Paradoxes and Contradictions in Halakha, Lesson 6

[Rabbi Michael Abraham] We dealt a bit with dichotomies, and after that I started talking a bit about paradoxes and their significance. I presented several examples of paradoxes, but what they really mean is that if they exist inside the system, then there’s a problem: the moment there’s a paradox inside the system, you can derive any answer from it. Any halakhic claim can basically be proven from within the system, if I’m talking about Jewish law, and that of course completely disrupts the system. And the question is how one deals with such a thing. So maybe I’ll mention one more thing that I’ll need later on. We talked a bit about positivism and about relating to a legal or halakhic system as some kind of normative system, as a kind of closed logical structure. There is a set of premises, rules of inference, and the system is only what can be derived from the premises by means of the rules of inference. That’s it, nothing more. That is a positivist conception, and that conception is what leads to the problematic nature of the existence of paradoxes, because if, as I said earlier, a paradox exists, then by logical tools I can derive any conclusion from it. And if I don’t see logic as the exclusive tool, the only thing that defines what is legal and what is not legal, or what fits the system and what does not fit the system, then this is less troubling. So wherever common sense says yes or no, okay, then you can maneuver as well; completely unreasonable answers, even if someone manages in a formal logical way to show them from within the system, will be thrown out. And that’s what people do in practice, of course. But in the positivist conception that becomes very problematic. And I think by the end I had already managed to move on to Jewish law and say that in Jewish law too there are the same hesitations, the same questions: can we really see Jewish law as some sort of axiomatic system, right? In a positivist form, even if we say that not everything is available to us. Meaning, there is no claim here saying that the halakhic decisor is a computer, or that he has a set of assumptions, he knows all the assumptions and the rules of inference, and he simply applies them mechanically in order to reach a result. I don’t think anyone would seriously describe the work of a halakhic decisor that way. The question is whether on the principled level such a system exists, one that really describes the halakhah. The true halakhah, right? What should have been if everything were in our hands. Could it be defined as some kind of axiomatic system of this sort? And to that I think many people would answer yes. That in principle, yes, there is some set of rules from which everything can be derived. True, we don’t know all the rules, not everything is formulated, not everything was received by tradition, so we have to work with more flexible tools as well. But that is only our limitation. I also mentioned, I think, the analogy to science. There too the same question arises: whether in the scientific world, in the scientific context, there is likewise some system of laws that describes all of reality completely, accurately, and fully, even though clearly today we do not have it. Nobody claims that today we know such a system and can use it to explain or predict absolutely anything. But the question is whether such a system exists in principle. In general, can the scientific world be described as some kind of axiomatic system, and we are merely uncovering more and more of it, discovering more and more of it.

[Speaker B] If we’re already talking about the general system as we know it, then you could always say that there’s an axiomatic system that is simply the list of all the particulars, all the existing halakhic rulings.

[Rabbi Michael Abraham] Yes, we talked about that in the series on intuition. There I discussed thinking with rules and thinking with particulars, and I spoke exactly about that. I said that yes, formally you can say that, but when we talk about rules, that’s not what we mean. We’re talking about something that exists in the meta-language. No, that’s not a halakhic statement; by definition it’s a meta-halakhic statement. Meaning, “women are exempt from all positive commandments that are time-bound” is not a halakhic statement. The halakhic statement is that women are exempt from sitting in a sukkah. That is a halakhic statement. When I move to the rule from which the halakhic statement is extracted, then I’m talking about Jewish law, not speaking Jewish law. When I speak Jewish law, I say something is forbidden, something is permitted, something is obligatory — that is a halakhic statement. When I talk about Jewish law, I’m describing it; my subject is Jewish law, not reality — what is forbidden and what is permitted. So… so basically if I return to Jewish law, I want to demonstrate what this means. Let’s take an example of some halakhic loop. Tosafot in tractate Bava Metzia asks a question based on several assumptions and on one of the opinions — not according to everyone, but you can construct a certain setup within which this question will arise. One’s own lost object, one’s father’s lost object, and one’s rabbi’s lost object. Right, there are three lost objects floating in the river, and I can save one of them. If I take one, the other two will float away on their path. So I can’t save them all; I can save one. The question is: whom do I save? My own lost object, my rabbi’s lost object, or my father’s lost object? So there are three rules there, right? My own lost object takes precedence over my father’s lost object — and again, this is some assumption, it doesn’t matter — that one is not commanded to honor his father at the expense of his own money. There’s a dispute in the Talmud, but in fact that is how Jewish law is ruled, and Rabbi Chaim is well known for this. Yes, the student who came to ask Rabbi Chaim whether he should travel by train to visit his parents — after all, that costs money — and he answered him: you are not supposed to honor your parents at your own expense, only at theirs, not yours.

[Speaker B] From his means. From his means, yes. Wait, the son does not honor — the son does not honor from his own money?

[Rabbi Michael Abraham] Yes, from his means no, from exertion yes. So he said, but traveling by train costs a lot of money. So he said, correct, then go on foot.

[Speaker B] But that will also cost money, because he’ll need food and drink for the whole way.

[Rabbi Michael Abraham] Food and drink he needs anyway.

[Speaker B] No, but he won’t be able to work. Fine, okay.

[Rabbi Michael Abraham] In any case, for our purposes: there is one principle that one’s own lost object takes precedence over one’s father’s lost object. The second principle is that one’s father’s lost object takes precedence over one’s rabbi’s lost object. And the third principle is that one’s rabbi’s lost object takes precedence over one’s own lost object. Non-transitive. Okay? Meaning this relation among every two lost objects does not satisfy the rule that if A is greater than B and B is greater than C, then A is greater than C. That’s the relation of transitivity, and here transitivity breaks down. So what do you do in such a case? Now this thing is — it’s hard to define it — it isn’t exactly a paradox, it’s a conflict. It’s a third thing. It’s a conflict because what’s the paradox here? Fine, no one is preferable to the other; it’s a non-transitive relation. We know quite a few non-transitive relations in reality. Being someone’s father is not a transitive relation. Right? Reuven is Shimon’s father and Shimon is Levi’s father — that doesn’t mean that Reuven is Levi’s father; he’s his grandfather.

[Speaker C] No, it’s a matter of priority; that’s something else. What? A is greater than B and B is greater than C, C is greater than A.

[Rabbi Michael Abraham] So as I said, who says? “Greater than” is a transitive relation, but here we’re not talking about “greater than”; we’re talking about a different relation, and that different relation doesn’t have to be transitive. There is nothing paradoxical about that in itself.

[Speaker C] Yes, it’s a kind of “greater than”; it’s more, it’s more than that.

[Speaker D] It’s like people in a circle; it’s the relation of being to the left of.

[Rabbi Michael Abraham] Yes, exactly. Being before someone is apparently not parallel to being greater than something. Meaning, okay, so what happened? There are relations that are not transitive. That by itself is not a paradox. What bothers us here is not the paradoxical nature of the matter but the inability to make a decision. And this is not the question of the consistency of the system; rather it’s the second question I mentioned, the completeness of the system. Meaning, is the system complete — can it give an answer to every question? Consistency is the question whether the answers will be consistent, whether you cannot say a proposition and its opposite from the same system. Here I’m talking about the second problem, the problem of completeness. And then — yes?

[Speaker B] If the problem is completeness, then how is this different from just another question where there’s simply no halakhah written on it, say?

[Rabbi Michael Abraham] No, essentially it isn’t different. Just a regular value conflict. Saving life and the Sabbath, or a positive commandment and a prohibition.

[Speaker B] After all, before they ruled on it. The halakhah — that it’s not simple, right.

[Rabbi Michael Abraham] Here the point is that in most regular conflicts, for example, there is a solution. Even if they are equal, there is a solution of passive omission is preferable. Why? Say a positive commandment versus a prohibition. But a positive commandment versus a positive commandment, or a prohibition versus a prohibition — ostensibly there’s no way to solve it. But there is a way. We have rules that also tell us what to do when there is no solution. One of them is: passive omission is preferable. And “passive omission is preferable” means that basically you just don’t fulfill the positive commandment; sit back and don’t fulfill the positive commandment. That’s “passive omission is preferable.” But with a positive commandment versus a prohibition there is a solution — meaning you don’t have to get there, because the positive commandment overrides the prohibition. As for positive commandment versus positive commandment, or prohibition versus prohibition, that is for example the solution. Now in this case there is no solution of that kind, because here you have to take one of the lost objects. So is “passive omission is preferable” — save none of them? You have to save one of them, and now there’s no way to decide which one. So here there is some problem of the completeness of the system. Meaning the question is whether the system really ought to have an answer for every situation. Should Jewish law tell us what is right or wrong in every situation? That too is a question. Many people will tell you yes, but honestly I’m not entirely sure I’m so troubled by this. So there’s no answer — what should I do? Draw lots. Drawing lots is also a solution.

[Speaker B] I think what troubles people in this question is not only that by comparison it’s a difficulty, it’s not only because it

[Rabbi Michael Abraham] looks like “greater than,” more than what Shmuel said earlier.

[Speaker B] because it’s something that really does look like a contradiction.

[Rabbi Michael Abraham] נכון, but as I said earlier, I think that the fact that it looks that way — it looks that way. Okay, and still it could be that it isn’t. It could be that this takes precedence over that because of one aspect, and it takes precedence over the other because of a different aspect. Maybe I’ll give another example. There is the paradox of judgment; maybe I mentioned it once. Reuven and Shimon come before the religious court, and Reuven sues Shimon, claiming that he violated something in the contract between them, that he did something the contract obligates him not to do, or that requires him to pay if he does it, doesn’t matter. Now three judges are sitting there and need to decide whether to accept this claim or not. So one of them says — there are two questions before them. One question is whether there was such a clause in the contract. Does the contract in fact require him not to do this or not? That’s a question of contract interpretation. The second question is factual: did he in fact do this illegitimate act, this improper act, or not? One is a question of interpreting the contract, and the other is a factual question. Now the distribution of opinions among the three judges was as follows. One judge said: there is no such clause. So it doesn’t matter what he did. There is no such clause. He didn’t violate the — there is no such clause. The second judge says: there is such a clause, and he violated it. And there is a third judge who says: there is such a clause, but he didn’t violate it, and everything is fine. So there are two who acquit him, right? The defendant. And one who holds him liable. So he should come out not liable. But if you do the calculation question by question, he comes out liable.

[Speaker B] Why liable? Doesn’t he come out tied?

[Rabbi Michael Abraham] No, no, he comes out liable.

[Speaker B] The first judge said he’s exempt. Make the table, right.

[Rabbi Michael Abraham] Make the table, but it comes out that he comes out liable. Meaning, when you check the factual question, you see that there are two against one that he did the act. You check the question of interpreting the contract, there are two against one that there is such a clause. Okay? So if there is such a clause and he did the act, then he is liable; he violated that clause in the contract. So if you examine it question by question, he is liable. If you examine it by the bottom line, no.

[Speaker E] Now the question is whether this thing—

[Rabbi Michael Abraham] By the way, there are many examples like this. Logicians love playing around with examples like this, and just now I saw several such examples in a book by Haim Shapira too. There are tourists’ paradoxes: what do you prefer, where do you prefer to be? The tour guide asks the group, and each person ranks the sites he would like to visit. Now you can offer certain packages — meaning, what do you think of one complete route versus several alternative routes — and you can show that contradictions can arise there. If you check site by site, it turns out that the chosen route is actually rejected by everyone. There are all kinds of such paradoxes where it’s not really a paradox. That’s why I bring these examples. All this means is that there is something non-transitive here. Non-transitivity in some broader sense. Something that, as Arik said earlier, at first glance looked as though it should preserve some transitive relation, but it doesn’t, it doesn’t have to preserve it. And regarding judges, really, okay, one can discuss the question of how one ought to decide. Is it correct to decide by the bottom line of each judge, or is it correct to decide question by question? There is logic to this, and there is logic to that, but there’s no problem here. Meaning, decide what seems more reasonable to you and act accordingly — it’s not a paradox.

[Speaker C] It will be the reasonable judge.

[Rabbi Michael Abraham] In any case, by the way, in the context of judges this really is an interesting point, because my own intuition says to go question by question. I don’t think there’s any system in the world that accepts that. They all go by the bottom line, including Jewish law.

[Speaker C] Why go question by question? If you want to look at it probabilistically… not probabilistically.

[Rabbi Michael Abraham] I’m saying if—

[Speaker C] there’s some dependency between the two in the same person, meaning the same person who says that. It’s written, but he didn’t do it — there’s some kind of dependence between the two…

[Rabbi Michael Abraham] No, there’s no dependence. I’m saying let’s assume there is no dependence. The way I presented the question, there’s not supposed to be any dependence. So what’s the problem? What is it that I really want to clarify? Whether the contract requires doing this or not doing this, and second— in the situation I described there’s no disagreement. Those are the two questions, and each of them expresses his opinion on both questions.

[Speaker C] Okay, so two rulings come out, one on the first question and one on the second. Yes, exactly.

[Rabbi Michael Abraham] That’s exactly what I want, exactly. And really, after all, we need to answer each of the questions when we come to the ruling. So what’s the problem? Then follow the majority on each of the questions.

[Speaker F] You can’t separate his opinion; in his opinion there’s a whole of the two things together, and they come as one unit.

[Rabbi Michael Abraham] Why as one unit?

[Speaker F] Because that’s how he reaches his conclusion.

[Rabbi Michael Abraham] No, the questions are independent; there is no connection between the questions. Interpretation of the contract is one question, and the factual question of whether I’m convinced he did it is a second question. What’s the connection? I have a view on both of those things, but there’s no connection between the two parts. So why does the bottom line matter?

[Speaker C] No, that’s understood, but that wasn’t the question. The dispute is whether, if you say that the judges—

[Rabbi Michael Abraham] I think there’s a lot of logic, a great deal of logic, in going that way. Wait — but neither Jewish law nor legal systems or anything like that—

[Speaker B] The authority of the judges is not to decide factual questions; their authority is to decide the ruling.

[Rabbi Michael Abraham] And for that purpose they need—

[Speaker B] to assess what the factual situation is. No, that’s not right. What do you mean?

[Rabbi Michael Abraham] Why are they there? To clarify reality and also to decide the law. What do you mean? The goal is the legal ruling. After all, they don’t have authority to determine

[Speaker B] what reality is independent of deciding reality.

[Rabbi Michael Abraham] I can’t go to court and ask them to determine for me what reality is. That’s true, but they do have authority to determine reality relative to the legal ruling. They are the ones authorized. All the experts in the world can come and say that this is not the reality; if the judges say it is, that’s what will count. On the legal level — not because they are right, but because on the legal level they are the ones authorized to determine what the reality was. What can you do?

[Speaker C] No, I think that’s true only if there is one decision-maker and three advisers. What? If there’s one decision-maker—

[Rabbi Michael Abraham] No, I’m talking about judges.

[Speaker C] three advisers… No, I mean actual judges. A decision-maker who doesn’t know what’s going on, he receives input from three advisers, then he’ll work the way the Rabbi says. He’ll take the majority here, majority there, and decide that way.

[Rabbi Michael Abraham] But if here it’s the bottom line, we are forced to ask—

[Speaker C] In the end, each judge, what are you— no, there’s no problem, let’s ask them what the bottom line is, that’s fine. So let them present it, and even if they say — that one knows, after all, and is convinced that the person didn’t do it. Why on earth should he be liable? But he’s convinced he’s innocent. He’ll acquit him, the second will also acquit him.

[Rabbi Michael Abraham] No, obviously. But I’m asking: why go by the bottom line rather than question by question?

[Speaker C] Because in the end, in order to ask each judge—

[Rabbi Michael Abraham] No, ask him, fine. But why go by the bottom line? You may ask them, but why go by what they say in the bottom line rather than by what they write?

[Speaker C] After all, if they are asking for declaratory relief—

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

[Speaker E] If the plaintiffs are asking for declaratory relief — what did the parties mean?

[Rabbi Michael Abraham] That the court should rule only on the interpretation of the contract?

[Speaker E] Exactly, so yes, as you said, declaratory relief could have been given, so—

[Rabbi Michael Abraham] So let’s break the ruling into declaratory relief.

[Speaker E] But they didn’t come to get declaratory relief. What difference does that make? They came in order to get a final decision.

[Rabbi Michael Abraham] Okay. Give them a decision, just build the decision by majority on each question, and so on. But who will issue them the decision?

[Speaker F] Who is this “build”? There is no religious court to “build.”

[Rabbi Michael Abraham] Only if it’s someone from outside. Why? The court will build it. What’s the problem?

[Speaker F] Who is that? When there are two against one, who is this one who decides?

[Rabbi Michael Abraham] The court.

[Speaker F] No, yes, fine, but again — when the majority comes in the final decision, the point is that in the end it goes according to the judge’s conclusion, not the particular question of how he got there.

[Rabbi Michael Abraham] But I understand that. The question is why. Why? What’s the logic in that? Fine, I don’t know. There is a matter of simplicity here, of course. Because in most cases it doesn’t proceed this way. In most cases there is a connection between the questions, and it’s much more complicated, and not everyone agrees that these really are the questions. Sometimes judges see the theoretical analysis of the situation before them differently. There’s practical logic to doing it that way. But if you ask me on the abstract level, right, about this specific question, which is well defined in a completely clean way — there I really don’t see the logic in it. And there is also probabilistic logic here.

[Speaker B] What do you mean, probabilistic logic? If I assume that the judges’ vote, their opinions, are evidence for—

[Rabbi Michael Abraham] No, the opinions at the bottom line are evidence for nothing. The opinions at the bottom line are simply the product of their opinions on the questions. So that has no significance in itself.

[Speaker B] No, he says—

[Speaker C] to me that it’s closer to being right.

[Speaker B] Do you agree with

[Rabbi Michael Abraham] the majority?

[Speaker C] What do you think?

[Speaker B] that it’s closer to being right. Two-thirds, the probability is two-thirds—

[Rabbi Michael Abraham] No, he wants to argue that the ruling has a higher probability because of—

[Speaker B] the multiplication of the probabilities, or perhaps a case in which that wouldn’t be probabilistic logic in that sense.

[Rabbi Michael Abraham] No, in my view it never would be.

[Speaker B] Here, if the chance is two-thirds for each question that the majority was right. Right? Okay. Then for both conditions to hold it’s four-ninths, which is two—

[Speaker C] thirds times two-thirds, and then it’s less than half. What do you mean?

[Rabbi Michael Abraham] That both conditions hold — what do you mean?

[Speaker C] For both conditions to hold, for that.

[Rabbi Michael Abraham] No, no, that’s not right. Wait, which question — I’ll do the calculation for you on paper. The same question, I think I once told this story, they asked me in the kollel — Rabbi Shabtai asked me. He was dealing with probability, studying probability from me, statistics. So he said to me, look, suppose each judge has a two-thirds chance of being right, okay? Why do we follow the majority? The probability that two judges are right is four-ninths — that is, multiply the probabilities. If each one is two-thirds, then the probability that both of them are right is two-thirds times two-thirds, four-ninths. That’s less than a half. So why follow the majority? There’s no logic in following the majority.

[Speaker E] Here that is certainly not correct.

[Rabbi Michael Abraham] Do you understand why it’s not correct? Because that’s not the question I’m asking. I’m asking the opposite question. Given that two judges said such-and-such, what is the probability that it is right? The question is not what is the probability that the judge is right — that’s the reverse probability. The question is: given that this is what the judges said, what is the probability that this is really what happened, or that this is the correct ruling? That is a conditional probability. And once you reverse it, you see there is no problem with following the majority. By the way, not always — it’s not trivial, it depends on the data and various things. And what is the reverse?

[Speaker C] What is the probability that they are wrong? Think in reverse. The probability that they are wrong, one minus—

[Rabbi Michael Abraham] Fine. Why go to the probability that they are wrong? That’s also not correct. The probability that they are wrong — the probability is one-third that each one is wrong, so what, that makes it one-ninth? The probability that both of them, who said something, are wrong is one-ninth? What do you mean?

[Speaker B] No, you have to take into account whether you’re checking those two fixed in advance, or the probability that there will be two. The relevant probability is the probability that there will be two, not—

[Rabbi Michael Abraham] Also the probability that there will be two. No, you’re already taking me back — the calculation has to be done more systematically. In the end, the probabilities don’t add up to one. Just think. If two judges is four-ninths, then what, the second judge is five-ninths? But we already said it’s two-thirds that he is right, not five-ninths. So two-thirds plus four-ninths doesn’t add up to one, so clearly something here is wrong. Meaning, we didn’t divide the space of possibilities correctly. Fine, this is Bayes, that’s what has to be done here, but I think here it’s the same thing in short.

[Speaker B] I don’t think so, I think it’s completely different.

[Rabbi Michael Abraham] Okay, in any case, I bring this example only to show that here too, for example, there is an uneasy feeling. It’s surprising to discover that such a situation exists. But then, when you think about it a second time, it’s really like transitivity. So what happened? What’s the problem? All I need to decide is which is more reasonable as a method of decision: to go by the bottom line or to go question by question. I said I tend to think that question by question is more reasonable, but it doesn’t matter; someone else may say not. But each such method has its own logic and is consistent and everything is fine. The fact that there is a dispute over what is more reasonable — there are many disputes. That’s not a paradox, and it shouldn’t bother anyone, okay? So many times there are things that trouble us at first glance, but afterward you see that there are things that don’t operate the way we expect them to operate — that is not called a paradox. If one calculation itself led to the opposite conclusion from what emerges from it, that would be a paradox. But here, in a certain sense, this is actually an anti-paradox, as I defined it. Right? There are two answers here, both of them consistent, both of them fine, and there is no way — suppose there is no way — to decide between them. So if anything, maybe it resembles an anti-paradox rather than a paradox. But even that, I don’t think this is an anti-paradox. These are just two possibilities, and one has to decide which is more reasonable. That’s all. So I return to our case. Basically, with one’s own lost object, one’s father’s lost object, and one’s rabbi’s lost object — what do you do in such a situation, at least on the practical level? There is a problem of completeness here, not a problem of consistency, but completeness too is a problem. What do you do in such a case? So we don’t have the solution of passive omission is preferable. I said earlier, first of all, you could draw lots. That’s also an option.

[Speaker B] You could say that he should do whatever—

[Rabbi Michael Abraham] he wants, which is what’s called the judges’ discretion, right? Decide whatever you feel like. Once there is no ruling, all of them are equally legitimate. But even that is not exactly true, because not all of them are equally legitimate. Because the moment you did something, clearly there is one step — another step — that is preferable to the step you took.

[Speaker B] Exactly, that’s the paradox, and that’s why it seems to me like something paradoxical.

[Rabbi Michael Abraham] Okay, but it’s not really a paradox. True, there is something preferable to what I did. So what happened? If I do that, there will also be something preferable to what I do. Why is that a paradox?

[Speaker B] Because clearly the step I’m required to take—

[Rabbi Michael Abraham] is—

[Speaker B] a step that I’m required, as it were, so it implies that—

[Rabbi Michael Abraham] No, there’s no step I’m required to take. That’s the whole point. So this is only a problem of completeness; it’s not a problem of consistency. The system can’t generate for me an answer as to what step I’m supposed to take. It can’t. So what do you do? Like I said before, do whatever you want, draw lots—but again, even if you draw lots, and even if you do whatever you want, those aren’t good solutions in a case like this. Because in a case like this, what it really means is that when you do that, you’re now choosing some particular step. You chose it by lot, you chose it arbitrarily because you felt like it, whatever it may be. And once you chose it, there’s another step that’s preferable to it—so why are you doing it? It doesn’t matter how you chose it; in the end, once you acted, well no. And it will decide, it will decide—obviously that’s the question.

[Speaker F] No, I’m saying it shouldn’t give you an unpleasant feeling, because that’s exactly the given situation. I’m the judge.

[Rabbi Michael Abraham] I’m saying, fine—and now I have to make a decision. I decide A. But if I say A, then B is preferable to it, so legally I can’t do A when there’s another side that the law tells me to do—do B, not A.

[Speaker F] So I relate to it as a set of questions where I know that whatever I do, there’s something preferable to it, and now in that situation you, as a judge, are asked to decide. So what?

[Rabbi Michael Abraham] And what am I supposed to do? What am I supposed to do—draw lots?

[Speaker F] For example, draw lots. Something along those lines.

[Rabbi Michael Abraham] Fine, but there’s no “closer”; it’s all completely symmetrical. Each one is preferable to the other to exactly the same degree, let’s say for the sake of discussion. What am I supposed to do? So draw lots, or choose whatever I feel like, or whatever. But I’m saying again: after I choose, I’m in a situation where I am issuing a ruling that goes against the law. Because the ruling I choose to issue—ruling A—the law says it is forbidden to do that, you must do B. B is preferable to A. But of course B will have the same problem. There’s something bothersome here. I don’t think it’s entirely, entirely innocent. I agree. I just don’t think it’s exactly a paradox. Meaning, it’s something a bit different. It’s not even a contradiction. Why is it a contradiction? There’s no contradiction here. No assumption contradicts any other assumption. There’s no statement that A is preferable to B and B is preferable to A—that would be a contradiction. There is a statement that A is preferable to B, and C is preferable to A, and B is preferable to C. So what’s the problem?

[Speaker B] So the question is only from the standpoint of the law, but with the always-binding standard that the children have to be provided for, and also not transferred to a school with safety defects.

[Rabbi Michael Abraham] No, there it’s a frontal contradiction. That’s exactly the point. If there were only two possibilities, and the law says A is preferable to B and B is preferable to A, that can’t be; it’s simply a contradiction in the law. It’s a contradiction.

[Speaker B] No, but it doesn’t say B is preferable to A. There’s no statement that if they won’t be brought to a school with safety defects, even if that means they won’t have any school at all—I don’t know if that’s a contradiction.

[Rabbi Michael Abraham] No, on the other hand there was a contradiction, because on the other hand there’s a law saying they must study even if there are safety defects. Those two laws both appear. That’s a contradiction.

[Speaker B] But it’s not written that way.

[Rabbi Michael Abraham] Why is it not written that way? It is written that way.

[Speaker B] What’s written is that there has to be compulsory education.

[Rabbi Michael Abraham] Right, right—if you’re telling me that that’s not what’s written, then you’re interpreting, and then there’s no problem. Okay. But I’m saying: assuming it says what it says, I take it as it is. No limitations, no interpretations. That’s what it says. It says: if A is preferable to B, that’s what’s written. I don’t know anything else—A is preferable to B. And on the other hand, elsewhere in the law book it says B is preferable to A. That’s a contradiction; that’s a problem of contradiction. What?

[Speaker G] There’s no such reality. What the Rabbi is describing—in law it isn’t “this is preferable to that.” It’s “this must be done” and “that must be done.”

[Rabbi Michael Abraham] No, no, there are rules of priority. Look, in Jewish law, for example, there are rules of priority. “His own lost object and his father’s lost object—his father’s lost object takes precedence.” That’s a law.

[Speaker G] But that’s derived from—

[Rabbi Michael Abraham] What isn’t derived?

[Speaker G] It’s derived from honoring one’s rabbi, and that’s derived from honoring one’s father. It’s not—no, no, that’s not a law. It’s not like there’s a compulsory education law and—why?

[Rabbi Michael Abraham] There is a law—open the Shulchan Arukh and it will say, “His own lost object and the lost object of—”

[Speaker G] “—his rabbi, his father’s lost object and his rabbi’s lost object.” I know. But I said it’s derived from something. Okay, if we’re going philosophically and going back to the underlying question, then it’s not a derivation—there isn’t such a law as “one must honor his rabbi” and “returning a lost object.” Rather, from those two, situations are derived that are… but they are not themselves laws. It’s not a law. It isn’t that one is preferable to his rabbi if we were going to the post-step, meaning the first step in the chain: his father’s lost object, his rabbi’s lost object, and returning a lost object, okay? Sorry, it wasn’t honoring his father, honoring his rabbi—there was only this case, his father’s lost object, his rabbi’s lost object.

[Rabbi Michael Abraham] There wouldn’t have been any problem.

[Speaker G] I’m saying: if those were the laws, then yes, then it could be set up that way. In Jewish law, because this isn’t presented as two laws, but those situations—his father’s lost object and his rabbi’s lost object—are themselves derivatives of other laws.

[Rabbi Michael Abraham] Fine. But there can be opposite derivations. What do you mean? Suppose there were only two people, not three.

[Speaker G] The derivations are the problematic thing, not the laws themselves. A derivation isn’t a law.

[Rabbi Michael Abraham] What follows logically from the laws is—

[Speaker G] But it’s not itself a law. What difference does it make? Does it become a law? But it doesn’t become a law. Absolutely not, because there could be opinions. Okay, I’m not speaking about Jewish law.

[Rabbi Michael Abraham] Leave it—you’re saying I derived it incorrectly. Assuming I derived it correctly.

[Speaker G] Right, and that’s why I’m saying such a problem can’t exist in reality.

[Rabbi Michael Abraham] Why not? Of course it can. Why not? I don’t understand why not. What’s the problem? The derivation is correct, and it comes out that his father takes precedence over his rabbi, and his rabbi takes precedence over his father, on the basis of two considerations—and I show you that. A paradox within the system. What’s the problem? Like Achilles and the tortoise: by one calculation I show you that Achilles catches the tortoise, and by another calculation I show you that Achilles doesn’t catch the tortoise.

[Speaker G] And that isn’t true.

[Rabbi Michael Abraham] What do you mean it isn’t true?

[Speaker G] In the end he does.

[Rabbi Michael Abraham] Fine, because you found a solution.

[Speaker G] Of course I’ll find one—there’s a solution to anything that exists in reality. That’s not true.

[Rabbi Michael Abraham] What? Not at all. It’s over—there can be a contradiction within the system. Of course. Are you assuming every system is complete? Of course not. There are systems that contain a contradiction. Look, in set theory in mathematics, after Bertrand Russell found the paradox of sets, he showed that the mathematical system—not a legal system, so all the more strongly—the mathematical system contains a contradiction. They had to throw it in the trash and rebuild it. We are not perfect people; we build systems, and systems can contain contradictions. No, that’s a very optimistic assumption. You’re saying maybe in reality a contradiction can’t exist—that’s what I heard before. Fine, I’m talking about legal systems. About reality itself there is such a claim, that in reality itself contradictions can’t exist. Meaning, if there’s a contradiction, that means you made a mistake somewhere in the calculation. But as far as the legislator is concerned, the legislator isn’t perfect; it is entirely possible that I didn’t make a mistake in the calculation and a contradiction was in fact discovered in the system. I believe one can find—and maybe they even gave examples of this, I don’t know—I assume there have been such things. Human beings are human beings; they can make mistakes. In any case, how do we still deal with this? So it seems to me that really the only way—I said at the beginning that you were aiming in this direction—it seems to me that the only way is really to abandon positivism. Meaning, you have to notice that the problem exists when I’m inside the system. I have a certain system that tells me A is preferable to B, B is preferable to C, and C is preferable to A. That’s it. Now, if that’s what I have before me and that’s all—nothing besides it—and all I can do is make a logical calculation from those premises and see what comes out, then I’m lost. I’m trapped in an infinite loop. But if I really can step outside the system and ask myself: wait, why is A preferable to B? And why is B preferable to C? And why is C preferable to A? But then not in order to find an error—if I find an error, then fine, that’s trivial.

[Speaker G] But the Rabbi himself gave a lecture on rabbinic law and Torah-level law, and when we searched for the source of the authority of the sages—looking for it within the authority of the sages itself obviously can’t work. Right. Okay, but I agree.

[Rabbi Michael Abraham] But you’re saying there’s an error in the derivation. I’m saying—

[Speaker G] “An error in the derivation” isn’t the point. If we treat the derivation as a law in itself, then you won’t be able to solve it there.

[Rabbi Michael Abraham] To go out—if so, then I agree with you.

[Speaker G] Let’s see where it came from, and then you’ll find that there’s a hierarchy of priority.

[Rabbi Michael Abraham] Maybe you’ll find one; at least it’s possible that you could find one. So if that’s what you mean, then I agree completely. Meaning, what I really want to say is that now, if I understand where the priority of A over B comes from, I can try, for example, to quantify it. To ask by how much it is preferable, or in what units that preference is measured, and maybe other priorities will look different. Now, if the units are different units, that won’t solve my problem; I still won’t be able to reach a decision—there are different perspectives. But there are situations in which I can step outside the system and suddenly see that when the only possible way, in the end—in the book I do this a bit more systematically—but the only way in the end to solve the problem is to say that these laws receive different weights in different situations. Meaning: the law that A is preferable to B is a law worth eight—I don’t know—strength eight. The law that B is preferable to C has strength five. And the law that C is preferable to A has strength two. Okay? Now, if I do B, then I ask myself what the cost of B is, right? I say: the cost of B is that I violated the law that A is preferable to B, and I fulfilled the law that B is preferable to C. To fulfill a law worth two and violate a law worth eight, that’s worth minus six, let’s say. Okay, so the value of such a step is minus six. Fine? And now you can make that sort of calculation when you step outside the system and try to weigh the principles themselves. But of course, when I come to weigh principles, that only works if there is… In such a situation I can reach a decision, and the way to reach it is to step outside the system. There is Gödel’s theorem in logic. It says that in every axiomatic system satisfying certain conditions, equivalent to arithmetic in some sense, there is a statement within it that is true but unprovable. Meaning, it is true and cannot be proven. And there is a proof of that theorem, of Gödel’s theorem. And if there is a proof of Gödel’s theorem—and the proof, by the way, is constructive, meaning it constructs that very statement that is true but unprovable—now that means that the statement was proven to be unprovable. How can you say it’s unprovable? At one time this bothered me enormously; I walked around with it for quite a while until I understood that “proof” in that context means proof inside the system. Meaning: there is an axiomatic system with rules of inference; inside the system you won’t be able to find a proof for that statement. That’s exactly the theorem. When you step outside the system and work in a meta-language, in the meta-language you can prove that the statement is true. That’s what Gödel did: he stepped outside the system and proved that inside the system it is impossible to prove that statement, but he also proved that it is true. So it’s a little subtle, but in order to say it you have to step outside the system, because inside the system you’re stuck. Something like that is what I’m saying here. When you’re inside the system, you have A preferable to B, B preferable to C, C preferable to A. That’s your system. As long as you stay inside that system, you won’t get out of it. The only way to extract some answer is to step outside the system and try to quantify the principles operating within it; then I can try to rank them and see what is preferable to what, and then actually reach a decision. But of course, when I rank principles, by definition that is done from a vantage point outside those principles. I can’t rank principles to which I myself am subject and within which I act; I need to look at them from the outside and then ask what the status of each one is. Maybe I’ll give an example.

[Speaker F] Wait—so is there a halakhic ruling in this case of the lost objects?

[Rabbi Michael Abraham] What do Tosafot say? I don’t remember anymore; you’d have to look there in the second chapter of Bava Metzia. Look there on page 33. The question is whether it makes a difference in a context where they don’t look at the value of the lost objects.

[Speaker C] Is it exactly the same lost object?

[Rabbi Michael Abraham] Let’s say the same value. Let’s say—we’re looking for a pathological case, so let’s draw it in the most perfectly pathological way possible. I don’t remember exactly what Tosafot say; you’d have to look there. But yes, there is some answer there. This whole paradox exists only according to a view that is not the accepted Jewish law—that I need not honor my father at my own expense, I think—and that isn’t the halakhic ruling.

[Speaker B] By the way, just generally—when there are two lost objects with very, very different values, do these rules still apply?

[Rabbi Michael Abraham] When the value itself also plays a role. I don’t know a source for that, but reasoning suggests yes. I don’t know a source that explicitly states it. It’s an interesting question, for example, what would happen in a case—take one of the rules: his father’s lost object versus his rabbi’s lost object, and that’s it, there’s no lost object of his own, just those two. But the value there is enormous. So maybe that would decide it.

[Speaker G] Does somebody there pay?

[Rabbi Michael Abraham] Ah yes. Okay, so actually what one could do—no, just do something simple—exactly, pay. Is that what you meant? I’m not following what you’re saying. In the end I agree after I understand. I thought you were telling a joke, but no, it’s a real answer. You could say that the rabbi’s lost object is worth a thousand shekels and the father’s lost object is worth ten shekels, and the father’s lost object takes precedence over the rabbi’s lost object according to the halakhic rules. Fine? So clearly what you need to do is save the rabbi’s lost object and have the rabbi pay the father ten shekels.

[Speaker B] Isn’t it the other way around? The rabbi takes precedence over the father, because the rabbi brings him to life in the World to Come?

[Rabbi Michael Abraham] It doesn’t matter, let’s say. If his father paid for him, then it isn’t like that—fine, okay. In any case, maybe I’ll bring an example that illustrates the matter. There’s a paradox mentioned by several later authorities (Acharonim); Elchanan Wasserman talks about it, and I once saw it in Avi Ezri and elsewhere—I don’t know who the original source is—about the paradox of matzah made from new grain. If we are now on Passover eve and we have to bake matzot for the night, then suppose we have old grain. “New grain” here means grain harvested after the Omer day. The waving day—so the waving day is—

[Speaker D] After Passover.

[Rabbi Michael Abraham] So actually grain from the new crop—not grain harvested after the waving day, sorry, grain that… new grain. New grain, meaning grain from this year, may be eaten only after the waving day. And when is the year?

[Speaker B] When does the year begin? On Rosh Hashanah?

[Rabbi Michael Abraham] No—when it grew, whenever it grew. Rosh Hashanah or something. No, actually not. Yes, it’s after Passover.

[Speaker G] It’s all grain that is before the waving, from that time until the next year—

[Rabbi Michael Abraham] This is called grain that I now have on Passover eve, and I’m forbidden to eat it because the waving day hasn’t arrived yet. It’s from when… maybe Rosh Hashanah, maybe Tu Bishvat or something—there’s a New Year for this too. So first of Shevat, I don’t remember anymore. In any case, there is a prohibition on eating grain from the new crop until the waving day, and the waving day is of course the day after Passover, while I’m talking about eating an olive-sized amount of matzah at the Seder night, which is on Passover eve, before that day. Fine. So there are a few rules we need to state first. One rule is that in order to fulfill a positive commandment, I am obligated to spend up to one-fifth of my property. Some say that above one-fifth I’m not only not obligated—it is even forbidden. For the sake of not violating a prohibition, I must spend all my property. That’s the second premise. The third premise: a positive commandment overrides a prohibition. Fine—those are the three premises we’ll need for this discussion. Now what is the situation? The situation is that I have before me two possibilities. I have grain from the new crop at a reasonable price, and I have grain from the old crop that is very expensive. Very expensive. So the question is what to do in such a case. I have three options, three possible moves. One option: not eat matzah. A second option: eat matzah from the new grain. A third option: eat matzah from the old grain. And let’s see that this is exactly such a non-transitive situation: A is preferable to B, B is preferable to C, and C is preferable to A. Why? Suppose I want to eat matzah from the old grain, which sounds the most sensible. Matzah from old grain means that for this positive commandment of eating matzah, I’m spending, let’s say, the old grain costs half my assets—I’m effectively spending half my property, which according to Jewish law I am not obligated to do, and maybe it’s even forbidden. Therefore clearly I don’t have to.

[Speaker C] Oh—does someone with a Mazda 3 have it parked at the entrance to our house downstairs?

[Rabbi Michael Abraham] On the tree somebody—

[Speaker C] said.

[Rabbi Michael Abraham] So one option is: first of all, matzah from old grain is the most sensible, and it’s permitted, and there’s a positive commandment—but it’s not something one is obligated, or perhaps one is even forbidden, to do. Fine. So what should we do? Not eat matzah at all. Why not eat matzah at all? Because there’s the prohibition of new grain. Matzah from old grain is expensive, and there’s the prohibition of new grain. So what’s the problem? Make matzah from the new grain, because a positive commandment overrides a prohibition. The commandment of eating matzah is a positive commandment, and it overrides the prohibition of new grain, so let a positive commandment override a prohibition, and eat matzah from the new grain. What do you mean, eat matzah from the new grain? Why?

[Speaker B] A positive commandment overrides a prohibition. Because the prohibition is eating? Yes.

[Rabbi Michael Abraham] So if I do that, what should I do? I’ll eat matzah from the new grain? Of course not—what license do I have to violate the prohibition of new grain? In order not to violate a prohibition I have to spend all my property. So I’ll eat matzah from the old grain, I’ll buy it for half my assets, and I’ll avoid the prohibition of eating new grain. So—matzah from the old grain. Matzah from the old grain? What do you mean—spend all that money? So don’t eat matzah at all. After all, for a positive commandment you don’t have to spend all that money. But if the alternative is not to eat matzah, then eat matzah from the new grain—a positive commandment overrides a prohibition—and so on. So this is exactly a non-transitive situation of that kind.

[Speaker B] I thought there was a solution. Because when I say that one has to spend all his property for the sake of a prohibition, that means that if I want to violate a prohibition—well, not want to—say I’m being forced.

[Rabbi Michael Abraham] What? Suppose I’m being forced, and I have to pay a lot of money in order not to violate a prohibition. A bribe to a coercer. He’s forcing me to violate a prohibition unless I pay him all my property.

[Speaker B] I have to pay him? Because I understood that the whole point of “I have to spend all my property” is that there I’m active—meaning I myself have to come and violate the prohibition—but if it’s coercion—

[Rabbi Michael Abraham] Coercion, no—but he isn’t forcing me to violate the prohibition; if I pay him, everything’s fine.

[Speaker B] You can prevent the violation of a prohibition—that’s something entirely different. You’re saying that’s included in this kind of case.

[Rabbi Michael Abraham] On this matter too, all the halakhic decisors I know say that even though I understand the distinction you’re making—I even somewhat agree with it—all the decisors say that even in such a case one still has to pay.

[Speaker B] One has to pay?

[Rabbi Michael Abraham] Yes. Now what happens here? It seems to me that here the solution—the natural solution, and that is indeed what at least some of the later authorities say; I no longer remember exactly what all of them say there—is that my obligation to spend all my property in order not to violate a prohibition applies when the prohibition stands on its own. But if I violate the prohibition because I have a positive commandment incumbent upon me, then the positive commandment overrides the prohibition. So it’s not true that here I am violating a prohibition in the ordinary sense, where in order to escape it I must pay all my property. If the alternative is simply to violate a prohibition for no reason, then yes, I must spend all my property in order not to violate it. But if I am violating the prohibition in order to fulfill a positive commandment, after all there is a halakhic rule that a positive commandment overrides a prohibition. So in such a case I do not have to spend all my property in order to avoid the prohibition. I have a justification—this is not a transgression. A positive commandment overrides a prohibition, so the prohibition is not a transgression—or not such a severe transgression; it is suspended or permitted or whatever. Okay? And therefore, what did I actually do here? You don’t find this in Jewish law as a formal principle. In Jewish law it says: a positive commandment overrides a prohibition; one may not spend more than one-fifth of one’s property for a positive commandment; one must spend all one’s property for a prohibition. Those are the rules. So what am I doing? I step outside the system and think about the rules. I ask myself: wait, it’s not logical that I need to spend all my property for a prohibition that I have justification to violate. The positive commandment overrides it. This is not an ordinary violation of a prohibition. So I make an interpretation—and how is that interpretation actually made? I look at this system of rules from the outside. If I were living inside this system of rules and subject to it, and not allowed to look at it from the outside or interpret it from the outside, to qualify it, then I couldn’t find a solution. And what actually happens is that I have to—what I did here, and I show this in the book—what I did here is basically assign a different weight to one of the rules in different situations. Meaning, this rule that says you may not violate a prohibition at any cost is indeed a rule with effectively infinite cost in an ordinary situation. But it has a different cost in a situation where it stands opposite the rule that a positive commandment overrides a prohibition. In such a case, when there is interaction between the rules, each of the rules already changes. Meaning, every rule when it stands alone is a binding rule, no qualifications, everything is fine. But when it encounters another rule, then not only do I have a problem of rule versus rule, but each rule itself receives a different weight in the presence of the other rule. It is not merely a matter of deciding between one rule and another; rather, the rule itself takes on a different meaning, a different weight, and only then can one decide. A decision in the sense of simply pushing one rule aside cannot work—you can prove it—it cannot work. Only in a situation where I can show that I am weighting the rules, and that the weight is different in each situation. In the presence of other rules, one of these rules has a different weight, and only then can I reach a decision here. Because then it means: A is preferable to B, B is preferable to C, C is preferable to A—but when A, B, and C are all present here, then in the presence of the rule that A is preferable to B, the rule that C is preferable to A already loses some of its significance. It doesn’t exist, or it is less strong, or things like that. Meaning, the rule “C is preferable to A” does not have universal weight. Its weight depends on the situation. In every situation it has a different weight. In such a case a decision can be reached. Almost all the decisions I can think of concerning such loops are made in this way—by means of situation-dependent weights. That’s the only way. And of course we always do this when we look at things from the outside, when we move beyond the system. A computer can’t do such a thing. Because with a computer, you feed it a system of rules and it calculates what follows from the rules. In that sense there is some action here that is very human. It is an action that steps out—just as a computer could not prove Gödel’s theorem. In order to prove Gödel’s theorem you have to step outside the axiomatic system and look at it from the outside, and then present the proof or construct the proof. That is exactly what a computer apparently—I don’t know, at least in the standard architecture I’m familiar with—cannot do.

[Speaker E] But then the rules—the rules themselves—are relative. Or suppose you say that women are exempt from positive time-bound commandments. Okay. And there are commandments that women are obligated in because “they too were part of that miracle,” and all those rules too are defined as relative.

[Rabbi Michael Abraham] No, that’s not relative. There is a rule that is more complex: they are exempt from those commandments, except if they too were part of that miracle. Okay? That’s the rule.

[Speaker E] Right, so here too—meaning…

[Rabbi Michael Abraham] No, but here I’m making an interpretation. It isn’t written in the Talmud that way. “They too were part of that miracle”—the Talmud itself says that here the rule that women are exempt from positive time-bound commandments does not apply. So from the outset the rule is a more complex rule. Complex rules are still rules, or qualified rules. But here I am making the qualification as my own interpretation; it is not in the sources of Jewish law. From the sources of Jewish law I received these three rules, but there I activate my own reasoning, I look at the rules, and I say: there is something illogical here because of the clash. No—the qualification you are speaking about is not because of a clash; rather, that qualification says this rule simply does not apply in certain cases.

[Speaker E] It could be that in the Talmud it was determined that way because in the Talmudic period there was a clash. Suppose for the sake of argument it was established that women are exempt from positive time-bound commandments. Okay. Then they say, wait a second—why? But we too were part of that miracle. Ah, you too were part of that miracle? Right. Therefore you are obligated.

[Rabbi Michael Abraham] That’s not a clash. That means that my logic immediately gives the solution. It isn’t a clash for which you found an interpretive solution. There is no clash. What clash? There is no rule that anyone who was in the same place is obligated in the commandment. Where is there such a rule? Rather, the Talmud simply says that this itself is the logic that qualifies the rule that women are exempt from positive time-bound commandments. It isn’t something brought in ad hoc to solve a problem.

[Speaker E] So what are you actually saying? That the same logic that was in the Talmud we continue in our own time. Okay.

[Rabbi Michael Abraham] No, I don’t think it’s exactly the same thing. So first of all that’s also true, but beyond that, what I’m saying is that the Talmud too made the same move that we are making. But I think there’s something more than that here, because there the Talmud says this rule is a qualified rule. Meaning, it doesn’t exist in such situations—not because of clashes or anything like that, but simply because here it doesn’t exist.

[Speaker G] I think the Talmud doesn’t even say that. Rabbi Pensen gave a lecture showing that they rely on that rule even in a place where it says “except.” Why? Because really there are no rules. All these situations are byproducts of the generalizations.

[Rabbi Michael Abraham] I’m saying: the moment that—exactly, and it’s the same thing. What I’m saying now is that I’m stepping outside positivist thinking.

[Speaker G] You’re saying—

[Rabbi Michael Abraham] Listen, every rule is open to interpretation and qualification. Don’t be naïve; don’t be a kind of mechanistic positivist about Jewish law. If you are mechanistic in that way, you’re stuck. But once you are willing to open up the system, then you can indeed find—

[Speaker G] Then you return to the real system.

[Rabbi Michael Abraham] Ah—now the question is, and it’s an interesting question—

[Speaker G] that you return to the real system, because you’re actually describing the real system through rules that make up the system.

[Rabbi Michael Abraham] That’s an interesting question: is what I’m describing here a move outside the system of Jewish law, or on the contrary? Is positivism the move outside the system, while the system of Jewish law is the more open, more flexible system? And positivism is just a bad approximation? Or not—is positivism the halakhic system, and I step outside the halakhic system in order to solve…

[Speaker G] But be consistent with that in your book, because with all the 3,200 aphorisms—there is no rule, that’s it.

[Rabbi Michael Abraham] Right, right. Okay, so that’s the example. Now I want to do something else in this context, and that is to talk about—try to analyze systematically a solution to logical loops. There is a systematic rule here that does this, but these are loops not of the type of collision between rules, but loops of other kinds. I’ll give an example. There is a Talmudic passage in tractate Gittin, page 83: “Rabbi Akiva spoke up and said”—the case is someone who divorced his wife on condition that she not marry a certain man. Meaning, Reuven divorced his wife on condition that she not marry Levi. Fine? Rabbi Akiva said: suppose she went and married one of the men in the marketplace—say, Shimon. She married Shimon. Everything is fine; she is allowed to marry Shimon, only Levi is forbidden. And then she had children from Shimon, and then she was widowed or divorced from Shimon, and she then went and married the one to whom she had been forbidden”—meaning, she now married Levi. “Once she married Levi, is it not found that the bill of divorce is void and her children are mamzerim?” So says Rabbi Akiva. Why? Because the condition was that the first divorce was given on condition that she not marry Levi. Now we are talking about a condition of “on condition that.” A condition of “on condition that” means—at least it is reasonable to assume this is such a condition, one can discuss it a bit—but in such a condition it means that once you violate the condition, the act is retroactively nullified. Meaning, from the time you performed the act, it is as if it was never done. So he gave her a bill of divorce; she went and married Shimon; at that time everything was fine, she was unmarried, she had received a bill of divorce, she married Shimon, had children. Shimon died. Now she goes and marries Levi, and Levi was the one regarding whom the first divorce was conditioned that she not marry him. So she violated the condition.

[Speaker B] The marriage to Levi… what? There’s also an internal paradox here. One could say that the marriage to Levi is not valid once—

[Rabbi Michael Abraham] Wait, I’m getting to that. So Rabbi Akiva says: when she married Levi, she violated the condition. If she violated the condition, then the divorce that had been given conditionally was nullified. So she is really still Reuven’s wife, right? Then her children from Shimon are mamzerim. Because the marriage to Shimon never took effect. The marriage to Shimon—

[Speaker F] No? That marriage did exist.

[Rabbi Michael Abraham] No! Because it turns out retroactively that when she married Shimon she was not actually divorced, right? That was clarified only now, but it turns out that that was already the case then. Because the divorce was nullified.

[Speaker F] Doesn’t that apply only with respect to her relationship with Levi?

[Rabbi Michael Abraham] No, no—of course not. The bill of divorce itself is conditioned on her not marrying Levi. If she marries Levi, the divorce is nullified. Now she is my wife. She never left me at all. It doesn’t matter whom she married. Okay? So the moment she marries Levi, the divorce is nullified; once the divorce is nullified, she is Reuven’s wife. So if she married Shimon while she was Reuven’s wife, then of course that marriage doesn’t take effect and her children are mamzerim. That is what Rabbi Akiva says. Tosafot ask the obvious question about this, of course: “What came first?” “And she went and married the one to whom she had been forbidden—is the bill of divorce then not found to be void?” And if you will say, Tosafot say: but the marriage cannot take effect, since she is forbidden to him as a married woman. The marriage to Levi also cannot take effect. Because if the divorce was nullified—

[Speaker D] then she is really Reuven’s wife.

[Rabbi Michael Abraham] So if that’s the case, when she comes to marry Levi, now after Shimon has already died and everything—yes, the whole tragedy has already happened—she marries Levi. She can’t marry Levi, because she is Reuven’s wife. So the marriage to Levi did not take effect. And if the marriage to Levi did not take effect, then the condition was not violated; the condition was fulfilled. So the condition itself basically cannot be fulfilled. Wait, we’ll see. It is dealing with an event of marrying a married woman, which is of course absurd, it doesn’t exist in this theory. Right. So he stipulated: if you marry Levi, the divorce is void. So he says: it turns out she married Levi, so the divorce was void. But once the divorce was void, then she didn’t marry Levi. And if she didn’t marry Levi, then the divorce stands, so she really is not a married woman. And if she is not a married woman, then she is in fact married to Levi. And this is basically this kind of paradoxical loop.

[Speaker D] This is a question that can be asked even without the whole situation of Shimon and the mamzerim. It’s already there in the very initial case. He says to her: “You are divorced except that if…” right?

[Rabbi Michael Abraham] But there’s no practical difference, so what difference does it make? But what difference does it make?

[Speaker D] No, the paradox is already there from the start.

[Rabbi Michael Abraham] You don’t know what her status is. But with children being mamzerim, you say either they are mamzerim or they aren’t—yes mamzerim, not mamzerim. In any case, that is what Tosafot ask. Now, on the face of it, this question too is hard to solve in the way I spoke about earlier. Try stepping outside the system and weighing principles—there aren’t clashing principles here. There’s cause and effect. There is cause and result. She violated the condition, so the divorce is nullified. The divorce is nullified, so there is a married woman. There is a married woman, so the betrothal doesn’t take effect. Come on—I don’t see a way to step outside the system, weigh the rules, and try to reach some decision. What?

[Speaker B] You can step outside the system and look at the interpretation of the condition.

[Rabbi Michael Abraham] What do you mean?

[Speaker B] Suppose that his plain intent was that it was on condition that she not marry Levi, and that even if she does marry—if she performs the wedding ceremony, then if it was done seriously, that would nullify the divorce.

[Rabbi Michael Abraham] She did it seriously—what do you mean?

[Speaker B] And then, accordingly, that would nullify the condition even if it wasn’t legally effective.

[Rabbi Michael Abraham] Then why make such a condition? Let him just divorce her outright. If he is saying to her: look, you can marry Levi, then what is the meaning of the condition that you not marry Levi?

[Speaker B] And that’s exactly my point—that his intention is that it’s enough for her to perform the wedding ceremony seriously; the ceremony itself—

[Rabbi Michael Abraham] Even if it doesn’t legally take effect, it still nullifies the divorce. Okay, so there are medieval authorities (Rishonim) who suggest that. Let’s say we arrive at the conclusion that this is not the correct interpretation of the condition. He says to her—he says, fine? I make this condition with the intention of placing you into a paradox. Fine, that’s also what I want; it’s my right.

[Speaker G] I can stipulate any condition I want. Except that then it isn’t a complete severance in the bill of divorce.

[Rabbi Michael Abraham] Huh? Then he didn’t fully sever the divorce, because there is another opinion among the tannaim there that such a divorce really isn’t valid because there is no complete severance here.

[Speaker B] Independent of the paradox, is there also such an opinion—

[Rabbi Michael Abraham] There is such an opinion too. There’s one opinion that it’s because of the paradox, and another opinion because he must permit her to everyone. By the way, that opinion was not accepted as the halakhic ruling. He does not have to permit her to everyone; he can exclude someone.

[Speaker B] Wait—and is there an opinion because of the paradox that appears explicitly in the Talmud, or is that an interpretation of the medieval authorities?

[Rabbi Michael Abraham] No, the medieval authorities, the medieval authorities. In any case, this paradox in itself seems insoluble by the methods we talked about earlier. Here Rabbi Shimon Shkop proposes a solution to this paradox in Sha’arei Yosher. He writes there as follows, in Gate Five. What?

[Speaker E] Isn’t this like the previous move?

[Rabbi Michael Abraham] That’s it. “And since these marriages cannot possibly take effect”—the marriage to Levi—“for if they were to take effect, they would be retroactively nullified because of the condition.” For if these marriages take effect, then it becomes clear that she violated the condition, and then the divorce is void. If the divorce is void, then the marriages do not take effect. So the very taking effect of the marriages cuts off the branch on which they themselves sit. “And so anything whose taking effect is impossible cannot take effect at all. Therefore the divorce remains valid and the marriage is void.” The divorce remains valid and the marriage is void. By the way, that is not what Rabbi Akiva says. Rabbi Akiva says: “is it not found that the divorce is void and her children are mamzerim?” Why did he choose specifically the version in which the divorce remains valid? One could go the other way: the divorce is invalid and there is no marriage because of the paradox. So that’s a point I’ll get to in a moment. So Rabbi Shimon basically wants—in yeshiva language they usually say it like this: any legal effect that strikes itself cannot take effect. That’s how they usually formulate Rabbi Shimon’s principle. What is he really saying? The betrothal to Levi is basically a legal effect such that, assuming it takes effect, it destroys itself. If it takes effect, then it doesn’t take effect. A legal effect of that kind does not take effect from the outset. It cannot take effect. Not because of the condition—it simply cannot take effect, because this is a legal effect that uproots itself. And then what is the result? That she is not married to Levi; consequently the divorce remains valid, the children are not mamzerim, and everything is fine. That is the conclusion. But according to Rabbi Akiva that doesn’t fit, because Rabbi Akiva says: “is it not found that the divorce is void and her children are mamzerim?” He says that as the Jewish law. Now when I make such a condition—

[Speaker D] Can she then go from Levi onward to another man without a bill of divorce?

[Rabbi Michael Abraham] Yes. Just as she did it before, she can do it afterward too. Yes—without a bill of divorce from Levi? Of course. She isn’t married to him. Yes.

[Speaker B] So wait—if she lives with Levi without marrying him, then there won’t be mamzerim? It won’t be with a married woman; it’ll just be an unmarried woman with an unmarried man.

[Rabbi Michael Abraham] But just a second—the man doesn’t have to be unmarried; that part doesn’t matter. But she is unmarried. Yes.

[Speaker B] Why doesn’t that matter?

[Rabbi Michael Abraham] Yes. But the point according to Rabbi Shimon is this: why not say the same thing about the divorce itself? This is a divorce that, if it takes effect conditionally, then violating the condition will nullify the divorce—so the divorce too should not take effect. Why do we say this only about the betrothal to Levi and not about the divorce?

[Speaker B] Why? You could say that the condition doesn’t take effect, not that the divorce—

[Rabbi Michael Abraham] that the condition did not take effect, and then the marriage to Levi is fine—not that the marriage to Levi is not fine. After all… this is a loop, right? Every link in this loop dismantles itself, right? That’s the meaning of the paradox. So why does he choose דווקא the last link, the marriage to Levi? He could take the condition, he could take the divorce document, he could take the marriage to Shimon, he could take a million things. There’s another point here. There’s an implicit assumption in the words of Rav Shimon Shkop. Beyond the principle that anything which, if it takes effect, then it does not take effect—so it does not take effect—which people always quote in his name, you have to assume one more thing. And that assumption says that although all these things happen at the same moment—she marries Levi, then the marriage takes effect, then the divorce is nullified—this is not happening in time, one moment after another. In other words, moment after moment. It’s a logical process, not a temporal process. It all happens at the same moment she marries Levi; all of this happens within that particular instant. Rav Shimon still assumes that there is an order to things here, even though everything happens at the same moment. That’s a very important assumption. Because now let’s do the calculation and see why he stops specifically at the betrothal to Levi. We’re basically looking at this as though the events happen moment after moment. We’re building a fictitious timeline. In reality it all happens at the same point on the timeline. But we build a fictitious timeline that rotates inside that time-point. Now we say like this: he divorces her conditionally. A conditional divorce does not uproot itself, right? If she never marries Levi, everything is fine. What’s the problem? There’s nothing here that by definition, simply by being done, uproots itself. Not at all. So that’s one point. There is no limitation here of something that if it takes effect then it

[Speaker D] doesn’t take effect, so it doesn’t take effect. The condition—if it is fulfilled? What?

[Rabbi Michael Abraham] The condition, if it is fulfilled, it

[Speaker D] uproots the divorce, so that’s something that cannot take effect. What can’t take effect? This divorce cannot take effect. Why not? She can marry someone else. She can marry someone else, everything is fine. The nullification of the divorce cannot take effect.

[Rabbi Michael Abraham] Fine, I don’t want to nullify it, I want to leave it in place. She is divorced—that’s what I’m saying. There’s no problem. Because here Rav Shimon’s problem does not exist, that the divorce uproots itself. When did I think I could ask the same question of Rav Shimon about the divorce too? When I heard that everything happens at the same moment. After all, there’s no order; it’s a loop that is entirely located at one point in time. Now I’m saying no, no—I opened the loop, I turned it into a line. Now there is a line in which the things are arranged one after another in a supposedly chronological order. It’s a logical order, but I treat the logic as if it were time, as if it were chronology. Now I say this: at the first point, I divorced the woman conditionally. There is no problem at all. That’s fine. There is no built-in problem here that will destroy itself. It may be destroyed, but in order to destroy it a priori, you have to say that built into it is that it destroys itself by definition. Once you say that, you know it is destroyed. That does not exist here. Okay? Now she goes and marries Shimon. That too is not problematic. If she does not later marry Levi, everything is fine. So that too takes effect. After that he dies, everything is fine, children were born and he died, and then she goes to marry Levi.

[Speaker F] Now that’s already a problem.

[Rabbi Michael Abraham] Why? Because the moment she marries Levi, that very act will bring about the uprooting of the divorce, and therefore also of the betrothal. So the divorce remains a divorce at the time I carried it out; there was no necessity then to dismantle it, the dismantling was not embedded within it. But with the betrothal to Levi, there the rule does exist: a legal effect such that if you apply it, it will destroy itself—you cannot apply it. Jewish law does not allow such a legal effect to be applied. That’s the rule. Okay? Therefore he applies it specifically to Levi’s betrothal. Now what’s nice about this insight… is that there are a great many such loops. I collected many such loops in the book. And all of them can be solved in this way, as I wrote there. It is basically a general method for solving halakhic loops. Of course, this method goes beyond positivism, because Rav Shimon is essentially proposing something that has no halakhic source. This rule that he invented—his rule that if you apply a legal effect that will destroy itself, you cannot apply such a legal effect. Where did you get that from? That’s your own decision. Fine, but he has to add something, because if you remain trapped inside the rules within which the loop is created, then it’s a loop, and you won’t be able to get out of it. You have to step outside the system. So he offers a very particular way of stepping outside the system, but one that solves an entire family of paradoxes. Tons of paradoxes. But there is one paradox that I found—at least one among those we went through—one that I found can be solved without Rav Shimon’s rule, just with Rav Shimon’s implicit assumption. Usually, in order to solve the

[Speaker B] rules, you can formulate new rules through reasoning.

[Rabbi Michael Abraham] No, the implicit assumption that you can spread out the loop into a linear timeline. That there is an order to things, and the logical order is treated as though it were chronological. And here I do it even without assuming the further assumption that any legal effect which, if we apply it, does not take effect—therefore does not take effect. So I’ll bring you the example, maybe I’ll give it to you—well, I don’t quite have enough time to do it now, maybe I’ll do it next time—but I’m already saying that there is such a thing, and therefore it’s important to pay attention to the additional assumption in Rav Shimon’s words, not only the assumption that he writes explicitly. There is another assumption behind what he says. And I’ll bring a few examples so you can see what kind of things are involved here.

[Speaker F] The condition of the divorce was that… what? The condition of the divorce was, “I divorce you on condition that you do not marry Levi.” Why can’t I interpret that to mean that from me you are not to go to Levi? That’s the condition. From the moment she married Shimon, then it’s already going from Shimon to Levi.

[Rabbi Michael Abraham] You can also interpret it the way Arik suggested earlier—that maybe if you perform an act attempting to become betrothed to Levi, that would nullify the divorce. Not if you succeed, but simply the attempt itself—I don’t want you even to try. But these are possible interpretations. What I’m saying is: let’s say he himself said that he didn’t mean that. After all, I’m looking for a situation—I’m constructing it—in order to create a paradox. Obviously if not, then you can interpret it in other ways. Let’s say he said it and clarified everything—and now what do we do?

[Speaker B] It could be that one would need to be more precise and word it differently. Say, if he says, “on condition that you do not marry Levi and that the marriage take effect,” then we’d have to say that this condition would activate only after the marriage takes effect, and then it wouldn’t matter.

[Rabbi Michael Abraham] I can suggest, I can define a case in which he says whatever is needed so that none of all these solutions will work. For every such solution, I’ll add one more clarification; when he states the condition, he’ll add another clarification. Fine? After all, I’m looking for a pathological situation. It doesn’t matter that there are other solutions and approaches that are solvable. So I’ll give you a few examples. Look, there’s one I found from Rabbi Yoav Rosenthal—I don’t know him—in the journal Beit Aharon VeYisrael. He says: Reuven divorced his wife on condition that she marry Shimon. Not that she not marry him, but that she should marry Shimon. The woman went and married Levi, and afterward she left Levi without a divorce document, okay? And married Shimon. What happens now? If she married Shimon, then the divorce is valid from the outset. It is valid. The divorce was on condition that she indeed marry Shimon.

[Speaker C] So

[Rabbi Michael Abraham] if the divorce is valid, then she was married to Levi, because she was free from the first husband.

[Speaker C] Does that only take effect from now on?

[Rabbi Michael Abraham] Only from the marriage? No, no, no—it works retroactively. It was a condition of “on condition that.” A condition of “on condition that” means that already from the moment the divorce document was given, it is valid. Because in the end it is valid, since she married Shimon.

[Speaker C] It was on condition that she marry Shimon.

[Rabbi Michael Abraham] Right. What? And once she later married Shimon, it became clear that when she married Levi she was not acting unlawfully? That only became clear to us later. But when she married Levi, was she acting unlawfully then?

[Speaker C] No, we didn’t know.

[Rabbi Michael Abraham] But she already knew in advance that she was going to marry Shimon. And even without a divorce document, so she’s not even dependent on Levi giving her a divorce document—she leaves without one. Fine? She marries Levi. Now what happens? She married Levi, then she fulfilled the condition. She fulfilled the condition, so the divorce takes effect. Right? No—she married Shimon, she fulfilled the condition. So she fulfilled the condition, and then the divorce takes effect, right? If the divorce takes effect, then her marriage to Levi is valid. And if her marriage to Levi is valid, and she left him without a divorce document, then she is Levi’s wife, so the marriage to Shimon does not take effect. But if the marriage to Shimon does not take effect, then it turns out that the divorce is nullified because she violated the condition. If the divorce is nullified, then she is not married

[Speaker C] to Levi, so

[Rabbi Michael Abraham] she is not married to Levi, so she

[Speaker B] can marry Shimon, and so on in a circle.

[Rabbi Michael Abraham] It’s the same solution.

[Speaker B] It’s another example. If you want another example, there are several here. For example, yes, someone from Get MiBeit Chinuch has a few like this. So one of them: a divorce document that was written not for her specific sake and on the Sabbath. He traced over it with a quill for her specific sake intentionally. He wrote a divorce document not for her specific sake? Exactly like that, not for her specific sake. Not for the sake of the woman? What else? For the sake of not-for-her-specific-sake. Don’t you write the woman’s name?

[Rabbi Michael Abraham] No, no—for the sake of the woman, intention. For the sake of this woman. The divorce document has to be written with the intention that it be for this specific woman, and if not, the divorce document is not valid.

[Speaker B] But why did he write it just like that?

[Rabbi Michael Abraham] What is he? He’s a divorce-document enthusiast. He was practicing writing a divorce document. Fine. Now the rule is that if you go over the document with a quill, writing again on top of the existing writing, that’s fine if you do it for her specific sake. Fine. Incidentally, the writing for her specific sake itself is what validates it, because if you were just writing ordinary writing on top of writing, it would not count as writing. But if you write the second writing for her specific sake, then there is something in the second writing that is not in the first, so it takes effect, and therefore it has significance too, okay? So this is something that builds itself. Never mind. In any case, what did he do? He wrote the second writing on the Sabbath intentionally. Now what happens? So he says: where is the law here? Behold, one who writes a bill of divorce intentionally on the Sabbath is an apostate, and it is not a valid divorce document, because if an apostate writes a divorce document, it is not valid. But what if it is not a valid divorce document?

[Speaker B] If he still got married and converted away from Judaism, yes, then is his wife chained and unable to remarry?

[Rabbi Michael Abraham] When he writes the divorce document, he is commanded to write the divorce document—that’s another issue. If he writes the divorce document, okay? He is not an agent; at that time he is not an agent.

[Speaker B] He can’t write a divorce document, but can he appoint an agent for a divorce document?

[Rabbi Michael Abraham] Maybe, yes.

[Speaker C] In any case, usually they don’t write it themselves; they take someone who writes.

[Rabbi Michael Abraham] Good question, I don’t want to get into it here. But here, if we say that it is not a valid divorce document and he is an apostate—well, if it is not a valid divorce document, then he did not desecrate the Sabbath at all, because writing on top of writing is permitted on the Sabbath. There is no Torah-level prohibition in writing on top of writing; you did not create any new writing. Only if this writing validates the divorce document—then this writing is significant writing, and so writing it on top of existing writing counts as desecrating the Sabbath. But that is only if it is a valid divorce document. But if it is a valid divorce document, then the writing counts as writing, so you are an apostate. If you are an apostate, then it is not a valid divorce document. If it is not a valid divorce document, then you are not an apostate, so it is a valid divorce document, and so on and so forth. We’ll talk about this a bit more next time and then we’ll finish.