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.

  1. That sounds like a caveman. 

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