Δ-Randomized Divergent Degree and Consistent Degree of Theories in ΠΔ

In quantitative logic, the Δ operator plays a crucial role in defuzzification by mapping multi-valued truth values to classical two-valued ones. This paper introduces and investigates the Δ-randomized divergent degree and consistent degree of theories in the continuous-valued product logic system ΠΔ. These notions measure, respectively, the probability that all formulas in a theory are simultaneously completely true and the degree of internal inconsistency of the theory. Basic properties including boundedness, monotonicity, additivity, and duality are established. The relationship between divergent degree and logical consequence is investigated, and the divergent degree between two theories is defined, with the triangle inequality proved, thereby constructing a pseudo-metric space on theories. As applications, a criterion for almost surely consistent theories is provided, and the existence of maximal consistent subtheories is proved with an illustrative example. This work enriches quantitative logic theory in product logic systems and provides a unified tool for measuring inconsistency in theories under uncertainty.