Bi-modal Gödel logic over [0,1]-valued Kripke frames

We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1] and prove strong completeness of Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of T, S4, and S5 obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras. In a previous paper [6], we have considered a semantics for Gödel modal logic based on fuzzy Kripke models where both the propositions and the accessibility relation take values in the standard Gödel algebra [0,1], we call these Gödel-Kripke models, and we have provided strongly complete axiomatizations for the uni-modal fragments of this logic with respect to validity and semantic entailment from countable theories. The systems G and G3 axiomatizing the -fragment and the 3-fragment, respectively, are obtained by adding to Gödel-Dummet propositional calculus the following axiom schemes and inference rules: G : (φ→ ψ) → ( φ→ ψ) ¬¬ φ→ ¬¬φ From φ, infer φ G3: 3(φ ∨ ψ) → (3φ ∨3ψ) 3¬¬φ→ ¬¬3φ ¬3⊥ From φ→ ψ, infer 3φ→ 3ψ Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia; [email protected] Departamento de Computación, Fac. Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina; [email protected]