Probabilities with Gaps and Gluts

Abstract

Belnap-Dunn logic (BD), sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes’ and Jeffrey updating and also suggest mechanisms for aggregating information from different sources.

References

  1. 1.

    Alchourrón, C.E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2), 510–530.

    Article  Google Scholar 

  2. 2.

    Anderson A.R., & Nuel, B.D. (1975). Entailment: The logic of relevance and necessity Vol. I. Princeton: Princeton University Press.

    Google Scholar 

  3. 3.

    Batens, D. (2001). A general characterization of adaptive logics. Logique et Analyse, 44(173-175), 45–68.

    Google Scholar 

  4. 4.

    Belnap, N.D. (1977). A useful four-valued logic. In Modern uses of multiple-valued logic (pp. 5–37): Springer.

  5. 5.

    Belnap, N.D. (2019). How a computer should think, (pp. 35–53). New York: Springer International Publishing. https://doi.org/10.1007/978-3-030-31136-0∖_4.

    Google Scholar 

  6. 6.

    Childers, T., Majer, O., & Milne, P. (2019). The (relevant) logic of scientific discovery. arXiv:2101.03593 [math.LO].

  7. 7.

    Christensen, D. (2007). Epistemology of disagreement: The good news. The Philosophical Review, 116(2), 187–217. https://doi.org/10.1215/00318108-2006-035.

    Article  Google Scholar 

  8. 8.

    da Costa, N. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15(4), 497–510.

    Article  Google Scholar 

  9. 9.

    da Costa, N., & Subrahmanian, V. (1989). Paraconsistent logic as a formalism for reasoning about inconsistent knowledge bases. Artificial Intelligence in Medicine, 1, 167–174.

    Article  Google Scholar 

  10. 10.

    Dunn, J.M. (1976). Intuitive semantics for first degree entailment and ‘coupled trees’. Philosophical Studies, 29(3), 149–168.

    Article  Google Scholar 

  11. 11.

    Dunn, J.M. (2010). Contradictory information: Too much of a good thing. Journal of Philosophical Logic, 39(4), 425–452.

    Article  Google Scholar 

  12. 12.

    Dunn, J.M., & Kiefer, N.M. (2019). Contradictory information: Better than nothing? the paradox of the two firefighters. In Graham Priest on Dialetheism and Paraconsistency (pp. 231–247): Springer.

  13. 13.

    Elga, A. (2007). Reflection and disagreement. Noûs, 41(3), 478–502. https://doi.org/10.1111/j.1468-0068.2007.00656.x.

    Article  Google Scholar 

  14. 14.

    Fagin, R., & Halpern, J.Y. (1991). Uncertainty, belief, and probability. Computational Intelligence, 7(3), 160–173.

    Article  Google Scholar 

  15. 15.

    Font, J.M. (1997). Belnap’s four-valued logic and de morgan lattices. Logic Journal of IGPL, 5(3), 1–29.

    Article  Google Scholar 

  16. 16.

    Halpern, J. (2017). Reasoning about uncertainty. Cambridge: MIT Press.

    Book  Google Scholar 

  17. 17.

    Jaskowski, S. (1948). Propositional calculus for contradictory deductive systems. Studia Logica, 24, 143–157.

    Article  Google Scholar 

  18. 18.

    Jøsang, A. (1997). Artificial reasoning with subjective logic. In Proceedings of the second Australian workshop on commonsense reasoning, (Vol. 48 p. 34): Citeseer.

  19. 19.

    Kelly, T. (2010). Peer disagreement and higher order evidence. In Goldman, A I, & Whitcomb, D (Eds.) Social epistemology: essential readings (pp. 183–217): Oxford University Press.

  20. 20.

    Klein, D., & Marra, A. (2020). From oughts to goals: A logic for enkrasia. Studia Logica, 108(1), 85–128. https://doi.org/10.1007/s11225-019-09854-5.

    Article  Google Scholar 

  21. 21.

    Klein, D., Majer, O., & Rafiee Rad, S. (2020). Non-classical probabilities for decision making in situations of uncertainty. Roczniki Filozoficzne, 4 (68), 315–343. https://doi.org/10.18290/rf20684-15.

    Google Scholar 

  22. 22.

    Kolmogorov, A.N. (2018). Foundations of the theory of probability. New York: Courier Dover Publications.

    Google Scholar 

  23. 23.

    Mares, E.D. (1997). Paraconsistent probability theory and paraconsistent bayesianism. Logique et analyse, 40(160), 375–384.

    Google Scholar 

  24. 24.

    Přenosil, A. (2018). Reasoning with inconsistent information PhD thesis, Charles University, Faculty of Philosophy.

  25. 25.

    Priest, G. (1979). Logic of paradox. Journal of Philosophical Logic, 8, 219–241.

    Article  Google Scholar 

  26. 26.

    Priest, G. (2002). Paraconsistent logic. In Gabbay, D.M., & Guenthner, F. (Eds.) Handbook of Philosophical Logic, (Vol. 6 pp. 287–393).

  27. 27.

    Priest, G. (2006). In contradiction. Oxford: Oxford University Press.

    Book  Google Scholar 

  28. 28.

    Priest, G. (2007). Paraconsistency and dialetheism. In Gabbay, D., & Woods, J. (Eds.) Handbook of the History of Logic, (Vol. 8 pp. 129–204).

  29. 29.

    Rescher, N., & Manor, R. (1970). On inference from inconsistent premisses. Theory and Decision, 1(2), 179–217.

    Article  Google Scholar 

  30. 30.

    Rodrigues, A., Bueno-Soler, J., & Carnielli, W. (2020). Measuring evidence: a probabilistic approach to an extension of belnap–dunn logic. Synthese https://doi.org/10.1007/s11229-020-02571-w.

  31. 31.

    Shafer, G. (1976). A mathematical theory of evidence Vol. 42. Princeton : Princeton University Press.

    Book  Google Scholar 

  32. 32.

    Zhou, C. (2013). Belief functions on distributive lattices. Artificial Intelligence, 201, 1–31.

    Article  Google Scholar 

Download references

Acknowledgments

We dedicate this work to the memory of J. Michael Dunn. This work started from an inspiring talk by him and greatly benefited from his valuable and generous insights and comments throughout. Besides, we would like to thank Timothy Childers, Johannes Korbmacher, Olivier Roy, Frederik Van De Putte, an anonymous reviewer and the audience of the MCMP logic colloquium, the Amsterdam LIRA seminar, Advances in Philosophical Logic 2019 in Lublin, the Prague workshop on non-classical epistemic logics, the Tulips seminar in Utrecht, and the logic seminar at the University of Maryland for valuable feedback and suggestions. The work of OM was supported by the Czech Science Foundation. trough the project Reasoning with Graded Properties [GA18-00113S]. The work of DK and SRR was partially supported by Deutsche Forschungsgemeinschaft (DFG) and Agence Nationale de la Recherche (ANR) as part of the joint project Collective Attitude Formation [RO 4548/8-1], by DFG through the project From Shared Evidence to Group Attitudes [RO 4548/6-1], by DFG through the network grants Simulations of Social Scientific Inquiry [426833574] and Foundations, Applications and Theory of Inductive Logic [432308570], and by the National Science Foundation of China as part of the project Logics of Information Flow in Social Networks [17ZDA026].

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dominik Klein.

Additional information

Publisher’s Note

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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Klein, D., Majer, O. & Rafiee Rad, S. Probabilities with Gaps and Gluts. J Philos Logic (2021). https://doi.org/10.1007/s10992-021-09592-x

Download citation

Keywords

  • Belnap-Dunn logic
  • First Degree Entailment
  • Non-standard probability theory
  • Probability theory
  • Bayes’ updating
  • Jeffrey updating
  • Probability aggregation