×

Three characterizations of strict coherence on infinite-valued events. (English) Zbl 1485.03058

Summary: This article builds on a recent paper coauthored by the present author et al., J. Symb. Log. 83, No. 1, 55–69 (2018; Zbl 1447.03001)]. 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.

MSC:

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

Citations:

Zbl 1447.03001
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Carnap, R. (1950). The Logical Foundations of Probability. Chicago: University of Chicago Press. · Zbl 0040.07001
[2] Chang, C.C. (1958). Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, 88, 467-490. · Zbl 0084.00704
[3] Cignoli, R., D’Ottaviano, I. M. L., & Mundici, D. (2000). Algebraic Foundations of Many-valued Reasoning. Trends in Logic, Vol. 8. Dordrecht: Kluwer. · Zbl 0937.06009
[4] Cignoli, R. & Marra, V. (2012). Stone duality for real-valued multisets. Forum Mathematicum, 24 (6), 1317-1331. · Zbl 1273.06006
[5] 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 . · JFM 57.0608.07
[6] De Finetti, B. (1974). Theory of Probability, Vol. 1. New York: Wiley. · Zbl 0328.60002
[7] Desiderata, F. P. & Shamos, M. I. (1985). Computational Geometry - An Introduction. New York: Springer-Verlag. · Zbl 0759.68037
[8] Ewald, G. (1996). Combinatorial Convexity and Algebraic Geometry. New York: Springer-Verlag. · Zbl 0869.52001
[9] 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. · Zbl 1352.03033
[10] Flaminio, T., Hosni, H., & Lapenta, S. (2018). Convex MV-algebras: Many-valued logics meet decision theory. Studia Logica, 106(5), 913-945. · Zbl 1475.06007
[11] Flaminio, T., Hosni, H., & Montagna, F. (2018). Strict coherence on many-valued events. The Journal of Symbolic Logic, 83(1), 55-69. · Zbl 1447.03001
[12] 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. · Zbl 1431.06005
[13] Gaifman, H. (1964). Concerning measures on Boolean algebras. Pacific Journal of Mathematics, 14(1), 61-73. · Zbl 0127.02306
[14] Hähnle, R. (1994). Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(3-4), 231-263. · Zbl 0856.03011
[15] Kelley, J. L. (1959). Measures on Boolean Algebras. Pacific Journal of Mathematics, 9(4), 1165-1177. · Zbl 0087.04801
[16] Kemeny, J. G. (1955). Fair bets and inductive probabilities. The Journal of Symbolic Logic, 20(3), 263-273. · Zbl 0066.11002
[17] Kroupa, T. (2006). Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems, 157(20), 2771-2787. · Zbl 1107.06007
[18] Kroupa, T. (2012). States in Łukasiewicz logic corresponds to probabilities of rational polyhedra. International Journal of Approximate Reasoning, 53, 435-446. · Zbl 1260.03050
[19] 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. · Zbl 1189.03076
[20] 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. · Zbl 1329.03062
[21] 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. · Zbl 1275.03099
[22] 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. · Zbl 0217.46702
[23] Mcnaughton, R. (1951). A theorem about infinite-valued sentential logic. The Journal of Symbolic Logic, 16, 1-13. · Zbl 0043.00901
[24] Mundici, D. (1994). A constructive proof of McNaughton’s theorem in infinite-valued logic. The Journal of Symbolic Logic, 58(2), 596-602. · Zbl 0807.03012
[25] Mundici, D. (1995). Averaging the truth-value in Łukasiewicz logic. Studia Logica, 55(1), 113-127. · Zbl 0836.03016
[26] Mundici, D. (2006). Bookmaking over infinite-valued events. International Journal of Approximate Reasoning, 43(3), 223-240. · Zbl 1123.03011
[27] Mundici, D. (2011). Advanced Łukasiewicz Calculus and MV-algebras. Trends in Logic 35, Springer, 2011. · Zbl 1235.03002
[28] Mundici, D. (2011). Finite axiomatizability in Łukasiewicz logic. Annals of Pure and Applied Logic, 162, 1035-1047. · Zbl 1248.03043
[29] Panti, G. (2009). Invariant measures on free MV-algebras. Communications in Algebra, 36(8), 2849-2861. · Zbl 1154.06008
[30] 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.
[31] Paris, J. B. & Vencovska, A. (2015). Pure Inductive Logic. Cambridge, UK: Cambridge University Press. · Zbl 1338.03001
[32] Shimony, A. (1955). Coherence and the axioms of confirmation. The Journal of Symbolic Logic, 20(1), 1-28. · Zbl 0064.24402
[33] Todorcevic, S. (1997). Topics in Topology. Lecture Notes in Mathematics. Berlin: Springer. · Zbl 0953.54001
[34] Weatherson, B. (2003). From classical to intuitionistic probability. Notre Dame Journal of Formal Logic, 44(2), 111-123. · Zbl 1069.60002
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.