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Bookmaking over infinite-valued events. (English) Zbl 1123.03011
A valuation is a function \(V\) from the set of formulas to the unit interval of real numbers such that \(V(\neg\phi)= 1- V(\phi)\), \(V(\phi\oplus\psi)= \min(1, V(\phi)+ V(\psi))\), \(V(\phi\odot\psi)= \max(0, V(\phi)+ V(\psi)- 1)\). Two formulas \(\phi\), \(\psi\) are equivalent if \(V(\phi)= V(\psi)\) for any valuation \(V\). A mapping \(s\) defined on the MV-algebra \({\mathcal L}_k\) of all equivalent classes is a state if \(s(1)= 1\) and \(s(f\oplus g)= s(f)+ s(g)\) whenever \(f\oplus g= 0\). Given a finite set of formulas \(\psi_1,\dots,\psi_n\) and real numbers \(\beta_1,\dots,\beta_n\in [0,1]\), the numbers arise from a state \(s\) (i.e. \(\beta_i= s(f_{\psi_i})\), where \(f_{\psi_i}\) is the class of \({\mathcal L}_k\) obtaining \(\psi_i\)) if and only if there are no real numbers \(\sigma_i\) such that \(\sum^n_{j=1} \sigma_j(\beta_j- V(\psi_j))< 0\) for every valuation \(V\). The result solves a problem of J. Paris [“A note on the Dutch book method”, in: G. de Cooman et al. (eds.), Proc. 2nd Int. Symp. on Imprecise Probabilities and their Applications, ISIPTA 2001. Ithaka, NY, USA: Shaker, 301–306 (2001)] and generalizes a result by B. Gerla [Int. J. Approx. Reasoning 25, No. 1, 1–13 (2000; Zbl 0958.06007)]. The author also extends the result for infinitely many formulas and he deals with the problem of deciding if a book is Dutch.

03B48 Probability and inductive logic
03B50 Many-valued logic
06D35 MV-algebras
60A05 Axioms; other general questions in probability
Full Text: DOI
[1] Cignoli, R.L.O.; D’Ottaviano, I.M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, Trends in logic, vol. 7, (2000), Kluwer Academic Publishers Dordrecht
[2] 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: Paola Monari and Daniela Cocchi (Eds.), Probabilitá e Induzione, Clueb, Bologna, 1993, pp. 291-321 · JFM 57.0608.07
[3] 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 Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources”, in: Henry E. Kyburg, Jr., Howard 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
[4] de Finetti, B., Theory of probability, vol. 1, (1974), John Wiley and Sons Chichester
[5] Ewald, G., Combinatorial convexity and algebraic geometry, (1996), Springer-Verlag Berlin, Heidelberg · Zbl 0869.52001
[6] Gerla, B., MV-algebras, multiple bets and subjective states, International journal of approximate reasoning, 25, 1-13, (2000) · Zbl 0958.06007
[7] Goodearl, K.R., Partially ordered abelian groups with interpolation, (1986), Amer. Math. Soc. Providence, RI · Zbl 0589.06008
[8] Horn, A.; Tarski, A., Measures on Boolean algebras, Transactions of the American mathematical society, 64, 467-497, (1948) · Zbl 0035.03001
[9] Lekkerkerker, C.G., Geometry of numbers, (1969), North-Holland Amsterdam · Zbl 0198.38002
[10] Mundici, D., Interpretation of AF C∗-algebras in łukasiewicz sentential calculus, Journal of functional analysis, 65, 15-63, (1986) · Zbl 0597.46059
[11] Mundici, D., Farey stellar subdivisions, ultrasimplicial groups, and K0 of AF C∗-algebras, Advances in mathematics, 68, 23-39, (1988) · Zbl 0678.06008
[12] Mundici, D., Averaging the truth value in łukasiewicz sentential logic, Studia logica, 55, 113-127, (1995), Special issue in honor of Helena Rasiowa · Zbl 0836.03016
[13] Paris, J., A note on the Dutch book method, (), 301-306, Available from:
[14] Riečan, B.; Mundici, D., Probability on MV-algebras, (), 869-909 · Zbl 1017.28002
[15] Rourke, C.P.; Sanderson, B.J., Introduction to piecewise-linear topology, (1972), Springer Berlin · Zbl 0254.57010
[16] Seidenberg, A., A new decision method for elementary algebra, Annals of mathematics, 60, 365-374, (1954) · Zbl 0056.01804
[17] Tarski, A.; Łukasiewicz, J., Investigations into the sentential calculus, (), 38-59, (Reprinted by Hackett Publishing Company, Indianapolis, 1983)
[18] Tarski, A., A decision method for elementary algebra and geometry, (1951), University of California Press Berkeley, California · Zbl 0044.25102
[19] Ziegler, G.M., Lectures on polytopes, (1995), Springer-Verlag New York, Berlin, Heidelberg · Zbl 0823.52002
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