On Rich Modal Logics

Abstract

This thesis is about the enrichment of modal logics. We use the term enrichment in two distinct ways. In the first of them, it is a semantical enrichment. We propose a fuzzy semantics to di erent normal modal logics and we prove a completeness result for a generous class of this logics enriched with multiple instances of the axiom of confluence. A curious fact about this semantics is that it behaves just like the usual boolean-based Kripke semantics for modal logics. The other enrichment is about the expressibility of the logic and it occurs by means of the addition of new connectives, essentially modal negations. In this sense, firstly we study the positive fragment of classical logic extended with a paraconsistent modal negation and we show that this language is su ciently strong to express the normal modal logics. It is also possible to define a paracomplete modal negation and restoration connectives that internalize at the level object-language the notions of consistency and determinedness. This logic constitutes a Logic of Formal Inconsistency and a Logic of Formal Undeterminedness.In such logics, with the objective of recovering lost inferences of classical logic, Derivability Adjustment Theorems are proved. In the case of the logic with one paraconsistent negation, if we remove the implication we still have a rich language, with both paranormal negations and its respective connectives of restoration. In this logic we study the minimal normal modal logic defined by means of a Gentzen calculus, differently of the others modal systems studied, which are presented by means of Hilbert calculus. Next, after we prove a ompleteness result of the deductive system associated to this calculus, we present some extensions of this system and we look for appropriate Derivability Adjustment Theorems.