Models for anodic and cathodic multimodalities

A system is classified as multimodal if its language has more than one modal operator as primitive, and such operators are not interdefinable. We extend the anodic and cathodic modal systems, introduced in [BS09a] and [BS10], to a class of the so-called basilar multimodal systems generating, in this way, the classes of anodic and cathodic multimodal logics. The cathodic multimodal systems are defined as extensions of positive multimodal systems (anodic multimodal systems) by adding degrees of negation plus consistency (and inconsistency) operators. In this way, cathodic multimodal systems are logics of formal inconsistency (the paraconsistent LFIs, as treated in [CCM07]) enriched with multimodal operators. We focus the attention on models for such classes of systems and discuss how modal possible-translations semantics, as well as possible-worlds (or Kripke semantics), can be defined to interpret basilar cathodic multimodal systems. While anodic systems are modeled by Kripke models only, we introduce the modal possible-translations models for cathodic systems. Such models, given by combinations of three-valued modal logics, besides their own interest, explain the role of non-trivializing contradictions in multimodal environment.