On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics

The aim of this paper is to explore the class of intermediate logics between the truth-preserving Lukasiewicz logic L and its degree-preserving companion L≤. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in L≤ and derivable in L and we characterize the corresponding intermediate logics. From a semantical point of view, we first consider the family of logics characterized by matrices defined by lattice filters in [0, 1], but we show there are intermediate logics falling outside this family. Finally, we study the case of finite-valued Lukasiewicz logics where we axiomatize a large family of intermediate logics defined by families of matrices (A, F ) such that A is a finite MV-algebra and F is a lattice filter.