Abstract

For quantifiers, being second-order relations, the relevant order for arguments is inclusion, and for values implication, and the general concept of monotonicity applies straightforwardly to quantifiers. However, for the quantifiers that show up as denotations of noun phrases (NPs) or determiners in natural languages, there are more fine-grained notions of monotonicity which play significant roles. In a way, Aristotle discovered logic by studying monotonicity. Each of the four Aristotelian quantifiers has strong monotonicity properties, and, moreover, many valid syllogisms can be seen as statements of these properties. This chapter discusses monotone quantifiers and some facts about the distribution of monotonicity in natural languages in the form of monotonicity universals. It also deals with smooth quantifiers, proves a result that sheds light on the semantics of quantified NPs with conjoined nouns, and looks at ongoing research into what monotonicity and other related properties have to do with the occurrence of (positive and negative) polarity items in natural languages.