Dependence Logic with Generalized Quantifiers: Axiomatizations

  • Fredrik Engström
  • Juha Kontinen
  • Jouko Väänänen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8071)

Abstract

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the sense that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as “there exist uncountably many.” Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.

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References

  1. 1.
    Engström, F.: Generalized quantifiers in dependence logic. Journal of Logic, Language and Information 21, 299–324 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Engström, F., Kontinen, J.: Characterizing quantifier extensions of dependence logic. Journal of Symbolic Logic 78(1), 307–316 (2013)MATHCrossRefGoogle Scholar
  3. 3.
    Mostowski, A.: On a generalization of quantifiers. Fund. Math. 44, 12–36 (1957)MathSciNetGoogle Scholar
  4. 4.
    Keisler, H.: Logic with the quantifier “there exist uncountably many”. Annals of Mathematical Logic 1(1), 1–93 (1970)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Clarendon Press (2006)Google Scholar
  6. 6.
    Kolaitis, P.G., Väänänen, J.A.: Generalized quantifiers and pebble games on finite structures. Ann. Pure Appl. Logic 74(1), 23–75 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hella, L., Väänänen, J., Westerståhl, D.: Definability of polyadic lifts of generalized quantifiers. J. Logic Lang. Inform. 6(3), 305–335 (1997)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Barwise, J.: On branching quantifiers in English. J. Philos. Logic 8(1), 47–80 (1979)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dahlhaus, E.: Skolem normal forms concerning the least fixpoint. In: Börger, E. (ed.) Computation Theory and Logic. LNCS, vol. 270, pp. 101–106. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  11. 11.
    Väänänen, J.: Dependence Logic - A New Approach to Independence Friendly Logic. London Mathematical Society Student Texts, vol. 70. Cambridge University Press, Cambridge (2007)MATHCrossRefGoogle Scholar
  12. 12.
    Henkin, L.: Some remarks on infinitely long formulas. In: Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959, pp. 167–183. Pergamon, Oxford (1961)Google Scholar
  13. 13.
    Kontinen, J., Väänänen, J.A.: On definability in dependence logic. Journal of Logic, Language and Information 18(3), 317–332 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Lindström, P.: First order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)MathSciNetMATHGoogle Scholar
  15. 15.
    Kaufmann, M.: The quantifier “there exist uncountably many”, and some of its relatives. In: Barwise, J., Feferman, S. (eds.) Perspectives in Mathematical Logic. Model Theoretic Logics, pp. 123–176. Springer (1985)Google Scholar
  16. 16.
    Makowsky, J., Tulipani, S.: Some model theory for monotone quantifiers. Archive for Mathematical Logic 18(1), 115–134 (1977)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kontinen, J., Väänänen, J.: Axiomatizing first order consequences in dependence logic. Annals of Pure and Applied Logic (June 6, 2013)Google Scholar
  18. 18.
    Engström, F., Kontinen, J., Väänänen, J.: Dependence logic with generalized quantifiers: Axiomatizations. arxiv:1304.0611 (2013)Google Scholar
  19. 19.
    Barwise, J.: Some applications of henkin quantifiers. Israel Journal of Mathematics 25(1), 47–63 (1976)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fredrik Engström
    • 1
  • Juha Kontinen
    • 2
  • Jouko Väänänen
    • 2
    • 3
  1. 1.Department of Philosophy, Linguistics and Theory of ScienceUniversity of GothenburgSweden
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiFinland
  3. 3.Institute for Logic, Language and ComputationUniversity of AmsterdamThe Netherlands

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