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Nested Sequents for the Logic of Conditional Belief

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Logics in Artificial Intelligence (JELIA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11468))

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Abstract

The logic of conditional belief, called Conditional Doxastic Logic (\(\mathsf {CDL}\)), was proposed by Board, Baltag and Smets to model revisable belief and knowledge in a multi-agent setting. We present a proof system for \(\mathsf {CDL}\) in the form of a nested sequent calculus. To the best of our knowledge, ours is the first internal and standard calculus for this logic. We take as primitive a multi-agent version of the “comparative plausibility operator”, as in Lewis’ counterfactual logic. The calculus is analytic and provides a decision procedure for \(\mathsf {CDL}\). As a by-product we also obtain a nested sequent calculus for multi-agent modal logic \(\mathsf {S5}_i\).

This work was partially supported by the Project TICAMORE ANR-16-CE91-0002-01 and by WWTF project MA 16-28.

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Notes

  1. 1.

    An equivalent definition of the simple belief operator is the following: \( Bel _i A := \lnot (\lnot A \preccurlyeq _i A)\) [10]. We choose a simpler formulation in terms of \( \top \), also from [10].

  2. 2.

    Refer to  [4, 9] for a detailed correspondence.

  3. 3.

    Refer to next section on \( \mathsf {S5}_i\).

  4. 4.

    Evaluating KA at a world x corresponds to evaluating \( \bot \preccurlyeq \lnot A \) in the outer neighbourhood of N(x) . For this reason, Lewis calls \( \mathsf {S5}\) the outer modal logic of \( \mathbb {V}\mathbb {T}\mathbb {A}\).

References

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Authors and Affiliations

  1. Aix Marseille University, Université de Toulon, CNRS, LIS, Marseille, France

    Marianna Girlando & Nicola Olivetti

  2. University of Helsinki, Helsinki, Finland

    Marianna Girlando

  3. Technische Universität Wien, Vienna, Austria

    Björn Lellmann

Authors
  1. Marianna Girlando
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  2. Björn Lellmann
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  3. Nicola Olivetti
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Correspondence to Marianna Girlando .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Francesco Calimeri

  2. University of Calabria, Rende, Italy

    Nicola Leone

  3. Mathematics and Computer Science, University of Calabria, Rende, Italy

    Marco Manna

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Girlando, M., Lellmann, B., Olivetti, N. (2019). Nested Sequents for the Logic of Conditional Belief. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_46

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  • DOI: https://doi.org/10.1007/978-3-030-19570-0_46

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  • Online ISBN: 978-3-030-19570-0

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