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Partially Undetermined Many-Valued Events and Their Conditional Probability

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Abstract

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic setting, and we find a logical characterization of coherence for assessments of partially undetermined events.

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References

  1. Aguzzoli, S., Gerla, B., & Marra, V. (2008). De Finetti’s no Dutch-book criterion for Gödel logic. Studia Logica, 90, 25–41.

    Article  Google Scholar 

  2. Blok, W., & Pigozzi, D. (1989). Algebraizable logics. In Memoirs of the American Mathematical Society (Vol. 396(77)). Providence: American Mathematical Society.

    Google Scholar 

  3. Burris, S., & Sankappanavar, H. P. (1981). A course in universal algebra. New York: Springer.

    Book  Google Scholar 

  4. Busaniche, M., & Cignoli, R. (2009). Residuated lattices as an algebraic semantics for paraconsistent Nelson’s logic. Journal of Logic and Computation, 19(6), 1019–1029.

    Article  Google Scholar 

  5. Canny, F. J. (1988). Some algebraic and geometric computation in PSPACE. In Proc. 20th ACM symp. on theory of computing (pp. 460–467).

  6. Chang, C. C. (1989). A new proof of the completeness of Łukasiewicz axioms. Transactions of the American Mathematical Society, 93, 74–80.

    Google Scholar 

  7. Cintula, P. (2006). Weakly implicative (fuzzy) logics I: Basic properties. Archive for Mathematical Logic, 45, 673–704.

    Article  Google Scholar 

  8. Coletti, G., & Scozzafava, R. (2002). Probabilistic logic in a coherent setting. Dordrecht: Kluwer.

    Book  Google Scholar 

  9. Czelakowski, J. (2001). Protoalgebraic logics. Dordrecht: Kluwer.

    Google Scholar 

  10. Dubois, D., & Prade, H. (1994). Conditional objects as nonmonotonic consequence relations. IEEE Transactions on Systems, Man and Cybernetics, 24(12), 1724–1740. IEEE.

    Article  Google Scholar 

  11. Flaminio, T., & Montagna, F. (2009). MV algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning, 50(1), 138–152.

    Article  Google Scholar 

  12. Hájek, P. (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer.

    Book  Google Scholar 

  13. Kroupa, T. (2006). Every state on a semisimple MV algebra is integral. Fuzzy Sets and Systems, 157(20), 2771–2787.

    Article  Google Scholar 

  14. Kroupa, T. (2005). Conditional probability on MV-algebras. Fuzzy Sets and Systems, 149(2), 369–381.

    Article  Google Scholar 

  15. Kühr, J., & Mundici, D. (2007). De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal of Approximate Reasoning, 46(3), 605–616.

    Article  Google Scholar 

  16. Montagna, F. (2000). An algebraic approach to propositional fuzzy logic. Journal of Logic, Language and Information, 9, 91–124.

    Article  Google Scholar 

  17. Montagna, F. (2005). Subreducts of MV algebras with product and product residuation. Algebra Universalis, 53, 109–137.

    Article  Google Scholar 

  18. Montagna, F. (2009). A notion of coherence for books on conditional events in many-valued logic. Journal of Logic and Computation. doi:10.1093/logcom/exp061. First published online: 17 September 2009.

  19. Mundici, D. (1995). Averaging the truth value in Łukasiewicz logic. Studia Logica, 55(1), 113–127.

    Article  Google Scholar 

  20. Mundici, D. (2006). Bookmaking over infinite-valued events. International Journal of Approximate Reasoning, 46, 223–240.

    Article  Google Scholar 

  21. Mundici, D. (2008). Faithful and invariant conditional probability in Łukasiewicz logic. In D. Makinson, J. Malinowski, & H. Wansing (Eds.), Trends in logic 27: Towards mathematical philosophy (pp. 1–20). Springer.

  22. Panti, G. (2008). Invariant measures in free MV algebras. Communications in Algebra, 36, 2849–2861.

    Article  Google Scholar 

  23. Papadimitriu, C. H. (1994). Computational complexity. Addison-Wesley.

  24. Paris, J. B. (2001). A note on the Dutch Book Method. In T. S. Gert De Cooman, & T. Fine (Eds.), ISIPTA ’01, Proceedings of the second international symposium on imprecise probabilities and their applications, Ithaca, NY, USA. Shaker.

  25. Sendlewski, A. (1990). Nelson algebras through Heyting ones, I. Studia Logica, 49, 105–126.

    Article  Google Scholar 

  26. Tsinakis, C., & Wille, A. (2006). Minimal varieties of involutive residuated lattices. Studia Logica, 83, 407–423.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini 44, 5100, Siena, Italy

    Franco Montagna

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  1. Franco Montagna
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Correspondence to Franco Montagna.

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Montagna, F. Partially Undetermined Many-Valued Events and Their Conditional Probability. J Philos Logic 41, 563–593 (2012). https://doi.org/10.1007/s10992-011-9185-3

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  • DOI: https://doi.org/10.1007/s10992-011-9185-3

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