What Is Jewish Law: The Rationales or the Bottom Line? (Column 257)

With God's help

When a dispute breaks out among the judges in a religious court, the rule is follow the majority. The opinion of the majority of judges determines the ruling. I have previously argued that, on a straightforward reading, the purpose of the majority in a religious court is to clarify the factual and legal truth,[1] as Sefer HaChinukh writes in commandment 78:

And the choice of this majority, it would seem, applies when the two opposing camps are equally versed in the wisdom of 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 wisdom is equal, or nearly so, the Torah informs us that a greater number of opinions will always agree with the truth more than a smaller number. And whether they in fact agree with the truth or not, in the view of the listener, the law requires that we not depart from the path of the majority. What I mean in saying that the choice of the majority always applies when the two disputing camps are equal in knowledge of the truth is that this is so everywhere except in the Sanhedrin. With them, we do not scrutinize, when they disagree, which side knows more; rather, we always act in accordance with the majority among them. The reason is that their number was mandated by the Torah, and it is as though the Torah explicitly commanded: after the majority of these, you shall conduct all your affairs. In addition, they were all great sages.

He assumes that the purpose of the majority is to come as close as possible to the truth, and therefore when there is a dispute between a small number of sages and a larger number who are less wise, it is the majority of wisdom that determines the ruling, not the majority of persons. To be sure, the decisors are divided on this issue, but it seems to me that virtually all commentators, including those who disagree with Sefer HaChinukh on this point, agree that the purpose of a majority in a religious court is to come as close as possible to the truth. This is unlike a democratic majority, whose function is to constitute the "truth" rather than uncover some objective truth.[2] It is precisely this point of departure that raises what is called the doctrinal paradox, which I want to discuss here.

Background: Following the Majority for Different Rationales

I will begin with the question of what the rule is when the opinions in the religious court are based on different rationales. The same applies to rulings about what is forbidden and permitted, where it is accepted in Jewish law (see Shulchan Arukh, Choshen Mishpat 25:2 and below) that one follows the majority of opinions among the halakhic decisors. What is the rule when the various decisors issue the same ruling, but on the basis of different rationales?

Rema, Choshen Mishpat 28:2, writes in the name of Maharik:

Likewise, if there is one individual against the majority, we follow the majority in every place (Rashba, siman 203/253), even if the majority do not agree for the same reason, but each has his own independent reason; since they agree regarding the legal ruling, they are called a majority, and we follow them (Maharik, shoresh 41/94).

The Shakh there, in subsection 19, qualifies this:

And even if the majority do not agree for the same reason, etc. In my work Siftei Kohen on Yoreh De'ah, siman 242 [in the rules of prohibition and permission], I argued that in a Torah-level prohibition we do not follow the majority if they agree to be lenient on the basis of two different reasons; see there. However, it appears that this applies specifically when we come to permit a ruling that is doubtful to us on the basis of the halakhic works. Therefore, since the majority are not before us, we are concerned that on this fundamental point the law may follow the decisor who is stringent, while on the other point the law may follow the second decisor who is stringent. But when the majority are before us, it is obvious that we follow them even though they do not agree for the same reason, for nowhere do we find that the Sanhedrin or judges need the majority to state one single reason. This also seems implied in the chapter "Echad Dinei Mamonot" [34a], where it says: Come and hear: the two court scribes write the words of those who acquit and the words of those who convict. Granted, the words of those who convict, etc.; but why the words of those who acquit? Is it not because if they later see another reason to convict, we do not pay attention to it? No; rather, so that two people should not state one reason from two verses, as Rabbi Assi asked Rabbi Yohanan: If two state one reason from two verses, what is the law? He said to him: they are counted only as one. From where is this derived? Abaye said: for Scripture says [Psalms 62:12], "God has spoken once; twice have I heard this"—one verse may yield several reasons, but one reason does not derive from several verses. This shows that if they stated two reasons, in any event we follow them, for otherwise it should have answered: No, so that two should not state two different reasons. Moreover, Abaye says that one verse may yield several reasons, etc. And the reason we do not say that two may state one reason from two verses is, as Rashi explained there [s.v. "they are not counted"], because certainly one of these verses was not said for this purpose, since we maintain that two verses are not written for one reason; therefore one of them is definitely mistaken. End quote. [Afterward I saw that in Hagahot Ashiri to the chapter "Echad Dinei Mamonot" [sec. 5] he also explicitly wrote as I did, and this is his language: Three who sat in judgment, and two acquit and one convicts, or two convict and one acquits—and indeed the two disagree with one another, so that this one's proof is not at all acceptable to his colleague, and likewise that one's to his colleague—it seems to me that since the two are equal in acquittal or in liability, they constitute a majority; and he wrote out a proof at length. From Or Zaru'a. End quote. Perhaps he intended the proof that I wrote.][3]

Even if there is room for debate regarding rulings about what is forbidden and permitted, with respect to a religious court whose judges sit together and deliberate, the matter is entirely agreed upon: the majority is determined by the bottom lines, not by the rationales.[4]

The Basic Difficulty: Rationales or Bottom Lines?

The conception I presented above, according to which the purpose of majority rule in a religious court is to come closer to the truth, raises a major difficulty with regard to the rule described here—that one follows the majority even when it rests on different rationales. Consider, for example, 23 judges sitting in a capital case, of whom 15 convict and 8 acquit. But each of the 15 judges who convict bases his view on a different rationale from the others. For example, each claims that the defendant's act falls under a different category of prohibited Sabbath labor (and let us assume, for purposes of the discussion, that he was warned with respect to all of them, or that no separate formal warning is needed for each labor): one judge sentences him to death because he engaged in sorting, the second because he kneaded, the third because he trapped, and so on. The dispute concerns the same act, of course, and the judges disagree over which prohibited labor is transgressed by performing it. As we saw, the ruling in such a case is that the defendant is executed by stoning as a Sabbath desecrator, because there is a sufficient majority among the judges for that.

But when we look at the level of the rationales, this is far from simple. Seemingly, against each of the rationales offered by those who convict, there is an overwhelming majority—22 judges who reject it. Twenty-two judges think he did not sort, 22 think he did not trap, 22 think he did not knead, and so on. Moreover, all eight judges who acquit agree that none of the rationales for conviction is valid. That means that the best-supported view is actually the minority view, namely, the view that the defendant committed no offense at all. If so, when we examine the rationales for the death sentence one by one, the conclusion is that none of them is correct (assuming that the majority is the criterion of truth). So how can one execute a person in such a situation?

If the purpose of the discussion is indeed to uncover the truth (and not to constitute it, as with a democratic majority), then it is only natural to expect that a majority based on different rationales would not count as a majority. If we are searching for the truth, we ought to follow the rationales and derive the conclusion from them, not seek a conclusion detached from rationales. The relevant question is whether that person trapped or did not, sorted or did not. Why does it matter that, according to a majority of opinions, he is liable to death, when each judge convicts him for a different reason? From the standpoint of the truth regarding the rational basis of the ruling, he is apparently not liable to death. And yet Jewish law accepts the majority ruling in such a case. The question is why.

Distinguishing Between Different Types of Dispute

In the past (see my article "Is Jewish Law Pluralistic" and the article on "The Price of Tolerance"), I pointed out that in most disputes among sages both sides are correct at the level of the rationales. I likened this to a dispute over whether to eat chocolate, where Reuven says one should because it is tasty, while Shimon says one should not because it is fattening. Both sides raise correct rationales, because there is no dispute that chocolate is both tasty and fattening. The disagreement concerns only the relative weight of the rationales: should considerations of taste (pleasure) take precedence over considerations of health (fattening), or vice versa? If the dispute in the religious court is of this type, it would seem that the difficulty I raised collapses. None of the judges disputes the other's rationale; they merely weigh them differently. The difficulty described above was based on the fact that the rationales raised by the judges are themselves disputed—that one judge does not agree with another at the level of the rationales.[5] To be sure, even in a dispute of the type described here there may still be room to see a difficulty. After all, at the bottom line one judge disagrees with another about the ruling or the punishment, even if the disagreement concerns only the relative weight of the rationales. This depends, of course, on further details, and I will not enter into that here.

In other cases, the core of the dispute is whether the act that was done crosses the Torah-level threshold or not.[6] Thus, for example, a decisor or a judge may think that the act performed does resemble sorting in a certain respect, but does not cross the Torah-level threshold (for example, someone who sorted food from waste, or by hand and not with a utensil; or someone who trapped a species not ordinarily trapped). But both agree that there is some element of sorting or trapping here, and that at least on the rabbinic level it should be prohibited. In such a case the difficulty remains in place, for so long as we have not crossed the Torah-level threshold there is no liability to punishment. Therefore, in such a case the dispute about the punishment or the ruling is genuine, and the difficulty I described can still arise.

A third case is where one judge thinks there is some chance that the other is right, but in the bottom line rules that he is not. In this kind of dispute, the difference is not one of substantive threshold, as in the previous type, but of the probability that the other is right. That is, the dispute is not over how much of the labor of sorting is present here, but over what the probability is that this is indeed the labor of sorting. In this case, the chances that the second judge is right are greater, because even the one who disagrees with him grants him a certain probability. In disputes of this type it is highly doubtful whether the difficulty I described exists at all (here too, of course, this may depend on how much probability each judge assigns to his colleague's rationale).

For our purposes here, this brief description will suffice.

The Doctrinal Paradox

There is an extreme case in which the difficulty I described is very easy to see, and it is called the doctrinal paradox. Consider a case in which two litigants who signed a contract come before the religious court. Shimon claims that Reuven breached the contract and sues him for payment, while Reuven claims he did not. In order to decide whether Reuven is liable to pay, the court must discuss two questions: 1. An interpretive/legal question—did the contract indeed forbid Reuven to do X? 2. A factual question—did Reuven do X? Since the two questions are independent, the distribution of opinions can be as in the following table:[7]

	Interpretive question	Factual question	Final ruling
A	1	1	1
B	0	1	0
C	1	0	0

If we analyze these opinions, we get a paradoxical result: a majority of the judges hold that Reuven is not liable (the final-ruling column in the table), and therefore the ruling issued by the court is that Reuven is not liable. But if we examine the opinion of the majority of judges on the interpretive question, we find that, according to the majority, the contract does indeed forbid doing X. On the factual question, we find that a majority of the judges think Reuven did X. If so, seemingly we should have ruled that Reuven breached the contract and is liable to pay. That is the ruling that emerges from analyzing the rationales and reasons, and it is of course contrary to the judgment that expresses the majority opinion at the bottom line. In other words, had we followed the rationales, the ruling would have been that Reuven is liable to pay, but in practice the ruling is that he is exempt.

The Doctrinal Paradox: A Real-Life Example

You probably think this is a case that I "tailored" (not I, of course; it is a well-known paradox) only in order to create a paradox—that is, a hypothetical situation that never actually occurs. To that I would say that I see no reason why there should not be masses of such cases involving contracts, or parallel situations in other legal contexts as well. I am sure there are quite a few such cases. A few months ago I myself encountered such a case, when I read a Supreme Court judgment dealing with the abduction of a girl by her mother. The case went through two judicial instances[8] and in the end the Supreme Court ruled that the girl would remain with her mother. The distribution of opinions in the Supreme Court was exactly as in the table above.

To understand this, I should preface that child abduction (by parents) is adjudicated under the Hague Convention (for the welfare of the child). Under the rules of the Convention, even if we conclude that there was indeed an abduction, if the other parent consented (at the time of the abduction, and only now is suing because he has regretted it) or acquiesced (retroactively, at some point after the abduction), there is no duty to return the child to the shared custodial arrangement. The views of the Supreme Court justices were as follows: all agreed that there had been an act of abduction and that the Convention's rules applied. But one of them held that there had been neither consent nor acquiescence. A second held that there had been acquiescence but not consent, while the third held that there had been consent but not acquiescence. As stated, the distribution of opinions was exactly as in the table above, and therefore it was decided by majority view that the girl should not be returned. But when we examine the question at the level of the rationales, with respect to whether there was consent, the majority view is that there was not; and the same is true regarding acquiescence. If so, this is a case of abduction without acquiescence and without consent—and seemingly the law should have required that the girl be returned to her father's custody.

Interim Conclusion: The Ruling Is the Bottom Line

The picture that emerges from here is that the opinions of a judge or justice, both in the realm of Jewish law and in the realm of secular law, are the bottom lines and not the rationales. Therefore, when opinions are counted and a majority is formed, what is counted is bottom lines, not rationales.

Perhaps one can see such an approach in a fairly widespread halakhic phenomenon, in which medieval authorities (Rishonim) rule in a dispute among Amoraim (Talmudic sages), or an Amora rules in a dispute among Tannaim (Mishnaic sages), and the ruling decides like one of the disputants in one case and like the other in another case. Seemingly, such a ruling contradicts both views, and if one assumes that an Amora cannot dispute with Tannaim or that a medieval authority cannot dispute with Amoraim, this should be impossible.

Let us take as an example the dispute between the Tannaim in Mishnah Ketubot 12b:

If one marries a woman and does not find signs of virginity, and she says, "After you betrothed me I was raped, and your field has been flooded," while he says, "No; rather, it happened before I betrothed you, and my transaction was a mistaken transaction"—Rabban Gamliel and Rabbi Eliezer say: she is believed. Rabbi Yehoshua says: we do not live by her word; rather, she retains the presumption of having had intercourse before she became betrothed and of having deceived him, unless she brings proof for her claim.

The Talmud there explains that the husband is making an uncertain claim (because he does not know exactly what happened to the woman), and according to Rabban Gamliel the woman is believed. Rabbi Yehoshua disagrees and maintains that she is not believed, because a definite claim against an uncertain claim cannot extract from the party in possession. The Talmud there presents a dispute among the Amoraim over whether a definite claim can extract from one in possession who advances only an uncertain claim, and the law is ruled in accordance with Rav Nahman, who holds that it cannot. In the course of the next passage in the Talmud (13a, before the Mishnah there), it becomes clear that Rabban Gamliel holds that the woman is believed in every case, while Rabbi Yehoshua holds that she is not believed in any case. But Rav Nahman rules (and so all the decisors ruled) like Rabban Gamliel when the woman has a migo (the ability to have advanced a stronger claim)[9] and like Rabbi Yehoshua when she does not have a migo.

Seemingly, Jewish law here was ruled against both Tannaitic views: it accords neither with Rabban Gamliel (who holds that a definite claim against an uncertain one can extract even without a migo) nor with Rabbi Yehoshua (who holds that a definite claim against an uncertain one cannot extract even if she has a migo). How can an Amora rule against both Tannaitic views, not in accordance with anyone ("like no one")? Necessarily, such a ruling is possible because each of the Amora's rulings is anchored in a Tannaitic view. The ruling that without a migo she is not believed is anchored in Rabbi Yehoshua, and the ruling that with a migo she is believed is anchored in Rabban Gamliel. To be sure, this is not in accordance with their rationales; that is, at the level of the rationales, the ruling contradicts the opinion of both Tannaim. And yet the bottom line does not contradict the views of all the Tannaim, and therefore this ruling is legitimate. Once again, we can see here that what binds us is the bottom line of the Tannaim, not their rationales.[10]

A Technical Explanation

And yet this phenomenon still requires explanation. If our aim is to rule in accordance with the truth, we ought to decide according to the rationales and not according to the bottom lines.

It seems there is no avoiding an explanation in terms of technical considerations. The rule that the majority at the level of the bottom line is determinative is certainly the simplest and most efficient way. One must remember that it is not always clear what exactly the dispute over the rationales is, or whether there even is one. Every judge would have to express his view about every rationale of his colleague, and then we would have to perform a probabilistic balancing and total up the support of all the judges for every rationale in order to reach a ruling. I remind you that I presented above three different types of dispute, and in each of them the aggregation of opinions at the level of rationales would have to be done differently. Once rationales and reasons are taken into account, the way to calculate what the ruling is becomes complicated and ill-defined, and perhaps that alone is enough to explain why Jewish law and secular law follow a majority of bottom lines. If so, it seems that the reason one nonetheless follows the bottom line and not the rationales is probably legal efficiency.

But in one of the lectures I gave on the subject, one of the listeners (my friend Shmuel Keren, to whom I am grateful) raised another possibility, a probabilistic one. He argued that perhaps precisely if we follow the majority of opinions at the level of the bottom line and not the rationales, this maximizes the likelihood that the ruling will be correct. I told him that this was an interesting proposal, and that I would check it. Here are the results of that examination.

A Probabilistic Calculation: Assumptions and Framework for the Discussion

In column 145 I presented calculations showing why the rule follow the majority, which establishes that in a case of dispute one follows the majority of judges, really does bring us closer to the truth. There I calculated the probability that the ruling hits the truth (= the quality of the ruling) on the basis of reasonable assumptions about the quality of the judges (the probability that each of them hits the truth), and I showed that with skilled judges the majority indeed gives the result closer to the truth. Here I will present a similar calculation concerning the doctrinal paradox presented above, and examine whether a decision according to the bottom lines is indeed less reliable (that is, less likely to hit the truth) than a decision in light of the rationales. I will do so, of course, under reasonable but as simple as possible assumptions, and with as simple a case as possible. Even so, the results are quite surprising.

As there, so here, let us assume that the quality of a judge is expressed in the probability that he hits the truth, whether factual or legal. That is, the conditional probability that if the truth is X, the judge will rule X. There I denoted the quality of the judge by the letter P. In our case, the relevant probability is of course the probability that the judge's rationale (I do not distinguish here between a legal rationale and a factual one) is correct, not that his bottom line is correct (as I assumed there).

In probabilistic terms, we will say that the quality of the judge is determined by the probability that if the true rationale is X, he will indeed determine that X (again, for the sake of simplicity I do not distinguish between a legal determination and a factual one). For our purposes, we will assume that all the judges who sat on the case and whose opinions are displayed in the table above have quality P. The meaning of this is that on the interpretive-legal plane, if the contract forbids the act in question, then with respect to each of the judges there is a probability P that he will say that the contract indeed forbids it. And likewise on the factual plane, if Reuven did in fact do X, the probability that each judge will determine that he did X is P.

It should be remembered that the quality of an arbitrary decision is 0.5, because even if we were to choose at random an answer to such a binary question (he did it or did not do it, the contract obligates or does not obligate), we would get the right answer with probability 0.5. Therefore, we will assume that the quality of the judges, that is, the probability P, is higher than 0.5. Otherwise we are not dealing with a skilled judge but with mere random lottery.

Now we look at the table before us and see what the judges said. The question before us is whether Reuven breached the contract or not. The determinations regarding the rationales (the interpretive and the factual) are only an instrument and have no importance in themselves. Therefore we must examine, given that these are all the data about the judges' views on the two questions, what the probability is that Reuven actually breached the contract. If that probability is higher than 0.5, then one should follow the bottom line. If the probability is lower than 0.5, then one should follow the rationales.

How can it even be that in such a situation it is preferable to follow the bottom line? Seemingly, the bottom line is nothing but the result of the rationales. Given the truth about the rationales, the final truth is derived from them in an unambiguous way. When we present the problem in this way, one can see that even if, with respect to each question separately, the majority is more likely to be right, there is still also the possibility that it is wrong about that question. And that possibility can combine with the possibility that the majority is wrong about the second question and together create a situation in which Reuven still did not breach the contract. In this way, it can in principle happen that it is precisely the bottom line that brings us closer to the truth.

At this point it is easy to guess that even if we discover that the bottom line really can bring us closer to the truth, this result will probably not be universal. It depends on the distribution of opinions, on the value of P (= the quality of the judges), on the question whether all the judges have the same quality, on how many different rationales there are and what the distribution of opinions regarding them is, and more. This can of course also depend on the nature of the disputes that arose in the religious court (as we saw in the three types described above). But here our concern is only demonstration, and therefore I will make do with analyzing the case in the table and with the simplest assumptions.

The Calculation

We must calculate the probability that Reuven did in fact breach the contract. We saw that this event is a conjunction of two claims: that the contract forbids doing X and that Reuven did X. I will assume here that these are independent events,[11] and therefore the probability of this is the product of the probabilities. Now note that the calculation with respect to each question separately is identical to the calculation we made in column 145 (what is the probability that the majority is right on some legal question, given the quality of each judge). There we found that in a case where three judges are divided (two against one), if these are skilled judges then the two are more likely to be right (that is, the probability Q of this is above 0.5). Given that we have a product of two such numbers, the answer becomes obvious. The probability that the majority is right with respect to the factual question is Q, and so too the probability regarding the interpretive question (it is the same calculation, and our simplifying assumption is that the skill of the judges is identical with respect to legal questions and factual questions).

Hence the probability that Reuven breached the contract in our case is Q². The question whether this quantity is greater than 0.5 receives a simple answer. As noted, it is only reasonable to assume that Q>0.5, for there we reached the conclusion that the majority is more likely to be right. And from this it follows that, if indeed we are dealing with a panel that reaches the correct decision on a single question with probability greater than 0.7, then Q²>0.5.

I remind the reader that the result we obtained at the end of column 145 was that under simple assumptions (for example, that the a priori probability of the two answers to the question is equal, and that all the judges are of the same quality) the probability that the majority is right is exactly Q=P; that is, the probability that the majority is right is exactly equal to the skill of the individual judge. Hence, under the assumption that these are judges whose skill is significantly above the baseline level (the random draw—P=0.5), namely judges whose skill is above 0.7, the legal result in the doctrinal paradox is that one should follow the rationales (that is, rule that he breached the contract) and not the bottom line (which leads to the conclusion that he did not breach the contract). With very skilled judges one should decide according to the rationales, but in the case of less skilled judges it is better to follow the bottom line.

The resulting picture is as follows:

If P is between 0.5 and 0.707 – one should follow the bottom line.

If P is between 0.707 and 1 – one should follow the rationales.

In other words, if the judges are not highly skilled, there is no paradox here: deciding according to the bottom line is also the decision that brings us closer to the truth. But with very skilled judges (significantly above the baseline level) there it is indeed preferable to follow the rationales.

In Practice

So what should we do now? Follow the bottom line or the rationales? Can such a calculation justify following the bottom line? To answer that, we must ask ourselves: if we want to establish a sweeping rule, what is the more sensible thing to do?

It seems to me that there is logic in establishing a sweeping rule that one follows the majority with respect to the bottom line. To be sure, this is not always correct (with very skilled judges it is not optimal), but as we saw above, in order to avoid complexity and achieve efficiency one needs a uniform decision for all cases and all types of panels. Since we have seen that a decision according to the bottom line is not as absurd as it appears at first glance, and that it is not even necessarily worse than a decision according to the rationales, legal reasoning says that one should establish a sweeping rule that always instructs us to act accordingly. The technical consideration I presented above now receives a probabilistic anchor and reinforcement. There remains a technical dimension in adopting such a rule (because factually it does not always bring us to the best decision), but now it is called for and reasonable, no less—and perhaps more—than the determination to follow the rationales. And as for the perfect judges, my apologies…[12]

[1] See my discussion of this in columns 66, 69, and 79.

[2] See the aforementioned columns.

[3] Regarding the Sanhedrin 34 passage from which he brings proof, see my article in Midah Tovah, 5766, at the end of Parashat Shemot (here, article 67).

[4] Although, as noted, there is also a dispute among the decisors even regarding a religious court, whether what determines the matter is the quantitative majority or the qualitative one.

[5] And that is apparently also the case when the sources they adduce in support of their words are different (as in the aforementioned Sanhedrin 34 passage).

[6] See a fuller discussion of this in our book On Fuzzy Logic in the Talmud, the twelfth in the Talmudic Logic series, and in Daniel Weil's article in issue 1 of the journal Higayon.

[7] A 1 means an affirmative answer to the interpretive/factual question (the contract forbids / the act was done). A 1 in the final-ruling column means that Reuven is liable to pay.

[8] See here and the legal opinion of the State Attorney's office here.

[9] She claims that she was raped after betrothal (and that does entitle her to a full ketubbah payment but disqualifies her from marrying a priest), but she could have claimed that she had lost her virginity accidentally (which also would not have disqualified her from marrying a priest). That is a better claim, and therefore she has a migo.

[10] Here is this principle in the form of a rule (Yad Malakhi, "Rules of the Talmud," sec. 40):

An amora has the authority to effect a compromise between the tanna'im, and this is not considered as ruling neither like this master nor like that master; and we indeed find this in several places, among them in the chapter "Ra'uhu Beit Din" and at the beginning of the chapter "Cheresh," in Mayim Chayim to Pri Chadash, fol. 32a; and he likewise wrote this in the laws of Passover, siman 442, se'if katan 1:

See also Gufei Halakhot 38; Halikhot Eli 519; and others.

[11] This is not precise, because on a straightforward view there seems to be dependence between them: assuming that the contract forbids the act X, it is reasonable to assume that Reuven would not do it. In any event, the probability that he did it is not equivalent to the probability that he did not. To be sure, Shimon too enjoys a presumption of integrity, and therefore the probability that he is lying should also be small. In practice this is of course complicated, and therefore once again we will assume simplifying assumptions here, and for our purposes we will assume that there is no dependence between the questions.

[12] One might perhaps suggest a consideration according to which, if the judges are perfect, it is less likely that such a paradoxical table will arise at all. The disagreements among such judges will be more minor, and presumably we will reach the correct result one way or the other. According to this, from the very fact that we are dealing with a distribution of opinions like the table of the doctrinal paradox, it is reasonable to infer that these are not perfect judges. That further strengthens the decision to follow the bottom line rather than the rationales. With perfect judges it will yield the correct result from both directions (both from the rationales and from the bottom line), and with imperfect judges it will yield the correct result because with them the truth is obtained by the consideration of a majority at the level of the bottom line.