×

Interpretation of De Finetti coherence criterion in Łukasiewicz logic. (English) Zbl 1180.03029

Let \(E\) be a set of \(n\) events and consider a set \(W\) of possible worlds, i.e. functions from \(E\) to \([0,1]\). The main theorem of the paper states that if \(W\) is a nonempty closed subset of \([0,1]^E\), then any other map \(\beta:E \to [0,1]\) that is “coherent” with respect to \(W\) (in the sense of De Finetti) is a state over the MV-algebra of McNaughton functions over \(n\) variables restricted to \(W\) (hence it is an integral with respect to a probability measure on \(W\)), and it is a convex combination of elements of \(W\) (and vice versa).
Further, it is shown that any closed set \(W \subseteq [0,1]^E\) is associated with a theory \(\Theta\) in Łukasiewicz logic given by the set of formulas taking value 1 by the (unique) extensions of elements of \(W\) to the set of all Łukasiewicz formulas with \(n\) variables. Then probability assignments that are coherent with respect to a closed \(W\subseteq [0,1]^n\) are convex combinations of Łukasiewicz valuations satisfying \(\Theta\).
Hence, when dealing with closed sets of possible worlds assigning values in \([0,1]\) to events, Łukasiewicz logic plays a prominent role.

MSC:

03B50 Many-valued logic
03B48 Probability and inductive logic
06D35 MV-algebras
60A05 Axioms; other general questions in probability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Aguzzoli, B. Gerla, V. Marra, De Finetti’s no-Dutch-Book criterion for Gödel logic, Studia Logica (special issue on Many-valued Logic and Cognition, Shier Ju et al. (Eds.)) 90 (2008) 25-41; S. Aguzzoli, B. Gerla, V. Marra, De Finetti’s no-Dutch-Book criterion for Gödel logic, Studia Logica (special issue on Many-valued Logic and Cognition, Shier Ju et al. (Eds.)) 90 (2008) 25-41 · Zbl 1165.03008
[2] Baioletti, M.; Capotorti, A.; Tulipani, S.; Vantaggi, B., Simplification rules for the coherent probability assessment problem, Annals of Mathematics and Artificial Intelligence, 35, 11-28 (2002) · Zbl 1006.68128
[3] Busaniche, M.; Mundici, D., Geometry of Robinson consistency in Łukasiewicz logic, Annals of Pure and Applied Logic, 147, 1-22 (2007) · Zbl 1125.03016
[4] Cignoli, R.; D’Ottaviano, I. M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, (Trends in Logic, vol. 7 (2000), Kluwer: Kluwer Dordrecht), Springer-Verlag, New York · Zbl 0937.06009
[5] Cignoli, R.; Elliott, G. A.; Mundici, D., Reconstructing \(C^\ast \)-algebras from their Murray von Neumann orders, Advances in Mathematics, 101, 166-179 (1993) · Zbl 0823.46053
[6] De Finetti, B., Sul significato soggettivo della probabilitá, Fundamenta Mathematicae, 17, 298-329 (1931), Translated into English as On the subjective meaning of probability, in: P. Monari, D. Cocchi (Eds.), Probabilitá e Induzione, Clueb, Bologna, 1993, pp. 291-321 · JFM 57.0608.07
[7] De Finetti, B., La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut H. Poincaré, 7, 1-68 (1937), Translated into English by H.E. Kyburg Jr., as Foresight: Its logical laws, its subjective sources, in: H.E. Kyburg Jr., H.E. Smokler (Eds.), Studies in Subjective Probability, Wiley, New York, 1964. Second edition published by Krieger, New York, 1980, pp. 53-118 · JFM 63.1070.02
[8] De Finetti, B., Theory of Probability, vol. 1 (1974), John Wiley and Sons: John Wiley and Sons Chichester · Zbl 0328.60002
[9] Emch, G. G., Mathematical and conceptual foundations of 20th century physics, (Notas de Matemática, vol. 100 (1984), North-Holland: North-Holland Amsterdam) · Zbl 0591.01020
[10] Flaminio, T.; Montagna, F., An algebraic approach to states on MV-algebras, (Štěpnička, M.; etal., Proceedings 5th EUSFLAT07 Conference, Ostrava, Czech Republic, vol. 2 (2007)), 201-206
[11] Grünbaum, B., Convex polytopes, (Graduate Texts in Mathematics, vol. 221 (2003), Springer-Verlag: Springer-Verlag New York) · Zbl 0163.16603
[12] Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer: Kluwer Dordrecht · Zbl 0937.03030
[13] T. Kroupa, States and conditional probability on MV-algebras, Ph.D. Thesis, Czech Technical University in Prague, Faculty of Electrical Engineering, Dept. of Cybernetics, 2005; T. Kroupa, States and conditional probability on MV-algebras, Ph.D. Thesis, Czech Technical University in Prague, Faculty of Electrical Engineering, Dept. of Cybernetics, 2005 · Zbl 1061.60004
[14] Kühr, J.; Mundici, D., De Finetti theorem and Borel states in [0,1]-valued algebraic logic, International Journal of Approximate Reasoning, 46, 605-616 (2007) · Zbl 1189.03076
[15] J. Łukasiewicz, A. Tarski, Investigations into the Sentential Calculus, in: [30]; J. Łukasiewicz, A. Tarski, Investigations into the Sentential Calculus, in: [30] · JFM 57.1319.01
[16] Mostowski, A., L’oeuvre scientifique de Jan Łukasiewicz dans le domaine de la logique mathématique, Fundamenta Mathematicae, 44, 1-11 (1957) · Zbl 0077.24217
[17] Mundici, D., Interpretation of AF \(C^\ast \)-algebras in Łukasiewicz sentential calculus, Journal of Functional Analysis, 65, 15-63 (1986) · Zbl 0597.46059
[18] Mundici, D., Farey stellar subdivisions, ultrasimplicial groups, and \(K_0\) of AF \(C^\ast \)-algebras, Advances in Mathematics, 68, 23-39 (1988) · Zbl 0678.06008
[19] Mundici, D., Averaging the truth-value in Łukasiewicz logic, Studia Logica, 55, 113-127 (1995) · Zbl 0836.03016
[20] Mundici, D., Bookmaking over infinite-valued events, International Journal of Approximate Reasoning, 43, 223-240 (2006) · Zbl 1123.03011
[21] Mundici, D., The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete and Continuous Dynamical Systems, 21, 537-549 (2008) · Zbl 1154.28007
[22] Mundici, D., Faithful and invariant conditional probability in Łukasiewicz logic, (Makinson, D.; Malinowski, J.; Wansing, H., Towards Mathematical Philosophy, Trends in Logic (2008), Springer: Springer New York) · Zbl 1163.03016
[23] Mundici, D.; Panti, G., Decidable and undecidable prime theories in infinite-valued logic, Annals of Pure and Applied Logic, 108, 269-278 (2001) · Zbl 1130.03019
[24] Mundici, D.; Tsinakis, C., Gödel incompleteness in AF C*-algebras, Forum Mathematicum, 20, 1071-1084 (2008) · Zbl 1163.46036
[25] Panti, G., Invariant measures in free MV-algebras, Communications in Algebra, 36, 2849-2861 (2008) · Zbl 1154.06008
[26] (Pap, E., Handbook of Measure Theory, vols. I, II (2002), North-Holland: North-Holland Amsterdam) · Zbl 0998.28001
[27] Paris, J., A note on the Dutch Book method, (De Cooman, G.; Fine, T.; Seidenfeld, T., Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications. Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications, ISIPTA 2001, Ithaca, NY, USA (2001), Shaker Publishing Company), 301-306, Available at http://www.maths.man.ac.uk/DeptWeb/Homepages/jbp/
[28] B. Riečan, D. Mundici, Probability on MV-algebras, in: [26]; B. Riečan, D. Mundici, Probability on MV-algebras, in: [26] · Zbl 1017.28002
[29] Semadeni, Z., Banach spaces of continuous functions, vol. I (1971), PWN, Polish Scientific Publishers: PWN, Polish Scientific Publishers Warsaw · Zbl 0225.46030
[30] Tarski, A., Logic, Semantics, Metamathematics (1956), Clarendon Press: Clarendon Press Oxford, Reprinted, Hackett, Indianapolis, 1983
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.