Qualified Syllogisms with Fuzzy Predicates

Abstract

A previous publication introduced the logic Q for qualified syllogisms, which formalizes arguments such as Most birds can fly; Tweety is a bird; therefore it is likely that Tweety can fly. This provided a probabilistic semantics for fuzzy quantifiers most, many, few, etc., fuzzy usuality modifiers usually, often, seldom, etc., and fuzzy likelihood modifiers likely, uncertain, unlikely, etc.. That work dealt only with crisp nonfuzzy predicates like Bird and CanFly, however. The present work proposes a simple and intuitively natural way to expand the semantics of Q so as to accommodate syllogisms such as Most Swedes are tall; Helge is a Swede; therefore it is likely that Helge is tall, where tall is a fuzzy predicate. To accomplish this, a formulation of the notion of the probability of a fuzzy event is proposed. It is shown that the new semantics validates the intended syllogisms as well as numerous other propositions and rules, including a rendition of the axioms and inference rules of classical first-order predicate calculus. These results may be viewed as steps toward future applications. A future work will follow the lines of the previous work and show how Q thus extended may be employed for nonmonotonic reasoning with fuzzy predicates.[a]