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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 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).

Note: This should not be confused with a negative universal quantification, which looks like this:

No X is Y.
∄ 𝑥 ∈ 𝕏 : Y
(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:

No dog exists that has exactly three legs.

Existential condition

While universal quantifications may refer to empty 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 syllo­gisms such as Modus Barbari  or Modus Calemos .

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 and mathematics, the symbol 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 ¬ (“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

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