Some Epistemic Extensions of G\"odel Fuzzy Logic

In this paper we prove soundness and completeness of some epistemic extensions of G\"odel 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 $\textbf{K}_\textbf{F}$ serves as a fuzzy variant of classical epistemic logic $\textbf{K}$, then by considering consistent belief and adding positive introspection and Truth axioms to the axioms of $\textbf{K}_\textbf{F}$, the axiomatic extensions $\textbf{B}_\textbf{F}$ and $\textbf{T}_\textbf{F}$ are established. To demonstrate the completeness of $\textbf{K}_\textbf{F}$, we present a novel approach that characterizes formulas semantically equivalent to $\perp$ and we introduce a grammar describing formulas with this property. Furthermore, it is revealed that validity in $\textbf{K}_\textbf{F}$ cannot be reduced to the class of all models having crisp accessibility relations, and also $\textbf{K}_\textbf{F}$ does not enjoy the finite model property. These properties distinguish $\textbf{K}_\textbf{F}$ as a new modal extension of G\"odel fuzzy logic which differs from the standard G\"odel Modal Logics $\mathcal{G}_\Box$ and $\mathcal{G}_\Diamond$ proposed by Caicedo and O. Rodriguez.