The Lattice of Extensions of the Minimal Logic

In this article, we survey the results on the lattice of extensions of the minimal logic $\mathbf{Lj}$, a paraconsistent analog of the intuitionistic logic $\mathbf{Li}$. Unlike the well-studied classes of explosive logics, the class of extensions of the minimal logic has an interesting global structure. This class decomposes into the disjoint union of the class {\tt Int} of intermediate logics, the class {\tt Neg} of negative logics with a degenerate negation, and the class {\tt Par} of properly paraconsistent extensions of the minimal logic. The classes {\tt Int} and {\tt Neg} are well studied, whereas the study of {\tt Par} can be reduced to some extent to the classes {\tt Int} and {\tt Neg}.