*By Mike Dorf*

In an important forthcoming article, my colleague Kevin Clermont argues that standards of proof and the law's method for combining probabilities of independent events can best be explained by "fuzzy logic" and "belief functions" rather than by the product rule of conventional bivalent logic. Confused? Let me explain--and then apply the principle to the First Circuit's recent decision invalidating Section 3 of the Defense of Marriage Act (DOMA).

Let's begin with what is sometimes called the "conjunction paradox." In order to prevail in a civil case, a plaintiff bears the burden of proving her claim by a preponderance of the evidence, i.e., of proving that it is more likely than not that she is entitled to relief. In instructing juries, judges routinely explain that this means that the plaintiff must prove

*each element*by a preponderance of the evidence. By way of illustration, suppose that the plaintiff is suing the defendant for battery. Let's imagine that the evidence shows that a masked assailant hurled a cherry pie in the direction of the plaintiff's face at the same exact time as a cherry pie came loose from a passing bakery truck. (Yes, I know this is an incredibly far-fetched scenario but the phenomenon exists in more realistic, albeit less vivid cases.) The evidence is contested on both fronts, so that there is some uncertainty as to whether the defendant was the masked assailant

*and*as to whether the pie that struck plaintiff's face came from the assailant or the truck. Let's suppose that after hearing all of the evidence, the jury puts the odds that the assailant was the defendant at 3:2 and also puts the odds that the pie that hit the plaintiff came from the pie-hurler at 3:2. In other words, there is a 60% chance that defendant was the assailant and a 60% chance that the plaintiff was struck by the intentionally-thrown pie. Under the judge's instructions, the plaintiff prevails, because the plaintiff has proved each element to be more probable than not (60>40). (If you're worried that the defendant's identity is not an "element," don't be. It's easy enough to give lots of other examples that clearly use elements.) But now we confront the paradox: Even though the plaintiff has proved each element to be more probable than not, conventional bivalent logic tells us that the likelihood that

*the defendant hurled a pie that hit the plaintiff*is the conjunction of two (we can assume) independent events, i.e., 0.6 x 0.6 = 0.36 < 0.5, and so the plaintiff should lose.

The legal academic literature mostly responds to this "paradox" in two ways. Some argue that the standard jury instructions are simply wrong. According to this view, juries should be instructed to apply the product rule to independent events. The alternative view says that the instructions are right but that's because the legal system isn't after truth; instead, the legal system seeks the most plausible narrative. Clermont takes a different approach entirely. He says that the legal system is indeed after truth but that it uses fuzzy rather than bivalent logic to get the truth.

Many of our concepts lack sharp boundaries. Consider the statement "Bill is tall" and suppose that Bill measures 6'1". Is the statement true? Well, yes and no. Bill is somewhat tall but not very tall. Fuzzy logic recognizes that tallness (and many other concepts) are fuzzy. Rather than saying that Bill either is or is not tall, fuzzy logic assigns

*degrees*to assertions of membership in a set. Thus we might say that Bill is a 0.6 member in the set of tall people (on a scale from 0--not at all a member--to 1--completely a member). At the extreme, fuzzy logic reduces to bivalent logic. E.g., Mugsy, who is 5'3" is 0.0 in the set of tall men, whereas Shaquille, who is 7'1", is a 1.0 member of the set of tall men.

The relevantly interesting fact about fuzzy logic for present purposes is that it does not use the product rule. Instead, it combines membership by applying the so-called "MIN" rule, which says that to figure out the degree of membership of x in sets A and B, we take the minimum value of A and B (and so on with sets C, D, E . . . .) So suppose that Bill, in addition to being a 0.6 member of the set of tall men, at the age of 61 is a 0.6 member of the set of old men. Bill's degree of membership in the set of "old tall men" is 0.6 -- not 0.36, as the product rule would say. Thinking about this without numbers, that seems about right. We would normally say that a 6'1" 61-year-old man is

*somewhat "*tall and old." We would not say he is

*mostly not*"tall and old" (as the product rule would lead us to say).

So here is Clermont's extraordinarily controversial claim in the paper: He says that figuring out whether some past event occurred, where we have conflicting evidence, is equivalent to assigning membership to a fuzzy set--even when the underlying event had an on/off character. In my hypo, the masked pie-thrower either was or was not the defendant, and the plaintiff was hit by either the pie-thrower's pie or the pie from the truck. These are

*not*quantum events as to which we could say that the plaintiff was hit by both-pies-and-neither-pie. Macroscopic reality is bivalent. But Clermont's claim is that given our imperfect knowledge, past macroscopic reality acts

*as though it were fuzzy.*And therefore, he concludes, the standard jury instructions for combining probabilities are right--because they employ the MIN rule. QED.

I am not persuaded of Clermont's basic claim, but I am also not entirely confident that he's wrong. Part of my mystification comes from my uncertainty about exactly what sort of claim Clermont is making when he says that we should regard uncertainty about past on/off events as equivalent to fuzziness. Is this a metaphysical claim? An epistemological one? Is it stipulative? I am sure that he has raised an extraordinarily important issue that warrants a lot more thought.

To close, I want to suggest that Judge Boudin's opinion in the First Circuit DOMA case bears an interesting relationship to the fuzzy/bivalent question. To oversimplify somewhat, the First Circuit says that DOMA Sec. 3 (which applies a federal one-man-one-woman definition to marriage, even when a same-sex couple is legally married under the law of the state in which they reside) raises issues of equal protection and federalism. As I read Judge Boudin, the level of scrutiny he applies is higher as a result of the combination of these two considerations than it would be if either consideration stood alone.

Another way to think about what the First Circuit did is this: In order for the government to win, it must defeat the plaintiffs' equal protection claim and it must also defeat their federalism claim; applying the product rule of bivalent logic, it will be more difficult for the government to defeat

*both*the equal protection and federalism claims than it would be to defeat either one standing alone.

This combination is unusual but not unprecedented: In

*Plyler v. Doe*, the Supreme Court mixed together equal protection and federalism concerns to invalidate a state law excluding children of undocumented immigrants from free public education. Likewise, in dicta in

*Employment Division v. Smith*, the Supreme Court said that "hybrid" claims mixing free exercise of religion and the right to direct the upbringing of children get greater solicitude than free exercise claims (or upbringing claims?) standing alone.

Yet such hybrid rules of law are generally disfavored. Why? Because we generally treat individual

*legal*claims as subject to the MIN rule (or its complement, the MAX rule) of fuzzy logic: If the legal tests can be laid on the same scale, then the plaintiff wins or not depending on whether he can get over the lowest threshold. If a plaintiff brings unrelated federalism and equal protection claims that invoke distinct tests, it's true that a third-party observer might accurately predict that the plaintiff has a better chance of prevailing than if the plaintiff brings only one or the other claim But that's quite different from the court itself combining the rules of law to actually employ a test more favorable to the plaintiff than the test given by either federalism or equal protection. Thus, whatever one thinks of fuzzy logic as applied to combinations of past facts, it does seem like it ought to apply to combining legal claims.

It's possible that I've misread the First Circuit. Near the end of the opinion, the court says this: "disparate impact on minority interests and federalism concerns both require somewhat more in this case than almost automatic deference to Congress' will . . . ." That language suggests that the court thought that either equal protection or federalism was by itself sufficient to produce the scrutiny it applied. But earlier in the opinion the court said something that looks a lot like a justification for a hybrid rule:

Although our decision discusses equal protection and federalism concerns separately, it concludes that governing precedents under both heads combine--not to create some new category of "heightened scrutiny" for DOMA under a prescribed algorithm, but rather to require a closer than usual review based in part on discrepant impact among married couples and in part on the importance of state interests in regulating marriage.That strikes me as a rather peculiar use of bivalent logic, i.e., an insufficiently fuzzy approach.

Finally, lest there be any doubt, I think the bottom-line result in the case is correct, but then I think that equal protection alone ought to protect a right to same-sex marriage, so for me this is an easy case.

## 17 comments:

To me, the court’s decision rests on the equal protection analysis; and federalism is merely a plus factor in this “unique case.” Obviously, the court did not think federalism concerns rose to the level of a 10th Amendment violation. Therefore, individual plaintiffs could have prevailed only on the EP ground – the sole claim they asserted in their lawsuit.

The court’s language makes clear that EP is the basis of its ruling. After stating that “[c]entral to this appeal is Supreme Court case law governing equal protection analysis” (p. 13), the court decided to scrutinize DOMA more closely than usual because “Supreme Court equal protection decisions have both intensified scrutiny of purported justifications where minorities are subject to discrepant treatment and have limited the permissible justifications.” (p. 15).

So, it seems that the First Circuit would have applied this “intensified scrutiny” to find an EP violation even where federalism concerns are absent.

I only skimmed the Clermont article, but I am thinking that either he is confused or you are (or both). Maybe Clermont argues for the strong conclusion you describe, but what he says appears to establish no more than that the law's treatment of the standard of proof in civil cases that involve more than one element is explained by fuzzy logic *only when (at most?) one of the elements is vague*. Fuzzy logic applies to *vague* terms, such as "tall" and "heap". Fuzzy logic is consistent with, and fully accommodates, both classical logic and classical probability theory in the case of non-vague terms. Now Clermont may think that vagueness is everywhere. But it isn't. In your hypo, for example, it is not vague whether the defendant was the masked assailant and it is not vague whether the pie that struck the plaintiff's face came from the assailant or from the truck. In such a case, classical logic and classical probability theory apply. So if the probability that the defendant was the assailant is .60 and the probability that the pie came from the assailant is .60, then the probability that the pie came from the assailant and that the assailant was the defendant is .36.

Clermont thinks that his MIN rule (which replaces the product rule) applies when there are two elements, one of which is vague. His example of a vague legal concept, I believe, is *fault*. But here I think the jury is out (forgive the pun). It may be that *fault* in many cases is not vaguely defined at all. The devil here is in the details. And then what are we supposed to do in cases in which only one element is vague and five elements are not? It seems to me that the product rule should apply to the non-vague elements.

Clermont's main thesis is descriptive, not prescriptive, namely that the law actually embraced the tenets of fuzzy logic before logicians caught up with the law. This strikes me as very likely mistaken. I find one of the following two hypotheses far more likely: (1) those lawyers, legal theorists and judges who rejected the product rule for independent elements were simply innumerate (lawyers are smart, but how many of them at that time had received rigorous training in logic and probability theory?), or (2) those who rejected the product rule for independent elements did so for purely pragmatic reasons, judging that it would be too difficult for the typical juror to apply the product rule, or that jurors might nullify if faced with a rule that delivers results that appear to them to be counterintuitive (because of their own innumeracy, or because of Kahneman-Tversky effects).

It seems to me that (2) is more likely. The reason is that the rule that jurors are asked to apply, wrongheaded as it is, (a) has a veneer of plausibility and (b) is far easier to apply in practice than the product rule. One reason for (b) is that all that a juror needs to determine about any one element is whether its probability is greater than .50; but the product rule requires the assignment of a numerical probability value (or a range of values) to each element.

Sam: Clermont is not confused, nor am I. I know that I am not confused about what Clermont is saying because I have read his paper carefully and talked to him about it. He confirms that I have correctly understood him. I know that Clermont himself is not confused because nothing you say in your comment contradicts what his paper says.

Now, although Clermont is not confused, he may very well be wrong. Indeed, I think he probably is wrong. His crucial claim--and the one that I thought I had described clearly in my post but apparently not--is that while classical probability theory applies to making prospective predictions regarding non-vague terms, fuzzy logic applies not only to vague terms but also to making retrospective assessments of non-vague terms so long as we can never truly know what their values will turn out to have been. As I said in the post, I am skeptical of that claim, but there is no confusion here.

Assuming that my skepticism is warranted and the product rule should apply, the question you raise (which I also raised in my post), how do we explain the failure of the legal system to employ the product rule? That question has generated a large literature, with some of the best contributions coming from Charlie Nesson, Ron Allen, and Alex Stein. They and others explore the possibilities you suggest--as well as others.

Mike: I apologize for saying that I thought that one or both of you was confused. What I should have said instead is that I am confused. If I am responsible for confusing myself, then I owe you a second apology. But let me share with you briefly why I am confused.

Here's a quote from Clermont's paper (p. 27): "For random uncertainty in a bivalent world, then the probabilistic operator will give the right answer...If one tries to deal with the variedly uncertain real world, a more inclusive approach to conjunction becomes generally appropriate. In a fuzzy world, the product rule retreats to a specialized role, applying only when the independent values of x and y happen to be randomly uncertain without being vague." [See also p. 36: "I therefore contend that the supposed paradoxes rest on applying a multiplicative rule of classical probability to a problem it cannot handle. It can handle only randomly uncertain estimates of independent events in a binary world, because it is built on the assumption of an excluded middle."]

Applying this to your hypo, we should say that the product rule, rather than MIN, applies. This is because whether the defendant was the assailant and whether the pie came from the assailant are not vague issues. Fuzziness does not apply, so the MIN rule does not apply.

On the other hand, on pp. 29-30, Clermont applies his thesis to the law. First, he considers simple cases involving one non-fuzzy element (identity) and one fuzzy element (fault). There he says that MIN applies. So far so good (at least on the assumption that fault is fuzzy, which is not clear to me, but never mind). But he then argues that MIN "still should apply even if both percentages measure only random uncertainty" (p. 30). Here's the argument.

"A 60% chance of the weakest link [here: identity] represents the chance that all the other elements are more likely than not to exist. Because a 70% chance of fault is good enough for liability, we should not further account for that chance of finding complete fault. To multiply the chances, getting 42%, would be double counting, as it represents the chances of fully establishing both identity and fault. Establishing every element to 100% is not what the law calls for, and so the chances of doing so are irrelevant."

There's something I'm not getting here. This argument assumes that what we are interested in is liability, not fault. But then the second element is whether the perpetrator is liable (not whether the perpetrator is at fault), and by hypothesis if a 70% chance of fault is sufficient for liability, then the probability of the perpetrator being liable is 1. So it's no wonder that the right answer here is .60 rather than .42. The product rule predicts this as much as the MIN rule does.

Generalizing, it's not clear from this example what Clermont would say about your hypo (where there is no translation of degree of fault into all-or-nothing liability). In your case, you really have two non-fuzzy issues to resolve, and, according to what Clermont writes on p. 27, it sure looks like the product rule applies. Does he have an argument (other than the one on p. 30) that MIN applies to it instead?

Again, maybe I'm missing something, but it's not clear to me whether, and if so how, Clermont's article applies fuzzy logic to the retrospective assessment of non-vague terms.

Thanks for the reading recommendations. I'll take a look at them when I have a free moment.

Sam: Thanks for the clarification. I agree that the paper only makes the point on which I'm focusing in passing, but I do think it's crucial. The relevant discussion is at p. 36, where he says: "If we can never convert the likelihood of a claim to one or zero, then all we can say is that the defendant is liable to a certain degree. Thus, when we can never know with certainty what happened, a likelihood of occurrence is not different from a degree of misfeasance: now likelihood of occurrence is not a classical probability, it is a fuzzy set."

I read that to be saying, as I characterized the point in the blog post, that necessarily imperfect knowledge of some past event is equivalent to membership in a fuzzy set. And for my Annie Hall moment: I've discussed this with Kevin and he agrees that this is what he was saying. But now he's not so sure it's right and is working on a narrower version.

Ah, I see, I missed it. I guess I thought the statement was limited to the example at hand, which involves vague terms (bad, breach). But yes, there does seem to be a move here from an epistemic claim about our relation to the past to a metaphysical claim about the past not being a bivalent world. If this is what was bothering you in your initial post, then I'm definitely on board with your worry. I guess I was more focused on parts of the paper that suggest that Clermont is not committing himself to the application of the MIN rule to non-vague cases.

Of course, I can't compete with Woody and Marshall.

what is wrong?

cheap jewelry storeBuy Cheap Windows 7 professional KeyI study that to be saying, as I recognized the factor in the short article, that actually partial understanding of some previous occurrence is comparative to regular member's program in a fluffy set. And for my Annie Lounge moment: I've mentioned this with Kevin and he confirms that this is what he was saying. But now he's not so sure it's right and is operating on a smaller edition.

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