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CONDITIONAL BELIEFS: FROM NEIGHBOURHOOD SEMANTICS TO SEQUENT CALCULUS

Published online by Cambridge University Press:  28 June 2018

MARIANNA GIRLANDO ,
SARA NEGRI ,
NICOLA OLIVETTI  and
VINCENT RISCH
MARIANNA GIRLANDO*
Affiliation:
Aix-Marseille University; Department of Philosophy, University of Helsinki
SARA NEGRI*
Affiliation:
Department of Philosophy, University of Helsinki
NICOLA OLIVETTI*
Affiliation:
Aix-Marseille University
VINCENT RISCH*
Affiliation:
Aix-Marseille University
*
*AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE and DEPARTMENT OF PHILOSOPHY UNIVERSITY OF HELSINKI HELSINKI, FINLAND E-mail: [email protected]
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF HELSINKI HELSINKI, FINLAND E-mail: [email protected]
‡AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE E-mail: [email protected]
§AIX-MARSEILLE UNIVERSITY UNIVERSITÉ DE TOULON CNRS, LIS MARSEILLE, FRANCE E-mail: [email protected]
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Abstract

The logic of Conditional Beliefs (CDL) has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics for CDL is defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization of CDL is sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus for CDL is obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of neighbourhood semantics for the analysis of epistemic/doxastic notions.

Keywords

03B4203F0554A99conditional doxastic logicepistemic logicneighbourhood semanticsplausibility semanticsAlexandroff correspondencelabelled sequent calculusstructural proof theorycompletenessdecidabilitycanonical models
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 736 - 779
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

BIBLIOGRAPHY

Alexandroff, P. (1937). Diskrete Räume. Matematicheskii Sbornik (NS), 2(3), 501–519.Google Scholar
Baltag, A. & Smets, S. (2006). Conditional doxastic models: A qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science, 165, 5–21.CrossRefGoogle Scholar
Baltag, A. & Smets, S. (2008). A qualitative theory of dynamic interactive belief revision. Logic and the Foundations of Game and Decision Theory (LOFT 7), 3, 9–58.Google Scholar
Baltag, A. & Smets, S. (2008). The logic of conditional doxastic actions. Texts in Logic and Games, Special Issue on New Perspectives on Games and Interaction, 4, 9–31.Google Scholar
Battigalli, P. & Siniscalchi, M. (2002). Strong belief and forward induction reasoning. Journal of Economic Theory, 106(2), 356–391.CrossRefGoogle Scholar
Board, O. (2004). Dynamic interactive epistemology. Games and Economic Behavior, 49(1), 49–80.CrossRefGoogle Scholar
Demey, L. (2011). Some remarks on the model theory of epistemic plausibility models. Journal of Applied Non-Classical Logics, 21(3–4), 375–395.CrossRefGoogle Scholar
Dyckhoff, R. & Negri, S. (2012). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51(1–2), 71–92.CrossRefGoogle Scholar
Gärdenfors, P. (1978). Conditionals and changes of belief. Acta Philosophica Fennica, 30, 381–404.Google Scholar
Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17(2), 157–170.CrossRefGoogle Scholar
Halpern, J. Y. & Friedman, N. (1994). On the complexity of conditional logics. In Doyle, J., Sandewall, E., and Torasso, P., editors. Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR’94). San Francisco: Morgan Kaufmann Publishers, pp. 202–213.Google Scholar
Halpern, J. Y. & Moses, Y. (1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54(3), 319–379.CrossRefGoogle Scholar
Hintikka, J. (1962). Knowledge and Belief: An Introduction to the Logic of the Two Notions, Vol. 4. Ithaca: Cornell University Press.Google Scholar
Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
Malcolm, N. (1952). Knowledge and belief. Mind, 61(242), 178–189.CrossRefGoogle Scholar
Marti, J. & Pinosio, R. (2013). Topological semantics for conditionals. In Dancak, M., and Punochar, V., editors. The Logica Yearbook 2013. London: College Publications, pp. 115–128.Google Scholar
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5–6), 507–544.CrossRefGoogle Scholar
Negri, S. (2017a). Non-normal modal logics: A challenge to proof theory. In Arazim, P., and Lávička, T., editors. The Logica Yearbook 2016. London: College Publications, pp. 125–140.Google Scholar
Negri, S. (2017b). Proof theory for non-normal modal logics: The neighbourhood formalism and basic results. IFCoLog Journal of Logic and its Applications, 4, 1241–1286.Google Scholar
Negri, S. & von Plato, J. (2001). Structural Proof Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Negri, S. & Olivetti, N. (2015). A sequent calculus for preferential conditional logic based on neighbourhood semantics. In De Nivelle, H., editor. Automated Reasoning with Analytic Tableaux and Related Methods. Cham, Switzerland: Springer, pp. 115–134.CrossRefGoogle Scholar
Pacuit, E. (2013). Dynamic epistemic logic I: Modeling knowledge and belief. Philosophy Compass, 8(9), 798–814.CrossRefGoogle Scholar
Pacuit, E. (2017). Neighborhood Semantics for Modal Logic. Cham, Switzerland: Springer.CrossRefGoogle Scholar
Stalnaker, R. (1996). Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12(02), 133–163.CrossRefGoogle Scholar
Stalnaker, R. (1998). Belief revision in games: Forward and backward induction. Mathematical Social Sciences, 36(1), 31–56.CrossRefGoogle Scholar
Stalnaker, R. (2006). On logics of knowledge and belief. Philosophical Studies, 128(1), 169–199.CrossRefGoogle Scholar
van Ditmarsch, H., van Der Hoek, W., & Kooi, B. (2008). Dynamic Epistemic Logic. New York: Springer Science & Business Media.Google Scholar