The Class of Extensions of Minimal Logic

In this chapter, we assign to every properly paraconsistent extension L of minimal logic an intermediate logic L _int and negative logic Lneg called intuitionistic and negative counterparts of L, respectively. It will be proved that the negative counterpart Lneg explicates the structure of contradictions of paraconsistent logic L. We show that both counterparts Lint and Lneg are faithfully embedded into the original logic L. Finally, we investigate a question: to what extent is a logic L ∈ Par determined by its counterparts? As a first step, we study paraconsistent extensions of the logic${\rm Le' : = Li } \cap {\rm Ln = Lj + }\left\{ { \bot \vee \left( { \bot \to p} \right)} \right\}.$

The class of extensions of this logic has a nice property that every logic L ∈ εLe′ ∩ Par is uniquely determined by its intuitionistic and negative counterparts.