Gentzen-type Refutation Systems for Three-Valued Logics

While the purpose of a conventional proof calculus is to axiomatise the set of valid sentences of a given logic, a refutation system, or complementary calculus, is concerned with axiomatising the invalid sentences. Instead of exhaustively searching for counter models for some sentence, refutation systems establish invalidity by deduction and thus in a purely syntactic way. Such systems are relevant not only for proof-theoretic reasons but also for realising deductive systems for nonmonotonic logics. In this paper, we introduce Gentzen-type refutation systems for two basic three-valued logics that allow to embed well-known three-valued logics relevant for AI and logic programming like that of Kleene, Łukasiewicz, Gödel, as well as three-valued paraconsistent logics. As an application of our calculus, we provide derived rules for Gödel’s three-valued logic, allowing to decide strong equivalence of logic programs under the answer-set semantics.