2019-04-22 – Between Midrash and Logic – Lesson 18

Table of Contents

• Absorbing refutations and Tosafot in Bava Kamma
• A microscopic-constraint refutation versus a column refutation
• Zevachim 10a and the refutation “written in the Torah”
• Rava’s attack at the end of Kiddushin: chuppah as “half”
• Rabbi Tarfon and the Sages, dayyo, and deciding between half and one
• A value continuum and a step function
• Deriving from what is itself derived, and an a fortiori argument from an a fortiori argument
• A double refutation as counter-proof

Summary

General overview

The speaker sums up and concludes the topic of the a fortiori argument and the binyan av through examples of unusual applications, and sets up a tabular-diagrammatic model that distinguishes between different refutations according to the structure they force on the completion. He presents the concept of absorbing refutations as defined by the medieval authorities (Rishonim), analyzes Tosafot in Bava Kamma on “their damage is common” as a microscopic-constraint refutation that gets absorbed into the a fortiori argument, and contrasts that with Tosafot in Zevachim 10a, which distinguishes between a refutation based on reasoning and a refutation “written in the Torah,” which is law and therefore is not absorbed. He then explains Rava’s attack at the end of the topic / passage in Kiddushin as the claim that the datum “chuppah completes” is not “one” but “half,” because chuppah works only on top of kiddushin, and he connects this to the law of dayyo in the dispute between Rabbi Tarfon and the Sages in Bava Kamma. Finally, he presents an extension to a continuum of values and to cases of “deriving from what is itself derived,” such as an a fortiori argument from an a fortiori argument, by means of tables with more than one empty square, and he ends with the claim that a double refutation from both the row and the column is not a refutation at all but counter-proof for a zero completion.

Absorbing refutations and Tosafot in Bava Kamma

The speaker defines absorbing refutations as a phenomenon that does not appear in the Talmud itself but is defined by the medieval authorities (Rishonim), and brings as an example Tosafot in Bava Kamma, chapter two, on Rabbi Tarfon and the Sages in the a fortiori argument of keren, shen, and regel between the public domain and the damaged party’s courtyard. He sets up a table of the data, explains zero completion and one completion, and the advantage of one completion from the standpoint of two-dimensionality and connectedness, and then presents Tosafot’s difficulty: “What is true of shen and regel, whose damage is common,” according to the one who says that half-damages are a fine, which is also the Jewish law. He explains that a fine is “when there is no monetary obligation by strict law” in order that “he should guard his ox,” and formulates the refutation as the claim that the stringency of “common damage” pertains to shen and regel, but not to keren.

A microscopic-constraint refutation versus a column refutation

The speaker states that presenting “their damage is common” as a halakhic / of Jewish law refutation is a mistake, because “that’s a parameter, it’s not a halakhic property,” and therefore this is not a column refutation but a constraint refutation on the table’s possible solutions by means of an additional variable that appears in shen and regel and does not appear in keren. He describes how the constraint forces the addition of another dimension in the one-completion, so that the dimensional advantage disappears, but the connectedness still leaves an advantage for one-completion. He explains that in the zero-completion the constraint does not force a new parameter, because there is already a parameter identified as “common damage,” so the constraint mainly “helps me identify parameter alpha.” He presents Tosafot’s conclusion—“And one can say that this is not a refutation… and this is the whole idea of an a fortiori argument”—as an absorption of the refutation into the very formulation of the a fortiori argument by means of that same microscopic parameter.

Zevachim 10a and the refutation “written in the Torah”

The speaker brings Tosafot in Zevachim 10a on an a fortiori argument between outside its proper time and not for its own sake in slaughtering and in slaughtering with intent to sprinkle, where the Talmud rejects the a fortiori argument because “What is true of outside its proper time, where karet applies.” He clarifies that Tosafot answers that “a refutation written in the Torah is not absorbed into the a fortiori argument,” and explains this as the claim that when the refutation is a Torah law, it is treated as an additional column and not as a microscopic constraint. He explains why Tosafot chose דווקא this example and not ordinary column refutations, because karet looks like a point-specific property of a certain square, and therefore the dilemma arises whether to view it as a constraint or as a column, and Tosafot rules that it is law and therefore a column refutation that cannot be absorbed.

Rava’s attack at the end of Kiddushin: chuppah as “half”

The speaker returns to the conclusion of the topic / passage in Kiddushin, which began with an a fortiori argument from money to chuppah, and presents Rava’s attack on Rav Huna: “Does chuppah complete at all except by means of kiddushin? And do we derive chuppah without kiddushin from chuppah with kiddushin?” He explains the attack as a different kind of refutation, one that does not add a row or a column and is not a constraint, but rather corrects the precision of the data: “You wrote a one here, but that’s not right—it’s a half,” because chuppah that effects marriage works only after money and not as “chuppah alone.” He states that the refutation undermines two intuitive formulations of the a fortiori argument, because even the claim that chuppah contains the more difficult case and therefore certainly the easier one falls apart once it becomes clear that chuppah’s efficacy depends on money.

Rabbi Tarfon and the Sages, dayyo, and deciding between half and one

The speaker compares the “half” refutation in the Kiddushin topic / passage to the original a fortiori argument in Bava Kamma, where the empty square can be zero, half, or one, and explains that zero is thrown out because the a fortiori argument requires some liability, leaving a dispute over half versus one. He presents the Sages’ ruling through the law of dayyo as choosing the minimum that has been proven: when both half and one “do the job,” you cannot prove more than half, and therefore “it is enough for that which comes from the law to be like that from which it is derived.” He explains that an intuitive presentation by columns might push toward one, while by rows it might push toward half, but in the full tabular model there is no “rotation” that decides the issue, so what remains is a minimal choice that matches dayyo; he states that since “the Jewish law follows the Sages,” Rava’s conclusion follows, that the a fortiori argument of chuppah cannot generate betrothal from chuppah that does not rest on kiddushin.

A value continuum and a step function

The speaker presents a generalization in which, instead of checking only zero and one, you raise the value continuously—“zero, epsilon, two epsilon, and so on”—and explains that a structural change takes place at a point like half, where the relations change. He describes this as a step function in which “from half onward nothing changes,” and formulates dayyo as stopping at the minimum point where the model starts to work, so that if, for example, “a third would do the job, then the result would be a third.”

Deriving from what is itself derived, and an a fortiori argument from an a fortiori argument

The speaker describes the topic / passage in Zevachim on “deriving from what is itself derived” in combinations such as an a fortiori argument from an a fortiori argument, a binyan av from an a fortiori argument, and a binyan av from a binyan av, and explains that he did not find in the Talmud an actual content-based example of an a fortiori argument from an a fortiori argument apart from theoretical discussion, except for one single place where there is a use that proves its validity in a loop. He argues that an a fortiori argument from an a fortiori argument requires a bigger table, such as a three-by-three one, with more than one empty square, and then the decision is made by comparing all the combinations of possible completions—such as four diagrams for two empty squares—according to measures of connectedness, changes of direction, and topology. He presents one example in which (1,1) “wins big,” and therefore an a fortiori argument from an a fortiori argument is accepted, and opposite that another example in which (1,1) is “the worst,” because a one-completion creates a refutation to the first a fortiori argument or to the second, and he concludes that you cannot always build an a fortiori argument from an a fortiori argument without solving the whole table all at once.

A double refutation as counter-proof

The speaker returns to an earlier claim about a regular column refutation, which in intuitive thinking does not necessarily break the “larger” relation, and then adds that if a refutation from the other side joins it, the result is different: the best completion becomes “zero-zero.” He states that a double refutation from the column and from the row is not “a refutation saying you didn’t prove it to me” but rather “counter-proof” that the completion must be zero, and explains that this is generally why you do not see in the Talmud refutations from both sides, because in such a case the move turns from a rejection into a positive determination against the higher completion.

Okay, what I want to do today is finish the topic of a fortiori reasoning and binyan av by means of a few examples of somewhat unusual applications. I moved this discussion up simply because I wanted to do it using the diagram that describes the conclusion of the passage. So I brought forward the general discussion about the meaning of these things, and now I’m going to do it one more time, and with that we’ll finish. The continuation will only be after Passover; I’ll talk about that separately in our framework. Fine. The first topic I want to deal with is the absorption of objections. There are several examples that need to be examined in light of this model. We’ll see how much we manage to get through. I’ll start with absorption of objections. What happens in absorption of objections? Absorption of objections is actually something that doesn’t appear in the Talmudic text itself. The medieval authorities (Rishonim) define it. There’s Tosafot in several places; there’s one in Bava Kamma in the second chapter there on Rabbi Tarfon and the Sages with the a fortiori argument. There are other places too. Let’s take the a fortiori argument in Bava Kamma. What happens in Bava Kamma? Let’s remind ourselves of the situation. Goring, tooth, and foot; public domain and the damaged party’s courtyard. Tooth and foot: exempt, liable. Goring: liable, here a question mark. All right? Now Rabbi Tarfon and the Sages are basically arguing about this question, which maybe I’ll touch on a bit later. Actually it’s half, right? A harmless ox in the public domain is liable for half, not full damages. Then the situation gets a bit more complicated, and on that… it is liable, but we don’t know whether it’s liable or not liable, right? What? Half is understood here? Obviously. The dispute between Rabbi Tarfon and the Sages arises because here there is a value that is not like here. The diagonal doesn’t have the same values. And then the argument is whether to put half here or to put one here. That’s really the dispute about dayyo, right? If you apply dayyo then there’ll be half here. How can you, as a value—I’m just asking in this specific example—but how can you put a value of half? What’s the problem? I’m putting in the data that we have. It’s true that this is a certain extension of the model, but at the principled level, even before the question of how I solve it within the model, first of all these are the data and this is the result. Now the question is whether my model knows how to handle that, so let’s see that in a moment. I’ll at least comment on it, all right? But first of all these are the data and this is the result. About this Rabbi Tarfon and the Sages argue. All right? Half or full? What? Could there also have been one there, no? That it would be liable for full damages. Why? What, in the case of a forewarned ox? Yes, we’re talking about a harmless ox. Right now we’re talking about a harmless ox. All right? But I’m ignoring that dispute at the moment; let’s talk about the regular a fortiori argument. The regular a fortiori argument says, as it were, that there’s a one here; you can look at it as though the liability is one, or look at it as though we’re really asking whether it is liable or exempt. We’re not asking how much it is liable for, so that too gives this table. All right? So when we look at the a fortiori argument in that way, then the a fortiori argument is built like this: if tooth and foot, which are exempt in the public domain, are liable in the damaged party’s courtyard, then goring, which is liable in the public domain, all the more so should be liable in the damaged party’s courtyard. Or of course in this direction. Then Tosafot asks—sorry, Tosafot here asks—that you can raise an objection to this. For this table we know what to do; maybe I’ll just write it here. So this is R and this is the damaged party’s domain and the public domain, and this is tooth and foot and goring. All right? So I say that in filling zero I basically have N and R independent, right? There’s no relation between them if the filling is zero. Then here I have alpha beta, okay, how do I do this so it doesn’t get mixed up with the… there, I wrote it backwards, one second, I’ll fix it later—alpha and beta. All right? That’s in filling zero, and in filling one… so there is from here, N is alpha. Yes. Backward. I’ll track it later. Alpha and beta. All right? That’s in filling zero. And in filling one then there is from here, N is alpha. Yes. Backward. Okay, that’s in filling one. All right? So alpha, that’s two alpha. All right? Therefore filling one is preferable because we know it’s also two-dimensional and also non-fit, meaning that’s a strong preference. All right? In favor of filling one. Now Tosafot comes and asks the following question. Why not object—maybe I need to add something here in filling zero. I’m just writing that S—what is S? It contains N and does not contain R, right? So S has alpha, and K in filling zero, K has only R, only beta. Right? That’s in filling zero. And in filling one we have S, it has alpha, and K has two alpha. No, here there’s no beta. Two alpha. All right? Those are the solutions for filling zero and filling one. Fine. Now Tosafot asks like this: “And if you should say: what about tooth and foot, for their damage is common? Will you say the same about goring, whose damage is not so common, since it stands under a presumption of being guarded, according to the one who says that half-damages are a fine?” Yes, because there is a dispute whether half-damages are compensation or a fine—what does that depend on there on page 15 in Bava Kamma? It depends on the question whether damage by goring is common, yes, whether its damage is frequent, or whether it is not frequent, meaning it is abnormal. All right? If its damage is frequent then basically it has ordinary liability, it’s compensation; and if it’s abnormal then it’s a fine. All right? Because then the person wasn’t supposed to know this would happen and isn’t so negligent; he wasn’t obligated to guard it. So that’s the dispute. So Tosafot says: let’s go according to the one who says that half-damages are a fine—which is also the Jewish law—that half-damages are a fine. He wasn’t obligated to guard it and therefore it’s a fine? Yes. Usually a fine is given when there is guilt. No, the opposite. When there is guilt, then you have monetary liability. You simply have to pay under the ordinary law of property damages. When there is no liability under the law, sometimes the Torah nevertheless imposes a fine, so that he’ll guard his ox. So that in any event he will guard it even though it’s not all that expected. All right? A fine is always when there is no monetary liability by law. So Tosafot says, then basically we can make an objection here that tooth and foot involve common damage. All right? Basically this is an objection like this, right? Common damage. Tooth and foot, its damage is common; and goring, at least according to the one who says half-damages are a fine, its damage is not common. So there’s an objection here. What do you say? What do you answer to something like that? One more line. No, no, those are the data. What do you answer to something like that? Is that an objection or not an objection? What do you say? Maybe both are more severe? Right, that it’s not relevant? Seemingly this is a full-fledged objection, right? We saw this table; this is exactly a case of an objection to an a fortiori argument, right? And it will refute the argument. We already saw that it refutes the argument. Tosafot says, “And one can say that it is not an objection, because this stringency is not effective to make it liable in the public domain, and this is the essence of the a fortiori argument. And what of tooth and foot, whose stringencies do not help make it liable in the public domain for full damages,” and so on. What is he saying? This stringency, that its damage is common, still is not enough to make tooth and foot liable in the public domain, right? So in effect this too is absorbed into the a fortiori argument, says Tosafot. What does that mean? Let’s formulate the a fortiori argument with the common-damage point. If tooth and foot, whose damage is common—even though their damage being common is not enough to make them liable in the public domain—nevertheless in the damaged party’s courtyard they are liable, then goring, which even without its damage being common is liable in the public domain—look how severe it is—then certainly in the damaged party’s courtyard it should be liable. The whole objection you can put into it as this kind of parameter. Oh—Tosafot immediately afterward asks: and in the first chapter of Zevachim regarding slaughtering for its sake—we’ll analyze that in a moment, I’m just mentioning it for now—there it brings an a fortiori argument with an objection, and then it says that a halakhic objection is not absorbed into an a fortiori argument. An objection written in the Torah—that’s what it’s called. An objection written in the Torah is not absorbed into an a fortiori argument. Now that’s a puzzle. Why is an objection written in the Torah not absorbed into an a fortiori argument, while an objection that comes from reasoning is? That itself is a puzzle in general, and a very simple thing. Let me explain to you now the whole move of Tosafot; look. Let’s start with this. First of all, this table is not correct. Because “common damage” is a parameter; it’s not a halakhic property. Right? Up to now, in all these tables of an objection to the common denominator—“for they involve benefit,” in intercourse and money, remember?—so the objection “for they involve benefit” I started as an additional column, and afterward I said, wait, wait, let’s stop. That’s not some law that characterizes money and intercourse, that they involve benefit. It’s a property of them. So basically it belongs to the parameters. There is some gamma that intercourse and money have and canopy and document don’t have. In a case written in the Torah, you can’t force it in as a parameter. Just one second. Before “written in the Torah,” first of all from the beginning. I’m going back now to the start of Tosafot. All right? So what exactly am I claiming here? After all, that it is liable in the public domain, liable in the damaged party’s courtyard—those are laws written in the Torah. Those are halakhic outcomes. This is not a halakhic outcome. It’s a microscopic property, it’s a parameter. One of those alphas, betas, gammas—the essence is that the damage is common. Right? Therefore it’s a mistake. It is not correct to present the objection like this. How should the objection be written? What do you say? We learned how we handle an objection that is microscopic. Simply bring it in in microscopic terms. How? How do you do that? After all, the table stays the same. We need to make different relations between alpha and beta. The table stays the same. We have different relations with alpha and beta. What do you mean, different relations? You introduce another quantity or variable. Meaning: there are constraints on the solution. That is to say, the table remains the same, the diagrams of course remain the same because they are a result of the table, but the solution—the alphas and betas that we mark on the diagram—is subject to constraints. And we saw that in the previous examples too. Now let’s see what kind of constraint we have here. That’s all this objection does. This is a constraint-objection. What does it say? What belongs to “what of tooth and foot, whose damage is common, whereas goring is not”? What does that mean? That we have another variable in the… exactly. It means that there is something else, some beta of some sort—let’s call it “the damage is common,” all right?—which tooth and foot have and goring does not. It isn’t gamma—we’ll soon see that it isn’t—but gamma, which tooth and foot have and goring does not. Right? And everything else remains the same. And it still changes nothing. Meaning: here, besides the fact that there is some stringency in goring beyond tooth and foot, as seen here in this column, there is also some stringency in tooth and foot that doesn’t exist in goring, right? So in fact what we’re really doing is saying this. In filling zero—let’s actually start with filling one because it’s easier for me to explain there. In filling one, the diagram is this, because the table stays the same. Right? What’s the solution? Add gamma to each… for now alpha and two alpha. I have no problem. Let’s see what that means here. What does that mean here? That S is alpha and K is two alpha. But here I have some problem. Because I have a constraint saying that S must also have beta. Beta, and here there is no beta. This is zero, yes? There’s no beta here, right? And there has to be some property in tooth and foot that doesn’t exist in goring, namely that the damage is common. Let’s call it beta for purposes of discussion. Then what comes out? That the solution with filling one is a two-dimensional solution. Here too suddenly there is a beta parameter. So the preference that before this was one dimension and this was two dimensions has now fallen away. Right? That’s how this objection works. But still there is the solution for the… why? Because of the connectivity. Exactly, I’ll get to that in a second. Good comment. But first of all, that’s how this objection is apparently built, right? So this objection basically says: look, true, I don’t see it in the table, but the constraints on the table—and this is what we saw also in previous microscopic objections—add parameters for me; I have no choice, I can’t satisfy the constraints without another parameter. Does that change anything about R and N? No, not necessarily. I have no reason to add beta here too. R and N can remain alpha and two alpha, because beta doesn’t affect them. What’s the proof? Look: in filling one, tooth and foot fail to produce the public domain, right? If they fail to produce the public domain, then that’s a sign that in the public domain it’s enough to have two alpha; I don’t also need beta in order to get the result, right? What happens in the damaged party’s courtyard? There they do manage to produce it. So in the damaged party’s courtyard beta must also change. Here too there must also be beta, right? No, sorry, no. You don’t need beta. Yes, right, that doesn’t have to be. Because if it has alpha and beta, beta is superfluous; it doesn’t matter, but it will succeed in containing N even if there is no alpha. So you can leave alpha and two alpha. Let’s see below; below too it’s the same, right? Beta doesn’t change anything. Here there is the required alpha, and here too there is the two alpha, which of course includes the required alpha. So in effect what happens is that the solution changes only with respect to the rows. You could of course also add some betas here in a way that won’t interfere, but you don’t need to. And in goring there will be beta, not alpha? What? Why? That goring also has alpha? No, in goring there is only alpha. Alpha, right, fine. So what does that mean? That a parameter has been added only in the operations; in the outcomes I have no need, the constraint doesn’t force me to add the parameter to the outcomes, right? But I still know that the model is two-dimensional. Never mind that the second dimension doesn’t affect R and N; it is certainly there. So once I already have two dimensions, now notice what happens here: in filling zero, the constraint is still that in tooth and foot there is something more severe than in goring, right? So look: in terms of alpha, tooth and foot are more severe than goring, right? In terms of beta, goring is more severe than tooth and foot. Agreed? What does that mean? That “common damage” is simply an identification of beta. There is no need to add a parameter here; I now know who beta is. Beta is the fact that tooth and foot… sorry, alpha. Alpha is the property of common damage. And indeed in terms of that property, tooth and foot are more severe than goring. Now the other question is how you use that afterward, the use Tosafot makes there in order to reject the objection. No, no, wait, let’s not jump ahead, I want to go step by step. Let’s go step by step; I’ll get to Tosafot. All right? So look, then, in filling zero, the constraint saying that in tooth and foot there is something more severe doesn’t force me to add another parameter, since I already have the parameter alpha representing that. Here I didn’t have a parameter with respect to which tooth and foot are more severe, right? Because the only parameter was one with respect to which goring was more severe, stronger, right? So that means that here I was forced to add another beta. But here, since there are two parameters, what the constraint did was only help me identify the parameter alpha. The parameter alpha is basically common damage. That’s all, right? Good. So what exactly does this mean? It basically means that in filling zero—what does filling zero mean? Filling zero means this is the proposal saying that the parameter of frequency of damage also affects the outcomes, and therefore here I don’t manage to contain this outcome. Here they would apparently need the frequency of damage too, and I don’t manage to contain this outcome. In the case of filling one, it means that this additional parameter does not affect the outcomes, but it still has to exist in the operations, right? So that is exactly the intuitive meaning of this constraint. So we get this exactly from the model. Just one point: what is this objection really supposed to do? It’s supposed to make filling zero equivalent to filling one, right? And seemingly it really does that, because now both are two-dimensional. But as you remarked earlier, that’s not correct. Because the preference in an a fortiori argument is not only about dimension—here dimension two and here dimension one—but also about fit. Here this is fit one and here fit two, right? Now if by means of this constraint I have equalized the dimension of filling zero to that of filling one, this still remains. Okay? Therefore in the end I’m still in a situation where filling one is preferable. What do we do? How do we understand this objection in our model? It requires serious analysis. After all, we have to remember that what Tosafot raises here is a difficulty, so it’s not the answer. Tosafot asks: why don’t we refute it by saying “what of tooth and foot, whose damage is common”? The answer is: because that is not an objection. That’s the answer. Now what does Tosafot present as the answer? What I said here—that this too is absorbed into the a fortiori argument. Meaning: even if we take into account that the change of “common damage” is an advantage of tooth and foot, it still won’t break the a fortiori argument. That is a general rule: whenever you find the phrase “this too is absorbed into the a fortiori argument,” it basically means that the microscopic parameter exists. Exactly. And that’s always. All the places where you find it are exactly the same structure, there’s no difference. It’s simply that the structure is the same structure, the same table, the same argument, everything the same. Just replace tooth and foot with other names. Why can’t you do that for every objection? Right, exactly. Now Tosafot comes and asks: okay, but there is a Talmudic text in Zevachim, and in the Talmudic text in Zevachim we have an objection that is not absorbed. What is the Talmudic text in Zevachim? So he writes as follows. He writes: “As in the first chapter of Zevachim, regarding one who slaughters for its sake in order to sprinkle its blood not for its sake, that it is invalid from an a fortiori argument from one who slaughters outside its proper time, which is valid,” and so on. In short, let me show you this on the board. The Talmudic text in Zevachim on page 10 says this. In sacrifices we have several similar disqualifications, and the subject is rather confusing. We have here outside its proper time and not for its sake, and here we have slaughtering and slaughtering with intent to sprinkle. That is to say, when I slaughter the offering outside its proper time, the law is that the offering is valid. But if I slaughter it while at the time of slaughter I am thinking that when I sprinkle it, I will sprinkle it outside its proper time, it is invalid. Or rather, it is invalid—it is even piggul. Like piggul of sprinkling with intent of outside its proper time; at least there’s a dispute in the Talmudic text, but also slaughtering with intent for a sprinkling outside its proper time is piggul. What happens with intent “not for its sake”? Slaughtering not for its sake is also invalid. For our purposes piggul and invalid are the same thing; right now I’m not distinguishing. Slaughtering not for its sake is also invalid. The Talmudic text says, if so, then slaughtering in order to sprinkle, where the sprinkling will be not for its sake, must also be invalid, yes? And that too is invalid. Let’s say invalid is one; there is invalidity B and invalidity A. Okay? Now the Talmudic text says: “What of outside its proper time, for it carries karet?” “Not for its sake” has no karet. I said it was piggul, outside its proper time; therefore it has karet. Okay? So Tosafot says: so here too we have an objection, and indeed the Talmudic text rejects the a fortiori argument. It does not say that this objection is absorbed into the a fortiori argument. What is different from here? asks Tosafot. So Tosafot says: an objection that is written in the Torah is not absorbed into an a fortiori argument. What does he mean? That basically this is not a parameter; if the Torah said it, then it came to teach something. So what—a new parameter? No. No, the Torah brought it as a law, unlike… It’s a law and not a property. This is a column-objection. It isn’t a constraint-objection on the microscopic parameters. I would say this even without Tosafot’s answer; it’s obvious. After all, if the objection is a column-objection, how can you absorb it into the a fortiori argument? We already analyzed a column-objection and got that it does indeed refute the a fortiori argument. The whole point here was that the objection is not a column-objection but rather a constraint on the solutions of the original table—a microscopic objection. A microscopic objection we saw really does not refute. But if I have an objection that is written in the Torah, what does “written in the Torah” mean? It means a law. “Its damage is common”—it is not written in the Torah that the damage of tooth and foot is common. That is my reasoning; it is a microscopic parameter. So maybe treat it as a parameter, right? It’s a parameter. If it is written in the Torah, it is a law, so just another column. Just as tooth and foot in the public domain is written in the Torah, and in the damaged party’s courtyard is written in the Torah, so too another law, another column. If there is a column-objection, then certainly you can’t absorb it into the a fortiori argument. That’s simple. The interesting question here is why Tosafot went specifically to Zevachim page 10—why didn’t he bring all the column-objections in the world? What was special for him about Zevachim page 10? Because karet is confusing, because karet looks like a property and not like yet another law and another force of the Torah. Exactly—even more than that, I would say. After all, the karet is found only here. It’s not a property of the whole row. After all, karet is a property of this cell—only here does karet operate. All right? So this is really not truly a column-objection; it’s an objection that relates to all this. Say, if we found something where tooth and foot are fully liable and goring is exempt, then that would be a regular column-objection. But here it’s something in between. Karet is basically a property of this cell: that this invalidity is not an ordinary invalidity, but an invalidity of karet. That’s the point. Isn’t slaughtering in order to sprinkle a property of slaughtering? What? Isn’t slaughtering in order to sprinkle a property of slaughtering? No, because you see that in ordinary slaughtering it is entirely valid—not just that there is no karet. When it is invalid, it is invalid at the level of karet. I would say—if I wanted to make a table—I might make the table like this, where here this is a strong invalidity. Basically the claim is that although karet is supposed to arise specifically in… not arise, but belong only to slaughtering in order to sprinkle outside its proper time, nevertheless we’ll accept that because it is… because it is a law. Exactly. Tosafot says this: the question is how to treat an objection of this sort. Is an objection of this sort really some other kind of constraint—for example, to turn the table into something like this, and when you turn the table into something like this the filling is indeed one, perhaps later on—or one or even two. What does two symbolize? Two, meaning a strong invalidity, an invalidity of karet, just as a symbol to remind me, it doesn’t matter right now. Or do I really see it as a column-objection—that’s the dilemma. So a regular column-objection Tosafot would never even have thought to ask about; that’s why he chose this example. A regular column-objection clearly cannot be absorbed into an a fortiori argument. Here Tosafot says: but there is some a fortiori argument where there was room to absorb it—look, how would I absorb it? I would say this: what of outside its proper time, that even though it is so severe that when it is invalid it even carries karet, and yet this stringency does not help it become invalid in ordinary slaughtering, nevertheless it becomes invalid in slaughtering with intent to sprinkle—then “not for its sake,” which is invalid even in ordinary slaughtering, certainly in “with intent to sprinkle.” That is an ordinary absorption into an a fortiori argument. The whole formulation of absorption could have been said here. In a regular column-objection you couldn’t make an objection like that; you couldn’t absorb it in this way. Therefore Tosafot raises the difficulty from the Talmudic text in Zevachim on page 10. But what Tosafot answers is that since this karet is a law of the Torah, it is not a characteristic, not a microscopic constraint, then you really do have to treat it as an additional column, and therefore it is an objection. That’s what Tosafot says. Let’s take another example. Not an example of absorption—now I’m moving to another case. The conclusion of the passage in Kiddushin. If you remember, the passage in Kiddushin began with an a fortiori argument from money to canopy. Times are H, and this is marriage and betrothal. Zero, one, one, and question mark. That was the beginning of the passage. All right? Now at the beginning of the passage we made an a fortiori argument and then objections and this and a common denominator and objections to the common denominator and so on. We finished everything, everything is fine, the a fortiori argument remains. Now at the end of the Talmudic text—which we still haven’t gone through—there’s another segment at the end where the Talmudic text suddenly goes back and says the following claim: “And furthermore,” Rava attacks Rav Huna. This a fortiori argument was made by Rav Huna. Now Rava attacks Rav Huna. All right, you showed me that this a fortiori argument works with all the diagrams and everything we did until now. “And furthermore, does canopy complete anything except by means of betrothal? And do we derive canopy without betrothal from canopy with betrothal?” What does he mean to say? Are we deriving canopy without prior betrothal from canopy used to complete the betrothal? That’s really what we asked here, no? He’s introducing a row-objection there. Money doesn’t produce marriage at all, does it? He said whether it’s possible… assuming one can betroth through canopy, then if the marriage-canopy includes also the… that’s the question. I’m asking: what objection came up here? A row-objection, no? There’s something else here. What? One column is canopy, one column is they are married, one column is marriage. If here we did marriage and betrothal. He’s speaking about—he wants, as it were, to add betrothal as… He says: look, the canopy—you’re telling me that it effects marriage; it’s stronger than money, right? That’s what you want to claim here. But the opposite—in betrothal it’s the opposite. How so in betrothal? He says that in betrothal, as it were, there is… No, not because of that. Canopy is more severe, stronger, than money; proof: canopy effects marriage and money does not. That’s how Rav Huna’s a fortiori argument begins, right? What’s the great cleverness here? Canopy comes after I already gave money. After all, I gave money and then betrothed her. After she is betrothed I perform canopy and now I marry her. So the canopy operates only after money has already been given here. Understand? It’s not canopy alone; it’s canopy plus money. Exactly. So obviously canopy plus money is stronger than money—that’s no great insight. But if you want to tell me that canopy itself can effect betrothal, here you want it to come instead of the money. Meaning that it itself should now effect the betrothal without there having been money before it. Who says canopy alone is stronger than money? He wants, as it were, to make a reverse a fortiori argument. What? He wants to do it in reverse. No, no, it’s not reverse; it just refutes this a fortiori argument. I don’t think there’s a reverse a fortiori argument here. He claims you cannot prove from this row—you want to prove from this row that canopy is stronger than money. He’s challenging our parameters. He’s introducing another parameter, dependence on time. In a second, in a second, we’ll see how we translate it. First I want to understand the objection. You want to prove from this row that canopy is stronger than money. He says: not true, because this datum is not really one. This datum is one because money was already given beforehand. One plus money. Exactly. No, canopy plus money gives the one. It’s one minus something. Money plus canopy. Money plus canopy gives one. So canopy gives me only part of this one. It doesn’t do the whole one, right? Basically if I needed to mark it in our model, I would write a half here. Right? You can also say that canopy has… doesn’t matter, a fraction. That it has to come after—no, this too has to come after, it depends on money. No, so dependence is another parameter. The question is whether we also need dependence here. That’s another option. If we get stuck on this, it may be that dependence too will have to play a role here. But even before dependence, first of all from the status itself there isn’t… The whole question of the passage is whether there is dependence or not. You can’t assume that. What? The whole question of the passage is whether there is… No, you can say: since there was dependence there, don’t prove anything from it. Because canopy, when it effects marriage, simply comes after money. So don’t prove anything to me about canopy alone. Yes, but that sounds like the first conclusion—that if Rav Huna wants to say this, then in his assumption he assumes let’s try to do it alone. But you don’t have to accept Rav Huna’s assumptions. Fine. I’m saying, at first pass there’s no reason to add more here; this is enough for me for now. Let’s see what it does. Okay? So basically what I want is this: I’m claiming we need to represent this objection like this. This is another kind of objection. It doesn’t add a column, doesn’t add a row. It’s not a constraint on the parameters. So what is it? It simply says: the data you wrote are not accurate. You wrote one here, but that’s not correct; it’s half. And now let’s do the calculation. Okay? That’s really the claim. Now let’s see for a moment whether this argument intuitively refutes the a fortiori argument. Seemingly yes, right? You can’t prove that canopy is stronger than money, because what canopy helps accomplish here is only because it comes on top of money. But what about this a fortiori argument? After all, what am I proving from here? I’m saying from here: if betrothal is stronger than marriage—meaning easier to effect than marriage—then if the canopy… wait, so then I say: then I’ve proved that this is easier to effect than this. So if canopy succeeds in effecting this, all the more so it will effect this. Does that stand, or is that too refuted? After all, there are two formulations here of the a fortiori argument in the intuitive formulation. Clearly that too is refuted. And this is canopy alone, without money. Yes. So… no, because… there are two… no, it’s refuted; this direction too is refuted. Why is this direction too refuted? Because from here we see that money, according to… true, betrothal is easier to effect than marriage. But then how do we continue? The continuation will tell us that if canopy succeeds in effecting the more difficult marriage, then certainly it will succeed in effecting the easier betrothal. But it is not true that it succeeds in effecting the more difficult marriage. Only with the kind assistance of the money does it manage to do that. And here you want it to come in place of the money. In place of the money it can’t do it. So this refutes both directions. All right? Now how does this find expression in the… or perhaps something else. Fine. Now how does this find expression in our model? So the objection is basically to change this one into a half. Now, how do I analyze that? How do you analyze such a thing? First of all, you need to know that this is simply exactly equivalent to the a fortiori argument of Rabbi Tarfon and the Sages, which I erased at the beginning, right? With tooth and foot and goring. I’ll remind you again: goring, tooth and foot, public domain and damaged party’s courtyard. Zero, one, half. That was the original a fortiori argument, right? Later I erased the one, turned the half, turned it into one, and asked whether there is liability or no liability without entering the question of how much liability there is. But the truth is that that is the a fortiori argument. That is the data table, right? Notice: exactly the same thing. Okay. Now what happens there? We already know; we have data on what happens there. What happens there? Rabbi Tarfon tells us—Rabbi Tarfon tells us that the filling here is one; he does not hold by dayyo, right? So the filling is one. And the Sages say half—dayyo, right? Jewish law follows the Sages. Right? So if Jewish law follows the Sages, then I already know what to do here. But what does a result of half-half mean? Wait, one second. So the result here is half, right? Good. Now here I’m discussing the question of how much to pay. He pays half the amount. But here, when they tell me that this does half the job relative to betrothal, what does that mean? That she would need a bill of divorce but she’s not married? No, it doesn’t manage to effect the betrothal. You didn’t do betrothal; there is no such thing as half-betrothal. You don’t manage to effect betrothal, that’s all. There’s some partial force. When you’re talking about money then you, then you, then the money says so pay half. But when you’re talking about betrothal, either you succeeded in effecting it or you didn’t. It is binary in its essence. How do you know to say it is specifically half? No, not half—a fraction, it doesn’t matter what fraction. And with money how do you know to say it’s half? There it really is half. Because there it really is half. The datum is that it’s half. There they pay half-damages. No, that I know—but if I say that in goring, if I reject because I can’t infer from one to the other… No, I can. I infer; I make an a fortiori argument and the result is half. In a moment I’ll get to that. The result is half, and I’ll show how we get to it. But in Rabbi Tarfon’s a fortiori argument he simply pays half. Yes, that’s the datum. But we also have that this depends on some other parameter of being forewarned and so on. No, I’m talking about a harmless ox, I’m talking about a harmless ox right now. Still, even if it wouldn’t be correct to do it, here we’ll be able to use this relativity for other things, so it will create implications and the like. It’s not only to change the payment here or something. So maybe it really will be relevant for more than that. Never mind. In principle yes, it could be that this half has a dramatic impact. It has a dramatic impact. It fails to effect the betrothal. After all, half the components of this table are like zero with respect to betrothal. So if we can—if there are all sorts of other things, where we expanded the table in the past, that this relativity will affect. Ah, so now we have to check what happens, yes, in the other places. It could be that this half will change all sorts of other data. Now we’ll have to—most of the other data, after all, were data given to us by the Torah. We did not fill them in. At the outset we filled only this cell. Only this one remained empty. No matter how big the table was, only this cell was empty. Only more and more surrounding data got added. But those data are fixed data, written in the Torah. So the whole question is just this: when this changed to half, what happens when you take that whole larger table and here there’s half? Will it still come out half? At least the assumption of the Talmudic text here is that yes—or at least not one. Less than one. All right? And in order to check that, of course, you have to do the work. But I’m saying, at the principled level I’m checking the basic a fortiori argument. All right? Now of course I still haven’t explained how one arrives at the result of half here. I only used the fact that I know that this is the result. How do we actually get there? Let’s do the analysis. But it needs to be a somewhat careful analysis. Look. The question is how, how, how. What could there be here in place of the question mark? Zero, half, or one. In principle you can also go on a continuum between zero and one—I’ll soon show you how to do that on a continuum—but in principle you have to check these three, right? It’s either zero or half or one. So let’s make the diagram, only this time we have three fillings to check and see which of them is best. Okay? So in filling zero, what comes out here? In filling zero, then we have here damaged party’s domain. What happens? What happens in filling half? Right? In filling half it goes into it. Okay? So if this is alpha, this is two alpha. Right? What happens in filling one? Clearly it goes into it. Right? Again alpha, two alpha. Right? What does that mean? That one and half are equivalent and both better than zero. Better than zero, just as a regular a fortiori argument works, right? We can throw out zero; zero is certainly not the result. The dispute of Rabbi Tarfon and the Sages is the question whether to do it with half or with one. Right? Because they are both equivalent. And how can we decide in such a situation? What? Ah! How can we decide? But the dispute of Rabbi Tarfon and the Sages—both agree that we have thrown out zero, because the basic a fortiori argument exists here; zero is not a result. Now the question is half or one. Now notice what happens here: this is presented beautifully. It’s not like the question whether you go from this side or from that side; it really is two results from both sides together. After all, in the model I no longer distinguish between the two directions. But the question is that half and one are equivalent. All right? So the Sages say—what will they say? The Sages say that the result is half. Why half? The lowest possible result. You have no proof that the filling there is more than half. Right? That is exactly the idea of dayyo. After all, what do the Sages do? The Sages say dayyo. What is dayyo? Dayyo says: take the minimum that you can prove. Anything beyond that you cannot prove. That is exactly what they say. If half does the job and one does the job, then for half you have proof. Half is definitely there. Anything beyond that—dayyo: it is enough for what comes from the law to be like the source. Therefore the Sages choose half. And you can’t argue with them? So then let’s formulate the a fortiori argument this way, the a fortiori argument that way. We asked there, when we learned this a fortiori argument in the intuitive formulation, I said: why do the Sages say half here because of dayyo? That’s not right. Look at the a fortiori argument this way and you’ll see that it should be one and not half. Yes, if the a fortiori argument is based on the fact that goring is stronger than tooth and foot. Okay? That you see here, right? Now let’s move over here. If the weaker tooth and foot is liable, then goring, which is stronger—how much is it liable? One, not half. In this formulation. Again. If I now try an intuitive formulation, if I formulate the a fortiori argument of the columns, then what do I say? From this column, goring is stronger than tooth and foot, right? I move to this column. The weaker tooth and foot is liable for full damages. Then goring, which is even stronger, full damages, not half. Right. But if you’re already saying according to Rabbi Tarfon it ought to be two, then that’s… No no no. According to Rabbi Tarfon it’s one. Why two? One. Since his requirement is not for the minimum, it is more than one. No. Not the minimum of this, but yes to one. Rabbi Tarfon doesn’t go beyond one. Why doesn’t he go beyond one? Like this. Because he says full damages. He doesn’t say two. Why not? Why does he… Because the dispute over dayyo is whether to make it half, not whether to make it one. Relative to one, even Rabbi Tarfon agrees that there is dayyo. And he agreed to make it one when there is half because you made the perspective balanced. If you do it in a… No, so wait. I’m now presenting two perspectives. Wait. In just a moment… So look, from the perspective of the columns, seemingly the naturally required result is one. Less than one is out of the question. If it’s less than one, then what do you see? Goring is more severe than tooth and foot here, and here suddenly it is less severe than tooth and foot. Something here doesn’t make sense. So it has to be one. If you looked from the perspective of the rows, then here what do we see? The damaged party’s courtyard is more severe than the public domain. Right? So if in the public domain goring is liable for half, what happens in the damaged party’s courtyard? Half. Why one? Half. Meaning the question is whether you go with this formulation or with this formulation, right? So now Rabbi Tarfon seemingly is correct against the Sages. Why? The Sages say to him, wait, wait, dayyo. Rabbi Tarfon says to them: what do you mean dayyo? Do the a fortiori argument this way and you’ll get one. One proof is enough to prove that it is one. Even if the second proof is not good enough, if I have a first proof that is good and was not refuted, why not put one? Rabbi Tarfon is right there. Right? Rabbi Tarfon should basically say to them: let’s rotate the a fortiori argument, and you’re done for. You refuted one proof for me with dayyo. But if I rotate the a fortiori argument, no dayyo will help you; there will be one here. Now notice what happens here in our formulation. Not true. Not true. One second. Let me just explain. In this formulation I’m saying, after all, I’m not rotating an a fortiori argument; I’m dealing with the entire data table. I’m searching for the minimal model. In the minimal model I see that half and one are equivalent. There is no significance here to whether I come from the rows or from the columns. This is a solution for the whole table, an explanation for the whole table. Now the question is whether it’s half or one. Zero is gone. This is a dispute between Wittgenstein and the Sages—and who is right now? The Sages. Since you have the possibility of half, the possibility of one, you cannot prove that there is more than half here. Dayyo: it is enough for what comes from the law to be like the source. That is the explanation for the law of dayyo according to the Sages. The big question is what Wittgenstein answers them if I go with this model. But here it already becomes a somewhat more complicated story; whoever wants can read it later in the article, yes. There you said that you can prove that it’s half and you can prove that it’s one. So you said that if I proved it’s one, that’s enough for me according to the view of Wittgenstein and the Sages. If I have two proofs for two different things, that means one of them… No—at least half and at least one. Not half and one. At least half and at least one. So at least half and at least one—the intersection is one. Because one does not contradict the proof that it’s half. The proof that it’s half means: the proof is that it’s at least half. One also satisfies that proof, no problem. By contrast, if you say it’s half, the proof that it’s at least one falls. All right? So basically that is exactly the explanation of the view of the Sages. Okay. Now if I return to our problem, our problem—I said that the objection is basically to put half here. The objection of marriage and betrothal is to put half here. Now I know that Jewish law follows the Sages, the model is this, the result is half. Therefore Rava is right when he objects to Abaye that marriage comes only after money, the canopy comes only after money. So what kind of a fortiori argument are you making here? An excellent objection. All right? Now Wittgenstein answers them—on what page is that? What? Now Wittgenstein answers the Sages—on what page is that? In the book? In the chapter, in the third part, in the chapter that deals with dayyo. Third chapter of the last section. Okay. So those are the examples. There is one more thing I want to show you, and with that I’ll finish the matter of the model. In principle I told you that I’d also do it for you with the continuum. In principle what I need to do here is take alpha and raise it from zero. All right? Put here zero, epsilon, two epsilon, and so on. Raise it continuously. When will something change? When I get to half. Right? If now you draw a diagram of the result in the cell—the empty cell—as a function of alpha. All right? Then I say: for alpha equals zero, that is filling zero. Alpha equals half changes the situation. From half onward, nothing changes anymore. This will be some kind of step function, yes? From half onward it changes. So dayyo says that I take the minimum. If there were a situation where, say, one third did the job, then the result would be one third. Therefore by means of this technique you can also deal with a continuum; it doesn’t have to be just zero and one. We relate it to what is in the box on the right, meaning that this is what defines for us the starting point? I’m drawing a diagram. No, I’m speaking quantitatively. You talked about half or one third. We relate to what is there, meaning down on the left. No, that’s what… what’s the point? How exactly do we define the point? So he says: this alpha is what I put here. Now I say: I’ll put zero here. Putting zero here gives me a bad diagram, right? I raise epsilon—the same diagram, nothing changes. I get to half. When epsilon equals what is to its right, we know that it starts becoming okay? Exactly. And that is precisely dayyo. Dayyo basically says that when it reaches here, you stop. The result is what is found here. Dayyo: it is enough for what comes from the law to be like the source. That is exactly dayyo. Okay. Now one final point I just want to show you very briefly—there are several interesting points here. The Talmudic text in Zevachim talks about deriving from something itself derived. A fortiori from a fortiori, binyan av from a fortiori, binyan av from binyan av, verbal analogy from a fortiori, all sorts of combinations like that. All right? So what is a fortiori from a fortiori? I searched through the whole Talmud for an example with actual content, with actual data, of a fortiori from a fortiori, and there isn’t one. Only the Talmudic text in Zevachim speaks at the theoretical level about whether one derives a fortiori from a fortiori. But I didn’t find anywhere that this is actually done, except in one place. Do you know where? The very passage that deals with a fortiori from a fortiori, the only one that uses this thing, and it proves that a fortiori from a fortiori is valid by means of an argument that is itself a fortiori from a fortiori. And that is really a loop, an amazing loop, right? Self-reference; a sort of Sabbath point. Okay? Now, in sum, I won’t go into the details. No, what are you talking about—the Talmudic text in Zevachim talks about whether one derives from something derived or not? No. Things learned from sacrificial law are not learned from something derived, but the question is in what way. Not every pair is not learned; it depends who from whom. Therefore the Talmudic text there goes through each pair and checks whether a fortiori after a fortiori, a fortiori after binyan av, verbal analogy after binyan av, all the combinations—it checks them one by one. Actually not all of them; general-and-particular and so on it doesn’t check. What we’ll see after Passover is why it doesn’t check that. General-and-particular can’t connect with anything else. So that’s a different conceptual world. But we’ll leave that for after Passover. For now I’ll show you the problem. Look. Here there is an a fortiori argument like this. A simple a fortiori argument looks like this, right? And the filling is one. Now I have another a fortiori argument; it too has to look like this. Now how do I connect them and create a table that will represent derivation of a fortiori from a fortiori? How can one build such a table? There would have to be one table on top of another. Let’s see—for example, let’s start with a table like this. This is a three-dimensional table. Wait, one second. Zero, one, one, question mark. Now I want that after I fill this in with an a fortiori argument, I will have another empty cell that I will also fill in by an a fortiori argument. Right? There is no way to do that like this. Agreed? It has to be three-dimensional. Not three-dimensional, but three by three, you mean. No, three-dimensional is a tensor. Here it doesn’t work in any way, right? There is no way to do it. So how can it be done? So I simply went to the passage there and took the data from the passage and just built the table out of the passage. And the data that came out there depend on a dispute. The first table that came out was this table, according to one side in the dispute. Something like this came out—three by three, as I said, and less than that is impossible. We have C, B, A. Analogy, verbal analogy—it doesn’t matter, some a fortiori argument; I marked it with three letters. Zero, one, one; one, these two are empty; zero, zero, one. These are the data from the passage. What is that? Doesn’t matter. They are derived from a fortiori, an a fortiori argument learned from an a fortiori argument, so all the more so that it should be learned from a verbal analogy or something. So here is the first hermeneutic rule and here is the second, or vice versa. That’s why I’m not writing it because it gets confusing. There are lots of complexities there in how to build this table. I’m not going into that now because it’s complicated. Because when you tell me that an a fortiori argument is learned from an a fortiori argument, that means there should also be symmetry. This a fortiori argument too should be learned from this a fortiori argument. There is some matrix symmetry here precisely because of the contents involved. But let’s leave that now. So basically here, why is this a fortiori from a fortiori? Because here we have a question mark; I’ll fill it by an a fortiori argument from here. After I put a one here, you see, I now have a quadruple here of an a fortiori argument, right? Then I’ll fill in another one here. Okay? That is a fortiori from a fortiori. Now you have to check whether it really works. How do you check it? In principle the best way is to do it like this. This is x and this is y. All right? Now I build: zero-zero, zero-one, one-zero, one-one. All the combinations. For each one I draw a diagram, right? And I check which of the four combinations comes out best. I compare the diagrams according to the parameters we defined: connectivity, changes of direction, topology, everything we defined. All right? I compare these four diagrams. I simply fill them in one by one and build a model. Zero-zero, build a model. Zero-one, one-zero, build a model. What comes out here? It turns out that one-one wins by a mile. That means there is a fortiori from a fortiori. Right? Because I can’t fill this one without filling that one; I have to do both together. So basically, a fortiori from a fortiori is simply a table with two cells that I have to fill in, not one. So what does that mean? That I have more diagrams to compare—not only one for zero and one for one, but for every pair, which makes four diagrams. If there are three empty cells, eight diagrams, and so on. Is it right, though, to look at this as a kind of vector of— the second a fortiori argument, is it right to look at it as a recurring column of E? No, no, you can’t. That’s exactly the point. Because there could be a problem here. Here, I’ll show you an example now of something that is problematic. Another example—look at this example. Zero one zero, one x y. Here: one zero one. I simply did it on the row instead of the column. Okay, here too apparently there is a fortiori from a fortiori, right? It comes out that here too there is an a fortiori argument—you fill in one here, right? Then when you fill in one here, there is again an a fortiori argument here and here too there is one, same thing. Check this and you’ll see that one-one is the worst. Zero-zero… You can’t always make an a fortiori from a fortiori. And why? It’s very simple why. Because think: if I fill in a one in place of this x, then the first a fortiori argument I made doesn’t work; there is an objection here. This is an objection to the first a fortiori argument. So you can’t fill in one here, right? If I want to fill in one here, then there is no objection, but there is an objection to this one. There is one here, right? Therefore it doesn’t work here. A fortiori from a fortiori does not always work. I have to deal with the whole table all at once. You can’t just do a fortiori from a fortiori casually. If I tried to formulate this without the table, I would say to you: what the analogy does not do to A, it does to B; it does not do to C, it does to D; the verbal analogy that does A, all the more so it does B. Then I’d take this square and again say to you: and what the a fortiori argument does not do to B, it does to C; the verbal analogy that does B, all the more so it will do C—you would not notice any mistake there. That would be an a fortiori argument that seemed to work, and it’s not true. Because you did not take into account the whole table; you took into account only the quadruple, the small two-by-two sub-matrix. But that’s not right. Since these two have to be solved simultaneously, not one after the other, in order to see whether this pair is really the best there is. Because after all, both of these are missing from the Torah, and I want to know what is simplest when I fill in both things. What is the most correct filling when I fill in both things. Fine, that—same thing I do with binyan av from binyan av, it doesn’t matter which; with binyan av from a fortiori, you can do everything the same way. Maybe one last point just to conclude—and this is what we started with at the very beginning. After all, I said that an a fortiori argument in the intuitive formulation, when you make an objection like this to it, it ought not to refute it at all. Like this: A B, zero one one question mark, one and zero. This is a regular column-objection. And I said that such an objection should refute the relation between little A and little B, but not between big A and big B. Right? Only the model showed us that in fact it still refutes. But in intuitive thinking it doesn’t refute. What happens if I now add an objection from here too? Little G, one zero—suppose here too there’s something, I don’t know what, I also don’t know what that is, it depends. Okay, there is another objection, both from here and from here. This of course should even intuitively refute the a fortiori argument, right? Here it will be something stronger. This again is a table of two cells, with these two—I don’t know, in principle all sorts of things could appear here. So it isn’t really a table of two cells; it’s a table where I’m interested only in this, but since all sorts of things can appear here, I have to solve it as though there were two empty cells here, and I see what the general filling is. It turns out that zero-zero is the best filling in such a table. What does it mean that zero-zero is the best? Notice—not that it is equivalent to one-one, in which case it would just be an objection. Rather, it is better than one-one. What does that mean? That a double objection, from the column and from the row, is a counter-proof, not an objection. It is a proof that the filling is zero. It is not an objection saying: you didn’t prove it to me; maybe it’s one and maybe it’s zero. It is a counter-proof. When there is a double objection, that is a counter-proof. You can read the a fortiori argument from the other side. What? Read the a fortiori argument from the other side. Right. That is basically to say: I can prove to you that the filling here is zero. Not to refute the proof that it is one. Once both of them—once the two numbers are the same, that is a proof. Here we had one and zero; now we have zero-zero. No, no, no. If here there had been one and zero, that too would have been a proof. When I have two pairs that are the same, that is an objection. When I have a preferable pair, that is proof that this is the correct pair. Really that there has to be zero-zero here. That isn’t merely an objection; it’s absolute proof. By “counter-proof” I mean: it is proof that the filling is zero; it is not an objection to the claim that the filling is one. It is a proof. Therefore generally in the Talmud you won’t see objections from both sides. After all, I asked: why doesn’t the Talmud bring objections from both sides? After all, an objection from one side can always be rotated. So first of all I showed that you can’t rotate—it was the model that showed that. Now I’m telling you something more than that. If there had been objections from both sides, that wouldn’t have been an objection. It would have been a counter-proof. Fine.