On Modal Expansions of Left-continuous T-norm Logics

Modal fuzzy logics is a research topic that has attracted increasing attention in the last years. Several papers have been published treating different aspects, see for instance [6] for a modal expansion of Lukasiewicz logic, [3, 4, 2] for modal expansions of Gödel fuzzy logic, [1] for modal logics over finite residuated lattices, and more recently [7] for a modal expansion of Product fuzzy logic. In this paper we extend the latter approach (based on infinitary proof systems) by considering the axiomatization problem of the modal expansion of the propositional logic of a left-continuous t-norm (with rational truth constants and Monteiro-Baaz ∆ operator) using crisp accessibility relations in the Kripke models over such tnorm. We provide explicit strongly complete axiomatizations solving such problem for a large family of t-norms (including all ordinal sums of Lukasiewicz and Product t-norms). Unfortunately, the proof is not general enough to deal with all such axiomatization problems, e.g., the problem remains open for Gödel t-norm. The technique used in the strong completeness proof involves building a canonical model, and such approach requires to previously know a strongly complete axiomatization of the propositional logic of a left-continuous t-norm (with rational truth constants and Monteiro-Baaz ∆ operator). Therefore, before jumping into the modal discussion we discuss several issues of this non modal logic. For an arbitrary left-continuous t-norm ∗, in [8] we introduced an axiomatic system L∗ that was proved to be strongly complete with respect to the algebra denoted by [0,1]∗ and consisting of expanding the MTL-chain [0,1]∗ (i.e., the one with domain the unit real interval and associated with ∗) with the ∆ operator and rational constants. The axiomatization there given is obtained adding to any known axiomatization of MTL∆ the usual book-keeping axioms for &,→ and ∆ connectives, and the following infinitary density rule