Consistency degrees of theories and methods of graded reasoning in n-valued R0-logic (NM-logic) Original Research Article

In the present paper the idea of Wang [G.J. Wang, Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theorem, Sci. China Inform. Sci. E 35(6) (2005) 561–569 (in Chinese)] is firstly extended to the n-valued R0-logic image and the concept of truth degrees of formulas in image is proposed. A limit theorem saying that the truth function τn induced by truth degrees converges to the integrated truth function τ when n converges to infinity is obtained. This theorem builds a bridge between discrete valued R0-logic and continuous valued R0-logic. Secondly, based on deduction theorem, completeness theorem and the concept of truth degrees of formulas in image, the concept of consistency degrees of theories is given. It is proved that a theory Γ over image is a useless theory(i.e., the deductions of Γ are all tautologies) iff the consistency degree consistn(Γ) of Γ is equal to 1, Γ is consistent iff image, and Γ is inconsistent iff consistn(Γ) = 0. Lastly, the concept of consistency degrees of theories is generalized and a method of graded reasoning in image is obtained.