Quantifiers ∗

Quantifiers are terms of generality. They are also among our prime examples of operators which take scope. The distinction between singular and general terms, as well as the ways that general terms enter into scope relations, are certainly fundamental to our understanding of language. Yet when we turn to natural language, we encounter a huge and apparently messy collection of general terms; not just every and some, but most, few, between five and ten, and many others. Natural language sentences also display a complex range of scope phenomena; unlike first-order logic, which clearly and simply demarcates scope in its notation. In spite of all this complexity, the study of quantification in natural language has made remarkable progress. Starting with a seminal trio of papers from the early 1980s, Barwise and Cooper (1981), Higginbotham and May (1981), and Keenan and Stavi (1986), quantification in natural language has been investigated extensively by philosophers, logicians, and linguists. The result has been an elegant and far-reaching theory. This chapter will present a survey of some of the important components of this theory. Section (I) will present the core of the theory of generalized quantifiers. This theory explores the range of expressions of generality in natural language, and studies some of their logical properties. Section (II) will turn to issues of how quantifiers enter into scope relations. Here there is less unanimity than in the theory of generalized quantifiers. Two basic approaches, representative of the main theories in the literature, will be sketched and compared. Finally, Section (III) will turn briefly to the general question of what a quantifier is.