Negation of quantified expressions

Question

Can someone explain the difference between ¬∀x P(x) and ∀x ¬P(x), and also why

¬∀xP (x) ≡ ∃x ¬P (x) can't be written as ¬∀x P(x) ≡ ¬∃x P(x) and vice versa.

Thanks.

Answer 1

Let the universe of discourse be bricks, that is, all our quantifiers are over the set of all bricks ($x$ is a brick). Let $P(x)$ be the statement that $x$ is red.

$\neg \forall x P(x)$: not all bricks are red,

$\forall x \neg P(x)$: all bricks are not red.

It should be apparent that the first statement is true but the second is false. However, the first statement is equivalent to

$\exists x \neg P(x)$: there is a brick that is not red,

but not

$ \neg \exists x P(x)$: there is not a red brick.

These statements are obvious once written out in plain English instead of possibly unfamiliar symbols.