Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism
Wesley H. Holliday
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0
2390,
Berkeley, CA 94720-2390, USA
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W. H. Holliday ( ) Department of Philosophy, University of California
, 314 Moses Hall
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 knowa 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.
1 Introduction
The debate over epistemic closure has been called one of the most significant
disputes in epistemology over the last forty years [45, 256]. The starting point of the
debate is typically some version of the claim that knowledge is closed under known
implication (see Dretske [22]). At its simplest, it is the claim that if an agent knows
and knows that implies , then the agent knows : (K K ( )) K ,
in the language of epistemic logic.
An obvious objection to the simple version of the claim is that an agent with
bounded rationality may know and know that implies , yet not put two and
two together and draw a conclusion about . Such an agent may not even believe
, let alone know it. The challenge of the much-discussed problem of logical
omniscience (see, e.g., Stalnaker [69]; Halpern and Pucella [29]) is to develop a good
theoretical model of the knowledge of such agents.
According to a different objection, made famous in epistemology by Dretske [19]
and Nozick [58] (and applicable to more sophisticated closure claims), knowledge
would not be closed under known implication even for ideally astute logicians
[19, 1010] who always put two and two together and believe all consequences of
what they believe. This objection (explained in Section 2), rather than the logical
omniscience problem, will be our starting point.1
The closure of knowledge under known implication, henceforth referred to as K
after the modal axiom given above, is one closure principle among infinitely many.
Although Dretske [19] denied K, he accepted other closure principles, such as closure
under conjunction elimination, K ( ) K , and closure under disjunction
introduction, K K ( ) (1009). By contrast, Nozick [58] was prepared to give up
closure under conjunction elimination (228), although not closure under disjunction
introduction (230n64, 692).
Dretske and Nozick not only provided examples in which they claimed K fails,
but also proposed theories of knowledge that they claimed would explain the failures,
as discussed below. Given such a theory, one may ask: is the theory committed to
the failure of other, weaker closure principles, such as those mentioned above? Is it
committed to closure failures in situations other than those originally envisioned as
counterexamples to K? The concern is that closure failures may spread, and they may
spread to where no one wants them.
Pressing such a problem of containment has an advantage over other approaches to
the debate over K. It appeals to considerations that both sides of the debate are likely
1Other epistemologists who have denied closure under known implication in the relevant sense include
McGinn [55], Goldman [27], Audi [4], Heller [34], Harman and Sherman [31, 65], Lawlor [47], Becker
[7], and Adams et al. [1].
to accept, rather than merely insisting on the plausibility of K (or of one of its more
sophisticated versions) (...truncated)