Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems

Abstract

The present paper carries out a deep analysis on the inconsistency of theories by means of deduction theorems, completeness theorems and satisfiability degrees of formulas, and introduces the concept of the degree of entailment of a contradiction from a theory in classical two-valued logic system, Łukasiewicz fuzzy logic system, Gödel fuzzy logic system, product fuzzy logic system and the R 0 -fuzzy logic system. It is proved that in classical two-valued logic system, Łukasiewicz fuzzy logic system and the R 0 -fuzzy logic system, respectively, the concept of consistency degrees of theories defined by the divergence degrees of theories in [X.N. Zhou, G.J. Wang, Consistency degrees of theories in some systems of propositional fuzzy logic, Fuzzy Sets and Systems 152 (2005) 321–331; H.J. Zhou, G.J. Wang, A new theory index based on deduction theorems in several logic systems, Fuzzy Sets and Systems 157(2006) 427–443] is reasonable and can accurately measure the consistency degrees of theories. The concept of the degree of entailment of a contradiction from a theory is generalized by replacing the contradiction with a general formula and then the generalized consistency degrees of theories w.r.t. general formulas in the above-mentioned logic systems is established.