[Math] Why negating universal quantifier gives existential quantifier

Question

Negating a universal quantifier gives the existential quantifier, and vice versa:

$\neg \forall x = \exists x \neg \\
\neg \exists x = \forall x \neg $

Why is this, and is there a proof for it (is it even possible to prove it, or is it just an axiom)? Intuitively, I would think that negating "for all" would give "none," or even "not for all," and that negating "there exists" would give "there does not exist".

Answer 1

The following two statements are equivalent:

"It is not true that all men have red hair."

"There exists at least one man who does not have red hair."

Hence $\neg\forall x\ \varphi$ is the same as $\exists x\ \neg\varphi$.

The following are equivalent:

"It is not true that some men have green hair."

"All men have non-green hair."

Hence $\neg \exists x\ \varphi$ is the same as $\forall x\ \neg\varphi$.

However, the form in which you've written them is not correct (as pointed out in Daniel Fischer's comment).