The Thought of Rabbi Gedaliah Nadel – Rov – Lesson 2
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- Following the majority and nullification by majority
- Dry mixed with dry and liquid mixed with liquid, and the dispute among the medieval authorities (Rishonim)
- Rav Chaim, the Sanhedrin, and Tosafot in Bava Kamma 27
- The example of “I do not know” in a religious court and its implications
- Chapter 4, page 47: an attempt to show that majority is not probability
- Chullin 11: a majority that is before us versus a majority that is not before us, a minor boy and girl, and an aylonit
- Rashi, Rabbi Shilat, and an alternative explanation: induction in a majority that is not before us
- The verse “follow the majority” as a novel permission to rely on majority despite the “cost” of prohibition
- Expected value, insurance, and the distinction between majority and expectation
- The St. Petersburg paradox and Pascal’s wager
- Rabbi Meir is concerned for the minority, and Tosafot in Yevamot and Bekhorot
- The law of fixed location: “anything fixed is treated as half-and-half” and “he lay in wait for him and rose against him”
- Criticism of the “scriptural decree” explanation in fixed cases, and the methodology of derivations
- A majority of judges and probability: the claim of minority superiority and clarification of a probabilistic mistake
Summary
General Overview
The text draws a distinction between following the majority and the law of nullification by majority, and argues that nullification by majority cannot be based on probability, because in a liquid-into-liquid mixture one certainly consumes prohibition, and nevertheless the permitted majority “nullifies” it. It examines the differences between a majority that is before us and a majority that is not before us, and presents Rav Chaim’s claim that even following the majority is not really a matter of probability. It then challenges the proofs for that claim from the passage in Chullin 11, from the law that anything fixed is treated as half-and-half, and from the argument about a majority of judges in a religious court. Along the way, it includes remarks about disputes among the medieval authorities (such as the Rashba), about the comments of Tosafot and Rashi, and about the methodology of derivations and distinctions between cases, together with probabilistic examples and expected-value examples (insurance, lottery tickets, the St. Petersburg paradox) meant to sharpen what probability does and does not explain.
Following the majority and nullification by majority
The Talmud in Chullin looks for a source for nullification by majority and connects following the majority to “follow the majority,” so that the majority decides not only in the Sanhedrin but also in cases like a piece of meat found in the street opposite nine kosher stores and one non-kosher store. The text argues that following the majority is presented as a statistical question, but nullification by majority is a different mechanism, because when one eats the whole mixture one certainly consumes prohibition, and nevertheless there is a rule that the minority is nullified in the majority. The text asks how nullification by majority can be attributed to “follow the majority” if “follow the majority” is understood as a probabilistic way of deciding who is right.
Dry with dry, liquid with liquid, and the dispute among the medieval authorities (Rishonim)
The text distinguishes between dry-with-dry mixtures, where the pieces remain separate and one can ask of each piece “what is it,” and liquid-with-liquid mixtures, where the substances are blended and every sample certainly contains both prohibited and permitted matter; flour is defined as like liquid-with-liquid in this respect. The text cites views among the medieval authorities (Rishonim) that in dry-with-dry mixtures the mechanism of permission resembles following the majority in a statistical sense, and attributes to the Rashba the view that if only two pieces remain, there is no longer a majority and it becomes forbidden to take one. The text emphasizes that in liquid-with-liquid, even if there is a permitted majority and the law is nullification, still in the cup one drinks there is certainly prohibited matter, so this cannot be a statistical explanation.
Rav Chaim, the Sanhedrin, and Tosafot in Bava Kamma 27
The text describes how Rav Chaim makes a distinction and argues that even following the majority is not probability, and tries to ground this in the case of a religious court: the Torah requires three judges, and when two are against one and we follow the majority, this implies that the minority judge somehow “joins” or is “nullified” so that the ruling counts as one of three. The text brings the proof Rav Chaim cites from Tosafot in Bava Kamma 27, which says “the minority is as though it is not there,” and notes that this is not the same as saying that the minority is absorbed into the majority; rather, it portrays the situation as though there is no third judge. The text questions the assumption that there must be three judges who all hold the opinion that was ruled, and argues that the Torah simply wants a panel of three even if one dissents.
The example of “I do not know” in a religious court and its implications
The text brings a case involving Rabbi Dichovsky in the Supreme Rabbinical Court, where Rabbi Sherman was in the minority regarding a conversion and said “I do not know” in order to trigger the addition of two more judges, because a judge who says “I do not know” leads to the addition of judges. The text considers whether the rule of “I do not know” strengthens Rav Chaim’s view that the ruling opinion needs three judges, but concludes that it does not provide clear proof for Rav Chaim, because “I do not know” can be interpreted as non-participation rather than proof that the minority flips into the majority. The text presents the move as expanding the panel rather than skewing the law, and ties the discussion to the question whether the ruling has to “come from the mouths of all of them.”
Chapter 4, page 47: an attempt to show that majority is not probability
The text shifts the discussion to chapter 4, page 47, and notes a change: at first it had been argued that nine stores and the Sanhedrin are both based on probability, but now the claim is that even following the majority is not learned from “follow the majority” as probability. The text arranges several proofs and arguments aimed against the probabilistic understanding, beginning with the passage in Chullin 11 that distinguishes between a majority that is before us and a majority that is not before us.
Chullin 11: a majority that is before us versus a majority that is not before us, a minor boy and girl, and an aylonit
The text describes the Talmud’s question in Chullin 11 about the source for following the majority, and the statement that from “follow the majority” one can learn only a majority that is before us, such as nine stores and the Sanhedrin, whereas the unresolved issue is where it is not before us, such as a minor boy and girl. The text brings the case of levirate marriage, where we worry that the young girl may turn out to be an aylonit, in which case there is no levirate marriage, and we rely on the majority of women who are not aylonit to permit it. The text quotes the argument that if the reason is probability, there is no reason to distinguish between before-us and not-before-us majorities, and even gives an example of 98% who are not eunuchs as against 90% in the case of nine stores, and then asks why the Talmud seeks a different source.
Rashi, Rabbi Shilat, and an alternative explanation: induction in a majority that is not before us
The text notes a comment by Rabbi Shilat in Meiri on page 55, and says that Rashi at the end of the passage offers two possibilities: either a majority that is not before us is also learned from “follow the majority,” or it is a law given to Moses at Sinai. Both possibilities are difficult in the plain sense of the Talmud, because the Talmud says neither of them explicitly. The text proposes an explanation that justifies the search for a source: in a majority that is not before us, induction and generalization from a sample are required in order to apply statistics to the case, whereas in a majority that is before us this is “clean” statistics based on a known group. The text argues that the Talmud thought a separate source was needed to allow the use of statistics built from generalization, and when no separate source was found and tradition clearly follows a majority that is not before us, Rashi explains that this too is learned from “follow the majority.”
The verse “follow the majority” as a novel permission to rely on majority despite the “cost” of prohibition
The text argues that the verse does not teach that statistics “work,” but rather that one is permitted to rely on them as a basis for halakhic conduct even when there is a serious cost, such as eating prohibited food. It illustrates that a person would not necessarily eat a piece of meat if there were a 49% chance it was non-kosher, because the cost on the side of prohibition is high even if the majority permits, so the verse gives permission to decide by majority. The text connects this to weighing “cost” against probability and claims this is what the verse adds beyond mere logic.
Expected value, insurance, and the distinction between majority and expectation
The text explains that in decision-making one does not always go by the “majority,” but rather by expected value and by weighing the cost of loss against gain, illustrating this with lottery tickets and insurance, where the insured person’s expected value is negative, yet insurance is still worthwhile in order to cover an event one cannot or does not want to absorb. The text gives a practical rule: “it is not worth buying insurance” except for an event one cannot bear, and offers a personal example of forgoing comprehensive insurance and keeping only third-party coverage. The text uses this to show that calculations of probability and expected value can mislead, and that one has to understand the distribution and not settle for the average alone.
The St. Petersburg paradox and Pascal’s wager
The text mentions Pascal’s wager and presents it as an expected-value argument in favor of faith, then argues that it contains a simple mistake that can be understood through the St. Petersburg paradox. The text describes a coin-toss game in which the payoff doubles by powers of two and the expected payoff comes out infinite, yet in practice people are not willing to pay a large sum to participate because the chance of reaching very large gains is tiny and the distribution is extremely broad. The text concludes that expected value is not a good criterion when the distribution is wide and the standard deviation is problematic, and that calculations of expectation and probability require caution.
Rabbi Meir is concerned for the minority, and Tosafot in Yevamot and Bekhorot
The text notes that Rabbi Meir is concerned for the minority in Chullin 11, and presents Tosafot in Yevamot and Bekhorot, who maintain that this is Torah-level and not merely a rabbinic stringency. The text reports that Tosafot say Rabbi Meir does not disagree regarding a majority that is before us, because the Sanhedrin and nine stores are learned from “follow the majority,” and that his dispute concerns a majority that is not before us. The text explains that this is no proof against probability, because one can argue that Rabbi Meir does not reject probability as such, but rather rejects probability based on generalization and induction.
The law of fixed location: “anything fixed is treated as half-and-half” and “he lay in wait for him and rose against him”
The text cites the rule “we hold that anything fixed is treated as half-and-half,” learned from the verse “he lay in wait for him and rose against him,” excluding one who throws a stone into a group, and illustrates it with a pit containing nine gentiles and one Jew, where someone throws a stone and kills the Jew. The text presents the distinction between “whatever separates is assumed to have separated from the majority” and “fixed” through the case of the nine stores: a piece found in the street is treated as having separated and is permitted based on the majority, but one who entered a store and took meat and does not know from which store it was taken is treated as a fixed case, and the meat is forbidden as a Torah-level doubt. The text argues that probabilistically there is no difference between a separated case and a fixed case if the chance is still one to nine, and therefore this serves as evidence against understanding “majority” as probability.
Criticism of the “scriptural decree” explanation in fixed cases, and methodology of derivations
The text argues that if one says fixed cases are merely a scriptural decree, then the rule should have been limited to capital cases, which is what the verse discusses, rather than generalized to all Torah prohibitions; the very fact that it is generalized suggests there is underlying logic and not mere arbitrariness. The text argues methodologically that when a law is not explicit in the verse and the Sages derive it by extension, there must be logic guiding the derivation, unless it is a traditional “supporting verse” for something like a law given to Moses at Sinai. The text adds that in the law of fixed cases there are distinctions such as “it separated before us and one took it,” and if there is no rationale for the law it is hard to understand how such distinctions are made between cases.
A majority of judges and probability: the claim of minority superiority and clarification of a probabilistic mistake
The text presents an argument that if a majority in a religious court were based on probability, one might say that it is less likely that many judges all “hit the truth” than that a few judges do, and therefore supposedly the minority should be preferable, and it brings a confusing example of a judge who is right 70% of the time. The text rejects that line of thought and explains that the confusion comes from switching the statistical question—from “if the event happened, what is the chance they would say this” to “if they said this, what is the chance the event happened”—and refers to this as a matter of conditional probability and Bayes’ formula. The text says it will return to the third proof next time and bring further examples, including one about a menstruant woman.
Full Transcript
[Rabbi Michael Abraham] So, following the majority. We began with the passage in Chullin, where the Talmud looks for the source for the rule that there is nullification by majority—that the majority nullifies the minority—and it learns it from “follow the majority.” That’s what the Talmud says. The Talmud says this about following the majority, that it is learned from “follow the majority.” Meaning that following the majority is not only in a religious court, where the majority determines the law, but also in a case like a piece of meat that we found in the street and we ask which store it came from—did it come from the kosher stores or from one of the non-kosher stores? So we follow the majority. But nullification by majority—and this is majority that is before us and majority that is not before us, we already made that distinction there—but nullification by majority is a different principle. In nullification by majority, it’s not a statistical question. Following the majority is, in principle, a statistical question, although he already said that later on he’ll say otherwise. But nullification by majority is definitely not a statistical question, because in nullification by majority, when I eat the whole mixture that contains forbidden food, it’s not because there was a high probability that I didn’t eat prohibition. I definitely ate prohibition. Rather, there is a law of nullification by majority, that the minority becomes nullified in the majority. How is that connected to “follow the majority”? “Follow the majority” is a statistical issue. The question is who is right, the majority or the minority; most likely the majority is right. Fine—but what does that have to do with nullification by majority? He mentioned the dispute among the medieval authorities (Rishonim) whether one may eat the entire mixture—we talked about this a little—or whether one has to leave something over at the end. Because in that dispute you can see that even in the question of nullification by majority, at least in dry mixed with dry, it’s a similar kind of question. In a mixture of pieces—dry mixed with dry means ordinary separate pieces of food, things that are distinguishable, not mixed together like liquids. Fine. Liquid mixed with liquid means actual liquid; flour too is considered like liquid. Since once forbidden flour gets mixed into permitted flour, when I take a spoonful from that flour, you can’t say maybe I took the forbidden part, maybe I took the permitted part—I definitely took both. That’s called liquid mixed with liquid. A dry-with-dry mixture means that if I took one item, it could be that I took only permitted food or only forbidden food, but it isn’t necessarily totally blended.
[Speaker B] Why does it specifically depend on whether it’s liquid or dry? Liquids mix. But dry things too—if it’s powder or something like that?
[Rabbi Michael Abraham] Right, I said that’s considered liquid mixed with liquid. So in dry mixed with dry, there is the Rashba’s view. I said there are views among the medieval authorities (Rishonim) that in a dry-with-dry mixture it’s like following the majority, not like nullification by majority. Because with every piece that I take, I can ask: what is its status? Is this a forbidden piece or a permitted piece? There is only one correct answer. And then it’s a statistical question. If most of the pieces in the mixture are permitted, and I took one piece, I can assume that most likely it is permitted. So it’s exactly like the stores. And then it really comes out that when only two pieces remain, the Rashba says you may no longer take one, because once you’re taking one out of two, you no longer necessarily have a majority of permitted pieces. Even though, of course, on the statistical level that isn’t true—on the statistical level there isn’t a greater chance at the beginning than at the end. It’s the same thing. But that view says that even nullification by majority is basically a mechanism like following the majority. It’s a statistical question. But that doesn’t really matter, because that’s only in dry mixed with dry. What about liquid mixed with liquid? There too there is nullification by majority. It’s a mixture of libation wine with ordinary wine, right? So if there is a permitted majority, we say it is nullified. If it’s nullified, then I can now drink, say, a glass of wine from that mixture. And in that glass there is definitely forbidden wine. So it’s no longer a statistical question. Therefore it doesn’t matter how you understand dry mixed with dry; in any case, the law of nullification by majority contains something beyond statistics—at least in liquid mixed with liquid. And I think we saw Rav Chaim, and there Rav Chaim made some distinction. He himself claims that even following the majority is not a matter of probability at all. We discussed the fact that in a religious court, when we follow the view of the majority and not the minority view, it is not—at least not simply speaking—not really a majority that is before us. Because it’s not like stores, a piece of meat, and the question which store it came from. There the question is which is more likely to be right, the majority view or the minority view. So that is actually a majority that is not before us, because we are talking about many, many courts, and the question is: in how many of them is the majority opinion mistaken? Fine? So there is really a problem even with deriving the issue from “follow the majority.” It can still be learned from there, because… yes, exactly. After all, the Torah says that you need three judges, for example in monetary law—you need three judges. Once there are two against one and the Torah tells us to go after the majority, that means we follow the opinion of the two. And if we followed the opinion of the two, then the court that ruled here did not consist only of two judges. The Torah wants three. We are forced to say that even the minority judge joins the majority opinion and is nullified, as if he changes his skin—he is considered as holding the majority view. And I said that he brought proof from Tosafot in Bava Kamma 27, where Tosafot says that the minority is as though it is not there—not that the minority flips into the majority, but rather as though there is no third judge here at all. That’s what Tosafot says, so Rav Chaim and Tosafot don’t exactly line up. And in truth, his assumption that there always must be three judges who hold the opinion that was ruled as Jewish law really does not look convincing, certainly not necessary, if at all. Because all the Torah wants is that a panel of three should sit. If two of them think one way and one thinks otherwise, what’s wrong with that? That’s fine too. Who said the ruling has to come from three judges? We once talked about that—I don’t think it was last time—about that ruling of Rabbi Dichovsky in the Supreme Rabbinical Court, where Rabbi Sherman was in the minority regarding conversion, and he made this formal claim of “I do not know,” in order to add two more judges. He was in the minority, and the majority had basically decided that it was a valid conversion, while he claimed that the conversion was invalid. So he said, “I do not know.” The moment one of the three judges says “I do not know,” they add two more judges. And then there was a chance to reverse the decision, because if those two would agree with him, then you’d have three against two. And whether that was proper or not—we discussed it a bit then—I think it was proper. But there I talked somewhat about this whole issue of the relationship between the judges. Meaning, does the ruling really have to emerge from all of them? And “you shall not go about as a talebearer among your people”—that you should not say, “I convicted and my colleagues acquitted,” so a person is not even allowed to say, from the standpoint of Jewish law, that he was in the minority. The ruling has to come out of everyone’s mouth.
[Speaker C] But that—fine—he’s skewing the judgment. What? He’s skewing… no, we discussed that there—
[Rabbi Michael Abraham] Let’s not go into that again now. I don’t think that’s right. He’s not skewing the judgment. He just wants a broader panel. If you have five judges and in fact three go in his direction—two more and him go in his direction—then what’s the problem? It’s a panel of five, there are three judges who think that way, so then everything is fine. And if that ruling really is not correct, no problem: the two who join will say what they think. If they think not, then the case is over. So what is wrong with adding two more? Okay, but we discussed that there.
[Speaker B] Wait—if someone says “I do not know,” then the other two can’t decide? So that really does look like what Rav Chaim said.
[Rabbi Michael Abraham] I said that apparently—but no, “I do not know” is… if he says against, then not.
[Speaker B] So that’s exactly what Rav Chaim said—that because, in other words, the ruling requires three judges, and only if he says “against” is he nullified, and if he says “I do not know,” then he… why not?
[Rabbi Michael Abraham] Right, that’s the question. It looks like Rav Chaim to the same extent that two against one also looks like Rav Chaim. I don’t think you gain anything here in favor of Rav Chaim. If you say that he flips, then he flips—and then you’d have to say that someone who says “I do not know” does not flip, for some reason. It isn’t clear to me why. But if you say he simply doesn’t belong, then fine—it could be, even without Rav Chaim, that this is not correct. Because if he doesn’t belong, then the panel here was not a panel of three. It wasn’t just that three did not decide the case—rather, there is someone saying “I do not know,” and he is outside the group. So three judges did not sit here—not only that three did not decide it. Anyway, I don’t think there’s proof for Rav Chaim from here.
[Speaker B] This law of “I do not know”—is that an agreed law? Yes. What? That if one judge says “I do not know,” they add more?
[Rabbi Michael Abraham] Yes. Fine. So we’re in chapter 4 on page 47. He wants to bring proof… he wants to bring proofs that following the majority is not based on probability—not based on probability. And then he says, let’s go back to the earlier matter. We said that nine stores can indeed be learned from following the majority in the Sanhedrin, because both are based on probability. After all, at the beginning, when he asked how we derive nullification by majority from “follow the majority,” he said that following the majority can be learned from “follow the majority,” but nullification by majority is something else, because following the majority is also a probabilistic issue—that’s how he presented it at first. Now he comes back and says no, that’s not correct—even following the majority cannot be learned from “follow the majority,” because it is not a probabilistic issue. So he brings a number of arguments. It seems he brings several claims against that understanding. First: we saw that the Talmud in Chullin 11 asks for the source of the law of following the majority, and says that from “follow the majority” one can learn only a majority that is before us, such as nine stores and the Sanhedrin. Because our question concerns where it is not before us, such as a minor boy and girl. We talked about the aylonit—whether a minor boy may enter levirate marriage with a minor girl, because it could turn out that she is an aylonit, and then it turns out that you have actually had relations with your brother’s wife—sorry, with your brother’s wife—when in fact there is no levirate marriage, because an aylonit cannot bear children. So now how do you know whether this little girl will eventually turn out to be an aylonit or not? We go after the majority. Most women are able to bear children, so he may enter levirate marriage with a minor girl on the assumption that although she may turn out to be an aylonit, most women are not aylonit. And regarding this the Talmud asks: from where do we learn the law of majority in such a case? Because that case is a majority that is not before us, not a majority that is before us, since it speaks about laws of nature, not about a group standing before us. But if the derivation from “follow the majority” is that one follows the greater probability and does not view such a case as an evenly balanced doubt, then what difference is there between a majority that is not before us and one that is before us? The probability that the minor before us will not turn out to be a eunuch may even be greater than the probability that the meat before us belongs to one of the nine stores. What is he basically saying? The Talmud says that from “follow the majority” we learn the law of following the majority in the case of a piece of meat and stores, but in the case of a minor boy and girl, in levirate marriage with a minor girl, you can’t learn it from there, and so the Talmud looks for another source. He asks: if the whole point is probability, then what’s the problem? Why can’t you learn it from there? It’s probability. Probabilistic considerations apply both to a majority that is before us and to a majority that is not before us. In both cases there is one chance in this direction and another chance in that direction, and if one is allowed to follow probability, then what difference does it make how the probability is calculated? A majority that is before us and a majority that is not before us—probability is probability. That is his claim.
[Speaker D] And mathematically that isn’t right, because in a majority that is not before us we don’t have the group.
[Rabbi Michael Abraham] What? In a majority that is not before us we don’t know the size of the group. So what? What’s the problem? Within our sample. We don’t know the size of the group, and it doesn’t belong to the group—it doesn’t belong to the group. In another second I’ll comment on that, but I agree with the basic point. I’ll comment in a moment. So that’s what he says: the probability that the minor before us will not turn out to be a eunuch can be even greater than the probability that the meat before us belongs to one of the nine stores. For example, if it is known from years and generations of experience that only two percent of the population are eunuchs, then the probability that this is not a eunuch is ninety-eight percent, as against a probability of ninety percent in the case of nine stores against one. So if we follow probability, then why—what room is there to distinguish between a majority that is before us and one that is not before us? Is it impossible to establish with certainty that there is a majority in a case where it is not before us? Surely prolonged and general experience, based on some natural cause—such as that most minors are not eunuchs—is no less certain than knowledge based on a random situation, that these stores sell slaughtered meat and that one sells carrion. Right? In both cases the reasoning is that it is more likely to be this way than that way. Are you allowed to go by that reasoning? So what’s the difference? And if you say: it is a scriptural decree to follow the majority only where the majority is before us, similar to the Sanhedrin—then you have no source even for a majority that is before us, because perhaps the scriptural decree applies only to the Sanhedrin itself. Why do you decide that it can be extended also to stores and to other cases of a majority that is before us? There the Torah established that the law must be decided, but regarding a doubtful prohibition like nine stores, perhaps one does not follow the majority. Here too there are two comments where I think he is not right. First, one comment on the plain meaning of the Talmud. This is a comment of Rabbi Shilat in Meiri, right at the end somewhere, on page 55 at the end of the chapter. In his footnote below he brings this Rashi. I didn’t quite understand what Rabbi Dahlia writes here, because Rashi is against him. At the end of the passage they don’t find a source for a majority that is not before us. They look for a source. A majority that is before us comes from “follow the majority.” And what about a majority that is not before us? They search and search, reject everything, find no source, and that’s it—they move on to the next passage. Rashi asks: wait a second, so do we follow a majority that is not before us? Of course we do, across the board. The Talmud looked for a source and didn’t find one, so what remains? So Rashi gives two possibilities. One possibility—I may already have mentioned this—is that this too is learned from “follow the majority.” And the second possibility is that it is a law given to Moses at Sinai. Both possibilities are difficult in the plain sense of the Talmud, because the Talmud says neither of them. It searches for a source, searches, doesn’t find one, and says nothing. But these are the two possibilities Rashi gives. Now, if it is a law given to Moses at Sinai, then fine. But if it is learned from “follow the majority,” then what does that mean? That according to the conclusion of the passage, this majority too is learned from “follow the majority”? Even a majority that is not before us? So it’s not true that only a majority that is before us is learned from there. Still, one could say: fine, but what was the initial reason to distinguish at all? Why was the whole process needed just to end up with that conclusion? From the start it should have been clear that if one follows a statistical majority, then what difference does it make what kind of statistical majority it is? Before us or not before us—why does the Talmud try to find another source, and only when it doesn’t find one does it derive it from “follow the majority”? That’s really how one can formulate his question. But I think there is another problem here. According to his definition and his approach, the whole discussion starts with this: why does the Talmud look for a source for a majority that is before us? From the simple consideration that one follows the majority. In all of life we make statistical judgments and follow the majority, right?
[Speaker B] Not always. Most of the time, I mean… many times we don’t settle for a majority.
[Rabbi Michael Abraham] You want to decide between two possibilities—you usually work, in the absence of knowledge, when you don’t have information, with a statistical calculation. What else are you going to do?
[Speaker B] A statistical calculation, yes, but not always a majority. A lot of times I’d say, if for example there’s a chance that some food is poisonous, I won’t go by the majority.
[Rabbi Michael Abraham] No, that’s something else. You’re talking about the question of cost. Fine, one second—I’ll get to that in a moment. But obviously, a decision between two possibilities is made through statistics when there is no information, okay? So why do we need a verse at all—“follow the majority”? After all, according to his approach, these are logical things. So why at all… fine, he already said once that the fact that something comes from a verse doesn’t mean it isn’t logical. Sometimes the verse teaches you the logical principle. But really, what does the verse teach? I think the verse comes to teach—still, it needs to add something. If it were just simple logic, then why call it reasoning? The fact that there is a verse does not mean there is no logic. But if there were logic sufficient on its own, then you wouldn’t need a verse. The verse has to tell me something about the logic. What does it tell me? Exactly what Arik said earlier. There are situations where the cost does not justify using statistics. Meaning, I have poison here, and there is a forty percent chance that it is poison. So should I go after the majority and drink it?
[Speaker E] But if there’s a one in two hundred chance, then you…
[Rabbi Michael Abraham] Fine, but at forty percent? Following the majority applies even in a case of fifty-one percent against forty-nine percent, or fifty plus epsilon. Right? Poison—you would drink it in such a case? Obviously not. Right? So you might say: maybe the prohibition in the Torah is harmful, or let’s not get into metaphysics at all—but it’s serious enough that we won’t rely on fifty-one percent. After all, there’s still a non-trivial chance you’re eating something prohibited; you have to be careful. Who says we follow the majority? For example, you found a piece of meat. There are a hundred stores in the city, fifty-one kosher stores and forty-nine non-kosher ones. Right? And how could you hear someone, if not for the verse? I can certainly imagine someone saying: listen, I’m not eating that. Forty-nine percent chance it’s prohibited. So what? True, it’s more likely to be permitted. If I absolutely had to decide, I’d decide. But that’s in a case where there’s a cost on both sides. Here there’s a cost on one side and no cost on the other side, so in my view even forty-nine percent is enough to refrain. Then the Torah comes and says “follow the majority”—now you may go after the majority. There is a novelty here. The novelty is not that statistics work; that we know even without the Torah. The novelty is that a statistical calculation can serve as a basis for conduct, meaning that I can actually act in that direction. By the way… in a second I’ll bring you an example from the day before yesterday, maybe, about this point. So that’s the novelty learned from “follow the majority.” But if that’s so, then I return to your comment—what was your name, remind me? Yoel. Yoel’s point was basically this: we saw in the previous lecture the difference between a majority that is before us and a majority that is not before us. A majority that is before us—I approach ten stores in town, I know all of them. I know that nine are kosher and one is non-kosher. Now I found a piece of meat, assuming that the piece didn’t come from somewhere else but from one of the stores in the city. Fine? Then it’s obvious that it came either from those stores or from that store. No tricks here. This is clean statistics. It’s a probability calculation of one to nine. Fine? So that is learned from “follow the majority.” What is learned there? That I’m allowed to eat the meat even though… I’m allowed to rely on a majority calculation, even though the prohibition of non-kosher meat is on the scale. But if I’m talking about the aylonit—the case of the minor boy and girl, where there is concern that the girl is an aylonit—there I don’t have… there I work with a sample. Meaning, in order to get to the statistics there I need something in addition to statistics. I need induction. Meaning, I have to take the sample and decide that the sample is representative—that this is really also the phenomenon in the world at large, including those women I didn’t sample—and then I have a probability. Now I can say: fine, if a young girl comes before me now and I ask whether she is an aylonit or not, the assumption is that she is not an aylonit because we follow the majority. But here one more step is required before I make the decision: the step of generalization, moving from the sample I know and establishing that there is some general rule here. Who said that one is permitted to do that in Jewish law? To go by statistics is permitted—but who said one may go by statistics that are the result of induction? Maybe only statistics that I know directly and have simply seen with my own eyes. But statistics that are the result of induction—who said one may use them? That’s what the Talmud is saying. The Talmud looks for a source for this. And therefore I argue that all of this can still be probability. It’s not because it isn’t probability—it can still be probability. But besides the probability in a case not before us, there is also induction involved. And because of that, you need a separate authorization or source that teaches me that this too is allowed—that one may also follow a majority that is not before us. In the end the Talmud doesn’t go that route; Rashi says: fine, learn this too from “follow the majority,” and now the whole matter is more understandable. At first the Talmud thought this was different, because you understand—I’m actually saying the opposite from the source you brought. In other words, the very passage you brought as proof in his favor actually goes against him. Because the Talmud initially assumed that a majority before us is learned from the verse, and a majority not before us is something else. So how can Rashi suddenly tell us at the end that a majority not before us is also learned from the verse? Because it was obvious to the Talmud that it is similar—that it is a majority issue. But there is still a distinction between them. This is statistics, and that is statistics plus induction, plus generalization from a sample. Fine? So the Talmud initially thought that this probably needs a separate source. It didn’t find a separate source, so then what? After all, we know that a majority that is not before us is accepted—there is a tradition that says we do follow such a majority. Apparently that too was learned from “follow the majority.” In other words, apparently this kind of statistics too is sufficiently similar to “follow the majority” to be learned from there. Because there really is a similarity. The distinction is an internal distinction, within the category. Both are probability; it’s just that here there is also probability built on induction. So there was room to say that maybe this is slightly different, and perhaps that cannot be derived from “follow the majority.” Once we didn’t find another source, and we know that we do follow a majority that is not before us, then Rashi says: apparently this too came from “follow the majority.” How did it come from there? Because it’s the same type. Because both are probability. And indeed, the other opinion in Rashi is that this is a law given to Moses at Sinai. There things are more open, at least. That’s not a proof against what I’m saying, but it also isn’t a proof against him. On that second opinion in Rashi, maybe he can lean.
What I said earlier is the issue of cost, right? When you make a statistical calculation, you don’t always go after the majority; it depends what is on the scale. What you really go by is expected value. Right? Meaning, someone who buys a lottery ticket—say they offer you a lottery of ten tickets, or a hundred tickets, each one costing a thousand shekels, and the prize is eighty thousand shekels. Fine? Say eighty thousand shekels. Now in terms of expected value, it is not worth buying the ticket, right? You invest a thousand shekels, and your expected payoff is eighty thousand divided by a hundred—eight hundred, sorry. Fine? The expected value is lower than the price of the ticket. Someone says: no, fine, but since if I win I’ll make eighty thousand shekels, while what I lose is only a thousand shekels, which isn’t a big deal for me, I’ll still enter the lottery. And that’s fine. We do this all the time with insurance, right? The expected value in insurance is negative. Otherwise insurance companies would shut down one by one, right? Where do they make money from? They make money from us. So if they profit, we lose. That’s just math; it’s zero-sum. So the expected value is negative. Why do we buy insurance? Because if something happens, the cost is so heavy that we either cannot or do not want to bear it. We’re willing to accept negative expected value in order to… protect ourselves against an event we don’t want to absorb, even though over the long run the calculation comes out as a loss. Meaning, in the end.
[Speaker F] It’s not zero-sum, because it’s built on spreading risk, and the company takes…
[Rabbi Michael Abraham] No—unless you know that you specifically are a risky person and the company doesn’t know it.
[Speaker F] It’s not against you, it’s against everyone. Meaning…
[Rabbi Michael Abraham] No, what does that have to do with it? Your expected value. I’m asking: what is the chance that you personally will have an accident this year, times the damage that will happen to you? That is less than the premium you pay. Right? That’s it. So your expected value is negative. Why are you making the deal?
[Speaker E] That’s why companies that have a whole fleet of vehicles…
[Rabbi Michael Abraham] Insurance companies do it because they have many people. Why? Even though vis-à-vis you too, their expected value is positive. But they make the opposite calculation. If there were only one customer, it could be that they’d lose a hundred thousand shekels, and they don’t want that. But they have many customers, and the law of large numbers says they’ll always profit. But I, specifically—just me against the company—in the transaction between us, it profits and I lose. My expected value is negative. So by the way, this is a rule about insurance that’s worth knowing. I think it’s a very sensible rule. Therefore—don’t buy insurance ever, except for an event that you are not prepared to absorb. You shouldn’t buy insurance. For years now I haven’t had comprehensive insurance, only third-party coverage. Third-party coverage—you could, I don’t know, hit some brand-new Mercedes, and you’d have to sell your house to pay for it. And although the chance of that is very small, if it happens you’ll be in a situation where you’d prefer to have paid every year in order to have peace of mind. But comprehensive insurance? Your bumper got smashed… fine, I’ll buy another one. Over the years I’ll come out ahead. Add up all my premiums over the years against what I got from the insurance—I’m at a loss. Everyone is, of course, almost everyone. Okay? So therefore it’s not worth doing, if the event is something you can handle. Because then, in the worst case, that event happened. You buy insurance only for an event you cannot handle.
[Speaker D] The problem is still with a majority that is not before us—you see how that gets smeared across everything. It doesn’t require anything from us. It’s like what Maimonides says about whether most events in the world are bad and the world is bad, or good. The world is good, we justify the bad events, and so on. How can you say, with a majority that is not before us, when you see that… without taking a sample that really shows it’s like it seems to you? Maybe your observation is wrong?
[Rabbi Michael Abraham] Fine, maybe—but the reasonable probability is that it is right.
[Speaker D] Why? That’s just how it works.
[Rabbi Michael Abraham] No, without doing—
[Speaker D] Any sample at all.
[Rabbi Michael Abraham] I’ll ask you a question: you’re walking down an unfamiliar street. You ask some random person how to get to such-and-such a place. He explains, and you go according to what he said. Why? Did you take a proper sample telling you how many liars there are among the people you meet on the street? No. You assume that a person generally—unless there’s something unusual going on—tells you the truth. And it turns out that this works in most cases, right? Considered… I personally sometimes ask three people about the same…
[Speaker D] Huh? I personally ask…
[Rabbi Michael Abraham] Fine, but let’s say you’re done, and then you find that the first one wasn’t lying?
[Speaker D] He wasn’t lying, but sometimes, with excessive confidence, a person…
[Rabbi Michael Abraham] Doesn’t matter. That’s not the point. I’m saying you find a statistic that works. It doesn’t matter if… so you want to be more certain, fine—but even if you didn’t do that, in most cases you’d still get there.
[Speaker D] But you don’t usually buy things based on an advertisement.
[Rabbi Michael Abraham] We don’t take a representative sample except when you need one… when you need a scientific result, then you take a representative sample, you test. And even there, it’s not a simple question what exactly a representative sample is. You can never know with certainty what a representative sample is. Fine—we work this way. But we do work this way, and it’s fine, it’s sensible and rational to work this way.
[Speaker D] Everybody works this way.
[Rabbi Michael Abraham] There are assumptions, simple assumptions that… You come across a die, you assume it’s fair. You say there’s a one-in-six chance it’ll land on five. Did you check? Did you roll it a thousand times to see how the outcomes are distributed? How do you know? Maybe it isn’t fair? Fine, but the assumption is that it’s fair, and that’s an assumption that will usually work. Sometimes it won’t. There’s no certainty in anything. But we can go by that; it’s reasonable to go by that. It’s not… yes.
[Speaker B] According to what the Rabbi is saying, if I have, say, a case where there are a hundred stores in a city and I didn’t go through one by one to check how many are kosher and how many aren’t kosher. I checked only ten and saw that nine…
[Rabbi Michael Abraham] No—if you heard from people who checked…
[Speaker B] No, I didn’t hear from people who checked. I checked nine and saw that nine are kosher, so that’s a majority not before us.
[Rabbi Michael Abraham] Of course. It’s like the whole world. What’s the difference between a city and the whole world? That’s exactly what the whole world does. So the example I most wanted to bring you—I wrote about this in the book—about the St. Petersburg paradox. In which book? The one on evolution. The St. Petersburg paradox—there’s a… I wrote it there as an introduction to Pascal’s argument. Pascal, of course, with Pascal’s wager on faith—he says expected value… and he uses the consideration of expected gain. He says, look, it pays to be a believer. Why? Because if you believe and it turns out to be nothing, then at most… at most you kept a few commandments, maybe ate a little less tasty food, and not even that much, it doesn’t demand all that much from you day to day, routinely. Whereas if…
[Speaker E] Except for a few kashrut supervisors? In our case, you supported the livelihood of a few kashrut supervisors.
[Rabbi Michael Abraham] Whereas if you weren’t religious and it turned out you were wrong, you’re going to get hit big-time. Or if you were religious and it turned out you were right, then you have a very great reward. In short, the expected gain significantly favors religious conduct. So let’s say the probability is fifty-fifty, or even if the probability is one in a hundred that the religious view is correct—still, in terms of expected gain, it’s very positive if you’re religious. That’s Pascal’s wager. Now where’s the mistake in that argument? Lots of people quibble this way and that, but there’s a simple mistake here. I think I’ve never actually seen anyone write it. People who invest in the stock market learn in courses about the St. Petersburg paradox. If someone offers you a lottery ticket for a game that works like this: you flip a coin. If it comes up heads, you get two shekels and the game ends. If it comes up tails, you flip again. Okay? If it comes up heads… wait, how does it go there? Right: if it comes up heads, you get four shekels and the game ends. If it comes up tails, you flip another coin. If it comes up heads—eight. Powers of two: two, four, eight, sixteen—or the game ends and you get two to the power of the number of flips, or else it comes up tails and you keep going. Okay? You can’t lose.
[Speaker B] Great way to make money. Right?
[Rabbi Michael Abraham] You lose the price of the ticket. So now, in order to participate in this lottery, you have to buy a ticket. How much money are you willing to pay for such a ticket? Theoretically, infinity. Right? Endlessly. Think for a second, just at first glance before doing calculations. I don’t know, maybe five shekels at most. How many times is it going to come up tails without heads? I need tails without heads a huge number of times, and then eventually heads, and then I’ll make a lot of money, right? What’s the chance of getting a run of more than three tails in a row, four tails in a row? Twenty shekels, if someone wants; I wouldn’t invest more than twenty shekels in such a game, right? What’s the expected value of such a game? Infinity. Infinitely many new shekels. It’s not practical. Infinitely many new shekels—why? Draw the probability tree, okay? You flip heads or tails. On the heads branch it says two. If it comes up tails, from that tails there’s another split. Heads—four, tails—another split. Heads—eight, tails—another split. Now when we sum the expected value, we need to sum the leaves of the tree, right? Every outcome and the chance that it happens. So there’s a one-half chance I’ll get two shekels, plus a one-quarter chance I’ll get four shekels, plus a one-eighth chance I’ll get eight shekels, plus a one-sixteenth chance I’ll get sixteen shekels, infinitely many such branches, right? That’s one—one-half times two shekels is one shekel. One-quarter times four shekels is another one shekel. One-eighth times eight shekels is another one shekel. Altogether I have an infinite expected gain. So where’s the mistake? But nobody is willing to invest even a hundred shekels. Why don’t people do it? The story is very simple: the expected gain really is infinite, but the chance of making a large amount is tiny. Let me give you an example. You outfit a ship to go treasure hunting, searching for treasure in the Pacific Ocean, on some island in the Pacific. You have no idea whether there’s treasure there. But how do you know—maybe once some pirate hid treasure there. There’s a one-in-a-million chance, and if the treasure is ten billion, is it worth investing a million shekels? A one-in-a-million chance. Fine, you’ll make something. That example doesn’t even have positive expectation. Never mind, let’s draw it so that it does. Why don’t you do it? Because even if the expected gain is positive, the chance of profiting is negligible. There’s one possibility—here it’s even better than St. Petersburg—where there are only two possibilities: either there’s treasure or there isn’t. So let’s draw two possibilities. One possibility is that with probability one in a million you’ll make one hundred billion, okay? And the probability is nine hundred ninety-nine thousand nine hundred ninety-nine that you’ll make nothing. Would you go on such an expedition? You’d be crazy. The expected gain is positive, but one in a million times one hundred billion is a very positive expected gain. So why don’t you go? Because the chance of making that huge amount—which creates the expectation—is tiny. And since the amount is so large, when you multiply it by a small probability you get a positive expected value, but the chance of it actually happening is practically zero. Are you crazy enough to invest even a penny in that? It’s terribly confusing, because expected value is not a good criterion when you have a problematic standard deviation. So when the distribution around the expectation is narrow, expected value is a good consideration for making decisions. But where you have a very wide distribution and the average is sitting here, and now you can land anywhere—are you crazy? There’s no point getting into such a thing at all. The expectation says nothing. The wider the distribution, the higher moments you need in order to make decisions. That is, yes, standard deviation is the second moment, so there’s… you need more and more knowledge of the distribution in order to decide. Expected value says nothing in a wide distribution. In St. Petersburg the distribution is, of course, very wide; it goes like two to the minus n, and it falls off with a very long tail. So in short, that means expectation—first of all, probability calculations are always confusing and you have to be very careful with them. You should follow expectation. But expected-value calculations are also very confusing, and you have to be very careful with them too. The truth is that you really need to calculate based on the distribution itself and not on the expectation, but that’s always complicated. So fine, let’s get back to our topic. So his claim—what distinction did the Talmud make between a majority that is before us and a majority that is not before us? He says you see that majority is not probability. That’s not true. Because majority is probability; it’s just that with a majority not before us there’s another element as well. And the Talmud thought that for the generalization—yes, induction—and for that additional element, a separate source was needed. That’s what the Talmud thought; therefore it looked for another source. That doesn’t mean it isn’t probability. More than that: Rabbi Meir is concerned for the minority. Yes, in that same passage in Chullin 11a, Rabbi Meir takes the minority into account; he does not follow the majority. Not precisely—sometimes he does follow it—but where it’s impossible to verify, he follows the majority. Where it is possible, then he doesn’t follow the majority. Where, as it were, it is possible to verify, or where you have a way to conduct yourself even while taking the minority into account—then he takes the minority into account. If there’s no such possibility, then he doesn’t. Yes—for example, you execute a person. Someone killed another person with a sword. Now two witnesses come, and Rabbi Akiva and Rabbi Tarfon, at the end of the first chapter of Makkot, say: did you see whether there might have been a perforation at the site of the sword wound? Maybe the murdered person was already mortally defective. Maybe he had a perforation in the lung exactly where the sword entered? And the sword did nothing; he would have died anyway. Someone with a perforated lung is supposed to die anyway. So they didn’t kill… he killed a mortally defective person; you can’t execute the murderer because he killed someone who was, in effect, already considered dead. Okay? So did you see? they ask the witnesses. No. Which means you can’t execute the murderer. About that the Talmud says that in such a situation even Rabbi Meir does not take the minority into account. Why? Because in such a case it would be impossible to try murderers. The Torah says to try murderers. If you take the minority into account in such a case, then you can’t try murderers. Why—one could say this would only be in a case where they couldn’t inspect… the medieval authorities (Rishonim) already ask that. No, it’s impossible; we’re talking about a case where they couldn’t inspect—a perforation inside the lung, an internal perforation. Oh, not an external perforation? No. Nachmanides wants to say there that it’s talking about an external perforation to resolve various things in the Talmud, no—the plain meaning of that Talmud is an internal perforation, and so it’s a terefah. Right. In any case, that’s Rabbi Meir, who is concerned for the minority. And Tosafot in Yevamot and Bekhorot maintain that this is at the Torah level—that Rabbi Meir does not accept the principle that one follows the majority on a Torah level. It’s not some rabbinic concern for the minority. Meaning, Rabbi Meir holds that by Torah law one must take the minority into account. And Tosafot say that regarding a majority that is before us, Rabbi Meir does not disagree, because concerning the Sanhedrin and the nine stores the Talmud said plainly that they are learned from “follow the majority,” and Rabbi Meir cannot disagree with that. Rather, Rabbi Meir’s dispute is specifically about a majority that is not before us. In parentheses—even though the flow of the Talmud implies that he also disputes a majority that is before us, never mind, that’s another discussion; there’s a dispute among the medieval authorities (Rishonim) there, but we won’t get into that. That’s what Tosafot say. Tosafot say that Rabbi Meir does not disagree in a majority that is before us, because a majority that is before us is learned from “follow the majority,” period. Certainly you follow the majority. With a majority that is not before us, which has no source—we learn that either from a law given to Moses at Sinai or also from “follow the majority,” according to Rashi—that, Rabbi Meir does not accept. Rabbi Meir takes the minority into account in a majority that is not before us. But if the reason for a majority that is before us is probability, then in a majority not before us there too there is probability that what is before us comes from the majority. So why is it obvious to Tosafot that Rabbi Meir distinguishes between a majority that is before us and a majority that is not before us? Perhaps he distinguishes between what is stated explicitly in the verse, namely the Sanhedrin, which… why? Because a majority that is before us and a majority that is not before us can both be probability. But in a majority that is not before us there is also induction. And when Rabbi Meir says, I do not follow the majority, he doesn’t mean I do not follow probability; rather, I do not follow probability when it is the result of a generalization. I don’t accept, essentially, what Yoel was talking about earlier—I don’t accept generalizations unless I have clear knowledge about the case before me. But if the case before me is not included in the knowledge base that I have, and I’m making some kind of generalization in order to do statistics about it—that Rabbi Meir does not accept. Wait, but in the case of the perforated lung, that too is a case where I’m making a generalization. What? The case of… of the one who was killed by… or something… it could always be that he had a perforation. That’s because it’s impossible. What? What do you mean? Ah, that’s because it’s impossible for another reason. Fine. In any case, for the same reason that I rejected the proof from the plain sense of the Talmud, I reject the proof from Rabbi Meir as well. I think that’s really not a proof at all, what he said. Another proof, proof B: We hold that anything fixed in place is considered like fifty-fifty. And the matter is learned from the verse “and he lay in wait for him and rose up against him,” excluding one who throws a stone into a group. Okay? What does that mean? There the Talmud discusses a situation where, say, there is some pit in which ten people are sitting, one of them a Jew and nine gentiles. Now someone throws a stone into the pit and kills the Jew, okay? The Jew. Now from the outset, when he threw the stone, he could not know that specifically the Jew would be killed; there’s a majority of gentiles there. And one is not liable to death for killing a gentile, okay? So the question is whether in such a situation one follows the majority. So the Talmud says that in that situation one does not follow the majority because this is called fixed. “And he lay in wait for him and rose up against him”—they learn from the verse that one does not follow the majority in such a case. And from this there emerges some kind of rule, a sort of unclear mysticism in Jewish law, that when things are fixed in place, one does not follow the majority. For example, when I enter—a city has nine kosher stores and one non-kosher one. I go into one of the stores, take a piece of meat, go home, and then suddenly remember that there’s one non-kosher store and I don’t know which store I was in, or I don’t know which of the stores is the non-kosher one, never mind. In such a case one does not follow the majority. In a doubt concerning prohibition, one rules stringently; it is forbidden to eat that piece. Not like a piece that separated off: if I found it in the street, I assume it came from the kosher stores. Why? Not why—but what’s the definition. Why, I don’t know why; but the definition is that this is what is called fixed. And when the matter is fixed in its place, because it did not separate off, rather while fixed in place—there the statistics don’t determine things. When the piece of meat belongs to one particular store, it doesn’t belong to one out of a group of stores; I just don’t know the character of that store—that is a question of fixed. And in a question of fixed it is like fifty-fifty in terms of Jewish law. It becomes a doubt. So what is fixed in the case of throwing a stone into a group? What? The people are fixed in their places. Since they are counted, they are… And if someone came out of the pit and I shot him, then that would be separated from the majority? What? No—people fixed in place. Even if there were… what you have here is a package that includes within it items of two types, okay? Now if my doubt arises in the place of fixedness, meaning in the very place where the mixture exists, then the treatment is fifty-fifty. If one element among those items separated off and now I ask what its character is, then I assign it to the majority. Therefore the Talmudic formulation is: anything that separated off, separated off from the majority. But if the thing did not separate off, then the law of majority is not applied to it. There is no law of majority. It’s a matter of place. Fixed is a matter of place. In the place of the mixture, in the place where the mixture is found and nothing has left it, it remains in its place within the mixture—there you do not follow the majority, there it is “anything fixed in place is considered like fifty-fifty.” And according to this, in principle there should be no rule of nullification by majority, and they already discuss that—the medieval authorities (Rishonim) already discuss how there can be nullification by majority if it is within the mixture. It becomes fifty-fifty; the minority receives the status of fifty-fifty, not of a minority. Fine, these are analytic complications. In any case, for our purposes, there is a difference between what separated off and what remains fixed in place. What remains fixed in place is fifty-fifty. If a particular item separated from a group, one follows the majority in order to determine the identity of the item. But if a person comes to the place where the items are fixed and there is doubt as to which one he came to, one does not follow the majority; the matter remains an evenly balanced doubt. In the example of the nine stores, if the meat is found in the city, the rule is: anything that separated off, separated off from the majority. A piece of meat that separated off came from the majority stores. And if most stores sell properly slaughtered meat, then it is kosher. But if a person entered one of the stores and took meat from there, and he does not know from which store he took it, anything fixed in place is considered like fifty-fifty, and the meat is forbidden because with a Torah-level doubt one rules stringently. A doubt here is an evenly balanced doubt. Because if it were not evenly balanced, then majority would apply. And if the logic of “follow the majority” is because of probability, then there is no difference at all for this purpose between separated and fixed. From the standpoint of probability, there’s no difference. If you go to the store and don’t know which store it is, probabilistically it’s still one in nine. In terms of probability calculation it’s the same thing. Right? Between a piece that separated off and a piece that is found in the place where it is fixed, in the store itself. What difference does it make? There are nine kosher stores and one non-kosher one, so the chance is one in nine. So why is there a difference between fixed and separated? That’s an excellent question—why is there a difference between fixed and separated—but I don’t know whether one can draw very many conclusions from it. Meaning, true, good question; there’s no good answer to it, and his answer isn’t good either. We’ll see later. There are so many examples that it doesn’t withstand the test of all the examples. And if the reasoning—yes, so… And if the reasoning of “follow the majority” is because of probability, then there is no difference for this purpose between separated and fixed, and in both there is the same probability. So what will we say? That it is a scriptural decree from “and he lay in wait for him” that the verse reveals to us regarding the law of fixed. Fine? What will you say? It’s actually a nice calculation. So what will you say—that there is a scriptural decree that with fixed cases one does not follow the majority. Majority really is probability, but we learn from the verse that with fixed cases one does not follow the majority. And here there is a very interesting consideration, one that yeshiva students do not always notice needs to be made, but it really is a correct consideration. He says: so what will we say? It is a scriptural decree from “and he lay in wait for him,” that the verse reveals to us in the law of fixed. But if so, we should have restricted it specifically to capital cases, which is what the verse is speaking about. That is, the Torah was lenient and said that one does not execute a person even if there was only one gentile in the group at whom he threw the stone. But from where do we derive a general rule from this for all prohibitions in the Torah, contrary to the rule of “follow the majority”? Think about it: if there is no logic in the law of fixed, let’s say the verse teaches us that when I throw a stone into a pit and it’s in the place of fixedness, it’s fifty-fifty, we do not follow the majority. That’s what the verse teaches. But the Talmud makes from this a rule that also applies to pieces of meat and mixtures, that fixed is always like fifty-fifty, that the law of majority applies only to separation, only to something that separated from the place of fixedness. How do you make that generalization? Maybe the Torah is speaking only about capital law? And there’s also some logic to be lenient there—because there’s one gentile, you already can’t execute the murderer. It’s fifty-fifty. In the place of fixedness it is considered fifty-fifty. Fine? So who told you to make this into a general principle? The fact that you make it into a general principle means that you understand that there is some logic here. Because otherwise, how did you reach the conclusion that specifically the set of cases we call fixed are the ones that should be treated as fifty-fifty? Maybe all cases involving people? Or all cases involving capital law? Or all cases where things are in a pit? I don’t know—you can make a hundred thousand generalizations. Why did you choose specifically this one? Presumably it seemed more logical to you than the other generalizations. So that means there is some logic in it. But probabilistically it isn’t true, because probabilistically it’s the same thing. Now why did I say that yeshiva students are not always aware of this point? Because among yeshiva students, many times when you say that something comes from a verse, that’s enough—it exempts them from having to find a rationale. Now that’s true when the thing is explicit in the verse. Or it could be. When the thing is explicit in the verse. Even then one could argue that there ought to be logic, but there you can still say: it’s written in the verse, I don’t know, maybe there’s another logic, maybe there isn’t, I don’t know, it’s in the verse. But most Torah-level laws that we know are not written explicitly in the verse. They are things that the Sages derived by means of some generalization, one exegesis or another. Now if there were no logic in it, how did they know to derive דווקא that? Why not derive something else? Why choose this rule or this principle rather than a hundred thousand other principles I could have attached to this verse? “The Lord your God shall you fear”—to include Torah scholars. Fine? Why not include cobblers? Because with Torah scholars there is somehow more logic in fearing them, somewhat like fearing the Holy One, blessed be He, than fearing cobblers. Right? There has to be some logic for why you choose one generalization, so even with Torah-level laws there has to be logic. It’s not speculation that maybe there ought to be logic, or some ideology that there has to be logic—it has to be there. Because everywhere the Sages are the ones who derived the Torah-level law, rather than it being written explicitly in the verse, you have to ask yourself how they derived specifically this law. Even though there is a source in the verse. And in order for that, you have to find the logic. Without the logic, what is the source? How can you derive it? So I think that’s a very correct consideration. It’s just methodology: if the Sages derived it from the verse, and it’s not written explicitly in the verse, then there must be logic, otherwise how did the Sages derive it? Unless they have a tradition—in which case the verse is just support and not the creator of the law—but a law given to Moses at Sinai is the thing that actually establishes the law. And furthermore, regarding the law of fixed, a number of distinctions were stated—such as if it separated before us, and if a gentile took it, and the like. And if it’s a scriptural decree without a reason, how can one distinguish between different cases when we do not know the reason for the law? Yes—if a piece of meat separated before our eyes from the store. We see it with our own eyes rolling from the store into the town square, okay? Now we take it. Fine? Is that separated or fixed? You could say that the whole city is the place and not the store. No—a piece of meat that separated without us seeing is certainly separated, even if it’s in the city. But now I’m saying: it separated before our eyes. We saw it ourselves rolling from the store into the city square. That is fixed. And the law of fixed—yes, and this is in public. Yes. And really, just like inside the store—what difference does it make? So he asks: fine, but if it’s all a scriptural decree, then how do you know maybe this is separated? After all, if there’s no logic behind the distinction between separated and fixed, then how do you determine rules about it? How do you decide that if something separated before our eyes it is considered fixed and not separated? You can determine it because your logic doesn’t get that far if it’s fixed; all you’re saying is you see it rolling, so you can say this is fixed and that is separated. But you’re not explaining the logic that says the law… But if you don’t explain the logic, then how do you know that this resembles fixed rather than separated? Similarity is context-dependent. Similarity always depends on the question: similar with respect to what? Human beings resemble one another in that they have two hands, but they do not resemble one another in skin color. It always depends on what you’re looking for. Why not in terms of probability? In short, fifty-fifty, like you said, a single event—fifty-fifty—that sounds sensible. No—would you think that probabilistically? Is that how you behave in life? It’s a single event. So what? Fifty-fifty. Not at all. You throw a die. Are you prepared to bet even money that it will come up five? A single event. We’ll throw the die once. Fifty-fifty—you’ll bet one against one that it will come up five? Of course not. Probability is made for a single event. Probability is realized when there are many events, because that’s the law of large numbers. But probability is the way to treat a single event. I ask: what’s the chance that it will come up five? The chance is one-sixth. When many events happen, then instead of chances I’ll ask how many times it landed on five in sixty rolls—but that actually happened, that’s not a chance. It materialized. How many times? Yes, because it materialized. But chance, calculation by probabilities, is stated about a single event. The theoretical probability is realized when many events occur. Say in St. Petersburg: if they offered you many such lotteries, then you would invest infinitely. Because if you do infinitely many such lotteries, eventually you’ll profit, then no problem—in that case you really would invest infinitely. Infinity in each lottery. Not infinity in total—infinity in each lottery. But that’s all if they guarantee you in advance that you have infinitely many games. But in one game, the chance of getting a large outcome is tiny, so it has no significance. Yes, so he says: you can’t say something is like fixed or like separated if you don’t know the difference between fixed and separated. Fine. There’s something to that claim, although again, there’s this feeling sometimes—I think maybe that’s what you wanted to say—that sometimes we characterize situations even if we don’t really understand the reason for the halakhic distinction between them. But if you ask me whether this is a case of fixed or a case of separated, I don’t know what the difference is, why Jewish law really sees fixed and separated differently. But if you ask whether this is a case of fixed or of separated, I say: this is a case of fixed. Even without understanding the reason why there is really a distinction between fixed and separated. Fine, so that’s a remark, but I think this proof is already better than the previous one as evidence that we’re not dealing here with probability. The very notion that following the majority of judges is due to the greater probability is not clear at all. Because when I discuss a particular event—whether it belongs to the majority or the minority, for example a piece of meat, whether it comes from properly slaughtered animals or from carcasses, and the majority in the city are properly slaughtered—then the probability that this too comes from the majority of properly slaughtered animals is greater than the probability that it comes from the minority of carcasses. But with judges you have the reasoning of the majority versus the reasoning of the minority, and how do we know that the reasoning of the majority is the truth? Is the majority always more correct than the minority? The majority doesn’t always go by probability, but never mind. After all, if we discuss speaking truth as a certain event, and speaking falsehood as the opposite event, and each of the judges… I don’t mean lying, but saying the correct thing, yes? Then seemingly it is less likely that all of the many judges succeeded in hitting upon the truth than that the few succeeded. That is certainly true. Huh? It is certainly true that the minority is more likely to be right than the majority. Right? What did he say? No, not certainly. Probability. Fifty-fifty. Probability. Suppose each judge hits the truth with probability seventy percent, okay? What’s the chance that two judges hit the truth? Forty-nine percent. What’s the chance that the judge opposing them—the minority opinion—hits the truth? Seventy. He rules Jewish law like the minority. No, but why? Forty-nine percent is only the chance that two specific judges hit the truth, not that two out of the three hit the truth. Well? You have two specific judges who tell you that Reuven murdered Shimon. Now the chance that one of them hits the truth is seventy percent, Reuven… That calculation is wrong. The correct calculation is: what is the chance that there will be two who say the truth? No, no. I’m talking about a specific case in front of me. What is the chance in this case that the majority is right? I’m saying the chance that the two hit the truth is lower than the chance that one hits the truth. Rabbi Shabtai asked me this question—Rabbi Shabtai Rappaport from the kollel at Bar-Ilan. He asked me this question about two months ago or something. By mistake, apparently; definitely by mistake. But I need a bit of time to explain the point—that’s Bayes’ formula. But superficially it’s a confusing question. He asked me another question there about a niddah as well; maybe we’ll talk about that next time, if there’s a little time to explain. And the two questions are unrelated, and they involve the same mistake. In any case, yes—so he says: if this were probability, then we ought to follow the minority, because the chance that the minority is right is greater than that… Everyone understands that’s not true. The only question is why. It’s tricky, but everyone understands that’s not true. I’ll already give a hint: when I ask, I’m not asking what is the chance that the two judges hit the truth. The question I’m asking here is completely the reverse, and in statistics and probability this is crucial—to define very, very carefully the question you’re asking. I’m asking the opposite question. If two judges say that Reuven murdered, what is the chance that he really did murder? The chance that they speak truth is a question like this: if he murdered, what is the chance they will say he murdered? Right? That they identified it. Now I’m asking the opposite question: they say he murdered—what is the chance that he really murdered? Right? After all, that is the question whether to follow the majority or not. Two judges say he murdered. Now I ask: what is the chance that this is really true, that he really murdered? Can one ask such a question? Certainly—why not? Does it depend on their opinions? Of course. After all, you have data. In seventy percent of cases each judge is right; in thirty percent he is wrong. Given those data, I can do the calculation—that’s Bayes’ formula, yes. You put the cart before the horse. What do you mean, put the cart before the horse? First the judges and then the case? We talked about Rabbi Shabtai’s case—there is a reality. I only want to know whether the judges preceded the reality or not. You didn’t reverse anything here. No, I don’t know what the reality was. After all, I have a lack of information. Now I ask: what’s the chance that reality was like this, and what’s the chance that reality was like that? I’m testing two hypotheses. You’re not examining what percentage of errors each judge has, that each judge hits the truth. No—suppose I know that in seventy percent of cases a judge hits the truth, just for the sake of argument. So is the chance that they are right really forty-nine percent? The whole thing is getting mixed up here. Meaning: the chance that if something happened, what is the chance that both of them will say it happened—that is forty-nine percent. But if both of them say it happened, what is the chance it really happened—that is a much higher probability. Got it? That’s reverse probability; that’s conditional probability. It’s not P(A|B), it’s P(B|A). And here you need the formula of total probability for that, but it’s not correct. Meaning, that’s the mistake: asking the wrong question. Now maybe I’ll bring a few examples of this next time too, including the niddah case, and that’ll be that, because it’s the same mistake, and it’s very confusing. The question you ask determines the answer. We’ll return to his third proof next time.