Complementary Logics for Classical Propositional Languages

In previous work, I introduced a complete axiomatization of classical non-tautologies based essentially on Łukasiewicz’s rejection method. The present paper provides a new, Hilbert-type axiomatization (along with related systems to axiomatize classical contradictions, non-contradictions, contingencies and non-contingencies respectively). This new system is mathematically less elegant, but the format of the inferential rules and the structure of the completeness proof possess some intrinsic interest and suggests instructive comparisons with the logic of tautologies.