Logical quantifiers, which one is correct for this question? The question is: Let G(x,y) be "X can love Y", Domain: "all people in the world".
How can I express: 
"There is no one who can love everybody"
My two guess are: 
$$\neg \exists x (\forall y G(x,y))$$
or
$$\neg \forall x (\forall y G(x,y))$$
Which one is the correct one? Thanks.
 A: The first one, $\lnot\exists x(\forall y(G(x,y))$, is correct. It reads "It is not the case that there exists a person $x$ such that for all people $y$, $x$ can love $y$."
Your second guess, $\lnot\forall x(\forall y(G(x,y))$, is incorrect. It reads "It is not the case that for all people $x$ it is the case that for all people $y$, $x$ loves $y$."
The main idea is that the second one states that not everyone can love everyone (leaving the possibility that there exists someone who can), but the first one states that there is nobody who can love everyone.
A: $~\neg\exists x~\operatorname{DoesIt}(x)~$ says "there is not somebody who does it" or equivalently "everybody does not do it," which is $~\forall x~\neg\operatorname{DoesIt}(x)~$.
$~\neg\forall x~\operatorname{DoesIt}(x)~$ says "there is not everybody who does it" or equivalently "somebody does not do it," which is $~\exists x~\neg\operatorname{DoesIt}(x)~$.
A: The first one is correct.
Judging from the result, it says everyone doesn't have the feature that... So the result should have "any" not "exist".
