2020/03/20 by Walter Carnielli, Marcelo E. Coniglio, Carnielli, Walter +3 · 1 citation
Computer Science · #03B45 #03B53 #03B62 #03G27 #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2003.09522
openalex publication_date 2020/03/20 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
It is customary to expect from a logical system that it can be algebraizable,\nin the sense that an algebraic companion of the deductive machinery can always\nbe found. Since the inception of da Costa's paraconsistent calculi Cn,\nalgebraic equivalents for such systems have been sought. It is known, however,\nthat these systems are not self-extensional (i.e., they do not satisfy the\nreplacement property). More than this, they are not algebraizable in the sense\nof Blok-Pigozzi. The same negative results hold for several systems of the\nhierarchy of paraconsistent logics known as Logics of Formal Inconsistency\n(LFIs). Because of this, several systems belonging to this class of logics are\nonly characterizable by semantics of a non-deterministic nature. This paper\noffers a solution for two open problems in the domain of paraconsistency, in\nparticular connected to algebraization of LFIs, by extending with rules several\nLFIs weaker than C1 , thus obtaining the replacement property (that is, such\nLFIs turn out to be self-extensional). Moreover, these logics become\nalgebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety\nof Boolean algebras extended with additional operations. The weakest LFI\nsatisfying replacement presented here is called RmbC, which is obtained from\nthe basic LFI called mbC. Some axiomatic extensions of RmbC are also studied.\nIn addition, a neighborhood semantics is defined for such systems. It is shown\nthat RmbC can be defined within the minimal bimodal non-normal logic E+E\ndefined by the fusion of the non-normal modal logic E with itself. Finally, the\nframework is extended to first-order languages. RQmbC, the quantified extension\nof RmbC, is shown to be sound and complete w.r.t. the proposed algebraic\nsemantics.\n