Skip to main content

Logics for Belief as Maximally Plausible Possibility

Studia Logica volume 108, pages 1019–1061 (2020)Cite this article

Abstract

We consider a basic logic with two primitive uni-modal operators: one for certainty and the other for plausibility. The former is assumed to be a normal operator (corresponding—semantically—to a binary Kripke relation), while the latter is merely a classical operator (corresponding—semantically—to a neighborhood function). We then define belief, interpreted as “maximally plausible possibility”, in terms of these two notions: the agent believes \(\phi \) if (1) she cannot rule out \(\phi \) (that is, it is not the case that she is certain that \(\lnot \phi \)), (2) she judges \(\phi \) to be plausible and (3) she does not judge \(\lnot \phi \) to be plausible. We consider four interaction properties between certainty and plausibility and study how these properties translate into properties of belief (e.g. positive and negative introspection and their converses). We then prove that all the logics considered are minimal logics for the highlighted theorems. We also consider a number of possible interpretations of plausibility, identify the corresponding logics and show that some notions considered in the literature are special cases of our framework.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Aumann, R., Agreeing to disagree, The Annals of Statistics 4: 1236–1239, 1976.

    Article  Google Scholar 

  2. Aumann, R., Interactive epistemology II: Probability, International Journal of Game Theory 28: 301–314, 1999.

    Article  Google Scholar 

  3. Balbiani, P., D. Fernández-duque, and E. Lorini, The dynamics of epistemic attitudes in resource-bounded agents, Studia Logica 107(3): 457–488, 2019.

    Article  Google Scholar 

  4. Baltag, A., and S. Smets, A qualitative theory of dynamic interactive belief revision, in G. Bonanno, W. van der Hoek, and M. Wooldridge, (eds.), Logic and the Foundations of Game and Decision Theory (LOFT 7), vol. 3 of Texts in Logic and Games, Amsterdam University Press, 2008, pp. 11–58.

  5. Battigalli, P., and G. Bonanno, Recent results on belief, knowledge and the epistemic foundations of game theory, Research in Economics 53: 149–225, 1999.

    Article  Google Scholar 

  6. Beddor, B., and S. Goldstein, Believing epistemic contradictions, The Review of Symbolic Logic 11: 87–114, 2018.

    Article  Google Scholar 

  7. Blackburn, P., M. de Rijke, and Y. Venema, Modal logic, Cambridge University Press, 2001.

    Google Scholar 

  8. Boutilier, C., Normative, subjunctive, and autoepistemic defaults: Adopting the Ramsey test, in Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR’92). Cambridge, MA, October 25–29, 1992, pp. 685–696.

  9. Chellas, B., Modal logic: an introduction, Cambridge University Press, 1984.

  10. David, C., Putting logic in its place: formal constraints on rational belief, Clarendon Press, 2007.

  11. Dietrich, F., and C. List, Reasons for (prior) belief in bayesian epistemology, Synthese 190: 787–808, 2013.

    Article  Google Scholar 

  12. Dylan, D., Belief and certainty Synthese 194: 4597–4621, 2017.

  13. Fermé, E., and S. O. Hansson, AGM 25 years, Journal of Philosophical Logic 40: 295–331, 2011.

  14. Fermé, E., and S. O. Hansson, Belief change: introduction and overview, Springer, 2018.

  15. Fiutek, V., Playing with knowledge and belief, Ph.D. thesis, University of Amsterdam, Institute for Logic, Language and Computation, 2013.

    Google Scholar 

  16. Foley, R., Working Without a Net, Oxford University Press, 1993.

  17. Foley, R., Beliefs, degrees of belief, and the Lockean thesis, in F. Huber, and C. Schmidt-Petri, (eds.), Degrees of Belief, vol. 342 of Synthese Library, Springer, 2009, pp. 37–48.

  18. Friedman, N., and J. Halpern, Modeling belief in dynamic systems. Part 1: foundations, Artificial Intelligence 95: 257–316, 1997.

  19. Gettier, E., Is justified true belief knowledge?, Analysis 23: 121–123, 1963.

    Article  Google Scholar 

  20. Goldszmidt, M., and P. Judea, Rank-based systems: A simple approach to belief revision, belief update, and reasoning about evidence and actions, in Proceedings of the Third International Conference on Principles of Knowledge Representation and Reasoning, KR’92, Morgan Kaufmann Publishers Inc., San Francisco, 1992, pp. 661–672.

    Google Scholar 

  21. Hintikka, J., Knowledge and belief: an introduction to the logic of the two notions, Cornell University Press, Cornell, 1962.

    Google Scholar 

  22. Klein, D., O. Roy, and N. Gratzl, Knowledge, belief, normality, and introspection, Synthese 195: 4343–4372, 2018.

    Article  Google Scholar 

  23. Kraus, S., and D. Lehmann, Knowledge, belief and time, Theoretical Computer Science 58: 155–174, 1988.

    Article  Google Scholar 

  24. Kyburg, H. E., Probability and the Logic of Rational Belief, Wesleyan University Press, 1961.

  25. Kyburg, H. E., Epistemology and Inference, Univ of Minnesota Press, 1983.

  26. Lamarre, P., and Y. Shoham, Knowledge, certainty, belief, and conditionalisation (abbreviated version), in Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufmann Publishers Inc., San Francisco, 1994, pp. 415–424.

    Google Scholar 

  27. Lehrer, K., and T. Paxson, Knowledge: undefeated justified true belief, Journal of Philosophy 66: 225–237, 1969.

    Article  Google Scholar 

  28. Leitgeb, H., Reducing belief simpliciter to degrees of belief reducing belief simpliciter to degrees of belief, Annals of Pure and Applied Logic 164: 1338–1389, 2013.

    Article  Google Scholar 

  29. Leitgeb, H., The Stability of Belief: How Rational Belief Coheres with Probability, Oxford University Press, 2017.

  30. Leitgeb, H., Stability and the lottery paradox, Tech. rep., Ludwig-Maximilians-University Munich, 2018.

  31. Lenzen, W., Recent work in epistemic logic, Acta Philosophica Fennica 30: 1–219, 1978.

    Google Scholar 

  32. Lenzen, W., Epistemic logic, in I. Niiniluoto, M. Sintonen, and J. Wolenski, (eds.), Handbook of Epistemology, Springer, 2004, pp. 963–983.

  33. Makinson, D., The paradox of the preface, Analysis 25(5): 205–207, 1965.

    Article  Google Scholar 

  34. Moses, Y., and Y. Shoham, Belief as defeasible knowledge, Artificial Intelligence 64: 299–322, 1993.

    Article  Google Scholar 

  35. Nelkin, D. K., The Lottery Paradox, knowledge, and rationality, The Philosophical Review 109(3): 373–409, 2000.

    Article  Google Scholar 

  36. Pacuit, E., Neighborhood Semantics for Modal Logic, Springer, Cham (Switzerland), 2017.

    Book  Google Scholar 

  37. Spohn, W., Ordinal conditional functions: A dynamic theory of epistemic states, in W. L. Harper, and B. Skyrms, (eds.), Causation in Decision, Belief Change and Statistics, vol. 2, D. Reidel, Dordrecht, 1988, pp. 105–134.

  38. Stalnaker, R., On logics of knowledge and belief, Philosophical Studies 128: 169–199, 2006.

    Article  Google Scholar 

  39. Swain, M., Epistemic defeasibility, American Philosophical Quarterly 11: 15–25, 1974.

    Google Scholar 

  40. van Benthem, J., Dynamic logic for belief revision, Journal of Applied Non-classical Logics 17: 129–155, 2007.

    Article  Google Scholar 

  41. van Benthem, J., Logical dynamics of information and interaction, Cambridge University Press, 2011.

  42. van Benthem, J., D. Fernández-Duque, and E. Pacuit, Evidence and plausibility in neighborhood structures, Annals of Pure and Applied Logic 165: 106–133, 2014.

    Article  Google Scholar 

  43. van der Hoek, W., and J.-J. Ch. Meyer, Epistemic logic for AI and computer science, Cambridge University Press, Cambridge, 1995.

  44. van Ditmarsch, H., W. van der Hoek, and B. Kooi, Dynamic epistemic logic, Springer, 2007.

    Google Scholar 

  45. Voorbraak, F., Generalized Kripke models for epistemic logic, in Proceedings of the 4th Conference on Theoretical Aspects of Reasoning About Knowledge, TARK ’92, Morgan Kaufmann Publishers Inc., San Francisco, 1992, pp. 214–228.

    Google Scholar 

  46. Williamson, T, Knowledge and its limits, Oxford University Press, 2000.

Download references

Acknowledgements

I am grateful to two anonymous reviewers for helpful and constructive comments. This research was supported by a grant from UC Davis.

Author information

Authors and Affiliations

  1. University of California Davis, Davis, CA, USA

    Giacomo Bonanno

Authors
  1. Giacomo Bonanno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giacomo Bonanno.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by Richmond Thomason

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bonanno, G. Logics for Belief as Maximally Plausible Possibility. Stud Logica 108, 1019–1061 (2020). https://doi.org/10.1007/s11225-019-09887-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-019-09887-w

Keywords

  • Knowledge
  • Belief
  • Certainty
  • Plausibility
  • Positive introspection
  • Negative introspection