Interpretation of De Finetti coherence criterion in Łukasiewicz logic

Annals of Pure and Applied Logic 161 (2):235-245 (2010)
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Abstract

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X1,…,Xm} of “events” occurring in an arbitrary set of “possible worlds”. In the particular case of yes–no events, , Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: a possible world of is a valuation satisfying Θ, β is coherent iff it is a convex combination of valuations satisfying Θ, iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, iff for some Borel probability measure μ on . Thus Łukasiewicz semantics, MV-algebraic states, and Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds

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References found in this work

La Prévision: Ses Lois Logiques, Ses Sources Subjectives.Bruno de Finetti - 1937 - Annales de l'Institut Henri Poincaré 7 (1):1-68.
Algebraic foundations of many-valued reasoning.Roberto Cignoli - 1999 - Boston: Kluwer Academic Publishers. Edited by Itala M. L. D'Ottaviano & Daniele Mundici.
Sul Significato Soggettivo della Probabilittextà.Bruno De Finetti - 1931 - Fundamenta Mathematicae 17:298--329.
Geometry of Robinson consistency in Łukasiewicz logic.Manuela Busaniche & Daniele Mundici - 2007 - Annals of Pure and Applied Logic 147 (1):1-22.

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