Lecture from 1 Iyar 5777

Table of Contents

• Opening and the plan for the lecture
• The a fortiori inference of Tooth and Foot, Horn, the public domain and the damaged party’s courtyard, dayo, and rotation
• Absorbing a refutation in Tosafot on Bava Kamma: common damage versus “half-damages are a fine”
• A formal analysis of a refutation of the “microscopic parameter” type, and why it is not really a refutation
• Tosafot’s distinction between absorbing a refutation and a halakhic refutation from Zevachim
• Formal logic, relevance, and the move to cumulative parameters in the sugya of chuppah in Kiddushin
• Abaye versus Rava: stage-dependence, combining forces, and alternative mathematical models

Summary

General overview

Thursday, 1 Iyar 5770, April 15, 2010, a lecture by Rabbi Michael Abraham, lecture number 24. The Rabbi continues the line of analysis of the kal va-chomer in the sugya of Tooth and Foot, Horn, the public domain, and the damaged party’s courtyard, and emphasizes that the intuition about “rotating” the kal va-chomer between rows and columns is misleading, because in the formal model it is the very same argument. He opens with the concept of “absorbing a refutation,” a formulation coined by the medieval authorities (Rishonim) in order to explain the Gemara, and shows in Tosafot on Bava Kamma how a refutation of the type “their damage is common” does not necessarily knock down the kal va-chomer, but is actually absorbed into it and even strengthens it, unlike “halakhic” refutations involving the addition of a column, which are not absorbed. He then moves to the sugya of chuppah in Kiddushin and presents Rava’s refutation of Rav Huna, that the dependence of chuppah on kiddushin changes the structure of the comparison; Abaye’s response, which tries to preserve the kal va-chomer; and the suggestion to interpret the distinction there as either a combination of parameters or as a tabular error stemming from dependence between stages, until the conclusion that “the Jewish law follows Rava” and “we’ve finally finished the sugya of Kiddushin.”

Opening and the plan for the lecture

Thursday, 1 Iyar 5770, April 15, 2010, a lecture by Rabbi Michael Abraham, lecture number 24. The Rabbi says he is adding more parameters to the discussion, and after that there will be “questions of deriving from it to teach from it,” such as a kal va-chomer from a verbal analogy, a kal va-chomer from a paradigm case, and another kal va-chomer. The Rabbi says he is beginning with absorbing a refutation, and after that there will apparently be a summary lecture and some other topics from the legal introduction.

The a fortiori inference of Tooth and Foot, Horn, the public domain and the damaged party’s courtyard, dayo, and rotation

The Rabbi recalls the Mishnah studied last time, about “the rotation of the kal va-chomer and dayo,” and the question whether the rotation does or does not deal with dayo. The Rabbi describes a comparison table of Tooth and Foot versus Horn, and the public domain versus the damaged party’s courtyard, in which Rabbi Tarfon fills in “one” and the Sages fill in “half,” and the Sages justify that through dayo, whereas for Rabbi Tarfon “he does not accept dayo in this context,” even though the Gemara says that this is a special case.

The Rabbi presents the intuition according to which, if you do the kal va-chomer through the rows, you get “one,” and if you do it through the columns, you get “half,” and then shows that in the Mishnah it does not work that way, because Rabbi Tarfon derives “one” even when beginning with a kal va-chomer that seems to lead to “half,” and the Sages respond, “It is sufficient for that which comes from an inference to be like that from which it is inferred,” even when you reverse the direction of the kal va-chomer. The Rabbi explains that Tosafot formulate this as “he said it to them according to their own view,” because in his own view “he does not accept dayo at all in a kal va-chomer,” and therefore the change of direction for Rabbi Tarfon is an attempt to persuade the Sages, not a change in his own conclusion.

He connects this to a formal model in which “a kal va-chomer of rows and a kal va-chomer of columns are equivalent arguments,” and emphasizes that in the Talmud there is no pattern in which, after one side is refuted, someone says, “Wait, but we still have the columns,” and this is the only place where they try a “rotation,” and even here it is rejected. The Rabbi describes the logic he presented at the end of the previous lecture, according to which, for the Sages, “if the two are equivalent, then take the minimal possible value” as a principle of dayo and “the burden of proof rests on the one who seeks to extract from another,” whereas according to Rabbi Tarfon, since he does not accept dayo in this context, “the problem is still open,” and therefore one moves to the structure that favors “one.”

Absorbing a refutation in Tosafot on Bava Kamma: common damage versus “half-damages are a fine”

The Rabbi states that the term “absorbing a refutation” does not appear in the Gemara itself, but was coined by the medieval authorities (Rishonim) in order to explain the Gemara, and he brings Tosafot on Bava Kamma. Tosafot ask: “What can you say about Tooth and Foot, whose damage is common; can you say the same about Horn, whose damage is not so common?” and they connect this to the view “according to the one who says that half-damages are a fine,” which assumes that animals are presumed guarded, meaning that ordinary oxen are not expected to gore, and therefore strictly speaking there would have been an exemption and the obligation is a fine.

The Rabbi spells out the logic of the one who says “half-damages are a fine,” namely that damage by an innocuous Horn is not a likely occurrence, and therefore “there is no liability” on the owner in principle, and when he is nevertheless required to pay half, “that means they’re fining me,” whereas the damaged party does not “receive a fine,” but simply is not entitled to full compensation for natural occurrences of damage. Into this he weaves a remark about the Rif at the beginning of Bava Kamma, who explains the exemption of Tooth and Foot in the public domain by saying that the damaged party has to be careful, because “oxen walk around in the public domain,” and he raises the possibility that Tosafot do not go with that rationale, and therefore ask specifically for a refutation of the type “common damage” as a source of stringency rather than leniency.

A formal analysis of a refutation of the “microscopic parameter” type, and why it is not really a refutation

The Rabbi presents a distinction between “conventional” refutations, which add a row or a column, and a new kind of refutation that seems “a bit different,” where one points to a stricter feature in Tooth and Foot that is not present in Horn, such as “common damage.” He translates this into a model of parameters, such as alpha and beta, and shows that when the additional parameter does not alter the required results in the table, its existence does not create equivalence between filling in “one” and filling in “zero,” and therefore it does not undermine the kal va-chomer.

The Rabbi explains that Tosafot also raise the possibility, “we derive from one domain to another domain,” in order to escape the refutation, and then reject that by saying, “Nevertheless, one can still raise the refutation properly,” and the Rabbi connects this to the principle that has already been clarified, that the rotation does not save one from a refutation because it is the same argument. He then cites the wording of Tosafot’s answer, “And one can say that it is not a refutation,” and explains that the stringency of “their damage is common” does not help impose full liability in the public domain for full damages, so the fact that despite that stringency Tooth and Foot are still more lenient than Horn actually absorbs the refutation into the kal va-chomer and strengthens its conclusion.

Tosafot’s distinction between absorbing a refutation and a halakhic refutation from Zevachim

The Rabbi concludes Tosafot by bringing the example “in the chapter in Zevachim, page 10” about a kal va-chomer concerning slaughter with proper intent and sprinkling with improper intent, where a refutation of the type “karet” is raised and is not absorbed. He explains that Tosafot conclude, “A stringency with which the Torah was stringent is different,” meaning that when we are dealing with a “halakhic” refutation that adds a substantive category accepted by the Torah, we do not say that the kal va-chomer absorbs it; rather, we are concerned that “since the Torah was stringent with this stringency, it may also have been stringent with another stringency,” and in the language of the model this parallels a refutation by adding a column, where the new parameter becomes relevant and changes the outcomes as well, so there is no absorption.

Formal logic, relevance, and the move to cumulative parameters in the sugya of chuppah in Kiddushin

The Rabbi emphasizes that the analysis so far does not depend on the content of oxen, damages, or impurities, but on a formal logical structure of tables and constraints, even though the determination of the “relevance” of parameters depends on the content chosen as a given. He announces a move to “parameters that operate cumulatively” through the sugya of chuppah in Kiddushin, and recalls the kal va-chomer built there: money, which can effect betrothal but cannot effect marriage, succeeds in effecting betrothal; therefore chuppah, which can effect marriage, certainly ought to effect betrothal, even though the Mishnah says, “A woman is acquired in three ways: by money, by document, and by intercourse.”

The Rabbi presents the basic difficulty: Rav Huna says, “Chuppah acquires by a kal va-chomer,” and that seems to go against the Mishnah. After discussions and attacks and defenses, the Gemara proposes Rava’s refutation: “Does not chuppah complete only through kiddushin? And can we derive chuppah without kiddushin from chuppah with kiddushin?” The Rabbi explains that Rava’s claim is that chuppah effects marriage only because kiddushin has already preceded it, and therefore one cannot derive from it an independent power to effect betrothal.

Abaye versus Rava: stage-dependence, combining forces, and alternative mathematical models

The Rabbi brings Abaye’s response defending Rav Huna: “Rav Huna too said precisely this: Just as money, which does not complete after money, nevertheless acquires, so chuppah, which does complete after money, is it not all the more so that it should acquire?” Abaye assumes that even after there is already betrothal, chuppah succeeds in effecting marriage whereas money does not complete it, and therefore chuppah is stronger than money even if it operates “after money,” and the Rabbi notes that this is a place where one can bring in various mechanisms that have already appeared earlier, such as “absorption,” “rotation,” “an additional index,” and “a microscopic external constraint.”

The Rabbi proposes a “more realistic” model in which one does not write “half,” but rather relates to the built-in dependence that chuppah for marriage joins with the money that has already been used for kiddushin, so the simple table is misleading and one has to build the diagram backward from the condition “chuppah plus money” in relation to marriage and not in relation to betrothal. He also presents another possible analysis in which Abaye divides the powers into different parameters, so that money gets alpha and chuppah gets beta, and thus chuppah is not the same kind of power as money but a different power that appears because of the combination of stages. The Rabbi concludes in the end that the Jewish law follows Rava, sums up with “More power to you,” and ends, “We’ve finally finished the sugya of Kiddushin.”

Thursday, 1 Iyar 5770, April 15, 2010, a lecture by Rabbi Michael Abraham, lecture number 24. Just a few more parameters that I also hope we’ll learn today, and after that there are questions of deriving one thing from another. Meaning an a fortiori inference by analogy, an a fortiori inference from a paradigm case, an a fortiori inference and so on. And with that, more or less, I’ll finish this. After that we’ll do some kind of summary lecture, and then a few other issues from the legal introduction. What I want to do today, as I said, is start with swallowing an objection, and after that I hope we’ll have time for something else. Okay, so what is swallowing an objection? For that I brought you the source on the sheet, in Tosafot on tractate Bava Kamma. It appears in a few other places too, but among the medieval authorities (Rishonim), this does not appear in the Talmud itself; the term “swallowing an objection” is something that was basically coined by the medieval authorities (Rishonim), of course always in order to explain the Talmud. Meaning, if they’re right, then it’s really already there in the Talmud, but the term, the conceptualization, was done by the medieval authorities (Rishonim). Okay, so basically I’ll just remind you briefly: we did the Mishnah last time, after the first source on the sheet. That’s the Mishnah where we talked about rotating the a fortiori inference and about the “it is sufficient” principle. How the rotation deals, or doesn’t deal, with “it is sufficient.” And now I’m moving on to Tosafot, which is really another aspect of that same a fortiori inference. And maybe I’ll just remind you what we’re talking about. We basically have an a fortiori inference — let’s do it on the side, rows and columns. Tooth and foot, horn, public domain, the injured party’s courtyard. That is basically the a fortiori inference that appears in the Mishnah, where here Rabbi Tarfon fills in one, and the Sages fill in half. And the Sages justify it with the “it is sufficient” principle, while Rabbi Tarfon does not have “it is sufficient” in this context. Meaning, the Talmud says this is a special case where Rabbi Tarfon does not apply “it is sufficient.” He also accepts the basic principle, but not here. In any case, there is a dispute here about how to fill in such a table. How do you fill it in? Do you fill in half here or one here? Seemingly, this depends — we saw that this depends — on the direction in which you make the a fortiori inference, at least that’s what we would have thought. Namely, we said you can do the a fortiori inference through the rows or through the columns. An a fortiori inference through the rows is one that starts from the right-hand column. In the right-hand column we see that horn is more severe than tooth and foot. Now we move to the left-hand column, and if the assumption is that horn here too is more severe than tooth and foot, then if tooth and foot is one, horn must also be at least one. Okay? And then the required conclusion is one. By contrast, if I make a comparison between rows — move away from here, kid — if I make a comparison actually by columns, sorry, if I do it in the upper row, in the upper row I see that the injured party’s courtyard is more severe than the public domain. Now I go down to the lower row, and I assume that here too the injured party’s courtyard is more severe than the public domain. So if here it’s half, then here it will be at least half. But here of course the result is half and not one. When we make comparisons between columns, the result is half. When we make comparisons between rows, the result is one. And what lies behind this? The comparison between columns means that I’m basically assuming the injured party’s courtyard is more severe than the public domain. That is the basic assumption of the a fortiori inference. And the a fortiori inference based on that assumption yields half. The a fortiori inference that compares rows is an a fortiori inference that basically assumes horn is more severe than tooth and foot, not that the injured party’s courtyard is more severe than the public domain. That horn is more severe than tooth and foot — that’s what we learn this way. So this should be more than that, and therefore we arrive at one. That’s the assumption we would intuitively have made. But in the Mishnah we saw that it doesn’t work that way. The Sages say half, Rabbi Tarfon says one. Rabbi Tarfon at first starts with an a fortiori inference that goes like this, which is very surprising. Specifically through this a fortiori inference he wants to prove to the Sages that here it should be one and not half, and then the Sages say: what are you talking about? It’s half — it is sufficient that what is derived by legal reasoning be like that from which it is derived. After all, if this is half and this is more severe than that, then an a fortiori inference like this is a comparison that works this way, so here it’s half. So Rabbi Tarfon now comes and rotates the a fortiori inference and basically learns a comparison like this. He grabbed the Sages’ argument, and therefore let’s complete it this way: if this is one, then here too it must be one. And the Sages, without batting an eyelid, tell him again: it is sufficient that what is derived by legal reasoning be like that from which it is derived. And they say that about this a fortiori inference too. So we saw there two things. First, for Rabbi Tarfon the result one does not follow only because he makes the a fortiori inference like this. Even in this a fortiori inference, in his view, the result is one. And that is really what Tosafot here — what I brought on the sheet — says: “He said it to them according to their own view,” because in his own view he doesn’t accept the “it is sufficient” principle at all in an a fortiori inference. He doesn’t have “it is sufficient” at all. Meaning that for him even in an a fortiori inference like this the result is one. Even when he changes the direction of the inference, that’s only to persuade them. Okay? And the Sages are the exact opposite by 180 degrees. Meaning the Sages say “it is sufficient,” and they say there is half here even in this a fortiori inference. Meaning Rabbi Tarfon says one even in this a fortiori inference, and the Sages say half even in this a fortiori inference. Which is very surprising from an intuitive point of view. But after we made the model, it really turned out that it indeed makes no difference even in this a fortiori inference, and whoever says half or says one will say it regardless of which side of the a fortiori inference you look at. Now that was the case of “it is sufficient” and of the connection. But if so, then what is the dispute in formulation between Rabbi Tarfon and the Sages? Because I think I got a bit confused about that earlier. According to the Sages, the simple assumption I showed at the beginning was that if you analyze this while ignoring the difference between half and one, liable or not liable, then it comes out that the possibility of filling in half here — we checked three possibilities: filling in zero, filling in half, and filling in one. The possibilities of filling in half and filling in one both come out equally according to the Sages. And then we always take the minimum? Then we always take the minimum, exactly, that’s the simple logic of “it is sufficient.” Meaning if both are equivalent, then take the minimal possible option. The burden of proof is on the one seeking to extract from another. More than that, you have to bring me proof. I agreed to half; more than that, bring proof. Rabbi Tarfon says that since he does not accept the principle of “it is sufficient,” at least in this context, then from his standpoint when half and one are both equivalent, the issue is still open. Precisely because he does not accept the principle of “it is sufficient,” the issue remains open. So with no choice, we moved to the rows, redefined how we get to the result of half, and then addressed the difference between half and one, and there indeed it came out that filling in one is the preferred filling. And there it came out categorically. So according to the Sages, by contrast… Right, now I said there — and then the question that came up at the end of the lecture was: so what do the Sages say? After all, what do the Sages need to explain? Why it says half here and not one, while relating to the differences you see in this table. So I said that here it’s open; I don’t know. There are several possibilities, and we don’t have additional passages to test it on. There is one additional passage whose structure is exactly like this, so certainly whatever we say here will apply there too. Meaning, the dispute between Rabbi Tarfon and the Sages really is whether we shift to the rows or not? And Rabbi Tarfon says let’s take it this way. According to the Sages, no, I’d say it differently. I’d say that according to the Sages, in order for it to be one, you need to get one in both directions. And according to Rabbi Tarfon, it’s enough that in one direction it comes out one for me to say the result is one. Understand? I don’t want to make a division between rows and columns, but there is some significance to the direction in which you look at the a fortiori inference. And that’s very intuitive because the matrix here really isn’t symmetric. So we would expect that if there is any difference at all between an a fortiori inference like this and one like that, it would be here. Can one say that Rabbi Tarfon sees the structure of an a fortiori inference as something where what I learn is more severe than the source from which I learn it? That’s true for everyone. No, that according to the Sages, their perspective is that the a fortiori inference means it will be at least the same. And therefore according to the Sages it will be half. You mean that according to Rabbi Tarfon it has to be actually more severe? Meaning not the same, but it has to be more? The very essence of an a fortiori inference means that if I say this is more severe than that, then it can’t be just equal all the time, can it? According to the Sages, the meaning of an a fortiori inference is that it is at least the same. I wouldn’t formulate it that way. I would formulate it as saying that for Rabbi Tarfon there is no preference for half over more. I don’t know whether Rabbi Tarfon requires that it must really be more. More by how much? Half plus epsilon. Fine, so it’ll be half plus a tiny bit. Why half or full? The next level is full. Why not? Why not? There are two people… Because if he damaged someone and they split the damage, that’s half. If he damaged someone and they don’t split the damage, that’s full. That’s enough. It’s divided because of the difference, because in his case I’m in doubt. Half is like “two holding a garment.” But then because I don’t know… No, or it’s an intermediate level. Not because I don’t know. No, there is an intermediate level, not because I don’t know. There is no intermediate level. Why not? Of course there are lots of intermediate levels, and that’s exactly the point. The whole advantage of “it is sufficient” is based on the fact that I’m not willing to accept intermediate levels. I go with the minimum. And Rabbi Tarfon, who doesn’t accept that… maybe yes, maybe no. I’m just saying that I think with Rabbi Tarfon I’d formulate it more strongly. Isn’t that what he says? What? He says it as full, doesn’t he? Yes, yes, he knows he says full, but the question is how he gets there. But I’m saying one could have formulated it that Rabbi Tarfon requires it to be actually more, as you suggested. He’s not satisfied with equality. I want to claim a weaker assumption in Rabbi Tarfon. I want to say that Rabbi Tarfon does not accept that half has priority. He doesn’t have to be twice as much, but I’m not willing to accept… Rabbi Tarfon says: it’s not true that the minimum has priority. For me it’s all the same, and now what determines it is the model. The half is problematic to begin with. What? Half, in relation to half-damages, is problematic. In relation to half-damages? What? In relation to half-damages? Fine, that’s just a question on the passage here. I’m not getting into it. But the plain sense is that it should be one. If anything, it should be one. Right, but here there is half. So you can’t ignore the fact that the Talmud here does play with the half. Of course, of course, the give-and-take does, but that half — why isn’t it zero? If you really relate to it in terms of severity, then the minimum here should have been zero rather than one. If you relate to the half as a fine, then really it should have been zero here. Sure, if Rabbi Tarfon didn’t consider it a fine. What? If Rabbi Tarfon didn’t consider it a fine. This half is not a fine. No, but the half is a fine. Yes, but if you say that nothing is learned from a fine… I didn’t say that, I didn’t say that. That’s monetary. He doesn’t recognize the concept of a fine. What? Come on, there’s a fine here. The Torah gave half; it waived half for him. Why did it waive half for him? That too is a fine. Why did the Torah waive it? The Torah created a legal creature. Why? Why? The Torah said nothing about that. Why assume it spoke specifically about the public domain? So everywhere, that too is a fine. After all, an innocuous horn in the injured party’s courtyard is also a fine. I have a harder time understanding it that way than Rabbi Tarfon’s view… Everyone can say what he understands, but I don’t know how to decide all these questions, because we have two examples and both are identical. We have nothing to test all these suggestions on. We have no more data. What’s the second example? Which one? What example? A passage in tractate Niddah that looks exactly the same. An a fortiori inference like this, they apply “it is sufficient,” they rotate it, there’s a dispute there between Rabbi Eliezer and the Sages, but it is completely parallel to Rabbi Tarfon and the Sages here. Exactly the same thing. In the analogy to gravitation, we only have a ball and a table. What? In the analogy to gravitation, we only have a ball and a table, and we don’t really know what happens to a pencil. Yes. So according to this approach it comes out that basically the dialogue between Rabbi Tarfon and the Sages in the Mishnah — until now I thought this explanation was just… they did it for the simpletons sitting there and watching. Obviously, it’s a pedagogical presentation. They, in the lab, had all the formulas and graphs and reached… their real dispute was there. This is just a pedagogical presentation of the issue. You attack in order to reject what the learner will think when he reads it. That’s exactly what I thought when I started reading the Mishnah: that the dispute is whether to learn the a fortiori inference this way or that way. So the Mishnah comes to teach me that this is not correct. How does it do that? Rabbi Tarfon attacks the Sages: come learn it this way, because this way it is certainly one. The Sages say: it is sufficient that what is derived by legal reasoning be like that from which it is derived. They teach you: don’t think it depends on the direction of the a fortiori inference. It doesn’t. But really it does depend. No, it doesn’t depend. It depends on what analysis you use, not on the direction of the a fortiori inference. That’s not the same thing. It’s not the same thing. What kind of answer is “it is sufficient that what is derived…” ? After all, if you tell him to learn it this way, then it must be this way. And what I’m saying is: no, this way and that way are the same thing. That’s exactly what I was trying to say all along. The Sages’ answer to Rabbi Tarfon — “it is sufficient that what is derived by legal reasoning be like that from which it is derived” — it doesn’t solve the… Why does it solve it? Because what the Sages are basically telling you is: if I learn from the public domain… that’s what they’re saying, that you don’t learn… Sorry, when they say “it is sufficient,” they’re saying that an a fortiori inference… there is no such thing as an a fortiori inference this way and an a fortiori inference that way. There is only one a fortiori inference. There isn’t an a fortiori… That’s what we said all along. An a fortiori inference of rows and an a fortiori inference of columns is an equivalent argument. They are not two different arguments, as it appears on the intuitive level. It’s the same argument itself. It’s the same argument itself because everything is based on the microscopic structure, on the chemical analysis. And in chemical analysis it makes no difference at all whether you look at the rows or the columns. And that is exactly what they’re telling you here too. Here too, the same thing. As the Rabbi said, there isn’t a single case in the Talmud where once they refuted the rows, they say, “Wait, but we still have the columns.” This is the only place where they try. And here too in the end they reject it. The Sages… What? The Sages reject it. Yes, meaning this is the only place where they even try a rotation, and here too in the end they don’t accept that attempt. That says something. The intuitive outlook immediately jumps to that.

Okay, so all that was just a summary of last time. Now look at Tosafot. “I do not derive horn from horn; he said it to them according to their own view, but according to my own view I do not accept that where there is an a fortiori inference…” We discussed that earlier. The opening only speaks according to the Sages’ view; he himself does not disqualify such a clear a fortiori inference — not this one. That is basically what Tosafot says, and that’s what we said earlier. “And if you say: what about tooth and foot, whose damage is common? Will you say the same of horn, whose damage is not so common, since it ordinarily stands under an assumption of being guarded, according to the one who says that half-damages is a fine?” There is a dispute in the Talmud regarding horn — whether half-damages for an innocuous horn is a fine or monetary liability. The details don’t matter right now. In any case, the one who says it is a fine basically says that by strict law he should not have had to pay. The Torah fined him to pay half; by strict law he should not have had to pay. Why should he not have had to pay? Because ordinary oxen are presumed to be guarded, okay? A person need not worry that his ox will gore unless it is already a forewarned ox. If it is forewarned, then the person needs to understand that his ox requires guarding, but an innocuous ox that never gored before — ordinary oxen are presumed to be guarded. If it causes damage by foot or tooth — because there is pleasure in the damage, or simply as it walks along — in that regard it is forewarned from the outset; those are damages an ox normally causes. But to gore is damage purely for the sake of harming, not for pleasure and not for anything else. I have no reason to assume my ox will do that unless it has done so before. Therefore the assumption is that ordinary oxen are presumed to be guarded; one need not guard them, and therefore in principle you should really be exempt so long as the ox is innocuous — the first three times. It’s a fine, yes, according to the one who says it is a fine. So according to the one who says it is a fine, he basically says ordinary oxen are presumed to be guarded, and therefore you really ought to be exempt; the Torah fined you so that you won’t be careless anyway, but it is essentially a fine. But for our purposes that doesn’t matter; the assumption behind the view that it is a fine is that ordinary oxen are presumed to be guarded, meaning it is rare for an ox to gore. By contrast, tooth and foot, as we said before — for tooth and foot one is liable for full damages already from the first act of damage; there is no stage of innocuousness and only later becoming forewarned. From the first act of damage one is liable for full damages. Why? Because tooth and foot are more common forms of damage than horn. Okay? Now if the term “fine” really isn’t all that clear to me — if someone totalled my car and my car is gone and I get only half-damages, whom exactly did they fine? Whom did they fine? I’m the one who got hit hard. You got hit hard, but what can you do? It’s in Heaven’s hands. So I suffered a total loss and they give me half, then I took the fine even though he ran into me? Wait, one second — if someone rammed his car into your car and he was driving a hundred and slammed the brakes and everything and he was empty-headed and took his car and smashed it into your car — does he have to pay you? No. That’s a good example — I think that’s exactly the point, because it’s the way of the world that an ox doesn’t… If he pays half, it depends how I look at it. Either he gets fined, because a person is always forewarned and does it knowingly. So both of them got fined? No, no. In “two holding a garment” there’s no fine at all as far as I can see. Why? Because regarding one half, he admits it to him — this one says “it’s all mine,” and the other says “half is mine,” so first give him the half on which they both agree. Fine. And with the remaining half they split it, therefore there one gets three-quarters. No, no, it doesn’t matter — “all” and “all” means half and half. Okay. Now the moment the Torah obligates one side to pay half, it depends from whose point of view who got fined. But that can also explain why in the public domain, horn — you walk there, you know there are cars, you know there are oxen, pay attention, watch yourself. That is the explanation for tooth and foot, why tooth and foot is exempt in the public domain. The Rif says at the beginning of tractate Bava Kamma that people walk there, and you need to know that oxen roam around in the public domain, so watch out. But you need to reckon with reasonable kinds of damage from oxen roaming in the public domain: they might eat, they might trample as they walk. You don’t need to reckon with damages where an ox suddenly starts going wild. If you have a wild ox, guard it — your ox — that’s not my problem, it’s yours. Don’t let it roam around alone in the public domain. That is the basic conception of why only for tooth and foot is there exemption, whereas for horn there is no exemption. But according to the one who says half-damages is a fine, he basically says: you know what? Even in horn, in principle you are exempt. They fined you so you wouldn’t be careless, but exempt. Now why is the fine a fine on the payer? That is exactly what he explained earlier. In the end the question is always whose fortune prevails. Meaning, if it were a natural disaster, you wouldn’t call that they fined you, right? Now an ox going wild and goring — from our standpoint that is a natural disaster, because it is not its ordinary way to gore. It’s not some negligence of the owner. The owner is not supposed to guard against that; he is not supposed to fear such a thing. If the ox suddenly started flying in the air and damaged a zeppelin up above, then okay, the owner never imagined his ox could also fly. Obviously that’s an exaggerated example, but that’s the idea: once it’s not something reasonable. Okay, gentlemen, Tuesday, goodbye… good, good. So these are considered natural disasters. Natural disasters happened to you; what can you do? It’s not a fine on you. You suffered, but what can you do — no one is really at fault. But although I’m not at fault, they told me to pay half. Why? I didn’t suffer anything. What happened? Nothing happened to me. The court obligates me to pay half; no natural disaster struck me. That means the fine is on me. Do you understand? It’s true that you also lose half, but that can’t be viewed as a fine. A natural disaster happened to you; there’s nothing to do. Instead of getting full compensation I get half — so I got punished too. You were not supposed to get full compensation; it’s a natural disaster. If lightning struck your ox, okay? It comes down from Heaven — you get nothing. Do you want to say they fined you? Who would pay you? The Holy One, blessed be He? Who? It’s a natural disaster. So it’s not that they fined you. If the responsibility is split half-and-half between the two of us, then it’s not a fine. No, that’s exactly the point — there is no responsibility here. An ox ordinarily does not gore — that’s the assumption of that view. And once it ordinarily does not gore, there is no responsibility. Nobody is responsible, neither me nor you. So what happened? You suffered damage. What should have been the law? You suffered damage. I can be sorry, but I have no responsibility. Now once they obligate me to pay half, that means they are fining me. You got half-compensation; say thank you it wasn’t zero compensation, because really nothing was owed to you. They fined me to pay half. That is the claim of the one who says half-damages is a fine. Maybe it’s just some kind of dispute, some kind of what? Social solidarity? That if an ox goes wild, it makes no difference whether it’s the public domain or the injured party’s domain. No, but here once it goes wild it’s the same thing, but you’re explaining some substantive reasoning, and I’m explaining the silence of the Sages — the Sages don’t respond to Rabbi Akiva. No, they do respond. No, that’s exactly the point. They don’t respond; they answer like… like autistic people — the same thing. Fine, but if they answer the same thing, that means they have a completely different reasoning here. Fine, okay, that’s possible. The Sages here are operating in the field of a fortiori inference and “it is sufficient.” If you were right, then the Sages really wouldn’t have had to enter into logical discussions with him at all. The Sages should have said to him: listen, your substantive reasoning is wrong. But what characterizes logic — and that was also part of why I gave all the introduction — what characterizes logic in general, and especially in these models, is that the conclusion does not depend at all on the contents involved here. It doesn’t matter whether we’re talking here about oxen, damages, public domain, half-damages; you could put here the impurity of a zav, leprosy impurity, and two other random things, and it would all be the same. It doesn’t matter. That’s what logic is about. Meaning that the contents play no role here. But there is something here… really… But regarding… it could be that in the hermeneutic rules by which the Torah is interpreted, or in the Sages’ derivations, the Sages already know the Jewish law is half. What you’re telling me about rules and a fortiori inference — that doesn’t interest me. I know the result. You come this way, so I’ll find reasons to reject your argument, as Tosafot says, that “he said it according to his own view”; they too said it according to his view. Fine, good, they said it according to his view, and I want to understand what they said. Why does he pay half? End of story. But what did they say according to his view? Fine, according to his view. But what did they say according to his view? Nothing. About everything. No, what do you mean “nothing”? “According to his view” has to hold water at least according to his own view. Otherwise it’s just… it’s like Rabbi Yohanan: “I ask a contradiction in pottery shards and don’t know what you’re saying.” Meaning, that is what should have been asked here. Fine, so you trapped him with pottery shards, but what is your own reasoning? Say: this is a tradition we have from our teachers, or “it is learned as a received Talmudic tradition,” or “a law given to Moses at Sinai.” Nobody says anything. They are operating here in the field of a fortiori inference and “it is sufficient.” No, the field of discussion here is not some specific reasoning tied to these contents. The issue of common or uncommon damage applies entirely in the public domain, and there it’s reversed — yes. Tooth and foot in the public domain is a leniency, and in the injured party’s domain it is a stringency. It comes out opposite. Meaning, in the public domain, tooth and foot is obvious — there’s an ox eating and all that — and then the person who suffered the damage needs to watch out. Whereas in the injured party’s domain it’s the reverse. He says: I brought my ox into your domain, but I never dreamed it would gore. Whereas that it would eat there — that’s certain. Meaning, tooth and foot in the injured party’s domain is more severe than horn in the injured party’s domain, as opposed to tooth and foot in the public domain. From the perspective of the damager it flips. No, so you’re saying that this objection Tosafot raises — common versus uncommon, which is tied to our passage — functions differently in these two boxes? Of course. That distinction basically says what? That it’s even more severe. Of course, because it’s obvious it would eat there if he brings it into the injured party’s domain, but it won’t gore, because that’s not common. Whereas in the public domain, why is the compensation zero? Because the person shouldn’t have put tomatoes in front of an ox. Oxen eat. So if he put tomatoes there, he gets no compensation. Whereas in the injured party’s domain, what can the injured party do? He’s got oxen there in his barn, and suddenly an ox comes in and gores them. The other fellow says, come on, I never dreamed it would gore. That’s an interesting comment. Tosafot and the other medieval authorities (Rishonim) didn’t understand it that way. No, because they understand — we’ll soon see what they answer — they understand that what counts as common plays the same role in both of these boxes. It plays into the a fortiori inference. We’ll soon see what they do with it. Maybe. Maybe you could say that according to the Rif — since the Rif basically suggested explanations for why tooth and foot in the public domain is exempt. Maimonides also goes that way, with a slight change in formulation, and others as well. But maybe Tosafot really didn’t learn that way, and therefore he doesn’t accept that explanation, because on the face of it there really is some room for it. And then maybe the Rif would not ask Tosafot’s question, because according to the Rif it’s obvious. Or perhaps even more than that, one could argue on substantive grounds: what does it matter whether the damage is common? Once you crossed the fence into my courtyard, you’re responsible for everything that happens there, period. What does it matter whether it’s common damage or uncommon damage? The very fact that you entered. What’s more severe? Common damage is at least more severe or equal; it certainly doesn’t make things easier. But in the public domain, common damage, as Yossi says, makes it easier, precisely. Because if the damage is common, then obviously I, as the injured party, need to watch out for it. After all, it’s the public domain. It’s a very common thing for an ox to eat by tooth or trample by foot, so certainly the duty of care falls on me, because the basic assumption is that for what they normally do in the public domain, they are exempt, because for that the duty of care is on me. So he’s correctly noting that the role played by “common” works one way in this box and the opposite way in that box. And maybe that really is evidence that Tosafot does not accept these explanations of the Rif. He apparently sees it as something formal, not because there are explanations. The Maharshal already comments on this. After all, we don’t generally derive Jewish law from the reason of the verse, and suddenly here the Rif offers an explanation of the reason of the verse, and the Maharshal asks that about him. What? The Torah says exempt, exempt — why are you going into all kinds of explanations? Because those explanations have halakhic implications. For example, a plank lying partly in the private domain and partly in the public domain — there they already discuss it. By the way, regarding this very page, I now remember, Tosafot brings the Rif on this page and disagrees with him. Which plank? A plank lying across, extending from the private domain into the public domain, and an ox walks in the public domain and steps on that plank, and the plank causes damage in the private domain. Then the question is whether that is considered damage in the private domain or damage in the public domain. If you understand the Rif’s conception, then the ox walked normally in the public domain, it… But if you understand it as a scriptural exemption in the public domain, then the damage occurred in the private domain, and you’re not exempt for that. So there that will come in — that will be the practical implication. And if I remember correctly, Tosafot indeed disagrees with the Rif on that issue. And if that’s so, then it actually fits very well, and therefore here Tosafot does ask this question. Truly, according to the Rif you’re right that the question apparently isn’t difficult.

All right. Tosafot says, yes, tooth and foot — its damage is common. So what if the damage is common? Tosafot’s assumption, of course, is that common damage is a reason, a characteristic, that is lenient or stringent? From the perspective of the damager, is it more stringent? Yes, yes. Is it easier to obligate if the damage is common, or harder? Easier to obligate. The less likely the damage is, the less we obligate — that’s what we saw in “half-damages is a fine.” Right? If it’s not likely that damage will occur, then we don’t obligate. We said tooth and foot are common damage. That is a reason to obligate, not a reason to exempt. Okay? Ignore Yossi’s comment for the moment — he’s right according to the Rif — but we’ll leave that aside for now; Tosafot apparently did not adopt that point. It doesn’t matter at the moment. I’m saying the assumption is that common damage is a reason that cuts in the stringent direction, meaning the more common the damage, the easier it is to obligate. So Tosafot asks: how can you say that horn is more severe than tooth and foot? After all, we found that horn is less common. Exactly — there is a stringency in tooth and foot, in that it is more severe than horn. So why are you telling me via an a fortiori inference that horn is more severe than tooth and foot? Right? Of course, we’ll immediately ask: then maybe let’s make an a fortiori inference like this — from one domain to another? Tosafot himself raises that possibility. Okay? We’ll see in a moment. So Tosafot asks like this: if the damage is common, then we have an objection: tooth and foot, its damage is common. Let’s see what we do with that objection in our model. How does that work? We’re used to objections of this type or that type, right? What kind of objection is this? Adding parameters? Maybe. Right? There is no addition of columns here. Common damage is a characteristic of the damager, of tooth and foot, not an additional law — not another domain in which, say, tooth and foot is liable and horn exempt, or another kind of damager that is liable in the public domain and exempt in the injured party’s courtyard. Those are two conventional objections. But here there is a slightly different objection. What? Yes, exactly, like the game with pleasure that we did, but here on a simpler level, on the level of a simple a fortiori inference. So on the level of a simple a fortiori inference, we now encounter for the first time a new kind of objection. There’s a column objection, there’s a row objection — those we already met, we saw them. In terms of our model, they are basically equivalent. Now a new objection: we know there is a parameter characterizing the side of tooth and foot at a higher value than horn. Say one in tooth and foot and zero in horn, just schematically. Okay? That’s a constraint. Let’s see what it does. So in the ordinary a fortiori inference — just to remind us, let’s ignore the objection for a moment. What happens in the ordinary a fortiori inference, and then we’ll see what the objection does. In the ordinary a fortiori inference, what happens? Say with filling in one. Then we have — maybe let’s mark it this way — tooth and foot is S, horn is K, public domain is R, injured party’s courtyard is N. Okay? So if N is filled in as one, then the table is this. This is alpha, this is two alpha. S contains N but not R. Meaning S has alpha, and horn contains… you know what, let’s ignore S for a moment, S doesn’t matter. In the ordinary a fortiori inference, let’s leave aside the complications for a moment: horn contains R, and the question is what… and if it contains this as well, then it’s two alpha. Right? That’s in the filling of one. What happens in the filling of zero? In the filling of zero, then here there is a disconnect. These two are not connected to one another, right? So if this is alpha, this is beta. And tooth and foot contains N, meaning it has alpha, and K has beta. Okay? That’s the ordinary a fortiori inference. Okay? Now what does the constraint, or the objection, tell us? The objection says that there is a quality in tooth and foot that horn does not have. A stringency-quality, of course. Right? We already said that the direction of the indices corresponds to a higher value, which means stringency. Okay? So that means that there is some parameter, let’s call it beta, which is present in tooth and foot and absent in horn. What? You used beta for R. No, I used R as R. Ah, beta I used for K. Okay, then alpha will be here. Here there is — leave that for a moment — I’m speaking now about filling in one, okay? Leave this aside; I’m talking about filling in one. Okay? There is a parameter beta that is in S and not in K. Right? That’s what we know; that’s the constraint. And we also know what beta is — common damage. Some force whereby tooth and foot is common and horn is not. Alpha is another parameter, and it is basically the parameter that determines the damage obligations. Beta apparently has no effect thus far. Notice that. Meaning, from the standpoint of these results there is no reason to insert beta. Right? Look. S contains the injured party’s courtyard, N, and that means it has alpha. And it does not contain the public domain… because here there is two alpha. Okay. Beta… there is no reason to add betas here in the diagram either. Likewise here. K contains R and also contains N. So that means it has two alpha. Beta here really has no role. Right? But we still know there is a parameter beta here. So if we ask ourselves what the dimension of the model is for filling in one, it is still two. Right? Because there is also beta. It’s true that in the operations, in the results, I don’t need to bring in beta, but there are two parameters in the field. Okay? What happens here? Why is beta not felt in the case of filling in one? It isn’t felt. The assumption of filling in one, what does it actually mean? This is an important point — pay attention. Filling in one is not considered separately from filling in zero. Filling in one basically says: there is a parameter beta, but it doesn’t affect anything, and therefore indeed the filling is one. Filling in one basically says that the existence of a parameter beta does not interfere with the a fortiori inference. The a fortiori inference remains; the result is still one. The fact that there is beta shouldn’t bother me. But why specifically beta? There could be, besides beta, delta and many others that don’t affect anything. No, obviously, but I have others too — after all, I’m taking the minimum possible. We could build models here with infinitely many parameters that would explain this. But I take the minimum. What forces the existence of beta? Beta is… the assumption of the a fortiori inference is… of the objection — that beta is a relevant parameter. The force of common damage. We know that, which is also why horn is liable for half, because it is less common. Meaning the fact that it is common is a relevant parameter. It turns out that if the filling is one, what does that actually reflect? It reflects that although this parameter is relevant, it nevertheless does not affect the laws. We’ll still see whether it’s true that it doesn’t affect them. We are checking this option against the second option. Okay, it’s very interesting, and the assumption is that despite its relevance, it does not affect. What does the assumption of filling in zero basically mean? Filling in zero is exactly what says that the parameter beta — sorry, I should have marked it the other way around — the parameter alpha here corresponds to filling in zero. You can just do filling in zero. What does that mean? This is the filling in zero. What? Filling in zero means that R and N are disconnected; there’s no relation between them. Right? Let’s call it, for consistency, this will be alpha and this will be beta. Fine? The sigma reaction. Tooth and foot is beta, right? Tooth and foot contains N but does not contain R, and K contains R but does not contain N. Okay? Now what does the objection do? It simply identifies beta. It changes nothing, right? It simply says: this beta that characterizes tooth and foot and is absent from horn — that is just common damage, that’s all. And why is that an objection? What? No, I didn’t say it was an objection; I’m now doing the analysis. Okay? So basically what we got here is that… sorry, I have a question. You said that if I fill in one, then the parameter beta is not relevant. Why? That was my critique. In parentheses, from a mathematical standpoint. Now I understood that the whole point of the objection is to tell you, my friend, there is beta here, and therefore the whole a fortiori inference — namely the filling in one — becomes irrelevant because there is beta. Filling in one is not irrelevant. Fillings aren’t a matter of relevance or irrelevance; I compare them to each other and see what’s true. The meaning of filling in one is that there is an a fortiori inference. Right. So beta tells you there is no a fortiori inference. No. Beta does not tell you there is no a fortiori inference. I now want to check — this is called refuting an a fortiori inference. Aha, and now I want to see whether indeed, in my model, this really refutes it. Tosafot says it refutes it; I am now analyzing whether in the model it actually looks like an objection. So let’s check. If it really comes out as an objection, what would that mean? It would mean that filling in one and filling in zero should be equivalent, right? Let’s see if that happens. In filling in one I did the analysis here; in filling in zero I did the analysis here. If those were equivalent, then it would be an objection, right? Do they come out equivalent? They do not, apparently. They do not. Why? No. This is two-dimensional and this is two-dimensional. This graph is stronger. Exactly, it’s a stronger graph. Precisely. And this thing still remains, and therefore filling in one still remains valid. This objection does not refute the a fortiori inference. The whole thing would have worked only if you had in fact had a strength in this table that could have served as an objection that blocked it here. Okay, that is exactly the claim. Only if we wanted a column objection — then, as we already saw, the graphs would change, and that really would be an objection. But it turns out that the objection Tosafot suggests is not an objection at all. That is Tosafot’s answer. So the objection Tosafot suggests is not an objection at all. Once we look at it this way, we say: okay, so something in my model is probably missing something, because this really does look like an objection, namely that tooth and foot is common damage. So then what does Tosafot say? “And one cannot say that we derive from one domain to another, and just as in the public domain, where the Torah was lenient regarding tooth and foot, it was stringent regarding horn…” And one cannot say that we’ll learn from one domain to another rather than from one damager to another — which is what we discussed, that then indeed this objection wouldn’t be interesting. Why? Because if I learn from one domain to another, domain to domain, then why should I care that tooth and foot is common damage? “Tooth and foot is common damage” refutes this hierarchy, but if I learn like this, then I assume that the injured party’s courtyard is more severe than the public domain. Why should I care that tooth and foot is common damage? It’s irrelevant. Right? So Tosafot seems to offer the obvious claim. What is he trying to do, basically? Rotate the a fortiori inference. But we already saw that rotating the a fortiori inference never helps. So Tosafot says: “And one cannot say we derive from one domain to another, and just as in the public domain, where the Torah was lenient regarding tooth and foot, it was stringent regarding horn, so all the more so in the injured party’s domain.” One cannot say that. Why? “Nevertheless the objection still applies properly.” Even if you rotate the a fortiori inference, the objection will bring it down. He doesn’t explain why, but he brings a proof, and the proof is some a fortiori inference on folio 6 — that doesn’t matter right now; we won’t get into it. Rotating it does not save it from objection. Not that I understand why he brings that a fortiori inference — all a fortiori inferences in the Talmud are like that. But no matter, that’s what he brings, and basically what he says here is that rotation cannot save you from an objection. In every a fortiori inference in the Talmud, when you raise a column objection against it, you don’t see them rotating it to rows and rescuing it. Tosafot — I don’t know what happened there — maybe there the objection really is a direct objection. No, if it’s a direct objection, then what is he saying? After all, he is trying to rotate the a fortiori inference in order to escape the objection. And then he says: let me show you that rotation doesn’t help. There is such-and-such an a fortiori inference there where rotation doesn’t help. I’m saying: that’s true everywhere. Everywhere we already saw that rotations change nothing, because it’s the same argument. It isn’t two different arguments. Intuition says that rotating creates a different argument. That’s not correct. Rotation leaves us with the same argument itself. So that’s just by the way.

Now let’s try to see the solution. But no — there is a certain similarity, because here you ignored the half and the one; I can’t explain it fully, but you ignored the half and the one, and now that you introduced the half, that a fortiori inference there also has half and one, like half-damages versus full. That’s the similarity to that a fortiori inference. Maybe. It doesn’t matter right now. Fine, but we already know that in every a fortiori inference it is true that rotation doesn’t change anything. Tosafot repeats that once again — again and again we see that rotations don’t save us from anything. And again and again we see why: because a rotated a fortiori inference is not a different argument, it is the same argument itself, contrary to what intuition says. And now Tosafot comes with his answer: “And one can say…” So where are we left? We are left with the fact that we do indeed have an objection: what about tooth and foot, whose damage is common? And we still remain, in parentheses, with the difficulty that this doesn’t refute anything at all, so why, right? Tosafot says: “And one can say that this is not an objection, because this stringency does not help to obligate him in the public domain. For this is how we reason in the a fortiori inference: just as tooth and foot, whose stringencies do not help to obligate him in the public domain for full damages, are nevertheless liable there for half-damages, so horn, whose stringencies do obligate him in the public domain, will certainly obligate him for half-damages.” What Tosafot is saying is very simple, even on an intuitive level. On an intuitive level, we found a characteristic in tooth and foot — maybe this connects to your point, removing half-damages under Tosafot’s assumption. But in connection with what Yossi said, this is exactly like with the half-damages. How do I know for certain that an ox, in the sense that its damage is common, is necessarily to the disadvantage of the ox’s owner? Who says that? If I’m walking along, then “its damage is common” is, for me, a warning sign. I’m driving a vehicle and suddenly I have a sign saying “its damage is common” — don’t take the turn at 120. That’s what Yossi means. But wait — regarding the half-damages? It contributes here and not there; in the public domain the duty of care is on the injured party. “Common damage” is my horn — when you come in, watch out. Who says? That’s not right. Wait, no — then it’s just like Tosafot: because it isn’t considered… it’s exactly like with the fine, half-and-half; who says the fine is on this one and not the other? In “common damage,” know that when you go out into the public domain, there is common damage there. Keep your eyes open. That’s what Yossi would say. Exactly that. So you can’t make any… But Tosafot does. Why? Because Tosafot apparently does not accept the Rif’s reasoning. Tosafot claims that the exemption in the public domain does not derive from the fact that this is how people normally go there and the duty of care is on the injured party. It is an exemption that the Torah newly instituted, and we do not enter into the reason of the verse. So there is no point now in playing with all those considerations, because you can’t say that common damage imposes even more of a duty on the injured party, since from the outset the exemption is not based on the duty of care being on the injured party. He is exempt. You can say it is a decree of Scripture. What? You can say it’s a decree of Scripture. Fine, but not for the reason that it is normal for him to walk in the public domain. I don’t know for what reason exactly, but it is an exemption in the sense that they do not impose the duty of care on the damager in the public domain — though for horn they do. What does Tosafot say? If you go out into the public domain, does he really spare you in the public domain? That’s not exact. In the court you don’t single out tooth and foot. That is what the Rif proposes to explain. So what is Tosafot actually saying? Look: suppose tooth and foot has some special stringency. Tosafot says: why does that undermine the a fortiori inference? The a fortiori inference remains as it was. What are you really saying? True, tooth and foot has a special stringency, and despite that special stringency you proved that it is still less severe than horn — we can see that here, right? Why should I care that it has a special stringency? Here too it should still be less severe than horn. And if here it is one, then here too it should be one. Meaning, the fact that you added a special stringency to tooth and foot does not refute the a fortiori inference. It… the a fortiori inference swallows the objection. That is what is called swallowing an objection. Meaning, you raise an objection — so what? That objection may actually strengthen the a fortiori inference all the more. Because that objection says that tooth and foot is so weak relative to horn, that even the stringent property it has — namely that it is common — does not help to obligate it in the public domain. Go learn from that how weak it is. And yet in the injured party’s courtyard it is liable. Meaning horn, which is so much stronger and more severe, will certainly be liable. So this may actually strengthen the a fortiori inference rather than refute it. Meaning the conclusion we drew earlier from the model — that an objection of this kind does not refute the argument — is perfectly fine, because it really doesn’t refute the argument. Right? The only difference is that if I work with this model, I wouldn’t even have asked Tosafot’s question, because it is obviously wrong. Tosafot thought about it intuitively, so intuitively it looked correct, and therefore he needed the answer. But that answer is really what we saw here, because what did we actually see here? What we actually saw here is that the parameter beta does not interfere with the a fortiori inference. I can leave the whole a fortiori inference as it is because beta does not affect it. Tosafot didn’t write this as a property; Tosafot just played the melody of an a fortiori inference and put it in a column or a row. Because it sounds like an objection, no? Without looking? What is an objection? Like a stringency, no? Like an objection? Yes, sorry, like an objection. Yes, he played the melody of how one makes an a fortiori inference, but he didn’t formalize it. But I’m saying that the very thing that tripped me up in understanding Tosafot’s question is actually a strong point in favor of this model. Because here is something that at first glance looks like an objection, and this model immediately shows you that it doesn’t refute anything. And it doesn’t become some matter of… that is, the fact that Tosafot didn’t necessarily fully grasp the way this model works — it still wouldn’t have broken the model. No — does it really refute? If it really does refute, then that is a sign that the model… after all, this whole model is trying to trace how human beings think. So if my assumption is that Tosafot thinks correctly — and usually I do assume that — then the model should follow Tosafot’s considerations and explain them. Or the Talmud, doesn’t matter. It could be that in extreme cases I’ll say I disagree with Tosafot; I’m not frightened by that either. But the simple assumption is that I should indeed follow what the Talmud and the medieval authorities (Rishonim) say, and the model should fit. So here, this question made us a bit uneasy, because we saw there is an objection that the model did not register. But then suddenly we understood that in fact, even in intuitive human thinking, if you think precisely, you see that this is not an objection — and that is perfectly fine. So on the contrary, this gives a certain reinforcement to the model. It means that for an objection based on a parameter, while you are assigning it, you also have to attach it to some existing relevance. In other words, you have to explain the result somehow. Exactly. In filling in zero, that really is what comes out. In filling in zero, the property beta has no effect on the results, because it does not change the difference. If you really want to get a grip on it, you want it to appear in both fillings. Exactly. So in filling in one you see that it changes nothing, so why should I care? So therefore what you were asking — if this really is the model, then beta is not a relevant parameter, because we proved that it has no effect. Right — that’s the answer. That is basically the microscopic expression in this model of Tosafot’s intuitive reasoning. It basically says that beta is not a relevant parameter. True, it exists there — so what? It was proven from the data in the table that it is not relevant. Okay? Now again, one has to pay close attention that all the… So you put beta there under that assumption, and you imposed the constraint that it should be as required. But here you did not show that constraint. Why? Which constraint? I wanted it to be in tooth and foot, and I don’t necessarily see the difference exactly. Why not? In the results — in M, in the results. In M you have one, right? In N there wasn’t supposed to be beta there? No. If alpha is enough and it has alpha, then it includes it. Why does it need beta? But what about the constraint that beta should be there just like alpha? No, it isn’t a constraint that it should have beta. The only constraint is that tooth and foot had to come out such that I would need to insert beta here too, and then it would have worked as an objection. Because I don’t actually know that “common” is an important parameter for obligating in the injured party’s courtyard; my intuition doesn’t tell me that. What it does tell me is that tooth and foot has this property, and the fact that this works — that is what I put into the model. Now I check whether as a result of that I’ll need to put it also in M and R. The answer is no. So if not, then it isn’t an objection. What happened in the model objection? To remind you, in the model objection we basically had another domain here, right? Where here it was one and here zero. Okay, what did that really do? After all, in the end what this does is also add another parameter, right? If you remember, it forced us, even in filling in one, to raise the dimension to two, to two parameters that need to play a role, and that even in filling in one. Therefore it comes out equivalent. It also tied the… the points to one another; it made it equivalent, and therefore it is an objection. But what lies behind that? When you analyze an objection — and we already did this; you have it in the summaries — an objection like that, a column objection, the additional parameter, that beta, also plays a role in the results. Meaning it has an effect on the results, and that is why it is an objection. And that is exactly the difference between a microscopic objection and a legal objection. A legal objection basically adds a parameter for me, but does not swallow it here; rather it places it here. And that means it is a relevant parameter, it has an effect, and then it can at least lead to an objection. A parameter like this, which changes nothing at all — the a fortiori inference remains as it was — will not be an objection. Objection: the reason that this parameter also changes M and R is because you added another column. Of course, of course. Meaning here you can ignore it because it changes nothing? Exactly. That’s why I said: that is exactly the difference. A legal objection is adding a column. The moment you add a column, another parameter beta gets added here, but that parameter also becomes relevant — it appears both in the operations and in the results. It becomes like another circle, say R inside M? Right, right. And that you have to do? Right, that is the formal explanation of what it means. What does it actually mean? It means that the parameter beta is relevant. By contrast, here this beta parameter is not relevant, and therefore it changes nothing. Therefore it is not really an objection. Okay?

Now look at the end of Tosafot. Tosafot basically says this: “And in the first chapter of tractate Zevachim, folio 10, regarding one who slaughters for its own sake in order to throw its blood not for its own sake, that it is invalid by an a fortiori inference from slaughtering with improper time intention, which is valid. And they ask: what is unique about improper time intention? It carries karet.” Even though this stringency does not help improper-time intention in itself to invalidate — in short, they bring an a fortiori inference from sacrifices. I don’t want to get into that now because I see our time is running short. They bring an a fortiori inference from sacrifices, and there too in that a fortiori inference they again raise some objection and do not swallow it, Tosafot says. Why don’t they swallow that objection? So Tosafot says — and this is of course not a difficulty for what we did here — “A stringency that the Torah itself imposed is different, because once the Torah imposed this stringency, it may have imposed another stringency as well.” What is Tosafot basically saying? That in a legal objection — a column objection — there is no swallowing of an objection. Swallowing an objection is a mechanism that exists only for a microscopic objection, of the sort we saw here. But in an objection like the one where we have another column, there is no swallowing of objection. Since the Torah was stringent in this way regarding tooth and foot, perhaps it was also stringent in another way here, and perhaps here too it is more severe and therefore the filling is zero. And in our language, we already saw why such a thing really is an objection, and why it is not true that one can swallow such an objection. So that is all Tosafot says at the end. It is the outcome of what we said earlier: an objection that adds a column is certainly not the kind of objection that can be swallowed. It is an ordinary objection, and we already saw that it refutes. That assumes that karet is a result and not a parameter. Right, right. Karet is a result, because the Torah established it — it’s an indication. Why is one liable to karet for it? Because apparently there is something extremely severe in its characteristics. Or can you say that something is severe because it carries karet? What? How could it be severe because of the karet? Why was karet given? Karet was given because apparently there is something very severe in its characteristics. So precisely for that reason it is legal and not microscopic. Okay? So that’s with regard to swallowing an objection.

And again, notice that all the analysis we did here did not depend at all on the contents involved in the process. It didn’t matter to us that these are damagers — tooth and foot, horn, public domain. It doesn’t matter. Everything follows from the logical structure of the issue. Once there is such a table and I have a constraint on a microscopic parameter, I don’t care what beta is, what alpha is, who these are and who those are. The mechanism gives the result. In that sense, there is logic here — formal logic. It doesn’t enter into the contents. It doesn’t enter into the contents, but every stage is really content, beginning with the relevance of things? No — but we said that anyway, that’s always true. And the fact that he says there is no relevance here, and there wasn’t this nothingness there, so what? Why does that matter? After all… No, it’s the same thing. The relevance of the things I put into the process is certainly a decision I make myself based on the contents. That’s obvious. But once I have the data I am working with, from that point onward it doesn’t depend at all on their character. We’ll later see things that do go beyond this boundary of formality. And what I want to do now is move on to parameters that operate cumulatively. Parameters that operate cumulatively — that is really the end of the passage about the wedding canopy in tractate Kiddushin. If you remember, in the wedding canopy passage there we started by learning an a fortiori inference. Money, which effects betrothal but does not effect marriage, nevertheless succeeds in effecting betrothal; so a wedding canopy, which effects marriage, surely should effect betrothal. And then they started raising objections from intercourse, a yevamah, hatred, and all that whole big mess. In the end it came out fine. Okay? Now the Talmud continues and says, after they finish everything and we’ve already organized and closed it all up — if you remember the big common-denominator argument, a common denominator that has a common denominator from one side and an a fortiori inference from the other side, and they make another common denominator out of the two, and then they go back, refute it, and reinforce it again — and that’s where it ends, and it comes out that the a fortiori inference really works. Now if that’s so, we have a problem. I don’t know if it’s a halakhic problem, because then it follows that a wedding canopy should in fact effect betrothal too, not just marriage. But that is not the Jewish law. According to Jewish law, money, document, and intercourse — not wedding canopy. “A woman is acquired in three ways”: money, document, and intercourse. Okay? Then the question truly arises — it is written in the Mishnah; this is not something you can debate. And suddenly Rav Huna says: a wedding canopy acquires by an a fortiori inference. What do you mean, it acquires by an a fortiori inference? That’s against the Mishnah. But on the other hand, after all the logical maneuvers we did there, he is right. So what do we do? So now Rava comes and says as follows: “And furthermore, does a wedding canopy complete anything except by means of prior betrothal? And can we derive a wedding canopy without prior betrothal from a wedding canopy with prior betrothal?” What is he basically saying? He is basically saying — let’s remind ourselves again of the original a fortiori inference in the wedding canopy passage. He says here that if there is no wedding canopy without prior betrothal, then you can’t make the a fortiori inference here. Right? We have here money, wedding canopy, marriage, marriage and betrothal. This is the final result. Yes, that’s the a fortiori inference. Okay, so again, the a fortiori inference works like this: if money, which does not effect marriage, does effect betrothal, then a wedding canopy, which does effect marriage, should certainly also effect betrothal. Okay, and therefore the result is one. After all the tricks, the result is one. But that can’t be right, because the Mishnah says that a woman is acquired in three ways — or three modes, yes? Money, document, and intercourse. And three ways. What? And three ways, yes. In the Mishnah they say “and,” in the Mishnah they rely on the verse. So that’s money, document, and intercourse. Fine. What about a wedding canopy? Rav Huna proved that a wedding canopy also acquires — here’s the a fortiori inference. So Rava now argues like this: “Does the wedding canopy complete anything except by means of prior betrothal?” After all, the wedding canopy comes at the stage after betrothal or kiddushin already exists. Right? First you betroth the woman, and now she is betrothed, and then the wedding canopy takes place and marriage is effected. Fine? That is stage two. And then Rava says: the whole wedding canopy operates only after I already created betrothal. So it’s no great feat that it succeeds in effecting marriage — it is assisted by the money that was given at the beginning in order to effect betrothal. Therefore you cannot learn from the fact that a wedding canopy effects marriage while money does not, that a wedding canopy is stronger than money. Why? Because the wedding canopy does this with the kind help of the money. Because there was money at the betrothal stage, and it effected betrothal, afterwards the wedding canopy can come and effect marriage, since money has already been given; the wedding canopy is only finishing the job. So you cannot assume that a wedding canopy is stronger than money. Okay? For the time being I’m just explaining this intuitively. Is this a kind of objection? Yes. It is basically the objection to Rav Huna, and this is how he explains the Mishnah. And therefore it’s not true — a wedding canopy does not acquire for betrothal — and so too in Jewish law: the law is that a wedding canopy does not acquire for betrothal. Okay? And here, in effect, this gets erased — or not erased, but Tosafot says that there in a second formulation it says “wedding canopy,” and we need to erase it and write “money plus wedding canopy.” So it’s not money? Right. It’s not wedding canopy at all; the wedding canopy doesn’t do anything here. There is no marriage without prior betrothal — there is already money. If there is no money, money effects betrothal, but a wedding canopy by itself cannot. Doesn’t the wedding canopy come together with the money? That’s what I’m asking you. No — the question is whether a wedding canopy can do the betrothal by itself. No, a wedding canopy by itself is absolutely useless. Only about that. In short, we need to write that there is dependence on something else. Wedding canopy in relation to marriage is not the same thing here. Like Yossi’s question in the previous a fortiori inference — here that really does apply. Because when a wedding canopy operates in relation to marriage, it comes after the money of betrothal. But a wedding canopy operating on betrothal works by itself without money. Therefore you cannot write here “money and wedding canopy.” I should have written here “money plus wedding canopy equals one.” That’s just wedding canopy, and here a question mark, maybe, or something like that. Okay? It has to be split, because in fact a wedding canopy functions differently for marriage and for betrothal. Now maybe you can erase the one under marriage and write there — that’s really what… maybe I would even write half, right? Meaning the first proposal that really occurred to me was to write half here. Because what does the wedding canopy do? It prepares half a marriage. Just thematically; I don’t care if it’s a third, a half, a fifth, or three-quarters. But it’s something between zero and one. Meaning, the wedding canopy manages to do only part of the job. And now let’s see what’s written here. That is basically the objection. The objection basically says: you shouldn’t have one written here; you should have half written here. Now what happens if, for example, it really is written as half? Well, that we already discussed last lecture, right? Then what comes out is actually the dispute between Rabbi Tarfon and the Sages. Right? Rabbi Tarfon says one, and the Sages say half. The Jewish law follows the Sages. So it comes out half. What does that mean? That a wedding canopy prepares half a betrothal. And you also need money in order to complete the betrothal. But when there is money, you also need the wedding canopy. So the money doesn’t do everything. What? Yes. So therefore when half is written here, that really means there is no betrothal, no kiddushin. Right? Unlike damages, where if it says half you pay half the damages — here there is no such thing as half-betrothal. Either she is betrothed or she isn’t. If you only succeeded in doing half, that means you didn’t succeed. So that is the Sages’ approach, and this is Rabbi Tarfon’s approach. Right. But then one rather problematic thing comes out — that according to Rabbi Tarfon, it really should still be one. Meaning, somehow it comes out that this objection doesn’t refute Rabbi Tarfon’s position. Now notice, immediately after this comes Abaye — that was Rava’s opinion. Rava raises this objection. Immediately afterward Abaye comes and says: “Abaye said to him: what you said, ‘Does the wedding canopy complete anything except by means of prior betrothal?’ — what do you mean by that, Rava? Rav Huna also meant just that. You haven’t refuted Rav Huna.” Abaye comes to defend Rav Huna. “If money, which does not complete after money, nevertheless acquires, then a wedding canopy, which does complete after money, is it not all the more so that it acquires?” What kind of objection is this, Abaye says to Rava? What are you basically saying? Let’s check what happens after betrothal. Betrothal has already taken effect. I gave money; there is betrothal. Fine. Let’s ignore the fact that betrothal can also be effected through intercourse and a document. Let’s say we effected it through money. Okay? Now after betrothal, a wedding canopy by itself works, it effects marriage, right? By itself. Money doesn’t manage to do that, even though there is already betrothal. So it is still true that a wedding canopy is stronger than money. Under certain conditions. What do you mean under certain conditions? Under the condition that money was already there. Why? Who says? No. Who says there was already money? In both places there was money. Before the wedding canopy, that’s it. Okay, and that is the point. So Abaye says: let’s compare a situation where she is already betrothed. When she is betrothed and I now want to marry her, I won’t be able to do it with money, but I will be able to do it with a wedding canopy. So that does prove that a wedding canopy is stronger than money. So it teaches me that it is stronger than money in that situation, but that’s not the situation I’m testing. But it’s always like that. In every a fortiori inference, you can tell me that a wedding canopy is stronger than money regarding marriage, but who says regarding betrothal? That’s always true. You’re saying that once it is stronger in one situation, my assumption is that it is stronger in all situations unless you prove otherwise. But the basic assumption is that if you demonstrated strength in one place, that strength applies everywhere. So if so, apparently Abaye is right. Right? Because it follows that even after Rava’s point about betrothal, the proof still remains intact. It’s a kind of swallowing, a spiral — everything appears here. Here you have half, combinations, swallowing — you can put all the mechanisms in here. I don’t even know which mechanism the Amoraim actually intended, because it is opaque. But you can put everything in here. The half is an assumption. What? The half is an assumption; maybe you need to write x there as an unknown, because it is known that on this there is a dispute of Rabbi Tarfon. The half is an assumption that doesn’t necessarily hold. I don’t know. Something between zero and one — that’s what I meant. You’re saying that if it were zero, then even Rabbi Tarfon would agree. Fine. So it’s not half and not one and not zero — maybe point eight, because they say it is in an entirely different dimension. No, that’s another matter — I’m getting to that now. Then it really wouldn’t be half but beta, some other index. Then we once again arrive at the previous chapter. So all the mechanisms we’ve checked up till now can be brought in here: swallowing, rotation, or an additional index, an external microscopic constraint. Why does he make the a fortiori inference from money? What? Why does he make the a fortiori inference from money? After all, money is not explicit. Okay. Money is learned from “when a man takes,” some specific source. If he had started from intercourse — intercourse itself is an objection. And that really is an objection, in the sense of… Anyway, what really comes out here? Therefore it seems to me that a more realistic model is the following: we are really adding another factor here, not half as we said before, but another parameter. It’s not half-step above, but some other parameter. And basically the principle is as follows. We are basically saying: this is our data table. We do remember that marriage happens after betrothal. That we remember. Okay? Now we say as follows: let’s check filling in one, okay? In filling in one, I have this: betrothal is alpha, marriage is two alpha. Wait — money is alpha, right? It contains betrothal but not marriage. And wedding canopy — what is it? Two alpha, alpha. Right? Because the wedding canopy comes after money. When the wedding canopy comes after money, it joins the money and there is two alpha, therefore it contains marriage. And still you have not succeeded in proving that a wedding canopy is stronger than money. By contrast, when the wedding canopy works by itself without the money, then it wants to effect betrothal, right? In filling in one it will succeed, because this is alpha and this is alpha. Okay. So what happens in filling in zero? That was filling in one. Wait, wait — what do we have here? I’m now checking filling in one. You gave alpha to betrothal — why doesn’t that effect betrothal? What? Because filling in one does. Filling in one effects betrothal. This is filling in one, I’m checking it now. Okay? Now what happens in filling in zero? In filling in zero I have betrothal as two alpha, marriage as three alpha, money as two alpha, and wedding canopy as alpha. Notice, very briefly: wedding canopy plus money gives me three alpha, and therefore it succeeds in effecting marriage. Right? I now want to explain filling in zero, and therefore I raised the value in that box. Okay? A wedding canopy by itself does not succeed in effecting anything. By itself it does not effect betrothal. It effects marriage only with the kind help of money. So therefore by itself it doesn’t succeed, right? That is the claim. By contrast, money succeeds in effecting betrothal even by itself. Marriage it does not effect — money by itself. Okay? So that is basically the explanation for filling in zero. But there is a problem here — at least from what I’ve understood until now. Whenever you had two separate things not dependent on each other, as in filling in zero here, there was always another parameter. There is another parameter. Okay. Here with money, that’s exactly the problem. If we go here, we cannot do an analysis that starts from the table and goes out to the diagram. Because the table is not correct. There is dependence here between the rows and the columns. The value of the wedding canopy relative to marriage has money joining it. Relative to betrothal, money does not join it. So the table is only a representation, but unlike the ordinary tables. I need to start from here and from here go back and draw the diagram. Not derive the diagram from the table. Understand? I actually need to say it that way. This fits all the data when I also take into account that the wedding canopy in relation to marriage comes after money. It’s a kind of activity with unknowns. The Rabbi basically wrote an equation: wedding canopy plus money is greater than marriage. Exactly. In relation to marriage and not in relation to betrothal. Exactly. And then the result is this, and now I can go back and draw diagrams. And what comes out in the diagrams, by the way? You can already see immediately: the same thing comes out. Right? Everything is with one parameter. Right? Betrothal and marriage, one inside the other; only that box is different, but we said that box isn’t important, right? So essentially it all comes out the same. In both cases the diagrams come out like this, and the dimension is one. Right? Therefore that is probably Rava’s ruling. Why do the diagrams come out like this? What? Why do the diagrams come out like this in zero too? Here too you have betrothal and marriage. Because… look, I am not deriving from the table. It is not correct to derive from the table, because the wedding canopy comes after money. I use this table to construct the model, and now I work backward from it to build the diagram that is the model. The table is only the expressive shorthand. Exactly. The table does not correctly express the data, because one has to remember that the wedding canopy comes after money. So regarding the graph — I want to use the topological parameters too, so I can’t use only the dimension. What about connectedness, number of points? And why in zero is it also such a graph? Same thing. Look: betrothal goes inside marriage. That’s the yellow part: betrothal and marriage. Betrothal goes inside marriage. And here too betrothal goes inside marriage. It is contained in it. Here in the basis, betrothal is contained in marriage? To a certain extent, yes. There betrothal is zero. That is exactly the point — it is forbidden to go from the table, and therefore I am not going from the table, because in the table the wedding canopy by itself does not contain betrothal, but the wedding canopy by itself also does not contain marriage. So really one should have written zero here. It also effects betrothal? Marriage. Only with the help of money does it effect marriage, therefore no — it is a mistake to rely on the table. One must go with this and then go back to the diagram. And then it really comes out equivalent, and apparently that is how Rava understood it, because Rava refuted Rav Huna’s a fortiori inference. How does Abaye understand it? Abaye understands what one of you said earlier; I’ll do it briefly. He basically understands a different filling in zero. He claims that there is another parameter here. Betrothal — this is Abaye — betrothal is alpha, this is alpha and also beta. This is alpha, alpha and also beta. Money is alpha, and wedding canopy is beta. Right? Because what is he saying? He says — and here the content enters, which is why I said this is not really pure logic, because here you already need to rely on content: what is the relation between money and wedding canopy? It’s no longer just the mathematics of the data set. And Abaye claims that once the wedding canopy functions after money, then obviously this is not the same kind of force as money. If money has alpha, then wedding canopy must have beta. It can’t be the same thing as money and then operate afterwards, because otherwise, what is he claiming? Otherwise money after money would also do the same thing. But that’s exactly what he says to him in the Talmud, right? “Come do the a fortiori inference after the money.” After money, can I prove that wedding canopy is stronger than money? Because he assumes that money after money also works. It doesn’t matter whether it’s one payment of money or two, right? So therefore, in his view, it is obvious that this is not just alpha; there is also beta here. And now you’ll see that this really does scramble the a fortiori inference — this filling in zero is weaker because it is two-dimensional. Therefore Abaye keeps the a fortiori inference, while Rava refutes it, and the Jewish law follows Rava. More power to you. We’ve finally finished the passage in tractate Kiddushin.