Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism

Abstract

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having "competently deduced" it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do not always believe, let alone know, the consequences of what they know-a fact that raises the "problem of logical omniscience" that has been central in epistemic logic. This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis's models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By "modal decomposition" I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study.

Lewis on iterated knowledge

“…In Sect. 1 I showed how we could both (a) explain the wide-spread impression that Lewis's account vindicates the iteration principles and (b) confirm that, in fact, Holliday (2015) is right to maintain that the account invalidates them both; the key is to be careful to distinguish which parts of the account describe the dependence of knowledge attributions on the attributor's context and which parts describe the dependence of knowledge attributions on the subject's situation. In Sect.…”

“…The resulting system is essentially a special case of Holliday's (2015) formalization of Lewis. 12 Simplifying slightly, Holliday's frames are, in our notation, the triples \W; E; A [ ; the rule of actuality is built in by requiring that w 2 AðwÞ. Our models are less general, because defining A in terms of S, R B , and C imposes additional constraints.…”

“…, Holton (2003), and Greco (2014), who all trace the claim to Lloyd Humberstone] maintain that Lewis's account validates an S5 epistemic logic, which would mean that it is committed to implausibly strong iteration principles for knowledge; by contrast, Holliday (2015) maintains that the knowledge iteration principles are invalid in Lewis's system. By distinguishing carefully between what is contributed by the conversational context of the agents attributing knowledge and what is contributed by the situation of the subject to whom knowledge is attributed, we can resolve this dispute in Holliday's favour: Lewis's system allows counterexamples to both the KK principle (that whenever someone knows something, they know that they know it) and what I will call the K:K principle (that whenever someone doesn't know something, they know that they don't know it).…”

Counterfactual theories of knowledge and the notion of actuality

“…Note that the proof is constructive. Given any (Holliday, 2012a)-model with a well-understood closeness relation between single worlds, one can construct a unique model with a closeness relation between world pairs, where the latter closeness relation is completely determined by the former closeness relation. Moreover, given any model with a closeness relation between world pairs, one can construct a unique model with a closeness relation between single worlds, where the latter is completely determined by the former.…”

Epistemic closure, assumptions and topics of inquiry

“…Yet, Jack does not have warrant for believing that he will not win the lottery just because of his modest salary and the little money in his bank account. 17 For a more detailed discussion, seeHawthorne (2004) andHolliday (2014).…”