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^{\leq}. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in \L^{\leq} 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.

}, author = {Marcelo Coniglio and Francesc Esteva and Llu{\'\i}s Godo} }