A connection between fuzzy quantifiers and the classical modalities

The classical modalities, “necessity” and “possibility,” are intuitively connected to probability theory by: proposition P is necessarily true if the probability of P´s being true is I, and P possibly true if the probability of P´s being true is greater than O. A prior work by the author introduced the logic QUAL, for reasoning symbolically with terms expressing fuzzy quantification, usuality, and likelihood. Within QUAL one can formalize syllogisms of the form “Most birds can fly; Tweety is a bird; therefore, it is likely that Tweety can By” and of the form “Usually, if something is a bird, it can fly; Tweety is a bird; therefore, it is likely that Tweety can fly,” as well as other formulas and inference rules expressing the interrelationships between these three concepts. The present work shows how QUAL can be extended to include the above versions of possibility and necessity. The extent to which aspects of the well-known modal logics K, S.4, and S.5 can be encoded in this system are also explored