Branching Quantifiers

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


This chapter is devoted to a particularly intriguing quantifier construction: branching. Branching interpretations of some natural language sentences are intractable, and therefore, their occurrence in natural language is far from obvious. I start by discussing the thesis formulated by Hintikka, which says that certain natural language sentences require nonlinear quantification to express their meaning. Then, I discuss a novel alternative reading for potentially branching sentences, the so-called two-way reading. This reading is expressible by a linear formula and is tractable. I compare the two-way reading to other possible interpretations and argue that it is the best representation for the meaning of Hintikka-like sentences. Next, I describe an experiment providing empirical support for the two-way reading. The basic assumption here is that a criterion for the adequacy of a meaning representation is its compatibility with sentence truth-conditions. This can be established by observing the linguistic behavior of language users. I report on experiments showing that people tend to interpret sentences similar to Hintikka’s sentence in a way consistent with the two-way interpretation.


Branching (Henkin) quantifiers Hintikka’s thesis Two-way reading Barwise sentence Negation normality Inferential test Symmetricity Existential fragment of second-order logic Intractability Experiments 


  1. K. Bach, Semantic nonspecificity and mixed quantifiers. Linguistics and Philosophy 4(4), 593–605 (1982)CrossRefGoogle Scholar
  2. J. Barwise, On branching quantifiers in English. Journal of Philosophical Logic 8, 47–80 (1979)CrossRefGoogle Scholar
  3. Beghelli, F., Ben-Shalom, D., & Szabolcsi, A. (1997). Variation, distributivity, and the illusion of branching. In A. Szabolcsi (Ed.), Ways of Scope Taking. Studies in Linguistic and Philosophy (Vol. 65, pp. 29–69). New York: Kluwer Academic PublisherGoogle Scholar
  4. Bellert, I. (1989). Feature System for Quantification Structures in Natural Language. Dordrecht: Foris PublicationsGoogle Scholar
  5. M. Dalrymple, M. Kanazawa, Y. Kim, S. Mchombo, S. Peters, Reciprocal expressions and the concept of reciprocity. Linguistics and Philosophy 21, 159–210 (1998)CrossRefGoogle Scholar
  6. J. Edmonds, Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)CrossRefGoogle Scholar
  7. M. Frixione, Tractable competence. Minds and Machines 11(3), 379–397 (2001)CrossRefGoogle Scholar
  8. D.M. Gabbay, J.M.E. Moravcsik, Branching quantifiers, English and Montague grammar. Theoretical Linguistics 1, 140–157 (1974)CrossRefGoogle Scholar
  9. N. Gierasimczuk, J. Szymanik, Branching quantification vs. two-way quantification. The Journal of Semantics 26(4), 329–366 (2009)CrossRefGoogle Scholar
  10. Godziszewski, M. T., & Kalociński, D. (2015). Computational complexity of Barwise’s sentence and similar natural language constructions. In Student Session at European Summer School in Logic, Language and Computation Google Scholar
  11. F. Guenthner, J.P. Hoepelman, A note on the representation of branching quantifiers. Theoretical Linguistics 3, 285–289 (1976)Google Scholar
  12. I. Heim, H. Lasnik, R. May, Reciprocity and plurality. Linguistic Inquiry 22(1), 63–101 (1991)Google Scholar
  13. J. Hintikka, Quantifiers vs. quantification theory. Dialectica 27, 329–358 (1973)CrossRefGoogle Scholar
  14. J. Hintikka, Partially ordered quantifiers vs. partially ordered ideas. Dialectica 30, 89–99 (1976)CrossRefGoogle Scholar
  15. Jackendoff, R. (1972). Semantic Interpretation and Generative Grammar. Cambridge, MA: MIT PressGoogle Scholar
  16. Janssen, T. (2002). Independent choices and the interpretation of IF-logic. Journal of Logic, Language and Information, 11, 367–387Google Scholar
  17. Janssen, T., & Dechesne, F. (2006). Signalling in IF games: A tricky business. In J. van Benthem, G. Heinzmann, M. Rebuschi & H. Visser (Eds.), The Age of Alternative Logics. Assessing Philosophy of Logic and Mathematics Today. Logic, Epistemology, and The Unity of Science, Chap. 15 (Vol. 3, pp. 221–241). Dordrecht, The Netherlands: SpringerGoogle Scholar
  18. Jaszczolt, K. (2002). Semantics and Pragmatics: Meaning in Language and Discourse. Longman Linguistics Library. London: LongmanGoogle Scholar
  19. R.M. Kempson, A. Cormack, Ambiguity and quantification. Linguistics and Philosophy 4(2), 259–309 (1981a)CrossRefGoogle Scholar
  20. R.M. Kempson, A. Cormack, On ‘formal games and forms for games’. Linguistics and Philosophy 4(3), 431–435 (1981b)CrossRefGoogle Scholar
  21. R.M. Kempson, A. Cormack, Quantification and pragmatics. Linguistics and Philosophy 4(4), 607–618 (1982)CrossRefGoogle Scholar
  22. Liu, F.-H. (1996). Branching quantification and scope independence. In J. van der Does & J. van Eijck (Eds.), Quantifiers, Logic and Language. Center for the Study of Language and Information (pp. 155–168)Google Scholar
  23. Lønning, J. T. (1997). Plurals and collectivity. In J. van Benthem & A. ter Meulen (Eds.), Handbook of Logic and Language (pp. 1009–1053). New York: ElsevierGoogle Scholar
  24. May, R. (1985). Logical Form: Its Structure and Derivation. Linguistic Inquiry Monographs. Cambridge: The MIT PressGoogle Scholar
  25. R. May, Interpreting logical form. Linguistics and Philosophy 12(4), 387–435 (1989)CrossRefGoogle Scholar
  26. Mostowski, M. (1994). Kwantyfikatory rozgałęzione a problem formy logicznej. In M. Omyła (Ed.), Nauka i język (pp. 201–242). Biblioteka Myśli SemiotycznejGoogle Scholar
  27. M. Mostowski, D. Wojtyniak, Computational complexity of the semantics of some natural language constructions. Annals of Pure and Applied Logic 127(1–3), 219–227 (2004)CrossRefGoogle Scholar
  28. Robaldo, L., Szymanik, J., & Meijering, B. (2014). On the identification of quantifiers’ witness sets: A study of multi-quantifier sentences. Journal of Logic, Language and Information, 23(1), 53–81Google Scholar
  29. Schlenker, P. (2006). Scopal independence: A note on branching and wide scope readings of indefinites and disjunctions. Journal of Semantics, 23(3), 281–314Google Scholar
  30. F. Schlotterbeck, O. Bott, Easy solutions for a hard problem? The computational complexity of reciprocals with quantificational antecedents. Journal of Logic, Language and Information 22(4), 363–390 (2013)CrossRefGoogle Scholar
  31. Sevenster, M. (2006). Branches of Imperfect Information: Logic, Games, and Computation. Ph.D. thesis. University of AmsterdamGoogle Scholar
  32. G. Sher, Ways of branching quantifiers. Linguistics and Philosophy 13, 393–442 (1990)CrossRefGoogle Scholar
  33. N. Soja, S. Carey, E. Spelke, Ontological categories guide young children’s induction of word meaning: Object terms and substance terms. Cognition 38, 179–211 (1991)CrossRefGoogle Scholar
  34. E. Stenius, Comments on Jaakko Hintikka’s paper ‘quantifiers vs. quantification theory’. Dialectica 30, 67–88 (1976)CrossRefGoogle Scholar
  35. J. Szymanik, Computational complexity of polyadic lifts of generalized quantifiers in natural language. Linguistics and Philosophy 33, 215–250 (2010)CrossRefGoogle Scholar
  36. N. Tennant, Formal games and forms for games. Linguistics and Philosophy 4(2), 311–320 (1981)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

Personalised recommendations