Many AI applications are based on some underlying logic that tolerates inconsistent information in a non-trivial way. However, it is not always clear what should be the exact nature of such a logic, and how to choose one for a specific application. In this paper, we formulate a list of desirable properties of `ideal' logics for reasoning with inconsistency, identify a variety of logics that have these properties, and provide a systematic way of constructing, for every n>2, a family of such n-valued logics.

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper, we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We first show that most of the logics that are based on properly non-deterministic three-valued matrices are not maximally paraconsistent. Then we show that in contrast, in the deterministic case all the natural three-valued paraconsistent logics are maximal. This includes well-known three-valued paraconsistent logics like P1, LP, J3, PAC and SRM3, as well as any extension of them obtained by enriching their languages with extra three-valued connectives.