Predicate logic with negation "not"

Question

I've really confused myself here: $$ P(x) = ``\text{x has a tail}" $$ How would I write: $$ ``\text{Not everything has a tail}" $$

would it: $$ ¬∀x P(x) $$

be correct?

Answer 1

"Everything doesn't have a tail" is not logically equivalent to ""Not everything has a tail."

Use instead $\lnot \forall xP(x)$, or equivalently $\exists x \lnot P(x)$, there is something that doesn't have a tail.