Advertisement

Logic and Probabilistic Update

  • Lorenz Demey
  • Barteld Kooi
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 5)

Abstract

This chapter surveys recent work on probabilistic extensions of epistemic and dynamic-epistemic logics (the latter include the basic system of public announcement logic as well as the full product update logic). It emphasizes the importance of higher-order information as a distinguishing feature of these logics. This becomes particularly clear in the dynamic setting: although there exists a clear relationship between usual Bayesian conditionalization and public announcement, the probabilistic effects of the latter are in general more difficult to describe, because of the subtleties involved in higher-order information. Finally, the chapter discusses some applications of probabilistic dynamic epistemic logic, such as the Lockean thesis in formal epistemology and Aumann’s agreement theorem in game theory.

Notes

Acknowledgments

The authors wish to thank Alexandru Baltag, Johan van Benthem and Sonja Smets for their valuable feedback on earlier versions of this chapter. Lorenz Demey is financially supported by a PhD fellowship of the Research Foundation–Flanders (FWO).

References

  1. 1.
    Alchourrón C, Gärdenfors P, Makinson D (1985) On the logic of theory change: partial meet contraction and revision functions. J Symb Logic 50:510–530CrossRefGoogle Scholar
  2. 2.
    Aucher G (2003) A combined system for update logic and belief revision. Master’s thesis, Institute for Logic, Language and Computation, Universiteit van AmsterdamGoogle Scholar
  3. 3.
    Aumann R (1976) Agreeing to disagree. Ann Stat 4:1236–1239CrossRefGoogle Scholar
  4. 4.
    Baltag A, Moss LS (2004) Logics for epistemic programs. Synthese 139:1–60CrossRefGoogle Scholar
  5. 5.
    Baltag A, Moss LS, Solecki S (1998) The logic of common knowledge, public announcements, and private suspicions. In: Gilboa I (ed) Proceedings of the 7th conference on theoretical aspects of rationality and knowledge (TARK ’98), pp 43–56Google Scholar
  6. 6.
    Baltag A, Smets S (2008) Probabilistic dynamic belief revision. Synthese 165:179–202CrossRefGoogle Scholar
  7. 7.
    Baltag A, Smets S (2008) A qualitative theory of dynamic interactive belief revision. In: Bonanno G, van der Hoek W, Woolridge M (eds) Texts in logic and games, vol 3. Amsterdam University Press, pp 9–58Google Scholar
  8. 8.
    van Benthem J (1983) Modal logic and classical logic. Bibliopolis, NapoliGoogle Scholar
  9. 9.
    van Benthem J (2001) Correspondence theory. In: Gabbay DM, Guenthner F (eds) Handbook of philosophical logic, revised 2nd edn, vol 3. Kluwer, Dordrecht, pp 325–408Google Scholar
  10. 10.
    van Benthem J (2001) Games in dynamic epistemic logic. Bull Econ Res 53:219–248CrossRefGoogle Scholar
  11. 11.
    van Benthem J (2003) Conditional probability meets update logic. J Logic Lang Inf 12:409–421CrossRefGoogle Scholar
  12. 12.
    van Benthem J (2007) Dynamic logic for belief revision. J Appl Nonclass Logics 17:129–155CrossRefGoogle Scholar
  13. 13.
    van Benthem J (2007) Rational dynamics and epistemic logic in games. Int Game Theory Rev 9:13–45CrossRefGoogle Scholar
  14. 14.
    van Benthem J (2008) Logic and reasoning: do the facts matter? Stud Logica 88:67–84CrossRefGoogle Scholar
  15. 15.
    van Benthem J (2011) Logical dynamics of information and interaction. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  16. 16.
    van Benthem J (2012) A problem concerning qualitative probabilistic update. p 10. (Unpublished manuscript)Google Scholar
  17. 17.
    van Benthem J, Gerbrandy J, Kooi BP (2009) Dynamic update with probabilities. Stud Logica 93:67–96CrossRefGoogle Scholar
  18. 18.
    Bonanno G, Dégremont C (2014) Logic and game theory. In: Baltag A, Smets S (eds) Johan van Benthem on logic and information dynamics. Springer, Dordrecht, pp 421–449 (Chapter 15 in this volume)Google Scholar
  19. 19.
    de Bruin B (2010) Explaining games: the epistemic programme in game theory. Springer, DordrechtCrossRefGoogle Scholar
  20. 20.
    Conradie W, Ghilardi S, Palmigiano A (2014) Unified correspondence. In: Baltag A, Smets S (eds) Johan van Benthem on logic and information dynamics. Springer, Dordrecht, pp 933–975 (Chapter 36 in this volume)Google Scholar
  21. 21.
    Demey L (2010) Agreeing to disagree in probabilistic dynamic epistemic logic. Master’s thesis, Institute for Logic, Language and Computation, Universiteit van AmsterdamGoogle Scholar
  22. 22.
    Demey L (2011) Some remarks on the model theory of epistemic plausibility models. J Appl Nonclass Logics 21:375–395CrossRefGoogle Scholar
  23. 23.
    Demey, L. (2013) Contemporary epistemic logic and the Lockean thesis. Found Sci 18:599–610Google Scholar
  24. 24.
    Demey, L. (2014) Agreeing to disagree in probabilistic dynamic epistemic logic. Synthese 191:409–438Google Scholar
  25. 25.
    Demey L (forthcoming) The dynamics of surprise. Logique AnalGoogle Scholar
  26. 26.
    Demey L, Kooi B, Sack J (2013) Logic and probability. In: Zalta EN (ed) The Stanford encyclopedia of philosophy, StanfordGoogle Scholar
  27. 27.
    van Ditmarsch H, van der Hoek W, Kooi B (2007) Dynamic epistemic logic. Springer, DordrechtCrossRefGoogle Scholar
  28. 28.
    Fagin R, Halpern J (1994) Reasoning about knowledge and probability. J ACM 41:340–367CrossRefGoogle Scholar
  29. 29.
    van Fraassen B (1984) Belief and the will. J Philos 81:235–256CrossRefGoogle Scholar
  30. 30.
    Gärdenfors P (1988) Knowledge in flux. MIT Press, CambridgeGoogle Scholar
  31. 31.
    Gerbrandy J, Groeneveld W (1997) Reasoning about information change. J Logic Lang Inf 6:147–169CrossRefGoogle Scholar
  32. 32.
    Halpern JY (1991) The relationship between knowledge, belief, and certainty. Ann Math Artif Intell 4:301–322 (errata in the same journal, 26:59–61, 1999)Google Scholar
  33. 33.
    Halpern JY (2003) Reasoning about uncertainty. MIT Press, CambridgeGoogle Scholar
  34. 34.
    Hintikka J (1962) Knowledge and belief: an introduction to the logic of the two notions. Cornell University Press, IthacaGoogle Scholar
  35. 35.
    Isaac A, Szymanik J, Verbrugge R (2014) Logic and complexity in cognitive science. In: Baltag A, Smets S (eds) Johan van Benthem on logic and information dynamics. Springer, Dordrecht, pp 787–824 (Chapter 30 in this volume)Google Scholar
  36. 36.
    Jeffrey R (1983) The logic of decision, 2nd edn. University of Chicago Press, ChicagoGoogle Scholar
  37. 37.
    Kooi BP (2003) Probabilistic dynamic epistemic logic. J Logic Lang Inf 12:381–408CrossRefGoogle Scholar
  38. 38.
    Lewis D (1980) A subjectivist’s guide to objective chance. In: Jeffrey RC (ed) Studies in inductive logic and probability, vol 2. University of California Press, Berkeley, pp 263–293Google Scholar
  39. 39.
    Lorini E, Castelfranchi C (2007) The cognitive structure of surprise: looking for basic principles. Topoi 26:133–149CrossRefGoogle Scholar
  40. 40.
    Meacham CJG (2010) Two mistakes regarding the principal principle. Br J Philos Sci 61:407–431CrossRefGoogle Scholar
  41. 41.
    Miller D (1966) A paradox of information. Br J Philos Sci 17:59–61CrossRefGoogle Scholar
  42. 42.
    Oaksford M, Chater N (2008) The probabilistic mind: prospects for Bayesian cognitive science. Oxford University Press, OxfordGoogle Scholar
  43. 43.
    Pietarinen AV (2003) What do epistemic logic and cognitive science have to do with each other? Cogn Syst Res 4:169–190CrossRefGoogle Scholar
  44. 44.
    Plaza J (1989) Logics of public communications. In: Emrich ML, Pfeifer MS, Hadzikadic M, Ras ZW (eds) Proceedings of the 4th international symposium on methodologies for intelligent systems: poster session program, Oak Ridge National Laboratory, Oak Ridge, pp 201–216 (Reprinted in: Synthese 158:165–179, 2007)Google Scholar
  45. 45.
    Romeijn JW (2012) Conditioning and interpretation shifts. Stud Logica 100:583–606CrossRefGoogle Scholar
  46. 46.
    Rott H, Girard P (2014) Belief revision and dynamic logic. In: Baltag A, Smets S (eds) Johan van Benthem on logic and information dynamics. Springer, Dordrecht, pp 203–233 (Chapter 8 in this volume)Google Scholar
  47. 47.
    Sack J (2009) Extending probabilistic dynamic epistemic logic. Synthese 169:241–257CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Center for Logic and Analytical PhilosophyKU Leuven—University of LeuvenLeuvenBelgium
  2. 2.Faculty of PhilosophyUniversity of GroningenGroningenThe Netherlands

Personalised recommendations