Vol. 9(5), 2009

In this paper we extend the anodic systems introduced in [BS09] by adding certain paraconsistent axioms based on the so called logics of formal inconsistecy, introduced in [CCM07], and obtain the classes of systems that we call cathodic. These classes consist of modal paraconsistent systems and this approach permit us to treat with certain kind of conflicting situations. Our interest in this paper is to show that these classes of systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics.

An interesting result that was obtained in [BS09] was that the class of frames which characterizes the anodic systems is not trivial, in the sense that the class of anodic systems has at least an incompletable system. In the similar way we also show that the class PI^_k,l,m,n of cathodic systems also includes an incompletable system.