2203) is used to indicate existential quantification. For example, the notation$\exists {n}{\in }\mathbb {N}:n\times n=25$represents the (true) statement

There exists some$n$in the set ofnatural numberssuch that$n\times n=25$.

The symbol's first usage is thought to be byGiuseppe PeanoinFormulario mathematico(1896). Afterwards,Bertrand Russellpopularised its use as the existential quantifier. Through his research in set theory, Peano also introduced the symbols$\cap$and$\cup$to respectively denote the intersection and union of sets.

Properties

Negation

A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The$\lnot \$symbol is used to denote negation.

For example, ifP(x) is the predicate "xis greater than 0 and less than 1 ", then, for a domain of discourseXof all natural numbers, the existential quantification "There exists a natural numberxwhich is greater than 0 and less than 1 "can be symbolically stated as:

$\exists {x}{\in }\mathbf {X} \,P(x)$

This can be demonstrated to be false. Truthfully, it must be said, "It is not the case that there is a natural numberxthat is greater than 0 and less than 1 ", or, symbolically:

$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)$.

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

$\exists {x}{\in }\mathbf {X} \,P(x)$

is logically equivalent to "For any natural numberx,xis not greater than 0 and less than 1 ", or:

$\forall {x}{\in }\mathbf {X} \,\lnot P(x)$

Generally, then, the negation of apropositional function's existential quantification is auniversal quantificationof that propositional function's negation; symbolically,

$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)$

(This is a generalization ofDe Morgan's lawsto 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:

$\lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)\not \equiv \ \lnot \ \forall {x}{\in }\mathbf {X} \,P(x)\equiv \ \exists {x}{\in }\mathbf {X} \,\lnot P(x)$

Negation is also expressible through a statement of "for no", as opposed to "for some":

$\nexists {x}{\in }\mathbf {X} \,P(x)\equiv \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)$

Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:

$\exists {x}{\in }\mathbf {X} \,P(x)\lor Q(x)\to \ (\exists {x}{\in }\mathbf {X} \,P(x)\lor \exists {x}{\in }\mathbf {X} \,Q(x))$

Rules of inference

Arule of inferenceis 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,

$P(a)\to \ \exists {x}{\in }\mathbf {X} \,P(x)$

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. If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then one can exit that sub-derivation 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 isnecessarily true,as long as it does not contain the name. Symbolically, for an arbitrarycand for a propositionQin whichcdoes not appear:

$\exists {x}{\in }\mathbf {X} \,P(x)\to \ ((P(c)\to \ Q)\to \ Q)$

$P(c)\to \ Q$must be true for all values ofcover the same domainX;else, the logic does not follow: Ifcis not arbitrary, and is instead a specific element of the domain of discourse, then statingP(c) might unjustifiably give more information about that object.

The empty set

The formula$\exists {x}{\in }\varnothing \,P(x)$is always false, regardless ofP(x). This is because$\varnothing$denotes theempty set,and noxof any description – let alone anxfulfilling a given predicateP(x) – exist in the empty set. See alsoVacuous truthfor more information.

As adjoint

Main article:Universal quantification § As adjoint

Incategory theoryand the theory ofelementary topoi,the existential quantifier can be understood as theleft adjointof afunctorbetweenpower sets,theinverse imagefunctor of a function between sets; likewise, theuniversal quantifieris theright adjoint.

See also

• Existential clause
• Existence theorem
• First-order logic
• Lindström quantifier
• List of logic symbols– for the unicode symbol ∃
• Quantifier variance
• Uniqueness quantification

References

• Hinman, P. (2005).Fundamentals of Mathematical Logic.A K Peters.ISBN1-56881-262-0.