I'm generally aware that there are logics beside classical predicate logic—fuzzy logic, modal logic, and paraconsistent logics, just to name a few.
I appreciate whenever I learn about how some aspect of the mathematics with which I'm aware is more rigorous than I had thought, which is why I appreciate the consistency (as we've thus far seen) of ZF(C) set theory.
The question here is whether the rules of the logic in question are considered part of the axiomatic foundation: for any given consistent mathematical foundation, could we use any logic we choose in combination with any collection of set-theoretic axioms?
My confusion began with learning classical logic and predicate logic, since, although it made sense to me, it also didn't seem to have concretely stated or summarizable axioms (like ZF(C) does). As far as I can tell, logic exists to express the conditions for both the truth and validity of a logical statement, and predicate logic exists to analyze the conditions under which indeterminates within a logical statement can change the truth value (but not the validity) of those statements. I generally understand inference rules and truth tables, but I don't know if there's one formal language in which I could sequentially write each step in the process of dealing with a logical problem.
How would results formulated in ZF(C) set theory would change if the logical foundation were shifted to some other non-classical logic? What are the axioms of predicate logic? And what is the relationship between these?
For reference, I'm just finishing up my first semester of Discrete Mathematics, but you don't have to go easy on me. I'd like to know what domain of math I should research to learn more on this line of questioning.
(P.S.: I'm using "ZF(C)" to acknowledge that not everyone accepts the Axiom of Choice.)