Doubt and Probability—In Halakha, Thought, and in General—Lesson 50 and Final—Rabbi Michael Abraham

All right,

[Rabbi Michael Abraham] let’s begin. I’m planning to finish the series today. A round number, lecture number fifty, is a good time to wrap it up. But I’ll open, maybe, in a slightly unusual way, with a Hebrew-language segment I heard on the radio today while driving. I just looked it up online now, found it — it’s three or four minutes. Listen to it with me for a moment. This thing just fell into my lap from heaven. What

[Speaker B] can we do for the Hebrew language?

[Rabbi Michael Abraham] Can you hear it? Yes, we can hear.

[Speaker C] Okay.

[Speaker B] Suppose we have thirty trees, and seventeen of them are fruit trees. Can I say that most of our trees are fruit trees? Seventeen out of thirty. Yes. Suppose nine out of our seventeen fruit trees are mango trees.

[Speaker D] Can I say that most of

[Speaker B] our fruit trees are mango trees?

[Speaker D] Yes, correct, nine out of seventeen. I like this segment, yes. Can I say that since most of our trees are fruit trees, and most of our fruit trees are mango trees,

[Speaker B] the majority of the majority of

[Speaker D] our trees are mango trees?

[Speaker B] Absolutely not.

[Speaker D] In other words, most of most of

[Speaker B] our trees — nine out of thirty — are mango trees.

[Speaker D] I just proved it to you, you agreed with me. Yes, up to that point I agreed with you. So where’s the problem? I don’t know, it doesn’t seem to me that nine out of thirty is a majority. Not just a majority, but the majority of the majority, right? The majority of the majority. Yes, I didn’t say the majority, I said the majority of the majority, most of most. Yes, I tend to agree with you. Literally, it’s correct. Right. But linguistically it sounds strange to us, right?

[Speaker B] When someone tells you that “most of most” of people thought such-and-such, what do you think?

[Speaker D] That most people thought such-and-such.

[Speaker B] Right, that the majority

[Speaker D] and not just a majority

[Speaker B] but an overwhelming majority, a large majority, a majority that’s closer to the whole. But we just saw that that isn’t true. Right. So what’s happening here? Why, when we say “most of most,” does it sound to us like an overwhelming majority, against logic? I think there are two explanations here. The first is a very simple explanation. When we say, for example, “all of it entirely,” I could make do with just “entirely” — it already includes everything. Why do I say “all of it entirely”? When do I use that? To emphasize. To emphasize, exactly. And the same thing if I say “deep inside,” to emphasize that it’s really inside. Right. “The very heart.” According to this, when we say “most of most,” we feel that this is the majority, the really large majority. We feel there’s emphasis here. In other words, we’re used to repetition creating emphasis, intensification, showing that it’s really that thing. Also in literary Hebrew: צדק צדק תרדוף (“Justice, justice shall you pursue”) — not just justice. Right. Also in everyday Hebrew, repeated words create emphasis. So that’s one explanation. A second explanation, which fits with this, from the linguist Gad Ben-Ami Tzarfati, is that he traced the source of this expression and came to the conclusion that the expression “most of most,” which was created in modern Hebrew, was created בעקבות an Aramaic expression, רובא דרובא — literally, the majority of the majority. That Aramaic expression too indicates an overwhelming majority. But it was created under the influence of another Aramaic expression, מיעוטא דמיעוטא.

[Speaker D] Meaning the minority of the minority.

[Speaker B] Now a minority of a minority really is a smaller minority, it really is a minority. So that explains it: it was created under the influence of מיעוטא דמיעוטא, but there isn’t real symmetry here.

[Speaker D] So in practice you might recommend not using דווקא this expression to emphasize a majority.

[Speaker B] That’s a good question. The question is whether we insist that language should always follow logic. I know people who insist on that very strongly. On the other hand, there are people who will say, fine, language has its own internal logic. For example here, the sound-pattern perhaps overcomes the logical, numerical, mathematical logic with which we began.

[Speaker D] One more thing before we finish, because we’ve reached most of the program now.

[Speaker B] We’ve reached most of the program. Maybe I’ll just point out that, alongside repetition that creates emphasis, there is also repetition that creates the opposite. For example, if we had a particularly threatening and large cat, according to that principle we should have called it a kitten? Right. And a large dog would become a puppy. But the man of letters Ze’ev Ya’avetz, based on a certain interpretation of the biblical words for reddish and greenish, where they mean light red and light green, coined this usage דווקא for diminution. By contrast, the classical interpretation is dark red and dark green.

[Speaker D] Tamar Katzir, from the Academy of the Hebrew Language, thank you very much.

[Speaker B] Thank you very much.

[Rabbi Michael Abraham] That’s the end of our Hebrew-language corner. They broadcast it today, this afternoon while I was driving, and it caught my attention because it’s exactly our topic. So what do you say about this expression רובא דרובא? That’s the topic we’ve been dealing with — רובא דרובא. What does the expression mean? It really does seem illogical at first glance. רובא דרובא can come out as a minority — nine out of thirty, in the example she gave before. So why is רובא דרובא understood as a stronger majority and not—

[Speaker F] So really it’s a subset of the majority — that’s “most of most.” Meaning, the majority of the majority.

[Rabbi Michael Abraham] Exactly. So it’s really a minority — that’s what she’s saying. So רובא דרובא isn’t a stronger majority, it’s a weaker majority, if it’s even a majority at all. With me? Can you hear me? Yes. Okay, there was some interruption here, I didn’t understand what that was. So what do you say — what is the meaning of רובא דרובא in this context? It’s not a very successful expression. It’s like the majority out of the majority, but it’s not — if anything, it’s the minority from the minority on the other side. In my view, when the Talmud says רובא דרובא, what it means is: apply the rule of majority twice. That’s stronger than applying it once. Why? Because basically you’re saying, let’s say there’s a majority of women, for example, who had relations before betrothal, just for the sake of example. That’s one majority. But that majority alone would not be enough to permit the woman; I want רובא דרובא. So what do I say? Even among the women who had relations after betrothal, which is really the minority, there is a majority who had relations under coercion and not willingly. Meaning, I’m applying the rule of majority twice. That is a stronger situation than applying the rule once, and therefore רובא דרובא…

[Speaker F] But she explained the logic of why that actually gives you a smaller group. She says they treat it like a product, and then if you multiply a number

[Rabbi Michael Abraham] smaller than one

[Speaker F] then you get an even smaller number.

[Rabbi Michael Abraham] Right, but that’s a mistake. Because she treats it as though I’m taking the majority out of the majority, but that’s not correct. רובא דרובא means applying the rule of majority twice, not taking the majority out of the majority. Rather, applying the rule of majority twice means making a stronger demand — not just one majority, but I want one majority on top of another. Now what is it really… But literally that’s not correct. One second. In practice this is really מיעוטא דמיעוטא, right? What I really want is that the minority should not be just one minority, but a minority of a minority. I have a majority for permission and a minority for prohibition — sorry, a majority for permission and a minority for prohibition. On the basic level, that’s one majority and that’s enough, because there’s a majority for permission. If you want something stronger, as we saw regarding permitting agunot, then even within the minority for prohibition there is still a majority for permission and a minority for prohibition. So when I get here, I really see that the reason to permit is much stronger; the majority really is stronger in such a situation. It truly is a stronger majority, but not because it’s a majority out of a majority; rather, it’s a majority on top of a majority. A majority on top of a majority means applying the rule of majority twice, so that it points in your favor twice. So that the rule leaves the minority pointing to prohibition even more reduced. In practice, this is really what she said — a minority of a minority. But the linguistic expression רובא דרובא — her mistake is that linguistically it doesn’t mean a majority out of a majority. That simply isn’t right. Rather, it means a majority upon a majority. In other words, applying the rule of majority twice.

[Speaker G] But literally רובא דרובא is the majority of the majority, not two majorities.

[Rabbi Michael Abraham] Not true. That’s if you look at “majority” as a Hebrew word talking about the statistic — eighty percent. Eighty percent of eighty percent is sixty-four percent. But that’s not the point. I’m claiming that the word “majority” here is the rule of following the majority; it’s not the majority itself. The rule of following the majority, when applied twice, is called רובא דרובא.

[Speaker G] But in Hebrew they should have phrased it differently. Whoever phrased it in Aramaic as רובא דרובא did not phrase it correctly.

[Rabbi Michael Abraham] Why? They phrased it perfectly.

[Speaker G] Because רובא דרובא is the majority of the majority.

[Rabbi Michael Abraham] The majority of the majority in Hebrew — you have

[Speaker G] a majority and then you

[Rabbi Michael Abraham] take the majority of the majority. No. You’re assuming that “majority” means the statistical majority, and I’m saying no — “majority” means the legal rule of following the majority. I want to apply this legal rule twice in the same direction. In other words, it should point twice to the permitted side, and then that’s much stronger than when I have one majority on the side of permission.

[Speaker G] They should have thought of a different wording. The wording is not successful. The Academy would not have phrased it that way if

[Rabbi Michael Abraham] if it understood it, it wouldn’t have phrased it that way. I think the wording is perfectly fine. It simply doesn’t understand the Talmudic context. In the Talmudic context, when you talk about a majority, you’re not talking at all about the fact that there is a majority; you’re talking about the rule of following the majority.

[Speaker G] It’s a legal rule. In the Talmudic context, the word “majority” simply marks the rule of following the majority. So the word “of” here, the connection, means “of” — the majority of the majority.

[Rabbi Michael Abraham] The majority of the majority—

[Speaker G] and not a majority of the minority added on top of the second majority.

[Rabbi Michael Abraham] You apply the rule of majority twice, רובא דרובא; twice you apply the rule.

[Speaker G] But the Rabbi says twice, but the “of” is not plus, it’s “of.” It’s a connector, a preposition meaning “of.” The majority of the majority. That’s incorrect Aramaic phrasing.

[Rabbi Michael Abraham] No, I disagree. “Majority upon majority” is also conveyed by that expression; that’s perfectly fine. Because in the end, you’re really saying they operate twice, and therefore in practice you get a better result. I don’t see any problem here. The logic is really the logic of מיעוטא דמיעוטא. And in that sense, the second explanation is really this explanation. I’m just claiming that on the linguistic level it really is correct that רובא דרובא is not simply a borrowed extension from minority, as she said — that since people say מיעוטא דמיעוטא, they then say רובא דרובא; that would just be a linguistic habit even though it’s really illogical. I’m saying no — it isn’t just borrowed from מיעוטא דמיעוטא. I’m not talking about the statistics of the majority but about the legal rule of following the majority. And you have two levers toward permission that you apply one after another.

[Speaker G] When I say “a minority of the minority,” what do I mean? A minority of the minority.

[Rabbi Michael Abraham] With a minority of the minority, you’re talking about statistics.

[Speaker G] And suddenly when I change it to a majority of a majority, now I mean something else?

[Rabbi Michael Abraham] Right, because there is no rule of following the minority.

[Speaker G] No, but linguistically I’m suddenly interpreting the “of” differently.

[Rabbi Michael Abraham] Right, because in מיעוטא דמיעוטא, “minority” means the percentage that is a minority. So there the “of” means a percentage out of a percentage. But with majority, there is a rule of following the majority; there is no rule of following the minority. So my claim is that the word “majority” is not parallel to the word “minority.” The word “minority” is the number; the word “majority” is the legal rule of following the majority.

[Speaker G] It actually seems much more reasonable, Rabbi, to understand that she is right: the intuition began with “a minority of a minority,” and then we feel that it’s not just so little but very little indeed, and then when people wanted to say what the Rabbi means, they got carried away by the language and said “the majority of the majority,” even though it isn’t linguistically precise.

[Rabbi Michael Abraham] I don’t think so, but again, it’s hard to prove.

[Speaker F] I agree — I think the Rabbi is right, because that’s also how we use it in everyday speech. When you say “most of most,” as you started this whole thing, you mean the larger part. I’m not satisfied with 51 percent; I want 90.

[Rabbi Michael Abraham] Everyone agrees about that — that’s not the issue. That’s obvious.

[Speaker F] Yes, but it also fits what you’re saying,

[Rabbi Michael Abraham] that “most of most” means a stronger majority. Correct. Everyone agrees that when you say “most of most,” you mean a large majority, an overwhelming majority. The only question is why. The debate is why; there is no debate that that is the meaning. Shmuel wants to argue, like she does, that this is only because, just as people say מיעוטא דמיעוטא, by extension they say רובא דרובא to mean a large majority.

[Speaker F] No, but she claims that this everyday usage of “most of most” is a mistake; it should be understood the opposite way. That’s exactly the point.

[Rabbi Michael Abraham] Right. And that’s what Shmuel is also saying — but by extension, people use “most of most.” She isn’t saying it’s a mistake; that’s how people use it, and it’s correct Hebrew. She’s only saying that logically it doesn’t fit. So she says, fine, it’s borrowed from the expression מיעוטא דמיעוטא. That’s her claim, and that’s fine; she has no criticism of people who say it, because everyone agrees that רובא דרובא means a strong majority. I’m saying no — it isn’t borrowed from מיעוטא דמיעוטא; it’s the rule of majority being applied twice. Not a majority out of a majority, but a majority after a majority, a majority on top of a majority. That is רובא דרובא. It really is “more majority” than a majority — as if you’re applying the majority rule twice, not once. Okay, fine, so that’s just an anecdote, but we’re arriving exactly at this from the other side of the coin. What do I mean? I want to bring two passages here.

[Speaker G] Especially since in the language, saying “a minority of a minority” is something we constantly use, right? Even in Hebrew today. “A tiny minority,” people say all the time. But to say רובא דרובא, people don’t say that, because they say: just say majority — why say it twice? Say overwhelming majority, crushing majority, sweeping majority. What’s this “most of most”?

[Rabbi Michael Abraham] Not true, people use it a lot. Definitely. What did she start from? From the Hebrew expression, not from רובא דרובא — “most of the public.” The Hebrew expression is “most of most.”

[Speaker G] Right, right.

[Rabbi Michael Abraham] So when you say “most of most,” that’s an expression people use.

[Speaker G] Right, right.

[Rabbi Michael Abraham] I’m claiming that the source of that expression is the Aramaic expression, and in the Aramaic expression it is not a corruption and not a mere borrowing, but has real meaning. That’s my claim. Fine, okay. It’s just an anecdote. But look at the other side of this anecdote.

[Speaker F] I want us to look at two passages.

[Rabbi Michael Abraham] Okay, so look. First example: there is a Mishnah in Yevamot. The Mishnah says: ספק בן תשעה לראשון, ספק בן שבעה לאחרון, יוציא והוולד כשר וחייבין באשם תלוי (“If there is doubt whether the child is a nine-month child of the first husband or a seven-month child of the second, he must send her away, the child is valid, and they are liable for a provisional guilt-offering”). Okay? Meaning, a woman’s husband died childless and she married his brother in levirate marriage. His brother performed levirate marriage with her. After seven months, she gave birth. Now that raises a doubt. Why? The assumption in the Talmud is that children are born either after seven months of pregnancy or after nine months of pregnancy. There are two possibilities: a short pregnancy and a long pregnancy. Okay? It’s not continuous — either seven or nine. No eight. That’s how they understood it then. I’m not sure why. Again, this isn’t technology, so if they said it maybe that really was the situation then. I have no idea. In any case, that is the Talmud’s assumption. Now, once a baby is born seven months after the levirate marriage, a doubt arises whether he is the child of the brother who performed levirate marriage, born after seven months, or the child of the dead brother, born after nine months. Suppose he performed levirate marriage with her two months after his brother died. Okay? So now the question is: is this child the child of the dead brother, born after a nine-month pregnancy — meaning she was already pregnant when the levirate marriage took place — or not? Was she not pregnant, and then she conceived from the surviving brother, and the child born after seven months is the child of the second brother, not the first? Now what practical difference does it make? Maybe first I should add one more thing. The rule is that one must wait three months of distinction in general. Any woman who remarries has to wait three months of distinction. What does that mean? When she marries someone after separating from her previous husband — either he died or they separated — she must wait three months. Why? Because if she waits three months, then when she gives birth, the child will be born at the earliest seven months after the second marriage, which is already ten months after separating from the first husband. So there is no chance that the child belongs to the first husband, and no doubts will arise. Therefore the Sages instituted waiting three months of distinction. Okay? One could discuss whether this is relevant today — we can do genetic testing and identify whose child it is if we are very worried. But fine, that’s not our issue right now. That’s the rule.

[Speaker H] Now here, one moment, wouldn’t you need to do a statistical test — what’s more likely? I didn’t understand. Statistically, which is more likely: that a baby would be born after seven months, or that she had already conceived earlier?

[Rabbi Michael Abraham] In a moment we’ll see that. There is a majority who are born after nine months, says the Talmud. We’ll see it in a second. The Talmud goes on. I just want to say that we’re dealing here with someone who did not obey the law — he didn’t wait three months; he performed levirate marriage after two months. And then the Talmud says: fine, then exactly what we expected happened — after seven months she gave birth. And then the question arises whether this is a nine-month child of the first husband or a seven-month child of the second. Now as far as the child himself is concerned, it makes no difference, right? Because if he is a nine-month child of the first husband, then he is the child of the first husband and everything is fine; he is fully valid. And if he is the child of the second husband, he is also fully valid; everything is fine. The problem arises with regard to the intercourse of the second brother with the woman. Why? Because if the woman really was pregnant by the first brother, then she was not actually obligated in levirate marriage, because she has a child from the first brother. And if she is not obligated in levirate marriage, then when his brother enters into levirate marriage with her, she is actually his brother’s wife — that’s a forbidden sexual relation, an incest prohibition, his brother’s wife. If he died without children, then there is an obligation of levirate marriage, but if there are children — if she is pregnant — then the incest prohibition returns and they are forbidden to marry. So the practical difference is whether she must leave the levirate husband and whether they transgressed a prohibition. The Mishnah says yes: he must send her away, and the child is valid. The child is of course valid, that’s obvious, but he has to send her away, they must separate. And they are liable for a provisional guilt-offering. Meaning, there is a doubt whether he had relations with her and violated an incest prohibition, because perhaps she was pregnant and perhaps not. And since there is doubt, wherever certainty would require a sin-offering for an inadvertent transgression, doubt requires a provisional guilt-offering.

[Speaker C] Rabbi, when it says “the child is valid,” does that mean that he’s not what?

[Rabbi Michael Abraham] Not disqualified?

[Speaker C] That he’s not a mamzer.

[Rabbi Michael Abraham] So now the Talmud there on page 37 says as follows: “If there is doubt whether he is a nine-month child,” etc. — that was the Mishnah. The Talmud says: Rava said to Rav Nahman, let us say: “Follow the majority of women, and most women give birth after nine months.” After all, there is a majority of women who give birth after nine months. Let us follow the majority of women. This is not an even doubt; there is a majority here. The Talmud says: he said to him, this is what I mean — I’m skipping a little, it doesn’t matter for us; this is the important part, or the beginning of the important part. He said to him: this is what I mean: most women give birth after nine months, and a minority after seven. You’re right, there is a majority of women who give birth after nine. And every woman who gives birth after nine months has a fetus that is recognizable after a third of her days — meaning after three months, one third of the pregnancy, you can already see that she is pregnant. Fine? By contrast, one who gives birth after seven is a different discussion. Now since the woman who gives birth after nine months has a fetus recognizable after a third of her days, and this woman, since her fetus was not recognizable after a third of her days, איתרע לרובא — the majority has been weakened. There is indeed a majority that women give birth after nine months, and therefore in practice the majority says that this is the first husband’s child, and indeed she must leave and they transgressed an incest prohibition. But there is another rule: if she gives birth after nine months, then the fetus is recognizable after a third of her days. That means we can see, one month after conception, whether her belly is showing or not. If her belly is showing after one month, we know that this is the first husband’s child, because three months have passed from the first husband and the fetus is recognizable. If it is not recognizable, then it is the second husband’s child. So since that is so — if her fetus had been recognizable after a third of her days, this whole discussion would not arise, right? The doubt here arises because we did not see her belly after a month. So because of that we hesitate. Thus he says: since her fetus was not recognizable after a third of her days, איתרע לרובא. So the majority has been weakened, and therefore he brings a provisional guilt-offering. The Talmud says: what are you talking about? If every woman who gives birth after nine months has a fetus recognizable after a third of her days, then since this fetus was not recognizable after a third of her days, he is certainly a seven-month child of the second husband. What are you telling me? That every woman who gives birth after nine months shows after three months. Now this woman did not show. So it’s one hundred percent certain that this child is the second husband’s child. This is not a doubt, not a provisional guilt-offering. He should not bring any provisional guilt-offering. He is fully valid and she need not leave, and he is the second husband’s child — because you told me that every woman who gives birth after nine months has her fetus recognized after a third of her days. And if this fetus was not recognized, then no. So it doesn’t make sense to say the majority was weakened. The majority disappears. There is no majority. He is certainly and definitely the second husband’s child. The Talmud answers: rather say this: most women who give birth after nine months have a recognizable fetus after a third of their days. Not every woman who gives birth after nine months — most women. And because this woman’s fetus was not recognizable after a third of her days, the majority has been weakened. What is it saying? It is saying that there is not a rule that every woman who gives birth after nine months visibly shows by a third of the pregnancy. No, it’s only a majority. Not certainty. Most of those who are pregnant for nine months — after a third of the pregnancy the fetus is recognizable. So it’s only a majority. Now what happens in that situation? In that situation, the original majority is undermined, right? It’s not certain that he is the second husband’s child, but there is also no longer a clear majority in favor of his being the first husband’s child, because if he were the first husband’s child, most likely her belly would have been visible. If her belly was not visible, that somewhat undermines the majority that women give birth after nine months. Therefore the Talmud says: that weakens the majority; this is a doubt, and he brings a provisional guilt-offering. That’s the first example. Now look at a completely parallel passage. A Talmud in Ketubot. Ravina said. The case is a woman standing before us, and we’re discussing whether she was a virgin when she married or not. Okay? The assumption is that no public report reached us that she married as a virgin. But on the other hand, most women who marry are virgins. Fine? Most marriages are to women who are virgins, but there is an assumption that when a woman marries as a virgin, word generally gets out, but no such report reached us. So the Talmud says as follows. Ravina said: because one may say that most women marry as virgins and a minority are widows. So if that is so, then most likely she married as a virgin, period. And every woman who marries as a virgin has public notice. But on the other hand, that’s the majority, yet if she really married as a virgin, why did no report reach us? הרי every woman who marries as a virgin has public notice, and this one, since she has no public notice, איתרע לרובא. The Talmud asks: if every woman who marries as a virgin has public notice, then when witnesses come, what difference would it make? Those witnesses are false witnesses. Exactly like the previous passage. What are you saying? That every woman who marries as a virgin has public notice? Well then, if she had no public notice, obviously she did not marry as a virgin. It’s not that the majority has been weakened. Rather, it turns out this majority is irrelevant here. Clearly she did not marry as a virgin, because otherwise public notice would have gone out. What does the Talmud answer? Exactly as in the previous passage. Rather, Ravina said: most women who marry as virgins have public notice. Not all of them. Most of those who marry as virgins have public notice. And this woman, since she has no public notice, the majority has been weakened. Okay? What is he really saying? Not that everyone who marries as a virgin has public notice, but rather that among those who marry as virgins, usually word gets out. So on the one hand there is a majority who marry as virgins; on the other hand, if she married as a virgin, I would have expected public notice, and here there was none. So that weakens the majority.

Now Rabbi Shmuel Rozovsky asks in his lectures on Yevamot as follows: on the face of it, who says that the second majority has the same ratio as the first majority? By way of example, if the first majority — that most women give birth after nine months — is in a ratio of four to five, for example, out of one hundred women giving birth, eighty give birth after nine months, then the second majority — that among those giving birth after nine months most have a recognizable fetus — is also in a ratio of four to five, meaning sixty-four out of the eighty who give birth after nine months. Yes, sixty-four out of the eighty who give birth after nine months have a recognizable fetus. I assume it’s the same proportion of majority — if this one is eighty percent, that one is also eighty percent. According to this it turns out that out of one hundred women there are twenty women who give birth after seven months and another sixteen women whose fetus is not recognizable. Okay? And if so, it requires examination,

[Speaker E] because this woman

[Rabbi Michael Abraham] certainly belongs to one of these minority groups, and not to the rest of the women who give birth after nine months and whose fetus is recognizable.

[Speaker E] And since that is so—wait, everyone—

[Rabbi Michael Abraham] I’m asking you to mute yourselves, at least be quiet if someone is talking, yes. Maybe you dropped down? I don’t know. Let’s keep order. Yes, so basically we are saying the following. Suppose there are one hundred women who married. Of them, eighty are virgins, because most women who marry are virgins. Okay? Twenty are not virgins. Now among those who marry as virgins, eighty percent have public notice, because generally there is public notice for women who marry as virgins. So how many is that? Sixty-four, right? Eighty percent of eighty. That means there are sixteen women who married as virgins but no public notice came out about them. And how many women married non-virgins with no public notice? Twenty, right? So how many women are there altogether with no public notice? Thirty-six. Right? Sixteen who married as virgins and had no public notice, and twenty who married as non-virgins and had no public notice. Thirty-six. Out of those thirty-six, sixteen are virgins and twenty are non-virgins. So now if the assumption is that for this woman no public notice came out that she married as a virgin, I now calculate: I’m discussing whether she married as a virgin or as a non-virgin. Notice that now we’re in the area of conditional probability, right? What is the probability that the woman married as a virgin given the fact that no public notice came out about her? If the probability were a priori, the general probability — say that eighty percent of women marry as virgins — then most likely she married as a virgin. But if I know that no public notice came out, that changes the whole picture, because among the women about whom no public notice came out, there are sixteen virgins — out of the eighty, sixteen had no public notice — and there are twenty non-virgins. So sixteen versus twenty: actually the twenty are the majority, right? So most likely she married as a non-virgin, not a virgin. Even though most women who marry are virgins, within the group about whom no public notice came out, the picture reverses. The majority there are women who married as non-virgins. This is exactly the mechanism of conditional probability we spoke about. I’m not asking what the chance is that the woman married as a virgin; I’m asking what the chance is that the woman married as a virgin given the fact that no public notice came out about her. That’s P of A given B, not just P of A. That probability is twenty out of thirty-six — more than fifty percent. And therefore the conclusion is that if indeed no public notice came out, then she probably married as a non-virgin. That’s the claim. And the same thing applies to childbirth. There is a majority of women who give birth after nine months, but among them there is a majority whose fetus is recognizable. Now if I am given that this woman’s fetus was not recognizable after three months, then I’m not asking the unconditional probability — what is the chance that the child was born after nine months, where there is a majority, eighty percent. I’m asking the question of conditional probability: given that her fetus was not recognizable, what is the probability that this child belongs to the first husband rather than the second? Here it’s conditional probability. Within that group, most likely he is the second husband’s child: twenty out of thirty-six, the same calculation. It’s exactly the same structure in both passages. Is what I’m saying clear? Fine, are you following, or is this…

[Speaker F] Yes, yes. Okay. Look—

[Rabbi Michael Abraham] Maybe I’ll show it in a diagram, similar to the diagrams we saw last time. Look at this ellipse. Okay? So on the left side of the ellipse, each one of these is a woman. Each little circle — or ellipse, whatever you want to call it — each one is a woman. The small part of the ellipse is the minority of those who did not marry as virgins. Here is the majority who married as virgins. Okay? Now among those who married as virgins there is a majority — sorry — among those who married as virgins, a majority had public notice. Most of them had public notice, and this is the minority who married as virgins but nevertheless no public notice came out about them. Now if I’m given that no public notice came out, then clearly the woman belongs either to this white group or to this white group. She is certainly not here, right? Because those are women about whom public notice did come out. If no public notice came out about her, then she belongs either to this group or to this group. Do you now see how the majority is distributed? Do you understand that now suddenly this becomes the majority, not that? But if we narrow the discussion only to the white area, removing the shaded group, then although this side of the ellipse was the majority a priori, a posteriori — if we already know that no public notice came out — then דווקא this side is the majority, the left side. Okay? All right? Yes, yes, understood.

Now in both these cases, notice, these two cases really are רובא דרובא in her sense, in Tamar Katzir’s sense — the one from the Hebrew-language segment. This is exactly the רובא דרובא she’s talking about, right? Because basically, notice the diagram again. Yes — because this is a majority within a majority. Most women who marry do so as virgins, and among them most have public notice. Right? Now that majority within a majority can in total turn out to be only a minority. Maybe not, maybe yes — it depends on the relation between the two majorities, okay? But in total it can turn out to be a minority. Therefore רובא דרובא is weaker than a single majority. This is רובא דרובא in her sense. רובא דרובא in my sense means looking at the two white pieces, at the relation between the two white pieces. Okay? That is רובא דרובא in my sense. But the literal רובא דרובא is these two passages, and this really is a case where איתרע לרובא — the majority becomes weaker, not stronger. When people talk about רובא דרובא, that’s not in the literal sense; there the majority is stronger. Okay? That is what I said before: it is applying the majority rule twice. And if I had a situation — the diagram for רובא דרובא in my sense would be this: I have a minority here, and within that minority there is a majority that had public notice. Okay? Suppose these all had public notice, and within these there is a majority with public notice, so what remains is those without public notice. That really is רובא דרובא in my sense, because it is a stronger majority than the previous majority. But here, what is drawn here is רובא דרובא in her sense. Okay? And here the Talmud really says that this is a weaker majority, not a stronger one — איתרע לרובא.

And now the interesting question arises. Rabbi Shmuel asks: let us assume, just for the sake of discussion — we don’t know, but assume for the sake of discussion — that these two majorities are of the same size. Say eighty percent. Eighty percent marry as virgins, and among those eighty percent, for eighty percent public notice comes out, while for twenty percent no public notice comes out. Okay? So now when I have to decide whether she married as a virgin or as a non-virgin, what’s the answer? Twenty out of thirty-six that she married as a non-virgin. Sixteen out of thirty-six that she married as a virgin, right? So there is a majority for the possibility that she married as a non-virgin. So why does the Talmud say that the majority has been weakened and he brings a provisional guilt-offering — meaning it is a doubt? It isn’t a doubt. On the contrary, the majority has reversed. So now you can decide that she married as a non-virgin. Why do you say it’s a doubt? The same with nine months and seven months, same thing. Why do you say there’s doubt whether he had forbidden relations, whether there was an incest prohibition there or not? That’s not right — most likely there was no incest prohibition. True, most children are born after nine months, but among those born after nine months most are visibly recognizable, and here this one was not recognizable; that flips the picture. So most likely this is the second husband’s child and everything is fine. So it isn’t a doubt; the opposite — there should be a majority in favor of the other side. So why does the Talmud say it’s a doubt? That is Rabbi Shmuel’s question, and he leaves it unresolved.

[Speaker I] What? How can that be? What, he has no answer to it? But statistically it seems pretty clear.

[Rabbi Michael Abraham] Why is it clear?

[Speaker I] No, because according to the calculation we just did, it’s clear that the majority has reversed.

[Rabbi Michael Abraham] Exactly! But the Talmud says we remain in doubt, not that the majority has reversed. He asks: why? The majority has reversed; it should be certain in the other direction — we should follow the majority. Why does the Talmud say it is a doubt and requires a provisional guilt-offering? When there is a majority, we do not bring a provisional guilt-offering. Right. What do you say? By the way, do the calculations for seventy percent.

[Speaker F] Seventy percent, okay?

[Rabbi Michael Abraham] Suppose seventy percent marry as virgins, and among those seventy percent, for seventy percent public notice comes out. Okay, so that’s forty-nine out of the seventy. There are twenty-one virgins with no public notice and thirty non-virgins. That still preserves the reversal, right? And that’s thirty out of fifty-one. But there are situations where it won’t happen. Suppose the majority of those marrying as virgins is ninety percent, and among them the majority with public notice is, I don’t know, sixty percent, fine? Then forty percent of ninety is thirty-six. Thirty-six is still a majority against the ten. So it doesn’t reverse the majority; it leaves the majority as it was. Rabbi Shmuel gave an example where the majorities are eighty percent, but in other examples it won’t come out that way. In the example of a ninety-percent majority — or two majorities of ninety percent — it already won’t come out that way. The majority won’t reverse at all; it will remain. His difficulty is still a difficulty, because if the majority remains, then it also should not be a doubt; rather, the majority should remain as it was. And if it’s eighty percent, the majority reverses. From eighty percent and down, the majority reverses. The borderline is probably somewhere around eighty-five percent, something like that. But from eighty percent and down, the majority reverses. Either way, this should not be a provisional guilt-offering. Either we decide in this direction or we decide in that direction. Why is it a provisional guilt-offering? What do you say? By now you should already be able to whistle the answer with your eyes closed. The majorities being discussed here are negative majorities. Nobody knows the numbers. Right? Nobody knows what the size of the majority is. The eighty-percent example is Rabbi Shmuel’s; it could also be seventy or ninety. I don’t know, right? It’s a negative majority. Now once it’s a negative majority, you really are in doubt.

[Speaker F] It could be a case where the majority remains,

[Rabbi Michael Abraham] and it could be a case where the majority reverses, and you don’t know which of the two possibilities is true. So you are in doubt. Therefore, precisely in— and I already said that the whole story of רובא דרובא and all these probabilistic multiplications always applies when the majority is a negative majority. If the majority were a positive majority, just do the calculation that Rabbi Shmuel did and the result would determine the law. There would be no need to multiply and do all sorts of things. Why should I care? Give me the bottom line. What is the probability? That’s all. All the discussions of רובא דרובא, double doubt, and all these things are only when the majority and the doubt are negative. Only there. Now once they are negative, then it really is a doubt. Therefore it is a provisional guilt-offering. Why? Because with negative doubts, the majority could be ninety percent and it could be seventy percent; you don’t know. So it is possible that the majority of virgins remains, but it is also possible that it reverses and now the majority is of non-virgins. Since you do not know the size of the majority, you do not know what the result will be. Therefore you are in doubt. If you are in doubt, you bring a provisional guilt-offering. And here you have yet another proof, or another implication, of the fact that all these discussions deal only with negative majorities. And Rabbi Shmuel is exactly the example, because as I told you, if I have the numbers, all these passages are irrelevant. Do the multiplications, arrive at the statistics, see whether you have a majority, and follow the majority. Why should I care whether this majority was created from multiplying two majorities, from one majority, from a double doubt? It’s irrelevant. Do the multiplication, you have the number, and you can know whether you have a majority or not. That’s all. All these rules — that I need רובא דרובא or a double doubt or probabilistic multiplications — what I called all the last lectures probabilistic multiplications — all these things are always when the probability is not known. The doubt is negative, or the majority is negative, as we saw with permitting an agunah. Okay? And therefore it’s clear — the Talmud is clearly right. It is a negative majority. And a negative majority really does lead to a provisional guilt-offering. Yes.

[Speaker F] Yitzhak, did you want to ask something?

[Rabbi Michael Abraham] Okay. So that’s regarding רובא דרובא and the anecdote from that language segment. Because what she describes is exactly this. It is a case where one majority offsets the second majority, and then it really comes out weaker if at all. It may even come out as a minority. In her calculation it’s nine out of thirty. Nine out of thirty is even a minority, not just not a majority. Fine? But in any case, it’s a weaker majority even if it remains a majority. But רובא דרובא in Jewish law is not like that. רובא דרובא in Jewish law means, of course, that the majorities work in the same direction.

Now I want to finish our series with a fairly well-known responsum of Havot Ya’ir. I already mentioned it once when we spoke about

[Speaker E] lotteries in Jewish law.

[Rabbi Michael Abraham] “In a study group that learns with me and listens to my voice, twelve householders dined together joyfully over wine. On Purim they cast lots over a large gilded silver cup, and each one contributed one Reichsthaler” — one Reichsthaler was the coin there — “and this was how the lot was conducted.” Meaning, what happened? They were sitting at a Purim feast, in good spirits with wine, and decided to make a lottery. Each one put in one Reichsthaler. They took twelve householders, collected twelve Reichsthalers, bought a large gold-colored cup — fine, a large gilded silver cup, whatever exactly that means — worth twelve Reichsthalers. And now they wanted to hold a lottery so that one of them would win the cup. Yes, they would all basically give him one Reichsthaler each. That was their way of making the lottery. “And this was the procedure of the lot: they brought two boxes or two vessels” — two bowls — “and placed twelve slips in one vessel, each with the name of one of them written on it. And in the second vessel they also placed twelve slips, on one of which was written ‘mazal tov,’ and the other eleven were left blank.” Meaning, there are two bowls, with twelve slips in each. In one bowl are twelve slips with the names of the twelve men. In the second bowl there are likewise twelve slips, one of which says “mazal tov” and eleven are blank. Okay? “Now the plan was that a little boy and a little girl would come, take one slip from this vessel and one from that vessel, and whoever came up with ‘mazal tov’ would win the cup.” Whoever got the “mazal tov” slip together with his name was the one who won the cup.

“And it happened in that lot that ‘mazal tov’ came up on the second draw. And this seemed wondrous to the onlookers, that the ‘mazal tov’ slip came up so quickly.” Yes, how did the ‘mazal tov’ slip come up already on the second draw? “And one of them checked and saw that there were only ten blank slips.” How many should there have been? Eleven. One ‘mazal tov’ slip and eleven blank slips. “But there were only ten, and there should have been eleven. And on two slips was written ‘mazal tov.’” Meaning there were twelve slips: ten blank ones and two that said ‘mazal tov.’ “And the one who won sought to say that however these two mistakes were made — deliberately, accidentally, or by way of joking and play — there is no diminution or increase of power for any one of us over the others. And if his luck caused him to win, then so too, had it been done correctly and properly, the hour would have stood for him.” What is he saying? The winner says: I don’t understand what you want from me. Still, each of us had the same chance of winning, right? Each of us had the same chance of winning. What difference does it make whether there were two ‘mazal tov’ slips or one? All it means is that the chance of drawing a ‘mazal tov’ slip increased, but relative to whose name it comes up with, that chance remains the same — one out of twelve, right? Because I draw one slip from the bowl of names and one slip from the bowl of ‘mazal tov’ slips. Okay? So in the bowl of ‘mazal tov’ slips, the chance of drawing such a slip is not one in twelve but one in six. But here I have a one-in-twelve chance for each of their names to come up, right? So in practice, each one’s chance of winning remains the same. So the man says: what complaint do you have against me? I won, and I would have won anyway. What difference does it make? Our chances of winning were the same. Why does he assume he would have won anyway? Because he assumes that the result of the lot comes from the Holy One, blessed be He. If God wanted me to win, He would have arranged for me to win anyway. But still, he says to them, your chances did not change. True, there was a glitch in the lottery, something unplanned, but it did not harm your chances, so you should have no claim against the result of the lottery.

[Speaker K] Meaning it simply changed the probabilities, but it changed them equally for everyone?

[Rabbi Michael Abraham] Exactly. It changed the probability of drawing a ‘mazal tov’ slip, but it still didn’t favor any one of the twelve over another. They all still had the same chance of winning. So why should I care? Even if there had been eleven ‘mazal tov’ slips there, or you know what — even twelve ‘mazal tov’ slips. If there had been twelve slips, all of them saying ‘mazal tov,’ what would each one’s chance of winning be? One in twelve. One in twelve — it makes no difference. I draw a slip, it’s certain to say ‘mazal tov,’ right? But I still have a one-in-twelve chance for each person’s name to come up. So it’s still one in twelve. In other words, your chance of winning does not depend on the probability of drawing a ‘mazal tov’ slip. There is no connection. And so this lottery was really conducted correctly. The second bowl is irrelevant. True, there is conditional probability here, but the conditional probability equals the prior probability. In other words, what is my chance of winning, given that the lottery was flawed? It is equal to my chance of winning regardless of the flaw. In other words, there is no dependence between my chance of winning and the flaws in the lottery. The flaws in the lottery did not change the winning chances. That means there is no dependence between them; the conditional probability equals the absolute probability. Therefore there is no problem with this lottery, and you should have no complaint. That’s what the winner told the people complaining there. Okay.

Havot Ya’ir says: “And I said simply that the lot is void.” He still claims the lot is void. Why? But nobody’s chances were harmed; the lot was still fair. So he says: “For we hold like Rav, who said, ‘the division is void,’ when another brother comes from overseas, in the chapter ‘Beit Kor.’” This is in Bava Batra 106. The Talmud there talks about a situation where two brothers divided their father’s inheritance. They made a lottery: say there were two fields in their father’s estate; they drew lots, one received field A and the other field B. Now a third brother arrives.

[Speaker F] They didn’t even know he existed; it turns out he is also a brother.

[Rabbi Michael Abraham] Right — start over, game reset. Rav says: “the division is void”; they do the lottery again. Shmuel says: “they adjust proportionally.” What does that mean? Each one gives him a third of his portion. You give him a third of your field, you give him a third of your field; he will have two thirds, and each of us will also remain with two thirds, and everything is fine. But Rav says: “the division is void.” Therefore Havot Ya’ir says: we see that if the lottery was conducted in a flawed way, the lottery must be canceled. What?

[Speaker I] Why should he agree to that? Who would agree? The third brother.

[Rabbi Michael Abraham] Why should he agree?

[Speaker I] To receive a third from this one and a third from that one.

[Rabbi Michael Abraham] He says no — that’s exactly what he means: the division is void. No, you have to cancel everything.

[Speaker I] No, but according to Shmuel’s opinion, that the division is not void — why isn’t it void? His right was harmed.

[Rabbi Michael Abraham] Why did Shmuel say the division is not void?

[Speaker I] I think maybe they compensated him — meaning he agreed.

[Rabbi Michael Abraham] The Talmud there ties it to whether a legal decision, once established, remains established. Once the judgment has been made, it stands. Meaning, as long as you had a chance to receive something — you get your third — then the legal decision stands. Maybe because of the dignity of the court or something like that. But Rav says the division is void, and we rule in accordance with Rav. By the way, that’s an interesting point, because in monetary law between Rav and Shmuel we usually rule like Shmuel. This is a monetary dispute here. On the face of it, yes, isn’t it? If so, why do we rule like Rav rather than Shmuel? The question whether this is really a monetary-law issue is itself interesting. But anyway, for our purposes, Havot Ya’ir says: you see that when there was a defect in the lot, the division is void, and therefore here too the division is void. What do you say about that argument?

[Speaker F] It’s a very strange argument.

[Rabbi Michael Abraham] Why? The reason the division is void there is really because each one can come and claim: if there had been a different lottery, I might have won a better share. I want the lottery redone. My rights were harmed. Okay? You can accept that, because the third brother who arrived, according to Shmuel, gets a third from me and a third from the other brother. Right? But perhaps he could have won the two-thirds that remained with me. Why are you forcing him to take a third from here and a third from there? Maybe he wants two-thirds of the same field.

[Speaker I] No, but suppose they did compensate him. I don’t understand. The one who got the less good half says: I’ll compensate him out of my own portion. Don’t worry, the third one agreed. He received for sure a half—

[Rabbi Michael Abraham] No, he doesn’t receive that. What do you mean? If he agreed, then maybe everything is fine. But he doesn’t agree. And now he has the right to demand that the division be voided.

[Speaker I] No, but apparently Shmuel’s novelty is that even if the third agreed, still one of the original brothers can say: let’s cancel everything.

[Rabbi Michael Abraham] No, the opposite — Shmuel says the division is not void. Rav says—

[Speaker I] No, I mean Rav, with whom the law follows, says the division is void even if the third agreed. Why? Because the one who got the less good half can cancel everything, and that’s why I’m making the analogy.

[Rabbi Michael Abraham] Wait — but if he agreed, what’s the problem? I don’t understand.

[Speaker I] No, because the loser — one of them probably got a better half than the other.

[Rabbi Michael Abraham] Right.

[Speaker I] He wants to redo everything.

[Rabbi Michael Abraham] Why? Because—

[Speaker I] He wants perhaps to win the better half—

[Rabbi Michael Abraham] You had a fair chance to get either the worse or the better. So what’s the problem? On the contrary, you actually had a better chance, because when there were two of us, you had— after all, in the end each one gets two-thirds, right? Suppose I won the better field — a third, right? Suppose I won the better field. No, two-thirds; there are two fields here. So each one gets two-thirds of a field. Okay. Suppose I won the better field and you’re the second brother. Then your chance of winning the better field was one-half, because we held a lottery between two brothers. So be grateful. Now you want a new lottery? You’ll only have a one-third chance to win it. You only lose from a new lottery.

[Speaker I] That’s true, and still he wants to redo it, because what does he have to lose? Better to redo it; at worst he ends up in the same place.

[Rabbi Michael Abraham] I don’t understand.

[Speaker I] Rav, according to whom the law follows, says the division is void. So he says despite the Rabbi’s argument, the one who lost can demand that the whole lottery be redone, and that’s the basis of the analogy made by Havot Ya’ir.

[Rabbi Michael Abraham] No, I’m saying: on the face of it, to void the division you need some consideration working to your detriment, right? If somebody says, “I got hurt here,” then that is not proof for our discussion, because in our case no one really got hurt, in Havot Ya’ir’s case. But if indeed, as we just said, one of the first two brothers wants to void the division — not the third, one of the first two brothers — then he may indeed be able to claim that his rights were harmed, because he says: if we had done the division equally, perhaps I would have won the better field. But that is a problematic claim, because statistically — no, it is not true. Statistically your chance decreased. On the contrary — exactly — your chance decreased, and still you want

[Speaker I] to void it, but the fact is that Rav voided it.

[Rabbi Michael Abraham] Fine, I’m saying there really is a problem in understanding it this way, when you tell me his chances did not worsen and yet Rav still voids it. So Havot Ya’ir says: ah, if even where my chances improved I can still void it, then all the more so if nothing happened I can void it. Right, okay, that’s the claim. So if it comes from the third brother, there is no proof here. But if it comes from one of the first two brothers, then it really is an a fortiori proof. Even in a case where my position improved, I can still claim that I want to cancel it. And why? Because in the end, if we had done that other lottery, perhaps I would have won. In fact I didn’t win. True, I had a better lottery, but in fact I did not win. Understand that this consideration also exists in Havot Ya’ir, because what does everyone who wants to cancel say? True, my chances were not harmed, but in practice I didn’t win. Let’s do the lottery again — maybe this time I will win. Fine? In other words, I want to cancel the lottery not because my chances were damaged, but because I don’t like the outcome. Once I’m allowed to cancel the lottery even though my chances were not lower, then I have a good reason to cancel it, because in practice I didn’t win, and if we do it again maybe I will.

[Speaker I] So why does Rav cancel the lottery?

[Rabbi Michael Abraham] I didn’t understand. That’s exactly it — Rav cancels the lottery… Again, I’m saying, if he cancels it because of the third brother, that’s not—

[Speaker I] Not because of the third brother.

[Rabbi Michael Abraham] I’m saying, if it’s because of the third brother, then he cancels it because his chances really were reduced. He makes a valid claim: you harmed me, I want a new lottery. But if it’s one of the first two brothers, then maybe Havot Ya’ir’s approach has some basis. But then it’s still unclear why you want to cancel the lottery — after all, you got a fair chance. What’s the problem? Just like in Havot Ya’ir.

So later on Havot Ya’ir really says that when the lottery was flawed, the Holy One, blessed be He, is not involved. Meaning, normally God takes care of who should win, and therefore everything is fine; the one who got it really deserved it. But where the lottery was conducted in a flawed way, there God is not involved. And that is the claim: let us do the lottery again, and then God will be involved. Okay?

[Speaker I] So maybe one could explain it this way: if we understand that the lottery really is… after all, we all understand that a lottery is not truly random; there is no real randomness here. Okay. We simply don’t know the data, but there is no real randomness. And we agreed on a certain lottery which from the outset the Creator already knew exactly who would come out in that lottery; it is deterministic. Okay. But that’s not the lottery we established, that’s not the lottery we decided on. We decided on a fair lottery.

[Rabbi Michael Abraham] The Creator also knew what was written on the slips, so He also knew that in advance.

[Speaker I] Fine, I’m saying—

[Rabbi Michael Abraham] Since formally that is not what we agreed to—He not only knew it, He also—

[Speaker J] caused it.

[Rabbi Michael Abraham] No, I’m not saying He caused it. Because when we discussed lotteries in Jewish law, I mentioned this Havot Ya’ir, and I said that the very notion of divine involvement — that God determines the results of the lottery — is precisely a conception that says there is no importance to the lottery being fair.

[Speaker I] Right, but exactly.

[Rabbi Michael Abraham] Even if the lottery were ninety-ten in your favor. Even if it were ninety-ten in your favor, God could still make sure that I, with my ten percent, would win.

[Speaker I] So that’s proof that it isn’t so.

[Rabbi Michael Abraham] The expectation of a fair lottery stems דווקא from the assumption that God is not involved. Right. But Havot Ya’ir does not think that way. Havot Ya’ir claims that God is involved only in a fair lottery.

[Speaker I] But Rav doesn’t think that way. Rav apparently thinks there’s no such involvement and no special responsibility, just not to rely on it. Havot Ya’ir follows Rav.

[Rabbi Michael Abraham] Havot Ya’ir follows Rav.

[Speaker I] Yes, but he doesn’t understand Rav correctly.

[Rabbi Michael Abraham] Maybe yes, maybe no, but he follows Rav and claims that this is the basis of what Rav said. Okay, right — obviously one can argue with that. Okay, now.

[Speaker F] Rabbi, but in the twelve-person lottery, just a question — it’s not really that it wasn’t fair, it was… so how can you say that there is—

[Rabbi Michael Abraham] that it was flawed, okay.

[Speaker F] a flaw in it, okay.

[Rabbi Michael Abraham] Once the lottery is flawed — not unfair,

[Speaker F] okay,

[Rabbi Michael Abraham] once the lottery is flawed, then God is no longer involved. That is basically his claim. Let’s get there in one second. Look, the continuation of the story — we need to finish soon, and I want to finish the series today. “And they went back and cast new slips in the above manner into two vessels.” They did another lottery. Fine, they voided the first one. He said the lot was void, Havot Ya’ir. So they voided the lot — they were his students, after all. This happened in his presence. They asked him; he voided the lot. So they did it again. Okay? And now, once again, two vessels, and one of them won on the sixth draw. So in the first lottery the ‘mazal tov’ came up on the second draw, which looked suspicious, so they checked the slips — how could it come up so fast? And indeed they found that there were two ‘mazal tov’ slips. Now it came up on the sixth draw, which is average, more or less reasonable. Okay? “But one of the remaining slips in the box, innocently, not suspecting anything, opened the tied-up slips that remained” — the slips they had not opened — “and from this he saw that one name of one of the contributors had been missing and had not been placed into the box. There were only eleven slips of householders.” These guys were apparently completely drunk. Even in the second lottery there was a flaw. It turns out there were only eleven name slips, not twelve. One of the householders was missing.

[Speaker J] So that’s obviously unfair. The missing name — that’s unfair.

[Rabbi Michael Abraham] Wait. “And the members of the group said that this second drawing too should be voided, and the one who won argued against them that there was no reason for their complaint to ruin the lottery. On the contrary, because one name was missing, they were closer to profit.” After all, each one of you who is complaining certainly has nothing to complain about. You each had a one-in-eleven chance of winning, not one in twelve. You only gained from it. “And the light is good” — wait, wait. “And concerning the missing person,” because the one whose name was missing definitely lost, “the winner sought to compromise and give him four Reichsthalers, and he was appeased.” He said: leave it, I’ll give you four Reichsthalers out of the twelve to compensate you. Are you satisfied? That’s beyond your expected gain, right? Your expected gain is one Reichsthaler: one-twelfth out of twelve Reichsthalers. You receive four. So you’re satisfied. I remain with eight, you have four, the others have no complaint, everything is fine, and the lottery stands.

Now look how strange this is. “And truly, at first glance, the matter seemed balanced to me, without a clear decision.” Here Havot Ya’ir actually hesitates, even though above he said certainly it is void. Very strange. Strange. “After a little reflection, the number of the word ‘garden,’ I ruled that here too the lot is void. For we see regarding two brothers who divided, and afterward another brother came from overseas, that even if there had been three fields equal in value, and one of them took a field and a half, and after the third brother came they drew again and wrote three slips and marked on them the three fields, and the foreign brother took one slip and it came up with the field that had been divided, and the third brother was satisfied with it — nevertheless each of the first two brothers can completely void the first lot. And these are the words of Ri in Tosafot there. And so too the Shulchan Arukh rules. Even though there is no reason and no logic and no ground for any of them to contest from the standpoint of reason.” What is he saying? He says: there were three fields. Now they thought there were only two brothers, so they drew lots and each one received one field plus half of the third field. One got field A and half of C, the other got field B and half of C. Then a third brother arrived. They redid a lottery, fine? Wrote three slips. The new brother got the field that had been split between the first two. Fine? So that brother says: you know what, I’m totally satisfied. I’ll take that field. Okay? Now either of the first two brothers can void the lottery entirely. Why? Their chances were only better when there was no third brother, right? So why can he void the lottery entirely? The third brother, who could have made claims, is satisfied. Everything is fine; he took his field and left. You remained each with a field, and your chances did not diminish. Why can you cancel it? He says: we see from here that even though there is no reason and no logic and no room for any of them to contest on rational grounds, nevertheless a lot conducted not properly is void. All the more so if the error is in the act of the lot itself. Yes? There, with the brothers, the lot itself was fine; they simply did not know about the third brother. In the cases in Havot Ya’ir, the very procedure of the lot itself was flawed. There that certainly means the lot is void.

And then he says — he says, “See the responsa… Even though there is no reason and no logic… see the responsa Sha’arei Teshuvah — nevertheless a lot is, even aside from this, especially suited to providence.” Here he brings God into it. “And see the responsa of the early geonim, that the lot comes only from heaven, as it says, על פי הגורל תחלק הארץ (‘By the lot shall the land be divided’), and one who violates the lot is like one who violates the Ten Commandments” — that is a quotation. “For we have seen from the Torah, the Prophets, and the Writings that they relied on the lot.” We saw that when I spoke about lots in Jewish law. So therefore everything — yes, he also brings Jonah here, and everything is by divine word — “because it is close to say that if the lot is conducted properly, higher providence adheres to it, as it says הבה תמים.” “But if the lot is flawed, there is no basis for saying that the one who won — this was from God — whether the flaw came through human contrivance or by accident. In any case, the lot is flawed, and each one can say: had the lot been done properly, perhaps my time would have stood for me, by my fortune or by my prayer, that success be granted to me in all my affairs.”

[Speaker J] And then he brings the Talmud in Bava Metzia, etc.

[Rabbi Michael Abraham] He claims that once the lot was flawed, even if it did not lower the chances, I can argue that here God was not involved, and if they had conducted the lot properly, God would have taken care of me, because I prayed that He should let me win. So I have a claim by which I can invalidate the lot in all these cases. Look how he concludes: “And something even greater appears to me: if a man acted slyly and placed two slips with his own name into the box, and someone else won with the ‘mazal tov’ slip, and afterward this sinner revealed it and they checked and found it so, nevertheless the others may void the lot — and even he himself may do so.” Even he can void the lot. “And this is what appears to me in my poverty.” What does he mean? He means: suppose we run a similar lot. Now I, who write the slips, insert thirteen slips instead of twelve, with two of them bearing my name. Meaning I increased my chance of winning. Now they drew the slip, and the winner was someone else, not me. Someone else won. Now who has a possible claim?

[Speaker K] Someone else, not you.

[Rabbi Michael Abraham] I can’t have a claim, right? Someone else can have a claim. Why? Because the other person says my chance of winning was one in thirteen, not one in twelve — a smaller chance. Right? But in fact, even though your chance was one in thirteen, the same was true for all the others except me, and I didn’t win anyway. So there’s room to hesitate even about that. But beyond that he says not only can the others void the lot — I myself can void it. I am the one who spoiled it; I gave myself a greater chance of winning, not a smaller one. I myself, with my own hands, arranged that my chances would be larger, and nevertheless I can still say: the lot was flawed; I demand that it be canceled. And that really seems very strange. This is a kind of mystical claim: if the lot is conducted in a flawed way, no matter what happened, it is void, and you redo it, whether intentionally or unintentionally.

[Speaker I] Why mystical? Why mystical? It’s not what was agreed upon. We agreed on a lottery, and now no lottery was actually performed. Something else happened, which also has the name “lottery,” but it isn’t the lottery that was agreed upon.

[Rabbi Michael Abraham] No, but what I said is that it did not harm your chances, so what difference does it make?

[Speaker I] Fine, but what does that have to do with it? That’s not what was agreed.

[Rabbi Michael Abraham] Why does it matter what was agreed? You only gained — more than what was agreed.

[Speaker I] No, but we agreed on a lottery, not on my—

[Rabbi Michael Abraham] a lottery with a greater chance.

[Speaker I] But that’s not the same thing called a lottery that was agreed on.

[Rabbi Michael Abraham] For example, can someone cancel the state lottery if the state lottery decides to let him win with two slips even though he bought only one?

[Speaker I] Again, if we understand that it’s deterministic and there is no real randomness — all the words “chance” and “lottery” are empty terms, just that we don’t know. We are simply in a reality of ignorance. We agreed on a certain lottery. That certain lottery, if it had been performed, its results were fixed already since the Big Bang fourteen billion years ago. True, but the chance could be… Now we don’t know, but we think our action is meaningful, while really it isn’t meaningful at all. Now something else happened entirely; the execution was completely different. So fourteen billion years— from fourteen billion years ago something completely different was already fixed. You can’t change it.

[Rabbi Michael Abraham] From fourteen billion years ago it was also fixed that the lottery would be flawed. Fine, but that’s a deterministic view.

[Speaker I] Fine, no problem, but right now we’re not talking about that. We’re talking about the claims. The claim is: we agreed on a lottery.

[Rabbi Michael Abraham] If you assume determinism, then the flaw in the lottery is also a deterministic result.

[Speaker I] Fine, but my claim is not about rights.

[Rabbi Michael Abraham] God decided that I would win, that’s all. It was decreed on Rosh Hashanah.

[Speaker I] Right, but the claim is not about rights or what God decided. The claim is what we agreed upon.

[Rabbi Michael Abraham] The agreement too was deterministic.

[Speaker I] Why does that matter? We… we’re not speaking about rights; we’re speaking about contracts, contract law. And in contract law, what was executed was not what was agreed. That’s all.

[Rabbi Michael Abraham] But you gained more than what was agreed. I gained?

[Speaker I] We’re entering the field of randomness — there is no randomness.

[Rabbi Michael Abraham] What was fixed was fixed. Why does it matter whether it’s random or not?

[Speaker I] You gained a greater chance.

[Rabbi Michael Abraham] I don’t see the difference. You gained a greater chance — so what?

[Speaker I] But I’m not making a claim that my rights were harmed. The word “chances” is unnecessary. There never was any concept of chance. Everything was fixed in advance. The question is what happened in reality. Was what was agreed actually carried out or not? No, what was agreed was not carried out. A different process was carried out, which also was not random at all and was fixed in advance.

[Rabbi Michael Abraham] Fine, if it’s not random then all these claims are strange, because then everything is fixed in advance, including the flaw in the lottery. So I don’t understand the discussion at all.

[Speaker I] Again, the Rabbi keeps returning to claims of “my rights were harmed.” The dispute is not about whether my rights were harmed; the dispute is whether what we agreed between us was carried out. Was the contract fulfilled or not?

[Speaker K] No, maybe this will help from the law of presumed benefit. Maybe one could say: fine, it’s clear that you would have agreed even to something like this.

[Rabbi Michael Abraham] Yes, that is really the claim — after all, you got a higher chance, so what do you want?

[Speaker K] No, but even the Rabbi agrees that you need the person’s actual consent in order to conduct the lottery itself. A person could say in principle: I don’t want the lottery.

[Rabbi Michael Abraham] If you need his actual consent, then that doesn’t help at all, because he certainly will not agree; he wants a repeat lottery so that maybe he’ll win this time.

[Speaker K] No, I mean clearly — before they did the lottery — clearly you need the consent of all the participants in the lottery.

[Rabbi Michael Abraham] Yes, that seems obvious.

[Speaker K] Fine, okay, so that’s—

[Rabbi Michael Abraham] an anecdote. One could certainly discuss this responsum much more. Even without Havot Ya’ir, suppose we don’t agree with him. We would still need to understand how we evaluate the alternative possibilities. Does the fact that I had a better chance of winning from the outset prevent me from making a claim? It may indeed be that logically, at the end of the day, in practice I did not win, and if we had done the lottery properly perhaps I would have won. That itself may be enough to justify the claim, even if I had received a better chance.

[Speaker E] There’s room here for hesitation on the conceptual level.

[Rabbi Michael Abraham] Okay, but that’s where we stop. This is the end of the series. We can raise a glass to the conclusion. We return to you, doubt and probability, and you return to us. May we not forget you, and may you not forget us, neither in this world nor in the world to come.

[Speaker I] There is no joy like the resolution of doubts.

[Rabbi Michael Abraham] There is no joy like the easing of doubts. As for actually resolving them, I’m not sure we managed that here. We permit prayer together with transgressors. Okay, any comments or questions?

[Speaker K] Thank you very much, Rabbi.

[Speaker I] Really, thank you. Thank you very much. Thank you, thank you very much. What is the next series? Wait, wait, what is the next series?

[Rabbi Michael Abraham] We’ll see, I still need to think. I was thinking maybe of speaking about artificial intelligence.

[Speaker I] Wow, wonderful.

[Speaker F] Sounds excellent.

[Rabbi Michael Abraham] We’ll see, I’ll still think about it, but I think I may go in that direction.

[Speaker I] In what sense?

[Rabbi Michael Abraham] Clarifications from various angles. Fine, so I’ll… there will be a message. Thank you.

[Speaker I] Thank you. Thanks to the Rabbi for all the effort, really. We are all full of gratitude. Our ability is too limited to express

[Rabbi Michael Abraham] our thanks and appreciation.

[Speaker I] our gratitude and appreciation.

[Rabbi Michael Abraham] Goodbye.