**Abstract**

Traditional epistemology of knowledge and belief can be succinctly characterized as justified true belief (JTB) epistemology, namely by the thesis that knowledge is justified true belief, i.e., K = JTB. Since Gettier’s (1963) classical paper, JTB-epistemology has come under heavy attack. The aim of this paper is to study JTB-epistemology and Gettier’s criticism of it in the framework of topological epistemic logic. In this topological framework, Gettier situations, for which knowledge does not coincide with true justified belief, occur for formal reasons, i.e., there are models for which K ≠ JTB. On the other hand, topological logic offers natural models of JTB, i.e., models for which knowledge coincides with true justified belief.Moreover, for every model of Stalnaker’s “combined logic KB of knowledge and belief” a canonical JTB model (its JTB doppelganger) can be constructed that is free of Gettier situations. In brief, the traditional JTB-epistemology can be shown to be a simplification of a more complex epistemological account of knowledge and justified true belief that assumes that these two concepts may differ. Further, for all models of Stalnaker’s KB-logic, Gettier situations turn out to be topologically exceptional events in a precise sense, i.e., they are nowhere dense situations. This entails that Gettier situations are doxastically and epistemologically invisible in the sense that they can be neither known nor believed with respect to the knowledge operator and the belief operator of the models involved. In sum, the version of topological epistemic logic presented in this paper leads to a partial rehabilitation of the traditional JTB-account: Gettier situations, where knowledge does not coincide with justified true belief, are characterized topologically as anomalies or exceptional situations. On the other hand, Gettier situations necessarily occur for most universes of possible worlds. Only for a special subclass of universes (epistemically characterized by a rather strong concept of knowledge and topologically characterized as the class of nodec spaces) can Gettier situations be avoided completely. This description amounts to the thesis that, in general, JTB-epistemology is false. JTB remains correct, however, for a special class of universes of possible worlds, namely, nodec spaces. Moreover, in a precise topological sense, any topological space whatsoever can be shown to be “almost” a nodec space. This fact renders the assertion plausible that the classical JTB account is “almost correct.”

**Keywords**

Topological epistemic logic, JTB-epistemology, Gettier problem, Justified belief, Epistemic and doxastic invisibility

**References**

- Aiello, M., Pratt-Hartmann, I., and van Benthem, J. (2007). Handbook of Spatial Logics, New York, Heidelberg, Berlin: Springer.
- Baltag, A., Bezhanishvili, N., Özgün, A., and Smets, S. (2017). The topology of full and weak belief. In H. Hansen, S. E. Murray, M. Sadrzadeh, and H. Zeebat. (Eds.), Lecture Notes in Computer Science 10148, TbiLLC 2015, 205–228.
- Baltag, A., Bezhanishvili, N., Özgün, A., and Smets, S. (2019). A topological approach to full belief, Journal of Philosophical Logic, 48, 205–244.
- Baltag, A., Bezhanishvili, N., Özgün, A., Smets, S. (2022). Justified belief, knowledge and the topology of evidence, Synthese, 200, 512.
- Bezhanishvili, G., Esakia, L., and Gabelaia, D. (2004). Modal logics of submaximal and nodec spaces, Festschrift for D. de Jongh, Dedicated on his 65th Birthday, University of Amsterdam.
- Blackburn, P., de Rijke, M., and Venema, Y. (2001). Modal Logic, Cambridge: Cambridge University Press.
- Bobzien, S. (2012). If it’s clear, then it’s clear, or is it? Higher-order vagueness and the S4 axiom. In K. Ierdiakanou, and B. Morison (Eds.), Episteme etc., Essays in Honour of Jonathan Barnes, Oxford: Oxford University Press, 189–221.
- Borges, R., de Almeida, C., and Klein, P.D. (Eds.) (2017). Explaining Knowledge: New Essays on the Gettier Problem, Oxford: Oxford University Press, 179–191.
- Carnap, R. (1950). Empiricism, semantics, and ontology, Revue International de Philosophie, 4, 20–40.
- Chellas, B. F. (1980). Modal Logic, Cambridge: Cambridge University Press.
- Fano, V., and Graciani, P. (2021). A working hypothesis for the logic of radical ignorance, Synthese, 199, 601–616.
- Fine, K. (2018). Ignorance of ignorance, Synthese, 195, 4031–4045.
- Gettier, E. (1963). Is justified true belief knowledge? Analysis, 23, 121–123.
- Ichikawa, J. J., and Steup, M. (2017). The Analysis of Knowledge (2017 edition), Stanford Encyclopedia of Philosophy.
- Jankovic, D. and Hamlett, T. R. (1990). New topologies from old via ideals, The American Mathematical Monthly, 97(4), 295–310.
- Kuratowski, K. (1922). Sur l’opération A- de l’analysis situs, Fundamenta Mathematicae, 3, 182–199.
- Kuratowski, K. (1966). Topology, volume I, New York: Academic Press.
- Kuratowski, K., and Mostowski, A. (1976). Set Theory. With an Introduction to Descriptive Set Theory, North-Holland, Amsterdam, New York, Oxford: North-Holland.
- Lenzen, W. (1979). Epistemologische Betrachtungen zu [S4, S5], Erkenntnis, 14(1), 33–56.
- McKinsey, J. C. C., and Tarski, A. (1944). The algebra of topology, Annals of Mathematics, Second Series, 45(1), 141–191.
- Mormann, T. (2023). Completeness and doxastic plurality for topological operators of knowledge and belief, Erkenntnis, online first.
- Njåstad, O. (1965). On some classes of nearly open sets, Pacific Journal of Mathematics, 15(3), 961–970.
- Reilly, I., and Vamanamurthy, M. K. (1985). On α-continuity in topological spaces, Acta Mathematica Hungarica, 45, 27–32.
- Sarsak, M., Ganster, M., and Steiner, M. (2006). On Si-metacompact spaces, Tatra Mountains Mathematical Publications, 34, 1–7.
- Steen, L.A., and Seebach, Jr, J. A. (1978). Counterexamples in Topology, Second Edition, New York, Heidelberg, Berlin: Springer.
- Sellars, W. (1975). Epistemic principles. In H. Castaneda (Ed.), Action, Knowledge, and Reality, Minneapolis: Bobbs-Merrill (reprinted in E. Sosa, J. Kim, J. Fantl, and M. McGrath (Eds.), Epistemology: An anthology, (2008), Oxford: Blackwell, 99–108).
- Stalnaker, R. (2006). On logics of knowledge and belief, Philosophical Studies, 128, 169–199.
- Steinsvold, C. (2006). Topological models of belief, PhD Thesis, CUNY. Graduate Center.
- Turri, J. (2012). Is knowledge justified true belief?, Synthese, 184(2), 247–259.
- van Douwen, E. K. (1993). Applications of maximal topologies, Topology and Its Applications, 51, 125–139.
- Willard, S. (2004). General Topology, New York: Dover Publications.
- Williamson, T. (2013). Gettier cases in epistemic logic, Inquiry, 56(1), 1–14.
- Williamson, T. (2015). A note on Gettier cases in epistemic logic, Philosophical Studies, 172(1), 129–140.
- Yap, A. (2014). Idealization, epistemic logic, and epistemology, Synthese, 191(14), 3351–3366.
- Zagzebski, L. (2017). The lesson of Gettier. In R. Borges, C. de Almeida, and P. D. Klein (Eds.), Explaining Knowledge. New Essays on the Gettier Problem, Oxford: Oxford University Press, 179–191.
- Zeman, J. J. (1969). Modal systems in which necessity is “factorable,” Notre Dame Journal of Formal Logic, 10(3), 247–256.