Skip to main content

Justification Logic with Confidence

Abstract

Justification logics are a family of modal logics whose non-normal modalities are parametrised by a type-theoretic calculus of terms. The first justification logic was developed by Sergei Artemov to provide an explicit modal logic for arithmetical provability in which these terms were taken to pick out proofs. But, justification logics have been given various other interpretations as well. In this paper, we will rely on an interpretation in which the modality \(\llbracket \tau \rrbracket \varphi \) is read ‘S accepts \(\tau \) as justification for \(\varphi \)’. Since it is often important to specify just how much confidence agents have in propositions on the basis of justifications, the logic will need to be extended if it is to provide a sufficiently general account. The primary purpose of this paper is to extend justification logic with the expressive resources to needed to do so. Thus, we will construct the justification logic with confidence (\({\mathsf {JC}}\)). While \({\mathsf {JC}}\) will be extremely general, capable of accommodating a wide range of interpretations, we provide motivation in terms of the notion of confidence deriving recent work by Paul and Quiggin (Episteme 15(3):363–382, 2018). Under this understanding, confidence must only correspond to a partial ordering. We axiomatise \({\mathsf {JC}}\) and provide a sound and complete semantics.

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

References

  1. Alston, W.P., Internalism and externalism in epistemology, Philosophical Topics 14(1):179–221, 1986.

    Article  Google Scholar 

  2. Artemov, S., Logic of proofs, Annals of Pure and Applied Logic 67:29–59, 1994.

    Article  Google Scholar 

  3. Artemov, S., Operational modal logic, Technical Report MSI 95-29, Cornell University, 1995.

  4. Artemov, S., Explicit provability and constructive semantics, The Bulletin of Symbolic Logic 7(1):1–36, 2001.

    Article  Google Scholar 

  5. Artemov, S., The ontology of justifications in a logical setting, Studia Logica 100(1–2):17–30, 2012.

    Article  Google Scholar 

  6. Artemov, S., and M. Fitting, Justification Logic, in E.N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, Winter, 2016.

  7. Artemov, S., and R. Kuznets, Logical omniscience as a computational complexity problem, in A. Heifetz, (ed.), Proceedings of the Twelfth Conference, TARK 2009, Stanford, 2009, pp. 14–23. Theoretical Aspects of Rationality and Knowledge, ACM.

  8. Artemov, S., and T. Straßen, The basic logic of proofs, in E. Börger, G. Jäger, H.K. Büning, S. Martini, and M. Richter, (eds.), Computer Science Logic, volume 702 of Lecture Notes in Computer Science, Berlin, Heidelberg, 1992, pp. 14–28. International Workshop on Computer Science Logic, Springer.

  9. Artemov, S., and T. Straßen, The Basic Logic of Proofs, Technical Report IAM 92-018, University of Berne, Switzerland, September 1992.

  10. Buchak, L., Risk and Rationality, Oxford University Press, Oxford, 2014.

    Google Scholar 

  11. Cohen, S., How to be a fallibilist, Philosophical Perspectives 2:91–123, 1988.

    Article  Google Scholar 

  12. Fagin, R., and J.Y. Halpern, Belief, awareness, and limited reasoning, Artificial Intelligence 34:39–76, 1988.

    Article  Google Scholar 

  13. Fan, T.-F., and C.-J. Liau, A logic for reasoning about justified uncertain beliefs. in Proceedings of the 24th International Conference on Artificial Intelligence, IJCAI’15, Buenos Aires, Argentina, 2015, pp. 2948–2954. AAAI Press.

  14. Fitting, M., The logic of proofs, semantically, Annals of Pure and Applied Logic 132:1–25, 2005.

    Article  Google Scholar 

  15. Fitting, M., Possible world semantics for first-order logic of proofs, Annals of Pure and Applied Logic 165(1):225–240, 2014.

    Article  Google Scholar 

  16. Ghari, M., Justification Logics in a Fuzzy Setting. ArXiv e-prints, 2014.

  17. Gödel, K., Eine interpretation des intuitionistischen aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums 4:39–40, 1933.

    Google Scholar 

  18. Halpern, J.Y., and R. Pucella, Dealing with logical omniscience. in Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge, TARK ’07, New York, NY, USA, 2007, pp. 169–176. ACM.

  19. Harman, G., Thought, Princeton University Press, Princeton, 1973.

    Google Scholar 

  20. Hintikka, J., Impossible possible worlds vindicated, Journal of Philosophy 4(4):475–484, November 1975.

  21. Holliday, W.H., Knowing What Follows: Epistemic Closure and Epistemic Logic, Dissertation. Stanford University, 2012.

  22. Humberstone, L., Heterogeneous logic, Erkenntnis 29(3):395–435, November 1988.

  23. Kokkinis, I., The complexity of non-iterated probabilistic justification logic, in Foundations of Information and Knowledge Systems - 9th International Symposium, FoIKS 2016, Linz, Austria, March 7-11, 2016. Proceedings, 2016, pp. 292–310.

  24. Kokkinis, I., P. Maksimovic, Z. Ognjanovic, and T. Studer, First steps towards probabilistic justification logic, Logic Journal of the IGPL 23(4):662–687, 2015.

    Article  Google Scholar 

  25. Kokkinis, I., Z. Ognjanovic, and T. Studer, Probabilistic justification logic, in Logical Foundations of Computer Science - International Symposium, LFCS 2016, Deerfield Beach, FL, USA, January 4-7, 2016. Proceedings, 2016, pp. 174–186.

  26. Kuznets, R., and T. Studer, Justifications, ontology, and conservativity, in T. Bolander, T. Braüner, S. Ghilardi, and L. Moss, (eds.), Advances in Modal Logic, volume 9, College Publications, 2012, pp. 437–458.

  27. Milnikel, B., The logic of uncertain justifications, in S. Artemov, and A. Nerode, (eds.), Logical Foundations of Computer Science: International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings, volume 7734 of Lecture Notes in Computer Science, Berlin, Springer Berlin Heidelberg, 2013, pp. 296–306.

  28. Mkrtychev, A., Models for the logic of proofs. in S. Adian, and A. Nerode, (eds.), Logical Foundations of Computer Science: 4th International Symposium, LFCS’97 Yaroslavl, Russia, July 6–12, 1997 Proceedings, Berlin, Springer, 1997, pp. 266–275.

  29. Moser, P.K., Knowledge and Evidence. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  30. Ognjanović, Z., N. Savić, and T. Studer. Justification logic with approximate conditional probabilities, in A. Baltag, J. Seligman, and T. Yamada, (eds.), Logic, Rationality, and Interaction. LORI 2017, volume 10455 of Lecture Notes in Computer Science, Berlin, Heidelberg, Springer, 2017, pp. 681–686.

  31. Pacuit, E., A note on some explicit modal logics, Technical Report PP-2006-29, University of Amsterdam. ILLC Publications, Amsterdam, 2006.

  32. Paul, L.A., and J. Quiggin, Real world problems, Episteme, 15(3):363–382, 2018.

    Article  Google Scholar 

  33. Pollock, J.L., Contemporary Theories of Knowledge. Rowman and Littlefield, Savage, 1986.

    Google Scholar 

  34. Quiggin, J., A Theory of Anticipated Utility, Journal of Economic Behavior and Organization 3:323–343, 1982.

    Article  Google Scholar 

  35. Quiggin, J., Generalized Expected Utility Theory. Springer, Dordrecht, 1993.

    Book  Google Scholar 

  36. Shafer, G., A Mathematical Theory of Evidence. Princeton University Press, Princeton, 1976.

    Google Scholar 

  37. Stalnaker, R., The problem of logical omniscience, i. Synthese 89(3):425–440, December 1991.

  38. Standefer, S., Tracking reasons with extensions of relevant logics. Logic Journal of the IGPL, forthcoming.

  39. Standefer, S., T. Shear, and R. French, Getting some closure with justification logic. Manuscript, 2019.

  40. Su, C.-P., T.-F. Fan, and C.-J. Liau, Possibilistic justification logic: Reasoning about justified uncertain beliefs, ACM Trans. Comput. Logic 18(2):1–21, 2017.

    Article  Google Scholar 

  41. Swain, M., Justification and the basis of belief, in G.S. Pappas, (ed.), Justification and Knowledge: New Studies in Epistemology, Springer Netherlands, Dordrecht, 1979, pp. 25–49.

  42. van Benthem, J., Reflections on epistemic logic, Logique et Analyse 34:5–14, 1991.

    Google Scholar 

Download references

Acknowledgements

This work was generously supported by ARC Discovery Project Grant #DP160100903. We are grateful to audiences at the 2017 Reasoning Club Conference in Torino, the Melbourne Logic Seminar, the Economics and Philosophy Workshop at the Australian National University, and the 2017 New Zealand Association of Philosophers Conference in Dunedin for valuable discussions. We would also like to individually acknowledge Guillermo Badia, Rohan French, Lloyd Humberstone, Fabio Lampert, Toby Meadows, Laurie Paul, Shawn Standefer, Joe Ulkowski, Greg Restall, Zach Weber, and two anonymous referees for their particularly helpful feedback.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ted Shear.

Additional information

Publisher's Note

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

Presented by Yde Venema

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shear, T., Quiggin, J. Justification Logic with Confidence. Stud Logica 108, 751–778 (2020). https://doi.org/10.1007/s11225-019-09874-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-019-09874-1

Keywords

  • Modal logic
  • Justification logic
  • Justification
  • Confidence
  • Bounded rationality