I can't think of any examples of ambivalent axioms in mathematics (two ideas are ambivalent if there are sometimes conflicts between them), so please let me humor you with a strange example. Suppose someone was working on a mathematical theory of economies and wanted to account for everyone's perspective and also minimize harm. So in addition to these two axioms, "harm" and "perspective" and "valid" would have to be defined:
1) Do no harm 2) All perspectives are valid
So, in the theory, suppose you come to a point where you want to apply axiom 2 but it conflicts with axiom 1, for instance, a portion of the people have a perspective such that if their will were carried out without being moderated by the other peoples' perspective, there would be harm caused to people, some other life on the planet, or the ecosystem. Then what do you do to resolve this conflict? One solution is to "split" the theory into two branches, one in which you ignore 1, and the other in which you ignore the perspectives of that harm-causing group. There are other ways to do this too, but this is just a stupid example anyway.
So with that scenario in mind, when there are ambivalent axioms, the theory branches into a tree of results in a way that is unlike conventional mathematics.
So my questions are, have I made a brain fart? If not, could ambivalent-axiom theories be transformed into conventional non-ambivalent axiom theories after all definitions given? Would this help mathematics at all or is this solely the territory of other sciences like psychology and psychohistory? Is this abuse of the beautiful, well-tested mechanisms of mathematics, and if there were some conflicting idea in a theory it would enter the stage in another way like an "if statement."
My apologies for not thinking this through myself. I'm not in a mathy perspective lately and am lazy :D