On extensions of minimal logic with linearity axiom

Abstract

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 lattices is finitely axiomatizable, has the finite model property, and is decidable. Finally, we prove that JC has exactly two pretabular extensions.