====== Existential Quantification ======

A statement about the existence of (at least) one element in a certain set of items.

For example:

> There exists dogs which have exactly three legs.

A //negative// existential quantification states that there are objects to which the description does //not// apply.

For example:

> There exist dogs which do //not// have exactly three legs.

===== Description =====

A //positive// existential quantification is a [[glossary:proposition|proposition]] that is true for some (or at least one) of the elements that it refers to.

> //Some// X //are// Y.

Or as logical formula:

> ''∃ 𝑥 ∈ 𝕏 : Y''
> (there exists [at least] one 𝑥 in the set 𝕏 for which Y holds true).

A negative existential quantification describes the case where the characteristic does not hold:

> ''∃ 𝑥 ∈ 𝕏 : ¬X''
> (there exists [at least] one 𝑥 in the set 𝕏 for which not-Y holds).

<aside box>
**Note:** This should not be confused with a negative universal quantification, which looks like this:

> //No// X //is// Y.
> <u questionable "negativer Allsatz">''∄ 𝑥 ∈ 𝕏 : Y''</u>
> (There exists //no// 𝑥 in the set 𝕏, which is Y)

The following would be a negative (although //false//) //universal// quantification in line with the examples above:

> <s invalid>No dog exists that has exactly three legs.</s>

</aside>

==== Existential condition ====

While [[glossary:universal_quantification|universal quantifications]] may refer to empty [[glossary:extension|extensions]], //existential quantifications// – as the name suggests – must always refer to an actually existing extension.

This can lead to the situation that when deriving existential from universal quanti­fi­ca­tions, it must first be proven that there exists at least //one// element of the extension set. Such auxiliary conditions are necessary, for example, for [[glossary:syllogism|syllo­gisms]] such as [[logic:inferences:modus_barbara:modus_barbari|Modus Barbari]] [[app>#barbari|Show in Syllogism-Finder App]] or [[logic:inferences:modus_celarent:modus_calemos|Modus Calemos]] [[app>#calemos|Show in Syllogism-Finder App]].

==== Natural language ====

There are many ways that the //existential quantification// can be formulated in natural language, which is also what gives this form so much flexibility. The most common forms are:

  * //Some// A //are// [//not// ] B.
  * //There exist// A //which are// [//not// ] B.
  * //There exists at least one// A //which is// [//not// ] B.
  * etc.

The properties of the //existential quantification// are independent from how it is formulated. E.g. also “some A are B” implies //existence//, etc.
===== Identifier =====

In both [[logic:index|logic]] and [[methematics:index|mathematics]], the symbol <span "there exists at least one …">''∃''</span> is used to denote existential quantification. This is usually pronounced as "there exists …". For example:

> ''∃ 𝑛 ∈ ℕ : 𝑛² = 25''
> (there exists at least one number 𝑛 which is Element of the set of natural numbers, for which it is true that 𝑛² is equal to 25)

To denote //negative// existential quantification, either the <span "not">''¬''</span> ("not") sign is used, or any other symbol expressing negation or inequality, e.g.:

> ''∃ 𝑛 ∈ ℕ : 𝑛² ≠ 25''
> (there exists at least one number 𝑛 which is Element of the set of natural numbers, for which it is true that 𝑛² is //not// equal to 25)

===== See also =====

  * [[glossary:categorical_statement|Categorical statement]]
  * [[logic:formal_fallacies:existential|Existential fallacy]]
  * [[glossary:universal_quantification|Universal quantification]]

===== More information =====

  * [[wp>Existential quantification]] on //Wikipedia//