Does luck have a place in epistemology?

Transcription

1 DOI /s Does luck have a place in epistemology? Nathan Ballantyne Received: 24 June 2013 / Accepted: 5 August 2013 Springer Science+Business Media Dordrecht 2013 Abstract Some epistemologists hold that exploration and elaboration of the nature of luck will allow us to better understand knowledge. I argue this is a mistake. Keywords Anti-luck epistemology Knowledge Luck Degrees of luck Significance Control account of luck Gettier cases 1 Introduction Philosophers have recently turned to explore the concept of luck. What is luck? The investigation has been driven by hopes of intervention into long-standing philosophical debates that involve luck debates about knowledge, morality, justice, and freedom that have in the past proceeded with only a sketchy thumbnail picture of luck. 1 I will here consider the place of luck in epistemology. Duncan Pritchard writes that very few commentators have explored what it precisely means to say that knowledge is incompatible with luck. In particular, no commentator, so far as I am aware, has offered an account of what luck is and on this basis identified what it means for a true belief to be non-lucky. Instead, the notion of luck...is largely taken as an undefined primitive...intuitively...if we could add some flesh to this platitude [i.e., that knowledge is incompatible with luck] via an elucidation of the notion of luck, then it ought to be possible to cast light on a number 1 For recent discussion of luck s nature, see Rescher (1995), Latus (2003), Pritchard (2005), Coffman (2007, 2009), Riggs (2007, 2009), Lackey (2008), Levy (2009, 2011), Driver (2012). N. Ballantyne (B) Fordham University, Bronx, NY, USA n.ballantyne@gmail.com

2 of central debates in epistemology...central to the anti-luck epistemological research project is, of course, an account of luck, since without this very little can be usefully said about epistemic luck. (2007, p. 278) Riggs (2009, 2007) has also developed accounts of knowledge that start with detailed pictures of luck s nature. The project of anti-luck epistemology has been taken up by others, and all involved assume that an account of luck is crucial to the success of this project. Even critics of anti-luck epistemology apparently leave intact the idea that the right account of luck can illuminate epistemological issues. Philosophers working on other luck-involved topics have also tried to discern luck s nature. For instance, in his recent book on freedom and moral responsibility, Neil Levy devotes an entire chapter to defending his account of luck, before applying the lessons to debates concerning free will. Levy says we need a detailed account of luck in order to make progress: Since some...debates, in particular those in which freedom is the focus, turn on arguments concerning whether a certain class of event is a matter of luck, we need to focus on luck itself in order to resolve them. We cannot rest content with a pretheoretical or intuitive notion of luck, because philosophers intuitions sometimes conflict, and because all of us sometimes encounter cases that we are at a loss to confidently classify. (2011, p. 12) Let us call philosophers like Levy, Riggs, and Pritchard interventionists. They think that exploration and elaboration of the concept of luck will help us to advance preexisting philosophical debates. (But don t mistake interventionists for all philosophers who talk about luck while exploring philosophical issues. Many philosophers make do with a pretheoretical notion of luck. Some epistemologists, for instance, talk about luck without thereby invoking any particular view about luck s nature: they re only interested in so-called safety or sensitivity analyses of knowledge, not luck itself. 2 ) Here is my hunch. Without the possibility of intervention, research on luck s nature itself would seem at least a bit less interesting. Understanding luck is largely a means to our philosophical ends central questions about knowledge, morality, justice, or freedom. It s the philosophical work that a robust conception of luck might accomplish that makes luck worth caring about. If the prospects for intervention look dim, then work on luck would seem less important. Or so I am inclined to suspect. 3 When it comes to luck s intervention in debates about knowledge, I am no interventionist. I doubt that long-standing debates about knowledge will be advanced by introducing a worked-out concept of luck. The reason is simple: the notion of luck is ill-suited to an account of knowledge. I will argue that basic facts about luck that it comes in degrees and that degree of luck can be determined by the so-called Significance Condition bring problems for luck s intervention in epistemology. If the argument succeeds, theorists will do better to turn attention away from the fullblooded notion of luck and towards notions better suited to grappling with knowledge. 2 See Black (2011), for instance. 3 Milburn (manuscript) writes: I am aware of no contemporary philosopher who gives an account of luck who is not motivated directly or indirectly by issues involving epistemic or moral luck.

3 That s not to say that recent discussions have taught us nothing about luck. Not at all. One lesson, I say, is that luck won t help us answer long-standing questions about knowledge. 4 2 Degrees of luck Our starting point is the idea that luck comes in degrees. Some lucky events are luckier than others. Imagine that you and a friend buy scratch and win lottery tickets. You both scratch and win, but your friend gets $2,000,000 while you get a measly $20. She is luckier than you are. Or imagine that you enter a 1,000-ticket lottery and your friend enters a 1,000,000-ticket lottery, and you each win the same cash prize. Here as well your friend is luckier than you are. That luck comes in degrees is a perfectly obvious part of the nature of luck. This has been widely recognized. 5 In this section, I am going to discuss how existing accounts of luck might explain the fact that luck comes in degrees. In the following section, I ll show how anti-luck epistemology faces a challenge in part because luck is a degreed notion. Here is an outstanding question. What fixes an event s degree of luck? What determines how lucky an event is? Plausibly, our answer will depend on what we think the necessary conditions for luck are. The standard options are as follows: Significance Condition: event E is lucky for individual X only if E is in some respect significant (good or bad) for X. 6 Modal Condition: event E is lucky for individual X only if E might not have happened had things been no more than slightly different just before E occurred. Control Condition: event E is lucky for individual X only if E is not something X did intentionally. 7 Exploitation Condition: event E is lucky for individual X only if X has not successfully exploited E for some purpose. 8 4 In a previous essay of mine (2011), I tried to challenge interventionists by arguing that one necessary condition on luck brings trouble for anti-luck epistemology. The basic idea was that once we think through the impact of a robust account of luck on anti-luck epistemology, we should think either that it s not really luck that precludes knowledge or that knowledge is sensitive to non-epistemic factors and thus that pragmatic encroachment is true. I figured that anti-luck epistemologists would for the most part want to resist pragmatic encroachment, so I hoped the arguments might encourage them to rethink the place of luck in their theories. That was my first effort on behalf of non-interventionism. Here I try something different. 5 Riggs notes that luck comes in degrees it is not an all-or-nothing concept (2007, p. 334). Pritchard observes that [i]t is certainly the case that sometimes events occur which are so fortuitous that they appear to constitute a greater degree of luck than is usual (2005, p. 142, note 11). See also Rescher (1995, pp ), Pritchard and Smith (2004, pp ), Levy (2011, p. 16), Ballantyne (2012, p. 320), and Church (2013: section 1). 6 This condition is articulated in different ways by Rescher (1995), Pritchard (2005, pp ), Coffman (2007, pp ), and Ballantyne (2012, pp ). 7 For recent discussion, see Lackey (2008), Levy (2009), Coffman (2009), and Riggs (2009). 8 See Riggs (2009).

4 Different philosophers have defended different sets of conditions. I will here focus on three leading accounts of luck. The Modal Account features the Modal and Significance conditions (Pritchard 2005, 2007). The Control Account brings together the Control, Exploitation, and Significance conditions (Riggs 2009). And the Mixed Account is the Modal, Control, and Significance conditions (Coffman 2007; Levy 2011; Riggs 2007). 9 The three accounts do not make it equally easy to account for degrees of luck, as I ll suggest. The Modal and Mixed accounts more naturally explain why luck comes in degrees than the Control Account does. Any good account of luck should explain how luck comes in degrees, so the Modal and Mixed accounts may enjoy an advantage over the Control Account. So what determines how lucky an event is? Sometimes, facts about the magnitude of some or other necessary condition fix the degree of luck. This is easiest to see with the Significance Condition and the Modal Condition, so suppose those conditions are true, in step with both the Modal and Mixed accounts. Those accounts may hold that an event s degree of luck is determined, at least, by facts about the magnitude of the Significance and Modal conditions (see Pritchard and Smith (2004, pp ) and Rescher (1995, pp )). First, the event can be more or less significant for someone, and the fact about how significant it is helps to fix the event s degree of luck. All things being equal, an event that greatly benefits someone is luckier than an equally likely event that benefits her only a little winning $1,000,000 in a lottery is a greater stroke of luck than winning $10 in a lottery of the same size. To deny this is to embrace the idea that we can t increase degree of luck by increasing significance. That seems implausible when we reflect on ordinary cases. (In Sect. 2 below, I ll say more about why significance determines degree of luck.) Second, a lucky event can be more or less likely to have happened, and the fact about how likely it was will help to fix the event s degree of luck. All things being equal, an event that happens in the actual world and in 1 percent of nearby worlds is less lucky than a similar event that happens in the actual world and in.001 percent of nearby worlds. 10 There is more to say here, but I won t say it. In general, certain facts about the magnitude of the Modal and Significance conditions will help to fix a lucky event s degree of luck It s worth noting that Levy only defends the Mixed Account for what he calls chancy luck. He suggests that there is another type of luck non-chancy luck and it calls for a different analysis. 10 Levy writes: the proportion of nearby worlds in which an event which occurred in the actual world failed to occur gives us a metric by which to measure the degree of chanciness, and thereby of luckiness, of the event. The larger the proportion of nearby worlds in which the event did not occur, the chancier its occurrence in the actual world, and other things equal the luckier its occurrence (2011, p. 16). See Church (2013: Section 1) for some complications for such a proposal for how the Modal Condition brings in degrees of luck. Since Church s discussion focuses solely on the Modal Condition, even if his proposal for how the Modal Condition admits degrees of luck is correct, more remains to be said about how luck comes in degrees on the Modal and Mixed accounts. That s because, on those accounts, the Significance Condition also helps fix degree of luck (see Pritchard and Smith (2004, pp ), for example). 11 Rescher (1995, pp ) gives a formula for computing the degree of luck. Degree of luck is a function of the event s improbability and its significance. Rescher s formula implies that a slightly improbable event that is rather significant could be equally lucky to a rather improbable event that is only slightly significant.

5 It is somewhat harder to see how degree of luck can be fixed given the Control and Exploitation conditions. Let me explain by discussing the Control Condition. Consider a pair of possible cases: (i) you win $1000 in a 100-ticket lottery and (ii) you win $1000 in a 1,000,000-ticket lottery. The second event is luckier than the first one. We can assume that the significance of (i) and (ii) is equal for you all that matters for you is the winnings, not the fist-pumping thrill of winning a really big lottery. We can also assume you did not bring about either lottery win intentionally, and so the Control Condition holds. How might the Control Condition help to explain why (ii) is luckier than (i)? This question is especially pressing for advocates of the Control Account, because they reject the Modal Condition. 12 To begin to see why, notice that the most natural explanation for (ii) being luckier than (i) is that (ii) is less likely than (i). In other words, (i) obtained in a larger set of nearby possible worlds than the set in which (ii) obtained, and unlikely events are normally luckier than more likely ones, all things being equal. But this explanation for why (ii) is luckier than (i) does not square neatly with the Control Account. The trouble is that proponents of the Control Account claim that the Modal Condition is problematic and so if they appeal to the relative likelihood of the lucky events to explain each event s degree of luck, then that would seem curious. After all, the Modal Condition says that lucky events are a kind of low probability event, taken in modal terms. If theorists reject the Modal Condition but must rely on something a lot like it to explain the relative luckiness of (i) and (ii), we might wonder why an appeal to probability is needed to explain degrees of luck but isn t required to explain luck itself. Why kick probability out of the account of luck only to bring probability back in to explain degrees of luck? This point seems to tell in favour of the Modal or Mixed accounts of luck over the Control Account, supposing we can t account for why (ii)is luckier than (i) by appealing to the Control Condition (or the Exploitation Condition). The Control Account doesn t naturally make sense of pairs of equally significant lucky events which have different degrees of luck. But things may be even worse. It is unclear whether the Control Condition can introduce degrees of luck in any case. Suppose the Control Condition is true. Imagine that an archer shoots an arrow that hits the target after a gust of wind blows the wayward arrow into the target. (Bullseye!) Call this whole event E. The archer didn t bring about E intentionally. The Control Condition says that an event is lucky for someone only if it is not something someone did intentionally, and so the condition implies, together with whatever else is necessary for luck, that E is lucky. Now let s consider one way we might expect the Control Condition to admit degrees of luck. By filling in the archery case in slightly different ways, we might think that E s degree of luck depends on further details about the obtaining of the Control Condition. (a) The archer did not intend E, but she recognized there was a gust of wind coming up and very nearly intended to use the wind to her advantage when taking the shot. (b) The archer did not intend E, and she was completely unaware of the incoming gust of wind. It s tempting to say that E is luckier in (b) than in (a). For in (a) the archer nearly hit the bullseye without any stroke 12 See Riggs (2009). Not all proponents of the Control Condition reject the Modal Condition: see Coffman (2007)andLevy (2011).

6 of luck whereas in (b) the archer was in no position to use the wind for the shot. If this is correct, then there are facts about the obtaining of the Control Condition that determine how lucky an event is. The relevant facts apparently concern how likely it is that the Control Condition has obtained. Specifically, the Control Condition almost didn t obtain in (a) but it was almost certain that the Control Condition obtained in (b). All things being equal, a higher likelihood that the Control Condition obtained means a lucky event will have a greater degree of luck. But I doubt this is correct. Notice that the judgment about the difference in degrees of luck between the two archery cases depends on this general thought: supposing that E was lucky for you and you were positioned to bring E about intentionally, if you hadn t been positioned to bring E about intentionally, then E would have been even luckier for you than it was. There are potential counterexamples to that general thought. We can begin with this case. Suppose you hold a ticket in a lottery that has been rigged in your favour. You can win in one of two ways: press a button which will make you the winner, or allow the lottery to proceed fairly. You could make yourself the winner, but you decide to refrain from pushing the button. In fact, it was unlikely you d exploit the set-up and make yourself an illegitimate winner you re no cheater. Lo and behold, you win the lottery, fair and square. 13 Now for a second case: you hold a ticket in an equally-large lottery that has not been rigged, and you win this non-rigged lottery, fair and square. Isn t the degree of luck you enjoy in the two cases equal? It seems so. If that is correct and I am inclined to think it is then it is false that failing to be positioned to intentionally bring about a lucky event makes that event even luckier for you than if you were positioned to intentionally bring it about. The general thought noted earlier is mistaken. 14 With that point in hand, we may return to the archery cases. In (a) the archer did not intend E, but she recognized there was an incoming gust of wind and very nearly intended to use the wind to her advantage when taking the shot, whereas in (b) the archer did not intend E, and she was completely unaware of the incoming gust of wind. Though it may be tempting to say that E is luckier in (b) than in (a), the pair of lottery cases calls that into doubt. I surmise that the judgment that E is greater luck in (b) than in (a) depends on the thought that E is more likely in (a) than in (b), not on how likely it is in the two cases that the Control Condition obtained. Here is what we ve got so far. Luck comes in degrees, but it is unclear how (and whether) this works on some particular accounts of luck. Degree of luck can be determined by facts about the magnitude of the Modal and Significance conditions. The Control and Exploitation conditions might introduce degrees of luck, too, but there are open questions about how these conditions can do that, and whether the Control Account can make good sense of degrees of luck without borrowing something a lot like the Modal Condition This case is borrowed from Coffman (forthcoming). 14 I am grateful to E.J. Coffman here. 15 There are of course different kinds of luck moral, epistemic, distributive, and so on. We can assume that any kind of luck is a species member of the genus luck. But why think that my partial characterization of the metaphysics of luck ideas concerning degrees of luck, how significance contributes to degrees

7 3 Against luck s intervention in epistemology Anti-luck epistemology is a product of reflection on justification and knowledge after Gettier (1963). Let s recount a classic Gettier-style case, due to Chisholm (1966, p. 23, footnote 22). Imagine that while looking over a field, you see what appears to be a sheep and come to believe there is a sheep in the field. And you re right: just beyond a hill in the middle of the field, there is a sheep. It s out of view, though, and you have no idea it is there. What you see is a dog, convincingly dressed up as a sheep. You have a justified true belief that there is a sheep in the field. Does that belief count as knowledge? Conventional wisdom says that you don t know that a sheep is in the field because your belief is true by luck (or accident or coincidence or some fortuitous or chancy event). A justified true belief is not enough for knowledge because of this luck. But philosophers have found never-ending trouble in figuring out what must be added to true belief to yield knowledge. In Gettier s trail, philosophers devised many examples of a belief true by luck (or accident or ) in order to challenge proposed definitions of knowledge. And it came to pass that epistemologists received their slogan: knowledge precludes luck. This truism about luck is a critical starting point for many contemporary discussions of knowledge. Anti-luck theorists try to understand knowledge by investigating the kind of luck that precludes knowledge in Gettier-style cases ( knowledge-precluding luck hereafter) and they begin by giving an account of luck simpliciter. According to me, luck won t help us intervene in debates over knowledge. The basic problem is simple. If luck can successfully intervene, anti-luck theories won t have absurd consequences; but those theories do have absurd consequences, owing to the special place they give to luck. The solution is to abandon luck. The take-home lesson from the arguments is that theorists need not clarify luck while seeking to understand the nature of knowledge. Anti-luck theories have moved the discussion away from the relevant phenomena by exploring luck. There are at least a couple ways to reveal anti-luck epistemology s absurd consequences. We can begin by assuming that degree of luck is determined by the Significance Condition together with whatever other conditions theorists favour. Since the Significance Condition is common theoretical ground, it s the only specific condition I ll focus on here. My case against interventionism depends on a pair of arguments. The first argument shows that a thinker can be gettierized by slightly increasing the significance of the thinker s true belief. Footnote 15 continued of luck, and so on accurately reflects each species in the genus? What if I have not identified traits of luck (the genus) but instead traits of a specific kind or kinds of luck? (Thanks to an anonymous referee for these questions.) I ll make two brief points. First, merely raising these sensible questions doesn t imply that the traits I ve noted do not describe the genus. So far, I see no reason to deny the traits are genus-level traits. Second, even if I am wrong and the noted traits only describe some or other species of luck, those traits seem hold for the species of luck we re interested in here: the luck some epistemologists claim to find in Gettier cases ( veritic luck).

8 A1 Suppose the degree of knowledge-precluding luck D for the event of having a true belief that proposition p is not enough to prevent knowledge but is almost enough. A2 Suppose some thinker has a true belief that p that is lucky to degree D. A3 So, the thinker can have knowledge. A4 Increase the degree of luck slightly above D by increasing the magnitude of the significance of the event. Call the resulting degree of luck D plus. A5 Degree of luck D plus is enough to prevent knowledge of p (given that D is almost enough to do so). A6 So, the thinker cannot have knowledge of p. AC But A6 is absurd. A1 follows from the idea that some degrees of luck are compatible with knowledge and others are not. This is a starting point for any anti-luck epistemology. Riggs writes, for instance: Knowledge is not incompatible with luck full stop. It is incompatible with luck of certain kinds to a certain degree (2007, p. 330, emphasis added). A3 follows from A1 and A2 a thinker with a true belief who is not in a Gettier case can be positioned, so long as some other conditions are satisfied, to have knowledge. A4 is supported by the idea that the degree of luck is determined by facts about the conditions for luck. We have assumed that the Significance Condition is necessary for luck, so increasing the significance of the event changes its degree of luck. A6 follows from A4 and A5. AC reports that it seems absurd that we could prevent a thinker from potentially knowing just by making a true belief slightly more significant for her. If A1 A6 hold, we can gettierize a thinker, and thus prevent the thinker from being positioned to know, by ever-so-slightly increasing the magnitude of the significance of the relevant event. That seems dubious. Something here must go. The argument presupposes at steps A1 and A5 that the epistemically-interesting property that precludes knowledge is luck. This is the source of the trouble, it seems to me. Suppose that theorists instead identify the knowledgeprecluding property with what is captured by the Modal Condition (or the Control Condition), leaving aside the Significance Condition. The resulting view wouldn t be a genuine anti-luck epistemology, to be sure, but at least it wouldn t imply an absurd conclusion. Before we try a second argument, let us consider several objections. Objection 1 The argument assumes that some degrees of knowledge-precluding luck are compatible with knowledge and other degrees are not. But any degree of luck will eliminate knowledge. 16 So, the argument rests on a mistake that is, A1 and thus does not challenge anti-luck theories. Reply 1 The objection seems to lead to a relatively radical skepticism. Riggs writes: We frail and causally inept humans are never 100 percent responsible for anything we accomplish, so any theory that requires the total absence of luck for knowledge is a recipe for instant skepticism Knowledge is not incompatible with luck full stop. 16 Church (2013) defends anti-luck infallibilism. Church says that a sufficient standard of analysis of knowledge cannot allow for knowledge that is even marginally lucky.

9 It is incompatible with luck of certain kinds to a certain degree (2007, p. 330). The objection says that theorists can embrace a kind of infallibilism, where any degree of luck precludes knowledge, in order to save interventionism. That s a lot to ask. Objection 2 The argument assumes that degree of luck can be partly determined by the Significance Condition. But that assumption can be rejected. Supposing the Modal Account is correct, for instance, why not say that degree of luck is exclusively a matter of the magnitude of the Modal Condition, not the Significance Condition? Then degree of luck merely tracks how unlikely a lucky event is. We can suppose that a minimum threshold for significance must be satisfied for an event to be a matter of luck rather than a merely unlikely event but significance won t help to fix the degree of luck. As a result, we can t increase the degree of luck by merely increasing a lucky event s significance. 17 Reply 2 The objection proposes that degree of luck is determined entirely by the Modal Condition, but it s notable that prominent accounts of luck reject that proposal. Nicholas Rescher says that degree of luck is a function of the event s improbability and its significance (1995, pp ). Duncan Pritchard and Matthew Smith, advocates of the Modal Account, regard one advantage of their view that both the Modal Condition and the Significance Condition help to fix degrees of luck (2004, pp ). 18 So, the objection won t square with prominent views. (As far as I know, no work in the literature on luck defends the view found in the objection.) But what s to be said for the objection s account of luck? We should reject the account, because it cannot accommodate three kinds of plausible judgments about luck. (i) Some lucky events involve greater luck than more unlikely lucky events. Suppose that you win $1 in a 1000-ticket lottery and your friend wins $100,000 in a 995- ticket lottery. Notice if degree of luck is fixed entirely by the Modal Condition, as the objection says, then a more unlikely lucky event must always be luckier than some more significant but relatively more likely event. So, the objection s account implies that you are luckier to win $1 than your friend is to win $100,000. But that is dubious. Plausibly, your friend is luckier than you are, despite the fact that it was more likely that her ticket would win. It makes little sense to brag about your better luck: I sure am luckier than my friend, even though her winnings are more significant than mine. This seems to misunderstand what luck is. (ii) Some equally likely lucky events are not equally lucky. Suppose, for instance, that you win $1 in a lottery and your friend wins $100,000 in a lottery of the same size. Clearly, it s a much bigger stroke of luck for your friend to win $100,000 than for 17 Thanks to Duncan Pritchard for suggesting an objection like this and to an anonymous referee for encouraging me to explore it in more detail. 18 Pritchard and Smith write: A further advantage of employing [the Significance Condition] as a condition on luck is that it can account for a second sense in which luck comes in degrees which is different from that accommodated by [the Modal Condition]. In the case of [the Modal Condition], we capture degrees of luck in terms of how many near-by worlds the event in question obtains. A second sense in which luck admits of degrees, however, concerns the significance involved (2004, p. 19). They illustrate their point: if a pair of equally modally fragile events could harm someone, but the second event would do more harm than the first, then the second would be greater (bad) luck than the first (2004,p.20).

10 you to win $1, though the odds for each lottery, and thus the chances of winning, are equal. Another example: an unlikely event luckily prevents Green from stubbing her toe and an equally unlikely event luckily prevents Brown from falling out of a helicopter to his death. Brown is luckier than Green here. Of course, if degree of luck is fixed entirely by the Modal Condition, then two equally unlikely lucky events must always be equally lucky, no matter differences in significance. But that is dubious. It badly strained to say, for instance: Green and Brown are equally lucky, even though Brown s life was saved and Green just avoided stubbing his toe. (iii) Sometimes good lucky events and bad lucky events cancel out and significance can play a role in determining when they do or don t. Imagine that two equally unlikely events happen to you: on the same day, hackers withdraw $100 from your bank account and the bank deposits $100 into your account in error. Plausibly, your bad luck and your good luck cancel out. We can perhaps explain how that works, assuming that degree of luck depends entirely on the Modal Condition: the degree of luck of the hackers bank transaction is equal to, and negated by, the degree of luck of the bank s error. But here is another pair of equally unlikely events: on the same day, hackers withdraw $1,000 from your account and the bank deposits $100 into your account in error. Your bad luck and good luck do not cancel out, because the significance of the bad event isn t balanced off with the good event. Yet supposing degree of luck is fixed entirely by the Modal Condition, we should apparently say that the bad and good events cancel out in both situations. But that is dubious. We should think that significance also determines degree of luck and thus helps us see when good and bad luck cancel out. These three types of judgments strongly suggest that the objection s view of luck is mistaken. We should think that significance can count toward degree of luck. Objection 3 So-called pragmatic encroachment views in epistemology hold that knowledge is harder to get when the practical stakes rise (see Fantl and McGrath 2002, 2007; Hawthorne 2004). Thus, if pragmatic encroachment is regarded as a plausible position, AC will seem doubtful, because slightly increasing the practical significance of an event can prevent knowledge. There is nothing absurd about A6, so the argument fails. 19 Reply 3 I expect that many anti-luck epistemologists will deny pragmatic encroachment. They ll say that knowledge is purely epistemic and so is fixed by truthdirected factors like reliability or evidence, not a thinker s practical situation. But then many anti-luck theorists won t warm up to this objection. Ignoring that, a question: is it plausible, given pragmatic encroachment, that we can create Gettier cases simply by raising the stakes for having a true belief? Pragmatic encroachment says that we can eliminate knowledge by raising the stakes, but it does not say that the knowledge is eliminated by creating Gettier cases. The issue is whether we can genuinely create Gettier cases by raising the stakes for having a true belief, and the objection holds that we can. We may concede the point. 19 Ballantyne (2011) discusses the relationship between anti-luck epistemology and pragmatic encroachment.

11 Even so, the objection won t work. Recall that A1 has us suppose that there s a degree of luck D that is not enough to prevent knowledge. A4 is the idea that we increase the degree of luck above D by slightly increasing the significance of the thinker s having a true belief, with a resulting degree of luck of D plus. The objection maintains that D plus will eliminate knowledge because D plus is partly fixed by greater practical stakes. Let me grant, for argument s sake, that D plus is possibly partly determined by practical stakes enough to eliminate knowledge. But that is not necessary:d plus might not be partly determined by stakes that would eliminate knowledge. To see why, notice that we can sensibly assume that the initial degree of luck D is partly fixed by extremely low practical stakes that is, stakes that are nowhere near enough to eliminate knowledge on pragmatic encroachment views. (Proponents of pragmatic encroachment typically argue that high stakes prevent a thinker from being in a position to know, where something important rides on the truth of the thinker s getting it right.) To make this more concrete, suppose that Green s having a true belief about Blaise Pascal s birthday is significant for her because $1 rides on her having the truth. Suppose also that Green has a true belief here that s lucky to degree D, partly because of its level of significance. Of course, we can boost the significance of the true belief for Green while keeping the stakes extremely low. Imagine that $2 now rides on the truth of her belief. So, the degree of luck is D plus, as she s luckier now to have a true belief. But it appears that D plus is not partly fixed by practical stakes that would eliminate knowledge. So, it seems that we can move from D to D plus without introducing practical stakes great enough to eliminate knowledge. Therefore, if we run the argument while mindful of that, the objection fails. Turn now to a different argument. It shows that anti-luck theories imply that we can de-gettierize a thinker by slightly decreasing the significance of the thinker s true belief. B1 Suppose that degree of knowledge-precluding luck D for the event of having a true belief that proposition p is just enough to prevent knowledge. B2 Suppose that some thinker has a true belief that p that is lucky to degree D. B3 So, the thinker does not have knowledge that p. B4 Reduce the degree of luck slightly below D by diminishing the magnitude of the significance of the event of having a true belief that p. Call the resulting degree of luck D minus. B5 Degree of luck D minus is not enough luck to prevent knowledge of p (given that D is just enough to do so). B6 So, the thinker s true belief that p is not lucky enough to be prevented from being knowledge. BC But B6 is absurd. B1 follows from the idea that some degrees of luck are compatible with knowledge and others are not. B2 is an assumption that would hold in any standard Gettier case. B3 follows from B1 and B2. B4 is supported by the idea that the degree of luck is determined by facts about the conditions for luck. Since we re assuming the Significance Condition is necessary for luck, diminishing the significance of the relevant event

12 changes the event s degree of luck. B5 follows from two claims: the idea that some degrees of luck are compatible with knowledge and some are not, and from B1 which notes that the degree of luck D is just enough to eliminate knowledge. BC reports that B6 is absurd because we can t de-gettierize a thinker just by making the truth of the thinker s belief slightly less significant for her. In other words, if B1 B6 hold, then we can eliminate knowledge-precluding luck by ever-so-slightly diminishing the magnitude of the significance of having a true belief. But that seems impossible. The argument presupposes at steps B1 and B6 that luck is the epistemicallyinteresting property that precludes knowledge. Once again, I suggest that anti-luck theorists should see the trouble stemming from that idea. If theorists instead say that whatever-it-is that precludes knowledge in Gettier cases does not require the Significance Condition, then there s no absurdity but no anti-luck epistemology either. Let us consider some objections. Objection 4 Suppose, as some philosophers have argued, that having a true belief is always significant for a thinker to some degree, no matter its content (see Lynch 2004; Kvanvig 2008). Thus, the base-level significance of a true belief can t be diminished without also changing that belief s truth-value. But then it s curtains for the argument. To see why, assume that degree of luck D is partly fixed by the base-level significance of the thinker s true belief along with facts about other necessary conditions for luck for example, the magnitude of the Modal Condition. B4 says that we can reduce the degree of luck from D to D minus just by diminishing the magnitude of the true belief s significance. But B4 is false: it is impossible to reach degree D minus by diminishing the magnitude of the true belief s significance without making that belief false. To move from D to D minus, we need to give the thinker a false belief. And so we cannot de-gettierize a true belief by diminishing its significance. Reply 4 This sort of objection hangs on a controversial thesis about the value of true belief (see Sosa 2003; Whiting 2013), and one that many anti-luck theorists will hope to avoid. But the objection fails even if we grant that thesis about true belief s value. Recall that the objection assumes that we can t move from degree of luck D to D minus without making the true belief false. This assumption is doubtful. To see why, suppose that degree of luck D is partly fixed by a degree of significance ever-soslightly above the base-level significance that s guaranteed by the thesis that true belief always has some value. Just because every true belief is valuable doesn t mean a particular true belief can t enjoy more value than the base-level. Now suppose that we understand B4 as diminishing the magnitude of the true belief s significance from D to D minus while leaving the true belief s significance at the base-level. This allows the belief s truth-value to remain fixed and shows how we can, given anti-luck theories, de-gettierize a true belief by diminishing its significance. Objection 5 The two arguments against interventionism should be rejected because similar arguments are sophistical. Here is one copy-cat argument. Let s assume that knowledge requires a belief-forming process with reliability of degree 0.7. Now suppose a thinker truly believes p with degree of reliability 0.7, and the other conditions for knowledge are met, but then the belief-forming process becomes ever-so-slightly less reliable to degree , say. Isn t it absurd to say the thinker no longer knows?

13 It appears so. But then, copying the two arguments against interventionism, we may conclude that knowledge does not involve reliability. That seems like a sophistical argument against reliabilism, but then the arguments against anti-luck theories must also be sophistical. Thus, it is doubtful that we can create or eliminate knowledgeprecluding luck by slightly changing the significance of having a true belief. 20 Reply 5 The objection alleges that a sophistical argument against reliabilism is just like the two arguments against anti-luck theories. But the argument against reliabilism is not sophistical. So, the objection fails. Let me argue this out. Here is the argument against reliabilism we find in the objection. Reliability is not required for knowledge because we can prevent a thinker from being positioned to know just by slightly decreasing the true belief s degree of reliability. But that, says the objection, is absurd. Though the argument is alleged to be sophistical, it isn t. Supposing there is a threshold of reliability required for knowledge, it is not absurd that we can prevent knowledge by slightly decreasing a true belief s degree of reliability. If knowledge demands some particular degree of reliability, any degree of reliability below it won t deliver knowledge, even when other conditions for knowledge are met. So, the objection s argument against reliabilism has a false step that it s absurd that we can prevent knowledge by slightly decreasing reliability. The basic idea behind the reply is that slight changes in reliability can eliminate knowledge even though slight changes in knowledge-precluding luck based on changes in significance cannot eliminate knowledge. 21 But why? If slight changes in reliability can get rid of knowledge, then why can t slight changes in significance do likewise? 22 The answer goes to the heart of the two non-interventionist arguments. It is this. Slight changes in significance of the agent s having a true belief are not an intuitively-suitable basis for a change in knowledge whereas slight changes in reliability are. Note well: I do not say that slight changes in knowledge-precluding luck based on slight changes to the modal profile of a true belief intuitively cannot eliminate knowledge. Indeed, they can. For example, just by looking, you can know there is a real barn before you when there s only one fake barn in the near vicinity. But if there are hundreds of papier-mâché facsimiles of barns all around and only one real barn, you arguably cannot know that the barn you re looking at is real (see Goldman (1976, pp )). With so many fake barns around, it s a modally fragile matter that you are looking at the real one. Thus, changes to the Modal Condition seem to be a good basis for eliminating knowledge just like changes in degree of reliability or changes in the strength of a thinker s evidence. Knowledge is sensitive to changes of certain kinds, but anti-luck theories let knowledge come or go with the wrong kind of change. 20 Thanks to Ian Evans for suggesting this objection. 21 See Objection 3 and Reply 3 above for an important caveat on the basic idea. In brief: supposing that pragmatic encroachment is true, some changes in significance can eliminate knowledge namely, changes that introduce high enough stakes. But I ve proposed that we think of the change in significance in the arguments as not being great enough to create high stakes. So, even given pragmatic encroachment, not every change in significance can eliminate knowledge. 22 I am grateful to an anonymous referee for this question.

14 Objection 6 Here is another copy-cat argument. 23 Call G the property, whatever it is, that precludes knowledge in Gettier cases. Suppose that a situation involving G to degree D is not quite enough to prevent knowledge of proposition p. And suppose that a thinker has a true belief in that situation. It follows that the thinker can have knowledge. But now let s change the situation so that it involves G to degree D plus by increasing the property that G tracks or is constituted by. It follows that degree D plus of G is enough to prevent knowledge of p. Thus, the thinker cannot know p. But that is absurd. Taking our cues from the two non-interventionist arguments, we should reject the idea that Gettier cases feature G, the property that precludes knowledge in Gettier cases. That is plainly incoherent, however. But notice we ve been led here by reasoning that mimics the non-interventionist arguments. So, since the argument against the idea that Gettier cases involve G is defective, it seems the arguments against interventionism are, too. Reply 6 The copy-cat argument is importantly different than non-interventionist arguments. First, for all the objection tells us, the getterizing property G may be categorical and so does not come in degrees, unlike luck. If G is a categorical property, the copy-cat argument fizzles out. And so if we are unsure whether G is a categorical property or a degreed property, the objection shouldn t move us. But let us overlook that and consider a second point. Although it is absurd that we can create or eliminate knowledge by certain slight changes in significance, it is not absurd that we can eliminate a thinker s knowledge by slightly increasing the property that precludes knowledge in Gettier cases. Increasing G above a threshold will eliminate knowledge. To be sure, not every alleged candidate for that property will actually eliminate knowledge. Imagine that a sleep-deprived philosopher claims the property in question is partly constituted by facts about glacier melt in the Himalayas or the number of smallmouth bass in Lake Simcoe. In general, we shouldn t expect changes of such properties to eliminate knowledge. Objection 7 The argument against interventionism proposes that a slight change to the significance of having a true belief will (absurdly) create or eliminate knowledgeprecluding luck. But this assumes there are sharp cutoffs between knowledgeprecluding luck and non-knowledge-precluding luck. Why not instead think the boundaries will be vague? But then we can t create or eliminate knowledge-precluding luck by slightly changing the significance of the relevant event. In other words, it takes more than a slight change in degree to move between knowledge-precluding luck and nonknowledge-precluding luck some degrees of luck in between are borderline cases, neither one nor the other. Thus, A4 and B4 are doubtful and the arguments don t go through. Reply 7 Even if luck is vague, the arguments will not be disarmed. To better grasp the objection, let us suppose that the vagueness is higher-order (see Sorensen 2010, for example). On this sort of popular picture of vagueness, it will be vague where the boundary between knowledge-precluding luck and non-knowledge-precluding luck 23 An anonymous referee suggested this objection.

15 begins. Here is the idea more fully. On the continuum of degrees of luck, there s no point before which there are clear cases of knowledge-precluding luck and immediately after it borderline cases, and no point before which there are clear cases of non-knowledgeprecluding luck and immediately after it borderline cases. But there are clear cases somewhere on both sides of the vague boundary. Think about a vague boundary separating two degrees of luck: D 1 and D 2. The first degree is enough to preclude knowledge whereas the second is not. If an increase in significance moves the degree of luck from D 2 to D 1, knowledge-precluding luck is created. And if a decrease in significance moves from D 1 to D 2, knowledge-precluding luck is destroyed. By recognizing that luck is vague we can see that not all slight changes are guaranteed to create or eliminate knowledge-precluding luck. As a result, some degrees of luck between D 1 and D 2 won t fuel the two arguments, because those degrees fall within the vague boundary. They are neither knowledge-precluding luck nor not-knowledge-precluding luck. I find no reason here for thinking a change from D 1 to D 2 can t be relatively slight. But then there is no reason to deny A4 or B4. So, the objection fails. The objection teaches us that maximally slight changes are not sure to create or eliminate knowledgeprecluding luck, given that the relevant transitions may be vague. Even if we cut talk of slight changes in the arguments, the arguments still succeed. The worry, as I ve emphasized, is that it s possible to create or eliminate being positioned to know by sliding the degree of luck between D 1 and D 2 by changing the significance of having a true belief. That seems dubious even if the distance between D 1 and D 2 is a little more than slight. Knowledge-precluding luck may be vague. If that is correct, anti-luck epistemologists should consider what their theories inherit for knowledge. For instance, if luck is vague, there will be cases of justified true belief that are neither Gettier cases nor not Gettier cases. This seems to imply that there are cases of justified true belief that are neither knowledge nor not knowledge. Perhaps that is curious. But it s just what we would expect if knowledge itself is vague. We ve seen two arguments against luck s intervention in epistemology and replies to a handful of objections. All of this shows that anti-luck theories make eliminating and creating knowledge-precluding luck absurdly easy. These absurd consequences press theorists to abandon the idea that it is luck that precludes knowledge in Gettier cases. 24 As I see it, the simplest diagnosis of the trouble for anti-luck theories is that the Significance Condition is necessary for luck. To avoid the absurdity, theorists need only say that knowledge is precluded by some property similar to luck that does not require significance. Plausibly, concepts that are closely related to luck for instance, accident and coincidence don t bring along significance, and so theorists might do better to 24 Steven Hales suggested that if the arguments against inventionism in Sect. 2 succeed, then there is a successful intervention of luck in epistemology those very arguments. By exploring luck s nature, we have come to see that particular theories about knowledge fail. Clever enough! But interventionism, as I ve used the term, is a view about how we can make progress on long-standing, luck-involving philosophical questions. If the arguments here sweep aside some recent attempts to understand knowledge, luck hasn t intervened in any long-standing discussion. So, the non-inventionist arguments are not self-defeating.