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STANDARD BAYES LOGIC IS NOT FINITELY AXIOMATIZABLE

Published online by Cambridge University Press:  22 March 2019

ZALÁN GYENIS
ZALÁN GYENIS*
Affiliation:
Department of Logic, Institute of Philosophy, Jagiellonian University and Department of Logic, Eötvös loránd University
*
*DEPARTMENT OF LOGIC INSTITUTE OF PHILOSOPHY JAGIELLONIAN UNIVERSITY KRAKÓW, POLAND and DEPARTMENT OF LOGIC EÖTVÖS LORÁND UNIVERSITY BUDAPEST, HUNGARY E-mail: [email protected]
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Abstract

In the article [2] a hierarchy of modal logics has been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to the modal logics of Medvedev frames it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case remained open. In this article we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces (these cover probability spaces that occur in most of the applications) is also not finitely axiomatizable.

Keywords

03B4203B4503A10modal logicBayesian inferenceBayes learningBayes logicMedvedev framesnonfinite axiomatizability
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 326 - 337
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal Logic. Cambridge: Cambridge University Press.Google Scholar
Brown, W., Gyenis, Z., & Rédei, M. (2018). The modal logic of Bayesian belief revision. Journal of Philosophical Logic, doi:10.1007/s10992-018-9495-9.Google Scholar
Chagrov, A. & Zakharyaschev, M. (1997). Modal Logic. Oxford: Claredon Press.Google Scholar
Diaconis, P. & Zabell, S. (1983). Some alternatives to Bayes’ rule. Technical Report 205, Stanford University.Google Scholar
Holliday, W. H. (2016). On the modal logic of subset and superset: Tense logic over Medvedev frames. Studia Logica, 105, 13–35.10.1007/s11225-016-9680-1CrossRefGoogle Scholar
Kechris, A. S. (1995). Classical Descriptive Set Theory. New York: Springer–Verlag.10.1007/978-1-4612-4190-4CrossRefGoogle Scholar
Łazarz, M. (2013). Characterization of Medvedev’s logic by means of Kubiński’s frames. Bulletin of the Section of Logic, 42, 83–90.Google Scholar
Medvedev, Y. T. (1996). On the interpretation of the logical formulas by means of finite problems. Doklady Akademii Nauk SSSR, 169, 20–24 (in Russian).Google Scholar
Prucnal, T. (1976). Structural completeness of Medvedev’s propositional calculus. Reports on Mathematical Logic, 6, 103–105.Google Scholar
Shehtman, V. (1990). Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable. Studia Logica, 49, 365–385.10.1007/BF00370370CrossRefGoogle Scholar
Skvortsov, D. (1979). The logic of infinite problems and the Kripke models on atomic semilattices of sets. Doklady Akademii Nauk SSSR, 245, 798–801 (in Russian).Google Scholar
Williamson, J. (2010). In Defence of Objective Bayesianism. Oxford: Oxford University Press.10.1093/acprof:oso/9780199228003.001.0001CrossRefGoogle Scholar