Abstract
Let \(A^*\) and \(B^*\) be finite sets of continuous events (e.g., physical observables, or random variables) represented by elements of semisimple MV-algebras A and B. Suppose \(\alpha :A^*\rightarrow [0,1]\) and \(\beta :B^*\rightarrow [0,1]\) are coherent books, i.e., maps satisfying de Finetti’s coherence criterion. Suppose all events in \(A^*\) are (logically) independent of all events in \(B^*.\) Let \(C=A\otimes B\) be the semisimple tensor product of A and B. We first prove that if \(a,a'\in A^*\) and \( b,b'\in B^*\) satisfy \(a\otimes b=a'\otimes b'\), then \(\alpha (a)\beta (b)=\alpha (a')\beta (b')\). Thus by setting \(\gamma (a \otimes b)=\alpha (a)\beta (b)\) we obtain a [0, 1]-valued function \(\gamma \) defined on the set \(C^*\) of pure tensors of C of the form \(a\otimes b\) for \(a\in A^*\) and \(b\in B^*\). We then prove that \(\gamma \) is a coherent book on \(C^*\). For the proofs we need the MV-algebraic extension of de Finetti Dutch Book theorem, Fubini theorem, and the Kroupa–Panti theorem (which in turn rests on the preservation properties of the \(\varGamma \) functor, the Stone–Weierstrass theorem and the Riesz representation theorem).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cignoli, R., I.M.L. D’Ottaviano, and D. Mundici. 2000. Algebraic Foundations of Many-Valued Reasoning, Trends in Logic, vol. 7. Dordrecht: Kluwer Academic Publishers.
de Finetti, B. 1993. Sul significato soggettivo della probabilitá, Fundamenta Mathematicae, 17 (1931) 298–329. Translated into English as “On the Subjective Meaning of Probability”. In Probabilitá e Induzione, ed. P. Monari, and D. Cocchi, 291–321. Bologna: Clueb.
de Finetti, B. 1980. La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut H. Poincaré, 7 (1937) 1–68. Translated into English by Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources”. In Studies in Subjective Probability, Wiley, New York, 1964, ed. H.E. Kyburg Jr., and H.E. Smokler, 53–118. New York: Second edition published by Krieger.
de Finetti, B. 2017. Theory of Probability, A Critical Introductory Treatment, Translated by Antonio Machí and Adrian Smith. Chichester, UK: Wiley.
Horn, A., and A. Tarski. 1948. Measures in Boolean algebras. Transactions of the American Mathematical Society 64: 467–497.
Kakutani, S. 1941. Concrete representation of abstract (M)-spaces, (A characterization of the space of continuous functions). Annals of Mathematics, (2), 42: 994–1024.
Koppelberg, S. 1989. General Theory of Boolean Algebras. In Handbook of Boolean Algebras (Vol.1), ed. by J.D.Monk with the cooperation of R. Bonnet, Elsevier, Amsterdam.
Kroupa, T. 2006. Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems 157: 2771–2782.
Kühr, J., and D. Mundici. 2007. De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal of Approximate Reasoning 46: 605–616.
Mundici, D. 1986. Interpretation of AF \(C^{*}\)-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis 65: 15–63.
Mundici, D. 1993. Logic of infinite quantum systems. International Journal of Theoretical Physics 32: 1941–1955.
Mundici, D. 1995. Averaging the truth value in Łukasiewicz sentential logic. Studia Logica, Special issue in honor of Helena Rasiowa 55: 113–127.
Mundici, D. 1999. Tensor products and the Loomis-Sikorski theorem for MV-algebras. Advances in Applied Mathematics 22: 227–248.
Mundici, D. 2009. Interpretation of de Finetti coherence criterion in Łukasiewicz logic. Annals of Pure and Applied Logic 161: 235–245.
Mundici, D. 2011. Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic, vol. 35. Berlin: Springer.
Mundici, D. 2017. Coherence of de Finetti coherence. Synthese 194: 4055–4063.
Panti, G. 2009. Invariant measures in free MV-algebras. Communications in Algebra 36: 2849–2861.
Sikorski, R. 1960. Boolean Algebras, 2nd edition, Ergeb. der Math. und ihrer Grenzgeb., vol. 25. Berlin: Springer.
Tao, T. 2010. An Epsilon of Room I: Real Analysis, vol. 117, Graduate Studies in Mathematics. Providence RI: American Mathematical Society.
Tao, T. 2011. An Introduction to Measure Theory, vol. 126, Graduate Studies in Mathematics. Providence RI: American Mathematical Society.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mundici, D. (2018). Coherence of the Product Law for Independent Continuous Events. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-98797-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-98796-5
Online ISBN: 978-3-319-98797-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)