Modalities and many-valued: modelling uncertainty measures and similarity-based reasoning and application to Fuzzy description Logics

The talk will began remembering results on papers [22, 15, 16] where the authors model uncertainty measures as fuzzy modalities on a many-valued residuated logics. the basic idea is that uncertainty measures is not a truth degree but could be interpreted as a truth degree of the fuzzy sentence ”the probability of φ”. The main properties being that the language does not allow nested modalities and their semantic is not defined as generalized Kripke models in strict sense. We will give the example of probabilities (possibilities, belief functions) over classical propositions and its generalizations to uncertainty measures on many-valued (fuzzy) propositions. But the main goal of the talk is modal many-valued fuzzy logics defined by Kripke models and some applications. Modal many-valued logics is a topic that has deserved few attention until the nineties. The first known papers are the papers [12, 13] of Fitting where the author defines a modal system over a logic of a finite Heyting algebra and give a complete axiomatization of them. In its late paper Fitting justify it in a very nice an elegant semantics based on a multi-agent system each one using classical logic and having a preference relation. He defined the modal operators based on semantics and his definitions is the ones used in later papers on the topic as generalizations modal operators on many-valued systems based on Kripke semantics (with many-valued worlds and many-valued accessibility relations). In these approaches Modal many-valued language is built taking the language of the many-valued logic (∧,∨,&,→&,¬&, 0, 1) plus at least one the usual modal operators (necessity 2 and possibility 3) and its semantic is defined by generalized Kripke models taking many-valued evaluation in each