Thirty years of generalized quantifiers

It is now more than thirty years since the first serious applications of Generalized Quantifier (GQ) theory to natural language semantics were made: Barwise and Cooper (1981); Higginbotham and May (1981); Keenan and Stavi (1986). Richard Montague had in effect interpreted English NPs as (type 〈1〉) generalized quantifiers (see Montague, 1974), but without referring to GQs in logic, where they had been introduced by Mostowski (1957) and, in final form, Lindström (1966). Logicians were interested in the properties of logics obtained by piecemeal additions to first-order logic (FO) by adding quantifiers like ‘there exist uncountably many’, but they made no connection to natural language. Montague Grammar and related approaches had made clear the need for higher-type objects in natural language semantics. What Barwise, Cooper, and the others noticed was that generalized quantifiers are the natural interpretations not only of noun phrases but also in particular of determiners (henceforth Dets).

This was no small insight, even if it may now seem obvious. Logicians had, without intending to, made available model-theoretic objects suitable for interpreting English definite and indefinite articles, the Aristotelian all, no, some, proportional Dets like most, at least half, 10 percent of the, less than two-thirds of the, numerical Dets such as at least five, no more than ten, between six and nine, finitely many, an odd number of, definite Dets like the, the twelve, possessives like Mary's, few students’, two of every professor's, exception Dets like no … but John, every … except Mary, and Boolean combinations of all of the above. All of these can – if one wants! – be interpreted extensionally as the same type of second-order objects, namely (on each universe of discourse) binary relations between sets. Given the richness of this productive but seemingly heterogeneous class of expressions, a uniform interpretation scheme was a huge step. Further, the tools of logical GQ theory could be brought to bear on putative Det interpretations, which turned out to be a subclass of the class of all type 〈1, 1〉 quantifiers with special traits. The three pioneer papers mentioned above offered numerous cases of novel description, and sometimes explanation, of characteristic features of language in terms of model-theoretic properties of the quantifiers involved.