¹ This is not Buchak's term.

² See Tribe, Laurence ‘Trial by Mathematics: Precision and Ritual in the Legal Process’ Harvard Law Review 84 (1971) 1329–1393Google Scholar; Cohen, JonathanThe Probable and the Provable (Oxford: Oxford University Press, 1977)CrossRefGoogle Scholar; Thomson, Judith Jarvis ‘Liability and Individualized Evidence’ Law and Contemporary Problems 49 (1986) 199–219CrossRefGoogle Scholar; Hock Lai Ho ‘The Legal Concept of Evidence’ Stanford Encyclopedia of Philosophy http://plato.stanford.edu/archives/win2015/entries/evidence-legal/ (Winter 2015); Redmayne, Mike ‘Exploring the Proof Paradoxes’ Legal Theory 14 (2008) 281–309CrossRefGoogle Scholar; Di Bello, MarcelloStatistics and Probability in Criminal Trials: The Good, the Bad and the Ugly Doctoral Dissertation, (Stanford University, 2013)Google Scholar; Enoch, David, Spectre, Levi, and Fisher, Talia ‘Statistical Evidence, Sensitivity, and the Legal Value of Knowledge’ Philosophy and Public Affairs 40 (2012) 197–224CrossRefGoogle Scholar; Smith, Martin ‘When Does Evidence Suffice for Conviction?’ Mind 127 (2018) 1193-1218CrossRefGoogle Scholar; Martin Smith ‘Accuracy, Proof and the Right Against Wrongful Conviction’ (ms); Clayton Littlejohn ‘Truth, Knowledge, and the Standard of Proof in Criminal Law’ Synthese (2017); Renee Bolinger ‘The Rational Impermissibility of Accepting (Some) Racial Generalizations’ Synthese (2018); Gardiner, Georgi ‘Legal Burdens of Proof and Statistical Evidence’ in Coady, David and Chase, James (eds.) The Routledge Handbook of Applied Epistemology (Routledge, 2018) 171–195Google Scholar; Michael Pardo ‘The Paradoxes of Legal Proof: A Critical Guide’ Boston University Law Review (2019); Julia Staffel ‘Three Puzzles about Lotteries’ in Igor Douven (ed.) Lotteries, Knowledge, and Rational Belief: Essays on the Lottery Paradox (Cambridge University Press, forthcoming). For a survey of proof paradox research, see Gardiner, Georgi ‘Legal Epistemology’ in Oxford Bibliographies: Philosophy, ed. Pritchard, Duncan (Oxford: Oxford University Press, 2019)Google Scholar.

³ See Morgan, E. M. ‘The Law of Evidence, 1941–45’ Harvard Law Review 59 (1946) 481–576CrossRefGoogle Scholar; Cohen op. cit. note 2, Thomson op. cit. note 2.

⁴ Pritchard, Duncan ‘Epistemic Risk’ Journal of Philosophy 113 (2016) 550–571CrossRefGoogle Scholar; Pritchard, Duncan ‘Legal Risk, Legal Evidence and the Arithmetic of Criminal Justice’ Jurisprudence 9 (2018) 108–119CrossRefGoogle Scholar. See also Pardo, Michael S. ‘Safety vs. Sensitivity: Possible Worlds and the Law of Evidence’ 24 Legal Theory (2018) 50–75CrossRefGoogle Scholar.

⁵ Pritchard, DuncanEpistemic Luck (Oxford: Oxford University Press, 2005)CrossRefGoogle Scholar; ‘Anti-Luck Epistemology’ Synthese 158 (2007) 277–297CrossRefGoogle Scholar; ‘Safety-Based Epistemology: Whither Now?’ Journal of Philosophical Research 34 (2009) 33–45CrossRefGoogle Scholar; ‘Anti-Luck Virtue Epistemology’ Journal of Philosophy 109 (2012) 247–279CrossRefGoogle Scholar. Although Pritchard's formulation divides nearby worlds into two discrete classes – nearby possible worlds and very close nearby possible worlds – this is best understood as a continuum: closer worlds are more significant for assessments of safety (2012: 255). For criticisms of Pritchard's safety-based approach to knowledge, see Comesaña, Juan ‘Unsafe Knowledge’ Synthese 146 (2005) 395–404CrossRefGoogle Scholar; Greco, John ‘Worries about Pritchard’s Safety’ Synthese 158 (2007) 299–302CrossRefGoogle Scholar; Hiller, Avram and Neta, Ram ‘Safety and Epistemic Luck’ Synthese 158 (2007) 303–313CrossRefGoogle Scholar; Pritchard (2009) op. cit. note 5; Miracchi, Lisa ‘Competence to Know’ Philosophical Studies 172 (2015) 29–56CrossRefGoogle Scholar; Gardiner, Georgi ‘Safety’s Swamp: Against the Value of Modal Stability’ American Philosophical Quarterly 54 (2017) 119–129Google Scholar; Priest, Maura ‘Why Anti-Luck Virtue Epistemology has No Luck with Closure’ Logos and Episteme 8 (2017) 493–515CrossRefGoogle Scholar.

⁶ Harman, Gilbert ‘Knowledge, Inference, and Explanation’ American Philosophical Quarterly 5 (1968) 164–173Google Scholar

⁷ Redmayne op. cit. note 2; Buchak, Lara ‘Belief, Credence, and Norms’ Philosophical Studies 169 (2014) 285–311CrossRefGoogle Scholar; Duncan Pritchard, ‘Risk’ Metaphilosophy (2015); Pritchard op. cit. note 4, and others explicitly state that the taxi examples closely resemble, or are identical to, lottery examples. As I argue in section five, the taxi vignettes exhibit significant epistemological differences from lottery vignettes. The prisoner version of the proof paradox more closely resembles the lottery vignettes, although there are nevertheless differences.

⁸ Pritchard, op. cit. note 4, 118.

⁹ Op. cit. note 7.

¹⁰ Op. cit. note 7, 292.

¹¹ Op. cit. note 7, 299.

¹² In reality, most prevalent diseases are multi-factorial and cannot be modelled simply; epidemiologically simple diseases are rare diseases. The American Heart Association reports cardiovascular deaths represent 30% of deaths globally. Nearly 50% African-American adults have cardiovascular disease. Cancer Research UK estimates the lifetime cancer risk is 50% for males, and 45% for females. I am grateful to Jelena Aleksic, Ilya Chevyrev, and Zoltán Molnár for helpful discussions.

¹³ The CDC reports >9.3% of the US have diabetes, and >33% US adults have prediabetes, where blood sugar levels are higher than normal but not high enough to classify as diabetes. 15–30% of prediabetics develop diabetes within five years. Black and Hispanic adults are about twice as likely to develop diabetes as non-Hispanic white adults.

¹⁴ Approximately 27% of males and 3% of females develop a groin hernia; Fitzgibbons, RJ Jr and Forse, RA ‘Clinical Practice. Groin Hernias in Adults’ The New England Journal of Medicine 372 (2015) 756–63CrossRefGoogle ScholarPubMed. The male-to-female ratio for colorectal cancer is 1.35 and for bladder cancer is 4.0; Dorak, M. Tefvik and Karpuzoglu, Ebru ‘Gender Differences in Cancer Susceptibility: An Inadequately Addressed Issue’ Frontiers in Genetics 3 (2012) 268–279CrossRefGoogle Scholar. Prostate and Skene gland cancer might offer a potential real-world analogue for disease B and – if the sexes were reversed – so might breast cancer.

¹⁵ Males smoke more (habit H) and contract corresponding diseases (disease C) at a higher rate; the male-to-female lung and bronchus cancer ratio is 1.52 (Dorak and Karpuzoglu op. cit. note 14). The male-to-female tuberculosis ratio approximates 2:1. In some nations smoking might explain up to one third of this sex difference; Nhamoyebonde, Shepherd and Leslie, Alasdair ‘Biological Differences Between the Sexes and Susceptibility to Tuberculosis’ The Journal of Infectious Diseases 209 (2014) s100–s106CrossRefGoogle ScholarPubMed.

¹⁶ Testosterone or cremaster and inguinal canal cells might be real-world analogues of M-molecules. On average, adult male testosterone levels are approximately eight times higher than adult female levels.

¹⁷ I argue that the prisoner vignettes, introduced below, are more similar to the Harman lottery vignettes and Kyburg lottery paradox than the taxi vignettes are.

¹⁸ Note, however, that Fiona's belief, based on bare statistical evidence, that ‘Jake committed the crime’ is insensitive. If the claim were false, S would nonetheless believe it. If intuitively Fiona's judgement is flawed, this result favours the sensitivity condition as an explanation for the inadequacy of bare statistical evidence.

¹⁹ Personal communication, April 2018. I am extremely grateful to Duncan Pritchard for many helpful conversations about these topics.

²⁰ An example: I ask my friend her grandmother's name. She replies truly and would not speak falsely in nearby worlds. Intuitively my resulting actual belief is safe and known; not easily could it be wrong. But this belief violates the more demanding proposed standard: Were I to instead ask many people, on many occasions, eventually likely someone would reply falsely. They would be mistaken or deceptive. Thus the proposed Pritchardian reply impugns many ordinary beliefs.

²¹ Nesson, Charles ‘Reasonable Doubt and Permissive Inferences: The Value of Complexity’ Harvard Law Review 92 (1979) 1187–1225CrossRefGoogle Scholar.

²² Smith op. cit. note 2; Pardo op. cit. note 2; Lai, Ho HockA Philosophy of Evidence Law (Oxford: Oxford University Press, 2008)Google Scholar; Redmayne op. cit. note 2; Pritchard op. cit. note 4; Buchak op. cit. note 7. Note a vignette can exhibit multiple epistemic errors. This means that Case One judgements could exemplify some shared epistemic faults, such as outright judgement on bare statistical evidence, whilst differing in other epistemic properties, such as whether the evidence is rendered inadequate by high error costs.

²³ Kyburg, Henry E.Probability and the Logic of Rational Belief (Wesleyan University Press, 1961)Google Scholar; Makinson, D. C. ‘The Paradox of the Preface’ Analysis 25 (1965) 205–207CrossRefGoogle Scholar.

²⁴ Smith op. cit. note 2.

²⁵ See also discussion of ‘capped convictions’ (Littlejohn, op. cit. note 2). As Smith (op. cit. note 2) notes, in the lottery and preface paradox vignettes once you endorse each individual claim, there is a further question, namely whether to conjoin individual verdicts and endorse the conjunction. The prisoner and taxi cases lack this option. Once you convict each individual, there is no isomorphic further question of whether to additionally convict the conjunction.

²⁶ Several interlocutors have suggested the crime distribution articulated in iPhone One says nothing about the probabilities concerning individuals in the building, but in Prisoner One the base rates do reveal something about the individuals. I demur, but this suggestion gestures at this difference between the iPhone and prisoner cases.

²⁷ Realistic prison populations exhibit subgroups, granted, but these reference classes are not essential to the case. By contrast, we cannot describe the taxi vignette without substantial assumptions about the relevant reference class. Thanks to Andy Mueller, Jon Garthoff, and participants in Sherri Roush's epistemology seminar at UCLA for helpful discussion.

²⁸ Granted, were a judge to often rule against taxi drivers on this evidence, she might eventually rule falsely. But this does not mean the original Taxi One verdict violates Pritchard's classic safety condition; it simply means if the judge continually repeated her method, one day she might judge falsely. As discussed above, many beliefs are safe, according to Pritchard's classic analysis, even though the method can potentially lead to false belief. This includes, for example, belief about a friend's grandparent's name, formed on the basis of asking that friend. Were you to ask many friends, you might one day form a similar belief falsely via a similar method. But this does not impugn the safety of your actual belief, which is modally secure. Put another way, if beliefs are unsafe merely because one day the method (might) result in a false belief, safety is too demanding.

²⁹ A further objection to Pritchard's treatment of the proof and profiling paradoxes is available to advocates of moral encroachment. The objection proceeds: Some vignette judgements are morally weighty, and so by treating the vignettes alike Pritchard thereby overlooks important aspects of epistemic and moral normativity. I describe moral encroachment in section six. For further criticisms of a safety-based approach to explaining the proof paradox, see Georgi Gardiner ‘Legal Evidence and Knowledge’ in Maria Lasonen-Aarnio and Clayton Littlejohn (eds.) The Routledge Handbook of the Philosophy of Evidence (Routledge, forthcoming).

³⁰ Cf. Enoch, Spectre, and Fisher op. cit. note 2; Enoch, David and Fisher, Talia ‘Sense And “Sensitivity”: Epistemic and Instrumental Approaches to Statistical Evidence’ Stanford Law Review 67 (2013) 557–611Google Scholar; David Enoch and Levi Spectre ‘Statistical Resentment' (ms). See also Roush, SherrilynTracking Truth (Oxford: Oxford University Press, 2007)Google Scholar.

³¹ Pace, Michael ‘The Epistemic Value of Moral Considerations: Justification, Moral Encroachment, and James’ ‘Will to Believe’’ Noûs 45 (2011) 239–268CrossRefGoogle Scholar; James Fritz ‘From Pragmatic Encroachment to Moral Encroachment’ Pacific Philosophical Quarterly (2017); Moss, SarahProbabilistic Knowledge (Oxford: Oxford University Press, 2018)CrossRefGoogle Scholar; Basu, Rima ‘Radical Moral Encroachment: The Moral Stakes of Racist Beliefs’ Philosophical Issues 29 (2019) 9–23CrossRefGoogle Scholar; Bolinger op. cit. note 2; Renee Bolinger ‘Varieties of Moral Encroachment’ (ms). For criticisms of moral encroachment, see Gardiner, Georgi ‘Evidentialism and Moral Encroachment’ in McCain, Kevin (ed.) Believing in Accordance with the Evidence (Springer, 2018) 169–195CrossRefGoogle Scholar.

³² Basu, Rima and Schroeder, Mark ‘Doxastic Wronging’ in Kim, Brian and McGrath, Matthew (eds.) Pragmatic Encroachment in Epistemology (Routledge, 2019) 181–205Google Scholar.

³³ Cf. Basu, RimaBeliefs That Wrong, doctoral dissertation, (University of Southern California 2018)Google Scholar.

³⁴ Op. cit. note 31. For criticism of Moss's moral encroachment view, see Georgi Gardiner ‘Relevance and Risk: How the Relevant Alternatives Framework Models the Epistemology of Risk’ Synthese (forthcoming).

³⁵ Thomson op. cit. note 2; Smith op. cit. note 2; Pritchard op. cit. note 4; Smith, MartinBetween Probability and Certainty: What Justifies Belief (Oxford: Oxford University Press, 2016)CrossRefGoogle Scholar.

³⁶ I am grateful to Mike Ashfield, Fabrizio Cariani, Jon Garthoff, Brian Hedden, Duncan Pritchard, Sherri Roush, Martin Smith, and Timothy Williamson for comments on earlier drafts. Thanks also to Kristoffer Ahlstrom-Vij, Jelena Aleksic, Boris Babic, Sven Bernecker, Renee Bolinger, Lara Buchak, Ilya Chevyrev, Julien Dutant, Amy Flowerree, Alan Hájek, Mareile Håland, Michael Hannon, Seth Lazar, Chad Lee-Stronach, Clayton Littlejohn, Silvia Milano, Zoltán Molnár, Sarah Moss, Andy Mueller, Julia Staffel, Katie Steele, and Brian Talbot for helpful discussions on these topics. I am grateful to audiences at the Ethics and Belief workshop at the University of Cologne, the Ethics and Risk workshop at the Australian National University, the Institute of Philosophy at the University of London, University of Edinburgh, and Oxford University. Finally, many thanks to Sherri Roush’s epistemology seminar at UCLA for fruitful discussions.