Express in logical form: There is exactly one person John likes.
Doing this out we can write it using the "There is exactly one" quantifier $\exists!$. Letting $L(x,y)$: $x$ likes $y$. This can be written as $\exists!xL(J,x)$. After seeing this I was wondering how we can negate it. If we were to not use the "There is exactly one" quantifiers it would become $\exists x\forall y(L(J,x)\land\lnot (L(J,y)\land y\neq x))$ Then if I were to negate it all we get $\forall x\exists y(\lnot L(J,x)\lor (L(J,y)\land y\neq x))$ Which to me reads as, John does not like everyone or John likes at least one person, and that person is not $x$. The negation that I have seems to lose the fact that John likes more than one person, regardless of if that person is $x$ or not. So how would I go about negating this statement? Is there a quantifier that is the opposite of $\exists!$?