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Dynamic Epistemic Logic as a Substructural Logic

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Johan van Benthem on Logic and Information Dynamics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

Dynamic Epistemic Logic (DEL) is an influential logical framework for reasoning about the dynamics of beliefs and knowledge. It has been related to older and more established logical frameworks. Despite these connections, DEL remains, arguably, a rather isolated logic in the vast realm of non-classical logics and modal logics. This is problematic if logic is to be viewed ultimately as a unified and unifying field and if we want to avoid that DEL goes on “riding off madly in all directions” (a metaphor used by van Benthem about logic in general). In this article, we show that DEL can be redefined naturally and meaningfully as a two-sorted substructural logic. In fact, it is even one of the most primitive substructural logics since it does not preserve any of the structural rules. Moreover, the ternary semantics of DEL and its dynamic interpretation provides a conceptual foundation for the Routley & Meyer’s semantics of substructural logics.

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Notes

  1. 1.

    We write \(\fancyscript{E},e\models \Box ^*_{AGT}\alpha \) when for all \(f\in \left( \underset{j\in AGT}{\bigcup }R_j\right) ^*(e)\), \(\fancyscript{E},f\models \alpha \). See for example [41] for a detailed study of the operator \(\Box ^*_{AGT}\) of common knowledge.

  2. 2.

    Note that Burgess [35] already proposed a ternary semantics for conditionals, but his truth conditions and his interpretation of the ternary relation were quite different from ours.

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Acknowledgments

I thank Olivier Roy and Ole Hjortland for organizing and inviting me to an inspiring workshop on substructural epistemic logic in Munich in February 2013. Also, I thank Johan van Benthem and Igor Sedlar for comments on an earlier version of this article. Finally, I thank Sean Sedwards for checking the English of this article.

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Authors and Affiliations

  1. University of Rennes 1 – INRIA, 263, Avenue du Gnral Leclerc, 35042, Rennes, Cedex, France

    Guillaume Aucher

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  1. Guillaume Aucher
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Correspondence to Guillaume Aucher .

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Editors and Affiliations

  1. University of Amsterdam Inst Logic,Language & Computation, Amsterdam, The Netherlands

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

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Aucher, G. (2014). Dynamic Epistemic Logic as a Substructural Logic. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_33

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