Two Semantical Approaches to Paraconsistent Modalities

Abstract

In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291-310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency, introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic. These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics. Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al.(Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and manyvalued logics, defining new blends of logics whose relevance we intend to show. Mathematics Subject Classification (2000). Primary 03B45, 03B53; Secondary 03B50, 03B62.

Models for anodic and cathodic multimodalities

“…First of all, we introduce the syntax of the anodic and cathodic multimodal systems and then show how these classes of multimodal systems can be characterized by means of multi-relational (Kripke-style) models. By using this result, it is also possible to obtain, for the basilar cathodic multimodal systems, a second semantical characterization by means of modal possible-translations semantics, extending the results for the monomodal cases obtained in [BS10].…”

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

“…If we agree with this, the notion of probability and its connection to logic are of fundamental importance. Of course, it is also possible to propose a modal paraconsistent approach to probability based on paraconsistent modalities (see [38]), following the lines of [39], but this has to wait for a better motivation.…”

LFIs Based on Other Logics

“…A more intimate relationship between LFIs and their modal versions is investigated by Bueno-Soler in [55], where the so-called anodic modal systems (purely positive modal systems) introduced in [56] are extended by adding certain paraconsistent axioms based on LFIs, defining a class of modal systems called cathodic modal systems. This class consists of modal paraconsistent systems, dealing with certain kinds of conflicting modal situations.…”

“…This class consists of modal paraconsistent systems, dealing with certain kinds of conflicting modal situations. A semantic interpretation is also given in [55], where it is shown that the cathodic systems can be semantically characterized in two different ways: by means of Kripke-style semantics and by means of modal possible-translations semantics.…”

“…(where m and ♦ n mean the iteration of modalities), are given in detail in [55]. Modal logics have also been built on top of other paraconsistent logics, as in [16], Nelson's logic N4 (analyzed in Sect.…”