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

Abstract

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.

Maximality in finite-valued Łukasiewicz logics defined by order filters

“…Remark 3. By using the techniques presented in [15], a sound and complete Hilbert calculus for each L i q (where i < q) can be defined from the one for L ≤ q+1 (the degree-preserving counterpart of L q+1 ) by adding additional inference rules. The negative feature of such approach is that these Hilbert calculi have "global" inference rules, that is, inference rules such that one of its permises need to be a theorem of L q+1 .…”

“…Remark 3. By using the techniques presented in [15], a sound and complete Hilbert calculus for each L i q (where i < q) can be defined from the one for L ≤ q+1 (the degree-preserving counterpart of L q+1 ) by adding additional inference rules.…”