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THE LOGIC OF JUSTIFICATION

Published online by Cambridge University Press:  01 December 2008

SERGEI ARTEMOV
SERGEI ARTEMOV*
Affiliation:
Programs in Computer Science, Mathematics, and Philosophy, Graduate Center CUNY
*
*PROGRAMS IN COMPUTER SCIENCE, MATHEMATICS, AND PHILOSOPHY GRADUATE CENTER CUNY 365 FIFTH AVENUE NEW YORK, NY 10016 E-mail:sartemov@gc.cuny.edu
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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.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 1 , Issue 4 , December 2008 , pp. 477 - 513
Copyright
Copyright © Association for Symbolic Logic 2008

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References

BIBLIOGRAPHY

Antonakos, E. (2007). Justified and common knowledge: limited conservativity. In Artemov, S., and Nerode, A., editors. Logical Foundations of Computer Science. International Symposium, LFCS 2007, New York, NY, June 2007, Proceedings, Volume 4514 of Lecture Notes in Computer Science. Berlin: Springer, pp. 1–11.Google Scholar
Artemov, S. (1995). Operational Modal Logic. Technical Report MSI 95-29. Ithaca, NY: Cornell University.Google Scholar
Artemov, S. (1999, August). Understanding Constructive Semantics. Spinoza lecture for European Association for logic, language and information. Utrecht.Google Scholar
Artemov, S. (2001). Explicit provability and constructive semantics. Bulletin of Symbolic Logic, 7(1), 1–36.CrossRefGoogle Scholar
Artemov, S. (2006). Justified common knowledge. Theoretical Computer Science, 357(1–3), 4–22.CrossRefGoogle Scholar
Artemov, S. (2007). On two models of provability. In Gabbay, D. M., Zakharyaschev, M., and Goncharov, S. S., editors. Mathematical Problems From Applied Logic II. New York, NY: Springer, pp. 1–52.Google Scholar
Artemov, S. (2008). Symmetric logic of proofs. In Avron, A., Dershowitz, N., and Rabinovich, A., editors. Pillars of Computer Science, Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Volume 4800 of Lecture Notes in Computer Science. Berlin: Springer, pp. 58–71.Google Scholar
Artemov, S., & Beklemishev, L. (2005). Provability logic. In Gabbay, D., and Guenthner, F., editors. Handbook of Philosophical Logic (second edition), Vol. 13. Dordrecht, The Netherlands: Springer, pp. 189–360.Google Scholar
Artemov, S., Kazakov, E., & Shapiro, D. (1999). Epistemic Logic With Justifications. Technical Report CFIS 99–12. Ithaca, NY: Cornell University.Google Scholar
Artemov, S., & Kuznets, R. (2006). Logical omniscience via proof complexity. In Computer Science Logic 2006, Volume 4207. Springer Lecture Notes in Computer Science. Berlin: Springer, pp. 135–149.Google Scholar
Artemov, S., & Nogina, E. (2004). Logic of Knowledge With Justifications From the Provability Perspective. Technical Report TR-2004011, New York: PhD Program in Computer Science CUNY.Google Scholar
Artemov, S., & Nogina, E. (2005). Introducing justification into epistemic logic. Journal of Logic and Computation, 15(6), 1059–1073.CrossRefGoogle Scholar
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
Boolos, G. (1993). The Logic of Provability. Cambridge, MA: Cambridge University Press.Google Scholar
Brezhnev, V. (2000). On Explicit Counterparts of Modal Logics. Technical Report CFIS 2000–05. Ithaca, NY: Cornell University.Google Scholar
Brezhnev, V., & Kuznets, R. (2006). Making knowledge explicit: how hard it is. Theoretical Computer Science, 357(1–3), 23–34.CrossRefGoogle Scholar
Dean, W., & Kurokawa, H. (2007). From the knowability paradox to the existence of proofs. Synthese. Submitted.Google Scholar
Dean, W., & Kurokawa, H. (2008). The knower paradox and the quantified logic of proofs. In Hieke, A., editor. Proceedings of the Austrian Ludwig Wittgenstein Society Vol. 31. Kirchberg am Wechsel, 2008.Google Scholar
Dretske, F. (1971). Conclusive reasons. Australasian Journal of Philosophy, 49, 1–22.CrossRefGoogle Scholar
Dretske, F. (2005). Is knowledge closed under known entailment? the case against closure. In Steup, M., and Sosa, E., editors. Contemporary Debates in Epistemology. Malden, MA: Blackwell, pp. 13–26.Google Scholar
Fagin, R., & Halpern, J. (1985) Belief, awareness, and limited reasoning: preliminary report. In Joshi, A. K., editor Proceedings of the Ninth International Joint Conference on Artificial Intelligence (IJCAI-85). Los Altos, CA: Morgan Kaufmann, pp. 491–501.Google Scholar
Fagin, R., & Halpern, J. (1988). Belief, awareness, and limited reasoning. Artificial Intelligence, 34(1), 39–76.CrossRefGoogle Scholar
Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning About Knowledge. Cambridge, MA: MIT Press.Google Scholar
Fitting, M. (2003). A Semantics for the Logic of Proofs. Technical Report TR-2003012, PhD Program in Computer Science. City University of New York.Google Scholar
Fitting, M. (2005). The logic of proofs, semantically. Annals of Pure and Applied Logic, 132(1), 1–25.CrossRefGoogle Scholar
Fitting, M. (2007, February). Intensional Logic. Stanford Encyclopedia of Philosophy Available from: http://plato.stanford.edu.Google Scholar
Fitting, M., & Mendelsohn, R. L. (1998). First-Order Modal Logic. Dordrecht, The Netherlands: Kluwer Academic.CrossRefGoogle Scholar
Frege, G. (1952). On sense and reference. In Geach, P., and Black, M., editors. Translations of the Philosophical Writings of Gottlob Frege. Oxford: Blackwell, pp. 56–78.Google Scholar
Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23, 121–123.CrossRefGoogle Scholar
Gödel, K. (1986). Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse Math. Kolloq., 14, 39–40, 1933. English translation In Feferman, S. et al., editors. Kurt Gödel Collected Works. Vol. 1. Oxford: Oxford University Press; New York, NY: Clarendon Press, pp. 301–303.Google Scholar
Gödel, K. (1995). Vortrag bei Zilsel/Lecture at Zilsel's (*1938a). In Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C., and Solovay, R. M., editors. Unpublished Essays and Lectures, Volume III of Kurt Gödel Collected Works. New York: Oxford University Press, pp. 86–113.Google Scholar
Goldman, A. (1967). A causal theory of knowing. The Journal of Philosophy, 64, 335–372.CrossRefGoogle Scholar
Goris, E. (2007). Explicit proofs in formal provability logic. In Artemov, S., and Nerode, A., editors. Logical Foundations of Computer Science. International Symposium, LFCS 2007, New York, NY, June 2007, Proceedings, Volume 4514 of Lecture Notes in Computer Science. Berlin: Springer, pp. 241–253.Google Scholar
Hendricks, V. F. (2003). Active Agents. Journal of Logic, Language and Information, 12(4), 469–495.CrossRefGoogle Scholar
Hendricks, V. F. (2005). Mainstream and Formal Epistemology. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Heyting, A. (1934). Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie. Berlin, Germany: Springer.Google Scholar
Hintikka, J. (1962). Knowledge and Belief. Ithaca, NY: Cornell University Press.Google Scholar
Hintikka, J. (1975). Impossible possible worlds vindicated. Journal of Philosophical Logic, 4, 475–484.CrossRefGoogle Scholar
Kleene, S. (1945). On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, 10(4), 109–124.CrossRefGoogle Scholar
Krupski, N. V. (2006). On the complexity of the reflected logic of proofs. Theoretical Computer Science, 357(1), 136–142.CrossRefGoogle Scholar
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
Krupski, V. N. (2006). Referential logic of proofs. Theoretical Computer Science, 357(1), 143–166.CrossRefGoogle Scholar
Kuznets, R. (2000). On the complexity of explicit modal logics. In Clote, P. G. and Schwichtenberg, H., eds. Computer Science Logic 2000, Volume 1862 of Lecture Notes in Computer Science. Berlin: Springer-Verlag, pp. 371–383.Google Scholar
Kuznets, R. (2008). Complexity Issues in Justification Logic. PhD thesis, CUNY Graduate Center. Available from: http://kuznets.googlepages.com/PhD.pdf. Accessed May 2008.Google Scholar
Lehrer, K., & Paxson, T. (1969). Knowledge: undefeated justified true belief. The Journal of Philosophy, 66, 1–22.CrossRefGoogle Scholar
Luper, S. (2005). The Epistemic Closure Principle. Stanford Encyclopedia of Philosophy.Google Scholar
McCarthy, J., Sato, M., Hayashi, T., & Igarishi, S. (1978). On the Model Theory of Knowledge. Technical Report STAN-CS-78–667. Stanford, CA: Stanford University Press.CrossRefGoogle Scholar
Meyer, J.-J. Ch., & van der Hoek, W. (1995). Epistemic Logic for AI and Computer Science. New York: Cambridge University Press.CrossRefGoogle Scholar
Milnikel, R. (2007). Derivability in certain subsystems of the logic of proofs is -complete. Annals of Pure and Applied Logic, 145(3), 223–239.CrossRefGoogle Scholar
Mkrtychev, A. (1997). Models for the logic of proofs. In Adian, S., and Nerode, A., editors. Logical Foundations of Computer Science ‘97, Yaroslavl’, Volume 1234 of Lecture Notes in Computer Science. Berlin: Springer, pp. 266–275.Google Scholar
Moses, Y. (1998). Resource-bounded knowledge. In Vardi, M., editor. Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge, March 7–9, 1988, Pacific Grove, California, Los Altos, CA: Morgan Kaufmann Publishers, pp. 261–276.Google Scholar
Neale, S. (1990). Descriptions. Cambridge, MA: MIT Press Books.Google Scholar
Nozick, R. (1981). Philosophical Explanations. Cambridge, MA: Harvard University Press.Google Scholar
Pacuit, E. (2005, July). A Note on Some Explicit Modal Logics. 5th Panhellenic Logic Symposium, Athens. University of Athens.Google Scholar
Pacuit, E. (2006). A Note on Some Explicit Modal Logics. Technical Report PP-2006-29. Amsterdam: University of Amsterdam. ILLC Publications.Google Scholar
Parikh, R. (1987). Knowledge and the problem of logical omniscience. In Ras, Z., and Zemankova, M., editors. ISMIS-87 (International Symposium on Methodology for Intellectual Systems). Amsterdam: North-Holland, pp. 432–439.Google Scholar
Rubtsova, N. (2005). Evidence-Based Knowledge for S5. Logic Colloquium 2005, Athens: University of Athens.Google Scholar
Rubtsova, N. (2006). Evidence reconstruction of epistemic modal logic S5. In Grigoriev, D., Harrison, J., Hirsch, E. A., editors.Computer Science—Theory and Applications. CSR 2006, Volume 3967 of Lecture Notes in Computer Science. Berlin: Springer, pp. 313–321.Google Scholar
Russell, B. (1905). On denoting. Mind, 14, 479–493.CrossRefGoogle Scholar
Russell, B. (1912). The Problems of Philosophy. London: Williams and Norgate; New York, NY: Henry Holt and Company.Google Scholar
Russell, B. (1919) Introduction to Mathematical Philosophy. London: George Allen and Unwin.Google Scholar
Stalnaker, R. C. (1996). Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12, 133–163.CrossRefGoogle Scholar
Troelstra, A. S. (1998). Realizability. In Buss, S., editor. Handbook of Proof Theory, Amsterdam, The Netherlands: Elsevier, pp. 407–474.CrossRefGoogle Scholar
Troelstra, A. S., & Schwichtenberg, H. (1996). Basic Proof Theory. Amsterdam, The Netherlands: Cambridge University Press.Google Scholar
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in Mathematics. Vols. 1, 2. Amsterdam, The Netherlands: North-Holland.Google Scholar
van Dalen, D. (1986). Intuitionistic logic. In Gabbay, D., and Guenther, F., editors. Handbook of Philosophical Logic. Vol. 3. Dordrecht, The Netherlands: Reidel pp. 225–340.CrossRefGoogle Scholar
von Wright, G. H. (1951). An Essay in Modal Logic. Amsterdam, The Netherlands: North-Holland.Google Scholar
Yavorskaya (Sidon), T. (2006). Multi-agent explicit knowledge. In Grigoriev, D., Harrison, J., and Hirsch, E. A., editors. Computer Science—Theory and Applications. CSR 2006, Volume 3967 of Lecture Notes in Computer Science. Berlin: Springer, pp. 369–380.Google Scholar