Some Epistemic Extensions of Gödel Fuzzy Logic

In this paper we prove soundness and completeness of some epistemic extensions of Gödel fuzzy logic, based on Kripke models in which both propositions at each state and accessibility relations take values in [0,1]. We adopt belief as our epistemic operator, acknowledging that the axiom of Truth may not always hold. We propose the axiomatic system K_F serves as a fuzzy variant of classical epistemic logic K, then by considering consistent belief and adding positive introspection and Truth axioms to the axioms of K_F, the axiomatic extensions B_F and T_F are established. To demonstrate the completeness of K_F, we present a novel approach that characterizes formulas semantically equivalent to ⊥ and we introduce a grammar describing formulas with this property. Furthermore, it is revealed that validity in K_F cannot be reduced to the class of all models having crisp accessibility relations, and also K_F does not enjoy the finite model property. These properties distinguish K_F as a new modal extension of Gödel fuzzy logic which differs from the standard Gödel Modal Logics 𝒢_ and 𝒢_ proposed by Caicedo and O. Rodriguez.