The topology of justification

Sergei Artemov, Elena Nogina



Justification Logic is a family of epistemic logical systems obtained from modal logics of knowledge by adding a new type of formula t:F, which is read t is a justification for F. The principal epistemic modal logic S4 includes Tarski’s well-known topological interpretation, according to which the modality 2X is read the Interior of X in a topological space (the topological equivalent of the ‘knowable part of X’). In this paper, we extend Tarski’s topological interpretation from S4 to Justification Logic systems with both modality and justification assertions. The topological semantics interprets t:X as a reachable subset of X (the topological equivalent of ‘test t confirms X’). We establish a number of soundness and completeness results with respect to Kripke topology and the real topology for S4-based systems of Justification Logic.


modal logic; justification Logic; topological semantics; Tarski

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Artemov, S., “Logic of proofs”, Annals of Pure and Applied Logic 67, 1 (1994), 29–59.

Artemov, S., “Operational modal logic”, Technical Report MSI 95–29, Cornell University, 1995.

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

Artemov, S., “Kolmogorov and Gödel’s approach to intuitionistic logic: current developments”, Russian Mathematical Surveys 59, 2 (2004), 203–229.

Artemov, S., “Justified common knowledge”, Theoretical Computer Science 357, 1–3 (2006), 4–22.

Artemov, S., and L. Beklemishev, L., “Provability logic”, pages 189–360 in D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, 2nd ed., vol. 13, Springer, Dordrecht 2005.

Artemov, S., J. Davoren, and A. Nerode, “Modal logics and topological semantics for hybrid systems”, Technical Report MSI 97-05, Cornell University, 1997.

Artemov, S., and E. Nogina, “Logic of knowledge with justifications from the provability perspective”, Technical Report TR-2004011, CUNY Ph.D. Program in Computer Science, 2004.

Artemov, S., and E. Nogina, “Introducing justification into epistemic logic, Journal of Logic and Computation 15, 6 (2005), 1059–1073.

Artemov, S., and E. Nogina, “On epistemic logic with justification”, pages 279–294 in R. van der Meyden (ed.), Theoretical Aspects of Rationality and Knowledge. Proceedings of the Tenth Conference (TARK 2005), June 10–12, 2005, Singapore, National University of Singapore, 2005.

Artemov, S., and E. Nogina, “On topological semantics of justification logic”, Algebraic and Topological Methods in Non-Classical Logics III (TANCL’07) Oxford, England, August 2007.

Bezhanishvili, G., and M. Gehrke, “Completeness of S4 with respect to the real line: revisited”, Annals of Pure and Applied Logic 131 (2005), 287–301.

Dabrowski, A., L.S. Moss, and R. Parikh, “Topological reasoning and the logic of knowledge”, Annals of Pure and Applied Logic 78, 1–3 (1996), 73–110.

Davoren, J., and A. Nerode, “Logics for hybrid systems (invited paper)”, Proceedings of the IEEE, No.7, 88(7) (2000), 985–1010.

de Jongh, D., and G. Japaridze, “Logic of provability”, pages 475–546in S. Buss (ed.), Handbook of Proof Theory, Elsevier, 1998.

Fitting, M., “A semantics for the logic of proofs”, Technical Report TR-2003012, CUNY Ph.D. Program in Computer Science, 2003.

Fitting, M., “The logic of proofs, semantically”, Annals of Pure and Applied Logic 132, 1 (2005), 1–25.

Gödel, K., “Eine Interpretation des intuitionistischen Aussagenkalkuls”, Ergebnisse Math. Kolloq. 4 (1933), 39–40. English translation in: S. Feferman et al. (eds.), Kurt Gödel Collected Works, vol. I, pages 301–303, Oxford University Press, Oxford, Clarendon Press, New York, 1986.

Gödel, K., “Vortrag bei Zilsel”, 1938. Pages 86–113 in S. Feferman (ed.), Kurt Gödel Collected Works, volume III, Oxford University Press, 1995.

Kremer, P., and G. Mints, “Dynamic topological logic”, Annals of Pure and Applied Logic 131 (2005), 133–158.

Kuratowski, K., “Sur l’operation a de l’analysis situs”, Fundamenta Mathematicae 3 (1922), 181–199.

Lewis, C.I., A Survey of Symbolic Logic, University of California Press, 1918.

Lewis, C.I., and C.H. Langford, Symbolic Logic, Dover New York, 1932.

McKinsey, J.C.C., and A. Tarski, “The algebra of topology”, Annals of Mathematics 45 (1944), 141–191.

McKinsey, J.C.C., and A. Tarski, “Some theorems about the sentential calculi of Lewis and Heyting”, The Journal of Symbolic Logic 13 (1948), 1–15.

Mints, G., and T. Zhang, “A proof of topological completeness for S4 in (0,1)”, Annals of Pure and Applied Logic 133, 1–3 (2005), 231–245.

Mkrtychev, A., “Models for the logic of proofs”, pages 266–275 in S. Adian and A. Nerode (eds.), Logical Foundations of Computer Science ‘97, Yaroslavl’, volume 1234 of Lecture Notes in Computer Science, Springer, 1997.

Nogina, E., “Logic of proofs with the strong provability operator”, Technical Report ILLC Prepublication Series ML-94-10, Institute for Logic, Language and Computation, University of Amsterdam, 1994.

Nogina, E., “Grzegorczyk logic with arithmetical proof operators”, Fundamental and Applied Mathematics 2, 2 (1996), 483–499 (in Russian, an English abstract is available at

Nogina, E., “On logic of proofs and provability”, Bulletin of Symbolic Logic 12, 2 (2006), 356.

Nogina, E., “Epistemic completeness of GLA”, Bulletin of Symbolic Logic 13, 3 (2007), 407.

Riesz, F., “Stetigkeitsbegriff und abstrakte mengenlehre”, in Atti del IV Congr. Internat. d. Mat., vol. II, Roma, 1909.

Sidon, T., “Provability logic with operations on proofs”, pages 342–353 in S. Adian and A. Nerode (eds.), Logical Foundations of Computer Science ‘97, Yaroslavl’, volume 1234 of Lecture Notes in Computer Science, Springer, 1997.

Slavnov, S., “On completeness of dynamic topological logic”, Moscow Mathematical Journal 5, 2 (2005), 477–492.

Yavorskaya (Sidon), T., “Logic of proofs and provability”, Annals of Pure and Applied Logic 113, 1–3 (2001), 345–372.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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