An Ehrenfeucht-Fraïssé Game for Inquisitive First-Order Logic

  • Gianluca GrillettiEmail author
  • Ivano Ciardelli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11456)


Inquisitive first-order logic, InqBQ, is an extension of classical first-order logic with questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. In this paper we describe an Ehrenfeucht-Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. We exploit this result to show a number of undefinability results: in particular, several variants of the question how many individuals have property P are not expressible in InqBQ, even in restriction to finite models.


  1. 1.
    Ciardelli, I.: Inquisitive semantics and intermediate logics. M.Sc. thesis, University of Amsterdam (2009)Google Scholar
  2. 2.
    Ciardelli, I.: Questions in logic. Ph.D. thesis, Institute for Logic, Language and Computation, University of Amsterdam (2016)Google Scholar
  3. 3.
    Ciardelli, I.: Questions as information types. Synthese 195, 321–365 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ciardelli, I., Groenendijk, J., Roelofsen, F.: Inquisitive Semantics. Oxford University Press, Oxford (2018)CrossRefGoogle Scholar
  5. 5.
    Ciardelli, I., Roelofsen, F.: Inquisitive logic. J. Philos. Log. 40(1), 55–94 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ehrenfeucht, A.: An application of games to the completeness problem for formalized theories. Fund. Math. 49(2), 129–141 (1961). Scholar
  7. 7.
    Fagin, R.: Monadic generalized spectra. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, 89–96 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fraïssé, R.: Sur quelques classifications des systèmes de relations. Publications Scientifiques de l’Université D’Alger 1(1), 35–182 (1954)Google Scholar
  9. 9.
    Frittella, S., Greco, G., Palmigiano, A., Yang, F.: A multi-type calculus for inquisitive logic. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds.) WoLLIC 2016. LNCS, vol. 9803, pp. 215–233. Springer, Heidelberg (2016). Scholar
  10. 10.
    Grilletti, G.: Disjunction and existence properties in inquisitive first-order logic. Stud. Logica (2018). ISSN 1572-8730
  11. 11.
    Hintikka, J.: Knowledge and Belief: An Introduction to the Logic of the Two Notions. Cornell University Press (1962)Google Scholar
  12. 12.
    Hodges, W.: A Shorter Model Theory. Cambridge University Press, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Immerman, N.: Upper and lower bounds for first order expressibility. J. Comput. Syst. Sci. 25(1), 76–98 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kolaitis, P., Väänänen, J.: Generalized quantifiers and pebble games on finite structures. Ann. Pure Appl. Log. 74(1), 23–75 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Połacik, T.: Back and forth between first-order kripke models. Log. J. IGPL 16(4), 335–355 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Punčochář, V.: Weak negation in inquisitive semantics. J. Log. Lang. Inf. 24(3), 323–355 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Punčochář, V.: A generalization of inquisitive semantics. J. Philos. Log. 45(4), 399–428 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Roelofsen, F.: Algebraic foundations for the semantic treatment of inquisitive content. Synthese 190(1), 79–102 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Väänänen, J.: Models and Games, 1st edn. Cambridge University Press, New York (2011)CrossRefGoogle Scholar
  20. 20.
    van Benthem, J.: Modal correspondence theory dissertation, pp. 1–148. Universiteit van Amsterdam, Instituut voor Logica en Grondslagenonderzoek van Exacte Wetenschappen (1976)Google Scholar
  21. 21.
    Visser, A.: Submodels of Kripke models. Arch. Math. Log. 40(4), 277–295 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationAmsterdamThe Netherlands
  2. 2.Munich Center for Mathematical PhilosophyLMUMunichGermany

Personalised recommendations