Beyond Degrees of Truth and the Axiomatization of Superintuitionistic Prime Mixed Models

Abstract

Introduction This thesis can be divided into two main parts. In the first part, after two chapters of overview of preliminary knowledge, we introduce a new family of logics, which we call order-based logics, and provide a Hilbert-style presentation for them. These logics are obtained by taking the axioms of a logic associated with a variety of bounded integral commutative residuated lattices and replacing the usual rule of modus ponens with a restricted version thereof. The motivation for introducing these systems is to study weaker counterparts of the well-known logics preserving degrees of truth associated with varieties of (bounded) integral commutative residuated lattices. The resulting systems form a family of paraconsistent logics for which we provide a natural matrix semantics. We also determine their position within the Leibniz and Frege hierarchies. In particular, we show that all order-based logics are fully selfextensional, while none of them is protoalgebraic and, all of them fail to be (fully) Fregean. Furthermore, we prove that the order-based logic associated with classical logic coincides with a language reduct of the well-known paraconsistent logic D2. As a consequence, this logic admits an interpretation into the global modal logic S5 as well as the local modal logic T via what Priest calls the Jaśkowski move in [11]. This result serves as the main motivation for the second part of the thesis.