Modal (logic) paraconsistency

According to the standard definition, a logic is said to be paraconsistent if it fails the (so-called) rule of ex falso: i.e., α, ¬α ∀ β.Thus, paraconsistency captures an important sense in which a logic is inconsistency-tolerant, namely when arbitrary inference is prohibited in the presence of inconsistencies. We investigate a family of notions of paraconsistency within the context of modal logics. An illustration comes from an epistemic version of the lottery paradox showing how important it sometimes is to distinguish between ordinary and higher order beliefs: A logic may fail to tolerate inconsistent beliefs at a given level while tolerating inconsistent beliefs across different levels. We identify a few properties arising from some well-known modal theorems and inference rules in order to classify modal logics according to their capacity to tolerate modalized inconsistencies. In doing so, we show various relationships among these logics.