Knowing What Follows: Epistemic Closure and Epistemic Logic
Wesley Halcrow Holliday

Abstract:
The starting point of this dissertation is a central question in epistemology and epistemic logic, statable roughly as follows: is it a necessary condition of an agent's knowing some propositions P_1,…,P_n that she has done enough empirical investigation of the world so that she could know any logical consequence of {P_1,…,P_n} without further empirical investigation? An affirmative answer amounts to a claim of full epistemic closure: the set of propositions that an agent knows or could know without further empirical investigation is closed under multi-premise logical consequence.

The idea of full epistemic closure creates a tension with an attractive idea of fallibilism about knowledge. According to fallibilism, for an agent to know a true empirical proposition P, it is not required that her evidence rules out every possible way in which P could be false and some incompatible alternative hypothesis could obtain. If such a feat were required, agents would know almost nothing. Yet full epistemic closure requires for knowledge of P that an agent does know--or could know without further empirical investigation--the negation of every such alternative hypothesis, assuming she knows that these hypotheses are incompatible with P. Although not a formal contradiction between closure and fallibilism, this is a tension to say the least.

In this dissertation, I explore the extent to which it is possible to make fallibilism compatible with closure. I begin by formalizing a family of fallibilist theories of knowledge in models for epistemic logic. Model-theoretic methods are used to characterize the closure properties of knowledge according to different fallibilist pictures, identify the structural features of these pictures that correspond to closure properties, transform models of one theory into models of another, prove impossibility results, and ultimately find a middle way between full closure and no closure for fallibilism.  

I argue that the standard versions of "Fallibilism 1.0" each face one of three serious problems related to closure: the Problem of Vacuous Knowledge, the Problem of Containment, and the Problem of Knowledge Inflation. To solve these problems, I propose a new framework for Fallibilism 2.0: the Multipath Picture of Knowledge. This picture is based on taking seriously the idea that there can be multiple paths to knowing a complex claim about the world. An overlooked consequence of fallibilism is that these multiple paths to knowledge may involve ruling out different sets of alternatives, which should be represented in our picture of knowledge. I argue that the Multipath Picture of Knowledge is a better picture for all fallibilists, whether for or against full closure. Yet I also argue that only by accepting less than full closure can we solve the closure-related problems that plague previous versions of fallibilism.