The generalized truth degree of quantitative logic in the logic system Ln∗ (nn-valued NMNM-logic system)

The concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic, from which the concept of logic metric in quantitative logic is derived. Logic metric takes an important role in quantitative logic, related to which are other concepts in quantitative logic such as divergence, consistency, etc. In the present paper, having combined the theory of generalized tautologies with the theory of truth degrees in quantitative logic, we have proposed the theory of ΣΓΣΓ-truth degrees of formulas related to theory ΓΓ in the logic system Ln∗ (nn-valued NMNM-logic system), and discussed some of its properties. With the help of the properties of ΣΓΣΓ-truth degrees: τΓ(A)+τΓ(A→B)≤1+τΓ(B)τΓ(A)+τΓ(A→B)≤1+τΓ(B), we have obtained the ΓΓ-logic metric on the set F(S)F(S) of formulas in the propositional logic system Ln∗ (nn-valued NMNM-logic system). By the work of this paper we can generalize the theory of quantitative logic in all-round way and establish an approximate reasoning’s framework related to theory ΓΓ in the logic system Ln∗ (nn-valued NMNM-logic system).