A note on measures of fuzziness applied to nonmonotonic fuzzy propositional logic

Abstract

Abstract We undertake a mathematical investigation of the problem of inferring a truth value(s) for a propositional sentence. given truth values for some other set of sentences. Interpreting such a set of sentences and truth values as an abstraction of a fuzzy knowledge base, this mathematical problem can be seen as an abstraction of the important problem of inference in fuzzy expert systems. We postulate axioms for nonmonotonic fuzzy propositional logic (inspired by Paris and Vencovska's axioms for ampliative probabilistic inference). We prove simple but new results regarding the consistency of the axioms. We consider one method of inference. Maximum Fuzziness — infer the truth value that is most fuzzy and yet consistent with the knowledge base. We demonstrate that under certain plausible assumptions about the measure of fuzziness, several of the axioms are not sound for Maximum Fuzziness. Finally, we show that computing close approximations to truth values inferred by Maximum Fuzziness is at least as difficult as monotonic deductive propositional logic.