Basic Generalized Quantifier Theory

This chapter is a crash course in generalized quantifier theory, which is one of the basic tools of today’s linguistics. In its simplest form generalized quantifier theory assigns meanings to statements by defining the semantics of the quantifiers, like ‘some’, ‘at least 7’, and ‘most’. I introduce two equivalent definitions of generalized quantifier: as a relation between subsets of universe and as a class of appropriate models. I discuss the notion of logic enriched by generalized quantifiers and introduce basic undefinability results and the related proof technique based on model-theoretic games. Then, I discuss a linguistic question: which of the logically possible quantifiers are actually realized in natural language. In order to provide an answer, I introduce various linguistic properties of quantifiers, including the key semantic notion of monotonicity.