Two-layer modal logics: from fuzzy logics to a general framework
Petr Cintula, Carles Noguera · EPiC series in computing · 2018
The idea of two-layer modal logics is inspired by the treatment of probability inside mathematical fuzzy logic, pioneered by Hajek and recently studied by numerous authors in numerous papers. Such logics are used in order to deal with a certain property of formulas of the base logic using a suitable `upper' logic (the seminal example being the probability of classical events formalized inside Lukasiewicz logic). The primary aim of this paper is to provide a new general framework for two-layer modal logics that encompasses the current state of the art and paves the way for future development. Diverting for the area of mathematical fuzzy logic, we show how one can construct such modal logic over an arbitrary non-classical logic (under certain technical requirements) with a modality interpreted by an arbitrary measure. We equip the resulting logics with a semantics of measured Kripke frames and prove corresponding completeness theorems. As an illustration of our results, we reprove Hajek's completeness result for Fuzzy Probability logic over Lukasiewicz logic.