> I'm not trying to say one is right and the other is wrong! These are just the names I learned for them in school.
Me too. My point was a bit different: I was saying that the poset-based definition of "not" that I sketched in [#34](https://forum.azimuthproject.org/discussion/comment/16758/#Comment_16758) applies to both classical and intuitionistic logic.
Category theorists tend to like intuitionistic logic, which includes classical logic as a special case: the internal logic of a topos is intuitionistic, but when the topos is "Boolean" the principle of excluded middle will hold, and the logic is classical. We may get to a bit of this later in the book.