Basic Generalized Quantifier Theory

  • Jakub SzymanikEmail author
Part of the Studies in Linguistics and Philosophy book series (SLAP, volume 96)


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.


Generalized quantifiers Definability, Ehrenfeucht-Fraïssé games Semantic universals Relativization Extensionality Conservativity Learnability Number triangle Monotonicity 


  1. Barwise, J., & Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4, 159–219.CrossRefGoogle Scholar
  2. van Benthem, J. (1986). Essays in Logical Semantics. Reidel.Google Scholar
  3. Frege, G. (1879). Begriffsschrift: Eine Der Arithmetischen Nachgebildete Formelsprache Des Reinen Denkens. Halle.Google Scholar
  4. Geurts, B. (2003). Reasoning with quantifiers. Cognition, 86(3), 223–251.CrossRefGoogle Scholar
  5. Geurts, B., & van der Slik, F. (2005). Monotonicity and processing load. Journal of Semantics, 22, 97–117.Google Scholar
  6. Gierasimczuk, N. (2009). Identification through inductive verification. Application to monotone quantifiers. In P. Bosch, D. Gabelaia, & J. Lang (Eds.), Logic Language, and Computation, 7th International Tbilisi Symposium on Logic, TbiLLC 2007. Lecture Notes on Artificial Intelligence (Vol. 5422, pp. 193–205). Tbilisi: Springer.Google Scholar
  7. Gierasimczuk, N., & Szymanik, J. (2011). Invariance properties of quantifiers and multiagent information exchange. In M. Kanazawa, A. Kornai, M. Kracht, & H. Seki (Eds.), Proceedings of 12th Meeting on Mathematics of Language. Lecture Notes in Computer Science (Vol. 6878, pp. 72–89). Berlin: Springer.Google Scholar
  8. Hodges, W. (1997). A Shorter Model Theory. New York: Cambridge University Press.Google Scholar
  9. Hunter, T., & Lidz, J. (2013). Conservativity and learnability of determiners. Journal of Semantics, 30(3), 315–334.CrossRefGoogle Scholar
  10. Icard III, T., Moss, L. (2014). Recent progress in monotonicity. Linguistic Issues in Language Technology, 9.Google Scholar
  11. Johnson-Laird, P. N. (1983). Mental Models: Toward a Cognitive Science of Language, Inference and Consciousness. Harvard University Press.Google Scholar
  12. Ladusaw, W. (1979). Polarity Sensitivity as Inherent Scope Relations. Ph.D. Thesis. University of Texas.Google Scholar
  13. Lindström, P. (1966). First order predicate logic with generalized quantifiers. Theoria, 32, 186–195.Google Scholar
  14. Montague, R. (1970). Pragmatics and intensional logic. Dialectica, 24(4), 277–302.CrossRefGoogle Scholar
  15. Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44, 12–36.Google Scholar
  16. Peters, S., & Westerståhl, D. (2006). Quantifiers in Language and Logic. Oxford: Clarendon Press.Google Scholar
  17. Tiede, H.-J. (1999). Identifiability in the limit of context-free generalized quantifiers. Journal of Language and Computation, 1, 93–102.Google Scholar
  18. Väänänen, J. (2011). Models and Games. Cambridge University Press.Google Scholar
  19. Väänänen, J., & Westerståhl, D. (2002). On the expressive power of monotone natural language quantifiers over finite models. Journal of Philosophical Logic, 31, 327–358.CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands

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