"Not Everybody likes everybody."
I have the problem that everybody is both the subject and object of that sentence.
I am not sure if that is correct: $\exists x \ \lnot \text l(x, x)$,
where x = everybody and $\text l(x, x) =$ "x likes x".
UPDATE 1: Second attempt
$\lnot \forall y \in x \ \forall z \in x \ \text l(y,z)$
UPDATE 2: There exists some person y that does not like every z:
$\exists y \ \lnot \forall z \ \text l(y,z)$
I disagree with user458276’s comment. You can’t say what $x$ is at the same time you use it as the variable for $\exists$. The variable after $\exists$ is “bound” and represents a particular but arbitrary element of a set (which you didn’t identify). For $x$ to take on a specific value, it must be “free” and able to be substituted with a value. You are also avoiding dealing with what “everybody” means by trying to let it be the value of a variable. If you had to write “Everybody likes cheese.” in propositional logic, would you let $C(x)$ mean “$x$ likes cheese” and write the sentence as $C(everybody)$?