On Some Systems for Two Versions of Many-valued Logics

The current research refers to the problem of constructing several proof systems for two versions of many-valued propositional logic and investigating of their properties . The generalization of Kalmar’s proof of deducibility for two-valued tautologies in the classical propositional logic gives us a possibility to suggest 1) a new method of proving the completeness of propositional proof system of three-valued logic of Lukasewicz that it is essentially simpler than other known proofs of completeness and can be easily modified into a proof of completeness for other versions of k-valued logics for k≥3 and even for fuzzy logic as well 2) a method of defining many traditional variants of proof systems for k-valued (k≥3) logics the completeness of which is easily proved directly without the usual immersion into two-valued logic. Most of all the introduced proof systems are “weak” ones with a “simple strategist” of proof search and we have also investigated the quantitative properties related to proof complexity characteristics in them.