14.1: Introduction

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As we noted in Chapter 13, sentences like those in (1a–c) seem to require some modifications to the simple rules of interpretation we have developed thus far:

(1) a. All men snore.
b. No women snore.
c. Some man snores.

Most of the sentences that we discussed in that chapter had proper names for arguments. We analyzed those sentences as asserting that a specific individual (the referent of the subject NP) is a member of a particular set (the denotation set of the VP). The sentences in (1a–c) present a new challenge because the subject NPs are quantified noun phrases, and do not refer to specific individuals.

Quantifier words like all, some, and no have been intensively studied by semanticists, and the present chapter summarizes some of this research. In §14.2 we present evidence for the somewhat surprising claim that quantifier words express a relationship between two sets. This insight, which we will argue follows from the general principle of compositionality, provides the critical foundation for all that follows. In §14.3 we show why the standard predicate logic notation that we introduced in Chapter 4 cannot express the meanings of certain kinds of quantifiers. We then introduce a different format, called the restricted qantifier notation, which overcomes this problem. In §14.4 we discuss two classes of quantifier words, cardinal qantifiers vs. proportional qantifiers, which differ in both semantic properties and syntactic distribution. §14.5 discusses an important property of quantifiers which was mentioned briefly in Chapter 4, namely their potential for ambiguous scope relations with other quantifiers (or various other types of expressions) occurring within the same sentence.)