A statement about the existence of (at least) one element in a certain set of items.
There exists dogs which have exactly three legs.
A negative existential quantification states that there are objects to which the description does not apply.
There exist dogs which do not have exactly three legs.
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).
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.
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 ◈.
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.
The properties of the existential quantification are independent from how it is formulated. E.g. also “some A are B” implies existence, etc.
∃ 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)
- Existential quantification on Wikipedia