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STRICT COHERENCE ON MANY-VALUED EVENTS

Published online by Cambridge University Press:  12 February 2018

TOMMASO FLAMINIO ,
HYKEL HOSNI  and
FRANCO MONTAGNA
TOMMASO FLAMINIO
Affiliation:
DEPARTMENT OF PURE AND APPLIED SCIENCES UNIVERSITY OF INSUBRIA VIA MAZZINI, 5 21100 VARESE, ITALY and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF MILAN VIA FESTA DEL PERDONO 7, 20122MILANO, ITALY E-mail:tommaso.flaminio@uninsubria.it
HYKEL HOSNI
Affiliation:
DEPARTMENT OF PURE AND APPLIED SCIENCES UNIVERSITY OF INSUBRIA VIA MAZZINI, 5 21100 VARESE, ITALY and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF MILAN VIA FESTA DEL PERDONO 7, 20122MILANO, ITALYE-mail:hykel.hosni@unimi.it
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Abstract

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We investigate the property of strict coherence in the setting of many-valued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap’s regularity condition, and (ii) a [0,1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal.

Keywords

Primary 03B48Secondary 60A0506D3506F2052B11Probability logicstrict coherenceMV-algebrasfaithful statesmany-valued logics
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 55 - 69
Copyright
Copyright © The Association for Symbolic Logic 2018 

References

REFERENCES

Bezhanishvili, G., Locally finite varieties. Algebra Universalis, vol. 46 (2001), pp. 531–548.Google Scholar
Bova, S. and Flaminio, T., The coherence of Łukasiewicz assessments is NP-complete. International Journal of Approximate Reasoning, vol. 51 (2010), no. 3, pp. 294–304.Google Scholar
Burris, S. and Sankappanavar, H., A Course in Universal Algebra, Springer-Velag, New York, 1981.Google Scholar
Carnap, R., The Logical Foundations of Probability, University of Chicago Press, Chicago, 1950.Google Scholar
Chang, C. C., Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, vol. 88 (1958), no. 2, pp. 467–490.Google Scholar
Cignoli, R., D’Ottaviano, I. M. L., and Mundici, D., Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.Google Scholar
Cignoli, R. and Marra, V., Stone duality for real-valued multisets. Forum Mathematicum, vol. 24 (2012), pp. 1317–1331.Google Scholar
de Finetti, B, Sul significato soggettivo della probabilità. Fundamenta Mathematicae, vol. 17 (1931), pp. 289–329.Google Scholar
de Finetti, B, Theory of Probability, vol. 1, John Wiley and Sons, New York, 1974.Google Scholar
Di Nola, A. and Leuştean, I., Łukasiewicz logic and Riesz spaces. Soft Computing, vol. 18 (2014), no. 12, pp. 2349–2363.Google Scholar
Ewald, G., Combinatorial Convexity and Algebraic Geometry, Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
Fedel, M., Keimel, K., Montagna, F., and Roth, W., Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Forum Mathematicum, vol. 25 (2013), no. 2, pp. 405–441.Google Scholar
Flaminio, T., Godo, L., and Hosni, H., On the logical structure of de Finetti’s notion of event. Journal of Applied Logic, vol. 12 (2014), no. 3, pp. 279–301.Google Scholar
Flaminio, T. and Kroupa, T., States of MV-algebras, Handbook of Mathematical Fuzzy Logic, vol. 3 (Fermüller, C., Cintula, P., and Noguera, C., editors), College Publications, London, 2015, pp. 1191–1245.Google Scholar
Flaminio, T., Hosni, H., and Lapenta, S., Convex MV-algebras: Many-valued logics meet decision theory. Studia Logica, forthcoming, doi: 10.1007/s11225-016-9705-9.Google Scholar
Gaifman, H., Concerning measures on Boolean algebras. Pacific Journal of Mathematics, vol. 14 (1964), no. 1, pp. 61–73.Google Scholar
Goodearl, K. R., Partially Ordered Abelian Groups with Interpolation, vol. 20, American Mathematical Society, Providence, RI, 2010.Google Scholar
Goodearl, K. R., On falling short of strict coherence. Philosophy of Science, vol. 35 (1968), no. 3, pp. 284–286.Google Scholar
Hosni, H. and Montagna, F., Stable non-standard imprecise probabilities, IPMU 2104, (Laurent, A., Strauss, O., Bouchon-Meunier, B., and Yager, R. R., editors), Communications in Computer and Information Sciences, 444, Springer, Cham, Heidelberg, New York, Dordrecht, London, 2014, pp. 436–445.Google Scholar
Kelley, J. L., Measures on Boolean Algebras. Pacific Journal of Mathematics, vol. 9 (1959), no. 4, pp. 1165–1177.Google Scholar
Kemeny, J. G., Fair bets and inductive probabilities, this Journal, vol. 20 (1955), no. 3, pp. 263–273.Google Scholar
Kroupa, T., Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems, vol. 157 (2006), no. 20, pp. 2771–2782.Google Scholar
Kühr, J. and Mundici, D., De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal of Approximate Reasoning, vol. 46 (2007), no. 3, pp. 605–616.Google Scholar
Lehman, R. L., On confirmation and rational betting, this Journal, vol. 20 (1955), no. 3, pp. 251–262.Google Scholar
Leuştean, I., Metric completions of MV-algebras with states: An approach to stochastic independence. Journal of Logic and Computation, vol. 21 (2011), no. 3, pp. 493–508.Google Scholar
Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I, North-Holland, Amsterdam, 1971.Google Scholar
McMullen, P. and Shephard, G. C., Convex Polytopes and the Upper Bound Conjecture, London Mathematical Society Lecture Note Series 3, Cambridge University Press, London-New York, 1971.CrossRefGoogle Scholar
Montagna, F., Fedel, M., and Scianna, G., Non-standard probability, coherence and conditional probability on many-valued events. International Journal of Approximate Reasoning, vol. 54 (2013), no. 5, pp. 573–589.Google Scholar
Mundici, D., Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis, vol. 65 (1986), no. 1, pp. 15–63.Google Scholar
Mundici, D., Averaging the truth-value in Łukasiewicz logic. Studia Logica, vol. 55 (1995), no. 1, pp. 113–127.Google Scholar
Mundici, D., Bookmaking over infinite-valued events. International Journal of Approximate Reasoning, vol. 43 (2006), no. 3, pp. 223–240.Google Scholar
Mundici, D., Advanced Łukasiewicz calculus and MV-algebras, Springer Verlag, Dordrecht, Heidelberg, London, New York, 2011.Google Scholar
Mundici, D., Coherence of de Finetti coherence. Synthese, vol. 194 (2017), no. 10, pp. 4055–4063.Google Scholar
Panti, G., Invariant Measures in Free MV-Algebras. Communications in Algebra, vol. 36 (2008), no. 8, pp. 2849–2861.Google Scholar
Paris, J., A note on the Dutch Book method, Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications (De Cooman, G., Fine, T., and Seidenfeld, T., editors), ISIPTA 2001, Shaker Publishing Company, Ithaca, NY, USA, 2001, pp. 301–306.Google Scholar
Paris, J., The Uncertain Reasoner’s Companion: A Mathematical Perspective, Cambridge University Press, Cambridge, UK, 1994.Google Scholar
Paris, J. B. and Vencovska, A., Pure Inductive Logic, Cambridge University Press, Cambridge, UK, 2015.CrossRefGoogle Scholar
Shimony, A., Coherence and the axioms of confirmation, this Journal, vol. 20 (1955), no. 1, pp. 1–28.Google Scholar