What do you mean with non-logic? I think that's hard to define. There are various approaches in logic-based AI that aim to somehow be more akin to how we handle multiple conflicting truths. A popular formalism is abstract argumentation theory, which is the study of evaluating the acceptability of multiple conflicting arguments. It has been shown that this is closely related to non-monotonic logics, such as LP with negation as failure and various types of defeasible reasoning. Proof theoretically, it replaces the notion of proof with that of a successful strategy in a two-person dialogue game.