2.1

Hawthorne notes the following intuitively appealing principle:

(Here, p and q are equivalent iff they have the same truth-value.)

2.2

Hawthorne, while explicitly addressing both Nozick and Dretske, glosses his arguments using just Nozick's Sensitivity condition on knowledge:

Dretske (Reference Dretske1971) does not endorse Sensitivity. Instead he requires that, in order to know p, one must have a conclusive reason to believe p. While the two requirements are similar, Dretske (Reference Dretske, Steup, Turri and Sosa2013: 38-9, n.4) himself is at pains to point out situations in which the requirements come apart. Hawthorne's (Reference Hawthorne2004) criticisms of Sensitivity, however, carry over to Dretske's conclusive reasons account (see Hawthorne Reference Hawthorne2004, Reference Hawthorne, Steup, Turri and Sosa2013), and their differences play no role in my arguments. I will follow Hawthorne (Reference Hawthorne2004) by concentrating on Nozick's Sensitivity account (though what I come to say about Nozick goes, mutatis mutandis, for Dretske also).Footnote ²

3.1

Hawthorne (Reference Hawthorne2004: 41) notes:

Nozick denies we can know (11) – one's belief in (11) isn't Sensitive. And so Nozick must either deny that we can know (9) (which would involve abandoning his account of knowledge) or else deny the (intuitive) conjunction of Distribution and Equivalence.

3.2

The argument generalises. Take any ordinary proposition p and some proposition, q, known to be entailed by p, such that one's belief in q isn't Sensitive. We thus know a priori p is equivalent to (p & q). By Equivalence, if one knows p then one is in a position to know (p & q). Then, by Distribution, one is in a position to know q. But, as one's belief in q is not held Sensitively, Nozick denies that q is known. And so Nozick is forced to deny either Distribution or Equivalence.

3.3

Note, however, that Distribution is effectively a restricted form of SPC: SPC claims knowledge is closed under competent deduction, whereas Distribution suggests knowledge is closed under conjunction elimination – a form of competent deduction. So, given that Nozick rejects SPC, it would be no major surprise were he to reject Distribution.Footnote ⁴ Indeed, as it stands, Sensitivity doesn't respect Distribution, and so Nozick rejects Distribution.

However, it is not satisfactory to respond to Hawthorne's argument simply by rejecting Distribution. First, Distribution is extremely plausible: it's hard to see how knowing a conjunction doesn't put one in a position to know each conjunct. Nozick (Reference Nozick1981: 228, 692, n.63) himself admits the plausibility of Distribution, although he does reject it. (Going further, Dretske (Reference Dretske1970: 1009) says ‘it seems to me fairly obvious that if someone knows that P and Q, ... he thereby knows that Q’.) And so if Nozick's account of knowledge were Hawthorne's chief target, an argument depending solely on Distribution – and not, additionally, Equivalence – might suffice to undermine it. Second, rejecting Distribution does nothing to undermine Hawthorne's claim that a subject who knows (9), also knows (12). But (12) is in all relevant respects similar to the kind of proposition that motivates the rejection of SPC (i.e. (11)), as there is some intuitive pull to the idea that we cannot come to know (12) via our senses and deduction. So rejecting Distribution is both implausible and fails to address a problem that Hawthorne poses for the intuitive motivation for rejecting SPC.Footnote ⁵

3.4

It might seem, then, that Nozick ought to reject Equivalence (though Equivalence, like Distribution, is extremely plausible, and so any rejection thereof will have associated costs). Like Distribution, Equivalence is effectively a restricted form of SPC. SPC, recall, claims knowledge is closed under competent deduction, whereas Equivalence suggests knowledge is closed under a priori known biconditional elimination – a form of competent deduction. So, again, given that Nozick rejects SPC, it would be no major surprise were he to reject Equivalence. But Hawthorne notes that Nozick's reasons for rejecting SPC don't carry over to rejecting Equivalence (although they do carry over to Distribution); and Nozick (Reference Nozick1981: 229, 690, n.60) agrees. For example, Nozick's rejection of SPC is based on the fact that a subject may Sensitively believe p without Sensitively believing some q, which that subject knows to be entailed by p. And since, for Nozick, Sensitivity is a necessary condition for knowledge, it can easily be shown that SPC fails for Nozick.Footnote ⁶ But, Hawthorne notes, this situation cannot readily arise when p and q are a priori equivalent. That is, we cannot get (~p □→ ~B_Sp)Footnote ⁷ & ~(~q ⩽ □→ ~B_Sq) when p and q are a priori equivalent, except, perhaps, in some recherché cases.

I will show, however, that, Sensitivity is (likely) unacceptable as it stands and should be modified for reasons independent of Hawthorne's argument. Moreover, my modification conflicts with Equivalence, thereby allowing Nozick to reject Equivalence in a principled way. So a plausible Sensitivity condition on knowledge does provide reason to reject Equivalence, just as Nozick's account of knowledge provides reason to reject Distribution. And so a plausible condition on knowledge does rule out Equivalence, even if, as Hawthorne notes, Nozick's original condition does not. Moreover, once Equivalence is rejected Nozick is able, I will show, to retain the attractive Distribution.

4.1

Consider the following case for reductio.Footnote ⁸ Suppose you have a headache. Now consider the following (ex hypothesi true) proposition:

Suppose you form the belief, (P), based solely on your evidence of having a headache. (To form the belief, (P), on this basis, is to display an epistemic failing: this is not an apposite method for forming beliefs of this general nature. But it is not incoherent to make the foregoing supposition. Next time you have a headache, try forming the belief, (P), solely on that basis. You will see for yourself that it can be done.) This is not to stipulate that having a headache is the only evidence you have; just that it is the only evidence on the basis of which you form the belief, (P). Thus, this leaves open that you have whatever evidence is necessary to have the conceptual/cognitive resources to form the belief, (P), provided such evidence is not playing a basing role. (For more on the basing relation, which is left undeveloped here, see Korcz Reference Korcz and Zalta2006.) Given your basing evidence, you will hold that belief Sensitively: in the closest world in which (P) is false (a world in which you fail to have a headache but retain all your limbs) you won't believe (P). (Assume, uncontroversially, Nozick's adherence condition on knowledge – were p true, S would believe p Footnote ⁹ – is met in this case.) But, your belief in (P) does not constitute knowledge.Footnote ¹⁰

And the case is generalisable: Stipulate that a subject's sole basing evidence is a headache, and construct a conjunctive proposition, which the subject believes on the basis of that evidence, in which the first conjunct is the statement that the subject has a headache and the second conjunct is a (ex hypothesi true) quotidian proposition for which the subject has no basing evidence,Footnote ¹¹ which comes out false only in worlds more remote than worlds in which the subject fails to have a headache.Footnote ¹²

4.2

So something has gone badly wrong with Nozick's account of knowledge (on which truth, belief, Sensitivity, and adherence are severally necessary and jointly sufficient) in relation to conjunctive propositions.Footnote ¹³ And we might think that the Sensitivity condition is to blame. But we need not abandon Sensitivity tout court. (And note the dialectic between Hawthorne and Nozick: it is not that Hawthorne is, in his arguments under consideration in this paper, denying that something like Sensitivity is necessary for knowledge. Thus to offer – as I come to do – principled modifications to the Sensitivity condition is not, to repeat, perforce to be a card-carrying Sensitivity theorist; it's simply to make Sensitivity as plausible as it can be.) Here is the obvious, and natural, fix. Supplement Sensitivity with an alternative (strengthened) necessary condition:

This supplementation straightforwardly enables us to avoid the reductio: in a world in which you're an amputee sitting alone basing your beliefs on your headache, you would still believe that you had all your limbs. Given you do not believe the second conjunct of (P) Sensitively, you cannot, on the modified account, know (P).

A word about applying CON (which also goes, mutatis mutandis, for DIS, my coming proposal with respect to disjunctions): We suppose Sensitivity is a necessary condition on knowledge. We then test propositions for Sensitivity as we check for knowledge. If a conjunctive proposition comes to be checked thus, we filter it off from unmodified Sensitivity to CON. And iff that conjunctive proposition meets CON's necessary condition on knowledge, (modified) Sensitivity is met.

This observation enables me to be a little more precise about what I mean when I talk of ‘modifying Sensitivity’ (where Sensitivity states a putative necessary condition on knowledge). I do not mean that I am modifying what it is to believe a proposition Sensitively. That is just what it has always been: it is a matter of meeting the condition that if the proposition were not true, one would not believe it. What I am instead doing is modifying the connection between Sensitivity and knowledge. For a conjunction, it is not required for knowledge of the conjunction that the conjunction should be believed Sensitively. Rather, it is required that each conjunct should be believed Sensitively. (Again, the same goes, mutatis mutandis, for DIS.)

4.3

Can I do better than simply contending that CON is the ‘obvious, and natural, fix’? Clearly, it would be better if I can. Here is one possible bolstering of CON: CON is true because belief in a conjunction just is belief in the conjuncts, and so knowledge of a conjunction just is knowledge of the conjuncts. Therefore, whatever account is plausible for knowledge of ‘atomic’ propositions, we can use that account to give an account of conjunctive propositions. Since a Sensitivity account is plausible for ‘atomics’, CON should be endorsed.

However, even assuming, pro tem, its validity, pretty quickly, two problems arise with the soundness of any such line of argument – a line of argument reliant, in its opening premise, on the following biconditional:

First, and more sketchily, it might be objected that belief in a conjunction somehow involves (in addition to believing its conjuncts) the concept of CONJUNCTION, whereas belief in the conjuncts does not. Second, and less sketchily, ‘lottery’ issues become salient. Take some p ₁, … , p _n, where one's rational credence for each p _i is very very high and the p _is are independent. If we make n large enough, the conjunction can have a very low probability – presumably one incompatible with belief.

These two objections share the goal of breaking down BEL CON* in the right–left direction. That is, each is designed to falsify the thesis that belief in each conjunct is sufficient for belief in the conjunction in question. And, insofar as they work – and I think they do –, they belie any efforts to bolster CON by reliance on BEL CON*.

But, even prior to consideration of these objections, one might well have had instinctive doubts about a proposal seeking to bolster CON – a principle only stating necessary conditions on knowing a conjunction – by so ambitiously trying to reach the interim conclusion that knowledge of a conjunction just is knowledge of the conjuncts (cf. n.23 infra). Indeed, one might wonder whether one needs an interim conclusion as strong as this to provide some bolstering of CON. Thus, one might – new proposal –, less ambitiously, seek to bolster CON by exploring weakening BEL CON* so as to be read in just the left–right direction:

I do not seek here to conclusively defend BEL CON (other than to note it is not counterexampled by the above objections to BEL CON*). It has some initial plausibility, however. Insofar as it is true, it could offer some extra support for CON beyond a bare appeal to its intuitiveness as a fix. Appeal could be made, that is, to BEL CON – a grounding principle concerning belief.

What, though, can be validly inferred from BEL CON? Specifically, can one infer that S knows a conjunctive proposition, p only if S knows each of p's conjuncts? It would seem not. (Likewise, even though I initially granted the validity of the original, more ambitious, proposal, I do not think one can infer that knowledge of a conjunction just is knowledge of the conjuncts, from the proposition that belief in a conjunction just is belief in the conjuncts.) All that can be validly inferred from BEL CON would appear to be that S knows a conjunctive proposition, p only if S believes each of p's conjuncts. (Likewise, all that can be validly inferred from BEL CON* would appear to be the immediately foregoing conditional and S knows each of p's conjuncts only if S believes the conjunctive proposition, p.) Thus, this new, less ambitious, proposal, with BEL CON as a first premise, does not promise to vindicate CON in the dramatic manner promised by the original, more ambitious, proposal.

Now, all of the foregoing is, concededly, fairly inchoate. Nonetheless, it seems we can conclude the following: Insofar as BEL CON (and, indeed, BEL CON*) offers some (additional) reason to accept CON (beyond a bare appeal to its intuitiveness as a fix), further exploration of it, its truth, what it supports, and what can be validly inferred therefrom, by the Sensitivity theorist will be worthwhile (cf. Williamson Reference Williamson2000: 283).

5.1

Recall Equivalence:

As I have just argued, CON is an independently motivated modification of Sensitivity. But if Sensitivity is modified with CON, we can see why Equivalence fails for Nozick. When you know a priori that p and q are equivalent and you know p, and where q is a conjunctive proposition the second conjunct of which you do not believe Sensitively (on the basis of your evidence for p), then you are not (confining yourself to your evidence for p) in a position to know q.Footnote ¹⁵

Moreover, supplementing Sensitivity with CON enables Nozick to maintain the attractive Distribution, since one can know a conjunction only if one Sensitively believes each conjunct. Nozick can thus accept Distribution but deny knowledge of (12) (since one does not Sensitively believe its second conjunct) which blocks the route to knowledge of (11).

5.2

And so, returning to Hawthorne's ‘Equivalence and Distribution’ pro-closure argument, the step from (9) to (12) is no longer legitimated. You can know (presumably on the basis of an experience as of a zebra): (9) that is a zebra. But, with Equivalence falsified, it doesn't follow that you can know: (12) that is a zebra and that is not a cleverly disguised mule. On the basis of your experience as of a zebra you won't believe the second conjunct of (12) Sensitively. With the route to (12) blocked there is now no route to Hawthorne's conclusion that you can know – pace Nozick – that is not a cleverly disguised mule.

With CON, we can follow Nozick in relinquishing SPC, not by relinquishing Distribution, but instead by relinquishing Equivalence, which fails when CON is added, as an alternative necessary condition, to a Sensitivity account as it needs to be. Given that this failure of Equivalence is independently motivated, the costs of denying SPC by rejecting Equivalence are not as high as Hawthorne suggests (more on this later, in 6.1).Footnote ¹⁶

5.3

Next, note Hawthorne has a second, ‘Equivalence and Addition’, pro-closure argument (2004: 39–41) resting on Equivalence and a second principle:

So from (6), by applications of AC and Equivalence, we can come to know (8). But according to Nozick we cannot know that we are not handless brains in vats.

5.4

As with Distribution and Equivalence, note that AC is also a form of SPC, claiming that knowledge is closed under disjunction introduction – a form of competent deduction. Given that Nozick rejects SPC it would be no major surprise if he were also to reject AC. But, as with Distribution, AC is extremely compelling. As Nozick (Reference Nozick1981: 230, 692, n.64) admits, rejection of AC ‘surely carries [things] too far. ... Surely our knowledge that p does not stand in such splendid isolation from knowledge of other things so closely connected to p’.Footnote ¹⁷ It would be a cost to Nozick if he were forced to reject it.

As it stands, however, Nozick does reject AC, since we can believe p Sensitively and yet fail to believe (p v q) Sensitively. For example, let p be (6) and let (p v q) be (7). Although, if I had no hands, I would not believe I had hands, it is not the case that if I were a handless BIV, I would not believe I had hands or was not a BIV. Rather, if I lacked hands and were a BIV I would, ex hypothesi, believe I had hands and was not a BIV. Put differently: I know, hence Sensitively believe, that I have hands. Now suppose that I deduce and come to believe that either I have hands or I am not a BIV. If that disjunction were false – i.e. if I had no hands and was a BIV – I would still believe that it was true. So the disjunctive belief is not Sensitive. So Nozick rejects AC.

5.5

But my introduction of CON to account for Sensitivity with respect to conjunctions, makes salient, and suggests an answer to, the question: What is the appropriate account of Sensitivity for disjunctive propositions? We might think that, if Sensitivity with respect to a conjunction requires that we Sensitively believe each conjunct, then Sensitivity with respect to a disjunction requires only that we Sensitively believe some disjunct. We would then need to add the following further modification to Sensitivity:

DIS* allows Nozick to say that a subject can know Hawthorne's disjunctions (e.g. (7)) and to endorse AC, just as CON allows Nozick to endorse Distribution.

5.6

DIS*, however, is not plausible in general. We can know a disjunction without knowing or believing any disjunct. For example, we can narrow down the epistemic possibilities: we can know that Danny is in the kitchen or he is in the lounge by knowing Danny is not in the bedroom. Similarly, suppose one knows a city is in France; one can then know that city is in either the north of France or the south of France. DIS*, then, is too strong.

What these cases show is that we can come to know a disjunction without knowing any disjunct. But when we infer a disjunction from one of its disjuncts, we know a disjunction in virtue of knowing one of its disjuncts. In such a case, our evidence for the disjunction is our evidence for the disjunct from which it is inferred. So, in this event, in order to know the disjunction we need only believe the disjunct from which it is inferred Sensitively rather than the disjunction itself. This suggests:

With DIS, Nozick can endorse AC. That is, we are countenancing a version of the principle Sensitivity that imposes a less demanding necessary condition on a disjunctive proposition when it is inferred from one of its disjuncts (see 4.2 supra for how to apply DIS). This will guarantee that modified Sensitivity does not present any obstacle to closure of knowledge under AC. Of course, this does not yet suffice to avoid Hawthorne's argument. Indeed the validity of AC is a premise of Hawthorne's argument. But, as we saw above, Nozick should reject Equivalence. And he should reject the instance of Equivalence in this case too. Although Sensitivity can be met with respect to (7) (i.e. (H v ~BIV)), since one can Sensitively believe (6) (i.e. H), it is not the case that Sensitivity can be met with respect to (8) (i.e. ~(~H & BIV)) even though this is equivalent to (7). And the reason for this is just as it is in Nozick's original rejection of SPC: one can know that one has hands via one's senses or via one's senses and deduction, but one cannot, in any such manner, know that one is not a handless BIV.Footnote ¹⁸

So by accepting DIS and endorsing AC, Nozick can once again reject Equivalence in a motivated way. The facts that lead to Nozick's original rejection of SPC are exactly the facts that should lead to his rejection of the transition from (7) to (8). So Hawthorne's second pro-closure argument does nothing to increase the costs that Nozick must face – once, that is, he accepts DIS.Footnote ¹⁹

5.7

Let me close by noting two (additional) costs of my proposal. Cost 1: What is the appropriate account of Sensitivity for negations of propositions? Negations of atomic propositions are to be treated just like the propositions they are negations of. That is, Nozick's unmodified Sensitivity condition suffices as a necessary condition for knowledge of the negation of an atomic proposition. But what about the negations of molecular propositions? In particular, what about the De Morgan equivalents of conjunctions and disjunctions? As we have just seen, on my proposed modification of Sensitivity, one can know (7) but not (8) – a negated conjunction. As a result, we cannot treat disjunctions and their De Morgan equivalents alike. But unmodified Sensitivity explains our failure to know (8), since if I lacked hands and were a BIV, I would still believe that it is not the case that I lack hands and am a BIV.

This might suggest that negations are subject to unmodified Sensitivity. But the reductio of Nozick's account of knowledge, which motivated CON, shows that we cannot apply unmodified Sensitivity to negations of disjunctions. If someone believes:

solely on the basis of their headache, we concluded that they would not know (P) despite satisfying unmodified Sensitivity. Similarly, the same considerations show that we do not know the De Morgan equivalent of (P) – viz. It is not the case that I neither have a headache nor all my limbs – even though we satisfy unmodified Sensitivity.

This all suggests the following, then. First, we should treat negations of disjunctions as conjunctions: one can know them only if one believes the negation of each disjunct Sensitively. Second, we should treat negations of conjunctions as subject to unmodified Sensitivity. Finally, note there will be no problematic interactions between CON and DIS, as we can never move directly from a disjunction to a conjunction or vice versa. Overall, however, it must be conceded that these manoeuvres may appear somewhat ad hoc.

5.8

Cost 2: Consider the following case which brings out an upshot of my proposal. Jones gets to know Peter, a paediatrician, by observing him medically treating children, and forms the Sensitive belief that Peter is a paediatrician on the basis of such observation. Nevertheless, Jones, who is in other relevant respects normal, holds the strange misconception that everyone is some kind of physician. So Jones believes the following conjunction: Peter is a paediatrician and Peter is a physician. By CON Jones does not know this conjunction, since his belief in the second conjunct is not Sensitive. But what about if Jones holds a belief concerning Peter involving, not a conjunction, but a complex (conjunctive) predicate? Specifically, what about if Jones believes: λx[x is a paediatrician and x is a physician](Peter)? Jones believes this proposition Sensitively since if Peter did not satisfy the predicate: λx[x is a paediatrician and x is a physician], Jones would not believe that he did.

In sum, we here have a case where Jones does not know:

But Jones does know:

Now it is clear that on some fine-grained and plausible way of individuating propositions (A) and (B) express distinct propositions. But how plausible is it to say Peter knows (B) but not (A)? Of course, the approach I have explored denies the general validity of Equivalence, but the link between (A) and (B) is, concededly, tight. Indeed, we usually do not distinguish between the two propositions.

(Perhaps, though, the cost of this case is not so bad. I have already allowed that Equivalence is not generally valid. I have also allowed that the very same proposition (a disjunction) needs to meet only a weaker Sensitivity condition, rather than the standard Sensitivity condition for knowledge, depending on whether the disjunction was arrived at by inference from one of its disjuncts. The relevant point here is that (B) is not in the form that allows immediate inference to ‘Peter is a physician’. And it is only when there is an inference from (p & q) to q, for a proposition that is not believed Sensitively and so not known, that it is important that CON should be applied to give the result that the conjunction (p & q) is not known either.)

Relatedly, consider the following case. Suppose that we have three birds: a male blackbird, a female bird that looks like a female blackbird but is of a different species, and a female sparrow. (Male blackbirds are black and easily recognizable, but females are brown and superficially similar to birds of other species.) Suppose S never makes mistakes identifying male blackbirds; but S would mistake the bird that looks like a female blackbird for a female blackbird; and S always mistakes female sparrows for male sparrows. Suppose that we pick a bird at random and show it to S; suppose we pick the male blackbird. And suppose, to simplify, that the picked-the-female-blackbird-lookalike world and the picked-the-female-sparrow world are equally close to the actual picked-the-male-blackbird world.

It seems that S knows, and Sensitively believes, that the bird is a male blackbird. If the bird had not been a male blackbird then it might have been the female blackbird lookalike or it might have been the female sparrow. So S might have believed that the bird is a female blackbird and might have believed that the bird is a male sparrow but, either way, S would not have believed that the bird is a male blackbird. However, it seems that S does not Sensitively believe that the bird is male. If the bird had not been male, it might have been the female sparrow, and so S might still have believed that the bird is male. Equally, it seems that S does not Sensitively believe that the bird is a blackbird. If the bird had not been a blackbird, it might have been the female blackbird lookalike, and so S might still have believed that the bird is a blackbird.

Nonetheless, as noted, when S sees a male blackbird, it seems that S comes to know, and Sensitively believe, that the bird is a male blackbird. Here is a preliminary way of summarising the point of the case: If the subject were to infer, from the seemingly known premise ‘That is a male blackbird’ either ‘That is a male bird’ or ‘That is a blackbird’, then the inferred conclusion(s) would not be known.Footnote ²⁰

What do I say about this case? The proposition believed by S – that the bird is a male blackbird – could involve a conjunction:

which, by CON, S does not know. Or it could instead (more naturally) involve a complex predicate:

which S knows. In sum, the paediatrician/physician case told us that Equivalence is to be given up in these cases where we have alternative, though closely related, parsings of a single sentence;Footnote ²¹ and now the male blackbird case tells us, moreover, that we can know a proposition involving a (complex) conjunctive predicate without knowing any of the corresponding propositions involving the simple predicates – and without knowing the equivalent conjunctive proposition. Overall, however, it must (again) be conceded that these upshots are a cost of my proposal.Footnote ²²