TABLE OF CONTENTS INTRODUCTION 623 I. WHY DOES THE CONJUNCTION PARADOX RAISE A PROBLEM? 629 II. PREVIOUS SOLUTIONS TO THE CONJUNCTION PROBLEM 635 A. Alternative Forms of Probability 636 B. Elements, Not Claims 640 C. The "Explanatory" or "Inference to the Best Explanation" Approach 642 1. Avoiding the Conjunction Problem (1): The Holistic Solution 645 2. Avoiding the Conjunction Problem (2): The Probabilistic Solution 647 D. Conclusion: Are the Cures Worse than the Disease? 654 III. NARROWING THE "PROBABILITY GAP": THE MULTIPLICATION RULE FOR DEPENDENT EVENTS 655 A. Probabilistic Dependence and Independence 656 1. Formal Definitions and Multiplication Rules 656 2. Logical v. Probabilistic Independence 657 3. The Multiplication Rules 658 B. The Conjunction Paradox and the Assumption of Independence 659 C. The Probabilistic Dependence of Most Elements of Litigated Claims 661 1. Overlapping or Shared Elements 663 2. Mutually Relevant Elements 666 D. Applying the Proper Multiplication Rule 667 IV. THE NATURE OF THE JURY FUNCTION 668 A. Why Do We Have "Each Element" Jury Instructions? 669 B. The "Each Element" Instruction as an Entailment Check 670 C. What Do Jury Instructions Actually Say? 673 1. Methodology and Interpretations 675 2. Conjunction-Problem Jurisdictions 678 3. Non-Conjunction-Problem Jurisdictions 681 4. Special Verdict Forms 686 D. Reconsidering the Conjunction Problem: Theory Meets Practice 687 CONCLUSION 691 INTRODUCTION

An important theoretical inquiry that has long engaged evidence scholars involves the nature of legal fact-finding. Because the reconstruction of past events in litigation is inherently uncertain, the legal system adopts decision rules, known as burdens of proof, to permit fact-finders to reach decisions in the absence of complete certainty. (1) The burdens of proof--"beyond a reasonable doubt" in criminal cases, "preponderance of the evidence" in most civil cases, and occasionally "clear and convincing evidence"--express probability thresholds that the law says must be reached to sustain a civil claim or criminal charge. (2) Burdens of proof can thus be seen as a particular application of a broader question that philosophers would characterize as falling within the subfield of epistemology. This question is "under what conditions is belief justified?" (3) In trials, we say in effect that belief in the defendant's guilt or liability is justified when the burden of proof is met (or vice versa). It is axiomatic in our legal system that meeting the burden of proof requires evidence, but a more elusive theoretical question remains: How do we know when the burden of proof has been met? While as a practical matter judges and juries almost daily render seemingly acceptable decisions that the burden of proof in a case before them has or has not been met, evidence theorists continue to debate the theoretical underpinnings of this question.

A major point of contention in this evidence scholarship is what role, if any, probability theory actually plays, or should play, in telling us when the burden of proof has been met. One recent article claimed that "[m]ost evidence scholars believe that adjudicative fact-finding is fundamentally incompatible with mathematical probability," and therefore the latter is not a usable guide to the burden of proof question. (4) This skepticism is driven largely by the belief that "[amplication of mathematical probability in the courts of law engenders paradoxes and anomalies that are not easy to avoid or explain away." (5) Judging by the attention paid to it, the most serious of these paradoxes, posing the greatest challenge to probability theory, is "the conjunction problem." (6)

As framed by evidence scholars, the conjunction problem (also known as the "conjunction paradox") goes like this: Suppose a plaintiff's civil case consists of two elements--for example, defendant's negligence and causation of plaintiff's damages. The jury will be instructed that it must find each element to the degree of probability defining the burden of persuasion in civil cases: more than 50%, or 0.5 on a probability scale of 0 to 1. The jury may also be told--and if not, it will simply be a systemic background sup-position (7)--that a verdict for the plaintiff can be rendered only if the plaintiff's overall claim meets or exceeds this same probability threshold.

So here is the problem: In basic probability theory, the "multiplication rule for conjunction" (also known as "the product rule") holds that the probability of co-occurrence of two or more events is based on multiplying the probability of each. If occurrences A and B are mutually independent, then the probability of both A and B occurring together equals the product of the probabilities of A and B each occurring independently: Pr(A & B) = Pr(A) x Pr(B). (8) Now suppose that the plaintiff proves each element to a 0.6 probability. The probability that both elements are true is 0.6 x 0.6 = 0.36. If the jury focuses on the probabilities of "each element" separately without multiplying them, the plaintiff wins even though the overall probability of her claim is less than 0.5. If the jury instead focuses on the overall probability (0.36), the plaintiff loses even though she has met the burden of persuasion under the jury instruction to prove "each element" by a preponderance of the evidence (here, 0.6). Either way, there is an apparently irreconcilable tension between the two formulations of the burden of proof: one based on proving each element, and the other based on proving the whole claim. (9)

The conjunction problem thus depends on the presence of two assumed conditions about how the law directs fact-finders to decide claims--specifically, the insistence both that the claimant wins by proving each element to the probability threshold (the "each element" condition), and that the claimant wins by proving her overall claim to the probability threshold (the "whole claim" condition). (10) Notably, these legal decision rules are not themselves compelled by probability theory. (11) But once they are both assumed, probability theory completes the conjunction paradox by imposing the axiom that the probabilities of the elements must be multiplied to determine the probability of the overall claim, which is their conjunction. (12) The further axiom that those probabilities will always be between 0 and 1 dictates arithmetically that the product of the probabilities will always be less than or equal to the lowest individual probability. (13) Putting the "each element" and "whole claim" conditions together with the probability axioms, some claims will necessarily fall into a gap between these two conditions: that is, they will meet the "each element" condition but not the "whole claim" condition. (14) We will call this the "probability gap." Expositors of the conjunction problem seem to believe that a lot of claims will get lost in the probability gap. Hence, the conjunction paradox becomes the conjunction problem for conceptualizing the burden of proof as a predetermined probability threshold (such as 0.5 in civil cases and 0.9 in criminal cases).

The prevailing view seems to be that the conjunction problem cannot be solved as a formal matter, and that it instead needs to be worked around by a high-level theoretical solution. (15) The most popular contemporary workarounds involve changing the rules of probability, banishing probability theory from the legal arena entirely, or abandoning the assumption that a claimant must meet a predetermined probabilistic threshold for his overall claim. (16) Each of these theoretical fixes creates potential difficulties and costs that should make us want to be sure that the disease is worse than the cure. (17)

In this Article, we argue that the conjunction paradox in fact presents a theoretical problem of little if any consequence. Dropping one aspect of the "each element" condition--the assumption that proving each element is a sufficient, as opposed to merely a necessary, condition for proof of a claim--makes the conjunction problem inconsequential. In a very brief symposium essay written over thirty years ago, Professor Dale Nance made this observation, and pointed out that nothing in logic or probability theory requires this assumption. (18) He observed further that its foundation in the rules of adjudication is ambiguous at best. (19) Unfortunately, Nance's argument has received relatively little attention, and has not been developed at length. (20) This Article takes up where Nance left off. We show that most jurisdictions treat proof of the elements in their jury instructions as merely a necessary (not a sufficient) condition for sustaining a claim, although the practice of using special verdicts in civil cases introduces a measure of ambiguity about this. (21)

With the removal of the condition that proving each element is sufficient for the plaintiff to win, all that is left of the conjunction problem is the above-described "probability gap." In this Article, we also show that the probability gap (again, as a theoretical matter) has been exaggerated by expositors of the conjunction problem, who have mistakenly presented the problem as if all or most elements of claims are probabilistically independent. (22) By recognizing the probabilistic dependence of most facts internal to a given claim and applying the correct multiplication rule for probabilistically dependent events, the probability gap may be considerably narrowed. (23) In any event, the probability gap presents a problem of perception. The intuitive belief that too many cases fall into the probability gap is not a logical problem nor a reason to abandon probability theory as a useful analytical tool in the context of adjudicative fact-finding. Moreover, as we show in Part IV, using the multiplication rule for dependent events helps to illustrate the actual function of the "each element" jury instructions that have so...