Determiners: A Relational Perspective

In this paper we examine from a semantic point of view a class of expressions variously referred to as determiners, quantifiers, or articles. Our treatment of the meaning of these expressions reflects an idea which has its origin in the work of Montague (1970), Lewis (1972), Geach (1972), and Cresswell (1973): determiners are to be interpreted as two-place relations between sets of individuals. This account, though valuable, is of course too general to produce any results of linguistic interest. For no assumption whatsoever is made concerning the content and structure of these binary relations. To remedy this, we shall suggest a number of principles restricting the range of admissible interpretations of natural language determiners. Next we examine some of the more important relational properties of determiners, provide alternative ways of characterizing several of these properties, and present a rather large class of non-existence results, many of which were first proved in van Benthem (1982). The significance of these results is that certain properties, and combinations of properties, are inconsistent with the principles that we shall adopt. By a determiner, semantically speaking, we mean a functor D which assigns to each universe E a binary relation DE between subsets of E. Familiar examples of determiners include all, no, some, and the three, and if we identify the determiner symbol with its interpretation, we have (for every universe E and every A,B E E)