Quantification, Witness Sets and Conservativity

Theories of semantic scope underspecification can be divided into two types, depending on whether they are based on the notion of satisfaction or on the notion of derivation.1 The central tenet of satisfaction-based theories of scope underspecification is that the relation between a representation SR with fully specified scope relations and a representation UR with underspecified scope relations is one of satisfaction: SR is a possible specification of underspecified representation UR iff SR satisfies UR.2 On the other hand, the central idea of derivation-based theories of underspecification is that the relation between a fully specified representation SR and an underspecified representation UR is one of derivation: SR is a possible specification of an underspecified representation UR iff SR can be derived from UR.3 What both types of theories have in common is that ultimately the denotation [[UR]] of an underspecified representation UR is the set of denotations {[[SR1]], . . . , [[SRn]]} of those fully specified representations SR1, . . . , SRn which (depending on the type of underspecification theory) either (i) satisfy UR or (ii) can be derived from UR. According to this perspective on underspecification, then, an underspecified representation denotes a set of interpretations (readings), while the (partial or full) specification of an underspecified denotation amounts to narrowing down the set of interpretations. This perspective on underspecification has, arguably, two shortcomings. First, since the denotation of the underspecified representation is a set of fully specified readings, it does not capture what the different readings have in common. That is, this type of denotation for underspecified representations does not capture the fact that all readings involve the same verb denotation, and the same assignment of semantic roles to the determiner phrase denotations. And secondly, the denotation of the underspecified representation is not in any sense part of the denotation of each specified representation; on the contrary, there is a clear sense in which the denotations of the specified representations are part of the denotation of the underspecified representation, being elements of the underspecified denotation. In this paper I will propose an alternative theory of quantification and scope underspecification, where (i) the denotation of the underspecified representation does indeed capture what the possible readings all have in common, and (ii) the specification of an underspecified denotation amounts to adding information, so that there is a clear sense in which the underspecified denotation is part of every specified reading. The outline of the paper is as follows. In section 2 I provide an alternative to the Generalized Quantifier analysis of determiner phrase denotations, by claiming that DPs denote pairs consisting of the restrictor and the set of witnesses. To illustrate the basic idea, the sets of witnesses for [[every man]], [[no man]] and [[at most one man]] are {[[man]]}, {∅} and {X : X ⊆ [[man]] ∧ |X| ≤ 1}, respectively, so