Can we have knowledge in the face of skeptical arguments? A defence of epistemic contrastivism

1 The Case for Contrastivism

1.1 Introduction

How can we have knowledge in the face of skeptical arguments? This central question of epistemology remains a topic of some considerable dispute, and this dissertation shall address this very problem.  Consider the following rudimentary skeptical argument:

1. If I know I have hands, then I know I am not a brain in a vat,

2. I don’t know I am not a brain in a vat,

C.   Therefore, I don’t know I have hands. 

This simple, compelling modus tollens argument seems to suggest that, because we cannot rule out skeptical hypotheses, then we cannot have everyday knowledge either. Two possible responses to this argument immediately avail themselves to the attentive philosopher. One is to deny that knowledge is closed under known entailment (for an example of this, see (Dretske, 2014)). If that is the case, then the above argument does not hold, since (1) would come out false. However, denying that knowledge is closed under known entailment is to deny that knowledge can be transmitted logically in a whole host of other cases. For example, if I know that A ⊨ B, and I know that A, surely, I ought to be able to say that I know B? If I know that the sky is blue, and that entails that the sky is not-green, then I should be safe in saying that I know the sky is not green, knowing also that it being blue entails it not being green? Retaining closure in this manner generates problems relating to how logic relates to epistemic states, such as any known proposition implying infinitely many disjunctive propositions. However, MacFarlane (2004) proposes a solution to this problem and for the purposes of this paper, it shall be assumed that such a solution is workable.

A second possible response is Moorean dogmatism, named for Moore’s famous Proof of an External World (Moore, 1939). The dogmatic response is to assert that the knowledge of my having hands is simply the sort of truth that can be known when compared to a skeptical claim to the contrary. The skeptical argument can be dismissed because the argument that we have hands carries a great deal more certainty than any skeptical argument to the contrary, despite the apparent appeal of such skeptical arguments. 

Both rejecting known entailment and Moorean dogmatism seem extreme responses, that require us to endorse questionable consequences. In the case of rejecting known entailment, as above, it appears we must reject means of reasoning that, intuitively, are unobjectionable. In the case of Moorean dogmatism we risk begging the question against the skeptic, by asserting the very kind of knowledge that the skeptic is calling into question. Though both responses have philosophical merit, I shall argue for an analysis of the knowledge relation that avoids both positions, while answering the skeptic’s position.

Therefore, my argument must retain known entailment, avoid Moorean dogmatism and answer the skeptic’s case while retaining everyday knowledge; these are the desiderata of the argument to be given. Further to these ends, I shall offer an analysis of the knowledge relation that is found in Schaffer, drawing on the work of Kvanvig, that suggests knowledge is a ternary relation, Kspq, between an agent, s, a known proposition p and a contrast proposition q, which should be a salient, mutually exclusive contrast which acts as a foil for the known proposition. This is contrastivism. In the remainder of section 1, I shall present the positive case for contrastivism, which meets the desiderata of retaining everyday knowledge while answering the skeptic. Then, in section 2, I examine contrastive closure, drawing on relevant literature and problems posed for it in the literature that suggest either closure can be retained, or dogmatism avoided, but not both. I then propose solutions to these problems, including one of my own devising, before giving a complete account of contrastive closure. Since my reason for rejecting skepticism, dogmatism and retaining closure were that the consequences of doing otherwise were unacceptable, in section 3 I shall examine some circumstances in which contrastivism seems to give rise to counterintuitive or otherwise unacceptable consequences, drawing on my own examples and the literature, and in each case provide a plausible account as to how such consequences might be avoided.

1.2.1 Contrastivism as a response to the skeptic

Consider again ‘I know I have hands,’ which is the antecedent of (1), above. According to contrastivism, this is an incomplete statement. To make it complete, we must add some contrast that is salient and mutually exclusive of having hands. (Kvanvig, 2008, p. 253). These conditions are required to retain closure, as shall be explored later, but for now shall be taken as necessary for counting as a contrast proposition in the q position. So, in this case, salient and mutually exclusive proposition might be ‘I have stumps.’ So, we could re-write the initial statement as ‘I know I have hands, rather than stumps.’ The consequent we might, similarly, re-write, ‘I know I am not a brain in a vat rather than in a BIV simulation.’ This, however, means that the argument is no longer sound. Though I shall return to the details of contrastive closure later, for now it is sufficient to note that the conditional is simply false. Just because I know I have hands rather than stumps, doesn’t seem to tell me anything at all about my knowledge of skeptical hypotheses like a brain in a vat scenario; once the contrasts are made clear, the skeptical argument becomes unsound. I may not know that I have hands rather than BIV-simulated hands, but I can know that I have hands rather than stumps, and the latter does not entail anything like the former.

What about the other desiderata? Section 2 is devoted to closure, so for now let’s focus on dogmatism. What can I say, as a contrastivist, about dogmatic assertions against the skeptic? It seems that I can simply deny them. ‘I know I am not a brain in a vat, rather than an embodied thing’ seems to come out as false. I can’t make any such assertion; after all, what possible justification could I possibly have for such a claim? That said, such an acceptance isn’t any threat to everyday knowledge either; my going about my everyday use of the term ‘know’ seems unaffected by there being a third place to the relation. I am committed to saying that the phrase ‘I know I have hands’ is incomplete, but it is not so uncommon for us to use shorthand in everyday speech. If I utter something like ‘the ball fell down,’ without further information we would take this to mean ‘the ball fell down toward the Earth’, however, it may also be taken to mean ‘the ball fell down from the counter’. It is not so uncommon for us to use incomplete statements in our everyday parlance. ‘I know I have hands,’ according to my argument, should be taken to be similarly incomplete. So, it seems that contrastivism offers us a way of avoiding both Moorean dogmatism about our common sense and avoiding skeptical conclusions about our knowledge by adding an explicit foil that limits our knowledge. In the next section I shall briefly offer a couple of examples to further explore why this might be desirable. 

1.2.2 Knowledge as ruling out

‘Does Ann know there is a goldfinch in the garden?’ (2012a, p. 354) Schaffer poses this question and answers it depends on the contrast. And surely that is right? When we ask if Anne knows if there is a goldfinch in the garden, aren’t we asking, in some real sense, what Anne can rule out? Isn’t knowledge, in some sense, a limit on the relevant possible alternatives? If we know something, then we have ruled out some set of salient possible alternatives. Let’s imagine that Anne has some knowledge of ornithology. She is looking out at her birdfeeder, and she knows that goldfinches are not commonly seen at bird-feeders. It seems reasonable to say that Anne knows the bird at her feeder is a goldfinch rather than a raven, but that she doesn’t know that it’s a goldfinch rather than an Evening Grosbeak, which is very similar. Assuming we take the knowledge relation to be factive, Anne can rule out the raven, but not the grosbeak, so we can say she knows one rather than the other. 

By way of further motivation, consider a pharmacist administering medicine. A pharmacist must know that any given medication is the correct medication. Doing so, surely, involves the elimination of the relevant alternatives. Let’s imagine a pharmacist who has a stock comprised of 1000 different medications. His assistant has fetched a batch of medication while he was at lunch and forgotten to label it. His assistant is now on lunch and cannot be asked. What does the pharmacist know about the medication? He has one set of medications in front of him. At this stage, pharmacist knows that the tablet before him is medication, rather than a placebo, since he doesn’t stock placebos. The tablet is small and white, which rules out all his 850 medications that are large or not white. The tablet has a line down the middle, so the pharmacist knows that the medication is one that is evenly distributed throughout the tablet and half the tablet is half the dose, ruling out another 120 medications that lack this feature. Therefore, he knows it is one of 30 medications he has in stock, rather than 970. The pills before him are stamped AIG, which means he knows that they are one particular substance, and can safely say he knows they are say, ibuprofen, rather than anything else. This process of elimination of the relevant alternatives is a good example of how we come to know things. We come to know because we have eliminated the relevant alternatives. In an everyday setting like this one, we need not eliminate skeptical hypotheses, like BIV-simulated tablets, because that alternative simply isn’t salient (see section 2.4 for an in-depth discussion of saliency) to the pharmacist. Yet, it doesn’t seem dogmatic to assert that the pharmacist knows the tablets are ibuprofen rather than acetaminophen. In some meaningful sense his knowledge of his medication stock justifies his knowledge that the tablets are ibuprofen, and not acetaminophen, in a way that isn’t the case with the BIV-simulation of tablets. 

What I hope to draw out by way of these examples is that it is intuitively plausible that knowledge, as we deploy that relation in everyday parlance, is a means of demonstrating that a knower has meaningfully eliminated a range of possibilities that are relevant, and that this can motivate a positive case for my position. 

2 Contrastivist Closure

One of the desiderata laid out earlier was the retention of known closure, an issue which shall now be addressed. Given the nature of the knowledge relation as viewed under contrastivism, a thoroughgoing account of how the relation operates with regards to logic is required. For the binary analysis of knowledge, the following account might suffice, although is not unproblematic:

1. Ksp1

2. p1 → p2

C)   Esp2

Here, Ksp1 means that s knows proposition p1, and E here stands for the ‘evidential component of K’ (Schaffer, Closure, Contrast and Answer, 2007), such that the knower in question has sufficient evidence to know p2 but does not thereby come to know automatically, which I take to be equivalent to propositional justification. Propositional justification here can be taken to mean justification rooted in evidence such that one may or may not believe it, as opposed to doxastic justification, which is the justification given for a belief already held (Ichikawa & Steup, 2017 Section 1.3.2). So, s has propositional justification for believing p2, though may not thereby come to know it without some sort of application of reason. No such straightforward picture is naturally available for the contrastivist account of the knowledge relation; one must be provided to preserve closure. Most important to handle will be knowledge when moving between different known propositions, as well as moving between differing contrast propositions. It seems intuitive that if I know I have hands rather than handless stumps, and having hands entails having carpal bones, then I have propositional justification for the belief that I have carpal bones rather than handless stumps. Similarly, if I know that I lack carpal bones, and lacking carpal bones entails having handless stumps, then intuitively it seems I have propositional justification for the belief that I have handless stumps. These exemplars are, of course, unsound, but they seem intuitively valid. The next section shall be devoted to formalising these intuitions.

In section 2.1 I shall give a mostly descriptive overview of Schaffer’s proposed principles of contrastive closure, and Kelp’s fatal objection to them. In section 2.2 I shall give an account of Kvanvig’s proposed principle, two problems from and my proposed to solution to these problems, and then give a summary of my view of contrastive closure. 

2.1 Schaffer and Kelp

2.1.1 Schaffer’s proposed principles

Schaffer’s proposes (2007) four rules of contrastive closure. I shall outline them below and explain what each means. 

1. (Expand-p) (Ksp1q ∧ (p1 → p2) ∧ {p2} ∩ {q} = ∅) → Esp2q)

This rule aims to handle when moving from knowing one thing to another with the same contrast. So, if Michelle knows p1 rather than q, and p1 entails p2, and p2 and q are mutually exclusive, then Michelle has propositional justification for p2 rather than q. By modus ponens, if p1 is the case then p2 must be the case also, so Michelle’s knowledge is entailed in the right sort of way.

2. (Contract-q) (Kspq1 ∧ (q2 → q1) ∧ {q2} ≠ ∅ → Espq2)

This rule handles our second need, which is to show how we may formally move between contrast propositions. If Mahmood knows p rather than q1, and q2 entails q1, and q2 is not the empty set (i.e., no proposition at all,) then Mahmood has propositional justification to know p rather than q2, since, if q1 is the case, then q2 must be the case by modus tollens.

3. (Intersect-p) (Ksp1q ∧ Ksp2q) → Es(p1 ∧ p2)q)

This rule simply permits that a knower can concatenate known propositions as conjunctions if the contrast is the same. Going back to Michelle, if she knows both p1 rather than q and p2 rather than q, she knows p1 and p2 rather than q. 

4. (Union-q) (Kspq1 ∧ Kspq2) → Esp(q1 ∨ q2))

This rule permits for the shortening of known statements when the same thing is known rather than two other things. So, going back to Mahmood, given that he knows p rather than q1, and p rather than q2, he knows p rather than q1 or q2. 

According to Schaffer, ‘contrastive knowledge extends by Expand-p, Contract-q, Intersect-p and Union-q’ (Schaffer, Closure, Contrast and Answer, 2007). Union-q and Intersect-p seem unobjectionable, and indeed are not much discussed in the literature and I cannot see anything objectionable about them, therefore I propose they should be retained. However, both Contract-q and Expand-p are problematic in their current form, and the problems highlighted in the literature are outlined below.

2.1.2 Kelp’s objection

Christoph Kelp (2011) uses a Cartesian deceptive genius to demonstrate a flaw in Schaffer’s principles. I shall paraphrase and explain his argument. First, let’s imagine that there’s an evil genius who can deceive someone completely about their sensory input. If this evil genius deceives someone into believing they are seeing a sheep when in fact they see a cow, then it is a cow before them. This is (q2 → q1). Now let’s imagine Guy. Guy is looking at a sheep and declares that he knows that it is a sheep before him, rather than a cow. Since guy knows that he is looking at a sheep rather than a cow, we can now use Contract-q, using the evil genius above, to conclude that Guy knows he is looking at a sheep rather than an evil genius deceiving him into seeing a sheep where in fact a cow stands. 

As Kelp points out, this can be generalised to cover many possible propositions (p. 290) that meet all the requirements Schaffer lays out. Though this example is incredibly simple, it seems that the contrastivist can either have closure or refrain from dogmatism but, crucially, cannot have both. In the next section, I shall discuss Kvanvig’s proposed closure rule, which can be used to supplement Contract-q and Expand-p by deploying some semantic notions, replacing (or strengthening) Schaffer’s proposed formalisation, before discussing two problems for contrastivist closure and my proposed solution to them. 

2.2 Kvanvig and Hughes

2.2.1 Kvanvig’s proposal for contrastivist closure 

Kvanvig’s proposal for a principle for contrastivist closure is as follows, with edits to the variables used to keep in line with the usage deployed thus far, (I shall refer to this as Kvanvig’s Principle, or KP):

‘If S knows p1 rather than q1, then if p1 entails p2, q2 is a salient contrast to p2 for S the possibility of which S has ruled out, and S is in a position to know p2 rather than q2 by having competently deduced p2 from p1, then S knows p2 rather than q2’ (p. 253).

Kvanvig’s Principle takes in both moving between contrast propositions and moving between known propositions in a single statement. Kvanvig also eliminates worries Kelp has raised about contrastive closure since Guy, in the example in 2.1.3, cannot possible rule out the possibility of being deceived by an evil genius. KP also gives the right results when we are thinking about Michelle and Mahmood from 2.1.1. Kvanvig’s deployment of ruling out and saliency in the contrast are the real manoeuvres here and they eliminate Kelp’s objection (as well as Kvanvig’s earlier objections, see (Kvanvig, 2007)), and in complement rather than replace Schaffer’s formal principles. For something to be a contrast proposition in the sense required for contrastivism, means it must be eliminated as a possibility for the knower in question, and it must be salient to the known proposition. If it meets these requirements, then Schaffer’s formulations will hold true, and I find I must endorse this position wholeheartedly. The next section shall discuss Hughes’ problem for KP, and my response to it.

2.2.3 Hughes’ conjunctive contrasts

Hughes’ strategy (2013) is to exploit a potential weakness in the Expand-p rule, while satisfying KP, to create dogmatic knowledge from everyday knowledge, in a manner he describes as analogous to Kelp’s method. He summarises his argument neatly, starting from S knowing ‘I have hands rather than stumps’ which I shall reproduce below:

‘(a) I have hands entails (I have hands and am not a BIV)

(b) I am an amputee is a salient contrast to (I have hands and am not a BIV) for S the possibility of which S has ruled out

(c) S is in a position to know ((I have hands and am not a BIV) rather than an amputee) by having competently performed the deduction expressed in (a)’ (p. 587).

This is a valid use off Schaffer’s Expand-p rule and meets all the criteria laid out by Kvanvig, above, and it appears we can derive a dogmatic assertion from everyday knowledge. This would certainly undermine my claim that contrastivism can retain closure without dogmatism. 

One possible response to Hughes’ objection is to deny (a) is the case; that a contrastivist would not take their having hands as entailing that they are not a brain in a vat. Note, however, that this is not a claim about what is known; it is a factual claim that ‘I have hands’ logically entails ‘I have hands and am not a BIV.’ Though there may be grounds to deny this, I shall instead deny (b), specifically, that ‘I have hands an am not a BIV’ is not a contrast to ‘I am an amputee,’ and thereby add to the picture of contrastive closure developed thus far. 

Though, logically speaking, Hughes conjunction (I have hands and am not a BIV) is mutually exclusive to being an amputee, and is salient in the right kind of way, it does not strike me as being a contrast in the right sort of way. When elucidating his proposed principles, Schaffer (2007) used probability space diagrams to show possible truth conditions for contrastive statements, and I am going to modify that method to highlight why I feel Hughes’ ‘I am an amputee’ is not a contrast for (I have hands and am not a BIV). Take the encircled space to be where the statement is true, and Hughes’ argument would look like figure 1. 

However, since (I have hands and am not a BIV) is a conjunction, it’s space can be subdivided, albeit in a somewhat unorthodox manner, as in figure 2. 

To summarise, one conjunct is not mutually exclusive with the known proposition, and at that point, it is my contention, it ceases to be a contrast. By way of further example, let’s imagine that Shelley takes herself to know the following: 

3. Tibbles is a cat rather than (Tibbles is a cat ∧ Tibbles is not a cat)

Let’s leave it as mysterious as to why Shelley might come to think this, and simply see whether we would want to permit this as a valid case of contrast. Since the contrast proposition is a contradiction, it must be false. And since it contains the negation of the known proposition, it seems like a candidate to be a contrast, as well as being salient to the known proposition. Yet, intuitively, I think we should reject this as a case of contrast, because one conjunct is identical with the known proposition. On what grounds should we do so? Since the contrast proposition meets all the requirements laid out so far, I propose adding something more. 

I propose that we supplement KP and Schaffer’s rules; that for any given conjunction to be a contrast to another statement, both conjuncts must meet the conditions of saliency and mutual exclusivity. Both Hughes’ example and (3) give us good grounds that to think there is something not quite right about the idea of a conjunct serving as a contrast unless it meets these conditions, and indeed, Schaffer’s Intersect-p gives ground for thinking that such a rule is acceptable. Recall:

(Intersect-p) (Ksp1q ∧ Ksp2q) → Es(p1 ∧ p2)q)

This implies that two known propositions, p1 and p2 must be known against the same contrast before they can be concatenated into a conjunction. Now, Hughes might argue that no such knowing is proposed in his argument, and I concede that point. However, I take this to imply that conjuncts within a complex proposition must be on a level playing field, as it were, with regards to the proposition they are being contrasted with, in order that the derivation be valid. This being so, it doesn’t seem at all unreasonable to insist that both conjuncts in a conjunctive proposition meet the criteria required of an atomic proposition. Therefore, I propose my own additional principle, OP: For a conjunction (A ∧ B) to stand in the contrast relation (as understood by contrastivism) to C, both A and B must stand independently in the appropriate saliency and mutual exclusivity relations to C.

To strengthen my point, I’d appeal to the semantic notion of contrast as being something wholly opposite to, and that in the case of complex propositions, that this opposition is entirely composed of the opposition of the parts from which that proposition is composed. In short, for a complex proposition to contrast with some simple proposition is simply for all its constituent parts to contrast with that simple proposition. For these reasons, I take Hughes’ argument to have failed to show that contrastive closure invites dogmatic knowledge from the everyday. 

2.3 Contrastive closure resolved

Before concluding this paper, I wish to tie together my proposed picture of contrastive closure. It is my assertion that, from the foregoing arguments, the following position is justified:

1. Schaffer’s Intersect-p and Union-q principles for concatenating contrastive knowledge propositions are acceptable

2. KP should be act as limitations on Expand-p and Contract-q, e.g., saliency and elimination requirements

3. OP: For a conjunction (A ∧ B) to stand in the contrast relation to C, both A and B must stand independently in the appropriate saliency and mutual exclusivity relations to C.

2.4 Saliency

Thus far, a discussion of saliency has not been given, but it is a key component to retaining contrastive closure. Consider the following case of Contract-q that ignores KP, given that a contradiction entails anything at all:

1. Shelley knows that Tibbles is a cat rather than a dog

2. (London is a city ∧ London is not a city) → Tibbles is a dog

C)  Shelley knows Tibbles is a cat rather than (London is a city ∧ London is not a city)

This seems problematic because it doesn’t look like Tibbles has anything to do with London, and yet, somehow, we have ended up there in the contrast. Consider the pharmacist example in section 1; what about this example means that skeptical scenarios are not salient? In this case we might appeal to what the pharmacist wants to achieve in that situation, and the skeptical hypotheses of the BIV-variety are not salient since they have no impact on the pharmacist’s desired outcome. If that is the case, then in some sense, isn’t the pharmacist’s desired outcome a limiting factor on his knowledge, and if that is so, can we really tolerate such limitations? Shouldn’t knowledge stand above and away from outcomes? Also, there may be skeptical hypotheses that would affect the pharmacist’s outcomes, such as the medication having been manufactured for the purpose of misleading the pharmacist, and I need an account of saliency that addresses such possibilities.

A thorough-going analysis of saliency could be a dissertation in its own right. However, a rough-and-ready way to think about saliency is to appeal to the psychology of the agent in question. Consider the pharmacist once more. A potentially salient skeptical hypothesis for the pharmacist is that the tablet before him has been secretly engineered by an evil pharmaceutical scientist to look like ibuprofen but is in fact, diamorphine. Now, if the pharmacist in question was presented with this possibility by a skeptical interlocutor, in an everyday circumstance, it seems plausible to suggest that the pharmacist would react with incredulity at the idea that such a possibility was salient to him, but not because the hypothesis being put to the pharmacist is irrelevant it the outcome. Generalising, what is salient is what the agent in question takes to be relevant in an everyday sort of way, based on the agent’s experience and psychological make-up. There are at least two possible objections to this account of saliency, and I shall address each. 

The first is that when the skeptical hypothesis is raised, it becomes salient. Going back to the pharmacist, when the skeptical interlocutor raises the skeptical hypothesis with the pharmacist, his initial reaction may be incredulity, but may on reflection, come to accept it as salient, and thereby lose the knowledge that they previously had. I cannot appeal to mere initial incredulity, since this would eliminate too many cases where we might be initially incredulous about salient possibilities that, on reflection, are salient. My best response is that, for my purposes, it is enough that such worries would arise on an agent-by-agent basis. This objection depends on the pharmacist, for example, coming to accept the skeptical hypothesis as salient on reflection, or perhaps under influence of skeptical argument unrelated to the structure of the knowledge relation itself, perhaps pertaining to this agents’ worries about the sorts of people who work as pharmaceutical scientists. As it relates to contrastivism, saliency can fulfil its role while permitting that some agents may come to take skeptical hypotheses as salient and thereby undermine their knowledge, but in so doing, these skeptical agents cannot thereby undermine the knowledge of another who does not take the skeptical hypothesis to be salient. Perhaps this is so much the better; since I have taken knowledge to fundamentally involve ruling out certain possibilities, then if there are those who take skeptical hypotheses to be salient to them, then they cannot possibly rule such possibilities out, and then perhaps it is right to say that they do not know. Nonetheless, these consequences would be limited to those agents who do in fact accept skeptical hypotheses as salient and need not encroach on everyday knowledge of agents whom are otherwise. 

The second objection is that, in principle, anything could be taken to be salient. Going back to Shelley, she might in fact take the contradiction to be salient to Tibbles being a cat, and we have no reason to gainsay here on the account given here. Strictly speaking this is correct, but since the desideratum is retention of everyday knowledge, it seems reasonable to take this as applying in everyday contexts, and thereby non-philosophers going about their business can retain knowledge in such contexts. Again, this might represent a weakening of the argument, but in the absence of a thorough examination of saliency, this must be accepted.

Understanding that this analysis is somewhat wanting, but lacking space to develop a more complex conception of saliency, I shall move on to consequences of the contrastivist analysis of knowledge.

3 Consequences of contrastivism

So far, I have discussed the positive case for contrastivism as preserving everyday knowledge, as well as proposing a series of principles for contrastive closure. Next, I shall discuss various consequences of contrastivism that may be drawn out, first of my own creation, and then in the literature. Since this was our grounds for rejecting other responses to the skeptical paradox, in each section I offer a contrastivist response to the problem posed such that Moorean dogmatism is and other intuitive or undesirable consequences are avoided, as well as everyday knowledge being preserved. Thereby, I aim to show that contrastivism remains preferable to the alternatives laid out in the introduction. 

3.1 Negative knowledge 

By way of a consequence that I considered, let’s consider how contrastivism handles negative knowledge, i.e., knowledge that something isn’t the case. At first glance, it isn’t obvious how we should cash out a case of negative knowledge contrastively. So, for example ‘I know that the sky isn’t green.’ What, in this case, should be the contrast proposition? And, does it make sense to cash out negative knowledge in this way? A lack of such an account would be a grave consequence for contrastivism indeed, since such knowledge is very much part of our everyday experience.

One possible response is simply to say the contrast proposition should be ‘rather than something else,’ but that won’t do. ‘Rather than something else’ is hardly salient, nor is it necessarily mutually exclusive. The sky could be green and blue polka-dot, and that would be ‘something else’ whilst not necessarily excluding the possibility of a partially green sky. So, what should the contrast be here? My best response to this line of thinking is as follows. When we talk of knowledge in the day-to-day running of things, we don’t usually include a contrast proposition. That means that we are frequently talking in short-hand about what we truly take ourselves to know or not know. We do this quite frequently with all sorts of things. When in conversation with someone, the use of indexicals or other context-sensitive language forms are short-hand for longer-form semantic content that it is just inconvenient to say. This, I think, illuminates how we should conceive of negative knowledge when we cash out our contrastive commitments in full. When I say, ‘I know the sky is not green,’ I’m saying something like ‘I know some fact about the sky, rather than it being green.’ 

So, my suggestion is that negative knowledge should be understood as placing the asserted knowledge into the contrast proposition, and that there is some positive statement that is being asserted but left unsaid. Though the obvious way to fill this is in ‘the sky is blue,’ this need not be so, however, this should give us a way of handling negative knowledge. Just as in the usual case I am leaving the contrasts unspoken (presumably because it would be impractical to enumerate all the possibilities I have eliminated,) so in the case of negative knowledge the known proposition is being left unspoken. Of course, however we fill out the known proposition here must still be mutually exclusive, salient to the contrast and true (if it is truly known,) so in this case, the obvious way to cash it out is ‘I know the sky is blue, rather than the sky is green.’ Doesn’t that, however, open us up to the charge of claiming to know too much? Aren’t we dogmatically assuming that I know the sky is blue, rather than simply knowing that the sky is not-green? I think not; ‘I know the sky is not-green’ could be filled out in multiple, albeit false, ways. I might be in fact asserting ‘I know the sky is purple, rather than green,’ or something similar. All that is required is that contrastivism be able to capture statements of negative knowledge in a way that mirrors its capture of positive statements, and it seems this method shall suffice. So much for negative knowledge, next I shall consider Rourke’s ‘Claret Case’ as a counterexample to contrastivism.

3.2 The Claret Case

The Claret Case and its supplemented form (Rourke, 2013) (I shall call these the CC and the CCs from here) claim to offer a counterexample to the contrastive account of knowledge by offering scenarios when the mutual exclusivity of salient contrasts cannot be counted upon, which would prove problematic for contrastivism given what was said in section 2. It would be a most unfortunate consequence for contrastivism if Rourke is right and contrastivism cannot handle such situations, since this would imply either a bifurcation of the knowledge relation or that there are types of everyday knowledge that contrastivism is incapable of capturing. 

Rourke’s initial case, CC (p. 639), can be paraphrased as follows:

1. Holmes hopes to answer the question ‘Who drank the claret?’

2. There are three people present, Lestrade, Hopkins and LeVillard and no others

3. Watson announces, ‘Holmes knows that Lestrade drank the claret’.

In (c), ‘Holmes knows Lestrade drank the claret,’ is not mutually exclusive of any other person present having drank the claret. So, Watson can’t claim ‘Holmes knows Lestrade drank the claret rather than Hopkins,’ since it isn’t the case that Holmes can rule out Hopkins having drunk the claret, and the same for LeVillard, as LeVillard and Hopkins may have also drunk the claret, regardless of whether Lestrade did or not. So, what are we to make of the contrastive claim here? We are assuming Holmes knows that Lestrade drank the claret, but how are we to analyse such knowledge? Let us assume for what follows that Watson speaks the truth in both cases and in fact Holmes does have such knowledge, since this seems to be implicit in Rourke’s discussion. I shall come on to Rourke’s own possible response in a moment, but it seems to me that straight away we can cash it out in the following way ‘Holmes knows that at least Lestrade drank the claret, rather than the claret going unconsumed.’ This clearly captures what Holmes knows, provides a salient and mutually exclusive contrast, and accounts for Holmes’ inability to rule out Hopkins or LeVillard having consumed the claret. Rourke’s response to save contrastivism (p. 639) is to suggest that Holmes is in fact proceeding through a series of questions, ‘Did x drink the claret?’ and that Holmes only possesses the answer to the question ‘Did Lestrade drink the claret?’ Thereby, the contrastivist can simply present the problem as a series of smaller problems, each of which can be answered with ‘Holmes knows that x drank the claret, rather than x not having drunk the claret.’ I find this solution to be inelegant, and my answer provides a perfectly adequate response to the case. However, Rourke then goes on to outline the case CCs, which he argues avoids these worries (p. 641). I shall outline it as follows:

4. Inspector Bradstreet consumes some claret before Holmes and Watson arrive

5. Holmes and Watson have reason to believe that the claret may be poisoned

6. A person already present having consumed the claret suffices to establish its not having been poisoned

7. Holmes seeks to answer the question ‘Who drank the claret?’

8. Watson announces, ‘Holmes knows that Lestrade drank the claret.’

Rourke suggests that the method of breaking it down into a series of questions here makes no sense, since what Holmes knows is sufficient to answer the question of the claret’s safety without pursuing further questions as to whether Hopkins or LeVillard have drunk the claret. Nonetheless, the account I offered should certainly suffice to explain what how Holmes stands in the knowing relation, even in this case. Even so, CCs is a bit strange. Surely, the question should not be ‘who drank the claret?’ but rather ‘has anyone drunk the claret?’ What Holmes knows there stands in the right sort of relation, and there is a salient, mutually exclusive contrast with the claret going unconsumed, and Holmes can answer the question quite easily. 

So, it would appear Rourke’s case does present a problem for contrastivism, however, the possibility of ineliminable alternatives does seem problematic for contrastivism. Are there ever any such cases? I can’t think of any, it doesn’t seem plausible that there are cases in the everyday sense whereby there are simply no contrasts that are salient and mutually exclusive, however, if one were found, it would be problematic for my case. Rourke’s general strategy certainly has the potential to generate problems for the contrastivist if cases can be found where no reasonable method exists for the contrastivist to respond to them, however, I cannot find any, nor does Rourke propose any others. In the next section, I shall consider arguments from Baumann that mathematical and logical knowledge lack contrasts and that such knowledge does not enter into a contrastive relation. 

3.3 Mathematical and logical truths

In his paper, Baumann (2008) argues that for contrastivism to be plausible, it must meet the following condition:

‘(Specificity) A contrastivist analysis of knowledge of some type is plausible only if there are for a given subject S a lot of triples of propositions, p q and r such that S knows that p rather than q but S does not know that p rather than r.’ (p. 191)

Baumann asserts that mathematical knowledge, understood contrastively, will violate this principle. When considering basic mathematical truths, such as 2 + 2 = 4, there doesn’t seem to be a contrast proposition such that a person knows 2 + 2 = 4 rather than 2 + 2 = 5 but doesn’t know 2 + 2 = 4 rather than 2 + 2 = ¼, or for any other plausible set of values. This, Baumann suggests, goes for logical truths as well. Consider A ⊨ A. What might the contrast be here such that S knows A ⊨ A rather than A ⊨ ¬A, but S doesn’t know A ⊨ A rather than A ⊨ ¬(A ∧ B)? If Baumann is right, then this is a serious consequence of contrastivism, since it would imply that either there are two analyses of knowledge that correct (which Baumann endorses), or, that there are types of everyday knowledge that cannot be captured by contrastivism without resorting to Moorean dogmatism. I profoundly disagree with Baumann, for the following reasons.

Baumann’s reasons for (Specificity) are not well spelled out, all he offers by way of justification is as follows:

‘Contrastivism has its real point when analysing normal cases where a subject knows p rather than q but does not know that p rather than r…. If that were not the case, contrastivism would lose its point and attractiveness.’ (p. 191).

Though I agree that contrastivism does indeed handle well everyday cases as Baumann suggests, it does not strike me as at all obvious that it ‘loses its point and attractiveness’ when applied to cases of the mathematical and logical variety, in the absence of (Specificity). Though it may seem unintuitive to apply to mathematical and logical knowledge, it does not appear to me the case that such application strips contrastivism of its ‘point,’ which as I have presented it is to conserve everyday knowledge and known closure while avoiding Moorean dogmatism. Nor are these aims unattractive; indeed, a great deal of epistemological research has been devoted to them. Therefore, Baumann’s assertion seems without merit. Nonetheless, let us also look at the consequences of taking (Specificity) seriously. (Specificity) explicitly precludes any form of necessary knowledge from a contrastivist analysis, since for any necessary proposition pn, there is no proposition r such that a knower knows pn rather than q but does not know pn rather than r, since for any necessary proposition that an agent knows, they can know it contrasted with any appropriate contrast. Since this is so, Baumann’s acceptance of the contrastivist analysis for perceptual knowledge but not for mathematical knowledge would lead to the bifurcation of knowledge into at least two different relations, one that is correctly analysed according to contrastivism, and one that that is otherwise. Since Baumann gives us no other good reasons for (Specificity), and that the consequence is that we are then compelled to postulate a plurality of knowledge relations, we should reject (Specificity) due to the principle of parsimony.

Given that we have ground to reject (Specificity), then we can simply respond in the case of mathematical knowledge that ‘I know 2 + 2 = 4 rather than 2 + 2 = z’, where z is some number other than 4 that I know. Now, this doesn’t work so readily for logical knowledge, since it isn’t so easy to cash out ‘some other value’ as it is in mathematics, but in the case of S knows A ⊨ A, the obvious contrast proposition is A ⊨ ¬A, and if we are rejecting (Specificity), that seems sufficient. To give a sense of a more plausible case, I am going to borrow from Schaffer (2012b, p. 412). Consider asking me the value of 27 x 513 x -1. I could say I know that the answer is negative rather than positive, and that I know the answer will end in a 1 rather than some other digit. Here, because the answer is unobvious, but I can make certain assertions based on my prior mathematical knowledge, the contrastive account becomes much more plausible. In cases like 2 + 2 = 4, the contrast proposition may not be obvious, may require us to cash it out in algebraic terms, but that is a cost I am willing to accept.  

4 Conclusion

In this paper I have argued that contrastivism can retain three desiderata in the face of the skeptical paradox: the preservation of everyday knowledge, the avoidance of Moorean dogmatism and the preservation of known entailment. I did this by showing first how contrastivism resolves the skeptical paradox while retaining everyday knowledge, as well as aligning with our intuitions about knowledge in other ways. Then, I proposed a series of principles for contrastive closure, drawing heavily on the work of Schaffer and Kvanvig, gave a rough picture of how I see the concept of saliency operating within contrastivism, and Hughes’ objection by proposing my own addition to these principles. Finally, I addressed consequences that flow from the contrastive account of knowledge, both of my own and from the literature, which seek to commit the contrastivist to failing to capture everyday knowledge, endorsing Moorean dogmatism, or the unparsimonious bifurcation of knowledge. Taken together, these arguments represent a defence of contrastivism as an analysis of the knowledge that meets the desiderata I set in the introduction. Much work, however, remains to be done in spelling out the nature of saliency, as it has been used here, and how exactly we should understand the notion of contrast, but that must be for future research. 

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