^1. Good accounts include Kaye, D.H., Clarifying the Burden of Persuasion: What Bayesian Decision Rules Do and Do Not Do, 3 Evidence & Proof1 (1999)Google Scholar; Hamer, D., Probabilistic Standards of Proof, Their Complements, and the Errors That Are Expected to Flow from Them, 1 U. N. Eng. L.J.71 (2004)Google Scholar

^2. See, e.g., L.J. Cohen, The Probable and the Provable (1977), ch. 5; Allen, R.J. & Jehl, S.A., Burdens of Persuasion in Civil Cases: Algorithms v. Explanations, 4 Mich. St. L. Rev.893 (2003)Google Scholar

^3. An excellent review of the literature is H.L. Ho, A Philosophy of Evidence Law: Justice in the Search for Truth (2008), at 135–143

^4. This example originates in Nesson, C.R., Reasonable Doubt and Permissive Inferences: The Value of Complexity, 92 Harv. L. Rev.1187 (1979)Google Scholar

^5. The example is taken from R.A. Duff, Dangerousness and Citizenship, in Fundamentals of Sentencing Theory: Essays in Honour of Andrew von Hirsch (A. Ashworth & M. Wasik eds., 1998)

^6. A variation on Summers v. Tice, 33 Cal.2d 80, 199 P.2d 1 (1948). In the actual case, the court found both defendants liable. This version is taken from J.J. Thomson, Liability and Individualized Evidence, in Rights, Restitution and Risk: Essays in Moral Theory (W.A. Parent ed., 1986)

^7. The Shonubi case gave rise to a number of different judgments. For an introduction to the litigation, see A.J. Izenman, Introduction to Two Views on the Shonubi Case, in Statistical Science in the Courtroom (J.L. Gastwirth ed., 2000)

^8. Wells, G.L., Naked Statistical Evidence of Liability: Is Subjective Probability Enough?, 62 J. Personality & Soc. Psychol.739 (1992)Google Scholar

^9. A useful review of the case law is Koehler, J.J., When Do Courts Think Base Rate Statistics Are Relevant?, 42 Jurimetrics J.373 (2002)Google Scholar

^10. 371 Mass. 469, 58 N.E.2d 754 (1945)

^11. United States v. Shonubi, 998 F.2d 84 (2d Cir. 1993); see also United States v. Shonubi, 103 F.3d 1085 (2d Cir. 1997)

^12. Wells, supra note 8

^13. See, generally, F. Schauer, Profiles, Probabilities, and Stereotypes (2003), ch. 3

^14. See Shonubi, 998 F.2d at 89

^15. See, e.g., Posner, R.A., An Economic Approach to the Law of Evidence, 51 Stan. L. Rev.1477 (1999), at 1509Google Scholar

^16. See Wells, supra note 8

^17. See, e.g., Posner, supra note 15, at 1510

^18. For an overview, see Redmayne, M., Objective Probability and the Law of Evidence, 2 Law, Probability & Risk275 (2003)CrossRefGoogle Scholar

^19. See, e.g., Finetti, B. de, Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science, 31 Erkenntnis169 (1989)Google Scholar

^20. Thus you could not also assign P (0.8) to “the moon is not made of cheese” or to “no large objects are made of cheese.”

^21. See M. Kaplan, Decision Theory as Philosophy (1996), at 85–88

^22. See, e.g., D. Gillies, Philosophical Theories of Probability (2000), at 119–125

^23. While the reference-class problem is generally associated with frequentist theories of probability, it can be molded to fit other theories of probability, too, with the exception of the extreme subjectivism rejected above. See Hájek, A., The Reference Class Problem Is Your Problem Too, 156 Synthese563 (2007)Google Scholar

^24. Allen, R.J. & Pardo, M.S., The Problematic Value of Mathematical Models of Evidence, 36 J. Legal Stud.107 (2007)Google Scholar; Colyvan, M., Regan, H.M. & Ferson, S., Is It a Crime to Belong to a Reference Class? 9 J. Pol. Phil.168 (2001)Google Scholar, reprinted in Probability Is the Very Guide of Life: The Philosophical Uses of Chance (H.E. Kyburg & M. Thalos eds., 2003)

^25. Allen & Pardo, supra note 24, at 109

^26. Colyvan, Regan & Ferson, supra note 24, at 172

^27. See Colyvan, M. & Regan, H.M., Legal Decisions and the Reference Class Problem, 11 Evidence & Proof274(2007), at 279Google Scholar, where the toll-collector argument is described as a “bad defence.” This is not as obvious in the original article

^28. Allen & Pardo, supra note 24 at 109

^29. See D.A. Schum, Evidential Foundations of Probabilistic Reasoning (1994), esp. ch. 3. See also Schauer, supra note 13

^30. They appear to allow that the reference-class problem can be overcome if the reference class is “homogenous”; it is not clear what this means, given that no human classes will ever be completely homogenous. In reality, there is some evidence to suggest that organized drug smugglers have little choice about the amount they smuggle, and this is likely to undermine the importance of factors personal to the smuggler in determining the amount swallowed; A.J. Izenman, Assessing the Statistical Evidence in the Shonubi Case, in Statistical Science in the Courtroom (J.L. Gastwirth ed., 2000), at 423–424

^31. See Gillies, supra note 22, at 122–125

^32. See A. Hájek, Conditional Probability Is the Very Guide of Life, in Probability Is the Very Guide of Life: The Philosophical Uses of Chance (H.E. Kyburg & M. Thalos eds., 2003)

^33. See, in particular, Cohen, supra note 2. Much of the debate sparked by this work can be found in Probability and Inference in the Law of Evidence (P. Tillers & E. Green eds., 1988), which collects papers originally published at 66 B.U. L. Rev. 377–952 (1986)

^34. See L.J. Cohen, The Dialogue of Reason: An Analysis of Analytical Philosophy (1986), ch. 3; see also the discussion, followed by commentary, in L.J. Cohen, Can Human Irrationality Be Experimentally Demonstrated?, 4 Behav. & Brain Sci. 317 (1981)

^35. See Cohen, Dialogue of Reason, supra note 34, at 165–192. See also Allen, R.J., On the Significance of Batting Averages and Strikeout Totals, 65 Tul. L. Rev.1093 (1991)Google Scholar

^36. Cohen, Dialogue of Reason, supra note 34, at 165.

^37. Id. at 180. Another way of making the point is that the factors are explanatory; explanation plays a significant role in Allen & Pardo's discussion of the reference-class problem. (Allen & Pardo, supra note 24); indeed, they would argue that the fundamental structure of proof is explanatory (see Pardo, M.S. & Allen, R.J., Juridical Proof and the Best Explanation, 27 Law & Phil.223 (2008))Google Scholar. This line of argument is not explored here, because in this context it adds nothing to causal accounts of the paradoxes

^38. Cohen, Dialogue of Reason, supra note 34, at 167

^39. Cohen comes close to accepting this: see id. at 166 for his “freshman” example; and discussion id. at 175, 177. But the precise point in the text is not explicitly discussed in Dialogue, where Cohen's emphasis is on psychological experiments that involve combining probabilities, and his argument is that one probability is more counterfactualizable than another and thus may rationally dominate calculations

^40. Cohen does argue that the noncounterfactualizability of this probability is significant. When discussing experiments in which subjects see colored chips drawn from a bag and are asked to state the probability that the bag is the one in which chips of a particular color predominate, Cohen criticizes psychologists for presuming that subjects should think that the draws are random. Subjects, he argues, are right to “allow for the effect of any bias in how chips lie, or are drawable from, a particular bag,” and thus should let the prior probability of picking a bag containing mainly, e.g., red chips dominate their calculations, because “the drawing of the bag is stated to have been at random” (id. at 169). This is a rather gerrymandered argument: one can raise the same concerns about bias in how the bags lie or are chosen; and in any case, the possibility of bias in the bag does not tell us whether the bias will be towards red chips or blue chips (though there is presumably more likely to be a pocket of balls of the predominant color)

^41. See T. McGrew, Direct Inference and the Problem of Induction, in Probability Is the Very Guide of Life: The Philosophical Uses of Chance (H.E. Kyburg & M. Thalos eds., 2003); Campbell, S. & Franklin, J., Randomness and the Justification of Induction, 138 Synthese79 (2004)Google Scholar; J.E. Adler, Belief's Own Ethics (2002), at 250–254

^42. In The Probable and the Provable (Cohen, supra note 2), Cohen's discussion is in terms of weight, but by the time of An Introduction to the Philosophy of Induction and Probability, he suggests that weight can be analyzed in terms of counterfactualizability (L.J. Cohen, An Introduction to the Philosophy of Induction and Probability (1989), at 109 n. 22)

^43. Wasserman, D.T., The Morality of Statistical Proof and the Risk of Mistaken Liability, 13 Cardozo L. Rev.935 (1991), at 943Google Scholar

^44. Id

^45. Id. at 949

^46. Duff, supra note 5

^47. See V. Tadros, Criminal Responsibility (2005), at 47–53. Duff discusses this theory of character in more detail in R.A. Duff, Criminal Attempts (1996), at 175–192

^48. Duff, supra note 5, at 155. In this context, the idea of a presumption of harmlessness seems to be due to J. Floud & W. Young, Dangerousness and Criminal Justice (1981), at 43–45. While Floud and Young view the presumption as an important moral constraint on incapacitation, they see it as a restraint on practice (only those convicted of an offence should be considered for incapacitation on the basis of likely future offending) and do not discuss the issue considered here: whether the presumption should constrain the sort of evidence we use to judge dangerousness

^49. Duff, supra note 5, at 156.

^50. Indeed, in one sense lawbreakers are more autonomous than the (literally) heteronomous law-abiding, and there is a certain romantic view of crime under which offending is an expression of freedom. Lurking here, however, are difficult questions about the meaning of autonomy. On a Kantian view of autonomy, the opposite conclusion would be reached, for lawbreakers lack autonomy because they cannot will their crimes as universal law. On this account of autonomy, see O. O'Neill, Autonomy, Coherence and Independence, in Liberalism, Citizenship, and Autonomy (D. Milligan & W.W. Miller eds., 1992). It is clear from context, however, that neither Wasserman nor Duff are using “autonomy” in this Kantian sense. Nor is there any need to settle debates about the meaning of autonomy for the purposes of the present discussion; Kantians are free to replace references to autonomy in the text with references to some other concept, such as independence.

^51. See J. Raz, The Morality of Freedom (1986), at 155–156; Patricia Smith, Liberalism and Affirmative Obligation (1998), at 87–88

^52. Colyvan, Regan & Ferson, supra note 24, at 171, emphasis in original; see also id. at 175. This echoes Wasserman's clearly moral point that in some cases where statistical evidence is used, D is disadvantaged “not by his general bad luck, but by the misconduct of his statistical ‘cellmates”’ Wasserman, supra note 43, at 946. See also Schauer, supra note 13, at 239, who, in an otherwise robust defence of the use of probabilities in Blue Bus, comments without elaboration that “the specter of imprisoning people because of the behavior of others or because of the aggregate behavior of a class in which they are placed is indeed frightening.”

^53. Peter Tillers also claims that a moral argument underlies Colyvan, Regan & Ferson's claims, though he suggests a slightly different one. See Tillers, P., If Wishes Were Horses: Discursive Comments on Attempts to Prevent Individuals from Being Unfairly Burdened by Their Reference Classes, 4 Law, Probability & Risk33 (2005)Google Scholar

^54. As they acknowledge; Colyvan, Regan & Ferson, supra note 24, at 175

^55. See, e.g., J.M. Doris, Lack of Character: Personality and Moral Behavior (2005). Cf. D.C. Funder, Personality Judgment: A Realistic Approach to Person Perception (1999)

^56. More, rather than as, suspicious because we have a larger and hence more reliable sample of the behavior of others

^57. See, e.g., I.H. Dennis, The Law of Evidence (3d ed. 2007), esp. ch. 2; P. Roberts & A. Zuckerman, Criminal Evidence (2004), esp. ch. 4; Ho, H.L., Justice in the Pursuit of Truth: A Moral Defence of the Similar Facts Rule, 35 Common L. World Rev. 51 (2006)Google Scholar. Cf. L. Laudan, Truth, Error and Criminal Law: An Essay in Legal Epistemology (2006), esp. chs. 7, 9

^58. Thomson, supra note 6

^59. See also Sorensen, R., Future Law: Prepunishment and the Causal Theory of Verdicts, 40 Noûs166 (2006)Google Scholar. Sorensen argues for a casual account of verdicts to explain why we do not prepunish crimes. His account differs from Thomson's in that he would not allow evidence of factors that cause (as opposed to being caused by) the crime, though he is prepared to allow some trade-off. Given that this would rule out motive evidence, it is problematic.

^60. Cf. Wright, R.W., Causation, Responsibility, Risk, Probability, Naked Statistics, and Proof, 73 Iowa L. Rev. 1001 (1998)Google Scholar, at 1059–1061

^61. Explaining just when a conclusion is due to luck is obviously difficult; see discussion in the following subsection; and J. Greco, Knowledge as Credit for True Belief, in Intellectual Virtue: Perspectives from Ethics and Epistemology (M. DePaul & L. Zagzebski eds., 2007)

^62. Thomson acknowledges Gilbert Harman for comments on the paper; Harman had already discussed the Lottery paradox in Gilbert Harman, Thought (1973), at 161

^63. J. Hawthorne, Knowledge and Lotteries (2004)

^64. The oddness can be accentuated by noting that if, after the draw, I read the number of the winning lottery ticket in the newspaper, I can claim to know which ticket won, even if the probability of a misprint is greater than the probability of a particular ticket winning

^65. Not because “truth” and “belief” are simple concepts that require no analysis, but because debates about these concepts are not central to the debates about knowledge surveyed here

^66. A good analysis of justification is S. Haack, Evidence and Inquiry: Towards Reconstruction in Epistemology (1993)

^67. See, further, Brewer, S., Scientific Expert Testimony and Intellectual Due Process, 107 Yale L.J. 1535 (1998), at 1596–1601Google ScholarPubMed

^68. The analysis of “inferential” accounts earlier in the paper could be thought of as exploring whether a particular type of justification, such as a causal one, is intrinsic to legal verdicts, but it is difficult to find support for any such requirement. The brief discussion below of contextualist approaches says a little more about the degree of justification required for a true belief to amount to knowledge

^69. See Gettier, E., Is Justified True Belief Knowledge?, 23 Analysis121 (1963)Google Scholar. A good discussion is Zagzebski, L., The Inescapability of Gettier Cases, 44 Phil. Q.65 (1994)Google Scholar

^70. See, e.g., R.A. Duff et al., The Trial on Trial: Volume Three: Towards a Normative Theory of the Criminal Trial (2007), at 89–91. For a skeptical response to this claim, see Beltrán, J.F., Legal Proof and Fact Finders' Beliefs, 12 Legal Theory293 (2006) at 301–303Google Scholar

^71. See Pardo, M.S., The Field of Evidence and the Field of Knowledge, 24 Law & Phil.321 (2005), at 322–323Google Scholar

^72. While the example used by Pardo (id.) is suggestive, there are more complex examples of Gettier cases, and it is not clear that these raise antiliability intuitions. For a skeptical view on importing the Gettier requirement to verdicts, see Sorensen, supra note 59

^73. This does raise the question of how high the justification criterion for knowledge should be. Beliefs can be more or less justified, but exactly how well justified does a belief have to be to count as knowledge? It is not easy to find a cutoff point, and some of the more radical analyses might allow something like weak knowledge that could be applied to civil cases; see S. Hetherington, Good Knowledge, Bad Knowledge: On Two Dogmas of Epistemology (2001). Contextualists also admit of variable standards for knowledge; contextualism is discussed briefly below, but it is doubtful that contextualists would go so low as the balance of probabilities.

^74. Nelkin, D.K., The Lottery Paradox, Knowledge, and Rationality, 109 Phil. Rev.373 (2000)Google Scholar. Nelkin's article also points to another way of analyzing the proof paradoxes. Nelkin draws parallels between Lottery and a similar paradox that involves belief rather than knowledge: you hold a ticket in a large lottery; it is initially plausible to suppose that you believe that the ticket will not win, as it is very improbable that it will. But you should then be prepared to form the same belief about any ticket in the lottery; the conjunction of your beliefs then entails that you believe that no ticket will win, hence the paradox. Nelkin resolves the belief version of the Lottery paradox by arguing that you do not in fact believe that your ticket will lose because, again, of the missing causal element. One might then argue that what is required for verdicts is belief rather than knowledge and that it is belief that is missing in the proof paradoxes. This possibility is not pursued at length here, largely because Nelkin's argument about the lack of belief in the belief version of the Lottery paradox has received little support. See, further, D. Christensen, Putting Logic in Its Place: Formal Constraints on Rational Belief (2004). For a skeptical view of a belief requirement for verdicts, see Beltrán, supra note 70

^75. Duncan Pritchard, Epistemic Luck (2005). See also the discussion in Duncan Pritchard, Knowledge, Luck and Lotteries, in New Waves in Epistemology (D. Pritchard & V. Hendricks eds., 2008)

^76. For objections to this account, see Hiller, A. & Neta, R., Safety and Epistemic Luck, 158 Synthese303 (2007)Google Scholar; Comesaña, J., Unsafe Knowledge, 146 Synthese395 (2005)Google Scholar; Lackey, J., Pritchard's Epistemic Luck, 56 Phil. Q.284 (2006)Google Scholar

^77. See Pritchard, Knowledge, Luck and Lotteries, supra note 75, at 46–48. Pritchard also suggests that I do not know that I will not be able to afford the expensive house if I own a lottery ticket. It is unclear whether Pritchard would stand by his claim in respect of Heart Attack if, for example, I have a history of sudden heart attacks in my family

^78. For an actual example, see R v. Smith, Court of Appeal Criminal Division (UK), 8 February 2000

^79. See Hiller & Neta, supra note 76

^80. See D. Nute, Topics in Conditional Logic (1980), at 65–73; R. Nozick, Invariances: The Structure of the Objective World (2001), at 148–155

^81. See, e.g., Hawthorne, supra note 63; Sosa, E., Relevant Alternatives, Contextualism Included, 119 Phil. Stud.35 (2004)Google Scholar

^82. Or at least, the responses of the philosophically literate. There have been no surveys of the responses of the general public to Lottery cases. Cf. Weinberg, J.M., Nichols, S. & Stich, S., Normativity and Epistemic Intuitions, 29 Phil. Topics429 (2001)Google Scholar

^83. See, e.g., J. Stanley, Knowledge and Practical Interests (2005); B. Weatherson, Questioning Contextualism, in Aspects of Knowing: Philosophical Essays (S. Hetherington ed., 2006); Davies, W.A., Knowledge Claims and Context: Loose Use, 132 Phil. Stud395 (2007)Google Scholar

^84. T. Williamson, Knowledge, Context and the Agent's Point of View, in Contextualism in Philosophy (G. Preyer & G. Peters eds., 2005); Williamson, T., Contextualism, Context-Sensitive Invariantism, and Knowledge of Knowledge, 55 Phil. Q.213 (2005)Google Scholar

^85. That said, quantification does sometimes seem to cause courts to refuse to convict even when the evidence is very strong: see, e.g., R v. Watters, Court of Appeal Criminal Division (UK), 19 October 2000. It may be that quantification alone makes doubt seem more vivid, and that particular forms of quantification (as in Blue Bus) have effects over and above this

^86. See Wells, supra note 8

^87. See Neidermeier, K.E. et al. , Jurors' Use of Naked Statistical Evidence: Exploring the Basis and Implications of the Wells Effect, 76 J. Personality & Soc. Psy.533 (1999)Google Scholar. See also Heller, K.J., The Cognitive Psychology of Circumstantial Evidence, 105 Mich. L. Rev.241 (2006)Google Scholar

^88. Koehler, J.J., The Psychology of Numbers in the Courtroom: How to Make DNA Match Statistics Seem Impressive or Insufficient, 74 S. Cal. L. Rev.1275 (2001)Google Scholar

^89. It also explains the embedded Lottery example given in discussion of Pritchard; it is easy to imagine that I might win even though that world is far off

^90. J.L. Austin, Other Minds, in Philosophical Papers (3d ed., J.O. Urmson & G.J. Warnock ed., 1979). It is not really accurate to describe Austin's account as a theory of knowledge; it hardly aims at completeness and is developed in the context of the problem of knowledge of other minds. However, Mark Kaplan is currently filling out the Austinian account. See M. Kaplan, If You Know You Can't be Wrong, in Epistemology Futures (S. Hetherington ed., 2006)

^91. Austin, supra note 90, at 85

^92. Id. at 82. One stream in the epistemological literature picks up on this insight and takes the nonexistence of “defeaters” as a condition for knowledge. Most tantalizingly, for our purposes, we find this comment: “A belief is epistemically justified for knowledge, according to the “criminal” standard that we endorse, when one has strong reasons in support of it, no undefeated epistemic reason to doubt it, and no undefeated epistemic reason to believe that one's evidence for it is unreliable”; E. Conee & R. Feldman, Evidentialism: Essays in Epistemology (2004), at 296. However, this does not really give us a clear answer to the puzzles that concern us: in Prisoners, what makes the slim possibility of the prisoner being innocent a defeater, when the same slim possibility of an eyewitness mistake is not?

^93. Austin, supra note 90, at 88. There are obvious connections here with contextualist approaches, noted in M. Kaplan, Deciding What You Know, in Knowledge and Inquiry: Essays on the Pragmatism of Isaac Levi (E.J. Olsson ed., 2006)

^94. See E. Craig, Knowledge and the State of Nature (1990), at 2; S. Haack, supra note 66, at 7

^95. See Kaplan, M., It's Not What You Know That Counts, 82 J. Phil.350 (1988)Google Scholar

^96. D can, of course, challenge the facts that he was near the crime scene, that the thread matched, and so on, but likewise, the blue bus company can challenge the fact that 60 percent of buses are blue. For actual cases that obviously call for conviction but where the evidence seems unchallengeable in the sense explored here, see R v. Straffen, [1952] 2 Q.B. 911; R v. Smith (1916) 11 Cr. App. R. 229; United States v. Veysey, 334 F.3d 600 (7th Cir. 2003). There are obvious connections here with Alex Stein's account of legal proof, though Stein arrives at the significance of challengeability by a different route; see A. Stein, Foundations of Evidence Law (2005); cf. Redmayne, M., The Structure of Evidence Law, 26 Oxford J. Legal Stud.805 (2006)Google Scholar; Nance, D.A, Allocating the Risk of Error, 13 Legal Theory129 (2007)Google Scholar

^97. See J.L. Kvanvig, The Value of Knowledge and the Pursuit of Understanding (2003)

^98. See J.L. Pollock & J. Cruz, Contemporary Theories of Knowledge (2d ed. 1999), at 13–15; Haack, supra note 66, at 7; Kaplan, supra note 95

^99. Pollock & Cruz, supra note 98, at 14

^100. And note that (see note 64 supra), our knowledge that the ticket lost may be less certain than our justified true belief that it did

^101. Cf. Weinberg, Nichols & Stich, supra note 82

^102. See, e.g., W. Lycan, On the Gettier Problem Problem, in Epistemology Futures (S. Hetherington ed., 2006), at 165 n. 22. Hetherington, supra note 73, at 104, describes the lottery proposition as “excellent, but not perfect, knowledge.” Olsson suggests that Isaac Levi is committed to the view that there is knowledge in Lottery, though he himself states that “our lack of knowledge [in Lottery] is as clear a part of common sense as any other claim I can think of”; E.J. Olsson, Levi and the Lottery, in Knowledge and Inquiry: Essays on the Pragmatism of Isaac Levi (E.J. Olsson ed., 2006), at 174

^103. P. Unger, Living High and Letting Die: Our Illusion of Innocence (1996)

^104. Sunstein, C., Moral Heuristics, 28 Behav. & Brain Sci.531 (2005)Google ScholarPubMed

^105. See also Spicer, F., Epistemic Intuitions and Epistemic Contextualism, 72 Phil. & Phenomenological Res.366 (2006)Google Scholar; Weatherson, B., What Good Are Counterexamples?, 115 Phil. Stud1 (2003)Google Scholar

^106. Thus note, for example, David Lewis's response to Unger in D. Lewis, Illusory Innocence? in Papers Ethics & Soc. Phil. (2000); and Barbara Fried's to Sunstein in Fried, B., Moral Heuristics and the Means/End Distinction, 28 Behav. & Brain Sci.549 (2005)Google Scholar

^107. A similar point is made in Shaviro, D., Statistical Probability Evidence and the Appearance of Justice, 103 Harv. L. Rev.530 (1989), at 543Google Scholar