Death of paradox: the killer logic beneath the standards of proof.

• Author: Clermont, Kevin M.
• Position: Introduction to II. Conjoining Assessments A. Fuzzy Operators, p. 1061-1106

The prevailing but contested view of proof standards is that fact-finders should determine facts by probabilistic reasoning. Given imperfect evidence, they first should ask themselves what they think the chances are that the burdened party would be right were the truth to become known, and they then should compare those chances to the applicable standard of proof.

I contend that for understanding the standards of proof the modern versions of logic--in particular, fuzzy logic and belief functions--work better than classical logic and probability theory. This modern logic suggests that fact-finders first assess evidence of an imprecisely perceived and described reality to form a fuzzy degree of belief in a fact's existence, and they then apply the standard of proof by comparing their belief in a fact's existence to their belief in its negation.

This understanding nicely explains how the standard of proof actually works in the law world. While conforming more closely to what we know of people's cognition, the new understanding captures better how the law formulates and manipulates the standards and it also gives a superior mental image of the fact-finders' task. One virtue of this conceptualization is that it is not a radical reconception. Another virtue is that it nevertheless manages to resolve some stubborn problems of proof including the infamous conjunction paradox.

TABLE OF CONTENTS INTRODUCTION I. ASSESSING EVIDENCE A. Theories 1. Psychology Theories 2. Probablity Theories 3. Zadeh's Fuzzy Logic B. Legal Application: Oradated Likelihood II. COJOINING ASSESSMENTS A. Fuzzy Operators 1. Maximum and Minimum 2. Product Rule Contrasted 3. Negation Operator B. Legal Application: Conjunction Paradox III. ANALYZING BELIERS A. Shafer's Belief Functions 1. Basics of Theory 2. Negation Operator 3. Lack of Proof B. Legal Application: Burden of Production IV. APPLYING STANDARDS A. Comparison of Beliefs B. Legal Application: Burden of Persuasion 1. Traditional View 2. Reformulated View 3. Implications of Reformulation CONCLUSION Le seul veritable voyage, le seul bain de Jouvence, ce ne serait pas d'aller vers de nouveaux paysages, mais d'avoir d'autres yeux, de voir l'univers avec les yeux d'un autre, de cent autres **

INTRODUCTION

We have made tremendous strides, albeit only recently, toward understanding the process of proof. The wonderful "new evidence" scholarship has made especial progress by shifting the focus of evidence scholarship from rules of admissibility to the nature of proof, while opening the door to interdisciplinary insights, including those from psychology. (1) Yet the new work has tended to remain either too wedded or overly hostile to subjective probabilities for evaluating evidence (2) and to Bayes' theorem for combining evidence, (3) and so caused the debates to become "unproductive and sterile." (4) In any event, the debates have left unsolved some troubling problems and paradoxes in our law on proof.

The "New Logic"

One specific diagnosis of this shortcoming is that the new evidence tended to neglect the contemporaneous advances in logic. (5) The new, so-called nonclassical logic looks and sounds much like standard logic but refuses to accept some critical assumptions. (6) Most commonly, the assumption rejected is that every proposition must either be true or be false, an assumption called the principle of bivalence. But if propositions are not bivalent, so that both P and not P can be true and false to a degree, then one can show that sometimes P equals not P--which is a rather disquieting contradiction. (7) Fashioning the new logic thus faced some challenges in its development.

The first move in the new logic of special interest to lawyers relates to and builds on the branch of modern philosophy, beginning with Bertrand Russell's work, that struggled with the problem of vagueness. (8) Work on vagueness addresses matters such as the famed sorites paradox of ancient Greece ("sorites" comes from the Greek word for heap):

Premise 1: if you start with a billion grains of sand, you have a heap of sand.

Premise 2: if you remove a single grain, you still have a heap.

If you repeat the removal again and again until you have one grain of sand left, then you will by logic still have a heap. But there is no heap. Thus, heap equals nonheap. Two true premises yield an absurd conclusion (or--to picture the paradox in another common way--start with Tom Cruise's full head of hair, and begin plucking hairs, yet Tom will by logic never become bald).

At some point the heap undeniably became a nonheap. Was there a fixed boundary? No, this is not a way out--at least according to most philosophers. A different path taken in the attempt to avoid the paradox leads to the embrace of many-valued logic. (9) This form of logic boldly declines the simplification offered by two-valued, or bivalent, logic built on a foundation of true/false with an excluded middle. It instead recognizes partial truths. Both a statement and its opposite can be true to a degree. In other words, sometimes you have neither a heap nor a nonheap, but something that falls in between, with the statement "this is a heap" being both true and not true. (10)

The second interesting elaboration of the new logic involves developments in the field of imprecise probability. (11) This field of mathematics provides a useful extension of probability theory whenever information is conflicting or scarce. The approach can work with many-valued logic as well as with two-valued logic. The basic idea is to use interval specifications of probability, with a lower and an upper probability. Despite its name, imprecise probability is more complete and accurate than precise probability in the real world where probabilistic imprecision prevails. In fact, traditional bivalent probability (within which I include the doctrine of random chance as well as the much newer subjective probability) appears as a special case in this theory. The rules associated with traditional probability, except those based on assuming an excluded middle, carry over to imprecise probability.

All this logic may be new, but it has an extended history of forerunners. Threads of many-valued logic have troubled thinkers since before Aristotle embraced bivalence, (12) even if their thoughts found more receptive soil in the East than in the West. (13) Imprecise probability goes back to the nineteenth century. (14) Nevertheless, the new logic has enjoyed a recent flowering, inspired by the development of quantum mechanics and instructed by those just-described advances in philosophy and mathematics.

"Fuzzy Logic"

The particular bloom known as fuzzy logic finds its roots in the seminal 1965 article by Berkeley Professor Lotfi Zadeh. (15) His critical contribution was to use degrees of membership in a fuzzy set running from 1 to 0, in place of strict membership in a crisp set classified as yes/no or as either 1 or 0. Yet fuzzy logic is not at all a fuzzy idea. (16) It became a formal system of logic, one that is by now highly developed and hence rather complicated. (17)

I do not mean to suggest that fuzzy logic resolves all the philosophical problems of vagueness (or that it is especially popular with pure philosophers). I am suggesting that fuzzy logic is a very useful tool for some purposes. Of course, it has become so well-known and dominant because of its countless practical applications, especially in the computer business and consumer electronics. (18) But its theory is wonderfully broad, extending easily to degrees of truth. It thereby proves very adaptable in imaging truth just as the law does. Indeed, of the various models for handling uncertainty, fuzzy logic seems to capture best the kinds of uncertainty that most bedevil law. (19) Accordingly, writers have previously voiced suspicions that it might relate to legal standards of proof. (20)

Herein, fuzzy logic will provide a handle on how to represent our legal understandings of likelihood. (21) But it is not an exclusive tool. "In order to treat different aspects of the same problems, we must therefore apply various theories related to the imprecision of knowledge." (22) Another, compatible theory will function herein as a description of how to make decisions based on those likelihood understandings.

"Belief Functions"

Useful for that purpose is Rutgers Professor Glenn Shafer's imposing elaboration of imprecise probability from 1976. (23) His work on belief functions built a bridge between fuzzy logic and traditional probability, and in the process nicely captured our legal decision-making scheme. (24) He used the word "belief' to invoke neither firm knowledge nor some squishy personal feeling, but rather the fact-finders' attempt to express their degree of certainty about the state of the real world as represented by the evidence put before them. (25) By allowing for representation of ignorance and indeterminacy of the evidence, he enabled beliefs to express uncertainty, again on a scale running from 1 to 0. (26) Indeed, his theory of belief functions rests on a highly rigorous mathematical base, managing to get quite close to achieving a unified theory of uncertainty. (27)

Belief function theory does not constitute a system of logic, unlike fuzzy logic. Instead, it is a branch of mathematics, like traditional probability. (28) Just as probability serves two-valued logic by handling a kind of uncertainty that the underlying logic system does not otherwise account for, belief function theory delivers mathematical notions that can extend many-valued logic. While probability treats first-order uncertainty about the existence of a fact, belief function notions supplement fuzzy logic by capturing and expressing the indeterminacy resulting from scarce information or conflictive evidence concerning the fact. Shafer's theory is thus similar to a scheme of second-order probability, (29) which admittedly has hitherto failed both the...