Let's say $P(x,y)$ means $x$ sent an e-mail to $y$.
If I want to say that no one has sent a message to Jean, then aren't there multiple ways to do this?
$\neg Ǝx(P(x, Jean))$
But I can also say
$∀x(\neg P(x, Jean))$ Aren't these equivalent? Or am I missing something?
Yes. The "negative" of "there is" is the same as "all" who "haven't."