# Course Content

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, 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 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. For a current bibliography of the subject see Generalized Quantifiers category at PhilPapers.

Day 1: Generalized quantifiers. In the first lecture I will introduce the notion of generalized quantifier. I will discuss different variants of its definition and show their equivalence. While doing that I will also introduce basic preliminary notions from logic and formal semantics as well as give a brief history of the development of generalized quantifier theory.

Day 2: Quantifier Universals. The second lecture will focus on linguistically central properties of generalized quantifiers: monotonicity properties, conservativity, extensionality, etc. I will discuss semantic universals that were proposed to delimit the class of all logically possible generalized quantifiers to the set of those that are realized by simple determiners in natural language. I will evaluate the proposal and discuss recent approaches to explain the linguistic importance of semantic universals in terms of learnability, evolution, and cognition.

Day 3: Monadic quantifiers. In this lecture I will focus on monadic quantifiers – the most important class of generalized quantifiers that captures the meanings of natural language simple determiners. In particular, I will introduce so-called semantic automata theory that associates each quantifier with a simple computational device. I will also discuss some classical definability results connecting expressibility with semantic automata. In doing that I will use fundamental notions introduced in the first two classes. I will present a number of experimental studies suggesting that the model of semantic automata is cognitively plausible.

Day 4: Polyadic quantifiers. This lecture will be concerned with polyadic quantification. I will give some examples how polyadic generalized quantifiers may be used to analyze the meaning of natural language sentences, e.g., multi-quantifier sentences and reciprocal sentences. I will explain how polyadic quantifiers result from semantically natural operations applied to monadic quantifiers, like iteration or cumulation. I will also discuss some definability issues, computational topics, and a few experiments related to polyadic quantification.

Day 5: Collective quantifiers. In the final lecture I will show how the standard generalized quantifier theory, originally designed to deal with distributive quantification, can be extended to cover collective quantifiers. I will discuss type-lifting strategies constructing collective readings from distributive readings for natural language quantifiers. I will also introduce the notion of second-order generalized quantifier that is a natural mathematical extension of Lindström quantifiers to the collective setting. This will lead to a more general methodological question about the bounds of everyday language and semantic constructions that may fall beyond natural language due to their inherent complexity.

Expected level and prerequisites. 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; questions and discussion are welcome during the lectures.

References. For an encyclopedia article see SEP entryPeters & Westerståhl 2006 is a thorough handbook treatment. During the course I will roughly follow the content of my PhD Thesis.

Slides from the class