I did not learn categorical logic as part of my training. I can reason through syllogisms just fine, but I find some of the terminology and standard examples counter-intuitive. (I recognize that this is just a coincidence. I'm sure it's intuitive to a person trained in it.) Some folks in my department include it in their logic curriculum so I get random student questions about it and I feel like an idiot.

Can someone explain the underlying logic to disjunction and conjunction. I feel like I am missing some history or practical context.

First, why do we call them this? It seems like a strict disjunctive just is a differently written conjunctive. Further it seems like in any other context "disjunctive" just means "mutally exclusive" which seems to be what conjunctives are. (For some reason the terms never bothered me while doing propositional and predicate logic.)

Second, for what practical reasons are common examples of conjunctives given as ~(A ∧ B)? I assume that this is because major premises like (A ∧ B) are unhelpful or uncommon in certain kinds of arguments common to categorical logic.

Can anyone recommend a good primer on the subject?