Signed Dual Tableau for Gödel–Dummett Logic

Abstract

In 1933 Kurt Gödel [Göd33] introduced a family of finitely many-valued propositional logics. His goal was to show that the intuitionistic logic cannot be characterized by a finite matrix. Dummett in [Dum59] generalized them to infinite-valued logics and presented their complete Hilbert-style axiomatization. It consists of the axioms of the intuitionistic propositional logic and the linearity axiom (φ → ψ) ∨ (ψ → φ). It is known that the set of tautologies of these logics is the same for any infinite set of truth values. Kripke-style semantics for these logics is determined by the intuitionistic Kripke models which are linearly ordered. The logic LC is an intersection of the sets of tautologies of all finite-valued Gödel logics. Gödel–Dummett logics have many applications both in logic and in computer science. Logic LC is employed in the investigations of the provability logic of the intuitionistic arithmetic [Vis82] and relevant logics [DM71]. It is applied to the foundations of logic programming [Pea99] and also it is considered as one of the most important fuzzy logics [Háj98].