On extensions of minimal logic with linearity axiom

The Dummett logic is a superintuitionistic logic obtained by adding the linearity axiom to intuitionistic logic. This is one of the first non-classical logics, whose lattice of axiomatic extensions was completely described. In this paper we investigate the logic $JC$ obtained via adding the linearity axiom to minimal logic of Johansson. So $JC$ is a natural paraconsistent analog of the Dummett logic. We describe the lattice of $JC$-extensions, prove that every element of this lattice is finitely axiomatizable, has the finite model property, and is decidable. Finally, we prove that $JC$ has exactly two pretabular extensions.