Generalized Quantifiers

Generalized quantifier theory studies the semantics of quantifier expressions, like, `every’, `some’, `most’, ‘infinitely many’, `uncountably many’, etc. The classical version was developed in the 1980s, at the interface of linguistics, mathematics and philosophy. In logic generalized quantifier are often defined as classes of models closed on isomorphism (topic neutral). For instance, quantifier “infinitely many” may be defined as a class of all infinite models. Equivalently, in linguistics generalized quantifiers are formally treated as relations between subset of the universe. For instance, in sentence `Most of the students are smart”, quantifier `most’ is a binary relation between the set of students and the set of smart people. The sentence is true if and only if the cardinality of the set of smart students is greater than the cardinality of the set of students who are not smart. 

Gottlob Frege was one of the major figures in forming the modern concept of quantification. In Begriffsschrift (1879) he made a distinction between bound and free variables and treated quantifiers as well-defined, denoting entities. However, historically speaking the notion of a generalized quantifier was formulated for the first time in a seminal paper of Andrzej Mostowski 1957, where the notions of existential and universal quantification were extended to the concept of a monadic generalized quantifier binding one variable in one formula, and later this was generalized to arbitrary types by Per Lindström 1966. Soon it was realized by Richard Montague 1970 that the notion can be used to model the denotations of noun phrases in natural language. Jon Barwise and Robin Cooper (1981) introduced the apparatus of generalized quantifiers as a standard semantic toolbox and started the rigorous study of their properties from the linguistic perspective.

For an encyclopedia article see Westerståhl 2008. For a survey of classical results we recommend: Keenan & Westerstahl 2011. Peters & Westerståhl 2006 is a thorough handbook treatment focused on definability questions and their applications in model theory and linguistics. For more computer science results consult, e.g., Makowsky & Pnueli 1995 . For a psychological perspective, see, e.g. Moxey & Sanford 1993. For a combination of formal work and cognitive science perspective, see, e.g., Szymanik 2016.