The Semantics of Determiners

Generalized quantifiers
With respect to theoretical approaches to determiners, an alternative and very influential view of existential, universal and other ‘quantificational’ DPs that is a small step away from the one just presented was developed in Barwise and Cooper (1981). The proposal is to treat determiners as expressing relations between two sets A and B, where, in our terms, the set A is given by the interpretation of the expression in the Restrictor, and the set B is given by the interpretation of the expression in the NS. For our simple examples, A is the set of students and B is the set of people who left. The relation denoted by the indefinite article a is that which obtains between two sets when their intersection is non-null. The extra requirement imposed by the definite article the is that A be a singleton. The universal determiner every, on the other hand, requires A to be a subset of B, expressed formally as A  B = A. The negative determiner no denotes a relation that holds between A and B just in case the intersection of A and B is the empty set, expressed formally as A  B = .

Interesting formal properties of determiners have been defined under this approach and universal properties of natural language determiners have been formulated. One intriguing such universal property is conservativity, a property that is equivalent to the claim that all quantification in natural language is restricted. Of particular use in characterizing the distributional properties of NPIs, mentioned above in connection with nici indefinites, is the monotonicity properties of determiners. Assume that a determiner D relates two sets, S and S’, monotone increasing and monotone decreasing determiners can be defined as in (12):

(12) a. A determiner is monotone increasing on S iff for every superset S” of S, if D

holds between S and S’, then D holds between S” and S’ as well.
b. A determiner D is monotone decreasing on S iff for every subset S” of S, if D
holds between S and S’, then D holds between S” and S’ as well.

The reader can verify that the determiner every is monotone decreasing on its Restrictor and monotone increasing on its NS; the determiner no is monotone decreasing on both its Restrictor and NS.

The monotone decreasing property has turned out to be useful in characterizing the contexts in which certain NPIs occur, particularly in English (see Ladusaw (1979) for the classic work on this topic.) The issue of the semantics of NPIs has developed into a subfield of its own, and since we cannot do it justice in this chapter we leave it out of its scope.