Quantifiers and their negation

Remark — Linguistic insight

In proofs the universal quantifier $\forall$ (“for all”) and the existential quantifier $\exists$ (“there exists at least one”) are negated by swapping places, while the inner proposition is negated: $\lnot(\forall x\,P(x))\equiv \exists x\,\lnot P(x), \qquad \lnot(\exists x\,P(x))\equiv \forall x\,\lnot P(x).$ Example: the negation of “every morning I have a coffee” is not “no morning I have a coffee”, but “there exists a morning on which I do not have a coffee” (Esty, ch. 3).

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Topics: Set theory
Concepts: Quantifiers