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.