LOGICS OF FORMAL INCONSISTENCY ENRICHED WITH REPLACEMENT: AN ALGEBRAIC AND MODAL ACCOUNT

Abstract

AbstractIt is customary to expect from a logical system that it can bealgebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa’s paraconsistent calculi, algebraic equivalents for such systems have been sought. It is known, however, that these systems are not self-extensional (i.e., they do not satisfy thereplacement property). More than this, they are not algebraizable in the sense of Blok–Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known asLogics of Formal Inconsistency(LFIs). Because of this, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization ofLFIs, by extending with rules severalLFIs weaker than$C_1$, thus obtaining the replacement property (that is, suchLFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum–Tarski’s sense by a suitable variety of Boolean algebras extended with additional operations. The weakestLFIsatisfying replacement presented here is calledRmbC, which is obtained from the basicLFIcalledmbC. Some axiomatic extensions ofRmbCare also studied. In addition, a neighborhood semantics is defined for such systems. It is shown thatRmbCcan be defined within the minimal bimodal non-normal logic$\mathbf {E} {\oplus } \mathbf {E}$defined by the fusion of the non-normal modal logicEwith itself. Finally, the framework is extended to first-order languages.RQmbC, the quantified extension ofRmbC, is shown to be sound and complete w.r.t. the proposed algebraic semantics.