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.

Answer 2

Several possible tips.

• The most straightforward formal version of a sentence that begins "nobody" will probably not begin with "there exists".

• After you've written a tentative formal answer, try reading it back into English as if you didn't know what it said. That might have helped you find your error here (though perhaps not helped you fix it).

• Try reformulating the English. In this case perhaps "If $x$ prosecutes case $y$ then $x$ does not judge case $y$" or "If $x$ prosecutes case $y$ and judges case $z$ then $ y \ne z$", These versions may lead to different (but equivalent) formal statements. You may find one of them easier to deal with. Together they help you grasp the underlying logic.

Answer 3

in my opinion when you have a sentence that starts like "everybody" or "nobody", then the sentence that follows is something true or false for everyone, so "everybody..." translates into $\forall x...$, while "nobody..." translates into $\forall x\neg(...)$. For example, the sentence that follows "nobody" is "is the judge for a case s/he is prosecuting", so there is a case y which is prosecuted by x and judged by x, so $\exists y(J(x,y)\wedge P(x,y))$, so the translation turns out to be $$\forall x\neg(\exists y(J(x,y)\wedge P(x,y)))$$which means for all x there is no case y in which x is both judge and prosecutor. Now, using that $\neg(\exists yA)\equiv\forall y\neg A$ and $B\vee\neg A\equiv A\Rightarrow B$, $\neg(\exists y(J(x,y)\wedge P(x,y))\equiv\forall y\neg(J(x,y)\wedge P(x,y))\equiv \forall y(\neg J(x,y)\vee \neg P(x,y))\equiv\forall y(P(x,y)\Rightarrow \neg J(x,y))$

notice that $\forall x\forall y\neg(J(x,y)\wedge P(x,y))$ is not the correct answer, but it is logically (from classical logic) equivalent to the "correct" solution. So if you find yourself with a translation that does not match the one given as a solution, try and look the two are logically equivalent.