ABSTRACT
Almost incredible is that no one has ever formulated an adequate model for applying the standard of proof. What does the law call for? The usual formulation is that the factfinder must roughly test the finding on a scale of likelihood. So, the finding in a civil case must at least be more likely than not or, for the theoretically adventuresome, more than fifty percent probable. Yet everyone concedes that this formulation captures neither how human factfinders actually work nor, more surprisingly, how theory tells us that factfinders should work.
An emerging notion that the factfinder should compare the plaintiff's story to the defendant's story might be a step forward, but this relative plausibility conjecture has its problems. I contend instead that the mathematical theory of belief functions provides an alternative without those problems, and that the law in fact conforms to this theory. Under it, the standards of proof reveal themselves as instructions for the factfinder to compare the affirmative belief in the finding to any belief in its contradiction, but only after setting aside the range of belief that imperfect evidence leaves uncommitted. Accordingly, rather than requiring a civil case's elements to exceed fifty percent or comparing best stories, belief functions focus on whether the perhaps smallish imprecise belief exceeds its smallish imprecise contradiction. Belief functions extend easily to the other standards of proof. Moreover, belief functions nicely clarify the workings of burdens of persuasion and production.
CONTENTS INTRODUCTION I. PROOF BY FACTFINDERS' BELIEFS A. Belief Functions 1. Basics of Theory 2. Negation Operator 3. Lack of Proof B. Comparison of Beliefs II. BURDEN OF PERSUASION A. Traditional View B. Reformulated View 1. Preponderance of the Evidence 2. Clear and Convincing Evidence 3. Beyond a Reasonable Doubt III. BURDEN OF PRODUCTION A. Traditional View B. Reformulated View IV. OVERVIEW OF STANDARDS OF PROOF A. Compatibility of Reformulated and Current Standards B. Application of Reformulated Standards to Multiple Elements CONCLUSION INTRODUCTION
The different standards of proof determine outcome. Empirical proof supports that point, as long as the standard applied to the empirical proof itself is not too demanding. (1) In any event, standards are definitely worth worrying about. A firmer understanding would affect the resolution of many legal issues that arise in connections with standards and burdens of proof. Almost incredible, however, is that no one has yet formulated an adequate model of proofstandard application. What does the law call for the factfinder to do?
The standard of proof often finds expression in terms drawn from traditional probability theory. (2) The formulation would be something along the lines that the factfinder must test whether the finding meets or exceeds the required standard on a scale of likelihood, albeit merely a nonnumerical scale with coarse gradations such as: (1) slightest possibility, (2) reasonable possibility, (3) substantial possibility, (4) equipoise, (5) probability, (6) high probability, and (7) almost certainty. Nonetheless, a yearning for more precision pushes many armchair theorists to use numbers in describing the scale.
Take as a prime example the usual standard of proof in civil cases, which calls for a probability of more likely than not. "As every firstyear law student knows, the civil preponderanceoftheevidence standard requires that a plaintiff establish the probability of her claim to greater than 0.5." (3) A moment's reflection, however, reveals all sorts of problems with such a formulation of proof. First, there are the routine objections to speaking of proof in numerical terms. (4) Not only are percentages of likelihood not how people normally think about legal cases, but also use of numbers can mislead the factfinder. (5) As soon as the theorist thinks more deeply about the nature of proof, those numbers produce all kinds of paradoxes. (6) Second, the civil standard seems impossibly difficult:
If the plaintiff must prove that some fact, X, is more probable than its negation, notX, then the plaintiff should have to show not only the probability that the state of the world is such that X is true, but also the probability of every other possible state of the world in which X is not true. This would mean that in order to prevail, plaintiffs would have to disprove (or demonstrate the low likelihood of) each of the virtually limitless number of ways the world could have been at the relevant time. This would be a virtually impossible task, and thus, absent conclusive proof, plaintiffs would lose. (7) But in recognition of inevitably imperfect evidence, the law allows recovery upon much less than a fifty percent showing of probability. (8) Third, the civil standard simultaneously seems an impossibly easy one. The factfinder supposedly starts in a state of perfect ignorance, wherein the plaintiffs claim has a fiftyfifty chance by the indifference principle. So, introduction of a feather's weight of evidence should suffice for victory over a silent defendant. But we all know that in such a case, the plaintiff would lose by directed verdict. A feather's weight might swing the burden of persuasion, but it does not satisfy the burden of production. The reality is that the law requires much stronger evidence. (9)
Consequently, it is abundantly clear that academics need to "let go of their love for p > 0.5." (10) Among the various proffered alternatives, (11) the most frequently ballyhooed way to let go of the love is the relative plausibility theory. (12) It builds on psychology's story model of holistic evidenceprocessing. (13) The relative plausibility theory posits that the factfinder constructs the overall story (or stories, in some variants of the theory) that the plaintiff is spinning and another story (or stories) that the defendant is (or could be) spinning. (14) The factfinder then compares the two stories (or collections of stories) and gives victory to the plaintiff if the plaintiffs version is more plausible than the defendant's. (15) This choice between alternative competing narratives is largely an ordinal process rather than a cardinal one. (16) Relative plausibility has advantages besides drawing on the currently prevailing psychological literature. It shows a nontraditional embrace of relative judgment by the factfinder, in preference to humans' weaker skills at absolute judgment of likelihood. (17) Also, by inventing a test to apply only at the end of a trial, it sidesteps many of the difficulties and paradoxes of using a numerical standard like greater than fifty percent. (18)
Yet, even as most of its proponents admit, relative plausibility theory has its own problems. (19) First, an ordinal comparison cannot easily explain standards of proof higher or lower than preponderance of the evidence. (20) Standards from a reasonable suspicion up to evidence beyond a reasonable doubt are hard to express as a comparison of stories. (21) Second, a more obvious difficulty is that it does not track well what the law tells its factfinders about how to proceed. (22) The law says to proceed elementbyelement and apply the standard of proof to each element, not to create holistic stories and compare them. (23) Third, it diverges from the law by compelling the nonburdened party, or at least imposing a practical obligation, to choose and formulate a competing version of the truth. (24) The law allows the defendant to stand mute and still prevail. (25) Fourth, comparing the plaintiffs story only to the defendant's favorite story, rather than to all versions of nonliability, will result in recovery by plaintiffs more often than normatively desirable. (26) The plaintiff should lose if liability is less likely than nonliability, regardless of which story the defendant prefers. (27) Fifth, the theory comes with baggage. (28) It requires, for example, acceptance of some holistic account of factfinding like the story model. (29)
There must be a better, and perhaps simpler, way to conceive the standard of proof. The mathematical theory of belief functions provides an alternative superior to traditional probability theory and to the newer approach of relative plausibility. Part I will explain belief functions and how they can help understand the idea of a standard of proof applied to a single element of a case. Part II uses the theory to explain both the burden of persuasion and its associated array of standards of proof. Part III then uses the theory to explain both the burden of production and its role in safeguarding certain process and outcome values. Part IV steps back from the theory to see how it will work in the real world, including how belief functions work when applied to the multiple elements of a case.

PROOF BY FACTFINDERS' BELIEFS
The first step on the journey is to realize that the key assumption of classical logic makes every proposition absolutely either true or false, an assumption called the principle of bivalence. (30) Multivalent logic instead allows propositions to be both true and false to a degree, so they can take on middle values of truth. (31) Consequently, classical logic has no tools for handling partial truths, propositions that will forever be uncertainly stuck partway between false and true. Traditional probability theory, a mathematical supplement to classical logic, treats only the random odds of a proposition turning out to be either false or true. Contrariwise, multivalent logic developed to handle partial truths. Fuzzy logic is one example of multivalent logic. (32) Deciding how to proceed in a world of persisting uncertainty (including how to combine partial truths) logically differs from predicting how uncertainty will resolve itself into certainty (including how to calculate the odds of multiple events occurring together).
Even in the absence of that fundamental...