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Unknown Truths and False Beliefs: Completeness and Expressivity Results for the Neighborhood Semantics

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Abstract

In this article, we study logics of unknown truths and false beliefs under neighborhood semantics. We compare the relative expressivity of the two logics. It turns out that they are incomparable over various classes of neighborhood models, and the combination of the two logics are equally expressive as standard modal logic over any class of neighborhood models. We propose morphisms for each logic, which can help us explore the frame definability problem, show a general soundness and completeness result, and generalize some results in the literature. We axiomatize the two logics over various classes of neighborhood frames. Most importantly, by adopting the intersection semantics and the subset semantics in the literature, we extend the results to the case of public announcements, which gives us the desired reduction axioms and has good applications related to Moore sentences, successful formulas and self-refuting formulas. Also, we can say something about the comparative merits of the intersection semantics and the subset semantics.

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References

  1. Ariew, R., and D. Garber (eds., and trans.). G. W. Leibniz: Philosophical Essays, Hackett Publishing Company, Indianapolis, 1989.

  2. Aristotle, De Interpretatione (On Interpretation), in R. McKeon, (ed.), The Basic Works of Aristotle, Random House, New York, 1941.

  3. Balbiani, P., and J. Fan, A complete axiomatization of Euclidean strong non-contingency logic, in Proceedings of 12th Tbilisi Symposium of Language, Logic and Computation (TbiLLC), Lagodekhi, Georgia, 2017, pp. 43–48.

  4. Brogaard, B., and J. Salerno, Fitch’s paradox of knowability, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, fall 2019 edition, 2019.

  5. Chappell, S.-G., Plato on Knowledge in the Theaetetus, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, Winter 2019 edition, 2019.

  6. Chellas, B. F., Modal Logic: An Introduction, Cambridge University Press, 1980.

  7. Fan, J. Logics of essence and accident, arXiv preprint 1506.01872, unpublished manuscript, 2015.

  8. Fan, J., Bimodal logics with contingency and accident, Journal of Philosophical Logic 48:425–445, 2019.

    Article  Google Scholar 

  9. Fan, J., A family of neighborhood contingency logics, Notre Dame Journal of Formal Logic 60(4):683–699, 2019.

    Article  Google Scholar 

  10. Fan, J., Strong noncontingency: on the modal logics of an operator expressively weaker than necessity, Notre Dame Journal of Formal Logic 60(3):407–435, 2019.

    Article  Google Scholar 

  11. Fan, J., Symmetric contingency logic with unlimitedly many modalities, Journal of Philosophical Logic 48(5):851–866, 2019.

    Article  Google Scholar 

  12. Fan, J., Logics of (in)sane and (un)reliable beliefs, Logic Journal of the IGPL, 2020. https://doi.org/10.1093/jigpal/jzaa052.

    Article  Google Scholar 

  13. Fan, J., and H. van Ditmarsch, Neighborhood contingency logic, in M. Banerjee, and S. Krishna, (eds.), Logic and Its Application, vol. 8923 of Lecture Notes in Computer Science, Springer, 2015, pp. 88–99.

  14. Fine, K., Ignorance of ignorance, Synthese 195:4031–4045, 2018. https://doi.org/10.1007/s11229-017-1406-z.

    Article  Google Scholar 

  15. Fitch, F. B., A logical analysis of some value concepts, The Journal of Symbolic Logic 28(2):135–142, 1963.

    Article  Google Scholar 

  16. French, R., and L. Humberstone, Partial confirmation of a conjecture on the boxdot translation in modal logic, The Australasian Journal of Logic 7:56–61, 2009.

    Article  Google Scholar 

  17. Gilbert, D., and G. Venturi, Neighborhood semantics for logics of unknown truths and false beliefs, The Australasian Journal of Logic 14(1):246–267, 2017.

    Article  Google Scholar 

  18. Gilbert, D. R., and G. Venturi, Reflexive-insensitive modal logics, The Review of Symbolic Logic 9(1):167–180, 2016.

    Article  Google Scholar 

  19. Heinemann, F. H., Truths of reason and truths of fact, The Philosophical Review 57(5):458–480, 1948.

    Article  Google Scholar 

  20. Hintikka, J., Knowledge and Belief, Cornell University Press, Ithaca, NY, 1962.

    Google Scholar 

  21. Holliday, W., and T. Icard, Moorean phenomena in epistemic logic, in L. Beklemishev, V. Goranko, and V. Shehtman, (eds.), Advances in Modal Logic 8, College Publications, 2010, pp. 178–199.

  22. Ma, M., and K. Sano, How to update neighborhood models, in International Workshop on Logic, Rationality and Interaction, Springer, 2013, pp. 204–217.

  23. Ma, M. and K. Sano, How to update neighbourhood models. Journal of Logic and Computation 28(8):1781–1804, 2018.

    Article  Google Scholar 

  24. Marcos, J., Logics of essence and accident, Bulletin of the Section of Logic 34(1):43–56, 2005.

    Google Scholar 

  25. Montague, R., Universal grammar, Theoria 36:373–398, 1970.

    Article  Google Scholar 

  26. Moore, G. E., A reply to my critics. in P.A. Schilpp, (ed.), The Philosophy of G.E. Moore, vol. 4 of The Library of Living Philosophers, Northwestern University, Evanston IL, 1942, pp. 535–677.

  27. Perner, J., S. R. Leekam, and H. Wimmer, Three-year-olds’ difficulty with false belief: The case for a conceptual deficit, British Journal of Developmental Psychology 5(2):125–137, 1987.

    Article  Google Scholar 

  28. Scott, D., Advice on modal logic, in K. Lambert, (ed.), Philosophical Problems in Logic: Some Recent Developments, Springer, 1970, pp. 143–173.

  29. Small, C. G., Reflections on Gödel’s ontological argument, in W. Deppert and M. Rahnfeld, (ed.), Klarheit in Religionsdingen: Aktuelle Beiträge zur Religionsphilosophie, vol. 3 of Grundlagenprobleme unserer Zeit, Leipziger Universitätsverlag, 2001, pp. 109–144.

  30. Steinsvold, C., Completeness for various logics of essence and accident, Bulletin of the Section of Logic 37(2):93–101, 2008.

    Google Scholar 

  31. Steinsvold, C., A note on logics of ignorance and borders, Notre Dame Journal of Formal Logic 49(4):385–392, 2008.

    Article  Google Scholar 

  32. Steinsvold, C., Being wrong: Logics for false belief, Notre Dame Journal of Formal Logic 52(3):245–253, 2011.

    Article  Google Scholar 

  33. Steinsvold, C., The boxdot conjecture and the language of essence and accident, Australasian Journal of Logic 10:18–35, 2011.

    Article  Google Scholar 

  34. van Benthem, J., What one may come to know, Analysis 64(2):95–105, 2004.

    Article  Google Scholar 

  35. van Benthem, J., Logical Dynamics of Information and Interaction, Cambridge University Press, 2011.

  36. van Ditmarsch, H., W. van der Hoek, and B. Kooi, Dynamic Epistemic Logic, Springer, 2007.

  37. Wellman, H. M., D. Cross, and J. Watson, Meta-analysis of theory-of-mind development: The truth about false belief, Child Development 72(3):655–684, 2001.

    Article  Google Scholar 

  38. Zvesper, J., Playing with information, PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 2010.

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Acknowledgements

The author would like to thank to Lloyd Humberstone and Hans van Ditmarsch and an anonymous reviewer of this journal for their invaluable comments. This research is financially supported by the Project 17CZX053 of National Social Science Fundation of China.

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Authors and Affiliations

  1. Institute of Philosophy, Chinese Academy of Sciences, Beijing, China

    Jie Fan

  2. School of Humanities, University of Chinese Academy of Sciences, Beijing, China

    Jie Fan

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Correspondence to Jie Fan.

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Fan, J. Unknown Truths and False Beliefs: Completeness and Expressivity Results for the Neighborhood Semantics. Stud Logica 110, 1–45 (2022). https://doi.org/10.1007/s11225-021-09950-5

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