Abstract
We study a modal logic \(\mathbf{K4_{2}^{C}}\) of common belief for normal agents. We discuss Kripke completeness and show that the logic has tree model property. A main result is to prove that \(\mathbf{K4_{2}^{C}}\) is the modal logic of all T D -intersection closed, bi-topological spaces with derived set interpretation of modalities. Based on the splitting translation we also discuss connections with \(\mathbf{S4_{2}^{C}}\), the logic of common knowledge.
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Notes
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- 2.
For the remainder of this section and later on for Theorem 9 we assume some familiarity with the modal μ-calculus. Lack of space hinders a fuller treatment, however for more details on the modal μ-calculus we refer to Blackburn et al. (2006, Part 3, Chapter 4); see also the discussion in van Benthem and Sarenac (2004).
- 3.
For a discussion of the splitting translation and its application in non-monotonic modal logics, see the authors’ (Pearce and Uridia 2011a).
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Acknowledgements
An earlier version of this chapter was presented at the conference on Agreement Technologies, AT2012, 15–16 October 2012, Dubrovnik, Croatia. The authors are grateful to anonymous reviewers whose comments helped to improve readability. This research has been partially supported by the Spanish Ministry of Science and Innovation through the AT project CSD2007-0022 and MCICINN project TiN2009-14562-CO5, by the Shota Rustaveli National Science Foundation project 52/05 (PG/72/4-102/13) and by SINTELNET, the European Network for Social Intelligence.
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Pearce, D., Uridia, L. (2015). The Topology of Common Belief. In: Herzig, A., Lorini, E. (eds) The Cognitive Foundations of Group Attitudes and Social Interaction. Studies in the Philosophy of Sociality, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-21732-1_7
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