1. This knowledge impediment might continue to hold even if the worlds where the table is not red—say, because unbeknownst to us there is a law forbidding other-colored tables under red lighting—are far enough from the actual world to render your belief safe. If so, this belief would be safe but insensitive.

2. Throughout, we apply Sensitivity and Safety to beliefs, findings, evidence, verdicts, and so on. In some contexts it may be important to be stricter on what it is that Sensitivity and Safety are meant to apply to, but not in ours, we think: we think that the most basic cases are that of sensitive (or insensitive) and safe (or unsafe) evidence. But all the other cases can be understood in the natural way—a belief is sensitive iff it is based on sensitive evidence, and so on. We acknowledge that this “natural way” may not be simple, and that complications may arise, but not ones, we think, that are crucial for our discussion here.

3. Note that both Safety and Sensitivity are factive conditions: if your belief is safe or sensitive, it is true. Otherwise the closest world in which your belief is false is the actual world (and presumably it is also false in most of the closest worlds where you believe that it is true).

4. Pardo, Michael, Safety vs. Sensitivity, 24 Legal Theory 50 (2018)CrossRefGoogle Scholar. All page references are to this paper, unless otherwise stated.

5. See Enoch, David, Spectre, Levi & Fisher, Talia, Statistical Evidence, Sensitivity, and the Legal Value of Knowledge, 40 Phil. & Pub. Aff. 197 (2012)Google Scholar. For more details engaging legal doctrines, see Enoch, David & Fisher, Talia, Sense and Sensitivity: Epistemic and Instrumental Approaches to Statistical Evidence, 67 Stan. L. Rev. 557 (2015)Google Scholar.

6. For a survey of attempts, see Enoch & Fisher, supra note 5, §1. For more recent attempts, see, for instance, Gardiner, Georgi, Legal Burdens of Proof and Statistical Evidence, in Routledge Handbook of Applied Epistemology (Chase, James & Coady, David eds., 2018)Google Scholar; Smith, Martin, When Does Evidence Suffice for Conviction?, 127 Mind 1193 (2018)CrossRefGoogle Scholar.

7. John Hawthorne's Knowledge and Lotteries (2004) has many Lottery-type puzzles and paradoxes. The particular one we focus on was first presented in Cohen, Stewart, Contextualist Solutions to Epistemological Problems: Scepticism, Gettier, and the Lottery, 76 Australasian J. Phil. 289 (1998)Google Scholar.

8. See, for instance, the thought experiment of having to choose the legal evidence regime under which your children will live. Enoch, Spectre & Fisher, supra note 5, at 213. We do not have a proof that in all possible circumstances a system that is hospitable to statistical evidence is more reliable than one that isn't. Still, in many circumstances this would be the case, and anyway, those who argue for the legal significance of some epistemological notion should be willing to defend it in the face of diminished reliability.

9. Enoch, Spectre & Fisher, supra note 5, at 213.

10. For the Normic account of the puzzle of statistical evidence, see Smith, supra note 6. We briefly discuss Smith's account in our Statistical Evidence, Sensitivity, and the Legal Value of Knowledge, supra note 5, where we ultimately reject it for the same reason we ultimately reject a Sensitivity-based account.

11. Sensitivity also has the advantage—because of its structural similarity with the incentive story we give—of accounting for the more favorable attitudes the courts have toward “cold hit” DNA evidence. This is an advantage that is not mirrored, to our knowledge, by any of the other epistemological accounts.

12. We do not want to pretend that legal doctrine is cleaner or more coherent—or less messy—than it actually is. While often (and across many jurisdictions) naked statistical evidence is not even admissible, in some cases it is, and the suspicion toward it is expressed in terms of weight or some such. For some detailed discussion of legal doctrine here, see Enoch & Fisher, supra note 5. All that's needed for the incentive story in the text to go through is that other things being equal, the more weight that is accorded to statistical evidence the more it weakens the relevant primary behavior incentives supplied by the substantive law.

13. Dretske, Fred, Conclusive Reasons, 49 Australasian J. Phil. 1 (1971)Google Scholar; Robert Nozick, Philosophical Explanations (1981); DeRose, Keith, Solving the Skeptical Problem, 104 Phil. Rev. 1 (1995)CrossRefGoogle Scholar. Derose embeds Sensitivity within a contextual account the result of which is an account that differs in important ways from the former two.

14. Enoch, Spectre & Fisher, supra note 5, at 202.

15. In David Enoch & Levi Spectre, Statistical Resentment (unpublished manuscript), we extend the discussion even further, to cover also cases of statistical resentment and many others. The results for Sensitivity, and indeed, for our incentive-based story, are mixed.

16. The extent to which this is a criticism of Pardo is mitigated by the fact that for him statistical evidence—and certainly, our discussion of statistical evidence—is not the main theme. Even if these points fail to engage our discussion of statistical evidence, they may still be relevant to his general discussion of the legal (in)significance of Sensitivity.

17. See, for instance, our discussion of DNA evidence, especially in Enoch & Fisher, supra note 5. In some cases, we argue, DNA evidence may be both statistical and good evidence.

18. Tyson Foods, Inc. v. Bouaphakeo, 136 S. Ct. 1036 (2016).

19. We are here echoing Kant on the different place that definitions have in mathematics and in philosophy. See Beck, Lewis W., Kant's Theory of Definition, 65 Phil. Rev. 179, 187–188 (1956)CrossRefGoogle Scholar and the references there.

20. Paraphrasing Justice Potter Stewart's famous phrase about the test for obscenity in Jacobellis v. Ohio, 378 U.S. 184 (1964).

21. See Nozick's discussion of why neighborhoods of possible worlds deliver a better heuristic for determining whether a belief is sensitive (or whether if the agent were to believe that p, p would be true—his fourth, Safety-like condition). Nozick, supra note 13.

22. See Sosa, Ernest, How to Defeat Opposition to Moore, 13 Phil. Persp. 137 (1999)Google Scholar.

23. We thank both anonymous referees for pressing us on related points.

24. This does not mean, as one referee suggested, that we're giving up on the relevance of Sensitivity, proceeding to do things in terms of (Sensitivity-unrelated) incentives alone. First, although in atypical cases the incentive story and the Sensitivity story come apart, still in the more paradigmatic cases they do not. Second, although the belief that the trash bag is in the basement is not in this case sensitive, beliefs in its vicinity are (say, that I tried to get the trash bag down to the basement), and such beliefs are relevant here both intuitively and for the incentive story.

25. Additional well-known counterexamples to Sensitivity being necessary for knowledge are Kripke's red fake barn (Kripke, Saul, Nozick on Knowledge, in Philosophical Troubles: Collected Papers Vol. 1 162–224 (2011)Google Scholar) and Williamson's brain in a vat climber (Timothy Williamson, Knowledge and Its Limits (2000)). To take Williamson's example: though I know I am not a brain in a vat who believes I'm climbing the Himalayas (as it doesn't even seem to me that I'm climbing the Himalayas), this belief is insensitive. In the closest worlds where that belief is false—that is, where I am a brain in a vat who believes he's climbing the Himalayas—I still have that belief (namely, that I'm not a brain in a vat who believes he's climbing the Himalayas).

26. One major issue is whether in applying Sensitivity to a belief we're holding the evidence fixed or not. In Sosa's example we arguably only get the result that the belief is insensitive because we're holding the agent's evidence—that he's thrown the bag down the chute—fixed. Otherwise, the closest world where his belief is false is one where he doesn't even throw the bag, in which case nor does he believe that the bag is in the basement, and the belief is sensitive after all. We could play with the relevant belief—that the bag is in the basement? That it's reached the basement?—but this looks like cheating. Whichever way we go on whether or not the evidence should be fixed seems to call for justification, and indeed to raise problems. We return below to the question of what is held fixed in checking for Sensitivity.

27. Nozick, supra note 13.

28. Williamson, supra note 25. See also Williamson, Timothy, Replies to Critics, in Williamson on Knowledge 280–384 (Greenough, Patrick, Pritchard, Duncan & Williamson, Timothy eds., 2009)Google Scholar.

29. See Pardo's footnote 48 for an attempt—one whose details are not clear to us—to explain what is meant by “reliability.” Also consider: “The reliability of evidence is distinct from whether a belief is sensitive. The reason for this—and this reason will be an important reason when assessing legal evidence—is that the closest possible world in which a belief is false may not be a close possibility” (11). The first sentence here is true but unproblematic, as we are about to argue. What follows it is unclear: there is, as far as we can tell, no straightforward connection between the modal property Pardo describes and reliability.

30. Pardo speaks of Sensitivity (or Safety) tracking reliability, but he can't mean by that more than just a claim about them not coming apart too often. So this is what we focus on in the text.

31. “Second, sensitivity does not track the risk of drawing erroneous inferences from evidence in close possible worlds” (10).

32. There is reason to suspect that even if Pardo means to be working with a reliability-supporting Safety condition, this intention must ultimately fail. The notion of Safety that Pardo is working with is Sosa's (supra note 22)—e.g., in his use of Sosa's Trash Bag example (146). This construal of Safety doesn't square well with Pardo's reliability claim. In response to the worry that modal conditions are not deductively closed, Sosa says: “Yes, in fact this is one reason why our account of Safety is only a first approximation. Here now is a closer approximation. What is required for a belief to be safe is not just that it would be held only if true, but rather that it be based on a reliable indication” (149). In other words, Safety certainly doesn't secure reliability—reliability must be added to it externally as an extra condition.

33. Pardo often accuses Sensitivity of being a poor guide to probative value, but it is unclear what he means by that phrase. Sometimes it seems that he just takes high probative value to be the property of good evidence. If so, accusing Sensitivity of having poor probative value is of course a serious accusation, but it needs substantiating (it's not the substantiation). At other times he seems to think of probative value as very closely connected to reliability. In that case, the discussion in this section applies. For a discussion of statistical evidence that utilizes one precise notion of probative value—that of likelihood ratio—see Marcello Di Bello, Trial by Statistics: Is a High Probability of Guilt Enough to Convict?, Mind (forthcoming).

34. We thank both referees for pressing us on related points.

35. Let us mention another kind of case that is relevant here. Cases may be described when the availability of the statistical evidence itself may depend on the defendant's behavior. For instance, a specific gatecrasher may be a social magnet, such that his gatecrashing may make it the case that many more gatecrash. In such a case, the statistical evidence against him may be sensitive after all (because had he not gatecrashed, the statistical evidence would not have been the same, in fact, it would have been much weaker, perhaps too weak for conviction even under an evidence regime that admits naked statistical evidence). For this kind of case, see Blome-Tillman, Michael, Sensitivity, Causality, and Statistical Evidence in Courts of Law, 4 Thought 102, 106–107 (2015)Google Scholar.

But either the fact finder knows that the person in front of them has this kind of social effect, or they don't. If they do, then the very statistical information—that so many people crashed the gates—is direct, nonstatistical, and unproblematic evidence against the defendant. And if the fact finder doesn't know about the defendant's social status, then, we want to suggest, it makes sense to hold fixed the percentage of those attending the stadium who crashed the gates, for reasons similar to the ones in the text.

36. Supra note 13.

37. It needs to be said, though, that on the most developed construal of the Safety condition (Williamson, supra notes 25 and 28), we might just as well talk about knowledge. This is not only because Safety is a sufficient condition for knowledge according to Williamson, it is also because in order to determine—specifically in complex cases—whether there are worlds (possibilities of error) that are too close for a belief to be safe, we will have to determine that by appealing to knowledge itself.

38. Pardo takes this example from Ernest Sosa, Judgment & Agency (2015), at 119. Interestingly, Sosa uses it to argue against Pritchard's Safety condition according to which for a belief to be safe, it needs to be true in all nearby worlds. On Sosa's Safety condition, it need not be true in all of them.

39. See Jonathan Kvanvig, The Value of Knowledge and the Pursuit of Understanding (2004), at 135.

40. For instance, Sosa, supra note 38—on whom Pardo often relies—thinks lottery propositions may be known, and attributes the opposite impression to a confusion between a Safety condition (which he endorses) and a Sensitivity one (which he rejects).

41. If Christian Piller (Beware of Safety, forthcoming in Analytic Philosophy) is right, this is not just a one-off mistake. He claims that the appeal of Safety, and the mistaken assumption that it is an epistemological property to begin with, is owed to its similarity with Sensitivity. And as already noted, Sosa, supra note 38, argues in the opposite direction that our temptation to resort to Sensitivity confuses us into thinking that we do not know that our lottery tickets will lose (when they do).

42. It's not entirely clear what to say of the eyewitness-testimony case here. Clearly if we believe that p based on the testimony of a 70-percent-reliable eyewitness, there are close worlds where we believe the witness but she is mistaken. So safe belief (that p) seems out of the question here, as is, for the Safety theorist, knowledge (that p). Perhaps the testimony suffices for weaker results (like knowledge that probably-p). However, we can get such knowledge from the market-share evidence as well. Whatever epistemic merits Safety might have, then, it seems ill-suited for the law even for the most basic case. See further discussion below.

43. Still, we want to emphasize that many of our points against Pardo earlier in the text are independent of this wider disagreement.

44. Pardo (14 n.63) is right, of course, that even if Sensitivity doesn't matter to the law, it doesn't follow that no other epistemological property is. Though this doesn't follow (and we never suggested that it did), we do believe—on independent grounds—the more general claim as well.

In the same footnote, Pardo accuses us of failing to appreciate knowledge's factivity—so that knowledge entails truth. He thinks that this failure on our part undermines the thought experiment we use to support the claim that no epistemic credentials justify any price in any other legally relevant value. But he is wrong here. First, as noted in the text, we do (of course!) acknowledge a role for truth. So what we say about knowledge is only about whatever else is needed for knowledge, on top of truth—this other stuff is not intrinsically legally significant. Second, and most troublingly, Pardo gets close to misquoting us. He says (of one of the worlds we ask the reader to compare) “World A in which jurors ‘only convict when they know,’…” noting that if they know that the accused is guilty, it follows that he is. Our text reads, however: “System A is epistemologically better: perhaps its courts only convict when they know (or think that they know) the accused is guilty, or perhaps they only convict based on sensitive evidence…” Enoch, Spectre & Fisher, supra note 5, at 212 (italics added). We don't know why Pardo dropped the bracketed comment, but this is misleading editing on his part. Obviously, knowledge is factive (so is Safety, by the way, which may spell further trouble for Pardo). Just as obviously, though, thinking-that-one-knows is not factive (and this, of course, is why we introduced the bracketed comment to begin with).

45. We draw here on Enoch, David, Comment on Yaffe's Attempts, 6 Jerusalem Rev. Legal Stud. 20, 32–33 (2012)CrossRefGoogle Scholar.

46. Here's David Lewis, taking for granted the point that we argue for in the text here (and elsewhere): “What if some far-fetched possibility is called to our attention not by a sceptical philosopher, but by counsel for the defence? We of the jury may wish to ignore it, and wish it had not been mentioned. If we ignored it now, we would bend the rules of cooperative conversation; but we may have good reason to do exactly that. (After all, what matters most to us as jurors is not whether we can truly be said to know; what really matters is what we should believe to what degree, and whether or not we should vote to convict.)” Lewis, David, Elusive Knowledge, 74 Australasian J. Phil. 549, 560 (1996)Google Scholar (italics added).