Structures of Oppositions in Public Announcement Logic

  • Lorenz DemeyEmail author
Part of the Studies in Universal Logic book series (SUL)


In this paper we study dynamic epistemic logic, and in particular public announcement logic, using the tools of n-opposition theory. Dynamic epistemic logic is a contemporary development of epistemic logic, which takes into account changes of knowledge through time. It studies, for example, public announcements and the influence they have on the agents’ knowledge. n-Opposition theory is a systematic generalization of the traditional squares of oppositions. In the paper we provide introductions to the intuitions behind and the formal details of both of these disciplines. We construct a square and a hexagon of oppositions for the structural properties of public announcement. We then generalize this to opposition structures for any partially functional process, and show how this generalization can be used to support the structuralist philosophy surrounding n-opposition theory. Next, we focus on the epistemic properties of public announcement, and construct an octagon and a (three-dimensional) rhombic dodecahedron of oppositions. Many of the techniques applied in this paper were originally developed by Smessaert (Logica Univers. 3:303–332, 2009); they are thus shown to be very powerful and widely applicable. Summing up: we establish a strong connection between public announcement logic and n-opposition theory, and show that this connection has definite advantages for both of the disciplines involved.


Epistemic dynamics Public announcement logic n-Opposition theory Square of oppositions Structure of oppositions 

Mathematics Subject Classification

03B42 03A05 03G10 03B70 68T27 


  1. 1.
    Baltag, A., Moss, L.: Logics for epistemic programs. Synthese 139, 165–224 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Bonanno, G., van der Hoek, W., Woolridge, M. (eds.) Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3, pp. 9–58. Amsterdam University Press, Amsterdam (2008) Google Scholar
  3. 3.
    Béziau, J.-Y.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–232 (2003) Google Scholar
  4. 4.
    Blanché, R.: Quantity, modality, and other kindred systems of categories. Mind 61, 369–375 (1952) CrossRefGoogle Scholar
  5. 5.
    Blanché, R.: Sur l’opposition des concepts. Theoria 19, 89–130 (1953) CrossRefGoogle Scholar
  6. 6.
    Blanché, R.: Opposition et négation. Rev. Philos. Fr. étrang. 167, 187–216 (1957) Google Scholar
  7. 7.
    Blanché, R.: Structures intellectuelles. Essai sur l’organisation systématique des concepts. Vrin, Paris (1966) Google Scholar
  8. 8.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  9. 9.
    de Pater, W.A., Vergauwen, R.: Logica: Formeel en Informeel. Leuven University Press, Leuven (2005) Google Scholar
  10. 10.
    Demey, L.: Some remarks on the model theory of epistemic plausibility models. J. Appl. Non-Class. Log. 21, 375–395 (2011) CrossRefGoogle Scholar
  11. 11.
    Demey, L., Smessaert, H.: Logical geometries emanating from the traditional square of oppositions. Manuscript (2011) Google Scholar
  12. 12.
    Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Kluwer, Dordrecht (1998) zbMATHCrossRefGoogle Scholar
  13. 13.
    Gerbrandy, J., Groeneveld, W.: Reasoning about information change. J. Log. Lang. Inf. 6, 147–169 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54, 319–379 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000) zbMATHGoogle Scholar
  16. 16.
    Hintikka, J.: Knowledge and Belief. An Introduction to the Logic of the Two Notions. Cornell University Press, Ithaca (1962) Google Scholar
  17. 17.
    Kozen, D., Parikh, R.: An elementary proof of the completeness of PDL. Theor. Comput. Sci. 14, 113–118 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lenzen, W.: Recent Work in Epistemic Logic. North-Holland, Amsterdam (1978) Google Scholar
  19. 19.
    Lenzen, W.: Glauben, Wissen und Wahrscheinlichkeit: Systeme der epistemischen Logik. Springer, Berlin (1980) zbMATHCrossRefGoogle Scholar
  20. 20.
    Lutz, C.: Complexity and succinctness of public announcement logic. In: Stone, P., Weiss, G. (eds.) AAMAS ’06: Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 137–143. Association for Computing Machinery, New York (2006) CrossRefGoogle Scholar
  21. 21.
    McNamara, P.: Deontic Logic. Stanford Encyclopedia of Philosophy (2010) Google Scholar
  22. 22.
    Moretti, A.: The geometry of logical opposition. PhD thesis defended at the University of Neuchâtel, Switzerland (2009) Google Scholar
  23. 23.
    Plaza, J.: Logics of public communications. In: Emrich, M.L., Pfeifer, M.S., Hadzikadic, M., Ras, Z.W. (eds.) Proceedings of the Fourth International Symposium on Methodologies for Intelligent Systems: Poster Session Program, pp. 201–216. Oak Ridge National Laboratory, Oak Ridge (1989). Reprinted in: Synthese 158, 165–179 (2007) Google Scholar
  24. 24.
    Pellissier, R.: Setting n-opposition. Logica Univers. 2, 235–263 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Sesmat, A.: Logique II. Les raisonnements, la logistique. Hermann, Paris (1951) Google Scholar
  26. 26.
    Smessaert, S.: On the 3D visualisation of logical relations. Logica Univers. 3, 303–332 (2009) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Smessaert, H., Demey, L.: On the fourth Aristotelian relation: subalternation versus non-implication. Manuscript (2011) Google Scholar
  28. 28.
    van Benthem, J.: One is a lonely number: logic and communication. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium ’02. Lecture Notes in Logic, vol. 27, pp. 95–128. Association for Symbolic Logic & AK Peters, Wellesley (2006) Google Scholar
  29. 29.
    van Benthem, J.: Open problems in logical dynamics. In: Gabbay, D., Goncharov, S., Zakharyashev, M. (eds.) Mathematical Problems from Applied Logic I. International Mathematical Series, vol. 4, pp. 137–192. Springer, Berlin (2006) CrossRefGoogle Scholar
  30. 30.
    van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non-Class. Log. 17, 129–155 (2007) zbMATHCrossRefGoogle Scholar
  31. 31.
    van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge (2011) zbMATHCrossRefGoogle Scholar
  32. 32.
    van Dalen, D.: Logic and Structure, 4th edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  33. 33.
    van Ditmarsch, D., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Berlin (2007) Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Center for Logic and Analytical Philosophy, Institute of PhilosophyKU Leuven – University of LeuvenLeuvenBelgium

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