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The Logic of Justification

  • Sergei ArtemovEmail author
Part of the Springer Graduate Texts in Philosophy book series (SGTP, volume 1)

Abstract

We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidence-based foundation for epistemic logic.

As a case study, we offer a resolution of the Goldman-Kripke ‘Red Barn’ paradox and analyze Russell’s ‘prime minister example’ in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning.

Keywords

Modal Logic Definite Description Epistemic Logic Constant Specification Justify True Belief 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author is very grateful to Walter Dean, Mel Fitting, Vladimir Krupski, Roman Kuznets, Elena Nogina, Tudor Protopopescu, and Ruili Ye, whose advice helped with this paper. Many thanks to Karen Kletter for editing this text. Thanks to audiences at the CUNY Graduate Center, Bern University, the Collegium Logicum in Vienna, and the 2nd International Workshop on Analytic Proof Systems for comments on earlier versions of this paper. This work has been supported by NSF grant 0830450, CUNY Collaborative Incentive Research Grant CIRG1424, and PSC CUNY Research Grant PSCREG-39-721.

References

  1. Antonakos, E. (2007). Justified and common knowledge: Limited conservativity. In S. Artemov & A. Nerode (Eds.), Logical Foundations of Computer Science. International Symposium, LFCS 2007, Proceedings, New York, June 2007 (Lecture notes in computer science, Vol. 4514, pp. 1–11). Springer.Google Scholar
  2. Artemov, S. (1995). Operational modal logic. Technical report MSI 95-29, Cornell University.Google Scholar
  3. Artemov, S. (1999). Understanding constructive semantics. In Spinoza Lecture for European Association for Logic, Language and Information, Utrecht, Aug 1999.Google Scholar
  4. Artemov, S. (2001). Explicit provability and constructive semantics. Bulletin of Symbolic Logic, 7(1), 1–36.CrossRefGoogle Scholar
  5. Artemov, S. (2006). Justified common knowledge. Theoretical Computer Science, 357(1–3), 4–22.CrossRefGoogle Scholar
  6. Artemov, S. (2007). On two models of provability. In D. M. Gabbay, M. Zakharyaschev, & S. S. Goncharov (Eds.), Mathematical problems from applied logic II (pp. 1–52). New York: Springer.CrossRefGoogle Scholar
  7. Artemov, S. (2008). Symmetric logic of proofs. In A. Avron, N. Dershowitz, & A. Rabinovich (Eds.), Pillars of computer science, essays dedicated to Boris (Boaz) Trakhtenbrot on the occasion of his 85th birthday (Lecture notes in computer science, Vol. 4800, pp. 58–71). Berlin/Heidelberg: Springer.Google Scholar
  8. Artemov, S., & Beklemishev, L. (2005). Provability logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 13, pp. 189–360). Dordrecht: Springer.Google Scholar
  9. Artemov, S., Kazakov, E., & Shapiro, D. (1999). Epistemic logic with justifications. Technical report CFIS 99-12, Cornell University.Google Scholar
  10. Artemov, S., & Kuznets, R. (2006). Logical omniscience via proof complexity. In Computer Science Logic 2006, Szeged (Lecture notes in computer science, Vol. 4207, pp. 135–149).Google Scholar
  11. Artemov, S., & Nogina, E. (2004). Logic of knowledge with justifications from the provability perspective. Technical report TR-2004011, CUNY Ph.D. Program in Computer Science.Google Scholar
  12. Artemov, S., & Nogina, E. (2005). Introducing justification into epistemic logic. Journal of Logic and Computation, 15(6), 1059–1073.CrossRefGoogle Scholar
  13. Artemov, S., & Strassen, T. (1993). Functionality in the basic logic of proofs. Technical report IAM 93-004, Department of Computer Science, University of Bern, Switzerland.Google Scholar
  14. Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University Press.Google Scholar
  15. Brezhnev, V. (2000). On explicit counterparts of modal logics. Technical report CFIS 2000-05, Cornell University.Google Scholar
  16. Brezhnev, V., & Kuznets, R. (2006). Making knowledge explicit: How hard it is. Theoretical Computer Science, 357(1–3), 23–34.CrossRefGoogle Scholar
  17. Dean, W., & Kurokawa, H. (2007). From the knowability paradox to the existence of proofs. Manuscript (submitted to Synthese).Google Scholar
  18. Dean, W., & Kurokawa, H. (2010). The knower paradox and the quantified logic of proofs. In A. Hieke (Ed.), Austrian Ludwig Wittgenstein society. Synthese, 176(2), 177–225.Google Scholar
  19. Dretske, F. (1971). Conclusive reasons. Australasian Journal of Philosophy, 49, 1–22.CrossRefGoogle Scholar
  20. Dretske, F. (2005). Is knowledge closed under known entailment? The case against closure. In M. Steup & E. Sosa (Eds.), Contemporary Debates in Epistemology (pp. 13–26). Malden: Blackwell.Google Scholar
  21. Fagin, R., & Halpern, J. Y. (1985). Belief, awareness, and limited reasoning: Preliminary report. In Proceedings of the Ninth International Joint Conference on Artificial Intelligence (IJCAI-85), (pp. 491–501). Los Altos, CA: Morgan Kaufman.Google Scholar
  22. Fagin, R., & Halpern, J. Y. (1988). Belief, awareness, and limited reasoning. Artificial Intelligence, 34(1), 39–76.CrossRefGoogle Scholar
  23. Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge: MIT Press.Google Scholar
  24. Fitting, M. (2003). A semantics for the logic of proofs. Technical report TR-2003012, CUNY Ph.D. Program in Computer Science.Google Scholar
  25. Fitting, M. (2005). The logic of proofs, semantically. Annals of Pure and Applied Logic, 132(1), 1–25.CrossRefGoogle Scholar
  26. Fitting, M. (2007). Intensional logic. Stanford Encyclopedia of Philosophy (http://plato.stanford.edu), Feb 2007.
  27. Fitting, M., & Mendelsohn, R. L. (1998). First-order modal logic. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  28. Frege, G. (1952). On sense and reference. In P. Geach & M. Black (Eds.), Translations of the philosophical writings of Gottlob Frege. Oxford: Blackwell.Google Scholar
  29. Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23, 121–123.CrossRefGoogle Scholar
  30. Gödel, K. (1933). Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse Math. Kolloq., 4, 39–40. English translation in: S. Feferman et al., (Eds.) (1986). Kurt Gödel collected works (Vol. 1, pp. 301–303). Oxford: Oxford University Press/New York: Clarendon Press.Google Scholar
  31. Gödel, K. (1995) Vortrag bei Zilsel/Lecture at Zilsel’s (*1938a). In S. Feferman, J. W. Dawson Jr., W. Goldfarb, C. Parsons, & R. M. Solovay (Eds.), Unpublished essays and lectures (Kurt Gödel collected works, Vol. III, pp. 86–113). Oxford University Press.Google Scholar
  32. Goldman, A. (1967). A causal theory of knowing. The Journal of Philosophy, 64, 335–372.CrossRefGoogle Scholar
  33. Goris, E. (2007). Explicit proofs in formal provability logic. In S. Artemov & A. Nerode (Eds.), Logical Foundations of Computer Science. International Symposium, LFCS 2007, Proceedings, New York, June 2007 (Lecture notes in computer science, Vol. 4514, pp. 241–253). Springer.Google Scholar
  34. Hendricks, V. F. (2003). Active agents. Journal of Logic, Language and Information, 12(4), 469–495.CrossRefGoogle Scholar
  35. Hendricks, V. F. (2005). Mainstream and formal epistemology. New York: Cambridge University Press.CrossRefGoogle Scholar
  36. Heyting, A. (1934). Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie. Berlin: Springer.Google Scholar
  37. Hintikka, J. (1962). Knowledge and belief. Ithaca: Cornell University Press.Google Scholar
  38. Hintikka, J. (1975). Impossible possible worlds vindicated. Journal of Philosophical Logic, 4, 475–484.CrossRefGoogle Scholar
  39. Kleene, S. (1945). On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, 10(4), 109–124.CrossRefGoogle Scholar
  40. Krupski, N. V. (2006). On the complexity of the reflected logic of proofs. Theoretical Computer Science, 357(1), 136–142.CrossRefGoogle Scholar
  41. Krupski, V. N. (2001). The single-conclusion proof logic and inference rules specification. Annals of Pure and Applied Logic, 113(1–3), 181–206.CrossRefGoogle Scholar
  42. Krupski, V. N. (2006). Referential logic of proofs. Theoretical Computer Science, 357(1), 143–166.CrossRefGoogle Scholar
  43. Kuznets, R. (2000). On the complexity of explicit modal logics. In Computer Science Logic 2000 (Lecture notes in computer science, Vol. 1862, pp. 371–383). Berlin/Heidelberg: Springer.Google Scholar
  44. Kuznets, R. (2008). Complexity issues in justification logic. PhD thesis, CUNY Graduate Center. http://kuznets.googlepages.com/PhD.pdf.
  45. Lehrer, K., & Paxson, T. (1969). Knowledge: Undefeated justified true belief. The Journal of Philosophy, 66, 1–22.CrossRefGoogle Scholar
  46. Luper, S. (2005). The epistemic closure principle. Stanford Encyclopedia of Philosophy.Google Scholar
  47. McCarthy, J., Sato, M., Hayashi, T., & Igarishi, S. (1978). On the model theory of knowledge. Technical report STAN-CS-78-667, Stanford University.Google Scholar
  48. Meyer, J. -J. Ch., & van der Hoek, W. (1995). Epistemic logic for AI and computer science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  49. Milnikel, R. (2007). Derivability in certain subsystems of the logic of proofs is \(\Pi _{2}^{p}\)-complete. Annals of Pure and Applied Logic, 145(3), 223–239.CrossRefGoogle Scholar
  50. Mkrtychev, A. (1997). Models for the logic of proofs. In S. Adian & A. Nerode (Eds.), Logical Foundations of Computer Science ‘97, Yaroslavl’ (Lecture notes in computer science, Vol. 1234, pp. 266–275). Springer.Google Scholar
  51. Moses, Y. (1988). Resource-bounded knowledge. In M. Vardi (Ed.), Proceedings of the Second Conference on Theoretical Aspects of Reasoning About Knowledge, Pacific Grove, March 7–9, 1988 (pp. 261–276). Morgan Kaufmann Publishers.Google Scholar
  52. Neale, S. (1990). Descriptions. Cambridge: MIT.Google Scholar
  53. Nozick, R. (1981). Philosophical explanations. Cambridge: Harvard University Press.Google Scholar
  54. Pacuit, E. (2005). A note on some explicit modal logics. In 5th Panhellenic Logic Symposium, Athens, July 2005.Google Scholar
  55. Pacuit, E. (2006). A note on some explicit modal logics. Technical report PP-2006-29, University of Amsterdam. ILLC Publications.Google Scholar
  56. Parikh, R. (1987). Knowledge and the problem of logical omniscience. In Z. Ras & M. Zemankova (Eds.), ISMIS-87 International Symposium on Methodology for Intellectual Systems (pp. 432–439). North-Holland.Google Scholar
  57. Rubtsova, N. (2005). Evidence- for based knowledge S5. In 2005 Summer Meeting of the Association for Symbolic Logic, Logic Colloquium ’05, Athens, 28 July–3 August 2005. Abstract. Association for Symbolic Logic. (2006, June). Bulletin of Symbolic Logic, 12(2), 344–345. doi: 10.2178/bsl/1146620064.Google Scholar
  58. Rubtsova, N. (2006). Evidence reconstruction of epistemic modal logic S5. In Computer Science – Theory and Applications. CSR 2006 (Lecture notes in computer science, Vol. 3967, pp. 313–321). Springer.Google Scholar
  59. Russell, B. (1905). On denoting. Mind, 14, 479–493.CrossRefGoogle Scholar
  60. Russell, B. (1912). The problems of philosophy. London: Williams and Norgate/New York: Henry Holt and Company.Google Scholar
  61. Russell, B. (1919). Introduction to mathematical philosophy. London: George Allen and Unwin.Google Scholar
  62. Stalnaker, R. C. (1996). Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12, 133–163.CrossRefGoogle Scholar
  63. Troelstra, A. S. (1998). Realizability. In S. Buss (Ed.), Handbook of proof theory (pp. 407–474). Amsterdam: Elsevier.CrossRefGoogle Scholar
  64. Troelstra, A. S., & Schwichtenberg, H. (1996). Basic proof theory. Amsterdam: Cambridge University Press.Google Scholar
  65. Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics (Vols. 1 & 2). Amsterdam: North–Holland.Google Scholar
  66. van Dalen, D. (1986). Intuitionistic logic. In D. Gabbay & F. Guenther (Eds.), Handbook of philosophical logic (Vol. 3, pp. 225–340). Dordrecht: Reidel.CrossRefGoogle Scholar
  67. von Wright, G. H. (1951). An essay in modal logic. Amsterdam: North-Holland.Google Scholar
  68. Yavorskaya (Sidon), T. (2006). Multi-agent explicit knowledge. In D. Grigoriev, J. Harrison, & E. A. Hirsch (Eds.), Computer Science – Theory and Applications. CSR 2006 (Lecture notes in computer science, Vol. 3967, pp. 369–380). Springer.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Graduate Center CUNYNew YorkUSA

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