Defuzzifying formulas in Gödel logic through finitely additive measures

Gödel logic is the fuzzy logic of the minimum triangular norm and its residuum. Using the functional representation of the Lindenbaum algebra of Gödel logic, we analyze the interaction between the integral operator and the logical connectives. On these grounds, we put forth a notion of finitely additive probability measure for Gödel logic. Our first main result shows that such measures precisely correspond to integrating the truth value functions induced by Gödel formulas with respect to a Borel probability measure on the real unit cube [0, 1]n. Our second main result shows that they also coincide with convex combinations of finitely many [0, 1]-valued assignments.