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Social Consolidations: Rational Belief in a Many-Valued Logic of Evidence and Peerhood

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Book cover Foundations of Information and Knowledge Systems (FoIKS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12012))

Abstract

We explore an interpretation of FVEL, a four-valued logic of evidence, where states represent agents, the propositional layer corresponds to the evidence available to these agents, and the relation corresponds to peerhood connections between them. Belief is determined based on the agent’s evidence, but also on her peers’ evidence. Consolidation functions are proposed, which map evidence situations to belief attitudes. We adapt some postulates of Social Choice Theory to our belief formation setting and, with them, we separate rational from irrational consolidations. We define a dynamic operator for addition and removal of evidence, which serves as a basis for some essential dynamic postulates and also for future developments on consolidations that take amounts of evidence into account. Our main technical result is a characterisation of a class of consolidations satisfying most of our rationality postulates.

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Notes

  1. 1.

    Some proofs at: https://github.com/ydsantos/appendix_scons/blob/master/proofs.pdf.

  2. 2.

    Notice that our language is non-standard in that even though a formula in \(\mathscr {L}_1\) has an evidential meaning (such as p meaning the agent has evidence for p), under the belief operator \(B\) these formulas are read as factual statements (e.g. \(Bp\) means that the agent believes p and not that the agent believes that she has evidence for p).

  3. 3.

    We chose \(B\) (belief) instead of K (knowledge) because we are working with imperfect evidence, which can be misleading. Therefore, our agents can form false beliefs, which violate factivity, a standard requirement for knowledge.

  4. 4.

    As a scientist investigating hypothesis H, you consider another scientist also investigating H to be your peer, but not if she committed fraud in the past.

  5. 5.

    Note, however, that we only make a loose connection to SCT here, not a formal one.

  6. 6.

    The function \({{\,\mathrm{Att}\,}}\) also depends on a model M, but this will be left implicit. We will usually write \({{\,\mathrm{Att}\,}}'\) if we are referring to another model \(M'\), \({{\,\mathrm{Att}\,}}''\) for \(M''\), and so on.

  7. 7.

    We denote by \(V|_s\) the restriction of a valuation V to \(At\times \{s\}\), with \(s\in S\).

  8. 8.

    We thank an anonymous reviewer for this suggestion.

  9. 9.

    See [1, 3, 4, 25] for modal logics with notions of counting.

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Acknowledgements

Special thanks to Barteld Kooi and Rineke Verbrugge for important suggestions. I would also like to thank the anonymous reviewers for very relevant comments. Research supported by Ammodo KNAW project “Rational Dynamics and Reasoning”.

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  1. University of Groningen, Groningen, The Netherlands

    Yuri David Santos

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  1. Yuri David Santos
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Correspondence to Yuri David Santos .

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  1. CNRS, University of Toulouse, Toulouse, France

    Andreas Herzig

  2. University of Helsinki, Helsinki, Finland

    Juha Kontinen

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Santos, Y.D. (2020). Social Consolidations: Rational Belief in a Many-Valued Logic of Evidence and Peerhood. In: Herzig, A., Kontinen, J. (eds) Foundations of Information and Knowledge Systems. FoIKS 2020. Lecture Notes in Computer Science(), vol 12012. Springer, Cham. https://doi.org/10.1007/978-3-030-39951-1_4

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  • DOI: https://doi.org/10.1007/978-3-030-39951-1_4

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