I'm trying to understand the following examples of expressing an English sentence using nested quantifiers.
Example One
"Everybody loves somebody else."
$\forall x \exists y ((x \ne y) \land L(x, y))$
I was told that this means that everybody loves at least one other person, but that it does not exclude the self: $L(x,x)$.
Example Two
"Everybody loves somebody other than self."
$\forall x \exists y (L(x, y) \land \neg L(x, x))$
Apparently this does exclude the self.
I don't see the difference between these examples. As far as I understand, $(x \ne y)$ is functionally equivalent to $\neg L(x, x)$. Because $(x \ne y)$ is always false at the diagonals, it seems like the first example should also exclude the self.
Any help would be greatly appreciated. :)
The clause "$(x \neq y) \wedge L(x,y)$" explicitly asserts the truth of $x$ loving $y$ for some $y$ that is different from $x$, but is completely silent on the truth of $L(x,x)$. (The inequality condition ensures that the clause cannot speak to the truth or falsity of $L(x,x)$.)
The clause "$L(x,y) \wedge \neg L(x,x)$" asserts both $L(x,x)$ is false and $L(x,y)$ is true. This forces $y \neq x$ and also explicitly assigns a truth value to $L(x,x)$.