Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 955–975 | Cite as

Structural completeness in propositional logics of dependence



In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogous result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic.


Structural completeness Dependence logic Inquisitive logic Intermediate logic 

Mathematics Subject Classification

03F03 03B55 03B60 03B65 


  1. 1.
    Burgess, J.P.: A remark on henkin sentences and their contraries. Notre Dame J. Form. Log. 44(3), 185–188 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, New York (1997)MATHGoogle Scholar
  3. 3.
    Ciardelli, I.: Dependency as question entailment. In: Abramsky, S., Kontinen, J., Väänänen, J. (eds.) Dependence Logic: Theory and Application, Progress in Computer Science and Applied Logic, pp. 129–181. Birkhauser (2016)Google Scholar
  4. 4.
    Ciardelli, I., Roelofsen, F.: Inquisitive logic. J. Philos. Logic 40(1), 55–94 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ebbing, J., Hella, L., Meier, A., Müller, J.S., Virtema, J., Vollmer, H.: Extended modal dependence logic. 20th International Workshop. WoLLIC 2013, Proceedings, Lecture Notes in Computer Science, vol. 8071, pp. 126–137. Springer, Berlin (2013)Google Scholar
  6. 6.
    Ghilardi, S.: Unification in intuitionistic logic. J. Symb. Logic 64, 859–880 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hella, L., Luosto, K., Sano, K., Virtema, J.: The Expressive Power of Modal Dependence Logic. In: Advances in Modal Logic 10, Invited and Contributed Papers from the Tenth Conference on Advances in Modal Logic, College Publications, London, 294–312 (2014)Google Scholar
  8. 8.
    Henkin, L.: Some remarks on infinitely long formulas. Infinitistic Methods. In: Proceedings Symposium Foundations of Mathematics, pp. 167–183. Pergamon, Warsaw (1961)Google Scholar
  9. 9.
    Hintikka, J.: The Principles of Mathematics Revisited. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  10. 10.
    Hintikka, J., Sandu, G.: Informational independence as a semantical phenomenon. In: Fenstad, R.H.J.E., Frolov, I.T. (eds.) Logic, Methodology and Philosophy of Science, pp. 571–589. Elsevier, Amsterdam (1989)Google Scholar
  11. 11.
    Hodges, W.: Compositional semantics for a language of imperfect information. Log. J. IGPL 5, 539–563 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hodges, W.: Some strange quantifiers. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht, Lecture Notes in Computer Science, vol. 1261, pp. 51–65. Springer, London (1997)CrossRefGoogle Scholar
  13. 13.
    Iemhoff, R.: On the admissible rules of intuitionistic propositional logic. J. Symb. Log. 66, 281–294 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Iemhoff, R.: Consequence relations and admissible rules. Tech. Rep. 314, Department of Philosophy. Utrecht University, Utrecht (2013)Google Scholar
  15. 15.
    Jeřábek, E.: Admissible rules of modal logics. J. Log. Comput. 15, 411–431 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kontinen, J., Väänänen, J.: A remark on negation in dependence logic. Notre Dame J. Form. Log. 52(1), 55–65 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kuusisto, A.: A double team semantics for generalized quantifiers logic. J. Logic Lang. Inform. 24(2), 149–191 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lohmann, P., Vollmer, H.: Complexity results for modal dependence logic. Stud. Logica. 101(2), 343–366 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Metcalfe, G.: Admissible rules: From characterizations to applications. In: Proceedings of WoLLIC 2012, LNCS, vol. 7456, pp. 56–69. Springer (2012)Google Scholar
  20. 20.
    Miglioli, P., Moscato, U., Ornaghi, M., Quazza, S., Usberti, G.: Some results on intermediate constructive logics. Notre Dame J. Form. Log. 30(4), 543–562 (1989)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Rozière, P.: Regles admissibles en calcul propositionnel intuitionniste. Ph.D. thesis, Université Paris VII (1992)Google Scholar
  22. 22.
    Rybakov, V.: Admissibility of Logical Inference Rules. Elsevier, Amsterdam (1997)MATHGoogle Scholar
  23. 23.
    Sano, K., Virtema, J.: Axiomatizing Propositional Dependence Logics Proceedings of the 24th EACSL Annual Conference on Computer Science Logic, 292–307 (2015)Google Scholar
  24. 24.
    Väänänen, J.: Dependence Logic: A New Approach to Independence Friendly Logic. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  25. 25.
    Wojtylak, P.: On a problem of H. Friedman and its solution by T. Prucnal. Rep. Math. Log. 38, 69–86 (2004)Google Scholar
  26. 26.
    Yang, F.: On extensions and variants of dependence logic. Ph.D. thesis, University of Helsinki (2014)Google Scholar
  27. 27.
    Yang, F., Väänänen, J.: Propositional logics of dependence. Ann. Pure Appl. Log. 167(7), 557–589 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Philosophy and Religious StudiesUtrecht UniversityUtrechtThe Netherlands

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