Unified theory of integral true degrees in NM theory

Based on the NM propositional logic system of nilpotent minimum logic, the concept of the integral truth degree of the formula is firstly proposed by integrating the function induced by the formula, and the MP rule and HS rule of the integral truth degree are proved by means of the integral invariance. Secondly, in NM propositional logic the integral similarity and integral pseudo-distance are introduced into the set of general formulas of the system, and some good properties about the similarity and pseudo-distance are proved. Finally, in terms of the concept of divergence degrees and diameter, a new membership function for reflecting the consistency degrees of theories in NM propositional fuzzy logic is proposed, which is proved that of inconsistent theories are equal to 0 and that of completely consistent theories are equal to 1.