On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics
Marcelo E. Coniglio, Francesc Esteva, Lluı́s Godo · Logic Journal of IGPL · 2016
The aim of this article is to explore the class of intermediate logics between the truth-preserving Łukasiewicz logic Ł and its degree-preserving companion Ł≤. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in Ł≤ and derivable in Ł 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 Łukasiewicz 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.