THE PARADOXES OF LEGAL PROOF: A CRITICAL GUIDE MICHAEL S. PARDO ABSTRACT For several decades, a variety of paradoxes have fascinated and frustrated legal scholars and courts discussing evidence, procedure, and legal proof. These paradoxes concern issues such as statistical evidence, burdens and standards of proof, and rules for jury verdicts. As with other types of paradoxes, the paradoxes of legal proof raise fundamental issues and assumptions. In the legal context, the issues and assumptions are not merely of theoretical interest; they also have significant practical implications at trial and, indeed, throughout the processes of civil and criminal litigation. At the same time, there remains very little agreement about any of the paradoxes, and the issues they raise remain highly contested. This Article explores the different types of legal-proof paradoxes and the connections between them. In analyzing the structure of the paradoxes, the Article makes three contributions. First, it explains the practical significance of the paradoxes and why they raise fundamental issues for evidence law and civil and criminal procedure. Second, it reveals the necessary connections between the different types of paradoxes and the ways in which one type of paradox has implications for the others. These unexplored connections help to explain why individual paradoxes have resisted consensus and continue to prompt discussion and disagreement. The paradoxes are too often treated as isolated problems to be “solved” without appreciating how the underlying issues relate to those raised by the other paradoxes. Third, the analysis clarifies the primary source of confusion for each type of paradox: namely, the popular but mistaken assumption that standards of proof are probabilistic thresholds. Abandoning this spurious assumption provides a number of salutary theoretical and practical consequences, including greater clarity of the legal issues underlying the paradoxes, a better explanation of legal doctrine, and an improved understanding of the manifold litigation issues dependent on the legal-proof process. Henry Upson Sims Professor of Law, Culverhouse School of Law, The University of Alabama. For helpful comments on previous drafts, my thanks to Ron Allen, Kevin Clermont, Shahar Dillbary, Meredith Render, Fred Vars, and participants at an Alabama faculty workshop. My thanks to Dean Mark Brandon and the Alabama Law School Foundation for generous research support. 233 234 BOSTON UNIVERSITY LAW REVIEW [Vol. 99:233 CONTENTS INTRODUCTION ............................................................................................... 235 I. THE PROCESS OF PROOF ........................................................................... 241 A. Central Features .......................................................................... 241 B. Two Theoretical Accounts ........................................................... 245 1. The Probabilistic Account ..................................................... 245 2. The Explanatory Account ...................................................... 249 II. THE STATISTICAL-EVIDENCE PARADOX .................................................. 253 A. The Examples and Their Assumptions ......................................... 253 B. Proposed Solutions ...................................................................... 256 C. Statistical vs. Individualized Evidence ........................................ 262 III. THE CONJUNCTION PARADOX .................................................................. 266 A. What Is the Conjunction Paradox? ............................................. 267 B. Proposed Solutions ...................................................................... 270 1. Denying Premise 1 (Elements) .............................................. 271 2. Rejecting Premise 1 (Elements) ............................................ 273 3. Alternative Probabilistic Accounts ........................................ 275 4. The Explanatory Account ...................................................... 279 C. The Significance of the Conjunction Paradox ............................. 280 IV. THE VERDICT PARADOX .......................................................................... 282 CONCLUSION ................................................................................................... 288 2019] THE PARADOXES OF LEGAL PROOF: A CRITICAL GUIDE 235 “[P]aradoxes raise serious problems. To grapple with them is not merely to engage in an intellectual game, but is to come to grips with key issues.”1 INTRODUCTION Paradoxes have played important roles in evidence scholarship.2 Since their introduction into the legal literature, paradoxes have provided a continuing source of both fascination and frustration—and, if recent publications are any indication, this trend shows no signs of waning.3 On one hand, the paradoxes 1 R.M. SAINSBURY, PARADOXES 1 (3d ed. 2009). 2 See Roger C. Park & Michael J. Saks, Evidence Scholarship Reconsidered: Results of the Interdisciplinary Turn, 47 B.C. L. REV. 949, 986-88 (2006) (discussing role of paradoxes in evidence scholarship). There have been numerous influential discussions over the past five decades. See generally L. JONATHAN COHEN, THE PROBABLE AND THE PROVABLE (1977) (discussing issues with using probability in judicial process); FREDERICK SCHAUER, PROFILES, PROBABILITIES, AND STEREOTYPES (2003) (discussing “Blue Bus” paradox and sufficiency of statistical evidence in judicial process); Ronald J. Allen, A Reconceptualization of Civil Trials, 66 B.U. L. REV. 401 (1986) (using paradoxes to explore limits on compatability between conventional probability theory and conventional views of trials); David Kaye, The Paradox of the Gatecrasher and Other Stories, 1979 ARIZ. ST. L.J. 101 (discussing paradoxes to expose misconceptions about burden of proof and value of probability estimates); Charles Nesson, The Evidence or the Event? On Judicial Proof and the Acceptability of Verdicts, 98 HARV. L. REV. 1357 (1985) (discussing relationship between probability in factfinding process and public acceptance of jury verdicts); Laurence H. Tribe, Trial by Mathematics: Precision and Ritual in Legal Process, 84 HARV. L. REV. 1329 (1971) (considering suitability of utilizing mathematical methods in legal process and possible dangers). 3 See generally DALE A. NANCE, THE BURDENS OF PROOF: DISCRIMINATORY POWER, WEIGHT OF EVIDENCE, AND TENACITY OF BELIEF (2016) (exploring nature of burdens of proof through discussion of balancing evidence against opposing evidence and total “weight” of evidence); Edward K. Cheng, Reconceptualizing the Burden of Proof, 122 YALE L.J. 1254 (2013) (arguing that conceptualization of preponderance standard as probability ratio eliminates difficulties posed by classical evidentiary paradoxes); Kevin M. Clermont, Common Sense on Standards of Proof, 48 SETON HALL L. REV. 1057 (2018) [hereinafter Clermont, Common Sense] (examining use of belief functions to eliminate conjunction paradox problem for factfinders); Kevin M. Clermont, Death of Paradox: The Killer Logic Beneath the Standards of Proof, 88 NOTRE DAME L. REV. 1061 (2013) [hereinafter Clermont, Death of Paradox] (arguing that modern systems of logic, such as fuzzy logic and belief functions, better reflect realities of legal factfinding as opposed to classical logic and probability theory); David Enoch & Talia Fisher, Sense and “Sensitivity”: Epistemic and Instrumental Approaches to Statistical Evidence, 67 STAN. L. REV. 557 (2015) (providing potential justification for inferiority of statistical evidence as compared to individualized evidence and implications stemming from this conclusion); Jason Iuliano, Jury Voting Paradoxes, 113 MICH. L. REV. 405 (2014) (applying two philosophical paradoxes to jury decision-making and exploring potential problems); David S. Schwartz & Elliott R. Sober, The Conjunction Problem and the Logic of Jury Findings, 59 WM. & MARY L. REV. 619 (2017) (arguing conjunction paradox is of little consequence and problems it raises are insufficient to abandon proability theory in adjudicative fact-finding); Mark Spottswood, Unraveling the Conjunction Paradox, 15 LAW PROBABILITY & RISK 259 (2016) (critiquing arguments about conjunction paradox and suggesting problems posed by paradox can be 236 BOSTON UNIVERSITY LAW REVIEW [Vol. 99:233 provide stimuli for new research, theories, and analyses of legal issues; on the other hand, when ignored, they provide potential stumbling blocks for discussions of evidence and proof. This Article provides a comprehensive account of the legal-proof paradoxes with two goals in mind. The first goal is explanatory: to analyze the different types of paradoxes, the issues they raise, and their significance for law. In addition to this groundwork, the second goal is to defend specific positions with regard to the paradoxes and the underlying legal issues. Before introducing the proof paradoxes, it will be helpful to briefly outline features they share with paradoxes in other fields as well as the context in which they arise in the legal literature. The paradoxes share three features with paradoxes in other fields. First, they are relatively “easy to state and immediately provoke one into trying to ‘solve’ them.”4 This feature no doubt explains some of their continuing fascination.5 Second, they arise because there appears to be an inconsistency between accepted assumptions or because accepted assumptions appear to lead to unacceptable conclusions.6 Third,