Possibility and necessity theories rival with probability in representing uncertain knowledge, while offering a more qualitative view of uncertainty. Moreover, necessity and possibility measures constitute, respectively, lower and upper bounds for probability measures, with the advantage of avoiding the complications of the notion of probabilistic independence.

On the other hand, paraconsistent formal systems, especially the Logics of Formal Inconsistency, are capable of quite carefully expressing the circumstances of reasoning with contradictions. The aim of this talk is to merge these ideas, by precisely defining new notions of possibility and necessity theories involving the concept of consistency (generalizing the proposal by (Besnard & Lang 1994)) based on connecting them to the notion of partial and conclusive evidence. This combination permits a whole treatment of contradictions, both local and global, including a gradual handling of the notion of contradiction, thus obtaining a really useful tool for AI and machine learning, with potential applications in logic programming via appropriate resolution rules.