Some quantifiers require two-predicate scopes

Abstract

Predicate calculus has long served as the basis of mathematical logic and has more recently achieved widespread use in artificial intelligence. This system of logic expresses propositions in terms of quantifications, restricting itself to the universal and existential quantifiers “all” and “some,” which appear to be adequate for formalizing mathematics. Systems that aspire to deal with natural language or everyday reasoning, however, must attempt to deal with the full range of quantifiers that occur in such language and reasoning, including, in particular, plurality quantifiers, such as “most,” “many,” and “few.” The logic of such quantifiers forces an extension of the predicate-calculus framework to a system of representation that involves more than one predicate in each quantification. In this paper, we prove this result for the specific case of “most.” Unlike some other arguments that attempt to establish the inadequacy of standard predicate calculus on the basis of intuitive plausibility judgements as to the likely character of human reasoning [11, 19], our result is a theorem of logic itself.