Most of us think we can safely enlarge our knowledge base by accepting things that are entailed by (or logically implied by) things we know. Roughly speaking, the set of things we know is closed under entailment (or under deduction or logical implication), so we know that a given claim is true upon recognizing, and accepting thereby, that it follows from what we know. This is not to say that our usual way of adding to our knowledge is simply to recognize and accept what follows from what we already know. Obviously much more is involved. For instance, we gather data and construct explanations of those data, and under suitable circumstances we learn from others. More to the point at hand, when we claim that we know, of some proposition, that it is true, that claim is itself subject to error; often, seeing what follows from a knowledge claim prompts us to reassess and even withdraw our claim, instead of concluding, of the things that follow from it, that we know that they are true. Still, it seems reasonable to think that if we do know that some proposition is true then we are in a position to know, of the things that follow from it, that they, too, are true. However, some theorists have denied that knowledge is closed under entailment. The arguments against closure include the following:

The argument from the analysis of knowledge: given the correct analysis, knowledge is not closed, so it isn't. For example, if the correct analysis includes a tracking condition, then closure fails.

The argument from nonclosure of knowledge modes: since the modes of gaining, preserving or extending knowledge, such as perception, testimony, proof, memory, indication, and information are not individually closed, neither is knowledge.

The argument from unknowable (or not easily knowable) propositions: certain sorts of propositions cannot be known (without special measures); given closure, they could be known (without special measures), by deducing them from mundane claims we known, so knowledge is not closed.

The argument from skepticism: skepticism is false but it would be true if knowledge were closed, so knowledge is not closed.

While proponents of closure have responses to these arguments, they also argue, somewhat in the style of G. E. Moore (1959), that closure itself is a firm datum—it is obvious enough to rule out any understanding of knowledge or related notions that undermines closure.

A closely related idea is that it is rational (justifiable) for us to believe anything that follows from what it is rational for us to believe. This idea is intimately related to the thesis that knowledge is closed, since, according to some theorists, knowing p entails justifiably believing p. If knowledge entails justification, closure failure of the latter might lead to closure failure of the former.

1. The Closure Principle

2. The Argument From the Analysis of Knowledge

2.1 Closure Fails Due to the Tracking Condition on Knowledge

2.2 Closure Fails on a Relevant Alternatives Approach

2.3 Closure and Reliabilism

3. The Argument From Nonclosure of Knowledge Modes

3.1 Knowledge Modes and Nonclosure

3.2 Responses to Dretske

4. The Argument From Not (Easily) Knowable Propositions

4.1 The Argument from Limiting Propositions

4.2 The Argument from Lottery Propositions

5. The Argument From Skepticism

5.1 Skepticism and Antiskepticism

5.2 Tracking and Skepticism

5.3 Safe Indication and Skepticism

5.4 Contextualism and Skepticism

6. Closure of Rational Belief

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1. Knowledge Closure

Precisely what is meant by the claim that knowledge is closed under entailment? One response is that the following straight principle of closure of knowledge under entailment is true:

SP: If person S knows p, and p entails q, then S knows q.

The conditional involved in the straight principle might be the material conditional, the subjunctive conditional, or entailment, yielding three possibilities, each stronger than the one before:

SP1: S knows p and p entails q only if S knows q.

SP2: If S were to know something, p, that entailed q, S would know q.

SP3: It is necessarily the case that: S knows p and p entails q only if S knows q.

However, each version of the straight principle is false, since we can know one thing, p, but fail to see that p entails q, or for some other reason fail to believe q. Since knowledge entails belief (according to nearly all theorists), we fail to know q. A less obvious worry is that we might reason badly in coming to believe that p entails q. Perhaps we think that p entails q because we think everything entails everything, or because we have a warm tingly feeling between our toes. Hawthorne (2005) raises the possibility that, in the course of grasping that p entails q, S will cease to know p. He also notes that SP1 is defensible on the (deviant) assumption that a thought, p, is equivalent to another, q, if p and q hold in all of the same possible worlds. Suppose p entails q. Then p is equivalent to the conjunction of p and q, and so the thought p is identical to the thought p and q. Hence in knowing p S knows p and q. Assuming that, in knowing p and q, S knows p and S knows q, then when S knows p S knows q, as SP1 says.

The straight principle needs qualifying, but this should not concern us so long as the qualifications are natural given the idea we are trying to capture, namely, that we can extend our knowledge by recognizing, and accepting thereby, things that follow from something that we know. The qualifications embedded in the following principle (construed as a material conditional) seem natural enough:

K: If, while knowing p, S believes q because S knows that p entails q, then S knows q.

As Williamson (2000) notes, the idea that we can extend our knowledge by applying deduction to what we know supports a closure principle that is stronger than K. It is a principle that says we know things we believe on the grounds that they are jointly implied by several separate known items. Suppose I know Mary is tall and I know Mary is left handed. K does not authorize my putting these two pieces of knowledge together so as to know that Mary is tall and left handed. But the following generalized closure principle covers deductions involving separate known items:

GK: If, while knowing various propositions, S believes p because S knows that they entail p, then S knows p.

Some theorists distinguish between something they call “single premise” and something they call “multiple-premise closure”. Such theorists would deny that K captures “single premise” closure, because K says that S knows q if S knows that two things are true: that p is true as well as that p entails q. The “single premise” closure principle is usually formulated roughly as follows (following Williamson 2002 and Hawthorne 2004):

SPK: If, while knowing p, S believes q by competently deducing q from p, then S knows q.

However, it is far from clear that one may competently deduce q from p without relying on any knowledge aside from p. Fortunately, it seems that nothing hinges on this possibility, except perhaps for people interested in whether we can identify something that can appropriately be labeled “single premise closure principle”.

Proponents of closure might accept both K and GK, perhaps further qualified in natural ways (but they might not: see the concerns about justification closure raised in section 6). By contrast, Fred Dretske and Robert Nozick reject K and therefore GK as well. They reject any closure principle, no matter how narrowly restricted, that warrants our knowing that skeptical hypotheses (e.g., I am a brain in a vat) are false on the basis of mundane knowledge claims (e.g., I am not in a vat). In addition to rejecting K and GK, they deny knowledge closure across instantiation and simplification, but not across equivalence (Nozick 1981: 227–229):

KI: If, while knowing that all things are F, S believes a particular thing a is F because S knows it is entailed by the fact that all things are F, then S knows a is F.

KS: If, while knowing p and q, S believes q because S knows that q is entailed by p and q, then S knows q.

KE: If, while knowing p, S believes q because S knows q is equivalent to p, then S knows q.

Let us turn to their arguments.

2. The Argument From the Analysis of Knowledge

The argument from the analysis of knowledge says that the correct account of knowledge leads to K failure. We can distinguish two versions. According to the first version, K fails because knowledge requires belief tracking. According to the second, any relevant alternatives account, such as Dretske's and Nozick's, leads to K failure. According to Dretske (2003: 112–3; 2005: 19), any relevant alternatives account leads “naturally” but “not inevitably” to K failure.

2.1 Closure Fails Due to the Tracking Condition on Knowledge

In rough outline, the first version involves defending say Dretske's or Nozick's tracking analysis of knowledge, then showing that it undermines K (versions of the tracking account are also defended by Becker 2009, by Murphy and Black 2007, and by Roush 2005, the last of whom modifies the tracking account so as to preserve closure; for criticisms of Rouse see Brueckner 2012). We can skip the defense, which consists largely in showing that tracking does a better job than competitors in dealing with our epistemic intuitions about cases of purported knowledge. We may also simplify the analyses. According to Nozick, to know p is, very roughly (and ignoring his thoroughly discredited fourth condition for knowledge, criticized, e.g, in Luper 1984 and 2009 and in Kripke 2011), to have a belief p which meets the following condition (‘BT’ for belief tracking):

BT: were p false, S would not believe p.

That is, in the close worlds to the actual world in which not-p holds, S does not believe p. The actual world is one's situation as it is when one arrives at the belief p. BT requires that in all nearby not-p worlds S fails to believe p. (The semantics of subjunctive conditionals is clarified in Stalnaker 1968, Lewis 1973, and modified by Nozick 1981 note 8.) On Dretske's view knowing p is roughly a matter of having a reason R for believing p which meets the following condition (‘CR’ for conclusive reason):

CR: were p false, R would not hold.

That is, in the close worlds to the actual world in which not-p holds, R does not. When R meets this condition, Dretske says R is a conclusive reason for believing p.

Dretske pointed out (2003, n. 9; 2005, n. 4) that his view does not face one of the objections which Saul Kripke (2011, 162-224; Dretske had access to a draft circulated prior to publication) deploys against Nozick's account. Suppose I am driving through a neighborhood in which, unbeknownst to me, papier-mâché barns are scattered, and I see that the object in front of me is a barn. I also notice that it is red. Because I have barn-before-me percepts, I believe barn: the object in front of me is a (ordinary) barn (the example is attributed to Ginet in Goldman 1976). Our intuitions suggest that I fail to know barn. And so say BT and CR. But now suppose that the neighborhood has no fake red barns; the only fake barns are blue. (Call this the red barn case.) Then on Nozick's view I can track the fact that there is a red barn, since I would not believe there was a red barn (via my red-barn percepts) if no red barn were there, but I cannot track the fact that there is a barn, since I might believe there was a barn (via blue-barn percepts) even if no barn were there. Dretske said that this juxtaposition, in which I know something yet fail to know a second thing that is intimately related to the first (there being a red barn, which I know, entails there being a barn, which I do not), “is an embarrassment,” and in this respect, he thought, his view is superior to Nozick's. Let R, my basis for belief, be the fact that I have red-barn percepts. If no barn were there, R would fail to hold, so I know a barn is there. Further, if no red barn were there, R would still fail to hold, so I know a red barn is there. So Dretske can avoid the objectionable juxtaposition. Still, it is surprising that Dretske cited the red barn case as the basis for preferring his version of tracking over Nozick's. First, Dretske himself accepted juxtapositions of knowledge and ignorance that are at least equally bizarre, as we shall see. Second, Nozick avoided the very juxtaposition Dretske discussed by restating his account to make reference to the methods via which we come to believe things (Hawthorne 2005). On a more polished version of his account, Nozick said that to know p is, roughly, to have a belief p, arrived at through a method M, which meets the following condition (‘BMT’ for belief method tracking):

BMT: were p false, S would not believe p via M.

If no red barn were there I would believe neither that there was a barn, nor that there was a red barn, via red-barn percepts.

Third, the red barn case is one about which intuitions will vary. It is not obvious that I do know there is a red barn in the circumstances Dretske sketches, which differ from those in Ginet's original barn case (where I fail to know barn) only in the stipulations that I see a red barn and that none of the barn simulacra are red. What is more, both Dretske's and Nozick's accounts have the odd implication that I know there is a barn if I base my belief on my red barn percepts yet I fail to know this if, in basing it on my barn percepts, I ignore the barn's color. Presumably the barn's color is not relevant to its being a barn.

The tracking accounts permit counterexamples to K. Dretske's well known illustration is the zebra case: suppose you are at a zoo in ordinary circumstances standing in front of a cage marked ‘zebra’; the animal in the cage is a zebra, and you believe zeb, the animal in the cage is a zebra, because you have zebra-in-a-cage visual percepts. It occurs to you that zeb entails not-mule, it is not the case that the animal in the cage is a cleverly disguised mule rather than a zebra. You then believe not-mule by deducing it from zeb. What do you know? You know zeb, since, if zeb were false, you would not have zebra-in-a-cage visual percepts; instead, you would have empty-cage percepts, or aardvark-in-a-cage percepts, or the like. Do you know not-mule? If not-mule were false, you would still have zebra-in-a-cage visual percepts (and you would still believe zeb, and you would still believe not-mule by deducing it from zeb). So you do not know not-mule. But notice that we have:

You know zeb

You believe not-mule by recognizing that zeb entails not-mule

You do not know not-mule.

In view of (a)-(c), we have a counterexample to K, which entails that if (a) you know zeb, and (b) you believe not-mule by recognizing that zeb entails not-mule, then you do know not-mule, contrary to (c).

Having rejected K, and denying that we know things like not-mule, Nozick also had to deny closure across simplification. For if some proposition p entails another proposition q, then p is equivalent to the conjunction p & q; accordingly, given closure across equivalence, which Nozick accepted, if we know zeb we can know the conjunction zeb & not-mule, but if we also accept closure across simplification, we will be able to know not-mule.

In response to this first version of the argument from the analysis of knowledge, some theorists (e.g., Luper 1984, BonJour 1987, DeRose 1995) argued that K has great plausibility in its own right (which Dretske acknowledged in 2005: 18) so it should be abandoned only in the face of compelling reasons, yet there are no such reasons.

To show there are no compelling reasons to abandon K, theorists have provided accounts of knowledge that (a) handle our intuitions at least as successfully as the tracking analyses and yet (b) underwrite K. One way to do this is to weaken the tracking analysis so that we know things that we track or that we believe because we know that they follow from things that we track (this sort of option has been turned against Nozick by various theorists; Roush defends it in 2005, 41-51). Another approach is as follows. Knowing p is roughly a matter of having a reason R for believing p which meets the following condition (‘SI’ for safe indication):

SI: if R held, p would be true.

SI requires that p be true in the nearby R worlds. When R meets this condition, let us say that R is a safe indicator that p is true. (Different versions of the safety condition have been defended; see, for example, Luper 1984; Sosa 1999, 2003, 2007, 2009; Williamson 2000; and Pritchard 2007.) SI is the contraposition of CR, but the contraposition of a subjunctive conditional is not equivalent to the original.

Let us suppose without argument that SI handles cases of knowledge and ignorance as intuitively as CR. Why say SI underwrites K? The key point is that if R safely indicates that p is true, then it safely indicates that q is true, where q is any of p's consequences. Put another way, the point is that the following reasoning is valid (being an instance of strengthening the consequence):

1. If R held, p would be true (i.e., R safely indicates that p)

2. p entails q

3. So if R held, q would be true (i.e., R safely indicates that q)

Hence, if a person S knows p on the basis of R, S is in a position to know q on the basis of R, where q follows from p. S is also in a position to know q on the basis of the conjunction of R together with the fact that p entails q. Thus if S knows p on some basis R, and believes q on the basis of R (on which p rests) together with the fact that p entails q, then S knows q. Again: if

(a) S knows p (on the basis of R), and

(b) S believes q by recognizing that p entails q (so that S believe q on the basis of R, on which p rests, together with the fact that p entails q),