Abstract
In this paper we investigate a probability logic which enriches propositional calculus with a class of conditional probability operators of de Finetti’s type. The logic allows making formulas such as CP ≥ s(β|α), with the intended meaning ”the conditional probability of β given α is at least s”. A possible-world approach is proposed to give semantics to such formulas. An infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. We prove decidability of the presented logic.
This research was supported by Ministarstvo za nauku, tehnologije i razvoj Republike Srbije, through Matematički institut, under grant 1379.
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References
Boole, G.: An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities, Walton and Maberley, London (1854)
Coletti, S., Scozzafava, R.: Probabilistic logic in a coherent setting. Kluwer Academic Press, Dordrecht (2002)
de Finetti, B.: Theory of probability, vol. 1. John Wiley and Sons, Chichester (1974)
Fagin, R., Halpern, J., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87(1/2), 78–128 (1990)
Fagin, R., Halpern, J.: Reasoning about knowledge and probability. Journal of the ACM 41(2), 340–367 (1994)
Flaminio, T., Montagna, F.: A logical and algebraic treatment of conditional probability. In: Zadeh, L.A. (ed.) Proceedings of IPMU 2004, Perugia, Italy, July, 4-9 (2004)
Gilio, A.: Probabilistic reasoning under coherence in System P ∗ . Annals of Mathematics and Artificial Intelligence 34, 5–34 (2002)
Hawthorne, J.: On the logic of nonmonotonic conditionals and conditional probabilities. Journal of Philosophical Logic 25(1), 185–218 (1996)
Hawthorne, J.: On the logic of nonmonotonic conditionals and conditional probabilities: Predicate logic. Journal of Philosophical Logic 27(1), 1–34 (1998)
Hoover, D.N.: Probability logic. Annals of mathematical logic 14, 287–313 (1978)
Keisler, H.J.: Hyperfinite model theory. In: Logic Colloquium 1976, pp. 5–110. North-Holland, Amsterdam (1977)
Keisler, H.J.: Probability quantifiers. In: Barwise, J., Feferman, S. (eds.) Model-theoretic logics. Perspectives in Mathematical Logic, pp. 509–556. Springer, Berlin (1985)
Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse Der Mathematik, 1933; translated as Foundations of Probability, Chelsea Publishing Company (1950)
Lukasiewicz, T.: Probabilistic Default Reasoning with Conditional Constraints. Annals of Mathematics and Artificial Intelligence 34, 35–88 (2002)
Marchioni, E., Godo, L.: A Logic for Reasoning about Coherent Conditional Probability: A Modal Fuzzy Logic Approach. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 213–225. Springer, Heidelberg (2004)
Marković, Z., Ognjanović, Z., Rašković, M.: A Probabilistic Extension of Intuitionistic Logic. Mathematical Logic Quarterly 49(5), 415–424 (2003)
Nilsson, N.: Probabilistic logic. Artificial inteligence (28), 71–87 (1986)
Nilsson, N.: Probabilistic logic revised. Artificial inteligence (59), 39–42 (1993)
Ognjanović, Z., Rašković, M.: Some probability logics with new types of probability operators. Journal of logic and Computation 9(2), 181–195 (1999)
Ognjanović, Z., Rašković, M.: Some first order probability logics. Theoretical Computer Science 247(1-2), 191–212 (2000)
Rašković, M.: Completeness theorem for biprobability models. Jornal of Symboloc Logic 51, 586–590 (1986)
Rašković, M.: Classical logic with some probability operators, vol. 53 (67), pp. 1–3. Publication de l’Institut Math, NS (1993)
Rašković, M., Đorđević, R.: Probability quantifiers and operators, Vesta, Beograd (1996)
Rašković, M., Ognjanović, Z., Marković, Z.: A probabilistic Approach to Default Reasoning. In: 10th International Workshop on Non-Monotonic Reasoning NMR2004, June 6-8, pp. 335–341. Westin Whistler, Canada (2004)
Rašković, M., Ognjanović, Z., Marković, Z.: Logic with Conditional Probabilities. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 226–238. Springer, Heidelberg (2004)
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Ikodinović, N., Ognjanović, Z. (2005). A Logic with Coherent Conditional Probabilities. In: Godo, L. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2005. Lecture Notes in Computer Science(), vol 3571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11518655_61
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