Abstract
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.
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Notes
- 1.
See, e.g., Hodges (1997) for a textbook exposition of model theory. Also consult the Appendix for a quick introduction.
- 2.
We will sometimes write \(R_i^M\) for relations to differentiate them from the corresponding predicates \(R_i\).
- 3.
For more undefinability results and an introduction to Ehrenfeucht-Fraïssé techniques, we suggest consulting, e.g., Chap. 4 in Peters and Westerståhl (2006). A reader interested in game-theoretic approaches to various extensions of first-order logic is encouraged to reach for the book by Väänänen (2011).
- 4.
Below we assume that a logic \(\mathcal {L}\) has the so-called substitution property, i.e., that the logic \(\mathcal {L}\) is closed under substituting predicates by formulas.
- 5.
However, it is actually conservative in the second argument.
- 6.
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Szymanik, J. (2016). Basic Generalized Quantifier Theory. In: Quantifiers and Cognition: Logical and Computational Perspectives. Studies in Linguistics and Philosophy, vol 96. Springer, Cham. https://doi.org/10.1007/978-3-319-28749-2_3
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