Typed Hilbert Epsilon Operators and the Semantics of Determiner Phrases

  • Christian Retoré
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8612)


The semantics of determiner phrases, be they definite descriptions, indefinite descriptions or quantified noun phrases, is often assumed to be a fully solved question: common nouns are properties, and determiners are generalised quantifiers that apply to two predicates: the property corresponding to the common noun and the one corresponding to the verb phrase.

We first present a criticism of this standard view. Firstly, the semantics of determiners does not follow the syntactical structure of the sentence. Secondly the standard interpretation of the indefinite article cannot account for nominal sentences. Thirdly, the standard view misses the linguistic asymmetry between the two properties of a generalised quantifier.

In the sequel, we propose a treatment of determiners and quantifiers as Hilbert terms in a richly typed system that we initially developed for lexical semantics, using a many sorted logic for semantical representations. We present this semantical framework called the Montagovian generative lexicon and show how these terms better match the syntactical structure and avoid the aforementioned problems of the standard approach.

Hilbert terms are rather different from choice functions in that there is one polymorphic operator and not one operator per formula. They also open an intriguing connection between the logic for meaning assembly, the typed lambda calculus handling compositionality and the many-sorted logic for semantical representations. Furthermore epsilon terms naturally introduce type-judgements and confirm the claim that type judgments are a form of presupposition.


Noun Phrase Choice Function Common Noun Natural Language Semantic Lexical Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asher, N.: Lexical Meaning in context – a web of words. Cambridge University Press (2011)Google Scholar
  2. 2.
    Asser, G.: Theorie der logischen auswahlfunktionen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (1957)Google Scholar
  3. 3.
    Bassac, C., Mery, B., Retoré, C.: Towards a Type-Theoretical Account of Lexical Semantics. Journal of Logic Language and Information 19(2), 229–245 (2010), CrossRefGoogle Scholar
  4. 4.
    Canty, J.T.: Zbl0327.02013: review of “on an extension of Hilbert’s second ε-theorem” by T. B. Flanagan (jsl 1975)Google Scholar
  5. 5.
    Egli, U., von Heusinger, K.: The epsilon operator and E-type pronouns. In: Egli, U., Pause, P.E., Schwarze, C., von Stechow, A., Wienold, G. (eds.) Lexical Knowledge in the Organization of Language, pp. 121–141. Benjamins (1995)Google Scholar
  6. 6.
    Evans, G.: Pronouns, quantifiers, and relative clauses (i). Canadian Journal of Philosophy 7(3), 467–536 (1977)Google Scholar
  7. 7.
    Girard, J.Y.: Une extension de l’interprétation de Gödel à l’analyse et son application: l’élimination des coupures dans l’analyse et la théorie des types. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, vol. 63, pp. 63–92. North Holland, Amsterdam (1971)CrossRefGoogle Scholar
  8. 8.
    Girard, J.Y.: The blind spot – lectures on logic. European Mathematical Society (2011)Google Scholar
  9. 9.
    von Heusinger, K.: Definite descriptions and choice functions. In: Akama, S. (ed.) Logic, Language and Computation, pp. 61–91. Kluwer (1997)Google Scholar
  10. 10.
    von Heusinger, K.: Choice functions and the anaphoric semantics of definite nps. Research on Language and Computation 2, 309–329 (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik, Bd. 2. Springer (1939), traduction française de F. Gaillard, E. Guillaume et M. Guillaume, L’Harmattan (2001)Google Scholar
  12. 12.
    Kneale, W., Kneale, M.: The development of logic, 3rd edn. Oxford University Press (1986)Google Scholar
  13. 13.
    Leisenring, A.C.: Mathematical logic and Hilbert’s ε symbol. University Mathematical Series. Mac Donald & Co. (1967)Google Scholar
  14. 14.
    de Libera, A.: La querelle des universaux de Platon à la fin du Moyen Âge. Des travaux, Seuil (1996),
  15. 15.
    Luo, Z.: Contextual analysis of word meanings in type-theoretical semantics. In: Pogodalla, S., Prost, J.-P. (eds.) LACL 2011. LNCS, vol. 6736, pp. 159–174. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Luo, Z.: Common nouns as types. In: Béchet, D., Dikovsky, A. (eds.) LACL 2012. LNCS, vol. 7351, pp. 173–185. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Mery, B., Moot, R., Retoré, C.: Plurals: individuals and sets in a richly typed semantics. In: Yatabe, S. (ed.) Logic and Engineering of Natural Language Semantics 10 (LENLS 10), pp. 143–156, Keio University (2013) ISBN 978-4-915905-57-5Google Scholar
  18. 18.
    Mery, B., Retoré, C.: Semantic types, lexical sorts and classifiers. In: Sharp, B., Zock, M. (eds.) 10th International Workshop on Natural Language Processing and Cognitive Science. Marseilles (September 2013),
  19. 19.
    Mints, G.: Zbl0381.03042: review of “cut elimination in a Gentzen-style ε-calculus without identity” by Linda Wessels (Z. math Logik Grundl. Math (1977))Google Scholar
  20. 20.
    Mints, G.: Cut elimination for a simple formulation of epsilon calculus. Ann. Pure Appl. Logic 152(1-3), 148–160 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Moot, R.: Wide-coverage French syntax and semantics using Grail. In: Proceedings of Traitement Automatique des Langues Naturelles (TALN), Montreal (2010)Google Scholar
  22. 22.
    Moot, R., Retoré, C.: The Logic of Categorial Grammars. LNCS, vol. 6850. Springer, Heidelberg (2012)Google Scholar
  23. 23.
    Moser, G., Zach, R.: The epsilon calculus and herbrand complexity. Studia Logica 82(1), 133–155 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Real, L., Retoré, C.: Deverbal semantics and the Montagovian generative lexicon \({\Lambda} \!\mathsf{{T}y}_n\). Journal of Logic, Language and Information, 1–20 (2014),
  25. 25.
    Retoré, C.: Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by “most”. Recherches Linguistiques de Vincennes 41, 83–102 (2012), CrossRefGoogle Scholar
  26. 26.
    Retoré, C.: Sémantique des déterminants dans un cadre richement typé. In: Morin, E., Estève, Y. (eds.) Traitement Automatique du Langage Naturel, TALN RECITAL 2013, vol. 1, pp. 367–380. ACL Anthology (2013),
  27. 27.
    Retoré, C.: The Montagovian generative lexicon ΛTy n: a type theoretical framework for natural language semantics. In: Matthes, R., Schubert, A. (eds.) 19th International Conference on Types for Proofs and Programs (TYPES 2013). Leibniz International Proceedings in Informatics LIPIcs, vol. 27, Dagstuhl Publishing, Germany (2014), Google Scholar
  28. 28.
    Russell, B.: On denoting. Mind 56(14), 479–493 (1905)CrossRefGoogle Scholar
  29. 29.
    Steedman, M.: Taking Scope: The Natural Semantics of Quantifiers. MIT Press (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christian Retoré
    • 1
    • 2
  1. 1.LaBRIUniversité de BordeauxFrance
  2. 2.MELODI, IRIT-CNRSToulouseFrance

Personalised recommendations