Mathematics recognizes that there are logics different from the propositional logic and the predicate logic. I have studied a few of them myself.
Then... why are all proofs in math textbooks based on propositional and predicate logics only?
I mean, why do these proofs assume that it's either $A$ or not $A$? Or... why do they assume that the negation of "for all $X$, we have that $P(x)$ holds true" is "there exists an $X$ such that not $P(x)$"?
Isn't this a flaw in the credibility of math itself (or say of all these texts)?
I guess it's not but I am not sure why.