Generalized Quantifiers

Abstract

The universal and existential quantifiers of predicate logic introduced in Chapter 7 are in two major respects inadequate for the semantic analysis of the rich variety of quantification in natural languages. First of all, as we have seen in translating from English to predicate logic and as was pointed out again in Chapter 13, the syntactic structure of quantified formulas in predicate logic is completely different from the syntactic structure of quantified sentences in natural language. Quantifying expressions of natural language are typically full NPs, where the noun (CN) and possibly additional relative clauses provide essential restrictions on a quantifier. Not just the determiner or specifier of an NP binds dependent arguments or pronouns, but from a semantic point of view the appropriate scope-defining and binding category is the entire NP. It will prove useful for linguistic purposes (too) to distinguish between quantifying over domains and binding arguments of predicates—two jobs conflated by the two standard first-order quantifiers of predicate logic. Secondly, many forms of quantification in natural language are not expressible or definable in terms of the first-order logical quantifiers. For instance, the NP more than half of the CN is not expressible in terms of just first-order quantifiers, since its interpretation requires a one-to-one mapping between two finite or infinite sets dependent on a well-ordering by cardinality (see Barwise and Cooper (1981) for a complete proof).