I feel like logic isn't as universal as it should be when you don't teather it to common sense at an axiomatic level. If we have $P(X)$, $Q(X)$ propostions over some object $X$ such that $P(X) \iff \neg Q(X)$ then we can use common sense to deduce that both properties can't act on $X$ at the same time, so we take that as an axiom.
But what if absurd statements like that aren't as absurd as we thought? It seems impossible due to the fact that we are working with axiomatic definitions of basic logical statements, but we only assigned those definitions because they made sense to us and because they were consistent together.

To a frog that lives in a world were time flows backwards there are many propositions about frogs which no longer apply to this particular, in fact, the negation of many statements we had for frogs are now applicable to him.
Is there a way to make logic as universal as possible for any scenario such that you don't have to specify the original axioms in order to spit out true statements? A sort of logic that's invariant under any circumstance?