Consistency degrees of finite theories in Łukasiewicz propositional fuzzy logic

The concept of divergence degrees of theories in mathematical logic can be used to grade the extent of consistency of theories. Based on the analysis of relationships among the properties of consistent, inconsistent, and fully divergent theories the present paper proposes a membership function for reflecting the consistency degrees of finite theories in Łukasiewicz propositional fuzzy logic. It is proved that the consistency degrees of consistent finite theories are in-between 12 and 1, and that of inconsistent theories are equal to 0, and that of completely consistent finite theories are equal to 1. Finally, a sufficient and necessary condition for theories being inconsistent is also proposed.