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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.

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