Journal of Logic, Language and Information

, Volume 19, Issue 3, pp 247–282 | Cite as

Probability Logic of Finitely Additive Beliefs



Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage’s postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ+ that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ+ iff it is satisfied in a finitely additive type space. Although we can characterize Σ+-theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics.


Probabilistic beliefs Type spaces Reasoning about probabilities Modal logic 


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  1. Aliprantis, C., & Border, K. (2006). Infinite dimension analysis. In A Hitchhiker’s guide (3rd edn). New York: Springer.Google Scholar
  2. Aumann R. (1999a) Interactive epistemology: Knowledge. International Journal of Game Theory 28: 263–300CrossRefGoogle Scholar
  3. Aumann R. (1999b) Interactive epistemology: Probability. International Journal of Game Theory 28: 301–314CrossRefGoogle Scholar
  4. Aumann, R., & Heifetz, A. (2002). Incomplete information. In: R. Aumann & S. Hart (Eds.), Handbook of game theory (pp. 1665–1686). Handbook Series in Economics.Google Scholar
  5. Blackburn P., de Rijke M., Venema Y. et al (2000) Modal logic. In: Abramsky S. (eds) Cambridge tracts in theoretical computer science (vol. 53). Oxford University Press, OxfordGoogle Scholar
  6. de Finetti B. (1975) Theory of probability. Wiley, New YorkGoogle Scholar
  7. Fagin R., Halpern J. (1994) Reasoning about knowledge and probability. Journal of ACM 41: 340–367CrossRefGoogle Scholar
  8. Fagin R., Megiddo N., Halpern J. (1990) A logic for reasoning about probabilities. Information and Computation 87: 78–128CrossRefGoogle Scholar
  9. Fishburn C. (1986) The axioms of subjective probability. Statistical Science 1: 335–345CrossRefGoogle Scholar
  10. Gerla G. (1994) Inferences in probability logic. Artificial Intelligence 70(1–2): 33–52CrossRefGoogle Scholar
  11. Gerla G. (2001) Fuzzy logic: Mathematical tools for approximate reasoning. In: Wojcicki R. (eds) Trends in logic. Kluwer, Studia Logica Library, Amsterdam, pp 1–1Google Scholar
  12. Goldblatt, R. (1993). Mathematics of modality (vol. 43). CSLI Lecture Notes. Stanford: CSLI.Google Scholar
  13. Goldblatt, R. (2008). Deduction systems for coalgebras over measurable spaces. Journal of Logic and Computation (to appear).Google Scholar
  14. Halpern J., Fagin R., Moses Y., Vardi V. (1995) Reasonign about knowledge. MIT Press, MAGoogle Scholar
  15. Harsanyi J. (1967) Games with incomplete information played by bayesian players, part one. Management Science 14: 159–182CrossRefGoogle Scholar
  16. Heifetz A., Mongin P. (2001) Probability logic for type spaces. Games and Economic Behavior 35: 31–53CrossRefGoogle Scholar
  17. Heifetz A., Samet D. (1998) Knowledge space with an arbitrary high rank. Games and Economic Behavior 22: 260–273CrossRefGoogle Scholar
  18. Heifetz A., Samet D. (1998) Topology-free typology of beliefs. Journal of Economic Theory 82: 324–341CrossRefGoogle Scholar
  19. Heifetz A., Samet D. (1999) Coherent beliefs are not always types. Journal of Mathematics and Economics 32: 475–488CrossRefGoogle Scholar
  20. Horn A., Tarski A. (1948) Measures in boolean algebras. Transactions of AMS 64: 467–497CrossRefGoogle Scholar
  21. Howson C., De finetti (2008) countable addivity, consistency and coherence. British Journal for the Philosophy of Science 59(3): 1–23CrossRefGoogle Scholar
  22. Los J., Marszewski E. (1949) Extensions of measures. Fundamenta Mathematicae 36: 267–276Google Scholar
  23. Machina M., Schmeidler D. (1992) A more robust definition of subjective probability. Econometrica 60(4): 745–780CrossRefGoogle Scholar
  24. Meier M. (2006) Finitely additive beliefs and universal type spaces. Annals of Probability 34: 386–422CrossRefGoogle Scholar
  25. Meier, M. (2009). An infinitary probability logic for type spaces. Israel Journal of Mathematics (to appear).Google Scholar
  26. Moss L., Viglizzo I. (2006) Final coalgebras for functors on measurable spaces. Information and Computation 204: 610–636CrossRefGoogle Scholar
  27. Ognjanovic Z., Raskovic M. (1999) Some probability logics with new types of probability operators. Journal of Logic and Computation 9(2): 181–195CrossRefGoogle Scholar
  28. Ognjanovic Z., Raskovic M. (2000) Some first-order probability logics. Theoretical Computer Science 247(1–2): 191–212CrossRefGoogle Scholar
  29. Ognjanovic Z., Perovic A., Raskovic M. (2008a) An axiomatization of qualitative probability. Acta Polytechnica Hungarica 5(1): 105–110Google Scholar
  30. Ognjanovic Z., Perovic A., Raskovic M. (2008b) Logics with the qualitative probability operator. Logic Journal of the IGPL 16(2): 105–120CrossRefGoogle Scholar
  31. Samet D. (2000) Quantified beliefs and believed quantities. Journal of Economic Theory 95: 169–185CrossRefGoogle Scholar
  32. Savage L. (1972) The foundations of statistics. Dover, New YorkGoogle Scholar
  33. Williamson J. (1999) Countably additivity and subjective probability. British Journal for the Philosophy of Science 50(3): 401–416CrossRefGoogle Scholar
  34. Williamson, J. (2002). Probability logic. In: D. Gabbay, R. Johnson, H. Jurgen Ohlbach, & J. Woods (Eds.), Handbook of the logic of inference and argument: The turn towards the practical (pp. 397–424). Studies in Logic and Practical Reasoning.Google Scholar
  35. Zhou, C. (2007). Complete Deductive Systems for Probability Logic with Application in Harsanyi Type spaces. PhD thesis, Indiana University, Bloomington.Google Scholar
  36. Zhou, C. (2009). A complete deductive system for probability logic. Journal of Logic and Computation (to appear).Google Scholar
  37. Ziemer, W. (2004). Modern real analysis. Online lecture notes.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Lab for Information Science and Technology, Department of Computer Science and TechnologyTsinghua UniversityBeijingChina

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