Presumption and prejudice in logical inference

Abstract

Two nonstandard modes of inference, confirmation and denial, have been shown by Bandler and Kohout to be valid in fuzzy propositional and predicate logics. If denial is used in combination with modus ponens, the resulting inference mode (“augmented modus ponens”) yields more precise bounds on the consequent of an implication than are usually called for in approximate reasoning. Similar results hold for augmented modus tollens constructed from confirmation and conventional fuzzy modus tollens. Two simpler modes of inference, presumption and prejudice, are also valid under the same assumptions as confirmation and denial. Prejudice imposes an upper bound on the truth value of the consequent of a fuzzy implication regardless of the truth value of the antecedent; presumption imposes a lower bound on the truth value of the antecedent regardless of that of the consequent. Some of the consequences of presumption and prejudice cast doubt on the suitability of fuzzy propositional and predicate logics for use in expert systems that are designed to process real-world data. A logic based directly on fuzzy sets is explored as an alternative. Fuzzy set logic supports fuzzy modus ponens and modus tollens but does not entail the more problematic modes of confirmation, denial, presumption, and prejudice. However, some of the expressive power derivable from the diversity of fuzzy propositional logics and their derivative fuzzy predicate logics is lost.