Is there any logic to required double negatives, or is that "just the way it is"?

All this demonstrates that you cannot apply simple predicate logic to language - and definitely not the rules of programming languages. The negative markers in different languages work according the internal rules of agreement.

Of course, this is just a matter of surface marking and convention. The scope of the negative is determined partly by syntax and partly by context and works differently in different languages. For example, languages with grammatical double negatives make it easier to distinguish between things like 'I have nothing' and 'I don't have anything' in the rare examples where you want to express the possession of nothingness. But even in English, it's possible to make that contextually clear.