From Inconsistency to Incompatibility

The aim of this article is to generalize logics of formal inconsistency (LFIs) to systems dealing with the concept of incompatibility, expressed by means of a binary connective. The basic idea is that having two incompatible formulas to hold trivializes a deduction, and as a special case, a formula becomes consistent (in the sense of LFIs) when it is incompatible with its own negation. We show how this notion extends that of consistency in a non-trivial way, presenting conservative translations for many simple LFIs into some of the most basic logics of incompatibility, thereby evidencing in a precise way how the notion of incompatibility generalizes that of consistency. We provide semantics for the new logics, as well as decision procedures, based on restricted non-deterministic matrices. The use of non-deterministic semantics with restrictions is justified by the fact that, as proved here, these systems are not algebraizable according to Blok-Pigozzi nor are they characterizable by finite Nmatrices. Finally, we briefly compare our logics to other systems focused on treating incompatibility, specially those pioneered by Brandom and further developed by Peregrin.