A new theory consistency index based on deduction theorems in several logic systems

Based on deduction theorems, completeness theorems and by means of the theory of truth degrees of formulas and the concept of divergence degrees of theories, the present paper proposes another new index reflecting the extent to which a general theory is consistent in five different logic systems, i.e., classical (two-valued) logic system C2, Łukasiewicz fuzzy logic system Łuk, Gödel fuzzy logic system Göd, product fuzzy logic system Π and the R0-fuzzy logic system L*. The concepts of normal consistency and almost inconsistency of theories in the above-mentioned logic systems are introduced, and sufficient and necessary conditions for theories being normal consistent or almost inconsistent are given. Finally, comparison with the existing concept of consistency degrees of theories is analyzed.