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

Abstract

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 R 0-logic L n ∗ and the concept of truth degrees of formulas in L n ∗ is proposed. A limit theorem saying that the truth function τ n 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 R 0-logic and continuous valued R 0-logic. Secondly, based on deduction theorem, completeness theorem and the concept of truth degrees of formulas in L n ∗ , the concept of consistency degrees of theories is given. It is proved that a theory Γ over L n ∗ is a useless theory(i.e., the deductions of Γ are all tautologies) iff the consistency degree consist n ( Γ) of Γ is equal to 1, Γ is consistent iff 1 2 ⩽ consist n ( Γ ) ⩽ 1 , and Γ is inconsistent iff consist n ( Γ) = 0. Lastly, the concept of consistency degrees of theories is generalized and a method of graded reasoning in L n ∗ is obtained.