Halakhic Positivism, Lecture 6

[Rabbi Michael Abraham] The board, because I’m using it today, so if you want to come a little closer, okay? Good. I’m going to go back to the session before the previous one. I started talking there about kal va-chomer, and I’ll also remind you of the context. Basically, what I’m trying to show is that even non-positivist thinking has some kind of ordered logic behind it. Meaning, you can show in a systematic way how we arrive at, how we infer our conclusions in non-deductive thinking—what in the traditional sense isn’t included under logic. I started with kal va-chomer and said that basically, we drew a two-by-two table there.

[Speaker B] We talked about horn—

[Rabbi Michael Abraham] —and tooth and foot, and the public domain and the injured party’s courtyard. And the tanna: in the public domain exempt, in the injured party’s courtyard liable. Horn is liable here—I’m ignoring the fact that it’s only half, for the moment I’m saying liable—and here basically the question is what we fill in. I’m just reminding you of what we already did. What I actually showed the first time is that behind kal va-chomer there sits a theory. In other words, kal va-chomer is not just a game played among data points; there is a theory behind it. We have three data points here, and we’re looking for the simplest explanation of those three data points, and from that explanation we derive the conclusion of what should be here. That’s really the basic philosophy. How do you do that practically? So what we did there was to set up two possibilities. There’s a possibility that the filling is one, and there’s a possibility that the filling is zero. We want to know which of the two is correct—whether it is liable in the injured party’s courtyard or not liable in the injured party’s courtyard. So let’s check. Assuming the filling is one, okay? Then what is the theory that can explain all these data? Zero, one, one, one. So the theory is the following: this is two alpha, this is alpha, this is two alpha, and this is alpha. What does that mean? The assumption is that there is some characteristic—what? Up above? No, it’s not reversed, because if this were alpha, then this alpha would have been enough, it would have made it one. In other words, the claim is that what’s going on—why isn’t it reversed? Because when you say two alpha for horn, that means horn is stronger. When you say two alpha for the public domain, that means it’s harder to impose liability there, not easier, meaning it’s weaker. In the domains it’s weaker; in the damaging agents it’s stronger. It’s always like that—the rows and the columns are always reversed. So the claim is that behind these laws there sits some characteristic, it doesn’t matter right now what it is, I’m not even identifying it. The claim is that it exists at a certain intensity in tooth and foot, and at a greater intensity in horn. And because of that, in the public domain you need the greater intensity in order to impose liability. Tooth and foot don’t have enough intensity—they only have alpha, and I need two alpha, so it can’t generate liability, okay? By contrast, here a single alpha is enough, so here you get a one, and the same thing here.

[Speaker D] Or either it isn’t, or it’s what’s required—yes, exactly.

[Rabbi Michael Abraham] So that’s the theory for one. Suppose the filling is zero. When the filling is zero, you immediately see there’s equivalence between the columns, so clearly you can’t do it without two parameters. Therefore the theory is the following: this is the red theory. The red theory basically says: here tooth and foot have the feature alpha; horn has the feature beta. And in order to impose liability in the public domain, you need beta, so this one can’t do the job, while this one has beta so it can do the job; and likewise in reverse here. Here you need alpha, so whoever has alpha does the job, whoever has beta doesn’t do the job. That’s the red filling. Now I’m saying these are the two possibilities standing before me.

[Speaker C] In the previous lecture you said there was another possibility—instead of working on this, to work on that—what? I mean, to make there the “has” and here the “needs”—does that help?

[Rabbi Michael Abraham] No, it doesn’t matter. No. At this stage I already showed that the lighter and heavier sides of the columns and of the rows are the same thing, because they both work that way. It’s universal. It doesn’t matter from which side you look at it.

[Speaker C] In the interpretation of “doesn’t need,” it basically flips, but it doesn’t matter—it’s the same thing.

[Rabbi Michael Abraham] Okay. So that’s exactly what I was trying to show there. Again, here I’m only getting to the bottom line; I’m not repeating the whole previous lecture. But if you remember, in the previous lecture I said there are two formulations of the kal va-chomer. You can learn from here that horn is more severe than tooth and foot, and then infer here, infer a one. And you can learn from here that the injured party’s courtyard is more… severe than the public domain, and then infer this here, infer this here. And on the face of it, these are two different claims, and therefore the refutations of them should also be different. So if we make a refutation against this kal va-chomer, it should still leave this kal va-chomer standing. But in the Talmud it doesn’t work that way. When they bring one refutation, whether against the columns or against the rows, the kal va-chomer falls. That’s the assumption. And basically all of this I tried to explain in order to account for that phenomenon. In other words, what I’m trying to show here is that no matter from which side you look, there is a theory behind it. And that theory—you can see that the same parameter is playing the game here and here. There is no difference between the columns and the rows. It’s the same thing. In other words, if I refuted alpha and two alpha here, then the relation between these two also collapses. It’s the same relation.

[Speaker C] Why does it have to be that way? These are two independent things, so why, if you—

[Rabbi Michael Abraham] What I’m saying is that they are not independent. They are not independent because behind the kal va-chomer there sits a theory. And once you understand that behind the kal va-chomer there sits a theory, then understand that whether you formulate it this way or formulate it that way, all you’re really doing is building these theories. It doesn’t matter in which direction. You’re building a theory that stands behind both the rows and the columns; these are the same parameters. And once you’ve toppled the theory, you’ve toppled it from both directions. And that’s exactly the point. And if that’s so, now I—again, all this was in the previous lecture. I don’t want to repeat all of it now, only the bottom line. The bottom line basically says that what I need to do is compare the theories that stand behind the inference in the case where this is zero and in the case where this is one. And how do I decide which is right? The simplest one, right? Which is simpler, the black or the red? The black. Because you can see that in the red there are two parameters, alpha and beta; in the black it tells me: I explain all the data with one parameter. That’s Ockham’s razor. Meaning, you take as few data points as you can, you don’t add data without a need for it. Okay. In a view that is closer to what we actually do, we are really looking for an explanation only of the three data points, not the fourth. We’re looking for how to explain these three data points, and after we have an explanation we fill in here. What I did is just mathematically easier, so I say: let’s fill in both possibilities there and look for an explanation for the whole table, and then think which of the two is simpler—but it’s the same thing, completely equivalent. Okay. So that basically means that we—now I’m making a somewhat more general statement, because it will be true of all inferences. What I’m really doing is writing the data in a table, including the question mark of the missing cell. I’m looking for the law in one case מתוך other laws that I know. Therefore these will always be tables with complete data except for one cell with a question mark. Sometimes people call it something else, it doesn’t matter, but right now I’m talking about one cell. What I need to do is look for the simplest explanation when the filling is one, the simplest explanation when the filling is zero, and see which of the two is simpler. And that’s what determines it. Now, if the result here is one, then the inference works—it’s valid. If the result here is zero, or if there is no preference for one over zero and no preference for zero over one, that is what will be called a refutation. Okay? For example, last time we even saw it. If, for example, we have some refutation here—say there is some other domain on the moon, and on the moon what happens is that tooth and foot are liable and horn is exempt. You can see that this reverses the relation, right? Here horn is more severe; here tooth and foot are more severe. So that is really a refutation of the kal va-chomer of the columns, right? It means that horn is not more severe than tooth and foot, right? You can see it here. Now why does that also refute the rows? That’s really what I’m showing. Because notice that together with this, if you now look for a theory—black and red—in both cases you’ll need two parameters. In other words, it won’t help. And therefore it refutes both directions of the kal va-chomer. Okay? We’ll get to it in a more orderly way, I just wanted to show the motivations. Now I’m going to jump quite a bit ahead, and I’ll try to show the same way of thinking with regard to the common denominator. Okay? So let’s take some Talmudic text as an example. A Talmudic passage in Bava Kamma—I’ve mentioned it in the past. The Mishnah brings there the four primary categories of damages: “the ox, the pit, the maveh, and the fire. The case of the ox is not like the case of the maveh, and the case of the maveh is not like the case of the ox, and neither of these, which have life in them, is like fire, which has no life in it”—meaning, they make a necessary differentiation among all the primary categories of damages. And then the Mishnah concludes: “Their common denominator is that it is their way to cause damage, and their supervision is upon you; they are forewarned, and your property and its supervision are upon you, and when they cause damage, the damager is liable to pay compensation for the damage from the best of the land.” In other words, at the end of the Mishnah there appears a rule: anything that is your property, whose supervision is upon you, when it causes damage, the damager is liable to pay compensation for the damage from the best of the land. So the Talmud on page 6 says, “their common denominator.” The Talmud asks: what does this come to include? Yes, once I joked a bit about this, because when we talked about rules—and it’s also connected to positivism—I said how strange this is, because Mishnayot usually give us examples, as you know. It’s casuistic—it works through cases, not through rules. And we have to formulate the rules; the Mishnah basically leaves us to do the work. The Talmud, or the medieval authorities (Rishonim), or the later authorities (Acharonim), or us—it doesn’t matter. Whenever we learn, we always try to understand the rules behind the cases. One time the Mishnah does us a favor and itself formulates the rule. Yes, the common denominator: your property, its supervision is upon you—a rule. What is common to all four primary categories? And then the Talmud comes along on page 6, and what does it say? What does it come to include? When you bring me the rule. For once they already did you a favor, and you ask about the rule why they brought the rule? Ask why they brought the details. Say: why are you bringing examples? You gave me the rule already, I can work with that, what’s the problem? For once they already did it right, and even there the Talmud asks: what does it come to include? So I said, I think the Talmud is right. Meaning, someone who sticks to rules makes mistakes a lot of the time; examples are stronger than rules. Fine. Anyway, here the Talmud asks: what does it come to include? The Talmud says: Abaye said, it comes to include his stone, his knife, and his burden, which he placed on the top of his roof, and they fell with a common wind and caused damage. In other words, he put some objects on top of the roof; a common wind came and knocked them down into the public domain, and they caused damage. The Talmud asks: what are the circumstances? This example is meant to tell me what we are looking for. We are looking, basically, for a damaging agent that cannot be learned from any one of the primary categories of damages by itself, and therefore I need the common denominator to teach it. That’s really what we are looking for. That one is still liable for. What?

[Speaker C] That one is still liable for.

[Rabbi Michael Abraham] Seemingly. In other words, the Talmud assumes that he is liable for it. Where does the Talmud get that assumption from? From the rule. Because the rule says that anything that is your property and whose supervision is upon you—when it causes damage, the damager is liable. And the Talmud says to that: ah, this rule says everything. Let’s go back to the examples and find a case that cannot be learned from the examples, and therefore I needed the rule to tell me that. Now what case is that? His stone, his knife, and his burden, which he placed on top of the roof, and they fell with a common wind and caused damage.

[Speaker C] But why do you need to bring this case too instead of the rule? What? If that’s the only thing?

[Rabbi Michael Abraham] Because there are four more cases, there are more cases that you wouldn’t gain. In other words, it’s only to demonstrate why the common denominator is needed. The Talmud itself brings four cases; there are more, but the Talmud itself already brings four cases. So the Talmud says: what are the circumstances? If they caused damage while moving—in other words, if they caused damage while they were falling from the roof—that’s fire. So it’s like fire. So yes, it can be learned from one of the damaging categories: fire. Why? How is fire different? Because another force is involved in it, and it is your property and its supervision is upon you. These too—another force is involved in them, and they are your property and their supervision is upon you. So it’s the same as fire. So you don’t need the common denominator for that. But if it was after they came to rest? In other words, what is the case? That they didn’t cause damage while flying, but after they came to rest below, and then it’s like a pit in the public domain. So the Talmud says: if he declared them ownerless, then whether according to Rav or according to Shmuel, that is a pit. How is a pit different? Because from the beginning of its creation it is for damage, and it is your property and its supervision is upon you. These too—from the beginning of their creation they are for damage, and they are your property and their supervision is upon you. But if he did not declare them ownerless—that also doesn’t work. In the end the Talmud says: really, it is a case where he declared them ownerless. It’s talking about a case where he declared them ownerless after they fell. Yes, we once talked about interpretive setups. Did he know they had fallen?

[Speaker C] Here he declared them ownerless.

[Rabbi Michael Abraham] Yes, he knew they had fallen and he declared them ownerless. Yes. So he despaired of them, as it were—they fell from the top of a roof, and he’s not going to retrieve them, someone will already take them, so he declares them ownerless. Right. So notice, we once talked about interpretive setups. His stone, his knife, and his burden: he put them on top of a roof, and then there’s a whole story—a common wind came and knocked them down, he sees they’re down there, he declares them ownerless, someone comes after he declared them ownerless, trips over them, and is injured. Yes, that’s the setup, and this is meant to teach the common denominator at the end of the Mishnah. You understand that it’s not really this case that it’s trying to teach; it’s trying to teach the principle that if it’s your property and its supervision is upon you, you have to pay. This case only reflects why the principle is needed and why it can’t be learned from any one of the details. We talked about this when we discussed interpretive setups—that the setup is not meant to teach the actual case with which the Mishnah is interpreted. So the Talmud says: really, it is a case where he declared them ownerless. So we’re talking about a case where he declared them ownerless. And they are not like a pit—you can’t learn it from a pit. Why? How is a pit different? Because no other force is involved in it. Will you say that here, where another force is involved in them? Here another force was involved in creating the damaging agent. A pit—I dug it myself, no other force was involved. And therefore there is a refutation; you can’t learn it from a pit. The Talmud says: fire will prove it. So fire will manage to prove it, why? Because in fire the wind is involved, another force is involved, so fire will prove it. The Talmud says: how is fire different? Because its way is to go and cause damage. But fire’s way is to go and cause damage—it damages while moving. Here, after they came to rest, they caused damage; they’re no longer moving. The pit will prove it. And the argument returns. So we learn it from pit and fire together, and that is basically the common denominator. This is really the classic case of learning by common denominator, what is called an analogy built from two verses. Okay? We have two teaching sources; neither of them alone manages to teach this, but together they do. Something very interesting happens here, and let’s try to see how it works. Every common denominator in the Talmud works like this. In other words, for example. We’re basically talking about two teaching sources—let’s say fire and pit, fire and pit. Let’s write here simply A and B, fire and pit. Okay. Now we want to learn—want to learn from his stone and his knife and his burden that he placed on the roof, with a common wind, and they came to rest and caused damage, okay? That’s C. Okay? Here it no longer works. I think both cases could be a case—it doesn’t have to be specifically a knife, a knife.

[Speaker C] Knife, fire? Knife. Okay.

[Rabbi Michael Abraham] So the derivation works like this: we start from pit, right? It’s similar to a pit. The Talmud says no, not from pit, because pit has a special property: no other force is involved in it. In these, another force is involved. So I say that pit has the property such that this one has X and this one doesn’t have X. Okay? No other force is involved in it; another force is involved in it. You can’t do that. We said: fine, so I’ll try to learn from fire. It says: no, not from fire either. Why? Because its way is to go and cause damage, right? Because its way is to go and cause damage. So this one has Y as well, and this one doesn’t have Y. You can’t learn it. Right? We try to learn from this one—the Y refutes it. We try to learn from that one—the X refutes it. And now something very interesting happens. By the way, it’s always obvious that here there is Y and here there is X, and not here. It’s always like this, everywhere in the Talmud, anywhere you look, it’s always like that. Why? How do I know that? Because if here there were X, and here there were Y, then there would also be a refutation of the common denominator, since there would be one shared property in the two teaching sources. Fine, I’ll explain in a moment. But already here I’m saying: it’s always like that. In other words, the feature that is absent here and present here will also be here—that is, Y.

[Speaker C] And the X—

[Rabbi Michael Abraham] The feature that is absent here and present here will be here. Okay? It’s always like that. It’s like genes, jeans. Yes, right, very similar. No, it really is similar. We once used that analogy. Okay. In any case, now what am I saying? I’m not adding any extra data. I’m saying: the derivation from here doesn’t work, the derivation from here doesn’t work, so therefore from the two of them together everything is fine. What does it mean, from the two of them together everything is fine? If neither of them alone can work, why should the two of them together work? You can go here in two ways.

[Speaker C] Without the rule, I couldn’t have inferred it. Because the Mishnah gave the rule, I can. That doesn’t mean the rule is true. The Mishnah says it.

[Rabbi Michael Abraham] Maybe. But how does it know? Let’s ask about the Mishnah, not about the Talmud. How does the Mishnah know that this rule is correct? It takes two cases and learns from them something general. Who says so? Let me sharpen that a bit more. Look, I have two possibilities here. And those two possibilities are apparently a dispute among the medieval authorities (Rishonim). Because on this passage, the Rosh brings a very interesting dispute. He says as follows: “Their common denominator,” and so on. Rabbi Alfasi did not bring in his work all these interpretive setups of those amora’im, where we conclude that we learn his stone, his knife, and his burden, which he placed on the top of his roof and they fell into the public domain with a common wind and caused damage after they came to rest, from pit and fire. Okay? Up to here, that is the conclusion of the Talmud. “And some of the great authorities wrote,” says the Rosh, “that one is liable only for what he is liable for in both of them, and they are exempt from damage to vessels and from causing human death like a pit, and from hidden items like fire, since they come by way of the common denominator, we give them the lighter law of the two.” The Rosh asks: we know that among the four primary categories of damages there are special leniencies for each category. For example, an innocuous ox pays only half. Fire is exempt for hidden objects. A pit is exempt for vessels—“an ox, but not a person; a donkey, but not vessels.” Okay? Every one of the primary categories has some sort of exemption unique to it. Now the Rosh asks: do the exemptions of the two teaching sources—namely, fire, which is exempt for hidden items, and pit, which is exempt for vessels and for a person—do they also apply to his stone, his knife, and his burden? Now what would we have said? On the face of it, what would you have said? Yes? That they do? Both kinds of exemptions. Both this one’s exemption and that one’s exemption. Why? What’s the idea behind that?

[Speaker C] If you need—

[Rabbi Michael Abraham] —both teaching sources in order to teach this one, then if you want to impose liability for, say, hidden items, you only have one teaching source. Because that one is exempt for hidden items. How can you impose liability for hidden items? After all, you need both teaching sources in order to derive liability for the learned case, right? Now in each area where there is an exemption either here or here, you don’t have two teaching sources, so you won’t be able to derive liability of his stone, his knife, and his burden for hidden items or for vessels, right? Therefore the obvious conclusion is that both exemptions should apply. That’s what the “some of the great authorities” quoted by the Rosh say. And what you’re really saying is that all the exemptions that exist in the teaching sources also exist in the learned case. Does the Talmud say otherwise? One second. The Talmud doesn’t say anything—it’s a dispute among the medieval authorities (Rishonim).

[Speaker D] When you learn from here that either X creates liability or Y creates liability—

[Rabbi Michael Abraham] Okay, soon. That’s already a third stage; I’ll get to that too. So the Rosh says—after bringing the “some of the great authorities,” he says, “and there are those who were uncertain about the matter.” A second opinion. What were they uncertain about? He doesn’t write it. What is the other side of the doubt? One side is like the “some of the great authorities”; what would you say the other side is? That there are no exemptions at all in the learned case, right? That’s how it appears. He doesn’t specify anything, only the opposite side, which says that there will be no exemptions. What’s the idea behind that? I can explain the idea behind that. The idea is basically the following: I take the common denominator that appears in the Mishnah—and that is the idea of the common denominator. The common denominator in the Mishnah says that anything that is your property and whose supervision is upon you, you must pay. Okay. Now I say this: his stone, his knife, and his burden that fell from the roof and caused damage—they are my property and their supervision is upon me. So now I don’t care about fire and pit; I’m not going back to the primary categories of damages. I’m now looking at the rule. After all, that’s why we learned the rule. So he is liable. Now I ask myself: is he exempt for hidden items? For that I do have to go back to the primary categories, because the rule doesn’t tell me how to determine who is exempt and when. Here I do have to enter the specific primary categories and see: if it’s similar to fire, he’ll be exempt for hidden items; if it’s similar to pit, he’ll be exempt for vessels and for a person, right? So for the very liability, I don’t need to look at the teaching sources. The very liability is: it’s my property, its supervision is upon me, therefore liable. Now I ask, okay, now let’s see: what’s the law for hidden items? What’s the law for vessels? Let’s see. Is it similar to fire? The answer is no, because there are differences here, so I can’t derive the rule about hidden items, so there won’t be an exemption for hidden items. And the same thing for vessels and a person. Then the other side of the doubt emerges, saying that there are no exemptions here at all. Because I’m really learning this as two levels. First I learn the very liability, and after that I need a new derivation in order to exempt for hidden items. Now the very liability is: anything that is your property and whose supervision is upon you—that’s the rule. And the specific primary categories were brought for their particular laws. By the way, this is in the Talmud on page 5; the Talmud says that—they were brought for their laws. So they were brought to teach when you’re exempt for hidden items, when you’re exempt for vessels, and so on. You won’t succeed in deriving that for his stone, his knife, and his burden, because it is similar neither to fire alone nor to pit alone. And then it comes out that you’ve learned the liability, but you don’t succeed in learning any exemption. Therefore the other side of the doubt is that he will be liable and will have none of the exemptions from the primary categories.

[Speaker C] Meaning, not that it doesn’t exempt, but as if you’re saying it isn’t enough in order to exempt.

[Rabbi Michael Abraham] Yes. So then I automatically have no exemption. First of all, he is liable; in order to exempt I need a derivation, I don’t have one, so he is liable. Okay? So that’s the doubt, right?

[Speaker C] The two different features are leniencies, so how is it that in general—

[Rabbi Michael Abraham] No, they are stringencies, not leniencies—always stringencies.

[Speaker C] When I go from A to C, A is more severe, right? So based on each of the features, C seems more severe, right? So how can I obligate from the two of them together?

[Rabbi Michael Abraham] I asked that earlier—I’m getting to it. That’s the question of questions here. The medieval authorities (Rishonim) already comment on it, but we’ll get to it soon. So that’s the doubt in the words of the Rosh. But the Rosh himself takes a third direction. He disagrees both with the “some of the great authorities” and with those who were uncertain, and he says: “and it seems to me that they have the full law of a pit.” Now he proves that from the flow of the Talmud; the details aren’t important. He claims that it’s a pit. How did the Rosh read it as a pit? Very simply. The Rosh says this: we really can’t learn it from fire; we learn it from pit, okay? And in fact it is similar to a pit, right? A pit in the public domain—it isn’t its way to go and cause damage; rather, someone bumps into it and is injured. That really is a pit. Except what? I have a refutation against learning from pit. Why? Because another force is involved in it. Fine. So fire shows that another force being involved doesn’t interfere. You see that in fire we do impose liability. But the essence still remains a derivation from pit, because it really is similar to a pit. I only have a technical problem because there is some difference between this and pit, so fire will show me that this technical problem is not a problem. It removed the obstacle. The primary derivation—I’ll define in a moment what the common denominator is, and then it will be clearer. The Rosh claims that this is an analogy from pit. Fire only helps me remove an obstacle to the analogy, that’s all. There was, as it were, a refutation of the analogy if I tried it. I make an analogy from pit: how is a pit different? Because no other force is involved in it. So fire proves that this doesn’t work. Finished. Basically, I didn’t really need to continue with “the argument returns,” and so on. And in the language of the Talmud too, the Rosh is a bit difficult, and in the Talmud it really does seem that we need “the argument returns,” meaning that we need the common denominator. But that’s how the Rosh read the Talmud. And then basically this isn’t a derivation by common denominator at all. So the question is: then what is it? How did the “some of the great authorities” learn it, since they understood it really is a common denominator?

[Speaker C] But is that considered regular? The Rosh has some derivation that’s like a common denominator—

[Rabbi Michael Abraham] No, it’s not a common denominator.

[Speaker C] Not a common denominator—it’s an analogy from pit—

[Rabbi Michael Abraham] Only that fire removes the obstacle.

[Speaker C] And from an analogy like that, can you build on it?

[Rabbi Michael Abraham] Yes, of course. The Talmud says so. No, one is liable for his stone, his knife, and his burden.

[Speaker D] And you—

[Rabbi Michael Abraham] also receives the exemption of a pit: he’ll be exempt for vessels and exempt for a person. Everything—it’s completely analogous to a pit. That’s the difference, because fire shows that the difference isn’t important. Okay? So what’s really happening here is that the second source only helps the first source do its job. Meaning, the first source had some problem, and the second source removes the problem. A common denominator is not that. In other words, some of the great scholars understood that what we have here is a derivation by common denominator, not a derivation by a foundational paradigm with a training wheel, but a derivation of common denominator. How does common denominator work? So they say like this: basically I want to derive from fire, and I claim that what determines things is X, okay? Or Y, it doesn’t matter, the presence of Y or that X determines it—then I make an analogy, I derive it from fire. Then I say no, here we see that there is no X and there is Y, and nevertheless the law still applies. So clearly it depends neither on X nor on Y-prime. Right? Because here there is no Y-prime and also no X. Therefore the law really depends neither on X nor on Y-prime. So fine, let’s derive from pit—maybe there it depends on X-prime and Y? Also not. Here it’s exactly the opposite, and still the law applies. “The law” meaning that one must pay damages. In both cases one must pay. So where does the obligation to pay come from? From X? No, because I see that here there is no X. From Y? Also not, because I see that here there is no Y. So apparently there is some Z that is common to both, and that’s why one is liable, and that Z exists here too. What is that Z? Anything that is your property and whose safeguarding is your responsibility—you are liable to pay from the best of the land. So apparently, if neither X obligates payment nor Y obligates payment, then probably Z does it, and Z exists here too. From X and Y you can’t derive from each one independently, but Z exists here too. That is the common denominator; that’s why it’s called the common denominator. It’s called the common denominator because it’s the only characteristic they share—the common denominator. Exactly, the shared denominator between them, and therefore I derive from the common denominator. Now look, I’ll bring an example of how the Talmud challenges a common denominator. What it usually challenges it with is this kind of refutation: here there is a feature W, and there too, in both of them, there is W. Why is that a refutation? Because X-prime or Y-prime is a refutation against each one individually, but the other one shows that it’s irrelevant. But if there is a refutation against both of them,

[Speaker C] then

[Rabbi Michael Abraham] it knocks out the common denominator. Why does it knock it out? Because who says the obligation to pay depends on Z, in which case that would also be true in our case—or maybe it depends on W-prime, but here there is W, and therefore one is not liable. You don’t know. And we said that a refutation is enough if it shows you’re not certain, because there are two possibilities. Exactly like an a fortiori inference. That’s how one challenges the common denominator. Why am I already introducing this here? In a moment I’ll come back to it again, to explain to you why I claimed that here there has to be X and here there has to be Y. Because if you understand—after all, I started by saying that from here you can’t derive because this is X-prime and this is X, and from here you can’t derive because this is Y-prime and this is Y. Why am I claiming that here there must be X and here there must be Y? Because if here there were X-prime, then there would be a refutation against the common denominator: both of them have X-prime. And if here there were Y-prime, then there would be a refutation against the common denominator from the Y side, because there is a shared feature in both of them. It has to be that way—and this is always true throughout the Talmud—in every common denominator you find, one side will have X and the other X-prime; one side will have Y and the other Y-prime. Always. Otherwise there is no common denominator; otherwise the common denominator collapses. Okay? Now the interesting question is the following. Basically I have two options, and here I return to the analogy I made to an a fortiori inference. I now have, in effect, this table—it’s sort of like the table there. Okay? I have two options, two theories, let’s call them. What’s going on in the derivation? Basically what I did here is very similar to what I did there with the alcohol and those things—these X’s and Y’s. That’s really the theory behind it. I’m claiming that behind this there is some characteristic that causes liability, and behind that there is some characteristic that causes liability. I construct the characteristics and try to see whether one can derive or cannot derive. Now some kind of miracle happens here: from this you can’t derive, and from that you can’t derive either. But from both together, without adding anything, from both together you can. I explained that miracle by saying: X is irrelevant, Y is irrelevant, so apparently there is some Z that is relevant. But it’s not that simple. Why not? Because there could be two… there could be a red theory that says either X or Y obligates payment. Then C would not be liable, right? This one has X and that one has Y, but this one—sorry, or… my combination is either X-prime or Y-prime, it doesn’t matter. So either X-prime or Y-prime—and this one has neither of them, right? Then it turns out that the common denominator won’t work. The second theory is the black theory: Z. Right? If the theory is that what creates liability is Z, that still explains both of my primary categories, so of course this one too would be liable, because it also has Z. Now the question is, wait a second—now the question is: how do I choose between the two theories? Which one is more correct? Simplicity. This one ties liability to one parameter, and that one ties it to one of two parameters. Exactly like with alpha and beta here. That’s why it’s exactly the same thing.

[Speaker D] Not exactly. Why? Because Z itself is composed of two… of several things, it has several components.

[Rabbi Michael Abraham] Wait, wait, wait.

[Speaker D] No, no, no.

[Rabbi Michael Abraham] I don’t identify that at all. I know—you already identify it because the Mishnah already did the work for us and identified it, but I don’t identify it, and I claim there has to be some Z that is neither X nor Y.

[Speaker D] No, but if Z itself is made up of several things, then you took the… you took the burden of one…

[Rabbi Michael Abraham] But who says it’s made up of several things? Because you identified it? I said I’m not identifying it. I’m saying: if not X and not Y, then there is another parameter, Z, that we don’t know. Second, I think that’s not true anyway, even if I do identify it. Why? Because when I say, for example, if there were X-prime and also Y-prime here, that wouldn’t make it more complex. “Or” is what makes it complex. Because if X-prime and Y-prime—call that Z—that’s one characteristic. It’s not either this or that. It’s one characteristic. Any characteristic—for example, you want it to be his property—well “his property” can also mean that he acquired it by lifting it, or acquired it by pulling it, or acquired it through his courtyard as a mode of acquisition. Okay? So is that also complex? No, because as long as I say that it comes together as one thing, then it’s one thing. It becomes complex only when you say either this or that; then it’s already a more complex theory. What’s the disadvantage in the theory here? What? It’s X or Y. I didn’t understand.

[Speaker C] If you do X or Y,

[Rabbi Michael Abraham] and X or Y is also an “or,” meaning they’re leniencies, so that can’t be.

[Speaker C] and that will lead you to a different conclusion.

[Rabbi Michael Abraham] But X or Y can’t be, because those are leniencies. We’re looking for stringencies that create liability, not leniencies that create liability. The fact that another force is involved is not a reason to obligate; it’s a reason to exempt.

[Speaker C] If you want there to be a conclusion…

[Rabbi Michael Abraham] No, that’s why I’m saying X-prime and Y-prime, not X or Y. To obligate, I mean X-prime.

[Speaker C] What if there is another Z?

[Rabbi Michael Abraham] Besides Z there is also S. Where is it found?

[Speaker C] It’s there. Fine, then let’s call it Z. No, no, no, maybe there are

[Speaker B] several types.

[Rabbi Michael Abraham] Call S “Z2,” it doesn’t matter.

[Speaker B] No, not Shakespeare whose cousin was also called Shakespeare.

[Rabbi Michael Abraham] So I’m saying that in the end here too, here too I have two theories, and I choose the simpler one, exactly parallel to what happens there. So here too, behind the matter there sits a theory. If we understand that behind the inference there sits a theory, a lot of things become clear to us. In a moment we’ll see more things that become clear.

[Speaker D] But how did the Mishnah know? Why did the Mishnah present the X and the Y? Meaning, the ox and the horn?

[Rabbi Michael Abraham] After we learn that this is really tooth and foot, those are their laws. Only for the sake of the exemption. The Talmud itself says this; it has nothing to do with me. The Talmud says on page 5 that the Mishnah taught them for their laws. Only for the sake of the exemption.

[Speaker C] But how does the Mishnah know the rule, according to what you’re saying? Maybe it’s X or Y, and then the exemption would be simpler.

[Rabbi Michael Abraham] No—according to the conclusion, what is simpler?

[Speaker C] What is simpler?

[Rabbi Michael Abraham] That there is one rule, one parameter that determines the characteristic it gave you—not either that another force is involved, or that its initial formation was for damage, or all sorts of things. No. “Either-or” is not simple; it’s complex. The simpler explanation was chosen. The Mishnah itself already did that mediation. The Talmud only decodes what the Mishnah means when it says: “their common denominator is that they are your property and their safeguarding is your responsibility.” The Talmud exposes what is already done in the Mishnah; this is really a move the Tannaim already made. So now look, first I’ll show how one goes from this presentation to that presentation. I’ll try to show you what a common denominator looks like in a table. Moon and stars—that’s even less important. Fine? Now let’s fill in the table. How does this thing actually work? We start with an a fortiori inference from tooth and foot to horn. Fine? We obtain the conclusion that here there is a one. Those are these four. Okay? Now there is a refutation from the moon. Right? Now I say, okay, then let’s—just throwing out another little thing again—derive from pit. How do we derive from pit? Again the same thing, zero and one. We make an a fortiori inference from pit. Right? Then we say there is a refutation, zero and one, another refutation on pit, right? So that doesn’t work. Now we make a common denominator. What do I fill in here? Exactly the consideration I made here. If here there is X, then here there must be X-prime. Meaning this is zero and this is always zero. Everywhere in the Talmud it will always be zero. Because if it isn’t zero, then there is a refutation against the common denominator. And if here there were a one, that would mean there is a refutation against the common denominator. So I’m really showing you: this is exactly like here, I’m just translating it into a table. Okay? So suddenly we see that the common denominator too is nothing but a data table in which I want to fill in this cell. And if so, then I can do exactly what I did with an a fortiori inference: look for a theory that explains all these data and see which theory is simpler, and that’s how to fill it in. We’ll do that. For now I’m just trying to show how this transition happens. Now I’ll say more than that: there are three types of common denominator in the Talmud. There cannot be more. That’s the nice thing—when you build it systematically, you don’t need to go through the whole Talmud to see it. There cannot be another type. That’s clear; I don’t have to scan the whole Talmud to know that. One type is where in both these cases there is a zero. These two cells are really the cells that can change. If both of them are zero, that’s a common denominator that begins with two a fortiori inferences. An a fortiori inference from tooth and foot to horn, and from pit to horn, two refutations, and then a common denominator. Now it could be that here there is a one, and then from tooth and foot to horn it’s a foundational paradigm. It’s not an a fortiori inference. There is a refutation, this refutation, and then one makes an a fortiori inference from pit. There is a refutation, and then a common denominator. Of course, if here there is a one and here a zero, that’s also possible, but it doesn’t matter, switch the rows—it’s the same thing. The third case is where there are two foundational paradigms on both sides, each is challenged, and then we make a common denominator. Fine? So there are really three types of common denominator in the Talmud, and all three exist, and only these three. No more. Okay? That will be important for me later. So I’ll say it again: there are three types—either in both places it’s zero, or in both places it’s one, or in one place it’s zero and in the other it’s one, it doesn’t matter which, upper or lower. Okay? These are the three types of common denominator, which basically begin with the question: what are the two derivations, what are these two arrows? If this arrow is an a fortiori inference and this one is an a fortiori inference, then it’s zero and zero. If this is an a fortiori inference and this is a foundational paradigm, then it’s zero and one. And if both are foundational paradigms, then it’s one and one. Okay? These are the three types that exist. There are, by the way, differences between them; the authors of the methodological rules claim there are differences between them. Now okay, so for the moment I’m leaving that. Now let’s move on to another Talmud passage.

[Speaker C] A Talmud passage in tractate Makkot, page 4.

[Rabbi Michael Abraham] This is something that recurs in several places, also in Ketubot 32, but here the medieval authorities are on it, so I’m bringing this passage. “This implies that Rabbi Yehuda holds that for a prohibition that does not involve an action, one is lashed.” Rabbi Yehuda holds—practically, according to Jewish law, one is not lashed, but Rabbi Yehuda holds that one is. From where does he derive it? Ulla said: he derives it from one who spreads a bad name. He derives it from motzi shem ra. Just as one who spreads a bad name involves a prohibition without an action, because it’s only speech, and yet one is lashed for a prohibition without an action, so too in every prohibition without an action one is lashed. Fine? We derive from motzi shem ra. Motzi shem ra is our A. Okay? The Talmud says: how can you compare it to motzi shem ra, for there he is both lashed and made to pay. True, he is lashed, but he also pays—so there is some stringency there from which one cannot derive. Okay? Rather, Reish Lakish said: he derives it from conspiring witnesses. That is B. Fine? We derive from conspiring witnesses. Just as conspiring witnesses involve a prohibition without an action—again, only speech—so too one is lashed for a prohibition without an action; likewise for every prohibition without an action. By the way, these two teaching sources are foundational paradigms, right? Not an a fortiori inference. Meaning: just as this is so, so too that is so. There is no a fortiori inference here, only: just as that is a prohibition without an action, so too this is a prohibition without an action. So the Talmud says: how can you compare it to conspiring witnesses, for they do not require prior warning. That is X-prime. Earlier it was Y-prime; now it is X-prime, namely that they do not require prior warning. So the Talmud says: motzi shem ra will prove otherwise. And the argument comes back around: this is not like that, and that is not like this; rather, the common denominator between them… What is the common denominator between them? That they are fines. Both conspiring witnesses and motzi shem ra are fines. Well, conspiring witnesses—is there not a tannaitic dispute whether that is a fine or not? The Talmud says: no difficulty, Rabbi Yehuda follows Rabbi Akiva, who holds that it is not a fine. Once you say that motzi shem ra is a fine and conspiring witnesses are not a fine, then that is not a refutation. Why is it not a refutation? Because it only means there is a refutation against one of the two teaching sources, so the other one proves otherwise. Add it in—even if there is another refutation, that doesn’t interest me, because the other source proves otherwise; it’s irrelevant. To challenge a common denominator, you need to find a refutation that exists in both teaching sources. The Talmud says: rather, what is the common denominator between them? That both contain a stringent side. What is the common denominator between them? That either X-prime or Y-prime exists—in each one there is a stringent aspect relative to the derived case, and therefore one cannot derive. Now this is absolutely astonishing. Why?

[Speaker D] Let’s arrange this again on the board.

[Rabbi Michael Abraham] Tosafot asks in Ketubot: according to this, then you can throw common denominator out of the Talmud, throw out a foundational paradigm from two verses from the Talmud—everywhere it’s like this. After all, that’s the structure, we said; that’s always the structure. I start from this and say no, here there is X-prime. I start from that and say no, here there is Y-prime. Fine? So what? Apparently not X is relevant, not X-prime is relevant, not Y-prime is relevant—apparently it’s Z. What do you mean? “How can you compare the two, seeing that they both have a stringent side”—this one has X-prime and that one has Y-prime. So the logic of the common denominator collapses here; this is something specific. In other words, it really undermines the logic of common denominator. Or in other words, what it says is: after all there is also the red theory, not just the black theory, right? That is really what it says. Maybe what determines things is either this stringent side or that stringent side—who told you it’s Z? Right? That is really what it says. But we already explained Ockham’s razor; after all, that’s the simpler theory, so what’s the problem? And if that exists, then what kind of refutation is this? Tosafot asks this in Ketubot, yes. And here the Ritva brings four approaches among the medieval authorities to explain the stopping point in Makkot 4b. So the Ritva brings four approaches here, and Tosafot in Ketubot also brings—there the Rosh writes that this is an especially unusual sort of side. I don’t know if—whatever, the answers of the medieval authorities are not convincing to me. I’ll tell you what I think the correct answer is. In light of what we saw earlier, the answer is simple.

[Speaker C] Look, is it simply because he doesn’t accept the principle that the simpler one wins? Is that the answer?

[Speaker D] And if he doesn’t accept it, then let him challenge every common denominator?

[Speaker C] No.

[Rabbi Michael Abraham] So then he has no common denominator at all? No, what that

[Speaker C] means

[Rabbi Michael Abraham] is that this challenge of a stringent side appears in maybe three or four places in the Talmud, no more. Common denominator appears in many places. So he doesn’t accept common denominator? Is he deleting a foundational paradigm from two verses from Rabbi Yishmael’s list?

[Speaker C] So that’s the difficulty, that’s the question.

[Rabbi Michael Abraham] So I’ll try to explain why it’s not difficult, to resolve it.

[Speaker C] To resolve Tosafot, as it were?

[Rabbi Michael Abraham] To resolve Tosafot, the Talmud, the question of Tosafot. Meaning: why according to the Talmud, if one challenges with a stringent side, then there is no such thing as common denominator at all? That’s what Tosafot asks, and what all the medieval authorities ask—the Ritva, the Ra’ah, all of them, Nachmanides, all ask this. In other words, how can there still be common denominator if one can challenge it with a stringent side? That’s what I’m trying to explain. And it’s simple.

[Speaker C] Wait, but does the Talmud actually say this—“a stringent side”?

[Rabbi Michael Abraham] Yes. “How can you compare the two, seeing that they both contain a stringent side?” That is the Talmud’s refutation. Look, there is a difference between the previous example I brought from Bava Kamma and this example. In Bava Kamma, what were the two refutations against pit and against fire? And in general, by the way, it’s a very unusual passage at the beginning of Bava Kamma. The refutations are: another force is involved in it, and its way is to go and cause damage. Right? Those are what I call factual refutations. What do I mean? They are not halakhic characteristics. It’s not that pit is exempt from this and that—that’s not a halakhic category but a factual one: a pit doesn’t move; a pit stands still, right? That’s a factual characteristic of the two teaching sources. The refutations in Makkot are halakhic refutations. Conspiring witnesses don’t require prior warning; motzi shem ra is lashed and pays. That’s not a characteristic of what motzi shem ra is, like moving one’s lips. If they had said, “How can you compare the two, seeing that in both cases they move their lips?”—that would have been a factual refutation. But here they bring a law that shows they are more stringent. Why is that important? Simple. If the halakhic refutations cannot be—the alphas and betas are always factual. Right? They must be factual. Because the law asks: what is there in tooth and foot that causes it to create liability—or in horn, or in the damaged party’s courtyard, right? You can’t tell me that what there is in tooth and foot causes it to create liability in the damaged party’s courtyard—after all,

[Speaker C] that’s the law—you’re asking what creates the liability.

[Rabbi Michael Abraham] Exactly. So it is always a factual characteristic. The laws are an indication that there is a factual characteristic, right? When I bring a refutation here, I’m saying: after all… tooth and foot are liable, horn is exempt. What does that mean? That’s a law: where here he is liable and there exempt—that is a halakhic refutation. But behind it sits gamma. In other words, behind it sits something saying: apparently there is some additional factual parameter at work here, and who knows what it is doing now. It’s making a mess here, reshuffling the cards. Okay? Now look. If my characteristics are factual characteristics—here there is one factual characteristic and there there is another factual characteristic—the common denominator between them doesn’t help. Because this is simpler than that. Why not? But if the refutations are halakhic refutations, then what are you saying? That here there is the law X-prime and there the law Y-prime. But it could be that the law X-prime here and the law Y-prime there are both expressions of the same factual parameter, alpha. The cause is the same cause. In a different context it comes out in the form of a different law. But what interests me is not the law. The law is only a sign. It could be that behind it stands the same type of stringency, which simply is not present there in exactly that form.

[Speaker C] And once that holds for two other cases,

[Rabbi Michael Abraham] yes—all the other cases, all the places in the Talmud where they make the challenge of a stringent side, it is always a halakhic refutation. By the way, generally speaking, refutations of common denominator are usually halakhic refutations and not factual ones. What does that mean? By the way, all the characteristics of the four primary categories of damages are factual characteristics.

[Speaker C] You didn’t ask us. The laws rest on the factual thing.

[Rabbi Michael Abraham] Exactly. But since only the laws are different, it could be that the fact which generates those laws is the same fact. The only thing is that, depending on the context, it has a different halakhic stringency. Now understand: I don’t need to prove that. It’s enough for me to raise the possibility for it to count as a refutation. And that is why it is a refutation. So the challenge of a stringent side does not mean that there is one factual characteristic standing behind the two laws; rather, since these are only two laws and not two facts, you still haven’t proved it. Maybe behind them there is one fact, and therefore you still haven’t proved your case. You see, for example, why this analysis is so important. Once I understand that behind all the laws there is a theory—in other words, I’m not speaking only about the laws, I’m saying that behind them there are alphas, betas, X’s, and all kinds of theories of that sort—suddenly many things fall into place. Things with which the medieval authorities got tangled up because they didn’t make this analysis. So look at the answers there: really unconvincing answers, both in the Ritva and in Tosafot in Ketubot. And I think that when one understands it this way, then it’s simple—it’s not even an answer, it’s obvious. It was so obvious to me that I don’t really understand why the rabbis disagree with Rabbi Yehuda and do challenge with a stringent side. There is a tannaitic dispute—I didn’t bring the continuation of the passage—there is a tannaitic dispute whether to challenge with a stringent side or not. In light of what I explained here, this is no longer an answer; it’s the plain meaning.

[Speaker C] But in Jewish law, what do we rule?

[Rabbi Michael Abraham] In Jewish law I think we do challenge with a stringent side, and the point is this. Wait, let me look at it with a little more confidence. The Jewish law follows the rabbis; let me see who says what.

[Speaker C] There’s a problem here with that.

[Rabbi Michael Abraham] Wait, I’ll get to it in a second. In any case they need to be explained, regardless of the practical ruling. We need to understand what they’re saying. In a moment. Rabbi Yehuda says that a stringent side is not a refutation—sorry, Rabbi Yehuda says that a stringent side is not a refutation, which means that the rabbis do challenge with a stringent side, and in practice the Jewish law does challenge with a stringent side. So yes, that is considered a refutation. And then I ask: what is the logic behind that? Why challenge with a stringent side? It seems to me that the logic behind it is that if it comes out in different forms in the two contexts, then it is likely not the same factual characteristic. Because if it were the same factual characteristic, it would create the same stringency. Suppose they had said that motzi shem ra does not require prior warning, just as conspiring witnesses do not require prior warning. If it really is the same characteristic, then why in motzi shem ra is the result that he is lashed and pays, whereas in conspiring witnesses the stringency is that they do not require prior warning? If it is the same factual characteristic in both, I would expect the halakhic expression also to be the same halakhic expression. Now there are places where the halakhic expression of one is not relevant in the other. If there they do not pay at all, then there is no point in saying that he should be lashed and pay there. But in the case of conspiring witnesses and motzi shem ra it is relevant. One could have said that motzi shem ra does not require prior warning, or that conspiring witnesses are lashed and pay, right? Why don’t they say that? So the sages say: apparently it is not the same factual characteristic, but two different ones, and therefore one does challenge with a stringent side. Fine? Therefore there is room to argue about it, but the one who does challenge with a stringent side, it seems to me that this is why he does it. And now even more than that—you could ask: so why not in every place in the Talmud where the refutations are halakhic refutations does the challenge of a stringent side arise? So as I just said, there may be places where it is clear that it could not emerge in the same form in both places, and therefore there is no room to challenge with a stringent side; and then it really is a matter of reasoning case by case. So what I’m saying is: I found a mechanism. How to apply it in each place—you have to sit with each topic and see. This is not mathematics where you can just hand it to a computer. Obviously you have to understand what is going on here. But once we understand these things, in the end it seems to me that we have a fairly orderly set of tools for dealing with this issue. And what I’ll do in the next stage—what I’ll do in the next stage—I’ll try to show…

[Speaker D] Alan bar Dor, Alan bar Dor, Alan bar Dor, Alan bar Dor.

[Rabbi Michael Abraham] What I’ll do next time is return to this table, okay? Or to these tables. And I’ll try to show how to do it systematically. How I construct a theory systematically for this table. I’ll do the three common-denominator tables, because from each of them something else interesting comes out. It’s a very interesting thing. And then I’ll summarize; I’ll simply show the most general form—on any table of any size you can do the same thing. And it always works, at least as far as I’ve checked. So what are you innovating here? How to fill in the cell in every situation.

[Speaker C] But then what? What are you trying to say that…

[Rabbi Michael Abraham] I’m trying to show the logic of non-deductive inferences—that there is a logic behind them. I’ll still talk about that. Okay.