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Logical Omniscience Via Proof Complexity

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Computer Science Logic (CSL 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4207))

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Abstract

The Hintikka-style modal logic approach to knowledge contains a well-known defect of logical omniscience, i.e., the unrealistic feature that an agent knows all logical consequences of her assumptions. In this paper, we suggest the following Logical Omniscience Test (LOT): an epistemic system E is not logically omniscient if for any valid in E knowledge assertion \(\mathcal{A}\) of type ‘Fis known,’ there is a proof of F in E, the complexity of which is bounded by some polynomial in the length of \(\mathcal{A}\). We show that the usual epistemic modal logics are logically omniscient (modulo some common complexity assumptions). We also apply LOT to evidence-based knowledge systems, which, along with the usual knowledge operator K i (F) (‘agent i knows F’), contain evidence assertions t:F (‘t is a justification for  F’). In evidence-based systems, the evidence part is an appropriate extension of the Logic of Proofs LP, which guarantees that the collection of evidence terms t is rich enough to match modal logic. We show that evidence-based knowledge systems are logically omniscient w.r.t. the usual knowledge and are not logically omniscient w.r.t. evidence-based knowledge.

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Authors and Affiliations

  1. CUNY Graduate Center, 365 Fifth Ave., New York City, NY, 10016, USA

    Sergei Artemov & Roman Kuznets

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  1. Sergei Artemov
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  2. Roman Kuznets
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Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Rovira i Virgili University, Tarragona, Spain

    Zoltán Ésik

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Artemov, S., Kuznets, R. (2006). Logical Omniscience Via Proof Complexity. In: Ésik, Z. (eds) Computer Science Logic. CSL 2006. Lecture Notes in Computer Science, vol 4207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11874683_9

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  • DOI: https://doi.org/10.1007/11874683_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-45458-8

  • Online ISBN: 978-3-540-45459-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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