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 quantifiers are often defined as classes of models closed on isomorphism (topic neutral). For instance, the 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 example, in the 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. Generalized quantifiers turned out to be one of the crucial notions in the development of formal semantics but also logic, theoretical computer science and philosophy.

This is a short self-contained and self-explanatory monographic course; reading the suggested bibliography prior to the course is not expected of the participants. However, if you are curious: for a current bibliography of the subject see the Generalized Quantifiers category at PhilPapers. For an encyclopedia article see SEP entry. Peters & Westerståhl 2006 is a thorough handbook treatment. For the previous similar courses see here and here.

The following manuscript may serve (partially) as the lecture notes (feedback is welcome!).

The course will consist of the following parts:

Day 1: General introduction to generalized quantifiers

Day 2: Semantic automata: monadic quantifiers

Day 3: Semantic automata: polyadic quantifiers

Day 4: Computational complexity and quantifiers

Day 5: Collective quantification: definability and complexity