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The logic of public announcements, common knowledge, and private suspicions. (English) Zbl 1386.03019
Arló-Costa, Horacio (ed.) et al., Readings in formal epistemology. Sourcebook. Edited with the assistance of Henrik Boensvang and Rasmus K. Rendsvig. Cham: Springer (ISBN 978-3-319-20450-5/hbk; 978-3-319-20451-2/ebook). Springer Graduate Texts in Philosophy 1, 773-812 (2016).
Summary: This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in [R. Fagin et al., Reasoning about knowledge. Cambridge, MA: MIT Press (1995; Zbl 0839.68095)]), but we reify aspects of the machinery in the logical language. Special cases of our logic have been considered in [J. Plaza, Synthese 158, No. 2, 165–179 (2007; Zbl 1126.03308); J. Gerbrandy, in: Logic, language and computation. Vol 2. Proceedings of the 2nd conference on information-theoretic approaches to logic, language and computation (ITALLC), London, UK, July 1996. Stanford, CA: CSLI Publications. 67–84 (1999; Zbl 0964.03018); Bisimulations on planet Kripke. Amersterdam: University of Amsterdam (PhD Dissertation) (1999); J. Gerbrandy and W. Groeneveld, J. Logic Lang. Inf. 6, No. 2, 147–169 (1997; Zbl 0873.03029)]. The latter group of papers introduce a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions, and by using a more general semantics. Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.
For the entire collection see [Zbl 1348.03005].

03B42 Logics of knowledge and belief (including belief change)
Full Text: DOI
[1] Kamin, S., & Levy, J. J. (1980). \(Two generalizations of the recursive path orderings\). Unpublished note, Department of Computer Science, University of Illinois, Urbana.
[2] Kozen, D., & Parikh, R. (1981). An elementary proof of the completeness of PDL. \(Theoretical Computer Science, 14\), 113-118. · Zbl 0451.03006
[3] Dershowitz, N. (1982). Orderings for term-rewriting systems. \(Theoretical Computer Science, 17\), 279-301. · Zbl 0525.68054
[4] Plaisted, D. A. (1985). Termination orderings. In D. Gabbay, et al. (Eds.), \(Handbook of logic in artificial intelligence and logic programming\) (Vol. I, pp. 273-364).
[5] Plaza, J. (1989). Logics of public communications. In \(Proceedings of the 4th international symposium on methodologies for intelligent systems\), Charlotte. · Zbl 1126.03308
[6] Halpern, J. Y., & Vardi, M. Y. (1989). The complexity of reasoning about knowledge and time. I. Lower bounds. \(Journal of Computer and System Sciences, 38\)(1), 195-237. · Zbl 0672.03015
[7] Buchholz, W. (1995). Proof-theoretic analysis of termination proofs. \(Annals of Pure and Applied Logic, 75\), 57-65. · Zbl 0844.03031
[8] Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995). \(Reasoning about knowledge\). Cambridge: MIT. · Zbl 0839.68095
[9] Gerbrandy, J., & Groeneveld, W. (1997). Reasoning about information change. \(Journal of Logic, Language, and Information, 6\), 147-169. · Zbl 0873.03029
[10] Moss, L. S. (1999). From hypersets to Kripke models in logics of announcements. In J. Gerbrandy, et al. (Eds.), \(JFAK. Essays dedicated to Johan van Benthem on the occasion of his 50th birthday\). Amsterdam: Vossiuspers, Amsterdam University Press.
[11] Gerbrandy, J. (1999b). \(Bisimulations on planet Kripke\). Ph.D. dissertation, University of Amsterdam.
[12] Baltag, A. (1999). \(A logic of epistemic actions, ms\). Amsterdam: CWI.
[13] Gerbrandy, J. (1999a). Dynamic epistemic logic. In L. S. Moss, et al. (Eds.), \(Logic, language, and information\) (Vol. 2). Stanford: CSLI Publications, Stanford University. · Zbl 0964.03018
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