Complexity of Collective Quantification

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


Generalized quantifier theory tends to focus on distributive readings of natural language determiners. In contrast, this last chapter is devoted to collective readings of quantifiers, e.g., “Most students played poker together”. I start by introducing the common strategy of formalizing collective quantification by using certain type-shifting operations. I show that all these lifts turn out to be definable in second-order logic. Next, I introduce an alternative approach to modeling collective quantification by means of second-order generalized quantifiers and develop a definability theory for them. I study the collective reading of the proportional quantifier ‘most’ and prove that it is not definable in second-order logic. Therefore, there is no second-order definable lift expressing the collective meaning of the quantifier ‘most’. This is clearly a restriction of the type-shifting approach. I finish by discussing various methodological interpretation of this result, touching again upon the issues of complexity in natural language and the semantic borders of everyday language.


Collective quantifiers Algebraic approach Type-shifting approach Invariance properties Second-order generalized quantifiers Definability Second-order logic Ristad’s thesis Semantic bounds 


  1. Andersson, A. (2002). On second-order generalized quantifiers and finite structures. Annals of Pure and Applied Logic, 115(1–3), 1–32.Google Scholar
  2. van Benthem, J. (1986). Essays in Logical Semantics. Reidel.Google Scholar
  3. van Benthem, J. (1988). The semantics of variety in categorial grammar. Report 83–29, Department of Mathematics, Simon Fraser University, 1983. In W. Buszkowski, W. Marciszewski, & J. van Benthem (Eds.), Categorial Grammar. Linguistic and Literary Studies in Eastern Europe (Vol. 25, pp. 37–55). Amsterdam: John Benjamins.Google Scholar
  4. van Benthem, J. (1991). Language in Action: Categories, Lambdas and Dynamic Logic. Cambridge: North-Holland, Amsterdam: MIT Press.Google Scholar
  5. Bartsch, R. (1973). The semantics and syntax of number and numbers. In J. P. Kimball (Ed.), Syntax and Semantics 2 (pp. 51–93). New York: Seminar Press.Google Scholar
  6. Ben-Avi, G., & Winter, Y. (2003). Monotonicity and collective quantification. Journal of Logic, Language and Information, 12(2), 127–151.Google Scholar
  7. Ben-Avi, G., & Winter, Y. (2004). A characterization of monotonicity with collective quantifiers. Electronic Notes in Theoretical Computer Science, 53, 21–33.Google Scholar
  8. Bennett, M. R. (1974). Some Extensions of a Montague Fragment of English. Ph.D. thesis. University of California, Los Angeles.Google Scholar
  9. Burtschick, H. J., & Vollmer, H. (1998). Lindström quantifiers and leaf language definability. International Journal of Foundations of Computer Science, 9(3), 277–294.Google Scholar
  10. van der Does, J. (1992). Applied Quantifier Logics. Collectives and Naked Infinitives. Ph.D. thesis. University of Amsterdam.Google Scholar
  11. van der Does, J. (1993). Sums and quantifiers. Linguistics and Philosophy, 16(5), 509–550.Google Scholar
  12. Frege, G. (1980). Philosophical and Mathematical Correspondence of Gottlob Frege. University of Chicago Press.Google Scholar
  13. Kamp, H., & Reyle, U. (1993). From Discourse to Logic: Introduction to Model-Theoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory. Studies in Linguistics and Philosophy. Berlin: Springer.Google Scholar
  14. Kontinen, J. (2002). Second-order generalized quantifiers and natural language. In M. Nissim (Ed.) Proceedings of the Seventh ESSLLI Student Session (pp. 107–118). Elsevier.Google Scholar
  15. Kontinen, J. (2004). Definability of Second Order Generalized Quantifiers. Ph.D. thesis. Helsinki University.Google Scholar
  16. Kontinen, J., & Szymanik, J. (2008). A remark on collective quantification. Journal of Logic, Language and Information, 17(2), 131–140.Google Scholar
  17. Kontinen, J., & Szymanik, J. (2014). A characterization of definability of second-order generalized quantifiers with applications to non-definability. Journal of Computer and System Sciences, 80(6), 1152–1162.Google Scholar
  18. Link, G. (1983). The logical analysis of plurals and mass terms: A lattice-theoretical approach. In R. Bäuerle, C. Schwarze, & A. von Stechow (Eds.), Meaning, Use, and Interpretation of Language (pp. 302–323). Berlin: Gruyter.Google Scholar
  19. Lønning, J. T. (1997). Plurals and collectivity. In J. van Benthem & A. ter Meulen (Eds.), Handbook of Logic and Language (pp. 1009–1053). Elsevier.Google Scholar
  20. Partee, B., & Rooth, M. (1983). Generalized conjunction and type ambiguity. In R. Bäuerle, C. Schwarze, & A. von Stechow (Eds.), Meaning, Use, and Interpretation of Language (pp. 361–383). Berlin: Gruyter.Google Scholar
  21. Russell, B. (1903). The Principles of Mathematics. Cambridge: Cambridge University Press.Google Scholar
  22. Scha, R. (1981). Distributive, collective and cumulative quantification. In J. A. G. Groenendijk, T. M. V. Janssen, & M. B. J. Stokhof (Eds.), Formal Methods in the Study of Language, Part 2 (pp. 483–512). Amsterdam: Mathematisch Centrum.Google Scholar
  23. Winter, Y. (2001). Flexibility Principles in Boolean Semantics. London: The MIT Press.Google 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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