When is a Substructural Logic Paraconsistent? Structural conditions for paraconsistency in ternary frames

In this work, we study structural, model-theoretical conditions that support paraconsistency in Substructural Logics. The idea is to follow the notion of Correspondence Theory from Modal Logics and apply it to Substructural Logics. Several logics in the family of Substructural Logics were initially defined with goals similar to those of Paraconsistent Logic. There are several possible ways of defining paraconsistency, but this work takes a neutral way towards all such definitions. We note that the formalization of such definitions vary according to the set of connectives present in the logical language, and also according to whether we view paraconsistency as the possibility to deny the principles of Non-contradiction or Trivialization. All this yields a number of possible definitions of paraconsistency. We propose a method that allows us to compute which effects a given definition may have upon the model theoretical structures of a Substructural Logic that adopt one such definition. It has been known since the work of Routley and Meyer [RM73] that binary logical connectives can be seen as modalities interpreted over Kripke frames (W, R) with a ternary accessibility relationship R subset of or equal to W x W x W. More recently, a correspondence theory was developed for substructural logics in analogy to the usual modal correspondence theory. In this a setting, we derive structural restrictions over ternary frames corresponding to the violation of a consistency condition, that is, an axiom. Such a process is performed on a fragment consisting of the connectives circle times (tensor product, also called multiplicative conjunction), -->(multiplicative implication), - (classical negation), similar to (intuitionistic negation) and boolean AND (classical conjunction).