How to translate "No husbands are wives" with binary predicates $H(x,y)$ , $W(x,y)$?

Question

How would you translate the sentence, "No husbands are wives," with these two binary predicates:

$\quad H(x,y)$ is defined to mean "$x$ is the husband of $y$."

$\quad W(x,y)$ is defined to mean "$x$ is the wife of $y$."

There is no predicate $H(x)$ that would mean just "$x$ is a husband."

Therefore, $\exists y\,H(x,y)$ would be the best translation for "$x$ is a husband."

My professor translates the sentence as $\forall x\,\forall y\,(H(x,y)\rightarrow¬W(x,y))$.

That looks to me like it means "No one is husband and wife to the same person," rather than the intended meaning, "If $x$ is the husband of some person, then $x$ is not the wife of any person."

I translated it as $\forall x\, ((\exists y\,H(x,y))\rightarrow(\forall y\, ¬W(x,y)))$.

Is there some way I am misunderstanding the statement $\forall x\,\forall y\,(H(x,y)\rightarrow¬W(x,y))$?

Answer 1

I completely agree with you. Yes, your professor's sentence means exactly what you say: No husband is the wife of the person he is the husband of. And that is not the same as saying that no husbands are wives (of anyone). And your translation for that is exactly correct!

For symmetry sake, you could do:

$\forall x (\exists y \: H(x,y) \rightarrow \neg \exists y \: W(x,y))$

which is of course by a single quantifier negation equivalent to your sentence.