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International Conference on Logic and Argumentation

CLAR 2020: Logic and Argumentation pp 3–21Cite as

Group Belief

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12061))

Abstract

While logical formalizations of group notions of knowledge such as common and distributed knowledge have received considerable attention in the literature, most approaches being based on modal logic, group notions of belief have received much less attention. In this paper we systematically study standard notions of group knowledge and belief under different assumptions about which properties knowledge and belief have. In particular, we map out (lack of) preservation of knowledge/belief properties against different standard definitions of group knowledge/belief. It turns out that what is called group belief most often is not actually belief, i.e., does not have the properties of belief. In fact, even what is called group knowledge is sometimes not actually knowledge either. For example, under the common assumption that belief has the KD45 properties, distributed belief is not actually belief (it does not satisfy the D axiom). In the literature there is no detailed completeness proof for axiomatizations of KD45 with distributed belief that we are aware of, and there has been some confusion regarding soundness of such axiomatizations related to the mentioned lack of preservation. In this paper we also present a detailed completeness proof for a sound axiomatization of KD45 with distributed belief.

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Notes

  1. 1.

    The concrete definition of the semantics of common belief in [8], as well as in many other works (e.g. [7, 9, 12, 13, 15,16,17, 22]), is that \((M,s) \,\models \, C_G\varphi \) iff \(\forall k\ge 1: (M,s) \,\models \, E^k_G\varphi \), where \(E^1_G\varphi \) stands for \(E_G\varphi \) and \(E^{k+1}_G\varphi \) for \(E_GE^k_G\varphi \). As noted by [8, Lemma 2.2.1] that definition is equivalent to using the transitive closure (for arbitrary models, not only S5 models).

  2. 2.

    In the terminology of [6], general, common and distributed belief all correspond to unanimous aggregation rules.

  3. 3.

    In the terminology of [6], general and distributed belief all correspond to neutral aggregation rules.

  4. 4.

    We refer here to the 1995 hardcover edition of [8]. The result appears to have been corrected in a later (2003) paperback edition; still without a proof of completeness however.

  5. 5.

    The necessitation rule N\(_D\) for distributed belief, i.e., “from \(\varphi \) infer \(D_G\varphi \)”, is provable via N, DB1 and DB2; hence omitted.

  6. 6.

    We refer to a modal logic textbook, say [3], for a definition of a (maximal) consistent set of formulas.

  7. 7.

    The two must be equal in a genuine model, but we cannot simply define \(R_{\{a,b,c\}}\) to be the intersection of all of its subsets, for that already makes a pseudo model to be a genuine model. The whole method collapses then: we encounter the very problem that the canonical model is not a genuine model (mostly because the intersection of relations is not modally definable), which violates the starting point of the canonical model method. This was discussed in more detail already in [18].

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Acknowledgments

We thank the anonymous reviewers for useful comments. Yì N. Wáng acknowledges funding support by the National Social Science Foundation of China (Grant No. 16CZX048, 18ZDA290).

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Authors and Affiliations

  1. Southwest University, Chongqing, China

    Thomas Ågotnes

  2. University of Bergen, Bergen, Norway

    Thomas Ågotnes

  3. Zhejiang University, Hangzhou, China

    Yì N. Wáng

Authors
  1. Thomas Ågotnes
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  2. Yì N. Wáng
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Correspondence to Yì N. Wáng .

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Editors and Affiliations

  1. Utrecht University, Utrecht, Utrecht, The Netherlands

    Mehdi Dastani

  2. Zhejiang University, Hangzhou, China

    Huimin Dong

  3. University of Luxembourg, Luxembourg, Luxembourg

    Leon van der Torre

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Ågotnes, T., Wáng, Y.N. (2020). Group Belief. In: Dastani, M., Dong, H., van der Torre, L. (eds) Logic and Argumentation. CLAR 2020. Lecture Notes in Computer Science(), vol 12061. Springer, Cham. https://doi.org/10.1007/978-3-030-44638-3_1

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  • DOI: https://doi.org/10.1007/978-3-030-44638-3_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-44637-6

  • Online ISBN: 978-3-030-44638-3

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