# Predicate logic with negation “not”

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?

• It isn't the same thing. The negation is there is something that doesn't have a tail. – André Nicolas Jul 10 '15 at 5:51
• I do not want to negate "Not everything has a tail" I'm confused as to how to write it out. – Feath Jul 10 '15 at 5:57
• That should be $\lnot\forall$, not $\lnot\exists$. (Remember that $\exists$ means "exists.") – Akiva Weinberger Jul 10 '15 at 6:04

Use instead $\lnot \forall xP(x)$, or equivalently $\exists x \lnot P(x)$, there is something that doesn't have a tail.
• I modified the answer to give you two alternatives. The simplest is $\lnot\forall xP(x)$. Then I gave an equivalent assertion using the existential quantifier. I suspect that one, for no good reason, may be preferred. – André Nicolas Jul 10 '15 at 6:06