Midrash and the Principles of Interpretation – Lesson 10

Table of Contents

• Biblical a fortiori reasoning versus Talmudic a fortiori reasoning
• Two formulations: “columns” and “rows,” and the claim that an a fortiori argument collapses under a refutation
• An indication from differing outcomes: half versus one, and the rule of dayo
• A parameter model, Occam’s razor, and understanding a refutation as an attack on the assumptions
• Why “columns” and “rows” are the same thing when the table is valid
• The physics-philosophy example, the psychometric exam, and constructing the table as belonging to one semantic field
• Rejecting an a fortiori argument a priori, and the mezuzah-tzitzit and impurity-damages examples
• Refutation versus proof, and the parallel to deductive logic
• The history, mathematics, and Descartes example: “cogito”
• The common denominator in Bava Kamma: “what does it come to include?” and Abaye
• Analysis of the Talmudic passage: fire, pit, “the law returns,” and the logic of Occam’s razor
• The common denominator as scientific induction, and the example of falling objects
• Ketubot: Ulla, monetary payment and lashes, and the difficulty of “a stricter side”
• A proposal: distinguishing between factual stringencies and halakhic / of Jewish law stringencies, and explaining the refutation of “a stricter side”
• The Rosh on his stone, knife, and load: vessels, concealed objects, and three approaches
• Conclusion and guidance

Summary

General Overview

The text presents an understanding of the Talmudic a fortiori argument as an inference based on a table containing three known data points and one missing conclusion, and explains that the key is the assumption of a hierarchy governed by the same relevant parameter. It argues that a refutation does not “turn around” the a fortiori argument but rather knocks down the assumptions that make it possible to construct the table as a single semantic field. Therefore, in the Talmud, an a fortiori argument collapses when faced with a refutation unless it can be rescued. It then applies this same principle to the structure of the “common denominator” in Bava Kamma, proposing an analysis of it as a choice of the simpler theory using Occam’s razor, and presents a difficulty of the type “what about the two source cases, which each have a stricter side?” in Ketubot, together with a proposal to distinguish between factual stringencies and halakhic / of Jewish law stringencies. Finally, it cites the Rosh on his stone, knife, and load, and presents three approaches, including a reading according to which the move is not really a “common denominator” but rather an analogy to a pit, with fire serving only to remove the refutation.

Biblical a fortiori reasoning versus Talmudic a fortiori reasoning

The biblical a fortiori argument rests on a single assumption and a hierarchical relation that allows one to infer from the lighter case to the more stringent one. The Talmudic a fortiori argument usually rests on three data points in a table and seeks to complete a fourth, missing datum. The structure is presented through an example from the laws of damages involving damaging agents—consumption and trampling versus goring—and domains—public domain versus the damaged party’s courtyard—where three cells are known and one cell is missing.

Two formulations: “columns” and “rows,” and the claim that an a fortiori argument collapses under a refutation

The “columns” inference produces a hierarchy among the damaging agents, according to which goring is more severe than consumption and trampling in the sense that it is easier to impose liability for it, and then projects that to the damaged party’s courtyard in order to impose liability for goring there. The “rows” inference produces a hierarchy among the domains, according to which it is easier to impose liability in the damaged party’s courtyard than in the public domain, and then projects that from goring in the public domain to goring in the damaged party’s courtyard. The text argues that at first glance these are two different arguments, because the hierarchy in the “columns” is among damaging agents while the hierarchy in the “rows” is among domains. But in practice, in the Talmud, a refutation knocks down the a fortiori argument, and one does not simply “turn it around” into the other formulation.

An indication from differing outcomes: half versus one, and the rule of dayo

The text points to a case in which goring in the public domain entails half-damages rather than full damages, and shows that the column formulation may lead to a result of one, while the row formulation may lead to a result of half because of the rule of dayo—that what is derived from the inference cannot exceed its source. It concludes that different outcomes emerge, and so the formulations appear to be different. It adds that in practice, “turning around” an a fortiori argument appears almost only in tractate Niddah and tractate Bava Kamma in contexts of tables involving half-values and dayo, whereas in symmetric tables there is no turning around.

A parameter model, Occam’s razor, and understanding a refutation as an attack on the assumptions

The text describes the a fortiori argument as an attempt to build a model that explains the three known data points through parameters such as alpha and two-alpha, and from that explanation infer the missing datum. It presents an alternative possibility of a two-parameter model—alpha and beta—that also explains the three known data points but yields a different filling for the missing cell, and argues that the choice of the correct model is made according to the simplicity of the theory, using Occam’s razor. It describes a mechanical implementation in which one fills the missing cell once with zero and once with one, looks for a model for each filling, and chooses the simpler model in order to determine the result.

Why “columns” and “rows” are the same thing when the table is valid

The text argues that each of the two formulations of the a fortiori argument actually assumes both hierarchies together, because in order to explain the three data points, the difference between the domains and the difference between the damaging agents must both be measured by the same relevant parameter. It explains that if the hierarchy among the domains were in terms of beta and the hierarchy among the damaging agents in terms of alpha, then the greater severity among the damaging agents would not be relevant to liability in the domains, and there would be no basis for the inference. It concludes that when a refutation forces a move to two parameters, it collapses the a fortiori argument in either formulation, so turning it around does not save it.

The physics-philosophy example, the psychometric exam, and constructing the table as belonging to one semantic field

The text illustrates an everyday a fortiori argument: if Shimon got 60 in physics and 80 in philosophy, then Reuven, who got 70 in physics, should get at least 80 in philosophy. It shows that the problem is that talent for physics is not necessarily the same talent as talent for philosophy. It presents a refutation by means of another subject, such as chemistry, which reveals a reversal in the hierarchy and teaches that there is more than one parameter, so one cannot infer philosophy from physics. It uses this to explain criticisms of the psychometric exam as a composite measure that does not necessarily predict success in every field, and argues that placing two domains into the same table reflects a hidden assumption that they are governed by the same parameters.

Rejecting an a fortiori argument a priori, and the mezuzah-tzitzit and impurity-damages examples

The text presents an a fortiori argument concerning mezuzah and tzitzit—a four-cornered garment versus a doorpost—and argues that it is rejected not because of a refutation but because the same parameters are simply not relevant across the two domains. It also presents an attempt to infer from domains in the laws of doubtful impurity to domains in damages, and argues that the parameters determining doubtful impurity are different from the parameters determining monetary liability. It states that a refutation attacks the very assumption that the data may be placed in the same table, not the “mathematics” of the inference once the table has already been set up.

Refutation versus proof, and the parallel to deductive logic

The text argues that deductive logic is not refuted except by attacking the assumptions, and compares this to the a fortiori argument and the argument from a paradigm case, where the refutation attacks the assumptions. It emphasizes an asymmetry: a refutation does not prove the opposite; it only shows that you did not prove your conclusion, because a single plausible alternative is enough to undermine the argument. It adds that the gap between deduction and “non-deductive” inferences is smaller than people think, because the necessity lies in what follows from the assumptions, while the uncertainty stems from the possibility of challenging the assumptions.

The history, mathematics, and Descartes example: “cogito”

The text illustrates this with history, by transferring from Africa to Europe a rule according to which “more soldiers win,” and argues that technology is an additional parameter that collapses the one-dimensional assumption. It distinguishes between mathematics, where the assumptions are definitions and are not attacked, and science, law, and Jewish law, where the assumptions are claims about the world and can therefore be challenged. It brings Descartes’ argument, “Cogito ergo sum,” and argues that the difference between that and “I walk, therefore I exist” lies not in the inference but in the validity of the assumption. There is also a reservation raised in the discussion, according to which the very knowledge “I think” relies on inner observation and is therefore not merely a definition.

The common denominator in Bava Kamma: “what does it come to include?” and Abaye

The text cites the opening of Bava Kamma: “There are four primary categories of damages,” and the end of the Mishnah: “The common denominator among them is that they are your property and their supervision is upon you…” It presents the Talmud’s question, “What does the common denominator among them come to include?” and explains that this stems from the casuistic character of the Mishnah and Talmud, which prefer cases over abstract rules. Abaye explains that the common denominator comes to include “his stone, his knife, and his load, which he placed on top of his roof and they fell in a normal wind and caused damage.”

Analysis of the Talmudic passage: fire, pit, “the law returns,” and the logic of Occam’s razor

The text follows the successive rejections: if they caused damage while moving, then this is fire; if they caused damage after coming to rest and he had declared them ownerless, then this is a pit. But it is argued that they are not similar to a pit, “because no other force is involved in it,” whereas here the wind participated in creating the hazard. It presents the comparison to fire, which proves that liability is possible even though “another force is involved in it,” and then fire is itself rejected on the grounds that “its way is to go and cause damage,” which is not true of his stone, knife, and load once they have come to rest. It interprets “the law returns” as an inference that those two special stringencies are not the determining parameter, and therefore there must be some other shared characteristic that explains the liability. The choice of that characteristic is made according to the simplicity of a one-parameter theory as against a two-parameter theory.

The common denominator as scientific induction, and the example of falling objects

The text compares the common-denominator move to scientific generalization: a book falls, a pen falls, and therefore “paper” and “roundness” are not the determining cause; rather, it is some shared property such as mass. It argues that this parallels the conclusion in damages that the relevant factor is “they are your property and their supervision is upon you,” and therefore his stone, knife, and load create liability even though they are not similar to fire or pit in their special stringencies. It notes that even here the full identification of the decisive factor may remain open, but the logical mechanism is the choice of the simpler theory.

Ketubot: Ulla, monetary payment and lashes, and the difficulty of “a stricter side”

The text cites Ulla: “Wherever there are both monetary liability and lashes, he pays the money and does not receive lashes,” and the attempt to derive this from one who injures his fellow, which is rejected because “he is liable for five categories of payment.” It then brings the derivation from conspiring witnesses, which is rejected because “they do not require prior warning,” and the conclusion: “rather, derive it from both of them,” in the structure of a common denominator. It presents the Talmud’s difficulty: “What about the two source cases, since they have a stricter side?” and Tosafot’s question that if this is a valid refutation, then “there is no common denominator anywhere in the universe.”

A proposal: distinguishing between factual stringencies and halakhic / of Jewish law stringencies, and explaining the refutation of “a stricter side”

The text proposes that there is a difference between stringencies that are factual characteristics—such as “its way is to go and cause damage” and “its initial formation was for damage”—and stringencies that are halakhic / of Jewish law characteristics—such as “they do not require prior warning” and “he is liable for five categories of payment.” It argues that behind a halakhic / of Jewish law stringency there stands some factual characteristic that explains why the law is more severe, and therefore it is possible that two different halakhic / of Jewish law stringencies reflect one shared severe factual characteristic. It concludes that in such a case there is real room for the refutation of “a stricter side,” because the two source cases may both be liable due to the same severe factual characteristic that does not exist in the target case. By contrast, where the stringencies are different factual characteristics, there is no such concern, and the refutation of “a stricter side” is not natural.

The Rosh on his stone, knife, and load: vessels, concealed objects, and three approaches

The text cites, in the name of the Rosh, three methods regarding the exemptions of fire and pit when one imposes liability for his stone, knife, and load through the common-denominator argument. It presents one opinion that “one is liable only for what both source cases are liable for,” and therefore he is exempt from damage to vessels, like pit, and from concealed objects, like fire, because “since they were derived through the common denominator, we assign them the lesser element of the two.” It presents another view of uncertainty between sweeping liability and sweeping exemption, together with the argument that in order to exempt one needs a source from both source cases, and for concealed objects or vessels there is no shared exemption. It presents the Rosh’s ruling, according to which there is exemption for vessels like pit, but no exemption for concealed objects, and explains this as a reading according to which the move is not a true common denominator but fundamentally an analogy to pit, with fire serving only to remove the difficulty of “another force is involved in it,” and not as a basis for establishing the exemptions of fire.

Conclusion and guidance

The text concludes by emphasizing that a refutation rests on “maybe,” and therefore is not a proof of the opposite but only the negation of a proof, and ends with a blessing: “Wishing you success on the exams. Thank you very much.”

Okay, last time we basically talked about a kal va-chomer, an a fortiori argument. And I said that a kal va-chomer usually—meaning, there is the biblical kal va-chomer, let’s call it the primitive one. Primitive not as an insult, but as the initial form. It’s based on one premise and a hierarchical relation. And from that one premise, which is the lighter case, together with the hierarchical relation, you can derive the conclusion about the more stringent case. Then the Sages come and say that there is an interpretive rule of kal va-chomer, and that interpretive rule, usually—in the context I call the Talmudic kal va-chomer as distinct from the biblical one—that kal va-chomer is based on three data points, not one. And we want to fill in the fourth data point. And usually the structure—let’s call it the spaceship structure. Okay, so like this: we had a kal va-chomer built, say, on damagers and domains. Damagers means tooth and foot and horn, and the domains are the public domain and the damaged party’s courtyard. Usually a kal va-chomer is built in such a way that here there’s zero, here there’s one, here there’s one, and this is what I don’t know. Okay? This is what I want to fill in. Now what do I do? So we explained that apparently there are two inferences, or two arguments, which on the face of it look different. The first argument—let’s call it the columns argument—is the argument that goes by the two data points in the right-hand column, extracts from them a hierarchical relation that horn is more severe than tooth and foot. “More severe” meaning it’s easier to obligate payment for it than for tooth and foot. Then I move to the left-hand column and say that if tooth and foot, for which it is harder to obligate payment, is liable in the damaged party’s courtyard, then horn, for which it is easier to obligate payment, will certainly be liable in the damaged party’s courtyard. That’s the columns inference. A kal va-chomer is basically an a fortiori argument. Right. Why do they call it kal va-chomer? “Chomer” is a noun—not a verbal noun, maybe an adjective, I don’t know what exactly to call it—the severe thing is called the chomer. Like chumra? No, no. Take, for example, the word “knowledge.” What is knowledge? Knowledge is seemingly an action, but on the other hand the knowledge that exists within me is the product of the action, right? It becomes a noun, a verbal noun. Okay, in this case it’s not exactly the same thing but it’s similar. “The chomer” is like the severe thing; that’s what it’s called. There is the light and there is the severe. I don’t know, that’s just how people say it. In any case, the second inference is the rows inference. I take the top row, and from it I infer that in the damaged party’s courtyard it is easier to obligate payment than in the public domain. Then I carry that over—that’s a hierarchical relation—and I apply that hierarchical relation to this data point. If horn, for which it is harder to obligate payment, is liable in the public domain—so in the public domain, where it is harder to obligate payment, horn is liable—then in the damaged party’s courtyard, where it is easier to obligate payment, horn will certainly be liable. On the face of it, this looks like two different arguments, because basically what I’m doing is taking the three known data points: one data point serves as the standalone premise, and from the other two data points I derive the relation—a hierarchical relation. And from that point on, you have the biblical kal va-chomer: one standalone datum, a hierarchical relation, and I derive a conclusion. Now the hierarchical relation is the core of the whole thing, but the hierarchical relation in the columns and in the rows is different. Right? In the kal va-chomer of the rows, the hierarchical relation establishes a relation between domains: that the damaged party’s courtyard is more severe than the public domain. In the kal va-chomer of the columns, the hierarchical relation establishes a relation between damagers: that horn is more severe than tooth and foot. Okay? Therefore, these arguments appear on the face of it to be completely different. But if that were so, then I would expect that a refutation of a column or a refutation of a row would not undermine the kal va-chomer; it would only force us to rotate it. Instead of working with the rows, we’d work with the columns, or vice versa, but basically the kal va-chomer would still stand. Okay? But in the Talmud we don’t find that. In the Talmud, once a refutation is brought, the kal va-chomer falls. They almost never rotate a kal va-chomer. I brought an indication for this, if you remember, that horn in the public domain—in truth it’s not one but half. And therefore there’s a difference between the arguments. If I go by the columns argument, then from here it follows that horn is more severe than tooth and foot. Now I move to the left column, so if tooth and foot, which is the lighter one, is liable for one, then horn, which is more severe, is certainly liable for one. The result is one, right? That’s the result of the columns kal va-chomer. But in the rows version, from here it follows that the damaged party’s courtyard is more severe than the public domain. And if I go down here, in the public domain horn pays half, so the damaged party’s courtyard has to be more severe than half, at least half, and then the rule of “it is enough for what is derived to be like the source” makes it half, so the result is half. And if the results are different, that means the arguments are different. Two identical arguments wouldn’t give two different results. There are several indications that these are different arguments, but as I said, the big problem is that in the Talmud we don’t find that they rotate a kal va-chomer. If you raise a refutation, that doesn’t make them rotate the kal va-chomer; it means the kal va-chomer has collapsed, unless you can save it somehow. But not by rotating it. The only two places where they do rotate it are tractate Niddah and tractate Bava Kamma regarding this kal va-chomer. And in both cases it’s simply a table of this sort that contains a half, a table in which the rule of “it is enough” applies. In symmetric tables they never rotate it. Why in this case they do rotate it—I’m not going to have time to get into—but why in the regular kal va-chomer cases they don’t rotate it, that I explained last time. Okay. I said that basically when I make such a kal va-chomer, I’m trying to build a model that will explain the three known data points. Forget the half for now, we don’t need it. Okay? A model that explains the three known data points, and from that model, or that explanation I’ve found, I can derive a conclusion regarding the missing datum. So in this case, say horn has alpha and tooth and foot have two alpha. Alpha and two alpha. In the public domain what do we have? Two alpha, right? That’s why tooth and foot aren’t liable there. Right? Tooth and foot aren’t liable there because tooth and foot have only alpha, and in order to obligate in the public domain you need two alpha. Whereas in the damaged party’s courtyard, alpha is enough. Okay? So that’s the model. Now once that’s really the model, then we look now—okay, so what will the law be for horn in the damaged party’s courtyard? No problem. Once here, with horn, it has two alpha, and in order to obligate in the damaged party’s courtyard one alpha is enough, if you have two alpha then of course it will be liable. The result is one. And I explain it just as in science. I take a scientific phenomenon, or scientific phenomena, okay? I propose an explanation for them: all bodies with mass attract each other. And then I can derive conclusions about a situation I haven’t observed. If there are two masses there, they’ll probably attract each other. I generalize on the basis of several facts I observed, and from that generalization I derive the missing conclusion, the fact I don’t know. So this is really just like in the… in the scientific context. And then I basically turned this into a more general technique, and what I said was that basically—how do I do it? I basically—well, before that, maybe one more thing: instead of putting two alpha here, I could have put beta. That too is a possible explanation. Right? That also explains everything. What does “everything” mean? It means these three data points here; here there’s a question mark, right? I don’t know here what to explain. That too is an explanation, right? If in the public domain you need beta in order to obligate payment, then tooth and foot, which have alpha, cannot incur liability, but horn, which has beta, can incur liability. If in the damaged party’s courtyard you need alpha in order to obligate payment, then tooth and foot, which have alpha, incur liability, but horn, which has beta—the answer here will be zero. It cannot incur liability. So now, in short, the result in the missing square depends on which theory I choose in order to explain the data. Now here I have two theories, each of which gives a different result. How do I choose which is the correct theory? I choose the simpler one. Ockham’s razor. The previous theory contained only one parameter; this theory contains two parameters. Okay? And therefore it is a less simple theory, and therefore I prefer the previous theory, and therefore the result is one. And that’s the basic principle. And how do you do it in practice? You take this table, fill it in once with zero and once with one. Okay? Fill it in once with zero and once with one, look for a model that explains the zero-filling, a model that explains the one-filling, and check which is simpler. Whichever is simpler is the correct one, and its filling is the result. That’s the same thing I did before. It’s just more convenient to implement on a computer or in a mechanical calculation. Okay? Then afterward I also showed how this explains the issue of the refutation, and from here we also got why these two arguments are really one and the same argument and not two different arguments, the columns and the rows. Because in both of them I’m really assuming a hierarchy on the same alpha parameter, whether between the domains or between the damagers, and therefore it really makes no difference whether I go by the columns or by the rows, because I have to explain all three data points. And in order to explain all three data points I have to assume that the hierarchy between the domains and the hierarchy between the damagers are based on the same parameter: alpha/two alpha and alpha/two alpha. Therefore, once I refute that, I already need two parameters, alpha and beta, so I’ve refuted the kal va-chomer entirely. Rotating it won’t help. That’s in brief. Now I want to move on to something else. I’m not going to continue with this algorithm now; rather, I’ll try to show a different type of inference. Meaning, we saw binyan av, we saw refutation, we saw that this whole business more or less works. And I said I didn’t give all the details because I didn’t have time to get into all of it. Just one clarification—I’m trying to remember—we said that we could create two different hierarchy rules, and then we said no. So what exactly is the point that links the two hierarchies, because these aren’t really two hierarchies? When you determine here—and now I’m returning to the model I chose, the simpler model—basically my claim is, okay, basically my claim is that this is the model. We ruled out beta; it’s less simple. Okay? So this is the model. So what do we see here? If I look at the hierarchy of the columns, I’m basically saying horn is more severe than tooth and foot, because it has two alpha versus alpha, and that isn’t enough. Also, when I make the kal va-chomer between the columns, I have to assume there is also a hierarchy between the rows of alpha and two alpha, otherwise there’s no explanation here. In other words, each formulation of the kal va-chomer assumes both hierarchies. These are not two independent hierarchies. Why? Because I have to assume that the difference between the domains is in the same relevant parameter as the difference between the damagers, otherwise what difference does it make? You tell me horn is more severe than tooth and foot. Why? Because it has two alpha and tooth and foot have only alpha. But if the severity between the public domain and the damaged party’s courtyard is actually in terms of beta—this is beta and this is two beta—then the severity between the damagers is irrelevant to the severity between the domains. So why should I care that this damager is more severe than that one? That doesn’t mean it will incur liability more in that domain. You have to assume that the hierarchy between the domains is also in the same parameter as the hierarchy between the damagers. It’s also alpha and two alpha. And once you’ve knocked that down, you’ve knocked down both the columns and the rows. When you assume a hierarchy in one column, you’re also assuming the hierarchy in the other column—in the row. If you assume a hierarchy between the columns, it doesn’t stand on its own. Suppose I wouldn’t assume anything… after all, how did I present it at the beginning? Let’s suppose there’s a hierarchy between tooth and foot and horn, and I’m saying nothing about the relation between the public domain and the damaged party’s courtyard. That doesn’t interest me, I don’t need it. Not true. Because if, with regard to the public domain and the damaged party’s courtyard, for example the relation between them is that the public domain is two beta and the damaged party’s courtyard is beta, and therefore the public domain is more severe than the damaged party’s courtyard, then with tooth and foot you wouldn’t be able to explain these three data points, the one, zero, and zero. Because they’re unrelated. Think about it in another way—I could say: Rabbi, suppose you brought an example here involving a tail. Suppose now there’s a variable here called a tail, and with the tail it behaves differently. Right, exactly. And that—doesn’t that undermine the… Exactly. No, a tail does undermine it, because there’s another parameter here. And once there’s another parameter here, you can no longer know which parameter is the important one. Let me give you an example that may be a bit more intuitive. Look. Let’s talk about a real-life problem, okay? Let’s talk about a profession… no… a person and a profession. Okay? Shimon and Reuven. Reuven works too. Here it’s physics and philosophy. Okay? Now I say this: I make a kal va-chomer. If Shimon, who got sixty in physics, got eighty in philosophy, then Reuven, who got seventy in physics, will get at least eighty in philosophy. Okay? A kind of kal va-chomer, right? If you like, call it fail and pass so it’ll be one and zero; it doesn’t matter at the principled level. So if Shimon, who failed physics, passed philosophy, then Reuven, who passed physics, all the more so will pass philosophy. Okay? This is the kind of kal va-chomer people often make. Now why is that problematic? Especially from today’s perspective. Because who said that the talents relevant to success in physics are the same talents relevant to success in philosophy, or vice versa? Maybe this requires a talent of type alpha and that requires a talent of type beta. Or in other words, I can say that the fact Reuven succeeded more in physics than Shimon is because Reuven has more of the alpha-type talent than Shimon. Right? But the question is whether that is the relevant talent for succeeding in philosophy. It could be that Shimon succeeded in philosophy not because of his alpha but because he also has beta. And Reuven doesn’t have beta, so it is not true that Reuven will succeed in philosophy. Who said the talent relevant to success in physics is the same talent relevant to success in philosophy, and vice versa? What does this really mean? It means that my alpha—that is what I call the talent or the factor causing the result we’re talking about—is not the same talent. Now in order to assume that there is a kal va-chomer, you have to assume that the talent needed for philosophy is also spoken of in terms of alpha. Or in other words, that the relation between physics and philosophy is that philosophy is alpha and physics is two alpha. And that too is spoken of in alpha terms, not in beta terms. Otherwise you can derive no conclusion at all. So once you assume that here too there is a hierarchy between alpha and two alpha, and here too, then it’s one kal va-chomer. Once I show that there is another kind of talent in play here—suppose I now bring you chemistry. Fine, chemistry: Shimon specifically succeeded and Reuven failed. Exactly the opposite of physics. Okay? That means Reuven has an advantage in alpha-type talent, and he probably has a disadvantage in beta-type talent. And now the question is which talent is important for philosophy—maybe it’s gamma altogether—but the question is whether it’s alpha or beta. And therefore you can’t know. So that is a refutation. Chemistry would be the refutation, because it introduces another kind of talent into the picture, and now you say: I don’t know which talent is responsible for success in physics. So therefore I have no way of knowing the answer here. Okay? I think here you can see it much more easily, but it’s the same thing in all the kal va-chomer cases I discussed earlier; it’s just that here it’s very intuitive. Think for example when we take a psychometric exam. Okay? What does the psychometric exam test? It tests certain abilities. But the abilities it tests are not the abilities you’ll need in your studies. In your studies you have physics, history, philosophy, chemistry, Aramaic, geography, who knows what, sociology, whatever it may be. Each of those requires a different kind of skill, and people who can be very good here will be less good there, and vice versa, right? So how can one test predict well the chances of success in all disciplines? You would need to test all types of talent. First of all that’s unnecessary. Why test my sociological talent if I’m going to study physics? Or vice versa. It’s simply unnecessary. So there’s no choice here; you need to choose some one simple test for everyone, because otherwise things get too complicated. So you try to create some kind of talent measure, or an index that more or less combines all the talents. But obviously it can miss, because it could be that in the combined talent I’m better than you—why? Because my talents in the exact sciences are far better than yours—while on the other hand you have better talents in the humanities, and maybe there the gap between us is somewhat smaller. So overall, on the psychometric exam maybe I’ll get a higher score than you, but that doesn’t mean I’ll do better than you in philosophy. It could be I won’t. And all the criticisms of the psychometric exam are based on exactly that. The criticisms of the psychometric exam are grounded in the fact that the talent for physics, which is alpha, is not necessarily the talent needed for philosophy; or what you test in the psychometric exam is not necessarily what you’ll really need once you actually begin studying different fields at the university. And therefore, when you make a kal va-chomer you are assuming some kind of assumption that there is only one talent at play throughout this whole field. But if you choose subjects like physics and mathematics, which are generally perceived as close, okay, then presumably that’s some kind of measure—I don’t know if it’s certain—but if I’m much better than you in physics I’ll probably also be better than you in mathematics. Or vice versa. But if you choose physics and literature, for example, then it’s really not convincing that if I’m better than you in physics I’m also better than you in literature, right? Which means that very often we have some kind of initial sense as to which two columns can be included in the same table. When you include something in the same table, you’re basically saying: “this belongs to the same field,” you’re talking about the same set of skills that are at play here. You can’t connect two columns in the same table when each one deals with a different talent, because then there is nothing to learn from one to the other. It’s irrelevant. Behavior in physics says nothing about behavior in literature, and it does say something about behavior in mathematics. Why? Because mathematics and physics require similar skills—not exactly the same thing, but much closer. By contrast, literature and physics are obviously not. So if it said “literature” here, for example, then in principle I could make my kal va-chomer, right? If Shimon, who got sixty in physics, got eighty in literature, then Reuven, who got seventy in physics, will certainly get at least eighty in literature. Here I would reject it even without a refutation. Not because of a refutation. I would reject it because intuitively I don’t think physics and literature really are similar in the skills they require. Meaning that when I build a table of this sort, I have all kinds of assumptions that I don’t even put on the table. They are assumptions that say: “this belongs to the same type of parameters governing the matter.” When you want to know what obligates tooth and foot, horn, public domain, damaged party’s courtyard—that’s all damages law. You understand that probably the same kind of parameters is responsible for the question whether you are liable or exempt. That’s an assumption. Now here you have to check that. There could be a refutation and it will turn out you assumed incorrectly, that there is more than one important parameter here. But my initial assumption is that it is so. If I find a simple solution, a model with one parameter, I’ll assume that’s the correct model. There are situations where I can find a model with one parameter, but it isn’t convincing because it’s obvious that physics and literature are not the same talent. There is no point in making a kal va-chomer from one to the other. Okay? So notice that sometimes I reject such an inference not because I have a refutation, I found a counterexample or a fact that teaches me something, but because I assume a priori that I am not allowed to analogize these two columns or these two rows. They are governed by different parameters. I gave you the example of obligating a doorpost in tzitzit. If a four-cornered garment, which is exempt from mezuzah, is obligated in tzitzit, then a doorpost, which is obligated in mezuzah, all the more so should be obligated in tzitzit. Or conversely, obligating a four-cornered garment in mezuzah, right? If a doorpost, which is exempt from tzitzit, is obligated in mezuzah, then a four-cornered garment, which is obligated in tzitzit, all the more so should be obligated in mezuzah. What’s wrong here? There are hundreds of such kal va-chomer arguments in the Talmud. Why is this not okay? What’s not okay here? Is it because there is a refutation? There is no refutation at all. No, no, what refutation is there here? I can’t think of one. It’s not logical because it’s obvious to you that the question whether something is obligated in mezuzah and whether a four-cornered garment is obligated in tzitzit is not determined by the same parameters; it’s irrelevant. The question whether something is ritually impure or pure, and the question whether you owe someone money or not owe him money—what connection is there between those two questions? If, in the public domain, in a case of doubt regarding impurity, the public domain is pure, and in a private domain it is impure, that means that in the public domain people are obligated less than in the damaged party’s courtyard, because in impurity we see that the public domain is lighter—in cases of doubt we are lenient. That’s not true. In damages law, after all, in the damaged party’s courtyard there is greater liability than in the public domain. Tooth and foot in the public domain are exempt, while tooth and foot in the damaged party’s courtyard are liable for full damages. Horn pays half in the public domain; in the damaged party’s domain there is a dispute whether it pays half or full, Rabbi Tarfon and the Sages. So how does that fit with impurity? Very simply: the governing factor, the parameter that matters in the domains for the question of what we do with doubt regarding impurity, is parameter alpha. The parameter that determines whether we obligate a damager to pay is parameter beta. What does that have to do with impurity? So I won’t put it into the same table at all. Okay? The construction of the table itself is already reflecting some assumption that this belongs to the same semantic field. This belongs to the same—these same parameters govern all the data in this table. Right, some kind of common denominator. Okay? Because if that weren’t the case, then you couldn’t make the kal va-chomer. And that is the reason why a kal va-chomer is an inference that is not deductive; it is not necessary. Once you have already determined—once you’ve put everything into the same table and now you have this table—from there onward it’s mathematics. If Shimon, who got sixty in physics, got eighty in literature, then Reuven, who got seventy in physics, all the more so in literature will get seventy? Something like that—it already depends. So this kal va-chomer is mathematics. I can explain why it is true with the alphas and the betas and all the rest, all that is fine. Why is it not necessary? Why might I still be mistaken? Why can there be a refutation here? A logical inference has no refutations. You can point out a mistake in the logical reasoning because I made a mistake, but if the logical reasoning is correct no example will refute it; there are no refutations. But kal va-chomer does have them. Why? What does the refutation do? The refutation tells me: you were not allowed to put this in the same table. The fact that you put it in the same table already involved some problematic assumption, namely that the same parameters govern these two columns or these two rows. Who says so? Maybe they are entirely different parameters, different talents, different parameters, all the examples we discussed earlier. Now, sometimes I don’t see that a priori. In literature and physics, say, I think many people would say from the outset: leave it alone, don’t put it in the same table, it’s not the same kind of talent at all. But say I had physics and mathematics—many people would make this kal va-chomer, right? And then I’d say to them, look, here’s Yossi. Yossi did extremely well in mathematics and not in physics, exactly the opposite of Reuven, who did well in physics and not in mathematics. What does that mean? That even physics and mathematics, which I a priori thought were indeed governed by the same talent or similar talent—no, that’s not true; I have a refutation showing that it’s not. It’s another kind of talent after all. Okay? That’s the role of the refutation. In other words, after I’ve already put it in the table and assumed a priori that it probably does belong to the same parameters, the refutation shows me that I was wrong. But sometimes I know this even without the refutation; I simply look at the subjects and see this is literature and this is physics, and I don’t put them in the same table at all. I don’t need refutations for that. Clearly, if you measured people you would discover refutations, because if physics and literature really do require different talents, then you’ll find people where one of them is better in literature than in physics and the other is worse in literature than in physics. Right? You’ll find all kinds of people; you won’t find a clear hierarchy. So the refutations would be an indication that literature and physics are different talents. But here I don’t need the refutations, because I know the fields and it’s clear to me that this is a different kind of talent. Sometimes I don’t know, and I do think it belongs in the same table—for that there is the whole world of refutations. Look at examples and see whether it really stands the test or not. The refutations will show you: look, there are people or domains or damagers in which the hierarchy reverses, so what you thought at the beginning when you put everything into the same table was simply mistaken, not true. There are different parameters governing the different columns or the different rows. And that is the meaning of a refutation. When I raise a refutation—here, let me raise one now—let’s talk in terms of… zero and one, okay, not fail and pass. Okay? So Shimon and Reuven: Shimon failed physics and passed literature. Reuven passed physics, and the question is what will happen to him in literature. Okay? Now if I make a kal va-chomer, then the assumption is one. Suppose someone thinks literature and physics are related, that this is the same kind of talent. So he’ll say one. Now I’ll come and tell him, look, maybe you’re right, I have a different intuition. Maybe you’re right and maybe you’re not. Let’s check. If we have no a priori way of deciding, then we test it. So let’s add, wait, one more example. Okay. Let’s take our Yossi. Okay. Yossi succeeded in physics and failed in literature. Okay. What does that mean? You thought that in literature one always does better than in physics—meaning, if you have alpha-talent, for literature that’s enough and for physics it isn’t enough. Okay? So if someone has two alpha talents, enough even for physics, then certainly for literature it will be enough, right? That was your previous assumption. But here, look at Yossi. Yossi succeeded in physics and not in literature. What does that mean? That success in literature is not determined based on your talent in physics. There is probably another parameter there, or an additional parameter, or whatever—something else is responsible for success in literature. Ah, if that’s so, then concerning Reuven too I can no longer infer conclusions. How would you know? I don’t know what the nature of Reuven’s beta-talent is. You cannot create a one-dimensional hierarchy; these are different variables. Okay? You can’t make it one-dimensional, right. There are different variables, exactly. Okay? Basically, the role of the refutation is to show me that when I put the two columns or the two rows in the same table, I was mistaken. I thought they were governed by the same parameters; not true, here is a counterexample. But sometimes I know this a priori. For example mezuzah and tzitzit, I won’t put them in the same table at all, because it’s obvious to me that these are entirely different parameters. But fine, that’s just because in principle I could put them in the same table and then look for examples. I know, say, if there is a doorway, then I don’t know what, it requires the form of an entrance in order to permit carrying inside. Okay? A garment has no such problem. Right? A garment has no issue of carrying within it if it doesn’t have the form of an entrance; that’s irrelevant. Okay? So I’ll bring you a refutation showing there are many different aspects between a door and a garment. So don’t latch onto one particular distinction you found and derive all conclusions from it. But in this case it’s unnecessary, because it was obvious to us from the start that these are two different topics. Very often the refutation comes to show us that what we thought was the same topic actually is not. Fine, these are two different things. That is basically what the refutation does. In other words, the kal va-chomer is really a necessary inference once I’ve built the table. If I know that this is the table, and it’s clear to me that this belongs to the same semantic field, that it is governed by the same parameters, then from there on it’s mathematics. No tricks—it’s straightforward. I can show you what the correct answer is for every such data table. The big problem is how I built the table—what assumptions underlie the very fact that I put a table here. The refutation attacks that, not the mathematics. Think of a logical inference: if it’s valid, exactly. Every human being is mortal, Socrates is a human being, conclusion: Socrates is mortal. Now I’ll show you, I don’t know, someone else who is a human being and not mortal. That’s a refutation, right? Can one refute it? The answer is: I’m not refuting the inference, I’m refuting the premise. The premise of the inference is that all human beings are mortal. Here, I brought you an example that is not so. Here is a human being who is not mortal. So the premise of the inference is not true. The derivation of the conclusion from the premises is necessary—you can’t refute that. No counterexample can refute it. The counterexamples, the refutations, attack the premises. Now in this respect, kal va-chomer and all these other inferences, binyan av and the like, are exactly the same as logical inference. When you bring a refutation, you are not attacking the inference of the kal va-chomer; the inference of the kal va-chomer is definitely correct. What you are attacking are the assumptions underlying the inference. The assumptions underlying the inference are that the two columns or the two rows are governed by the same parameters. The refutation attacks that. And if you attack that, fine, you can remove the conclusion, but not because the argument is invalid. And in this sense you can really see that non-deductive inferences—analogies, inductions, and so on—are actually the same thing as logical inferences. The inference, the derivation of the conclusion from the premises, is necessary. The refutation attacks the premises. If the premises are not true, then fine—even though the conclusion follows from them, the conclusion is not necessarily true, because the premises are not true. Okay? And therefore the great gap we talked about between analogy, induction, and deduction—the great gap that people supposedly see between deduction and analogy and induction, that these are non-necessary inferences—is a much smaller gap than people think. At the end of the day, even in deduction the conclusion is not necessary. What is necessary is the derivation of the conclusion from the premises; but the conclusion itself is not necessary, because I can attack the premises. In that sense, kal va-chomer and binyan av can also be viewed in that way. That is why this is logic just like any other logic. What we saw here about kal va-chomer—and it can be extended much more—I told you this is the basis of all non-deductive reasoning, in science, in law, in Jewish law, everywhere. Okay? I can show you an orderly logical calculation, completely mechanical, exactly as in deductive logic, and I’ll derive for you all the legal, historical, scientific conclusions, everything, in a completely deductive manner. I turned it into mathematics. So what, is it really mathematics then? Is history mathematics? No. Why not? Because the counterexample will refute the premise of the historical argument. Okay? You think that winning a war in Europe is the same thing as winning a war in Africa? Suppose in Africa the army with more soldiers wins, so in Europe too—that would be a historical thesis or a military doctrine. Let’s examine two battles in Africa. Actually not even two, just one battle. Which of the two armies won? The one that had the larger number of soldiers. And the one with fewer soldiers lost. Okay? I move to Europe; I have another battle between two armies, many soldiers and few soldiers. Can I infer from Africa that in Europe too the army with many soldiers will win? No. Why? Because in Europe there is much more technology than in Africa, and sometimes technology can help a small smart army defeat a large foolish army. And in Africa there isn’t that technology, so there the number of soldiers determines the outcome. But when you want to transfer that to Europe—who says? So what failed here? The binyan av or the kal va-chomer? No. What failed here is the assumption that only the number of soldiers is the determining parameter. No: there are additional parameters that determine things, such as technology. In Africa there isn’t any, the beta is zero in Africa, there is no technology. In Europe there is beta, there is technology. And that can change the relations determined by alpha, which is the number of soldiers. Okay? So therefore, as you can see, conclusions in history, conclusions in law, conclusions in Jewish law, conclusions in science—all these things are built on the same forms of reasoning. And those forms of reasoning can be formalized in a completely mechanical mathematical way, just like deductive logic. The same thing. That doesn’t mean it’s mathematics. It isn’t. Because you can attack the premises. In mathematics you can’t attack the premises, because the premises are what I stipulate—they are not claims about the world. There you can’t attack premises. That is the real difference between mathematics and science or Jewish law or law generally. You cannot attack the premises—not because the inference is necessary; there too the inference is necessary. It’s just that here you can attack the premises, because who said the army with the larger number of soldiers wins? In mathematics you simply define: a point is something with zero dimensions; a triangle is something with three sides. You defined it—how are you going to attack that? What, will you bring me a triangle that does not have three sides? If it doesn’t have three sides, it’s not a triangle. That’s a definition. So mathematics is kind of… you know, this reminds me of Descartes’ cogito argument. Cogito ergo sum, right? I think, therefore I am. People often make a mistake and ask: what is Descartes really saying? I think, and if I think that means there is someone thinking, therefore I exist. He derives the conclusion from the fact that he thinks to the fact that he exists. Now why not derive the conclusion “I walk, therefore I exist”? After all, if I walk, there is somebody walking, no? So that means I exist. What is special about “I think”? We already talked about it, that thought itself is necessary. Ah, we already talked about it? I don’t remember anymore. Could be. The difference is not in the inference; the inference is correct in both cases. The difference is in the premise. The premise “I think” is… the premise itself is necessary from within itself. The premise “I walk” is not. Or I’m mistaken and I’m not walking. Or I think I’m walking and I’m not walking. And I’m not walking. I think that I think. Exactly. If I think that I’m not thinking, that too is a thought, so I’m again thinking. Basically just as Descartes sets it up here. Meaning? The very fact that he says he thinks, itself means that he thinks. Yes, right. And therefore the difference between “I think, therefore I exist” and “I walk, therefore I exist” is not in the inference. In both cases the inference is correct. If I walk, then certainly I exist. If I think, certainly I exist. Both those inferences are equally correct. The difference is in the premise—whether you can attack the premise. The premise “I think” cannot be attacked. The premise “I walk” can be attacked. Maybe I’m mistaken and I’m not walking. Okay? That’s the difference. Very often we make a mistake and think: this is a necessary inference, that is a non-necessary inference. Every inference is necessary. The only question is whether you can attack its premises. That’s the difference between the fields. Okay. So I was sure that this—so the Rabbi explained to me that what you wrote in some article in a column, too. So if I’m not mistaken, what I’m saying is that in the end even the very fact that I understand that I’m thinking is a kind of observation that I receive by intuition from the senses. Not from the senses, but from a kind of inner observation. That itself—if I’m not relying on the observation—it’s not a definition, it’s a claim. Exactly, so then there is no certainty. Right. Fine. In any event, that’s regarding kal va-chomer. Now I want to move on—and again, here I’m not going to do the mathematics of the matter, because it starts getting a little complicated and you need to learn a few more things along the way—but I just want you to get an impression of the principle. I want to talk about the common denominator, the tzad ha-shaveh. Okay? What is the common denominator? Let’s look, for example, at the Talmud in Bava Kamma 2a. Okay. The Talmud says there, the Mishnah says at the beginning of Bava Kamma: there are four primary categories of damages—the ox, the pit, the grazer, and the fire. Okay? Now the Talmud… and at the end of the Mishnah it concludes: the common denominator among them is that they are your property, and their safeguarding is upon you, and when they cause damage, the damager is obligated to pay compensation for damage from the best of his land. Now each one of these primary categories has its own characteristics. A pit is created from the outset for damage; fire has another force mixed into it; it goes and causes damage; horn intends to cause damage; it doesn’t matter—each one of these types of damagers has certain characteristics, and therefore you can’t derive one from another, that’s what the Mishnah says. If you try to derive one from the other, you won’t succeed, because there are differences between them. And then the Mishnah says at the end: the common denominator among them is that they are your property, their safeguarding is upon you, and when they cause damage… The Talmud asks: “The common denominator among them”—what is that coming to include? Why did you add this sentence, “the common denominator among them”? What is it teaching me? What is it adding? Now this itself—I think I already talked about it—this itself is already a strange question. The Mishnah gives four examples of damagers: ox, pit, grazer, and fire. Okay? Then it gives me the rule: anything that is your property and whose safeguarding is upon you, if it causes damage you are liable to pay. Right? Now, if I were reading such a Mishnah, I would ask: why did you bring the examples? There is the rule. You gave me the rule, I already know everything. The examples are just particular cases. But the Talmud says, wait—you brought the examples, who needs the rule? What is this? What is the logic? It’s the opposite of the logic. The fact is that in the Mishnah and the Talmud they don’t like using rules. When there are rules, they immediately ask “what is this coming to include?” They much prefer to speak in examples rather than in rules. That is why the Talmud has what is called a casuistic character. Casuistic—by cases. The Mishnah and the Talmud do not deal in sweeping rules. What is the law when you benefited and… no. If someone entered his fellow’s courtyard without his knowledge, does he have to pay him or not? That’s a case, and I’m talking about the case. Talking about the case from which afterward you can derive principles, but we are talking about cases. British law is basically casuistic. It’s changing somewhat in recent years, but basically it is casuistic. Continental law, say German law for example, is more positivistic. It proceeds on the basis of principles, rules, from which the cases are derived deductively from the general principle. Now the Talmud is casuistic. That’s its nature—Talmud, Mishnah, the whole Talmudic corpus. It has a casuistic character. It is more… it’s more natural, more human, you could say, no? Yes, one could say a lot about that. I think there is a lot of wisdom in not being positivistic. I think positivism is a childish approach. But leave that aside for now; that’s not our topic. Here the Talmud says—but this is the Talmud’s language—the Talmud says, “The common denominator—what is it coming to include?” You gave me the rule—for what? I’ve got four examples, everything is fine. Why do I need the rule? The Talmud says: Abaye said, it comes to include his stone, his knife, and his load, which he placed on top of his roof and they fell by a common wind and caused damage. Right, we’re talking about someone who placed some object—for example his stone, his knife, and his load—some object on the roof, the wind came, blew it off the roof, and it caused damage. Fine. This you wouldn’t have been able to derive from any one of the primary categories that appear in the Mishnah; that is what the common denominator comes to teach. The Talmud asks: what are the circumstances? If they caused damage while moving, then that is just fire. If they cause damage while in flight—when they fall from the roof they hit someone or something and cause damage—then they are really just like fire; you can derive it from fire. You told me you were looking for something that can’t be derived from any individual primary category, and for that you need the common denominator, right? But this could have been derived from fire alone. Just as fire has another force mixed into it and carries it along in its destructive movement, these too have another force mixed into them and carries them. Rather, it is after they came to rest. So what are we talking about? After they came to rest. They were thrown to the ground, came to rest there, and then someone tripped over them, say, or something like that. The Talmud says: if he declared them ownerless, then according to both Rav and Shmuel this is just a pit. It’s like a pit, so once again there is nothing unique here. The Talmud says: actually he did declare them ownerless. We’re talking about a case where they fell from the roof and I declared them ownerless, and then they caused damage. Actually he did declare them ownerless. And they are not similar to a pit. You told me this could be learned from a pit—no, it is not similar to a pit, because there is no other force mixed into a pit. Here the medieval authorities explain: what does it mean that no other force is mixed into it? These too have come to rest; there is no other force mixed into them now, they are resting here like a pit and causing damage. But how was the pit created? In a pit too, when it causes damage I didn’t directly cause the damage, the pit caused it. Why are they coming to me with claims? Because I dug the pit. Now who dug this pit, this stone, knife, and load that fell? The wind. Not me. I was negligent in putting it on the roof. But the wind also has a role in the matter. Therefore you can’t derive it from a pit. Because with a pit, my responsibility for what the pit does is complete—I dug it, I’m responsible for the fact that there is a pit here. In the case of my stone, knife, and load that came to rest there, I have only partial responsibility; the wind is also part of the story. So if only pit had been written, I would not know this. So the Talmud says: right, what is true of a pit, where no other force is mixed into it—can you say the same of these, where another force is mixed into them? Fire will prove it. We see that fire goes together with the wind and yet one is liable. So derive it from fire alone. The Talmud says no: what is true of fire, which normally goes and causes damage? Once fire exists as a damager, its way is to spread, to go and cause damage. But after my stone, knife, and load are down below, they’re already down below, they don’t move. Their way is not to go and cause damage; the injured party comes to them, not they to the injured party. Okay? And therefore it can’t be derived from fire either. And then the law returns. “And then the law returns” is a structure which, simply understood, looks like the common denominator. Why? Let’s see how we translate this structure. So look, let’s put here, right. I have like this. I have one source-teacher, A, fire. The second source-teacher is B, pit. Elegant English. Fine? So this is fire and this is pit. Fine? Now I want to derive my stone, knife, and load, let’s call it C. This doesn’t work well with the letters, but let’s call it C. My stone, knife, and load. Right—the point is something I placed on top of the roof, it’s not important, that’s just the case the Talmud gives—something I put on the roof and it fell, okay? And afterward it came to rest and caused damage. Now I say this: how does it begin? I begin by trying to learn it from fire, right? Let’s derive it: if fire is liable, then my stone, knife, and load are also liable. That’s how it starts, right? So they tell him, no—fire has a property that its way is to go and cause damage. Right, exactly. Its way is to go and cause damage—let’s call that property X. Fine, so this one has X and C does not have X. Okay? And therefore you can’t. X is always a stringency, right? So if A has a stringency, then maybe what makes one liable for fire is precisely that particular stringency, and my stone, knife, and load do not have that stringency, and therefore maybe they are not liable. So he says, no, then the pit will prove it. Right? Let’s derive it from pit. No, pit has another stringency, it has stringency Y. Okay? And my stone, knife, and load don’t have stringency Y. That roof-on-top thing means there is no stringency Y. Okay? And now what? Then they say, “and then the law returns.” What does that mean, “and then the law returns”? You can’t derive it from A and you can’t derive it from B. So now what—what does “return” mean? They didn’t add any data; they just say, okay, “and then the law returns,” all is well. What do you mean all is well? You can’t derive it from A and you can’t derive it from B. So let’s derive it from the two of them together. Right? That’s the common denominator—let’s derive it from both together. What does “from both together” mean? Look. First of all, here property Y is always absent, and here property X is always absent. This is always the rule in the common denominator. Okay? “Its way is to go and cause damage,” “another force is mixed into it”—not present. That always has to be so. Because if there were X here and also X here, then I could simply derive from X that X is liable. The special property that exists in A must not exist in B, and vice versa. Therefore in a common-denominator argument there is always a need for each one. What exists in A does not exist in B, what exists in B does not exist in A, and only then do I make the common denominator. There has to be mutual necessity. What do they both have? They both have a common property Z, which they both have, and C has it too. Okay? Then I say, look: basically it can’t be that X determines liability. If X determined liability, then in C you couldn’t obligate liability, right? Maybe the property X of A is our alpha, essentially, right? A has some alpha property because of which one is liable, and therefore in my stone, knife, and load, which don’t have the alpha property, one would not be liable. No, fine. So you can’t derive from fire. Let’s derive from B. What do we derive from B? It has property beta, Y, because of which one is liable. But then C doesn’t have the beta property—it’s roof-Y or whatever—therefore C also would not be liable. Fine, so let’s derive from both together. What do you mean “both”? You can’t derive from each one individually, and now suddenly together you can derive? How does that happen? Very simple answer. They have a common property Z, which C also has. And now in fact I have two possible theories. I basically want to learn liability in damages. Let P be liability in damages. Okay? P is liability in damages. Now I know that fire and pit are liable in damages, and I want to know whether my stone, knife, and load also incur liability in damages. Right? That’s basically what I want to know. Right. So I say this: I start trying to learn from A to C. He says no, with A it may be that X determines it, X is the parameter that governs P, governs liability in damages. Then I say, so perhaps only in A, where there is X, is there liability, but in my stone, knife, and load, where there is no X, there is no liability. Fine, so let’s try from B. Maybe the determining parameter is Y. No—not that it can’t be, but if it is Y, then again you can’t derive to C because C doesn’t have Y. He says—but it can’t be either X or Y. Why? Because if X were the determining factor, then B would not be liable; only A would be liable, but B would not. So X is not the relevant parameter. You see? This is exactly like a refutation. So X is not the relevant parameter. Fine, so maybe Y is the relevant parameter? That can’t be either. Because if Y were the relevant parameter, then I understand why pit is liable, but then fire would not be liable, because fire doesn’t have Y. Therefore X and Y cannot be the relevant parameters. Apparently there is some other parameter, Z, that exists in both, and that is the relevant parameter. And lo and behold, my stone, knife, and load also have Z. Therefore from A and B together I can derive that C is also liable. I can derive the property P from A and B together, even though from each one separately I cannot derive it. And why? Because I really have two theories: either what governs P is either X or Y—right, that symbol here means “or” on this board—or what governs P is Z. Right? Two theories. Agreed? One theory says whoever has either X or Y is liable. According to that theory, what follows? That A is liable, B is liable, but C is not, because C has neither X nor Y. Right? That’s one theory. The second theory: what governs liability is Z. That also explains the data. A is liable because it has Z, and B is liable because it has Z. But here the result is that C too will be liable, because C also has Z. You see that this is exactly like the table. I’m just not doing it by the… it’s exactly the same thing. I have two theories, one with alpha and beta—that’s the X and Y here—and one only with alpha. From the alpha-and-beta theory, it would follow that horn in the damaged party’s courtyard would be exempt, just as here C would be exempt. From the alpha-only theory, it would follow that horn in the damaged party’s courtyard is liable. That’s my C. Right? I have two theories. How do I choose which theory is correct? Question: how do I know C has Z? Suppose I know it, let’s say I know it—it’s not important right now. The simplest theory. Right, the simplest theory. Which is simpler, this theory or this theory? Fewer variables. Right, fewer variables, and therefore I choose this theory, because it is simpler. And from that it follows now that from A and B together I can derive C, even though from each one separately I cannot derive C. That’s how this miracle happens. There is some sort of miracle here. I try to derive from A; they say no, can’t do it, from A you can’t because it has a stringency. Let’s derive from B—no, from B you can’t either because it has a stringency. Fine, so both are gone, finished—what can I do? Impossible. No, impossible from either one separately; but together, without adding any new datum, together it suddenly can be derived. Why? What’s the idea? The idea is Ockham’s razor. It is exactly the same idea as the kal va-chomer. I can make a table here and show you—it is constructed in exactly the same way. And these very X and Y here are the alpha and beta parameters we talked about. The question is whether this is a two-parameter theory, X or Y, or a one-parameter theory, just Z. And since I prefer the simpler theory, the one-parameter theory, the result is that in the case of my stone, knife, and load there is liability. Now understand that this inference I spoke about here is scientific generalization—it is scientific induction. Let’s look. Suppose I take this book, let go of it, and whoosh, it falls to the ground. Okay? Fine. Now I ask whether this bag too will fall to the ground. But no, maybe this book is made of paper, therefore it falls to the ground. Right? Fine, so let’s see—maybe this pen will fall? Whoosh. This pen is not made of paper, right? And it falls, so apparently this bag too will fall. What suddenly? This pen is round. The bag isn’t round. Right? Now this isn’t round, right? So this is round but not made of paper, and this is made of paper but not round. Do you see the X and Y? Right? So I say, apparently roundness is not relevant, because otherwise this wouldn’t have fallen. And paper also is not a relevant parameter, because otherwise this wouldn’t have fallen. So what is? There is some additional other property shared by both, Z. What is that? They both have mass, right? And that is what causes them to fall. Ah, look, this too has mass, so probably it too will fall. Wait, but that doesn’t mean… that doesn’t mean there is Z, but it doesn’t tell us what Z is. Right. But X and Y too I don’t necessarily know. Did I tell you what alpha and beta are in the kal va-chomer? I didn’t tell you what alpha and beta are there either. I assume there are such things. Sometimes I know what they are, sometimes I don’t. Fine? But there is a difference in midrash… that the simple thing is probably what causes the… Once I have a situation where two different factors, each on its own, seem to produce the phenomenon, that is not likely. Apparently there is something in the background, one thing present in both those objects, that causes the phenomenon. The Z. Yes, but for that Z one could come up with all kinds of ideas that explain the… Right, and then I can look for what Z is. Fine? For example in our Talmudic passage, anything that is your property and whose safeguarding is upon you—that is the Z. Right? The ox, the pit, the fire, all of them are really my property and their safeguarding is upon me. That is the Z common to them both. Now I say, the special properties of fire and pit are not interesting. If it is my property and its safeguarding is upon me, I am liable to pay. So look: my stone, knife, and load don’t have the special properties of pit and don’t have the special properties of fire, but they are my property and their safeguarding is upon me. So I am liable to pay. That is the Z. But one could have refuted the Talmud and invented some other criterion here that… Find a criterion that sounds relevant, because we always test. If you don’t find one, apparently not—the Mishnah says no. The criterion is that it is your property and its safeguarding is upon you. You see? It’s exactly the same as scientific generalization. Exactly the same thing. All of this logic is basically the logic responsible for our inferences in every area of life except mathematics. Now I have a riddle for you. Let’s look at a Talmudic passage in Ketubot. “This shows that Ulla holds: wherever there is monetary liability and lashes, he pays money and is not lashed.” Right? If there is monetary liability and lashes, you pay and are not lashed for the same offense. Okay. “From where does Ulla derive this?” The Talmud says: he derives it from one who injures his fellow. Just as one who injures his fellow, where there is monetary liability and lashes, pays money and is not lashed, so too in every case where there is monetary liability and lashes, he pays money and is not lashed. Fine—one who injures another must pay and is not lashed. The Talmud says: what is true of one who injures his fellow, who is liable for five types of payment? There is a special stringency in one who injures his fellow, because he doesn’t pay only the damage; he pays humiliation, pain, healing, loss of livelihood. There is a special stringency there, unlike damage caused by one’s property, for example. Okay? Therefore you can’t derive from there—this has a special stringency. The Talmud says—and I’m skipping the parenthetical passage because it only complicates things here—rather, he derives it from conspiring witnesses. So apparently you can’t derive it from one who injures his fellow. So he derives it from conspiring witnesses. Just as conspiring witnesses, where there is monetary liability and lashes, pay money and are not lashed, so too in every case where there is monetary liability and lashes, one pays money and is not lashed. Fine, conspiring witnesses, where there is monetary liability and lashes for “you shall not bear false witness,” they pay and are not lashed—that is his source. The Talmud says: what is true of conspiring witnesses, who do not require prior warning? Conspiring witnesses have a stringency: they are punished without prior warning. You can’t derive from conspiring witnesses; that’s an especially severe case. Okay? Rather, he derives it from both together. So he learns from both together. What is the common denominator between them? That where there is monetary liability and lashes, one pays money and is not lashed, so too in every case where there is monetary liability and lashes, one pays money and is not lashed. Exactly like the structure we saw before, right? I have two source-teachers: conspiring witnesses and one who injures his fellow. Each of them has its own special stringency, remember? X and Y—this one doesn’t have Y and that one doesn’t have X. You can’t derive from either one individually to the general case, but I say: in both there is Z, namely that money is paid and lashes are not given—that’s what I derive as a general rule from my two source-teachers. Now comes the bombshell. The Talmud says: what about the fact that the two source-teachers each have a severe aspect? What is that? Let’s return for a moment to the diagram we drew before. Think of it now as follows: this is one who injures his fellow, and this is conspiring witnesses. Fine? And I want to derive some other case in which there is monetary liability and lashes. Fine? Some C, it doesn’t matter. Now we saw that each of them has its own special property. Here, conspiring witnesses don’t require warning; one who injures his fellow involves the five payments—that’s the Y. This one doesn’t have the five payments, that one doesn’t require warning, this one does require warning. Okay? So each of them has its own stringency. Then I say there is Z, and therefore I derive from Z, and all is fine. Right? That is the classic common-denominator structure. Okay? Except that now suddenly the Talmud raises something that really seems strange. “What about the fact that the two source-teachers each have a severe aspect?” A has severe aspect X and B has severe aspect Y. True, it’s not the same severe aspect, but the fact is each of them has some severe aspect, and C has neither of those severe aspects, so therefore you can’t derive it. That is how the Talmud objects. Tosafot there in Ketubot asks: if there is a refutation like this—“what about the two source-teachers, each of which has a severe aspect?”—then there is no common denominator anywhere in the universe! Every common denominator is like that. We saw it—that is the structure of the common denominator. If you refute with “what about the two source-teachers, which each have a severe aspect,” you can throw every common denominator in the Torah into the trash. For example the one we saw in Bava Kamma 6, and all the rest. The medieval authorities in Makkot 4 ask similarly; there is a long Ritva there who brings several positions of the medieval authorities on this matter. How can you make a refutation of this sort? So I’ll suggest to you a proposal—there are various answers in the medieval authorities, and some of them may mean what I’m saying here—but it seems to me what I’m saying is very reasonable. There is a difference between a situation where the severe aspects X and Y—my alpha and beta, it doesn’t matter—are legal rules, and a situation where they are factual characteristics. Suppose I’m talking about a pit, whose creation from the outset is for damage—that is a factual characteristic of a pit. It is not some special law regarding a pit, right? If I say that someone requires prior warning, that is a legal characteristic. Or that he pays five forms of payment—that is a legal characteristic, a legal stringency. In pit or in fire, where another force is mixed into it, or where its way is to go and cause damage, or where its creation is for damage—all these are factual characteristics. Right? Fire physically moves by means of the wind. Okay? That is a factual matter; it is not a legal innovation that the Torah introduces. Why is that important? I want to make the following claim. Behind every legal rule—this is the whole idea of all these schemas we are learning now—behind every legal property there are characteristics that cause that legal property. When I say horn is liable in the public domain and in the damaged party’s courtyard, and tooth and foot and so on—behind that sits some alpha. That alpha is what is responsible for why horn is liable or exempt. Horn being liable or exempt is a legal rule. Behind that sits a characteristic. There is something special about horn—I don’t know, maybe it has intent to cause damage, so one has to be very careful, therefore I am stringent with horn. Fine? Intent to cause damage is a factual characteristic, not a legal one. It is simply that an ox that gores intends to damage. Okay? So the assumption behind all the models I’m talking about here is that behind the legal properties, or at the basis of the legal properties, sit factual characteristics. Those are the alphas and betas; sometimes I know them, sometimes I don’t, but there are certain factual characteristics because of which Jewish law determines what it determines. Okay? It is always like that. Therefore, in kal va-chomer, when I want to know whether someone is liable or exempt, I need to look at which characteristics he has. Does he have alpha, beta, two alpha—what are his characteristics? That is what determines whether he is liable or exempt. Right? That’s how it is in Jewish law in general. We also talked about this in previous lectures. I said there is a presumption that a person does not repay before the due date. That is a fact. As a result, you are not believed when you say that you repaid. That is the law. The law is always based on some factual characteristic of the situation. There is a situation, and a legal norm is applied to it. Okay? What happens if X and Y are legal characteristics rather than factual ones? Then they are not alpha and beta. So when I say, with fire—sorry, not fire—with conspiring witnesses, they don’t require prior warning, right? Let’s say A is conspiring witnesses. Conspiring witnesses do not require warning; that is their stringency, that is X. Okay? I ask myself, fine, but why don’t they require warning? There is apparently something severe about them because of which the law is stringent with them. Sorry—not “they require warning”; their stringency is that they do not require warning. They are punished even without warning. Okay? Why? Because apparently—I don’t know—their plot to harm, or whatever exactly it is, is very severe. Maybe it undermines the judicial system. Witnesses who come and decide to lie in court undermine the entire possibility of judging justly. So Jewish law views that very severely, and therefore even without warning they receive punishment. Okay? Meaning that when I speak of a legal stringency, behind it sits some factual explanation. Why is the law stringent in this way? Because there is some particular factual characteristic there because of which one must be stringent. Right? Then there is a difference between putting a stringency or factual characteristic there, in which case there is nothing behind it—that is the factual characteristic itself—and saying there is a legal stringency. If there is a legal stringency for A or B, I need to keep digging, because behind that legal stringency sits some factual characteristic. So here, in the case of conspiring witnesses, there is an accumulation of characteristics, basically? Exactly. That’s what I want to suggest. I want to claim this: if A is conspiring witnesses and B is one who injures his fellow, A has the legal characteristic that it does not require warning. Conspiring witnesses are punished even without warning—that is X. Okay? One who injures his fellow has the legal characteristic that he is liable for five types of payment. There are five payments—damage, pain, loss of livelihood, healing—he is liable for five things. I ask myself: okay, but what is it in one who injures his fellow that causes me to be so legally stringent with him? There is something in one who injures his fellow that is very severe. Likewise with conspiring witnesses. What is so severe in conspiring witnesses that causes them not to require warning, that they are punished even without warning? It could be that the factual characteristic is the same factual characteristic in both. I don’t know what it is. Maybe. And if so, then this really is a refutation. Because if there is a factual characteristic shared by A and B because of which they are so severe, and perhaps that characteristic does not exist in C, then I really cannot derive from A and B to C. If the X and Y characteristics in A and B are factual characteristics, then I can see that this is X and this is Y—they’re not the same thing. But if the characteristics are legal characteristics, then it could be that at their base sits a single factual characteristic that expresses itself in different legal ways in the two contexts—these are different contexts. Here I am stringent in that warning is not required; here I am stringent in that he pays five forms. In the case of conspiring witnesses I can’t say, “You also owe loss of livelihood, humiliation, and healing.” Why? Because there is no loss of livelihood, humiliation, and healing there. You simply can’t impose those payments. So it expresses itself in a different stringency; there I punish you even without warning. Fine? Therefore it could be that because of the context the legal expression looks different. But the factual characteristic may be the same factual characteristic. And if there is the same factual characteristic in the two source-teachers, that is certainly a refutation. Right? Because then it may be that both are liable not because of Z but because of that same factual alpha shared by both, and that alpha is absent in C. C has Z, but not alpha. Therefore this is a refutation. Therefore, whenever the common denominator is characterized by legal characteristics, then the refutation of “severe aspect” can arise. There is a Tannaitic dispute whether such a refutation exists or not. And there can be a view that one does make the refutation of severe aspect. But if the characteristics are factual, as in the Bava Kamma passage we saw, no one raises a refutation of severe aspect there. Because there we are dealing with factual characteristics, so I can see that they are different characteristics in fire and pit. In a pit no other force is mixed into it; in fire it is. A pit is created for damage from the outset; fire is not; and so on. In other words, these are different factual characteristics. So there’s no point in taking one more step and asking: wait, maybe this is really the same characteristic? It’s not the same characteristic. They are different characteristics. And two characteristics, as we already said, is a less simple theory than one shared characteristic. Therefore a theory of two characteristics is not one I accept. But if the stringencies are legal stringencies, there could be a theory with a single characteristic responsible for it. Then Ockham’s razor no longer decides it. It could be that that one characteristic is the determining factor, in which case you can’t derive anything. And this really sharpens for us the importance of understanding that all these inferences are always built on the fact that when you make a legal determination, it actually rests on a factual characteristic, on the circumstances. The law always applies to circumstances. There are factual circumstances because of which the law is what it is. Okay? Now look at something nice, and with this I’ll finish. The law is shaped by the circumstances? Yes, it applies to the circumstances. When there are more severe circumstances, the law determines that the norm here will be more severe. Here you’ll pay more, here you’ll pay in more cases, or even without warning, or things like that. But it comes out of something more severe in the situation. It’s not just caprice. Like among Ashkenazi Jews, where in matters of martyrdom under decrees of persecution they were much stricter than Sephardic Jewry, because there the situation was more… Could be, yes. We can use that example. And in the refutation of severe aspect, do you need to identify the factual characteristic behind it? Is it enough just to say “maybe there is such a thing”? Yes. A refutation is always based on “maybe.” That is the difference between a refutation and a proof. A proof has to tell me that this is definitely one. The refutation says maybe it’s zero—you haven’t proved it. After all, I’m not trying to infer the conclusion that it’s zero; I only want to tell you that you haven’t proved it’s one. That’s exactly the difference. There is a built-in asymmetry between a refutation and a proof. A refutation is not a proof that horn is exempt in the damaged party’s courtyard. A refutation says you haven’t proved that it is liable. I have one theory that says it is liable, and another theory that says it is exempt, and those are two equivalent theories—you haven’t proved anything. Returning to the Talmud in Bava Kamma, there is a very interesting dispute there. We won’t really have time. The Rosh brings there a very interesting dispute. He brings the common denominator there in Bava Kamma. “And some of the great authorities wrote that one is liable only for what is liable in both, and exempt from damage to vessels and from death of a person like a pit, and from hidden objects like fire, because since they come by means of the common denominator, we give them the lighter element of both.” What does that mean? Let’s return for a moment—I didn’t share? Okay, I didn’t share on Zoom, with God’s help. Let’s return for a moment to the picture here. Now I go back to fire and pit, okay? Fine, fire—they have a common denominator that it is your property and its safeguarding is upon you, and therefore they are liable. But fire is exempt regarding hidden property. If fire burns a stack of grain and inside it there are vessels, I am liable for the stack but exempt for what is inside it. That is a special feature of fire, entirely unique to fire. A pit is exempt for damage to vessels—“an ox and not a person, a donkey and not vessels.” If an ox or donkey falls into it, the Talmud derives “ox and not a person, donkey and not vessels.” Never mind the details now. Now what about my stone, knife, and load that caused damage? They came to rest there and caused damage. They damaged vessels. My stone, knife, and load came to rest on the ground; some animal came by, I don’t know, fell, and the vessels on its back were damaged. Am I, the owner of the stone, knife, and load, liable to pay? So look: if this were an ordinary pit, I’d be exempt; a pit is exempt for damage to vessels. But fire is liable for damage to vessels. What about my stone, knife, and load—liable for damage to vessels? Same with hidden property: my stone, knife, and load—suppose there was a box that fell and what was inside the box broke. And that was hidden inside the box. So if it had been fire, you would be exempt for hidden property, but pit is liable even for hidden property. What do you do with the fact that we derive it from both? The Rosh brings three opinions. First opinion: yes, no, maybe. First opinion: it has both the exemptions of pit and the exemptions of fire. It is exempt both for hidden property and for vessels. Why? Very simple logic. Why? Because after all you need both source-teachers in order to derive it. If even one of the two doesn’t work here, you can’t obligate liability, right? That’s what we saw. If you want to obligate for hidden property from fire, you can’t derive that—only from pit. But from pit alone you can’t derive liability for my stone, knife, and load. You need both source-teachers. Therefore anything that doesn’t pass through both source-teachers won’t generate liability in C either. Therefore it will be exempt both for vessels and for hidden property. That’s the first opinion. Second opinion: there are those who were uncertain in the matter. Uncertain between what? Simply understood, on what? On vessels, on hidden property, on both? After all, we said the first opinion says you’re exempt for vessels and exempt for hidden property. Exempt from both, because hidden property due to resemblance to fire, vessels due to resemblance to pit. You resemble both fire and pit; you come from both, right? Exempt both for vessels and for hidden property. Now there is an opinion that is uncertain. What does that mean, uncertain? What are the two sides of the doubt? Simply understood, either you are liable for both vessels and hidden property, or exempt for both vessels and hidden property. Right? There is a side that says you would be liable for both, both vessels and hidden property. What is the logic behind that? Leave aside the X and Y; I have Z now. Anything that is your property and whose safeguarding is upon you, I am liable to pay. Do my stone, knife, and load fall into that category? Liable to pay. Now what do you want? You want to exempt vessels? From where will you derive exemption for vessels? From pit? But you aren’t completely like a pit; you are also like fire. You can’t exempt vessels. You want to exempt hidden property? What, do you want to derive that from fire? But you aren’t completely like fire; you’re also like pit. So you have nowhere from which to derive the exemption. For hidden property or vessels, you are liable for everything, because it is your property and its safeguarding is upon you—that is the Z. Now you want to derive that vessels are exempt—you can derive that from pit. But after all it’s not completely like pit; it’s also like fire. You can’t derive exemption for vessels, and you can’t derive exemption for hidden property. Therefore it would be liable both for vessels and for hidden property. That is one possibility. The opinion is uncertain between those two possibilities. Wait, but hidden property here—what does hidden property have to do with my stone, knife, and load? We said: suppose they damaged a box and inside the box there was something hidden inside the box, something else, and it was damaged. Am I liable to pay for it? Inside the… inside the injured party’s box. My stone, knife, and load are lying on the ground, right? An animal passes by. Fine? On the animal there is a box. Inside the… inside the box there are vessels. Fine? When the animal falls, the box breaks, the vessels break. For the box I am liable to pay; the question is what about the vessels? The vessels were hidden inside the box. So if the fire had burned the box, I would be exempt for the vessels. Because the vessels are hidden inside the box, and fire is exempt for hidden property—that’s the rule. No, with fire you are exempt for the vessels—let me say it again—with fire you are exempt for hidden property, okay? That is hidden property, hidden inside the box; what do you mean it’s not hidden? Yes, outside the box that’s something else. Yes, yes, outside the box is something else. There too it’s something else. Look, I’m not going to argue with the law—that’s the law. You can debate why it is so, but that’s the law. Now the question is what happens with my stone, knife, and load. Exempt for hidden property? The first opinion the Rosh brings says yes, and also for vessels, because you need what is liable in both source-teachers; only that can create liability, because you need both in order to create liability. The opposite opinion says no, you would be liable, you wouldn’t be exempt for anything, because in order to exempt you need both source-teachers. But hidden property is not exempt in both source-teachers, only in fire. Vessels are exempt only in pit, not in fire. So you can’t exempt anything. First of all you are liable because it is your property and its safeguarding is upon you, and you have no way to exempt either vessels or hidden property; therefore you are liable for everything. And the Rosh himself has a third view, which says: he is exempt for vessels like a pit, but not exempt for hidden property. Here the symmetry breaks. What is the idea behind that view? And I’ll do this very briefly, in conclusion. The Rosh understands that apparently we are not dealing here with the common denominator at all. This is basically a pit. Look at the logic. It falls onto the ground—that’s a pit. Meaning, whoever trips over it, that’s like tripping over a pit, right? So what’s the problem? I have a problem because another force was mixed into the creation of the pit. So you can’t derive it from a regular pit. That I will learn from fire. But what does fire teach me? It’s not really similar to fire. It didn’t cause damage while moving; then it would have been similar to fire. It caused damage after it had already come to rest on the ground. It damages like a pit. It’s just a pit with a flaw, namely that another force was involved in it. And in fire I see that this flaw doesn’t matter. The fact is that in fire one is liable even though another force is mixed into it. So it goes back to being just a regular pit. Fire only removed a certain problem for me. It looks unlike a pit because another force is mixed in. See from fire that that isn’t a problem. Regarding what issue? That another force is mixed into it. The pit here was created not only by me; the wind also participated in creating the pit. So look at fire: in fire too the wind is involved, and that does not exempt one. Aha, so the involvement of another force does not exempt. We removed the side problem, and we are left with the fact that my stone, knife, and load are simply a pit. But with fire, hidden property is exempt, no? Right. But this isn’t like fire. It’s like a pit; it damages like a pit, therefore it is exempt for vessels like a pit. But it will not be exempt for hidden property, because it’s not like fire. Fire only shows me that the existence of another force mixed in does not exempt. And now it returns to being fully analogous to a pit. So it will be exempt for vessels like a pit, and not exempt for hidden property like fire, because it isn’t like fire. That’s how the Rosh understands it. Again, I made that point briefly—I don’t have enough time here to explain it fully. But what the Rosh is really claiming is that this is not truly a common denominator. It is an analogy to a pit, and there is something else that removes the refutation. But basically it remains similar to a pit. I did not learn it from… I learned it from X. This is what obligates X. They tell me, fine, but there is also Y here. Look over there and see that Y doesn’t have to bother you. And then remain with the fact that you’re similar to X. Okay? That’s the… Okay. He says this is not a common denominator. Right, it’s not a common denominator—you’re not learning from the shared Z of fire and pit. You’re learning from the X of pit. Fine? Okay, we’ll stop here. I wish you success on the exams. Thank you very much. You’re welcome.