Doubt and Probability—in Halakha, in Thought, and in General—Lesson 41 – Rabbi Michael Abraham

[Rabbi Michael Abraham] Okay. So we’re starting the last chapter in this series, and I want to talk a bit about double doubt and two majorities, basically probabilistic multiplications. We touched on certain things along the way, but here I want to give a somewhat more systematic view. So maybe I’ll start with a definition of doubt and double doubt. We know that the rule in Jewish law is that a Torah-level doubt is treated stringently and a rabbinic-level doubt leniently. According to most views, a Torah-level doubt treated stringently is itself a Torah law. According to Maimonides, it’s a rabbinic rule, meaning that the obligation to be stringent in a Torah-level doubt is itself only a rabbinic obligation. But I’m ignoring that point for the moment. Bottom line: with a Torah-level doubt you have to be stringent. What happens when there’s a double doubt? Double doubt means I have one doubt, and on top of it another doubt. In that situation, even in a Torah-level matter we’re lenient. Yes, if we have a double doubt then even in a Torah-level matter we’re lenient. Now I want to clarify the situation a bit more. So maybe I’ll start with an example. Let me share this for a second. It’s simple, but still, I think the visualization here will help us later on. So look here at the diagram, at the chart here. In practice this is the chart: let’s say there’s this node here, the upper node basically splits into two possibilities, A and B. That’s a case of doubt. If possibility B itself splits into two more possibilities, that’s already a double doubt. Example: a husband who got married comes to his wife and finds that she is not a virgin. So yes, the husband claims, “I found an open entrance.” These are the discussions at the beginning of Ketubot. Now we have a doubt about when that intercourse happened. Meaning, assuming it’s really true that the entrance is open, we have one doubt about when that intercourse happened. If it happened before the betrothal, then the woman does not become forbidden. So that doesn’t bother me. Only if it happened after the betrothal and the woman had intercourse with someone else does she become forbidden. So if I have a doubt whether it happened before or after the marriage, then because of doubt the woman becomes forbidden—Torah-level doubt treated stringently. But even if it happened after the marriage, there is still room for doubt about what exactly happened. Meaning, the fact that the hymen was torn could result either from intercourse or from an injury. Yes, she got some kind of blow, fell, something like that, and that is what tore the hymen. In that situation we’re basically in a case of double doubt: doubt whether it happened before the betrothal or after the betrothal, and even if it happened after the betrothal, doubt whether it was intercourse or whether it was injury. So you see—let’s go back to the diagram for a second—we basically have a doubt whether she had intercourse before the betrothal, which is A, or after the betrothal, yes, that’s B. If we’re at this node, she becomes forbidden. At this node she is permitted. So if we had only the upper part, doubt between A and B—there are noises here—doubt between A and B. I’m muting everyone, and I ask you not to unmute unless you want to speak. One second. Okay. So yes, we basically have a doubt whether she had intercourse before the betrothal or after. We still have a doubt whether it was intercourse or whether she was injured. Therefore it’s a double doubt. Here too, in principle, if it happened before the betrothal, we could also have a doubt whether it was injury or intercourse, but that doubt isn’t relevant because in any case, either way, she doesn’t become forbidden. Meaning, in principle I could also have drawn two more branches here. Exactly the same two branches that I have here, I could have drawn them here too. All right? But that doesn’t matter, because in both of them the woman ends up permitted. So it basically turns out that in a situation of this kind, notice, we actually have three possible paths. One path is this, the second path is this, and the third path is this. This path means she had intercourse after the marriage and it was intercourse. So here she becomes forbidden to her husband, right? If we went down this path, she becomes forbidden. If it was after the marriage but it was an injury and not intercourse, then she is permitted. Right? Here she is permitted. If it was before the betrothal—not the marriage—if it was before the betrothal, then she is also permitted. Meaning, here she is permitted, here she is permitted, and here she is forbidden. We have three paths, in one of which she is forbidden and in two of which she is permitted. So let’s say each such split is fifty-fifty. Then what is the chance that she is forbidden?

[Speaker D] Zero point seventy-five? Twenty-five.

[Speaker E] Twenty-five.

[Rabbi Michael Abraham] The chance that she is forbidden, not the chance that she is permitted. The chance that she is forbidden is only this path, which is one half, right? Here we went down here, that’s half, multiply that by another half, which is zero—

[Speaker G] Point twenty-five, a quarter out of a hundred. Yes.

[Rabbi Michael Abraham] So a quarter that she is forbidden. What happens here? There is a quarter that she is permitted. What happens here? Here it’s half that she is permitted, right? Not a quarter, because the chance that she had intercourse before is half. So half plus a quarter is three quarters that she is permitted and a quarter that she is forbidden. If I had made a diagram where these two branches also appeared here, meaning yes, injury or intercourse also before the betrothal, then visually it would have been clearer, because then it would come out that we have four possibilities, each one twenty-five percent, and in three of them she is permitted and in one she is forbidden. Right? Therefore, in a case of double doubt, the distribution is one out of four, one out of four for prohibition. Okay, that’s basically the assumption. So whether you draw the split here or not doesn’t really matter. In the end, the probability is twenty-five percent, meaning one out of four. Now, I want to make a point that will also be very important later on. I already talked about this, but I want to mention it again. The doubts we’re dealing with in this case—yes, doubt whether she had intercourse before or after, and doubt whether it was injury or intercourse—we assume this is a doubt. Doubt means fifty-fifty. That is, if the two possibilities are seventy-thirty or sixty-forty, then it’s not doubt, then it’s a majority. Let’s say if there were only one drawing—yes, suppose there were only this drawing, I’m going up to the picture above for a second—yes, suppose there were only this drawing, only the upper node splitting into two, all right? Then there would be a doubt and she would be forbidden. But what happens if the split is not fifty-fifty, but say sixty-forty? In that case she would of course be forbidden because that’s a majority, right? If it were forty-sixty, then she would be permitted. In principle, you follow the majority, right? When is it considered a doubt, and with doubt we go stringently? When the mutual chances are fifty-fifty. Okay. Same thing in the second part. Here too it’s fifty-fifty. Now, we need to understand why the possibility that she had intercourse before the betrothal and after the betrothal are equal in weight. Why do I assume it’s fifty-fifty? I’d say even more than that: after all, if she already had intercourse, then most likely it happened before the betrothal. Why assume she’s wicked and committed this transgression after the betrothal? There’s even some logic in preferring the possibility that she had intercourse before the betrothal. But let’s leave that aside for the moment—just generally, why do I decide it’s fifty-fifty? Why isn’t it seventy-thirty? Same thing with the distribution between injury and intercourse. There too, why do I assume it’s fifty-fifty? Why not eighty-twenty?

[Speaker D] If you don’t know anything about the sample, then that’s the assumption, I guess. Okay, why assume otherwise? I don’t know, if you want to assume otherwise, you need to bring a reason.

[Rabbi Michael Abraham] Meaning that what we are basically encountering here—and this is a very important point, a point I think many people miss—is what I previously called negative doubt. Negative doubt means I don’t have information. If I have no information and I have two possibilities, then it’s fifty-fifty. So it’s not really fifty-fifty because I know the odds are equal on both sides. It’s fifty-fifty due to ignorance, not due to knowledge. I could arrive at fifty-fifty because I simply know that the percentage of women who had intercourse before and after the betrothal is fifty percent, say, for the sake of discussion. If I knew that, it would be positive doubt. But usually that’s not the case in these situations. Rather, the doubt is a negative doubt. Since I have no information, I know nothing. There are two possibilities here, and I have no reason to prefer one over the other, so the assumption is that as far as I’m concerned it’s fifty-fifty.

[Speaker G] By the way, why don’t you bring in the probabilities of the things themselves? I mean, injury is very, very rare. Definitely something—

[Rabbi Michael Abraham] And intercourse with a woman who had already had intercourse is also very, very rare.

[Speaker G] Intercourse with a woman who had already had intercourse, that’s not—

[Rabbi Michael Abraham] You meant a married woman, yes.

[Speaker G] Is that rare? Yes.

[Rabbi Michael Abraham] So? There’s no way to bring that in. I don’t think there’s a way to bring that in.

[Speaker G] Yes, but on the other hand, in Jewish law for example we never rule—

[Rabbi Michael Abraham] One at a time, one at a time. I—

[Speaker G] I’m saying that for example, if I put that aside right now and I relate to the Jewish law as written at the time of marriages and wedding canopies and so on, and we see this all the time, we treat it as doubt, as full doubt, even if she already had intercourse and so on, and we say that the person supposedly doesn’t want to pay the ketubah, so we just put it into the category of doubts. That’s what’s ruled. I didn’t understand. Meaning, today, right, when they do a wedding canopy, a person comes to the rabbinate, wants to marry a woman and so on, they don’t ask her whether she’s a virgin or not.

[Rabbi Michael Abraham] Okay, that’s—

[Speaker G] It’s not a matter of the rabbinate and so on. In the ketubah they’ll write in any case that she is an excellent bride and so on and the whole story and so on. Meaning, from the outset, deliberately, even if they lived together for several years before that—

[Rabbi Michael Abraham] Fine, but how is that relevant to us? What’s the question?

[Speaker G] No, because there’s a probability here.

[Rabbi Michael Abraham] Meaning, we don’t get into it, we don’t ask the question so as not to embarrass her or for all sorts of reasons, it doesn’t matter right now, but that’s not statistical. I’m talking right now about the statistical calculation, all right? So I’m saying that here we are dealing—

[Speaker H] Rabbi, you’re saying it’s very rare that a betrothed woman would have intercourse with someone else, okay, but why not apply a majority not physically present before the betrothal?

[Rabbi Michael Abraham] What do you mean?

[Speaker H] I don’t know. No, I mean there’s a majority, or let’s say that most girls have had intercourse or have not had intercourse before the betrothal, and therefore we shouldn’t begin with an even doubt at the outset. It won’t be a negative doubt where we know nothing.

[Rabbi Michael Abraham] No, that’s a mistake, a statistical mistake. Relative to—after all I know that she had intercourse.

[Speaker H] What?

[Rabbi Michael Abraham] I know that she had intercourse!

[Speaker H] No, there’s knowledge here. No, I’m saying before, before the betrothal, why not apply a majority?

[Rabbi Michael Abraham] No, it doesn’t matter, I know she had intercourse. Right now, when she comes, the current situation is that I know she had intercourse before the marriage. Marriage, not betrothal. Now the question is only how long before the marriage. So you can’t use the statistics of women, the general statistics of women who do not have intercourse before the betrothal, because among the women who did have intercourse before marriage, you don’t know how many of them it was before the betrothal. Do you understand what I’m saying?

[Speaker H] Conditional—like conditionally given.

[Rabbi Michael Abraham] Exactly, it’s conditional probability. Like we discussed, if you remember, we discussed the presumption that what is in a person’s possession belongs to him. And seemingly people usually understand that most objects are in the hands of their owners, but that’s not true, because that’s true only regarding—meaning, is the plaintiff really, in most cases, the liar? In most cases is the plaintiff the liar and not the defendant? I don’t see a reason to assume that. So one always has to be careful with statistics that we make about a large group when we already have information that focuses us on a subgroup. And it’s not clear that with respect to that subgroup, the distribution is the same as the distribution in the larger group, because this sample may not be representative. Okay? And therefore in this case too, we have information here that this woman had intercourse before marriage. Meaning, that’s given information. So if that’s the case, we’re already inside a subgroup out of all the women in the world. Inside that subgroup, I no longer know that most women did not have intercourse before the betrothal. On the contrary, as I said earlier, I would say the opposite. If she already had intercourse, why assume she did it unlawfully after the betrothal? It’s more likely that she did it before the betrothal.

[Speaker G] Maybe that’s what we need to say? What? Maybe that’s what we need to say?

[Rabbi Michael Abraham] Fine, maybe. The Talmud doesn’t say that. There is really room to wonder why not. But certainly, certainly there isn’t the opposite majority. Yes.

[Speaker D] Rabbi, but if you’re saying you always need fifty-fifty, and once you have no information about the sample or don’t know about the outcomes, I mean, about what you’re dealing with, then you assume it’s fifty-fifty, then it comes out that doubts exist only in a situation where you really don’t have information? Because to arrive at exactly fifty-fifty is impossible.

[Rabbi Michael Abraham] Fine, because you can’t narrow it down, you mean. Fair enough. Yes.

[Speaker D] I’m asking whether fifty-fifty means specifically exactly fifty-fifty, or whether it’s—

[Rabbi Michael Abraham] Look, there are intermediate situations where I have some fairly clear information, but it’s not really something you can narrow down. It’s between 51-49 and 49-51, something in between. Say I see a person in front of me, and the question is whether the person is male or female. So I know that the distribution of women and men in the world is fifty-fifty plus a bit in favor of women, okay? So it’s not—so you can’t narrow it down—but here I’d be willing to hear someone tell me that this is considered an evenly balanced doubt in a positive sense. Okay?

[Speaker I] Yes, meaning it’s not really fifty-fifty, it’s more like something—

[Rabbi Michael Abraham] Yes, you can never know with certainty. Even if you could narrow it down and in a sample you found that it’s exactly fifty-fifty, in the general population who says the sample represents it exactly? All right? So in the general population you say maybe it’s 52-48. So the whole concept of narrowing it down is irrelevant altogether to this kind of calculation, because at most it can appear in a sample, but not in the general population that I am sampling from.

[Speaker D] I see. But isn’t there some threshold where you say from here you have to stop? Say 48-52, you’d tell me that doesn’t count?

[Rabbi Michael Abraham] I don’t think so. Look, there isn’t some general quantitative threshold like that. There’s some sort of perspective of the reasonable person, I don’t know. A reasonable person sees the probability of male versus female as fifty-fifty, or heads versus tails in a coin toss, where there too it isn’t really fifty-fifty, yet it’s accepted to think of it as fifty-fifty.

[Speaker D] Why not? I mean, a fair coin—why not fifty-fifty?

[Rabbi Michael Abraham] No, it’s not fifty-fifty. No, what is a fair coin? There just isn’t any coin that is fair.

[Speaker D] There isn’t any fair coin? Why?

[Rabbi Michael Abraham] What do you mean, why isn’t any coin fair? Because the heads side and tails side have different designs. And I think I mentioned this in one of the previous meetings: I read some article a few months ago, or a year ago, I don’t know how long, not so long ago, I read some article where someone checked it. There isn’t any coin in the world that’s exactly fifty-fifty. How can you know?

[Speaker D] But you’d have to sample it a billion times.

[Rabbi Michael Abraham] He did it over a very large number of tosses, and none of them came out fifty-fifty. Really? Fine, of course it’s always possible that if you continue to infinity it will come out fifty-fifty, but he apparently found a fairly consistent convergence. You can check it. By the way, yes.

[Speaker G] And why do you treat doubt specifically as fifty-fifty? You said before that if it’s a majority then it’s a majority, so there’s no doubt here, but I think that in every question where there’s some small doubt, it depends on how you relate to the doubt. For example, I’ll give a clearer example: if we’re in a place where Jews are serving food, then there’s reason to be lenient in checking kashrut, and if you’re in a place with non-Jews, even if they tell you it’s kosher, there’s reason to be stringent—for what reason? Because of the doubt in the serving of the food. Or let’s say in the writing of a Torah scroll and so on, we have many doubts about the person who wrote it and we have many reasons to be lenient. So what’s the question? No, the rabbi said that in doubt it always has to be fifty-fifty. But that depends; it’s not really like that. If you have a question where there is twenty-five or thirty percent doubt, it depends on how you relate to that doubt in order to determine whether it’s a doubt or a majority.

[Rabbi Michael Abraham] I don’t understand. I don’t understand the point. I don’t understand what you’re saying.

[Speaker G] No, I asked why with doubts it always has to be fifty-fifty.

[Rabbi Michael Abraham] It doesn’t have to be—doubt is defined as fifty-fifty. If it’s not fifty-fifty, then it’s a majority.

[Speaker G] A majority determines that it’s not a doubt?

[Rabbi Michael Abraham] Yes. We follow the majority—go after the majority—in Jewish law as a matter of principle. Again, there are stringencies in many places where we don’t rely on the majority, but fundamentally, in Jewish law we follow the majority, yes.

[Speaker G] Wait, for example, just a general question—regarding blood appearances? Well. There we don’t go that way.

[Rabbi Michael Abraham] So I’m saying, there are places where we are stringent and do not follow the majority. Fundamentally, in Jewish law we do follow the majority. That, yes. But no, a majority is fifty-one percent. Fifty-fifty plus epsilon—that’s a majority. That’s the definition. I understand, yes.

[Speaker G] I just didn’t understand why it has to be specifically fifty, because there are many times we don’t follow the majority and we still treat it—

[Rabbi Michael Abraham] No, so there are many places where we’re stringent; with lineage too we are stringent, all right, we are stringent in many places, but that stringency is rabbinic. At the Torah level we follow the majority. So in short, to return to our issue: the doubts that appear in this case—whether she had intercourse before the betrothal or after the betrothal, or whether it was injury or intercourse—those are negative doubts. Negative doubts, for our purposes, are considered balanced doubts. I call it fifty-fifty even though it’s not really fifty-fifty, but we treat it as an even doubt, because we have no other information and no reason to prefer this over that. If you remember, I gave several examples of this in the past. I talked about a situation where, say, I toss a coin and I don’t know whether the coin is fair or not. Okay? If you say that I know the coin is fair, then the chance of heads is fifty, the chance of tails is fifty, and here there’s a doubt whether it will land on heads or tails—a positive doubt, because it stems from information. I know that the coin is balanced, fair. But what happens with a coin about which I have no idea? I don’t know whether it’s fair, I know nothing, anything could be. Now someone asks me what to bet on, heads or tails, or what probability I assign to heads. The answer is fifty percent. Why? Because I have no other information; I can’t prefer tails over heads or vice versa, and therefore in a negative way I treat it as an even doubt, fifty-fifty. You can see this from another direction in Jewish law. They distinguish between a doubt involving one piece and a doubt involving one piece out of two pieces, where the prohibition is fixed in place. What does that mean? I have one piece that is possibly forbidden fat, possibly permitted fat. Forbidden fat is prohibited to eat, permitted fat is allowed to eat if you’re not on a diet. And here you have a doubt whether it is forbidden fat or permitted fat. That’s a doubt involving one piece. Now you don’t know, you have no information about the chance that it is forbidden fat or permitted fat. It’s a doubt due to ignorance. You don’t know whether it’s forbidden fat or permitted fat, and therefore for you it’s an even doubt, either forbidden fat or permitted fat. What happens in a situation where you have one piece out of two pieces? Meaning, there are two pieces here in front of you; you know that one is forbidden fat and one is permitted fat, you just don’t know which is which. Now if you took one, and now you ask yourself whether you may eat it, the answer is no. Why not? Because there is a fifty percent chance that it is forbidden fat. But that fifty percent is positive, not negative. Because after all you know that there is one piece of forbidden fat and one piece of permitted fat here, that one of them is forbidden fat and one is permitted fat; you just don’t know which one is which. That parallels a fair coin. The fifty-fifty distribution here is a positive distribution, not a negative one. A doubt involving one piece is a negative doubt, like a coin about which you know nothing. Even then the assumption is that I treat it as an even doubt, as fifty-fifty, but it’s a negative doubt, not a positive doubt, and we’ll see later that this distinction is very important, and many people get confused about it, including halakhic decisors—they get confused about it, meaning they make mistakes about it. So we spoke about the relation between doubt and double doubt, and I said there is negative doubt and positive doubt. In most cases the doubts we deal with are negative doubts. In most cases. Negative doubts, because we don’t have any solid information that the distribution really is fifty-fifty; on the contrary, usually we don’t know anything, and in any event the assumption is that there are two equally weighted possibilities, and therefore for us the distribution is fifty-fifty. Now, that’s regarding the doubts we saw. Now I want to talk about a majority upon a majority. What happens when we’re in a situation like the one we saw in the diagram—let’s go back to it for a second—but the nodes we are dealing with are not split fifty-fifty; rather there is a majority here. So look: let’s say there is a greater likelihood that she had intercourse before the betrothal, a minority likelihood that she had intercourse after the betrothal. And even if she had intercourse after the betrothal, meaning it happened after the betrothal, there is a greater likelihood that it was injury, and a minority likelihood that it was intercourse. This is called a majority upon a majority. What does that mean? There is a double majority here for permission, right? Meaning, one majority that she had intercourse before the betrothal, say eighty percent for the sake of discussion, that eighty percent means she is permitted. Out of the twenty percent for which she is forbidden, still not all of that twenty percent is actually prohibition, because sixteen percent of that would be injury after the betrothal, and only four percent—twenty percent out of twenty percent, only four percent—is that she had intercourse after the betrothal. So in that situation, notice, I have ninety-six percent that she is permitted and four percent that she is forbidden. Meaning the majority becomes a very solid majority, a very significant one. True, even with a single majority we follow the majority, but with a majority upon a majority it’s obviously even stronger. There are situations where with one majority we are stringent, but with a majority upon a majority we are lenient.

[Speaker G] Isn’t there a logical contradiction here? Hear me?

[Rabbi Michael Abraham] It’s not—

[Speaker G] Isn’t it a logical contradiction?

[Rabbi Michael Abraham] Why?

[Speaker G] I’ll explain. If most likely she had intercourse before the betrothal, then she is already permitted on that basis, from the fact that she had intercourse before the betrothal. Now we’re adding some other data point, injury, but there’s no need for that data at all because she—

[Rabbi Michael Abraham] In this case we would permit her even on the basis of a single node; the eighty percent is enough to permit her. It adds nothing. Doesn’t matter. But I’m saying that still, in Jewish law there is a case called a majority upon a majority.

[Speaker G] No, that’s true, it’s just that here—

[Rabbi Michael Abraham] And this case, the case I described here, is called a majority upon a majority. Now, in a large number of cases a single majority is enough to permit, and therefore it doesn’t matter that we’re dealing here with a majority upon a majority, because even a single majority already permits. But you yourself commented earlier that there are situations in Jewish law where a majority alone isn’t enough; we are stringent despite there being a majority for permission. In such situations, sometimes with two majorities, with a majority upon a majority, we are lenient.

[Speaker G] Agreed, absolutely. It’s just that I said that injury at the time she had intercourse already contradicts it somewhat logically. What do you mean?

[Speaker E] I didn’t understand. If—

[Speaker G] If the woman, if she had actual intercourse, was actually with a man, then she isn’t an injury case.

[Rabbi Michael Abraham] If she had intercourse, then she isn’t an injury case.

[Speaker G] Right, exactly, that’s what I said.

[Rabbi Michael Abraham] No, but that’s not the formulation. The formulation is that the question is whether it happened before the betrothal or after it, and on the side that it happened after the betrothal, the question is what happened—was it intercourse or was it injury. All right? Obviously if she had intercourse then she isn’t an injury case. That’s not the—

[Speaker D] Rabbi, but with a majority upon a majority, meaning if you have a single node with ninety-six percent and four percent, is that treated differently?

[Rabbi Michael Abraham] We’ll get to that, we’ll get to that in a moment, we’ll get to that in a moment. In any case, I’m not brushing you off—I’m just about to get to it, and I want to do it step by step. So the point is that a majority upon a majority—and again I treat all these things as probabilistic multiplications. In probability they literally draw trees like this, and the probability of each path is the product of the probabilities. So if here there is fifty percent and here fifty percent, then the probability of ending up here is twenty-five percent. Here it’s twenty percent and another twenty percent, so the probability of ending up here is four percent. 0.2 times 0.2. Okay? Meaning that in a probability tree we always multiply the probabilities. If we go down this path, for example, then the probability of ending up here is sixteen percent. Eighty percent, and you multiply that eighty percent by twenty percent, so that gives sixteen percent. All right? Here it’s only one step, so it’s eighty percent—you don’t multiply it by anything. Okay? So these are the multiplications; that’s basically our way of relating to the different situations. We’ll see later that these multiplications don’t really have probabilistic meaning, but it’s convenient to think about them in probabilistic terms. So that’s regarding a majority upon a majority for permission. But notice that once the element of majority enters here, it can also work the other way around. Let’s say that the probability of A is twenty percent and the probability of B is eighty percent. Then here, if there were only a single doubt, only this node, then the woman would be forbidden, right?

[Speaker A] And her ketubah would be one hundred.

[Rabbi Michael Abraham] So then she actually would not—sorry, her marriage settlement would not be one hundred; she wouldn’t receive a ketubah at all. So if she had relations—if the likelihood that she had relations, and that this happened after the betrothal, is eighty percent, and now here too there is another distribution: it could be, for example, that eighty percent—sorry, that twenty percent is actual intercourse and eighty percent is an injury caused by an external blow. All right? So eighty times twenty is sixteen percent that she is forbidden, and eighty-four percent that she is permitted, right? It’s not four percent, it’s sixteen percent, but still that’s a minority, right? Overall it’s still more likely that she is actually forbidden—sorry, that she is permitted. Okay, what happens if this was eighty percent, and this too was eighty percent, and here it was twenty percent? Then the chance that she is forbidden suddenly becomes sixty-four percent, meaning most likely she is forbidden. Meaning that when these junctions are not evenly balanced, that is, when there is a majority here in one direction, the situation gets complicated. In other words, there can be a situation where we arrive—when I speak about a double majority, that is always when the first majority is already a majority toward permissibility, and on top of that there is another majority toward permissibility. That is called a majority of a majority. Meaning that the permitted possibility is always the one with the higher likelihood. But all the other situations in a diagram like this, where the distributions are majorities but not majorities in the same direction of permissibility, that will no longer be a majority of a majority. Right? Here the cases become much more complicated. In the simple sense, if we make a probabilistic calculation here, then at the end of the day, once we multiply, we get whatever probability comes out. If it comes out to sixteen percent, then there is a majority here; if it comes out to sixty-four percent, then there isn’t a majority here. I don’t care that it came from multiplication; in the end I either have a majority or I don’t. Okay. Now the important point to understand here is that there is a difference between a majority and a doubt. What is the difference? With doubt, I made a distinction between positive doubt and negative doubt. There are

[Speaker C] situations

[Rabbi Michael Abraham] where I have no information whether she had relations before or after, so I assume it is evenly balanced, fifty-fifty. I called that negative doubt. There is a doubt where I know that the distribution is fifty-fifty; that is positive doubt. And I said earlier that the doubts we are talking about here are negative doubts. What happens with a majority? Usually there is no such thing as a negative majority. If I know it is eighty percent—that is, if I know nothing, then it is not a majority, it is an evenly balanced negative doubt, but still an evenly balanced doubt. When I assume that it is a majority, that means I already know, because if I didn’t know it would be an evenly balanced doubt. So I assume that I know. Now if I know there is a majority here, then the majority here is a positive majority. There is no such thing as a negative majority, because anything negative is not a majority, it is a doubt. Therefore when I speak about a majority, by definition it is a positive majority. With a majority, apparently there is no such thing as a negative majority. All right? Do you understand what I’m saying?

[Speaker A] But that sort of can’t work together?

[Rabbi Michael Abraham] What? A majority by definition is positive, because if it were negative it wouldn’t be a majority. Because what does negative mean? Negative means I have no information. If I have two possibilities and I have no information, then I relate to them as if they are equal. That is doubt.

[Speaker G] So if the two

[Rabbi Michael Abraham] possibilities are not equal, meaning that there is a majority here and not a doubt, that means I have information, so it is not negative.

[Speaker D] Fine, so it depends on your question. Say I don’t know, maybe you want, say, ninety percent and you have eighty percent.

[Rabbi Michael Abraham] No, in Jewish law it’s either fifty percent or a majority. Okay.

[Speaker G] So it comes out that negative is minus and positive is plus, right?

[Rabbi Michael Abraham] I didn’t understand.

[Speaker G] Negative always drops down to the minus side, to doubt, and positive is plus, which is…

[Rabbi Michael Abraham] Negative doubt and positive doubt are both doubts. There is doubt because of lack of information, and there is doubt because of information. But with a majority there is always information, because if there were no information it wouldn’t be a majority, it would be doubt. But mathematically,

[Speaker G] if I treat it as zero—if I don’t have it, negative means I go toward the minus direction, that means I have nothing, I remain in doubt. With positive, when I rise with some information, it turns into a majority.

[Rabbi Michael Abraham] No, no. There is also positive doubt. Not everything positive is a majority. Take a doubt that can be positive. A doubt about a coin that I know is fair—the doubt whether it comes up heads or tails is positive doubt. I have information, but it is still doubt.

[Speaker G] Which is to say that any information you get beyond zero makes it plus. If

[Rabbi Michael Abraham] you have information beyond balance. Yes. Fine. But when I spoke about positive, I meant information in general. Why specifically beyond balance? There is information that gives me the very balance itself. That is positive doubt. All right? But what I just said is not precise. Meaning, all these things will be very important later. So I want you to pay close attention to these distinctions. They are very simple, but you need to keep them in mind, because afterward people get very confused in these topics. With a majority there is also a case of negative majority, contrary to what I said before. Apparently with a majority there is always information, and therefore a majority by its essence is always positive and not negative. But that is not true, not precise. Why? Because many times there can be a situation where I clearly assess that one of the options is more likely, but I don’t have a calculation of how much of a majority it is. Seventy percent, ninety percent, eighty-four percent—I don’t know how to calculate it. But it is clear to me that A is much more likely than B. For example, a person fell into the sea and I did not see him come out, all right? So most likely he drowned, right? But there is some chance that he came out somewhere and I didn’t notice. In water with no visible boundary, for example—we’ll get to that later. Okay? Now I don’t know how to quantify the probability here, but I still know how to say that he most likely drowned. What does that mean? That I have a majority, but it is still negative. It is negative in the sense that I don’t have numbers describing how large that majority is. Eighty percent, seventy percent, ninety-three percent—I don’t know how to calculate it.

[Speaker J] But Rabbi, isn’t that just because of ignorance?

[Speaker H] What? You’re switching the definition of negative now, because at first… Right, right, obviously.

[Rabbi Michael Abraham] No, that’s obvious. There is some information here, but not complete information. Why is this important? Because look, if for example we go back to this tree, and we have a majority—as you said before, someone asked earlier—for example we have a majority of eighty-twenty. All right? And now again here I have eighty-twenty. So now I actually have four percent versus ninety-six percent, right? That’s the calculation we made before. So what difference does it make? Both eighty-twenty and ninety-six-four are a majority, and in Jewish law we follow the majority. So why should I care whether it’s one majority or a double majority? But there are situations where things are not so simple. Because true, there is a majority, but I don’t know how to multiply these probabilities. Look, for example, imagine a situation where here there is twenty percent and here there is eighty percent, all right? But here too there is twenty percent and here there is eighty percent.

[Speaker A] Okay,

[Rabbi Michael Abraham] so eighty times twenty is sixteen percent here. But what happens if the probability here is that this is eighty and this is twenty, and here it is forty percent versus sixty? Then it comes out to thirty-two percent, right? What happens if here it is sixty-forty, all right? And here it is ninety-ten? So sixty times ninety… fifty-four percent, right?

[Speaker C] But

[Rabbi Michael Abraham] if it is sixty times eighty, then it is forty-eight percent, which is already less than fifty percent. Right? Meaning, when I have two majorities here, in the end the situation is not clear. There could be a majority toward permissibility, and there could be a majority toward prohibition; it depends on the relation between the majorities. Does one majority offset the second majority, or not offset it? But I can say all this only if I have numerical data for how much each majority is worth. This one is sixty-forty and that one is ninety-ten. But if these are negative majorities—if this is a negative majority, where I can’t say what the probability of A is against the probability of B, but I know that B is more likely—then I can no longer multiply numbers in a

[Speaker J] simple way, right?

[Rabbi Michael Abraham] So it’s not complete clarification. I

[Speaker J] didn’t understand. If there is a majority, isn’t that clarification?

[Rabbi Michael Abraham] It is partial clarification. Meaning, I know it is not doubt—that information I have. But I don’t know how big the majority is. So that is a certain kind of clarification. Relative to a majority, I call that a negative majority. All right? Okay. Now in such a situation, in such a situation, if I were to tell you: look, there is a majority here for B over A, all right? And there is a majority for B-one over B-two. Now I ask whether in such a situation it is permitted or forbidden. Permitted. No. Why? That’s not right. Because it depends on the question of what each majority is. For example, if this is eighty versus twenty, all right? And this is sixty versus forty. Eighty times sixty percent is forty-eight percent. It comes out here that there is

[Speaker F] a majority toward permissibility. I’m not dealing with the math for a moment. I’m saying this: from the moment there is at the second junction a majority toward one B, that is, toward injury from an external blow, then all the—no no no, sorry.

[Rabbi Michael Abraham] Yes, one B is injury from an external blow; I’m talking about when there is a majority in favor of intercourse.

[Speaker F] No no, that’s clear. But I’m talking about when there is a majority in favor of injury from an external blow. As far as I’m concerned, even if there were one hundred percent that it was after the betrothal, she should be permitted.

[Rabbi Michael Abraham] Of course, but that is mathematics. Why do you say it’s not mathematics? One hundred percent times a majority—it will always come out less than fifty percent for prohibition.

[Speaker F] No, the one hundred percent, the one hundred percent is against me, against the woman.

[Rabbi Michael Abraham] Obviously, but within that there is a minority toward prohibition. Right. So practically, after multiplication it comes out that there is a minority toward prohibition here.

[Speaker G] Wait, Rabbi,

[Speaker F] but I want to define it differently. I want to define it that from the moment that at the second junction there is a majority toward injury from an external blow, I am no longer interested in the earlier data.

[Rabbi Michael Abraham] Right, because here one majority is enough. I said that. But if there is—but if there is a majority in favor of intercourse, for example,

[Speaker H] if

[Rabbi Michael Abraham] there is a majority in favor of intercourse, then the law is not clear. That is correct. If from B over A there is eighty percent, and B-two is sixty percent versus B-one, then the chance that she is forbidden is forty-eight percent. Meaning there is a majority toward permissibility. Even though at both junctions there was a majority toward prohibition. At each of the junctions there was a majority toward prohibition.

[Speaker F] That I don’t understand how you get.

[Rabbi Michael Abraham] Here, I’ll explain again. Say there is twenty percent that she had relations before the betrothal, and eighty percent that she had relations after.

[Speaker F] Right. Now there is forty percent that she was injured by an external blow

[Rabbi Michael Abraham] and sixty percent that she had intercourse. Okay. In both cases there is a majority toward prohibition, right? Sixty-forty. Right. But what is the actual chance that she is forbidden? Eighty percent that it happened after the betrothal, times forty percent that she had intercourse and was not injured by an external blow—that means it is thirty-two. Forty-eight.

[Speaker H] Or forty-eight.

[Rabbi Michael Abraham] No, thirty-

[Speaker H] two.

[Rabbi Michael Abraham] Eighty times sixty is forty-eight.

[Speaker H] Rabbi, why are you formulating the first line B as “she had intercourse”? No—that she is not one who had intercourse. And the next two lines are either injury from an external blow or intercourse. Why? Because B is that the event happened after the betrothal.

[Rabbi Michael Abraham] I don’t know what the event was.

[Speaker H] But the event happened after the betrothal. Yes. But then you can’t say that she is forbidden. It still depends on these two possibilities.

[Rabbi Michael Abraham] Certainly. If it were a single doubt, if there were no further doubt afterward.

[Speaker F] Right, but

[Rabbi Michael Abraham] since we have a doubt whether she had intercourse or was injured by an external blow, it depends on the proportions. So I’m drawing your attention to the fact that even if at each junction there is a majority toward prohibition, in this kind of majority-upon-majority case, it is still not certain that there is a majority toward prohibition. It depends on how large this majority is and how large that one is.

[Speaker F] Do again the—just do the mathematical calculation again for a second. You said eighty-eighty.

[Rabbi Michael Abraham] Right. Here eighty, all right? And here there is forty versus sixty. Right. So now twenty percent that she had relations before the betrothal. Yes. What is the chance that she was injured by an external blow afterward? Eighty times forty, which is thirty-two percent. Okay. Right? Thirty-two plus twenty is fifty-two. What remains is forty-eight percent.

[Speaker F] Now in order to forbid—but I think there is some kind of, at least from a logical standpoint, something that doesn’t seem reasonable. I stop first of all at the first junction and say: according to what is happening here now, she is forbidden. No. She is forbidden because it happened after.

[Rabbi Michael Abraham] No, only if she had intercourse is she forbidden; but it could be that she was injured by an external blow.

[Speaker F] Meaning, they depend on each other. You can’t stop at the first junction because you actually don’t know the meaning of “I found an opening,” because first you have to clarify what its source is, so you can’t get to any—

[Speaker K] after the time, after the time.

[Rabbi Michael Abraham] You can. If you ask what the probability is that it was after the time, the answer is that most likely it was after the time.

[Speaker F] Right, but what

[Rabbi Michael Abraham] happened—but what happened—we are being asked what happened there after the time.

[Speaker G] No, the line of reasoning here is very clear, very clear, unequivocal. Okay. Because the first junction gives us nothing.

[Rabbi Michael Abraham] Yes. In short, the ruling has to be on this whole story. By the way, I think we spoke about the doctrinal paradox, didn’t we? I think I mentioned it. What, factories? No, the doctrinal paradox is that judges rule by following the majority. What?

[Speaker D] Did we talk about there being a hundred?

[Rabbi Michael Abraham] Right, this is exactly the situation here, basically. Meaning, if you were to decide each question separately, you would reach a different answer than if you decide the two questions together. Yes, the doctrinal paradox is, say, that two people signed a contract with each other. Now Reuven claims that Shimon breached the contract because he did X. And Shimon makes two claims: A, I did not do X; B, even if I did X, the contract does not forbid doing X. The first claim is factual, practical, and the second claim is legal. Okay? He makes the two claims in the alternative, all right? Now the court has to decide. One judge says he did not do act X, but the contract really does forbid doing it. Meaning, he rules in his favor factually and against him legally. A second judge rules against him legally and in his favor factually. And a third judge rules against him both factually and legally.

[Speaker C] So if you go after

[Rabbi Michael Abraham] the bottom lines of the three judges, it will come out that the majority says, let’s say, that he is liable—you need to see the chart in front of your eyes, but let’s say that. But if you decide the factual question by majority, and afterward decide the legal question by majority, and then combine the answers, you will reach the opposite conclusion. That is the doctrinal paradox. And by the way, I brought an example of this from an actual case in the Supreme Court here in Israel; I found a case that was exactly like this.

[Speaker C] A case that somehow I was

[Rabbi Michael Abraham] indirectly involved in without even knowing. Some couple whose betrothal I annulled, and then suddenly my son showed me that these people had already gone through a long legal saga including the Supreme Court, and then suddenly I discovered this amusing Supreme Court decision that is literally the chart of the doctrinal paradox.

[Speaker G] Yes, because each time they have a different majority on the question.

[Rabbi Michael Abraham] Yes.

[Speaker F] But what happened in practice?

[Rabbi Michael Abraham] In practice, you follow the bottom lines, both in Jewish law and in civil law. No, what happened there?

[Speaker F] Do you know what the issues were?

[Rabbi Michael Abraham] Yes, yes, I can send a link if you want; I have a column about it.

[Speaker F] Yes, I’d be happy to.

[Rabbi Michael Abraham] I describe everything there with sources; you can also look at the sources. Which number? I’ll look for it later, I don’t remember exactly. It’s interesting. In any case, what I want to say is that in the end, here too as in the doctrinal paradox, we judge the whole case, not each junction separately. Because in the end what we want to know is whether she is forbidden or permitted; we are not interested in whether she had relations before or after—that has no practical implication.

[Speaker A] If it happened before, or if there were two testimonies in a religious court or something like that—would that be an option where we would split it into two? I didn’t understand. Is there any possible situation where we would calculate this as two doubts?

[Speaker F] No, you can’t.

[Rabbi Michael Abraham] I don’t see how. Testimony is not doubt; testimony is certainty. Okay. So in the end, I’m saying it is very worthwhile to look at this in terms of multiplication, because that clarifies the structures before us. But you need to remember very well that with doubt too there is a case of negative doubt, and there we still make multiplications as though it were really fifty percent. But with a negative majority you can no longer make multiplications. Because with a negative majority, if you don’t have the numbers, you can’t multiply. If you do have the numbers, you can multiply, and it is not clear what you will get; it depends on the numbers. All right? Now, however,

[Speaker L] even in the doctrinal paradox do we follow the final line?

[Rabbi Michael Abraham] That’s an interesting question. I really discuss it there; it’s a hard question. A hard question. But apparently that’s how it is all over the world, as I understood,

[Speaker G] at least.

[Speaker K] There are cases where judges do split each issue on its own, as if—they say on this issue and that issue the majority view agreed, and on this issue and that issue it didn’t.

[Rabbi Michael Abraham] No, you can write that on this issue we agreed and on that issue we didn’t agree, but the ruling itself is determined by majority in the bottom lines. I read the Supreme Court decision; they wrote the reasons, I read them, I know. They wrote that there was a majority here this way and a majority there that way, but the decision was a majority of bottom lines. Okay, in any case, so how much is a majority?

[Speaker K] What? From what point is it a majority? From 51?

[Rabbi Michael Abraham] What?

[Speaker G] Two against one.

[Speaker K] From what point is it considered a majority?

[Rabbi Michael Abraham] Yes, any fifty plus epsilon.

[Speaker K] And any doubt has to be exactly 50 percent?

[Rabbi Michael Abraham] Yes. And that is why I say that usually when you speak about doubt, there

[Speaker K] is never such a case.

[Rabbi Michael Abraham] Usually when you speak about doubt, it is negative doubt, because positive doubt almost never happens—although, as I noted above, it can happen, like women versus men or a fair coin, where it’s not exactly 50-50 and still you can’t narrow it down. But in the normal way a reasonable person looks at it as 50-50. But usually doubts are really negative doubts, not positive doubts, and also with a majority.

[Speaker F] But in criminal law it could be that you wouldn’t need to get to the bottom lines; you would stop even before that if you had facts whose significance is that he did the act. Then you would only need to go on in order to see whether he did it intentionally or not, because the second junction would no longer affect the act itself.

[Rabbi Michael Abraham] It wouldn’t affect the act itself, but say if he did the act unintentionally, then let’s say he is acquitted. So it is exactly the same thing, because what do I care that he did the act if he did it unintentionally and is therefore acquitted? And there is something similar in criminal law—I also once wrote a column about that. I don’t think it was the same column; it was a bit different—about Zadorov. Because what happens there is that in criminal law you need proof beyond a reasonable doubt. Right. Now what happened in the Zadorov trial? There were two against one in one of the appeals—I no longer remember at what stage—two against one to convict him. And everyone wondered: if there is one judge who thinks he should be acquitted, why does that itself not constitute reasonable doubt? After all, you need near certainty; you need beyond any reasonable doubt in order to convict. Now here is a judge, a legal professional, who saw the evidence, who is deeply immersed in the matter, and one out of three said that in his opinion the man is innocent. So how can such a thing not be enough to constitute reasonable doubt, and therefore to acquit him, even though two judges convicted him?

[Speaker D] Did they say he was innocent?

[Speaker F] Because reasonable doubt—reasonable doubt—is about the case itself, not about opinions on the case itself. What do you mean?

[Rabbi Michael Abraham] But if one judge was not convinced by the case itself, that means the case itself contained reasonable doubt.

[Speaker F] For that judge. Right. Fine, but the fact that one qualified person

[Rabbi Michael Abraham] one out of three was not convinced by the facts—doesn’t that mean that the facts were not sufficiently unequivocal?

[Speaker F] I don’t think you can assess doubt based on a judge’s opinion. You assess doubt within the case itself; otherwise you would never be able to convict by a two-to-one vote.

[Rabbi Michael Abraham] Right. Exactly.

[Speaker F] You would never convict.

[Rabbi Michael Abraham] Right. In criminal law, if you want beyond a reasonable doubt, then conviction would have to be unanimous. Right, that is exactly the argument.

[Speaker F] No, no, but again I’m saying that “beyond a reasonable doubt” is an analysis of the case itself, not an analysis of what people think about the case.

[Rabbi Michael Abraham] What do you mean? That’s just a formal distinction. What difference does it make? If there is one judge who was not convinced by the facts, that means the facts are not unequivocal.

[Speaker D] No, but couldn’t it simply be that this one judge just holds that the facts—meaning, he thinks you need ninety percent and you reached eighty-five percent?

[Rabbi Michael Abraham] Ah, the only answer I found—and in my opinion it doesn’t fit all the cases, obviously not all the cases—is in a situation where according to the two convicting judges it is beyond a reasonable doubt—which is obvious, always, otherwise they would not convict—but the acquitting judge does not acquit because he is convinced that the defendant is innocent; rather, as far as he is concerned there is still reasonable doubt. He too thinks one should convict, but there is reasonable doubt. In such a situation I can understand that this is called a case where there is no reasonable doubt, because according to the majority of opinions there is no reasonable doubt here. But if the third judge truly acquits—not because he still has reasonable doubt; he acquits not on account of doubt, he acquits completely—there, in my opinion, that is called reasonable doubt. To not view that as reasonable doubt, I don’t really understand why that would be correct—what is the logic in that? That is somewhat similar to what we spoke about earlier. All right, but that is a different discussion. Fine.

[Speaker D] In any case, back to our issue—but one question for a moment about majority. If in the end you are not relating to it—that is, you ask whether she had intercourse or did not have intercourse, and in the end that’s not the question; in the end you relate to whether she is forbidden or not forbidden—then in what sense is this a majority? You don’t really have a majority. Meaning, it isn’t correct to define it as a majority.

[Rabbi Michael Abraham] Which thing? What is “this”?

[Speaker D] The question, say, that you made there with the probability tree. At the top you say I have a majority on the question

[Rabbi Michael Abraham] whether it happened before the betrothal or after the betrothal. Right. And if it happened after the betrothal, the question is whether it was injury from an external blow or intercourse. Right. And I have a majority both in the first part and in the second part. What is the problem?

[Speaker D] Yes, no, I mean, you have a majority, but not on the prohibition itself. Meaning that you make a prohibition of what?

[Rabbi Michael Abraham] But these are the questions that determine the prohibition; on each of them I have a majority.

[Speaker D] That’s not true, no, but when you said, for example, that with doubt that is negative, only then I—I mean I do sort of go according to the doubt, I do rule according to negative doubt and not according to negative majority—it’s simply because

[Rabbi Michael Abraham] I didn’t say that, I didn’t say that. What I said is that even in negative situations I still look at it probabilistically. Meaning, when I have negative doubt, then I actually know nothing, but I assume there is fifty-fifty here, and the statistical calculations are done by multiplying probabilities. So if I have a double doubt, then half times half is a quarter, even though actually I have no information at all. That is negative doubt, but despite its being negative I assign a probability and multiply probabilities. And what I’m claiming about negative majority is that with negative majority you cannot multiply, because you don’t have numbers.

[Speaker D] Because you have a sort of range from fifty-one to ninety, and that—

[Rabbi Michael Abraham] Exactly. And the results can change, as I showed you in the example; it depends how large the first majority is and how large the second is, and that can reverse the whole picture. Clear. Therefore, when the majority is a negative majority, you cannot multiply.

[Speaker D] I understand, simply because you don’t know.

[Rabbi Michael Abraham] Exactly. Therefore the concept of a majority of a majority has to be applied very carefully in a case of negative majority. Negative. And almost all majorities are negative. There are very few cases where I know the statistics. Today, by the way, in our generation, when we are more accustomed to checking things in the field and conducting surveys and drawing statistical conclusions, we may also have numbers on certain things. But certainly in the time of the Sages it was not like that. They did not know how many women are infertile by nature out of all the women in the world. They only knew that it is a minority. But it could be that we would do statistics and even have a number.

[Speaker D] Yes, no, I mean, if you know it’s eighty-eighty, then if you have some indication of a certain majority then yes, you can do it.

[Rabbi Michael Abraham] I’m saying, if it is a positive majority, then certainly you can multiply.

[Speaker D] No, I mean even if it is a negative majority, but a negative one where you know more information. Meaning, you don’t know fifty-one to ninety-nine; you know it is between seventy-five and ninety.

[Rabbi Michael Abraham] Okay, fine, these are gray cases, obviously. Meaning, if you can create bounds and they still give you an unequivocal answer, then fine, do that. In any case, that is regarding the types of doubt and the types of majority, negative and positive, and the multiplications we make as a result. That is the general introduction. Now I want to move on to a discussion about agunot—women whose husbands are missing and who therefore cannot remarry. Because we will come back to these things; I will want to apply them to agunot and to other cases, double doubt in other cases, double majorities. So I want to—I’ve finished this introduction about double doubt and double majorities and things like that. Now an introduction to agunot. There are many leniencies that we know in permitting agunot. Yes, a woman whose husband disappeared—we don’t know what happened to him—so the woman is in a very difficult situation. On the one hand she is a married woman, because we don’t know what happened to her husband, so her husband is still possibly alive or presumed alive, and therefore we cannot permit her to remarry. On the other hand, the husband is not here. Meaning the woman is basically sitting alone in her house; she cannot remarry and she has no husband. She both has no husband and is not free to remarry. That is the condition of being an agunah. Therefore even today in our era, a husband’s refusal to give a divorce is also called aginut, because in truth the situation is similar. The husband is here, but in effect she has no marital relationship from the husband, and she also cannot remarry because she is a married woman, so it is like aginut. In such a situation the Sages adopt very far-reaching leniencies. For example, they permit a woman to remarry on the basis of a single witness. Even though this is a matter of forbidden sexual relationships, and in such matters one matter of forbidden sexual relationships is not established by fewer than two witnesses, so by strict law it should require two witnesses. But the Sages determined that one witness is enough. I won’t go into it now; in the simple understanding this is a rabbinic rule, but that doesn’t matter at the moment, because there is the principle that the woman checks carefully before remarrying, and there are disputes in Maimonides’ view, and so on. The accepted view is that this is a rabbinic enactment. But they permit a woman to remarry on the basis of one witness. Other leniencies: for example, they permit a woman to remarry even on the basis of hearsay testimony. In regular testimony, hearsay is invalid; that is secondhand testimony. Yes, you heard it from someone else—that is not testimony. In testimony for a woman, hearsay is valid; they permit her to remarry.

[Speaker F] The testimony is about the husband’s death? Yes, yes.

[Rabbi Michael Abraham] Now other leniencies: for example, testimony of a woman, a slave, a minor, a non-Jew, all kinds of things like that—at least a non-Jew speaking casually without intent to testify—then too they permit the woman to remarry. Meaning, all invalid testimonies and all the restrictions on the laws of testimony are basically set aside here. The Sages were extremely lenient, extremely lenient with the woman in a very radical way.

[Speaker E] If I remember correctly, not regarding inheritance, right? No?

[Rabbi Michael Abraham] No, no, regarding permission to remarry. That is splitting the statement.

[Speaker E] So not in monetary matters?

[Rabbi Michael Abraham] Yes, for money that is something else. They permit her to remarry, but they do not allow descent to inheritance, as the Talmudic text in tractate Yevamot says. In any case, that is basically the case of splitting the statement. In any case, these leniencies on the one hand are very impressive. On the other hand, we know that the Sages are very hesitant to permit an agunah to remarry. Yes, the case of Ron Arad, for example, which is very well known—it was as though the whole world was already convinced that the man was completely dead, that there was no chance he was alive, and nevertheless the rabbinate did not permit Tami Arad to remarry until a very, very late stage. I don’t know exactly when, but for years and years and years, when it was already completely clear to everyone that the man was dead, the rabbinate did not permit her to remarry. And so it is in many, many cases. Meaning, there is a very great stubbornness on the part of the Sages, the halakhic decisors, the rabbinical judges, not to allow an agunah to remarry. And the question always arises: how does this fit with the far-reaching leniencies that we find in testimony for a woman regarding an agunah? A second question that arises here is: how can the Sages uproot something from the Torah? To permit a married woman—one of the gravest prohibitions there is—the Sages uproot something from the Torah, and they uproot it through positive action. They permit the woman to remarry even though by strict law she is a married woman. How can one do such a thing, uproot something from the Torah through positive action? A third question: there are all kinds of very well-known stringencies in permitting agunot. Not only leniencies, but also stringencies. For example, they require seeing the face with the nose specifically. You have to see his face directly, with the nose—not only this part or that part—and only then do they believe you when you say that the husband died. If you say, listen, I know him, I’m telling you it’s him, that doesn’t help. Only if you saw him in a very unequivocal and direct way. Meaning, in short, there are quite a few extreme stringencies in permitting a woman to remarry. Now it is very clear why to be lenient, because the woman is in severe distress. It is also very clear why to be stringent, because at the end of the day you are permitting a married woman to the general public, which is one of the gravest prohibitions. But then decide—why? Then go with strict law. What is the difference? When are you lenient and when are you stringent? And in the end, as a result, yet another question: how can the Sages uproot something from the Torah through positive action at all? That is the question. Those are the three questions. And I want to say in the simplest way possible—perhaps one more stringency that will be important for us later—the law of water with no visible boundary. Water with no visible boundary: a person drowned in the Atlantic Ocean, all right? Now there is—I was there, the man didn’t come out anywhere, he drowned, no question. I would stake my head on it that he drowned, but they do not permit his wife. Why? Water with no visible boundary means that I do not see the edge of the body of water, so perhaps he came out somewhere I did not see, and therefore they do not permit his wife to remarry. By the way, according to most opinions this is a rabbinic rule, but they do not permit his wife to remarry. But in water with a visible boundary—say a swimming pool—someone drowned in a pool, they don’t find his body, never mind, if I see the boundaries, all the boundaries—by the way you have to see all of them—there I permit his wife. Why? Because if he had come out, I would have seen; I can see the whole shore, okay? But in water with no visible boundary, it is possible that he came out without my seeing him; therefore, even though it is basically clear that he drowned, we do not permit his wife to remarry. This too is a very extreme stringency in the laws of aginut.

[Speaker D] When you say “clear,” you mean because the chance that he actually got out and you didn’t see is negligible.

[Rabbi Michael Abraham] Yes, very small. So now I want to say in general—and one can argue about the applications—but in my opinion the general principle is the following, and this is a great principle in the laws permitting agunot. All the leniencies in permitting agunot never deal with the probability that the husband died. Meaning, if there is any chance that factually the husband did not die, nobody will permit the woman. All these leniencies were stated in a situation where factually it is clear to me that the husband died, and all that is holding me back are merely formal halakhic limitations, such as the need for two witnesses rather than one. That is a scriptural decree: “By the testimony of two witnesses a matter shall be established.” But there are many situations in which a trustworthy person comes and tells me, listen, I saw him, he is dead, there’s nothing to do about it, and clearly he has no interest in lying—so on the factual level I have no doubt that the husband died. But there is a rule in Jewish law that in matters of forbidden sexual relationships you need two witnesses.

[Speaker J] Is that rule because of a factual issue?

[Rabbi Michael Abraham] Wait a second. I have a formal limitation. Where there is a formal limitation, the Sages waive it. Or when the witness is invalid, for example, a woman testifies. With a woman, there is no reliability problem with the woman. Two women come and testify—there are two witnesses—but the witnesses are invalid because they are women. Now there is no problem with the reliability of the testimony; it is simply a scriptural decree that disqualifies the testimony of a woman. Therefore we are lenient and accept the testimony. Why? Because factually it is clear to me that the husband died. All that is holding me back here is a formal limitation. With formal limitations, the Sages say: do not take them into account; be lenient, so long as you are convinced that factually the husband died. When is there stringency? Stringency exists in the same place where there is some chance that in fact the husband did not die. There, even if the chance is small, I will be stringent and under no circumstances permit the woman. And I think this is the line dividing the stringencies in testimony for a woman from the leniencies in testimony for a woman. The leniencies in testimony for a woman are a waiver of formal limitations. The stringencies in testimony for a woman are concern for any small possibility that factually the husband did not die. Now of course the boundary is not sharp. In water with no visible boundary they were stringent. Why? You’ll say: factually, de facto, it is clear that he drowned. But no—since there is some factual possibility that he didn’t, that he came out and we did not see him, therefore even though the chance is small, we do not permit the woman. On the other hand, one witness comes and testifies that the husband died, and he saw the face with the nose, saw everything directly and clearly—we permit her to remarry. What, is there no chance that the witness made a mistake? Here too there is some small chance that factually the husband is alive. So I say: the boundary is not sharp, but still the approach is always a commonsense one. On the commonsense level, if a trustworthy person comes and tells me that the husband died, then factually the husband died. There is always some chance that not; even when two valid witnesses come, there is still some chance that not—we spoke about statistical evidence in law; we discussed that at length there. So even when two witnesses come—but from the standpoint of a reasonable person, we regard this as one hundred percent. If we regard it as one hundred percent factually, then the requirement for two witnesses or for valid witnesses is a formal limitation. The claim of the Sages is that in water with no visible boundary we do not view it as one hundred percent. True, the probability is very high, but it is not one hundred percent. We do not regard it as one hundred percent. And that is a matter of perspective; nowhere is it one hundred percent. So the question is how one views it, what a reasonable person sees as one hundred percent. Therefore in water with no visible boundary they are indeed stringent—by the way, only rabbinically, meaning that by Torah law she really is permitted, but rabbinically they are nevertheless stringent. Yes, what was the question?

[Speaker D] You are assuming here, right, I…

[Speaker C] He says, as a matter of fact—sorry, yes, right.

[Speaker D] You’re assuming there’s really no greater chance that there would be two witnesses—that two witnesses tell the truth more than one witness, right? You’re basically saying it’s the same thing.

[Rabbi Michael Abraham] It could be that there’s a greater chance, but one witness is also good enough. The fact is that in matters of prohibitions we accept one witness. To permit eating pork, that’s good enough.

[Speaker D] So why aren’t you stringent? Okay, you’re saying that’s the fine line, kind of?

[Rabbi Michael Abraham] Yes. I’m saying I really don’t know how the Sages determined when a probability counts as formally halakhic and when it’s a probability at the factual level. Notice that we made this same distinction in the lesson, in the lessons about statistical evidence in law. There too we talked about the prison—remember? A hundred prisoners in prison. One of them didn’t participate. There’s a one percent chance that the person standing before me didn’t participate. A ninety-nine percent chance that he did participate, and still we don’t convict. But if two witnesses come and say he murdered someone, there’s still a chance they were mistaken. Even, say, a two percent chance they were mistaken—and there we do convict. So we talked about the fact that there are situations where the assumption is that this is the factual truth. There’s always some theoretical chance that there was an error here, but the reasonable person sees this as factual truth. Understood.

[Speaker D] And there is

[Rabbi Michael Abraham] a situation where it’s not a statistical difference; the probability isn’t smaller, but the reasonable person sees it as a certain situation. Understood.

[Speaker D] But with regard to uprooting something from the Torah, it’s like…

[Rabbi Michael Abraham] So now I’m saying: this distinction answers all three questions you asked. It explains the extreme leniencies, it explains the extreme stringencies, and we’re left only with the question of how the Sages can uproot something from the Torah.

[Speaker F] No, no, I don’t—I don’t have a clear picture.

[Rabbi Michael Abraham] My answer is that here the Sages are not uprooting anything from the Torah, because they did not permit a married woman to the public. They permitted an unmarried woman to the public. Because we know as a matter of fact that her husband is not alive. So she isn’t a married woman. Why is that factual? Because factually we became convinced that he died.

[Speaker A] But the Torah says that factually it’s only two witnesses.

[Rabbi Michael Abraham] No, the Torah doesn’t say factually; the Torah says halakhically. Factually the Torah says nothing. Facts I know on my own even without the Torah. Facts are determined by me, not by the Torah. The Torah determines what Jewish law requires. That’s the claim. And the claim is that if, from my perspective, he is factually dead, then I’m not permitting a married woman to the public. I’m only being lenient in the laws of proof, in the laws of testimony. That’s entirely passive omission. I’m basically saying: leave it, I don’t need the two witnesses. The permission for the woman is permission as an unmarried woman, so this is not an active permission of a forbidden sexual relationship. I permitted something in the laws of proof; I didn’t permit a forbidden woman to the public, because she isn’t forbidden, because factually she isn’t a married woman—her husband died. Therefore this is not considered that the Sages are uprooting anything. Uprooting something from the Torah is by positive action.

[Speaker K] Maybe we should sharpen it and say: factually maybe she isn’t a married woman, but halakhically she is. Meaning that the Sages permitted something prohibited by the Torah.

[Rabbi Michael Abraham] No. If factually she isn’t a married woman, then what does halakhically mean? Halakhically she isn’t a married woman, because halakhically I need to accept that on the basis of two witnesses, and that is what the Sages canceled.

[Speaker K] But halakhically, right now, as long as there aren’t two witnesses, she is a married woman. So the Sages canceled that?

[Rabbi Michael Abraham] No. Halakhically, as long as there aren’t two witnesses, I can’t claim that she is

[Speaker K] not a married woman.

[Rabbi Michael Abraham] Right, so does that mean she is a married woman?

[Speaker F] No, it doesn’t mean that, it

[Speaker K] means I can’t claim that she isn’t a married woman. That’s from doubt. Maybe it becomes a doubt; maybe you can explain it simply: it becomes a doubt, and according to Maimonides a Torah-level doubt is treated stringently by Torah law. No, no, that’s how it is. Even according to the one who says a Torah-level doubt is treated stringently only rabbinically, that’s how it is. Is one witness considered a doubt? What does it mean that one witness is considered a doubt? Does one witness, by Torah law, have enough weight to make this doubtful? Why?

[Rabbi Michael Abraham] Because what—does it not remove something from its established status? It obligates him to take an oath.

[Speaker K] So halakhically there’s a problem here.

[Rabbi Michael Abraham] Why is that a problem?

[Speaker K] Because halakhically, as long as you don’t have two witnesses—

[Rabbi Michael Abraham] No. As long as I don’t have two witnesses, I can’t say she isn’t a married woman. It’s not true that she is a married woman. Factually she isn’t a married woman. Okay, I can’t behave that way. Fine. So the Sages say: you can behave that way. But it’s not that I permitted a married woman to the public.

[Speaker D] To uproot that halakhic limitation—as if the Sages have the power to do that.

[Rabbi Michael Abraham] Yes, that’s the claim.

[Speaker A] But why is one witness, factually, considered enough for me to know that it’s true?

[Rabbi Michael Abraham] So I said: because one witness is also enough to let you eat pork based on his word, right?

[Speaker G] Because he is believed in matters of prohibitions.

[Rabbi Michael Abraham] Exactly. So you see that it’s good enough even for serious prohibitions, serious Torah-level prohibitions.

[Speaker G] And that’s actually a good combination with uprooting something from the Torah, because it says, “By the mouth of two witnesses a matter shall stand,” and then the Sages come and say one witness is believed in matters of prohibitions.

[Rabbi Michael Abraham] The question is why we shouldn’t say that one witness is believed in matters of prohibitions because “a matter” applies only to monetary law and matters of forbidden sexual relations.

[Speaker G] Right, so—

[Rabbi Michael Abraham] Why did the Sages make things lenient here in matters of prohibitions? There? It’s Torah law that one witness is believed in matters of prohibitions.

[Speaker H] But with regard to a chained woman, not all prohibitions. With regard to a chained woman, the Sages say that in a case of potential chaining, regarding one witness, they don’t say halakhically or factually—they say that giving up on the requirement of two witnesses is a leniency because of the chaining.

[Rabbi Michael Abraham] Correct. Because of the chaining, they forgo the laws of proof, but not that they permit a married woman to the public. Of course they’re being lenient here. The leniency is because of the chaining; it’s just that the leniency is a halakhic leniency—they are giving up halakhic limitations—but not that they are permitting a married woman to the public because of chaining.

[Speaker G] It’s also true with a woman whose bill of divorce is delayed. Someone who signs an authorization on behalf of the Rabbinate can also be just one person, even though usually they have someone else sign too, but even one person can sign it.

[Rabbi Michael Abraham] Sign what? I didn’t understand.

[Speaker G] An authorization for a bill of divorce.

[Rabbi Michael Abraham] Well, what do you mean, before one person? Yes. Ah, fine, okay.

[Speaker G] Yes, no, it’s the same thing, practically it’s the same thing. It’s a formality that has to be maintained.

[Rabbi Michael Abraham] Okay. Yes, someone who brings a bill of divorce from overseas as well—that’s also one witness: “It was written in my presence and signed in my presence.”

[Speaker A] And the fact that the Sages permitted reliance on one witness in matters of prohibitions—that isn’t factual.

[Rabbi Michael Abraham] The Sages didn’t permit it; the Torah permitted it.

[Speaker A] The Torah, sorry. So again, that’s not factual, it’s halakhic. So what’s the proof that…

[Rabbi Michael Abraham] Of course. But obviously, if there were a factual concern that I was eating pork, the Torah would not permit it.

[Speaker A] So is the Rabbi mixing fact and Jewish law?

[Rabbi Michael Abraham] I’m saying that in the direction of permission, obviously—after all, if they tell me that on the basis of one witness I may eat this, then it’s obvious to the Torah that factually this is not pork, otherwise it would not permit it halakhically.

[Speaker A] And if the Torah says that you need—again, if the Torah says, “By the mouth of two witnesses a matter shall stand”?

[Rabbi Michael Abraham] Then that’s for stringency, for stringency with respect to permission. The Torah says that even if it’s factually clear to you that the woman is not a married woman, treat her as if she were a married woman if there were not two witnesses.

[Speaker G] Yes, that comes out completely as a stringency. Meaning it’s not a leniency. It comes out that even if you factually know something is true, you’re still bound by a certain formality.

[Rabbi Michael Abraham] You know, in the formal sense this is an absurd thing. Suppose two witnesses come and testify that a certain man died. We permit her to remarry, and then he comes back—that’s a story in the Talmud—and if he comes back, she is his wife. It doesn’t help that the religious court told her that she could remarry, because factually she is his wife. Whether she is his wife depends on whether he is factually alive, not whether he is halakhically alive.

[Speaker D] But that would mean that if, say, just now an invalid witness testifies, for example, about a certain woman whether she is a married woman or not, and according to Jewish law, let’s say, that doesn’t count—her husband is alive. Okay, so okay, so let’s say one person testifies that her husband is alive.

[Rabbi Michael Abraham] No, sorry, I—no, her husband is alive.

[Speaker D] Yes, but I want to understand cases without a chained woman.

[Rabbi Michael Abraham] Meaning, say, a case of a matter of forbidden sexual relations with fewer than two.

[Speaker D] No, but if in the end you’re saying that one witness really does in practice clarify things for me, then I should always relate to it that way, even without a chained woman, because in the end—

[Rabbi Michael Abraham] I know that halakhically it is limited.

[Speaker D] Okay, but what? Will you kill a person because of that halakhic limitation even though in practice there’s really nothing there?

[Rabbi Michael Abraham] What does it mean to kill?

[Speaker D] I don’t know, say you punish him, I don’t know, let’s find some example—

[Rabbi Michael Abraham] That someone had relations with this woman and then I say: you had relations with a married woman. Why? Because she is halakhically a married woman, even though factually I think she isn’t a married woman.

[Speaker D] Say that, yes.

[Rabbi Michael Abraham] Fine. So first of all, maybe I wouldn’t kill him. And second, a person is not executed unless he was warned and accepted the warning upon himself, and then maybe he does deserve to die. Understood. Okay, let’s stop here. We’ll continue next time.

[Speaker G] Wait, how many lessons do we still have on doubt and probability?

[Rabbi Michael Abraham] I don’t know, maybe three more. Two, three.

[Speaker G] Oh, because at the beginning of the lesson you said this was the last chapter.

[Rabbi Michael Abraham] Last chapter, but a chapter is a unit of several lessons.

[Speaker L] Rabbi, doesn’t the adjudication paradox prove—or at least support—that all the rational considerations we give ourselves and others aren’t really, in the end, what determines things, and aren’t really the important ones?

[Rabbi Michael Abraham] In the end—

[Speaker L] In the end, the feeling and the general sense of what’s going on and what should be done are much more important.

[Rabbi Michael Abraham] The consideration behind why we follow the majority, bottom line, is probably a matter of legal simplicity, that’s all. No need to take it too far. Because you can always break the discussion down into many, many sub-questions, and for each one start checking whether there is a majority among the judges or not, and there could be different ways of breaking down the case. It becomes very complicated if we go by those considerations, and therefore it definitely makes sense to determine it more simply—that the majority in the bottom line is what decides.

[Speaker L] But that’s something after the fact. In a perfect world, supposedly, we really should go into it.

[Rabbi Michael Abraham] Could be, yes.

[Speaker L] But if not, then it strongly supports a causal approach in everything the Rabbi presented here once. Fine. Rabbi?

[Rabbi Michael Abraham] Yes.

[Speaker G] In the legal system in this country, is there such a thing as a jury?

[Rabbi Michael Abraham] I don’t think so, not that I know of.

[Speaker G] Meaning every criminal case here is decided only by judges. Judges? Yes. Not like in America, for example.

[Rabbi Michael Abraham] No, here—as far as, again, there are legal experts here, I mean—but as far as I know, there are no juries in Israel. Rabbi?

[Speaker A] Yes. Do you have an email where I can send the Rabbi questions, just in general?

[Rabbi Michael Abraham] There is a responsa section on the website.

[Speaker A] On the website, but I’m on NetFree, so I don’t know if it’ll work for me.

[Rabbi Michael Abraham] So there is an email: mikyav.

[Speaker A] How—miky what?

[Rabbi Michael Abraham] M-I-K-Y-A-B.

[Speaker G] It’s written on the website. There’s an email there.

[Rabbi Michael Abraham] Yes, but he can’t get into the website. At gmail dot com. It’s time to get rid of your NetFree.

[Speaker A] Why?

[Rabbi Michael Abraham] Because it’s political filtering, it’s not filtering—

[Speaker G] It really doesn’t

[Speaker A] filter

[Rabbi Michael Abraham] any

[Speaker G] thing.

[Speaker A] In practice it protects you, doesn’t it?

[Rabbi Michael Abraham] Then get some other protection instead of the political Haredi protection. Get protection that protects against the things you actually need protection from. The Haredi protections mainly protect the companies that run the filters.

[Speaker L] What, the Rabbi’s website is blocked on NetFree?

[Rabbi Michael Abraham] That’s what I’ve heard from many people, yes.

[Speaker G] Not only that—there are many kosher sites that are blocked. There’s no justification for it, all this supposed protection.

[Rabbi Michael Abraham] No, that’s a question of the quality of the filter. But there is agenda-driven filtering there, filtering against opinions, filtering against arguments. It’s not filtering only against pornography, violence, or things that the other filters deal with. It’s Haredi filtering. Yes, that’s what I said. There are many very nice, very good websites where every child ought to hear all the arguments on every issue for himself. Definitely. Certainly no one else should decide for him. Okay, that’s a discussion we can have separately.

[Speaker A] I hope no stumbling block comes from it.

[Rabbi Michael Abraham] Of course a stumbling block comes from it. A stumbling block comes from everything. More stumbling blocks come from NetFree than from the other filters. NetFree produces foolish people, uneducated people. People who are captive to their opinions, people who don’t form an independent position—is that not a stumbling block? Fine, understood.

[Speaker K] On the subject of separating the combination of doubts and all that—if in each question on its own there is a practical difference, then do we separate the doubts, or do we still combine them in the bottom line?

[Rabbi Michael Abraham] If the questions are different questions, then there is room to discuss separating them, yes.

[Speaker K] What do people usually do? It’s a very broad topic.

[Rabbi Michael Abraham] The whole method of combinations… I don’t know how to give a general answer.

[Speaker K] Very complicated questions. I don’t know how to give a general answer. The method of combinations in the Mishnah Berurah, where they combine from this doubt the majority that exists here and from that doubt—

[Rabbi Michael Abraham] Yes. This concept of combining opinions is something a bit different. It’s not exactly a double doubt. These are supporting factors for leniency; it’s not exactly a double doubt. Yes, but that’s actually an interesting question. Maybe we should devote a lesson to that too.

[Speaker G] Maybe, maybe—

[Rabbi Michael Abraham] really to these supporting factors for leniency.

[Speaker K] That’s right, right.

[Speaker G] It’s actually a very nice question regarding double doubt and, for example, majority upon majority. If it ever comes up in the laws of niddah, these are day-to-day things; there are so many accumulations there. Maybe it really is worth talking about. Why not? That too is Torah.

[Speaker K] I’m saying it’s like when sometimes the Mishnah Berurah takes two unrelated topics and there’s a majority. Now once that topic was decided by a majority of opinions, why are you now taking the minority there that wasn’t ruled like—and combining it with some other minority… I said, these are supporting factors.

[Rabbi Michael Abraham] The concept of supporting factors for leniency is a very interesting question. You can look at my article on the website. There’s an article called “What Is a Leniency?”

[Speaker K] “What Is a Leniency” with an h or with an aleph?

[Rabbi Michael Abraham] “What is” with an h, “leniency” with an aleph. Something like that. Search for it and you’ll find it. If not, ask me and I’ll send it to you. There I talk a bit about these things.

[Speaker K] Thank you.

[Speaker G] Don’t forget to send the column about law, what we talked about in the lesson.

[Rabbi Michael Abraham] Okay, בלי נדר.

[Speaker G] Fine, yes, in the WhatsApp group if possible, okay. Just out of interest. Yes.

[Rabbi Michael Abraham] Which group?

[Speaker A] Just a question—does the Rabbi know Rabbi Dovaleh Davidovitch? Do people appreciate Rabbi Dovaleh Davidovitch? No. Okay.

[Rabbi Michael Abraham] Okay, fine, friends, happy Purim—

[Speaker G] Good night, thank you very

[Speaker C] much.