From Inconsistency to Incompatibility

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