Existential quantification

Existential quantification

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

In

This is a single statement using existential quantification. It is roughly analogous to the informal sentence "Either ${\displaystyle 0\times 0=25}$, or ${\displaystyle 1\times 1=25}$, or ${\displaystyle 2\times 2=25}$, 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 its domain of discourse to be the natural numbers, not, for example, the real numbers.)

This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce the true statement ${\displaystyle 5\times 5=25}$. It does not matter that "${\displaystyle n\times n=25}$" is true only for that single natural number, 5; the existence of a single

In contrast, "For some

${\displaystyle n}$

${\displaystyle n\times n=25}$

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

is

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

The

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

There exists some ${\displaystyle n}$ in the set of natural numbers such that ${\displaystyle n\times n=25}$.

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 ${\displaystyle \cap }$ and ${\displaystyle \cup }$ to respectively denote the intersection and union of sets.^[5]

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

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:

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

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

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

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

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

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

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

Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,

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

(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:

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

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

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

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

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

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,

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

The formula ${\displaystyle \exists {x}{\in }\varnothing \,P(x)}$ is always false, regardless of P(x). This is because ${\displaystyle \varnothing }$ 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 for more information.

In

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