Inference in Extensions of Intuitionistic Logic as Fuzzy Computing

Mathematical fuzzy logic focuses on fuzzy functions for logical formulas whereas the core of fuzzy systems is fuzzy facts and rules specifying properties of concrete predicates and functions. The gap between them is bridged by representing fuzzy facts and rules as nonlogical axioms in intuitionistic and intermediate sequent calculi. Inference in these calculi extended with fuzzy nonlogical axioms is sound with respect to fuzzy Godel models. If a formula is derivable, then a lower bound of its truth value is calculated from the derivation tree. Inference in these calculi is complete for atoms whose truth values are positive. Inference in Godel-Dummett calculus remains complete for valid formulas even in presence of nonlogical axioms.