JTB Epistemology and the Gettier Problem in the Framework of Topological Epistemic Logic

Abstract

Traditional epistemology of knowledge and belief can be succinctly characterized as 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: 1 Independent Researcher, Tsukuba/Japan. 2 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".