Signed Dual Tableau for Gödel–Dummett Logic

Abstract

AbstractIn 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].KeywordsLogic ProgrammingIntuitionistic LogicPropositional VariableRelevant LogicBranch StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.