With God’s help

In columns 66 and 69 I dealt with following the majority (in the first, the question was whether the majority determines, and in the second whether the majority is right). We saw there that, unlike a democratic majority, which is grounded in the citizen’s right to influence his fate, a majority in a religious court is intended to come as close as possible to the truth. In this column I will return to these two questions: the first from the perspective of Jewish law, and the second again from the probabilistic perspective.

The first question I will address here is what the source is, if any, for the claim that the majority is more likely to hit the truth than the minority—that is, who says that the majority determines. This question depends on the question of what that claim means, namely, in what sense the majority is right: is this a law of nature, or perhaps a probabilistic claim? At first glance this sounds reasonable, but as I will try to show, the matter is not so simple.

The purpose of the majority in a religious court

In column 69 I cited the words of the author of Sefer HaChinuch, who wrote:

And choosing to follow the majority on this basis of similarity applies when the two disputing groups are equal in their knowledge of the wisdom of the Torah, for one cannot say that a small group of sages should outweigh a large group of ignoramuses, even if they were as numerous as those who left Egypt. But when the wisdom is equal, or nearly so, the Torah informs us: that the greater number of opinions will always agree with the truth more than the minority. And whether they agree with the truth or do not agree with it, in the opinion of the listener, reason dictates that we should not depart from the path of the majority. And what I say—that choosing the majority always applies when the two disputing groups are equal in their knowledge of the true wisdom—this is so everywhere except with the Sanhedrin. In their case, when they disagree, we do not scrutinize which group knows more; rather, we always act in accordance with the view of the majority among them. The reason is that their number was fixed by the Torah, and it is as though the Torah explicitly commanded: after the majority of these, conduct all your affairs. Moreover, they were all great sages.

The words of Sefer HaChinuch sound reasonable at first glance, but as we saw in column 69, the assumption that the majority generally hits the truth is not so simple. True, he explains that this is said only about a situation in which the judges are roughly on the same Torah level, and in such a situation the truncated Gaussian considerations I brought there are not necessarily relevant (although a narrow Gaussian of the distribution of levels among the judges here is still possible). But according to most views, following the majority is obligatory in every religious court, even if there are differences in level among the judges.

If so, according to the accepted approach, the question is very strong (as I explained in column 69). But even if we adopt, if only for the sake of discussion, the approach of Sefer HaChinuch, it is still not clear whence we really know that the majority usually hits the truth better. It is important here to sharpen the question further. Clearly, Sefer HaChinuch does not mean to claim that the majority is always more right than the minority, but only that if one has to establish a general principle, it is better to choose following the majority than following the minority, since the chance that the majority is right is greater. My question here is: from where do we draw this seemingly reasonable assumption? Is it learned from our experience? Or is it an a priori assumption?

To understand the question better, let us first present the distinction in Jewish law between two types of majority: rov de’ita kaman and rov de-leita kaman. After that we will return and see the implications for our issue.

Rov de-leita kaman and rov de’ita kaman

A rov de’ita kaman is a majority that is present before us. The clearest example of such a situation is a city in which there are ten shops selling meat, of which nine sell kosher meat and one sells non-kosher meat. In such a case, if a person finds an unidentified piece of meat in the street and it is unclear whether it is kosher or not, then according to Jewish law he may rely on the majority of the shops, and therefore he is permitted to eat it.

Jewish law distinguishes this type of majority from another type, rov de-leita kaman, that is, a majority that is not present before us. It is commonly explained that this is a majority grounded in the nature of the world. An example of such a majority is the Talmud’s assumption that most women give birth at the end of the ninth month (and not at the end of the seventh). Or that most borrowers repay their debt after the due date and not before it. Or that most people buy an ox for plowing and not for slaughter, or that most women are not aylonit (that is, women who cannot give birth). All of these are assessments about the nature of the world or of human beings. This is a majority rooted in the nature of the world or of human beings in general, and not in the accidental circumstances before us—unlike rov de’ita kaman, which is an accidental majority. There is no law of nature that, somewhere, most shops are kosher or non-kosher. This is a specific situation that follows from some piece of information we have about that place and time. There is no law of nature here.

What is the difference between these two types? A rov de’ita kaman is based on a collection of contingent facts (depending on which city is involved). This is not a law of nature but an accidental state of affairs, grounded in observation of those concrete facts. We have information about this particular situation through direct acquaintance with the facts that obtain in it. By contrast, a rov de-leita kaman speaks about the ordinary way the world functions, that is, about a general collection of cases in every place and at every time. This is a law of nature that does not depend on specific information about a particular situation. How is such a law formed? Usually one begins with observations of a representative sample, from which one creates a generalization into a general law, and finally applies the general law to the case before us. For example, among the women I know, most are women who can give birth. At the second stage I assume that this is a representative sample, generalize from it, and formulate from it a general law of nature that most women in the world (most of whom I do not know) can give birth. Finally, I return and apply this to the specific case that comes before me as a judge or halakhic decisor. This process is similar to the creation of laws of nature in science. These are obtained from measurements on a representative sample, which serve as a basis for generalization into a general law that is then reapplied to specific situations.

Let us look at this distinction from another angle. Try to think about the problem of the shops. Suppose we wanted to examine the question with statistical tools. How would we conduct the experiment on the sample group? Can one examine a collection of cases of pieces that separated from shops and see whether the separation is in fact uniformly distributed among the shops? If not, then there is no basis for the assumption that the probability follows the percentage of kosher shops. We have an a priori assumption that there is no difference in the chances of pieces of meat being lost, and on that basis we apply our information about the shops and infer from it the probability that the piece before us is kosher or not. There is no generalization here on the basis of a representative sample. By contrast, a law of nature can be tested empirically. We can take several women (a random and representative sample), check how many of them gave birth, and see whether most really can give birth or not. From here we can draw a conclusion about all the women in the world. A rov de-leita kaman, like laws of nature, is inferred from observations on a representative sample and from generalization of them, whereas a rov de’ita kaman is formed from a priori assumptions without recourse to a sample.

In another formulation, rov de’ita kaman is based on full information about the situation. We know all the information about what is happening in this particular city, and the question that arises regarding the piece of meat is what the status of this particular piece is, not what the city is like. By contrast, in rov de-leita kaman the question is what the world itself is like, since we do not have full information about it. We have to take partial information (about a sample) and generalize from it about the nature of the world. In rov de’ita kaman the question is what the nature of the case before us is, when the general situation itself is known (we have full information about it). By contrast, in rov de-leita kaman the question is about the nature of the general situation, and the case before us will be derived from the general situation. This is a condition of lack of information, which is completed by generalizing from partial information (about a sample).

It is important to clarify that even in rov de-leita kaman our generalizations are based on common-sense assumptions (for example, that the sample of women known to us is representative and that there is no bias there or selection of a special subgroup). But after those assumptions (which themselves must be examined by various regression analyses), we are doing scientific work of elimination, which brings us to a representative sample and to generalization from it. There is an empirical-scientific element here, although like all science it is based on foundational assumptions of common sense. In rov de’ita kaman there are only the assumptions of common sense. There is no empirical-scientific component here at all of observations on a sample and generalization from them.

Following the majority

Let us now return to following the majority in Jewish law. From where do we learn that Jewish law recognizes following the majority? The source of the rule of following the majority is discussed in the passage in Hullin 11a–12a:

From where is this matter derived, that the Rabbis said: Follow the majority? From where do we know it? For it is written: “Follow the majority”! A majority that is present before us, such as nine shops and the Sanhedrin—we do not need to ask about that. Our question is about a majority that is not present before us, such as a minor boy and minor girl—from where do we know it?

A distinction is made here between rov de’ita kaman and rov de-leita kaman. Rov de’ita kaman is learned from the verse follow the majority (“follow the majority”), which is said about a religious court, whereas rov de-leita kaman cannot be learned from there (for some reason), and therefore the Talmud seeks a separate source for it. As noted, the conclusion of the passage is that we have no source (at least in a place where there is also the option of following the minority).

Just to complete the passage, I will cite Rashi there on 12a, who writes:

With regard to the Passover offering—since the Merciful One said (Exodus 12), “And they shall eat the flesh”; and with regard to peace-offerings—since the Merciful One said (ibid. 29), “And they shall eat those things by which atonement was made for them,” teaching that atonement also depends on eating—what can be said? Rather, certainly one may eat, relying on the majority where it is impossible otherwise. And even so, he disputes the case of a minor boy and minor girl because for us it is possible. So too for us, how do we derive from it? Rather, certainly this is a law given to Moses at Sinai: that we rely on the majority even where it is possible [to investigate further]. Alternatively, “follow the majority” (Exodus 23) implies both a majority that is present before us and a majority that is not present before us, for what difference is there between this case and that one? And on this principle we rely, and we do not inspect all eighteen kinds of trefot; only for a punctured lung, because a defect is common there, do we inspect. But where it happened that a lung was separated [from the animal] and was not inspected, it may be eaten, for we rely on this principle and against Rav Huna, who said above (Hullin 9): once slaughtered, it retains its presumption of permissibility. And the matter is not publicized.

Two explanations are brought here: either one learns this from the verse follow the majority, or it is a law given to Moses at Sinai. His words are very puzzling in light of the course of the Talmudic discussion, but in truth the passage itself is puzzling (for in the conclusion no source at all is brought for following a rov de-leita kaman, even though throughout the Talmud and among the halakhic authorities it is clear that one follows such a majority as well).

A side note: which majority is stronger?

Rabbi Shimon Shkop notes that the Talmudic passage seemingly implies that rov de’ita kaman is stronger, for even after the Talmud learned it from the verse, it still is not willing to infer that one may rely on rov de-leita kaman as well. At first glance this sounds reasonable, since rov de’ita kaman is based on full information. There is no component of generalization there, which by its very nature contains a speculative dimension (David Hume’s problem of induction). But Maimonides’ view (and that of other medieval authorities) is that rov de-leita kaman is stronger.

This can perhaps be understood through an example from a dispute that once took place in the Likud central committee between David Levy and his rivals. Levy argued that although he headed a faction that included a substantial portion of the central committee (let us say, for the sake of discussion, 30%), they had no representation at all in the executive positions. His opponents argued that those positions are filled by democratic vote, and the members of the committee choose them. That is democracy. Levy argued in response that there was a built-in distortion here, since every position is filled by a vote and the majority determines. Therefore a very small majority is enough to win all the positions. 100% of the positions are filled by 51% of the members. Democracy is a poor method in this respect, and proportional representation is fairer. The same is true in the Supreme Court: even in the period when it had one religious judge, its defenders argued that this was the proper proportion relative to the number of religious people in the population (which in itself, of course, was not factually correct). But there is another problem here too, namely that 100% of the decisions on matters of religion and state are made on the basis of the majority of judges on the panel (which is always secular). Therefore the outcomes are not proportional to the religious share of the public.

From here one can understand that rov de-leita kaman has stronger legal significance, since for every woman who comes before us we will decide regarding her that she gives birth according to the laws of nature (that is, in the ninth month). We will ignore the minority who do not give birth in the ninth month. By contrast, in rov de’ita kaman there is room to be concerned about the minority, since it is clear to us that in the situation under discussion there is also a minority.[1]

The possible ways of understanding the nature of the majority in a religious court

Is the majority in a religious court a rov de’ita kaman or a rov de-leita kaman? At first glance, this is a majority that is present before us (three specific judges), and therefore the Talmud is right to compare it to rov de’ita kaman.

But above we saw that according to Sefer HaChinuch, following the majority in a religious court is based on the assumption that the majority is usually more right than the minority. Rabbi Shimon Shkop (Sha’arei Yosher, Gate 3, Chapter 1) notes that according to the explanation of Sefer HaChinuch, this comes out as rov de-leita kaman, since we are relying on the reasoning that generally (in most panels) the majority is right. Our sample space is the collection of panels in different places and times in which the judges disagreed with one another, and there is a law of nature that in most cases it is the majority that was right. The majority under discussion is not the majority of judges on this specific panel before us, but the majority among all the panels in which the majority was right. If so, the words of Sefer HaChinuch are seemingly contradicted by the passage in Hullin, for there it is evident that a majority in a religious court is rov de’ita kaman. From this Rabbi Shimon Shkop concludes that the explanation of following the majority among judges must be different from that of Sefer HaChinuch (and he proposes there a formalist explanation of his own).

The question is what the author of Sefer HaChinuch himself held. Can his words be reconciled with the Talmud?

A proposal for explaining the view of ‘Sefer HaChinuch’

We saw above that rov de’ita kaman is not based on a sample and generalization, since there is no way to test it experimentally. This is unlike rov de-leita kaman, which is a scientific generalization on the basis of a representative sample (like the claim that most women give birth in the ninth month). Now let us think about the majority in a religious court according to the explanation of Sefer HaChinuch. He explains that the rule to follow the majority is based on the assumption that the majority is usually right. What is that assumption based on? Is it the result of a sample and generalization? Clearly not, if only because there is no way to carry out tests on such a case and collect information about sample cases.

Suppose we want to test this assumption empirically. We would take a representative collection of judgments decided by majority against minority in some sample of cases and panels, and ask in how many of them the majority was right and in how many the minority. If the result were that the majority was usually right, then we would have obtained our law of nature. But we have no way at all of knowing who was right in any of those cases. If we look at a specific case and examine whether the majority of judges was right or rather the minority, how will we know? We have no more information than the judges themselves, and we too do not know whether there really was a murder, or a loan, or not. If we had a sample in which, in each of the cases, we had a verified answer, we could generalize from it to the totality of cases. But there is no way to create such a sample, because there is no way to determine, in any specific case, whether the majority was right or the minority. Consequently, the general law is not a generalization based on such a sample either.[2]

So what, then, is the conjecture based on that the majority is usually right? Apparently on common sense, which can be represented in a probabilistic calculation. One consideration, for example, says that if one assumes that a judge is right with probability P (which must be above 50%, otherwise it would be better to conduct a blind lottery), then when two judges support one of the sides there is a higher probability that that side is the correct one than that the other side is. But this is a common-sense consideration, based on the optimistic assumption that the judge really is right in most cases (P>1/2). In any event, this is not an inference from a representative sample but an a priori assumption. Therefore, even if we adopt the assumption that the majority is usually right, this will not be a conclusion obtained by generalization from a sample, but a common-sense assumption.

We have thus found that following the majority in a religious court actually resembles rov de’ita kaman and not rov de-leita kaman. At first glance this is indeed a situation with the character of rov de-leita kaman, because we are not dealing with a specific case that is present before us, but the validity of this majority is based on common sense (like the validity of rov de’ita kaman) and not on generalization from a sample, as happens with laws of nature and rov de-leita kaman. It is therefore no wonder that the Talmud in Hullin determines that the majority in a religious court is similar to the majority of shops and is rov de’ita kaman. This is quite understandable even according to the view of Sefer HaChinuch.

Probabilistic calculation[3]

A certain Torah scholar once asked me the following question about the assumption that the majority of judges is usually right. Suppose the probability of such a judge being right is P (as noted, this presumably needs to be more than 1/2). The probability that two judges are right is P², which is of course lower. If, say, P=0.7, then P²=0.49. The conclusion is that the minority is usually right and not the majority.[4]

This is, of course, a mistake. An immediate indication of this is that according to that same calculation, the probability that both are mistaken is ²(1+P-), and in our example we get 0.09. The sum of the two possibilities is not 1, and therefore it is clear that this calculation is incorrect. A correct calculation shows that the majority is indeed right. One can see it this way: the probability that such a panel errs is composed of several possibilities: a. if all three judges were wrong—³(1+P-). b. three cases in which one is right and two are wrong—3P(1-P)². The total probability of error in a panel of three judges is the sum of the probabilities: 1+2P)(1-P)²). If these are very good judges, for example P=0.9, then the probability of error in a panel of one judge is 0.1, and in a panel of three the formula above yields a probability of error that is much smaller: 0.028. To be sure, adding judges, if they are worse, can also make things worse. For example, if there is a judge who is always right, then clearly any addition of imperfect judges will only worsen matters. This also happens in the case of a good but not perfect judge with two judges who are significantly worse than he is.[5]

These calculations lead to the conclusion of Sefer HaChinuch that if the judges are of similar quality, the majority improves the chance of reaching a just judgment. But these are a priori considerations and not the results of generalization from observations of a sample. Therefore, even according to Sefer HaChinuch, the majority among judges is not rov de-leita kaman, contrary to Rabbi Shimon Shkop’s claim.

[1] This can explain what the halakhic authorities wrote regarding a mixture of one forbidden piece of meat among a majority of permitted pieces. According to Jewish law, when only two pieces remain, it is already forbidden to eat from them, since now the assumption that there is a majority of kosher pieces is no longer necessarily correct (a doubtful majority is not a majority). In rov de’ita kaman one cannot completely ignore the minority that is certainly present in the situation.

[2] An exception is a case in which the factual truth became known to us in some way after the ruling (in a legal dispute rather than a factual one, there is never any way to know this). For example, when DNA was introduced as admissible evidence in the American legal system, tests were carried out on offenders who had been convicted, and it turned out that there were quite a few wrongful convictions. There are additional indications that try to suggest ways to examine whether a given judicial ruling is just or not, and Dr. Boaz Sangero discussed this in his book, Wrongful Convictions in Israel and Worldwide, published by Resling. Still, it is difficult to regard those indications as well-founded research, since there are not enough cases to examine scientifically the probability of reaching the truth in judicial decision-making.

[3] See an interesting discussion of this in my friend Nadav Shnerb’s book, Keren Zavit, in the essay on the Torah portion Devarim. I will only note that he ignores there the dependence on the probability that the defendant is in fact guilty.

[4] See also column 69 on this.

[5] Nadav Shnerb, in his article, gives as an example a case of one judge at 0.9 with two judges at 0.6, where the probability of error is 0.208, that is, significantly greater than the 0.1 obtained for the single judge.