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THREE CHARACTERIZATIONS OF STRICT COHERENCE ON INFINITE-VALUED EVENTS

Published online by Cambridge University Press:  04 October 2019

TOMMASO FLAMINIO
TOMMASO FLAMINIO*
Affiliation:
IIIA - CSIC, Campus de la Universidad Autònoma de Barcelona
*
*IIIA - CSIC CAMPUS DE LA UNIVERSIDAD AUTÒNOMA DE BARCELONA S/N 08193 BELLATERRA, SPAIN E-mail: tommaso@iiia.csic.es
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Abstract

This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

Keywords

Primary 03B48Secondary 60A0506D3506F2052B11de Finetti’s coherencestrict coherencefaithful statesMV-algebrasŁukasiewicz logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 3 , September 2020 , pp. 593 - 610
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Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Carnap, R. (1950). The Logical Foundations of Probability. Chicago: University of Chicago Press.Google Scholar
Chang, C.C. (1958). Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, 88, 467–490.CrossRefGoogle Scholar
Cignoli, R., D’Ottaviano, I. M. L., & Mundici, D. (2000). Algebraic Foundations of Many-valued Reasoning. Trends in Logic, Vol. 8. Dordrecht: Kluwer.CrossRefGoogle Scholar
Cignoli, R. & Marra, V. (2012). Stone duality for real-valued multisets. Forum Mathematicum, 24 (6), 1317–1331.CrossRefGoogle Scholar
de Finetti, B. (1931). Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17, 298–329. Translated into English as “On the subjective meaning of probability.” In Monari, P., and Cocchi, D., editors. Probabilità e Induzione. Bologna: Clueb, pp. 291–321, 1993 .Google Scholar
de Finetti, B. (1974). Theory of Probability, Vol. 1. New York: Wiley.Google Scholar
Desiderata, F. P. & Shamos, M. I. (1985). Computational Geometry – An Introduction. New York: Springer-Verlag.Google Scholar
Ewald, G. (1996). Combinatorial Convexity and Algebraic Geometry. New York: Springer-Verlag.CrossRefGoogle Scholar
Flaminio, T., Godo, L., & Hosni, H. (2014). On the logical structure of de Finetti’s notion of event. Journal of Applied Logic, 12(3), 279–301.CrossRefGoogle Scholar
Flaminio, T., Hosni, H., & Lapenta, S. (2018). Convex MV-algebras: Many-valued logics meet decision theory. Studia Logica, 106(5), 913–945.CrossRefGoogle Scholar
Flaminio, T., Hosni, H., & Montagna, F. (2018). Strict coherence on many-valued events. The Journal of Symbolic Logic, 83(1), 55–69.CrossRefGoogle Scholar
Flaminio, T. & Kroupa, T. (2015). States of MV-algebras. In Fermüller, C., Cintula, P., and Noguera, C., editors. Handbook of Mathematical Fuzzy Logic - volume 3. Studies in Logic, Mathematical Logic and Foundations, Vol. 58, Chapter XVII. London: College Publications.Google Scholar
Gaifman, H. (1964). Concerning measures on Boolean algebras. Pacific Journal of Mathematics, 14(1), 61–73.CrossRefGoogle Scholar
Hähnle, R. (1994). Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(3–4), 231–263.CrossRefGoogle Scholar
Kelley, J. L. (1959). Measures on Boolean Algebras. Pacific Journal of Mathematics, 9(4), 1165–1177.CrossRefGoogle Scholar
Kemeny, J. G. (1955). Fair bets and inductive probabilities. The Journal of Symbolic Logic, 20(3), 263–273.CrossRefGoogle Scholar
Kroupa, T. (2006). Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems, 157(20), 2771–2787.CrossRefGoogle Scholar
Kroupa, T. (2012). States in Łukasiewicz logic corresponds to probabilities of rational polyhedra. International Journal of Approximate Reasoning, 53, 435–446.CrossRefGoogle Scholar
Kühr, J. & Mundici, D. (2007). De Finetti theorem and Borel states in [0, 1]-valued algebraic logic. International Journal of Approximate Reasoning, 46(3), 605–616.CrossRefGoogle Scholar
Marra, V. (2014). The problem of artificial precision in theories of vagueness: A note on the rôle of maximal consistency. Erkenntnis, 79(5), 1015–1026.CrossRefGoogle Scholar
Marra, V. & Spada, L. (2013). Duality, projectivity and unification in Łukasiewicz logic and MV-algebras. Annals of Pure and Applied logic, 164(3), 192–210.CrossRefGoogle Scholar
McMullen, P. & Shephard, G. C. (1971). Convex Polytopes and the Upper Bound Conjecture. London Mathematical Society Lecture Note Series, Vol. 3. London: Cambridge University Press.Google Scholar
McNaughton, R. (1951). A theorem about infinite-valued sentential logic. The Journal of Symbolic Logic, 16, 1–13.CrossRefGoogle Scholar
Mundici, D. (1994). A constructive proof of McNaughton’s theorem in infinite-valued logic. The Journal of Symbolic Logic, 58(2), 596–602.CrossRefGoogle Scholar
Mundici, D. (1995). Averaging the truth-value in Łukasiewicz logic. Studia Logica, 55(1), 113–127.CrossRefGoogle Scholar
Mundici, D. (2006). Bookmaking over infinite-valued events. International Journal of Approximate Reasoning, 43(3), 223–240.CrossRefGoogle Scholar
Mundici, D. (2011). Advanced Łukasiewicz Calculus and MV-algebras. Trends in Logic 35, Springer, 2011.CrossRefGoogle Scholar
Mundici, D. (2011). Finite axiomatizability in Łukasiewicz logic. Annals of Pure and Applied Logic, 162, 1035–1047.CrossRefGoogle Scholar
Panti, G. (2009). Invariant measures on free MV-algebras. Communications in Algebra, 36(8), 2849–2861.CrossRefGoogle Scholar
Paris, J. (2001). A note on the Dutch Book method. In De Cooman, G., Fine, T., and Seidenfeld, T., editors. Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications. ISIPTA 2001. Ithaca, NY: Shaker Publishing Company, pp. 301–306.Google Scholar
Paris, J. B. & Vencovska, A. (2015). Pure Inductive Logic. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Shimony, A. (1955). Coherence and the axioms of confirmation. The Journal of Symbolic Logic, 20(1), 1–28.CrossRefGoogle Scholar
Todorcevic, S. (1997). Topics in Topology. Lecture Notes in Mathematics. Berlin: Springer.CrossRefGoogle Scholar
Weatherson, B. (2003). From classical to intuitionistic probability. Notre Dame Journal of Formal Logic, 44(2), 111–123.CrossRefGoogle Scholar