A fuzzy logic with interval truth values

Abstract The Kenevan Truth Interval Fuzzy Logic, in which truth values of propositions are represented as subintervals of the real unit interval that contain the single truth value rather than the truth value itself, is described herein. Inference rules are presented and proven to be correct, consistent, and as strong as possible. The classical propositional logic is shown to be a special case of the truth interval fuzzy logic in which the truth intervals are restricted to [0.0,0.0] and [1.0,1.0]. If the logic is restricted to propositions and formulas whose truth intervals are contained in the interval [x 0 ,x 1 ], 0.5 0 ⩽x 1 ⩽1.0 , or its negated equivalent, the full logic is reduced to the Zadeh's dispositional logic. A truth interval refinement proof method is described along with proof methods based on a fuzzy analog of the classical concept of logical equivalence for the dispositional logic. The latter are proven to be complete and sound. The computational complexity of the truth interval refinement proof method is shown to be Θ (2 3∗N ) ; that of the proof methods based on fuzzy logical consequence, Θ (2 4∗N ) ; and all of the proof methods are shown to be decidable and fair. A predicate truth interval fuzzy logic is also introduced.