Gapping for the Full Power of Binary Quantification

Recent work in generalized quantifier theory has greatly increased our understanding of the types of quantification expressible in natural language. Keenan and Moss (1985) prove an expressive completeness result concerning standard unary quantifiers in English. Since Barwise (1979) proposed a semantics for "branching quantifiers, " various linguistic expressions for binary (generally, polyadic) quantifiers have been observed and discussed in the frame of generalized quantifier theory (Keenan 1987, 1992; van Benthem 1989, among others). Here we prove English has the full expressive power for binary quantifiers, which is revealed by gapping constructions. For the last decade or so, generalized quantifier theory has been developed to bear fruitful results on the meaning of noun phrases in natural language. Two classical works are Barwise and Cooper (1981) and Keenan and Stavi (1986). In this tradition, NPs are interpreted as type <1> functions from properties (sets of objects) into truth values, i.e., [P(E) 2].1) For example, the English NP more than ten students denotes a Type <1> (or unary) function which maps a property P to True iff there are more than ten students who have the property P in the model. These "standard" generalized qunatifiers are extended to denote functions from binary relations to properties, i.e.