Translating predicate logic to english

Question

I'm super confused about translating from predicate logic to english and vice versa. I can't find any good explanation that I can generally follow.

Here is an example:

"Nobody is the judge for a case that s/he prosecutes."

I would choose

J(x,y) = x is a judge of case y

P(x,y) = x is the prosecutor of case y

Now I translated it as follows:
$∃x∀y(P(x,y) → ∀z¬J(z,y)$

So there exists someone that for any case the person prosecutes, there is nobody that will be the judge for that case.

Now the correct answer was:
$∀x∀y(P(x,y) → ¬J(x,y))$

I just don't understand why and how you translate these with certainty. I would really appreciate some tips here.

Answer 1

You're not reading that sentence in the way that was anticipated. You're imagining that I'm pointing at a lawyer and saying "That's Jim. Nobody is the judge for a case that he prosecutes." The interpretation they intended is that no case has a judge and a prosecutor who is the same person.