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.
Highlights
The debate over epistemic closure has been called “one of the most significant disputes in epistemology over the last forty years” (Kvanvig 2006, 256)
While some consider Fact 8.7 to be a serious problem for sensitivity theories, Fact 8.9 seems even worse for subjunctivist-flavored theories in general: according to the ones we have studied, it is possible for an agent to know the classic example of an unknowable sentence, p ^ ¬Kp (Fitch 1963)
We have investigated an area where epistemology and epistemic logic naturally meet: the debate over epistemic closure, involving two of the most influential views in contemporary epistemology—relevant alternatives and subjunctivism
Summary
The debate over epistemic closure has been called “one of the most significant disputes in epistemology over the last forty years” (Kvanvig 2006, 256). According to a different objection, made famous in epistemology by Dretske (1970) and Nozick (1981) (and applicable to more sophisticated closure claims), knowledge would not be closed under known implication even for “ideally astute logicians” (Dretske 1970, 1010) who always put two and two together and believe all consequences of what they believe This objection (explained in §2), rather than the logical omniscience problem, will be our starting point.∗. Welcome.∗ The theories’ structural features responsible for these closure failures lead (in §8) to serious problems of higher-order knowledge, including the possibility of knowing Fitch-paradoxical propositions (Fitch 1963) Analysis of these results reveals (in §9) that two parameters of a modal theory of knowledge affect whether it preserves closure. Readers who wish to focus on logical ideas should be able to step from definitions to lemmas to theorems, reading the exposition between steps as necessary
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