Fun with negation and idioms

This is trivial but kind of fun.

In my old analysis textbook, there was a problem to take old idioms and turn them into logical statements.

• “All that glitters is not gold” → “There exists an object that glitters and is not gold”.

• What makes the following sentence ambiguous: “a death row prisoner can’t have too much hope” → can’t could mean “can’t hope too much” or “shouldn’t hope too much”.

There was also one on negating implications.

“They will sink unless they swim”. In logic¹: if not swim then sink.

A simple definitional shift makes finding the negation trivial.

A handy identity is \(p \implies q \equiv \neg p \lor q\). The negation of that is \(p \land \neg q\). By rewriting the idiom as a disjunction, we get they can not swim and not sink.