quality:corruption, information:extraction, quality:correction, linguistic:translation
Negations of quantifiers

In first-order logic, there are two quantifiers, one is the universal quantifier ∀ {\displaystyle \forall } (means "for all") and the other is the existential quantifier ∃ {\displaystyle \exists } (means "there exists"). The negation of one quantifier is the other quantifier (¬ ∀ x P (x) ≡ ∃ x ¬ P (x) {\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)} and ¬ ∃ x P (x) ≡ ∀ x ¬ P (x) {\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}). For example, with the predicate P as " x is mortal" and the domain of x as the collection of all humans, ∀ x P (x) {\displaystyle \forall xP(x)} means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is ¬ ∀ x P (x) ≡ ∃ x ¬ P (x) {\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}, meaning "there exists a person x in all humans who is not mortal", or "there exists someone who lives forever".