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Studia Logica

, Volume 101, Issue 2, pp 293–322 | Cite as

Lottery Semantics: A Compositional Semantics for Probabilistic First-Order Logic with Imperfect Information

  • Pietro Galliani
  • Allen L. Mann
Article

Abstract

We present a compositional semantics for first-order logic with imperfect information that is equivalent to Sevenster and Sandu’s equilibrium semantics (under which the truth value of a sentence in a finite model is equal to the minimax value of its semantic game). Our semantics is a generalization of an earlier semantics developed by the first author that was based on behavioral strategies, rather than mixed strategies.

Keywords

Logic with imperfect information Independence-friendly logic Dependence-friendly logic Dependence logic Equilibrium semantics Compositional semantics 

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References

  1. 1.
    Blass A., Gurevich Y.: Henkin quantifiers and complete problems. Annals of Pure and Applied Logic 32, 1–16 (1986)CrossRefGoogle Scholar
  2. 2.
    Cintula, P., and O. Majer, Towards evaluation games for fuzzy logics, in O. Majer, A.-V. Pietarinen, and T. Tulenheimo, (eds.), Games: Unifying Logic, Language, and Philosophy, vol. 15 of Logic, Epistemology, and the Unity of Science, Springer, 2009, pp. 117–138.Google Scholar
  3. 3.
    Enderton H.B.: Finite partially-ordered quantifiers. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 16, 393–397 (1970)CrossRefGoogle Scholar
  4. 4.
    Gale, D., and F.M. Stewart, Infinite games with perfect information, in H.W. Kuhn, and A.W. Tucker, (eds.), Contributions to the Theory of Games, vol. 2, Princeton University Press, Princeton, 1953, pp. 245–266.Google Scholar
  5. 5.
    Galliani, P., Game Values and Equilibria for Undetermined Sentences of Dependence Logic, MSc thesis, ILLC Publications, MoL-2008-08, 2008.Google Scholar
  6. 6.
    Henkin, L., Some remarks on infinitely long formulas, in Infinitistic Methods. Proc. Symposium on Foundations of Mathematics, Pergamon Press, Oxford, 1961, pp. 167–183.Google Scholar
  7. 7.
    Hintikka, J., Language-games for quantifiers, in Studies in Logical Theory, vol. 2 of American Philosophical Quarterly Monograph Series, Basil Blackwell, Oxford, 1968, pp. 46–72.Google Scholar
  8. 8.
    Hintikka J.: Quantifiers vs. quantification theory. Dialectica 27, 329–358 (1973)CrossRefGoogle Scholar
  9. 9.
    Hintikka J.: The Principles of Mathematics Revisited. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  10. 10.
    Hintikka, J., and J. Kulas, The Game of Language: Studies in Game-Theoretical Semantics and Its Applications, D. Reidel Publishing Company, Dordrecht, 1983.Google Scholar
  11. 11.
    Hintikka, J., and G. Sandu, Informational independence as a semantic phenomenon, in J. E. Fenstad, I. T. Frolov, and R. Hilpinen, (eds.), Logic, Methodology and Philosophy of Science VIII, Elsevier, Amsterdam, 1989, pp. 571–589.Google Scholar
  12. 12.
    Hintikka, J., and G. Sandu, Game-theoretical semantics, in J. van Bethem, and A. ter Meulen, (eds.), Handbook of Logic and Language, Elsevier, Amsterdam, 1997, pp. 361–410.Google Scholar
  13. 13.
    Hodges W.: Compositional semantics for a language of imperfect information. Logic Journal of IGPL 5, 539–563 (1997)CrossRefGoogle Scholar
  14. 14.
    Hodges, W., Some strange quantifiers, in J. Mycielski, G. Rozenberg, and A. Salomaa, (eds.), Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht, no. 1261 in Lecture Notes in Computer Science, Springer, 1997, pp. 51–65.Google Scholar
  15. 15.
    Kuhn, H.W., Extensive games and the problem of information, in Contributions to the Theory of Games III, Princeton University Press, Princeton, 1953, pp. 193–216.Google Scholar
  16. 16.
    Mann, A.L., G. Sandu, and M. Sevenster, Independence-Friendly Logic: A Game- Theoretic Approach, no. 386 in London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2011.Google Scholar
  17. 17.
    Peirce, C.S., Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898, Harvard Historical Studies, Harvard University Press, Cambridge, Massachusetts, 1992.Google Scholar
  18. 18.
    Sevenster, M., Branches of Imperfect Information: Logic, Games, and Computation, Ph.D. thesis, Institute for Logic, Language and Computation, Universiteit van Amsterdam, 2006.Google Scholar
  19. 19.
    Sevenster M., Sandu G.: Equilibrium semantics of languages of imperfect information. Annals of Pure and Applied Logic, 161, 618–631 (2010)CrossRefGoogle Scholar
  20. 20.
    Väänänen, J., Dependence Logic: A New Approach to Independence Friendly Logic, no. 70 in London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2007.Google Scholar
  21. 21.
    von Neumann J.: Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, 295–320 (1928)CrossRefGoogle Scholar
  22. 22.
    Walkoe, W.J., Jr., Finite partially-ordered quantification, Journal of Symbolic Logic 35:535–555, 1970.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.FNWI, ILLCUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Department of MathematicsColgate UniversityHamiltonUSA

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