"Everybody Loves a Lover" Isn't Z part of Everybody? I have a short circuit when I try to think of this formula meaning "Everybody Loves a Lover":
$\forall x \forall y ~[ \exists z (L(x,z) \land L(z,x)) \Rightarrow L(y,x)]$
my problem is that if I divide the universe of all people in $X,Y,Z$. Then we say that for all $X$ and $Y$, $Y$ will love $X$ only if $X$ is lover with atleast one $Z$, this does not tell me that also all of the $Z$ will love every lover $X$.. so it is incomplete in defining "everybody loves a lover" because everybody is missing the $Z$ people. But if $Z$ is just a different name to define the all of the poeple and $Y$ is another way to define the all of the poeple then it is ok. So point is that $X,Y$,$Z$ are the same set right? but we use them separately so to define diverse role in between the different people at different "times"? 
Thanks!
 A: $$\forall x~\forall y~\Big(\exists z~\big(L(x,z)\wedge L(z,x)\big)\to L(y,x)\Big)$$
The formula says: "$\text{Any $x$ will$\underbrace{\text{ be loved by }}_{\text{may be unrequired}}$any $y$ if that $x$$\underbrace{\text{ loves and is loved by }}_{\text{mutual love}}$some $z$.}$"
Or compactly: "Anyone who is a lover, is loved by everyone."
It does not claim that every lover loves everyone.

However, the PreNex form is indeed:
$$\forall x~\forall y~\forall z~\Big(\big(L(x,z)\wedge L(z,x)\big)\to L(y,x)\Big)$$
"For any $x,y,z$, then $y$ will love $x$ if $x$ loves and is loved by $z$."
A: You can't say that $X$, $Y$, and $Z$ are sets of people, but then also say things like '$Y$ will love $X$ if ...', because at that point you are treating $X$ and $Y$ as individual persons again, rather than as sets of people. Indeed, when you say '$Y$ will love $X$ if ...', then your $X$, $Y$, and $Z$ are no different than variables $x$, $y$, and $z$.
What is going on, though, is that $x$, $y$, and $z$ are all quantified, and when we use several quantifiers in a sentence, all quantifiers quantify over the exact same domain, which in this case is all people. 
And so yes, what you say in the end is captures the right idea correct: since the $z$ comes from the exact same domain as the $y$, it will in the end be true that whatever you picked for $z$, which is some person, will at some point also be picked for $y$ (since $y$ is going to range over all persons), and hence also that $z$ will end up loving everybody, as desired, even though you do not explicitly state that about $z$ in the formula.
