Lecture dated 24 Nisan 5777

Table of Contents

• The table model and the definition of simplicity
• Advantage, refutation, and equivalence between different fillings
• The question of halakhic rulings by decisors, monetary law, and the presumption of permissibility
• Microscopic and macroscopic refutations and factual characteristics
• Offsetting refutations and extending the model to more than two values
• Horn damage on the injured party’s property: a table of zero, half, and one, and rotating the a fortiori argument
• Dayo, absorption of refutations, and equivalence between half and one on the microscopic plane
• Actions and results, reversing arrow directions, and topology that distinguishes between half and one
• Rabbi Tarfon’s ruling versus the Rabbis, and the absence of additional test cases

Summary

General Overview

Thursday, the 24th of Nisan, 5770—April 8, 2010—a lecture by Rabbi Michael Abraham presents a model for analyzing an a fortiori argument as the solution to a “data table” with one missing cell, by comparing competing models and choosing the simpler one. Simplicity begins as a smaller number of parameters, but topological parameters of the diagram are also required: connectedness, changes of direction, and number of points, together with dimension. The assumption is that if there is a clear advantage in one parameter, that decides the issue; and if each side has advantages, that creates a refutation and equivalence. The discussion moves on to microscopic and macroscopic refutations and to the distinction between factual characteristics such as “benefit” and halakhic characteristics. Then specific implications are examined: offsetting refutations, absorbing refutations, and moving from a binary a fortiori argument to a field of three values—zero, half, and one—through the dispute between Rabbi Tarfon and the Rabbis regarding horn damage on the injured party’s property and the application of the principle of dayo.

The Table Model and the Definition of Simplicity

The model assumes a data table in which all the cells are filled except for one, and the missing cell is determined by comparing different possible fillings and constructing a model that explains all the data. Rabbi Michael Abraham defines the work of building the model as a kind of “chemical analysis” that breaks actions and results down into components that explain the table. Simplicity is determined not only by the number of parameters but also by three topological parameters: the connectedness of the diagram, the number of directional changes relative to the arrows, and the number of points. Combined with dimension, these yield four indices that define simplicity.

Advantage, Refutation, and Equivalence Between Different Fillings

The basic assumption is that an advantage in one parameter in favor of a certain filling decides in its favor, but advantages in opposite directions create a state of refutation. A refutation is not proof that the alternative filling is correct; rather, it refutes the claim that the original filling is correct. So the two possibilities remain equivalent, with no way to measure one advantage against the other. Rabbi Michael Abraham emphasizes that the distinction between “proof” and “refutation” is the distinction between a positive decision and leaving two possibilities as equivalent, and that there is no essential significance to labeling something as zero versus one.

The Question of Halakhic Rulings by Decisors, Monetary Law, and the Presumption of Permissibility

In the discussion of the tendency of halakhic decisors, the possibility is raised that there may be additional considerations beyond pure logic, such as giving weight to the last speaker in the Talmudic passage. But Rabbi Michael Abraham does not confirm such a pattern without examples, and he distinguishes between a refutation and an interpretive limitation. In monetary law, the rule is highlighted that “the burden of proof rests on the one who seeks to extract payment from another,” so that in a case of refutation the practical result is non-liability. But the explanation is not that there is proof that no law applies; it is that there is no proof that a law does apply. Rabbi Michael Abraham presents the view that Jewish law is built on a presumption of permissibility: everything is permitted unless it has been proven forbidden. He cites, in the name of the Klausenburger Rebbe of blessed memory, the opposite statement in order to stress that it does not reflect the spirit of Jewish law.

Microscopic and Macroscopic Refutations and Factual Characteristics

Rabbi Michael Abraham distinguishes between a “microscopic” refutation and a “macroscopic” or “halakhic” refutation, and also speaks about a “refutation of the characteristics.” In the example of the common denominator in tractate Kiddushin, “benefit” turns out to be a factual characteristic of money and intercourse, not a halakhic characteristic. Therefore it acts as a constraint on the model, requiring a parameter shared by money and intercourse that does not exist in a document. This constraint creates effects parallel to a column refutation, but it is not treated as an additional column in the table.

Offsetting Refutations and Extending the Model to More Than Two Values

The lecture continues from the general description to applications, and presents a situation in which the Talmud offsets refutations and distinguishes between different strengths of refutation, so that an a fortiori argument can “survive” when one refutation is perceived as weaker than another. To represent this, Rabbi Michael Abraham introduced more than two values into the table, and now he turns to cases in which the field itself has three values, and not just as a way of representing offsetting. This move opens up the possibility of a filling that is not only zero or one but also “half,” and requires comparison among three possible fillings.

Horn Damage on the Injured Party’s Property: A Table of Zero, Half, and One, and Rotating the A Fortiori Argument

The Mishnah in Bava Kamma presents a dispute regarding a harmless horn on the injured party’s property: Rabbi Tarfon obligates full damages, while the Rabbis obligate half damages. On public property, a harmless horn pays half damages, while tooth and foot are exempt; and on the injured party’s property, tooth and foot pay full damages. A table is constructed with two domains and two types of damagers, and the missing cell is horn on the injured party’s property. Here, for the first time, the table is not binary but contains zero, half, and one. Rabbi Michael Abraham explains that two ways of formulating the a fortiori argument—comparing columns versus comparing rows—seem capable of yielding different results in a non-symmetrical table. So one might expect that the dispute of the Tannaim would revolve around the direction of the a fortiori argument. But according to the plain sense of the Mishnah, both sides do not make the law depend on this rotation: for Rabbi Tarfon the result is always one, and for the Rabbis the result is always half. The Rabbis respond, “It is sufficient that what is derived by law be like that from which it is derived,” and they always formulate it as: “Just as on public property it is half damages, so too on the injured party’s property it is half damages.” Rabbi Tarfon tries the move: “I too will not infer horn from horn… I will infer horn from foot,” but even to that the Rabbis return to the formula of dayo.

Dayo, Absorption of Refutations, and Equivalence Between Half and One on the Microscopic Plane

Within the model, a zero filling creates independence—there is no connection between the columns—whereas a half filling and a one filling can appear to have the same microscopic diagram, and therefore both are preferable to zero but equivalent to each other. From that, a result is described that mirrors the principle of dayo: when half and one are equivalent in the model and there is therefore no way to decide between them, the minimum is chosen, and so the result is half, in accordance with the Rabbis. Rabbi Michael Abraham presents this as a “microscopic reflection” of dayo that does not necessarily depend on a sharp verbal identification of “what is derived by law” and “that from which it is derived” within the table, but rather on choosing the minimum among equivalent possibilities after zero has been ruled out.

Actions and Results, Reversing Arrow Directions, and Topology that Distinguishes Between Half and One

The discussion becomes sharper when it is necessary to explain not only the order-relations but also why the value “half” actually appears, and the possibility is considered of a model in which a parameter appears in different intensities—such as “two alpha”—so that a partial lack in the parameter produces a result of half. Rabbi Michael Abraham shows that in different fillings there is a need for structures like “two alpha” versus “alpha and beta” in order to reconstruct the four data points, and he notes that with actions and their intensity, the direction of the arrow and the intuition are reversed in comparison with results and thresholds of application. Here the topology of the graph becomes relevant, so that rotating the perspective between rows and columns can change the shape of the graph even if the microscopic array of parameters remains similar. Therefore there may indeed be a difference between a half filling and a one filling.

Rabbi Tarfon’s Ruling Versus the Rabbis, and the Absence of Additional Test Cases

In the analysis attributed to Rabbi Tarfon, the one filling is preferable to the zero filling because of connectedness and topology, whereas in the zero filling one gets disconnected units. Rabbi Michael Abraham suggests that the Rabbis may rely on only one direction, or may focus on the binary stage of liability as such and ignore the quantitative difference between half and one. But he notes that there are not enough additional cases in the Talmud beyond the two parallel cases to test and decide among the different possibilities. The lecture ends by saying that the dispute in the three-value table remains dependent on how one understands dayo, on whether consistency is required between the two directions of inference, and on the effect of the topology of the diagram on deciding the missing filling.

Thursday, 24 Nisan 5770, April 8, 2010, a lecture by Rabbi Michael Abraham. We’re basically returning to the model based on the attempt to solve a data-table problem. Data: all the cells are filled with data except for one, and the question is how we fill in the missing datum. And the basic claim is that we fill it in by comparing two possible fillings, either a possibility of zero or a possibility of one. For each such possibility we try to find what I called a chemical analysis, the components present in each action and in each result that could explain all the data in the table. You can explain the data of filling in zero with one model, filling in one by means of one model, and filling in zero by means of a second model, and see which model is simpler. And that’s what determines which filling is the correct one. The big question, of course, is what “simpler” means. What is simple? So at first the obvious proposal was that the simpler model is a model containing a smaller number of parameters. We saw that this explains a large part of the inferences but not all of them. We also had to add topological parameters. We added three: connectedness — connectedness means that when we draw the diagram, all the points are connected to each other, meaning there is only one unit, the diagram is not divided. If there are two units, then the connectedness is two, and so on. The smaller the connectedness, the better, the simpler. So connectedness is one, changes of direction is the second — meaning if we go along a path from point to point on the graph, do we have to reverse direction against the arrows — and the number of points. The number of points is the third parameter. And the assumption is that out of the four parameters — the three topological parameters and the parameter of the number of points, or the dimension — these four indices determine the simplicity, define the simplicity of the model. The assumption is the following: if one of the fillings has an advantage in one parameter, then that is the preferable filling. If there is an advantage in certain parameters for one filling and in other parameters for the second filling, that means we are in a situation of refutation. It doesn’t matter right now how many parameters there are in each direction, because that really does seem to me to be our intuition surrounding this matter of refutation. “Refutation” means there are certain advantages to one filling, certain advantages to filling in zero; we have no way to measure those advantages against each other, and therefore once there are advantages on both sides, we are basically in a situation of refutation. And a situation of refutation, as a reminder, is not a proof that zero is the correct filling; rather, it is a disproof of the claim that one is the correct filling. It means that filling in zero and filling in one are equivalent; we have no way to determine which is preferable. That is called refutation. Because proving that filling in zero is correct is the same thing as an inference in the direction that one is correct — after all, it doesn’t matter what we mark as zero and what we mark as one. Therefore proof versus refutation is always positive proof versus leaving the two fillings as two equivalent possibilities. Okay, last time we saw — sorry. Yes.

In a lot of cases I’ve run into a situation where it seemed that the halakhic decisors tended to follow the method of refutation because of some consideration like the editor of the Talmudic text gave him the right to the last word on the page or something in that style, even though…

They tended to follow the method of refutation? Meaning, the assumption that the refutation came to refute means it is necessarily not correct, and therefore the opposite is correct? In other words, it’s not that the refutation only operated logically to leave you in doubt, but because they gave him…

The ukimta…

Refutation means that both possibilities remain open. As a pure logical question, it really is as you say, but there are additional considerations, as though whoever spoke last on the page, as though his opinion was accepted.

No, but even whoever spoke last on the page, as though his opinion…

The last one who spoke did not say that zero is correct; rather, he raised the refutation.

Say the first one claimed claim A; it’s not that the second proved not-A, but rather he proved not-A necessarily.

Right.

Meaning A and not-A are both…

But from the way you describe that the halakhic decisors say if they gave him more, then as though — it’s not clear. If you have examples, I’d be happy to see them; I don’t remember an example at the moment.

There are thousands of examples like that where they gave a solution called an ukimta. And in such cases, very often Maimonides rules — there’s a discussion on a Mishnah, the Talmudic text concludes with a solution by way of an ukimta, and Maimonides ignores the Talmudic text and rules according to the plain Mishnah, but the Rif and the Rosh rule according to the ukimta, interpreting the Mishnah according to the ukimta. Is that what you meant?

No, he was talking about refutations; he’s not talking about ukimtot. An ukimta is also a kind of refutation like that, in truth — a case like that, it doesn’t belong here.

No, an ukimta is not a refutation. An ukimta is an answer.

But you didn’t succeed in proving that it isn’t so.

Refutation in our context is something entirely different. I don’t know. If you have examples, I’d be happy to see them. It’s obvious that, say, in a monetary context, if you’re talking about payment obligations in financial damages — an a fortiori inference, that of Rabbi Tarfon, which we dealt with a bit — there, “the burden of proof rests on the one seeking to extract from another.” So it’s clear that de facto once there is a refutation, the answer is zero, because you aren’t obligated. But that’s only in monetary law.

Okay. Every a fortiori inference comes to innovate?

What?

Every a fortiori inference comes to innovate?

To innovate a law. In damages, generally the innovation of the law is that you must pay. By the way, not only in monetary law, really — if the a fortiori inference innovates some law and it is refuted, then the result is that the law indeed is not there. But the law is not there because we proved it isn’t there; rather, because we didn’t prove that it is there. And the assumption is that as long as you haven’t proved some law, it does not exist. So maybe in that sense you’re right. An a fortiori inference always comes to innovate some law, and as long as you didn’t innovate it, then the law isn’t there. We don’t conduct ourselves in laws of doubt when we have no basis for doubt. You need a basis in order to doubt. And if we didn’t prove בכלל that there is such a law, then it isn’t there, so there isn’t. The assumption probably — I once heard in the name of the Klausenberger Rebbe of blessed memory — he said that he would not cut the challah on the Sabbath even if it were not written in the Shulchan Arukh that this is permitted. That’s some sort of conception that says everything is forbidden unless it has been proved to be permitted. But obviously that’s not correct. In Jewish law everything is permitted unless it has been proved to be forbidden. You need a source in order to forbid something, not in order to permit something. Everything is permitted unless the Torah forbade it. So let’s prove that the thing is forbidden. Good.

That’s the spirit of Jewish law.

What?

That’s the opposite of the whole spirit of Jewish law.

Okay. The matter of the a fortiori inference of an unpaid guardian and a paid guardian.

Yes, so that we finished. Yes, what I really want — basically what I’m doing now, after having completed the general picture of the model, we saw two kinds of refutations. We saw a refutation — just to remind you, because we’ll come back to this — what I called a microscopic refutation and a macroscopic refutation, or a halakhic refutation and a refutation of characteristics. And we talked about the fact that, for example, if you remember in the common denominator argument there was “what do intercourse and money have in common? They involve pleasure.” At first we treated that as though it were another column in the table. But afterward we saw that this really isn’t right: the pleasure is one of the factual characteristics of intercourse and of money, and not a legal characteristic. And therefore it is actually some kind of constraint on the model we are building. This model has to contain a certain parameter that will be present in money and in intercourse and absent from a document. We identify that parameter as pleasure. This constraint will do various things parallel to a refutation, to a column-refutation. But we’ll come back to that in a bit. In any case, last time I started looking at more specific implications of the model. Up to now we dealt with the general analysis.

And you talked about there being more than two values.

That I haven’t talked about yet.

You did say it.

No, that was the result with the two. Today I’m going to get to what happens with more than two values, because until now that was just a mode of presentation. Meaning, what I talked about last time was offsetting refutations. How we don’t offset refutations. Meaning, if there is a refutation saying that A is more severe than B — bang — the assumption of the a fortiori inference is that A is more severe than B. The refutation says that B is more severe than A. So the assumption is that we are in a refutation; it’s equivalent. And we saw a case like reed and not oath, “the body itself falls under an oath.” We saw there a situation where the Talmudic text offsets refutations. This refutation is weaker than that one and therefore they leaned… A is still more severe than B and the a fortiori inference survives. And we tried to explain that with the model, because there I really inserted more than two values into the table in order to represent such a situation. Today we’ll see a case where we really are simply over a field of three values, not two. So that was the first result — offsetting refutations, which I tried to explain there. The second result that I want to examine is what is called absorbing refutations, absorbing refutations into an a fortiori inference. Take the page, take the page. Maybe leave one for me. Okay, does everyone have a page? The two passages on this page are a Mishnah and Tosafot. And I’ll need both of them for this topic and for the next one. So I’ll start with the Mishnah, although our first discussion here is actually in Tosafot. In the Mishnah it says as follows — we’ve already seen these things, actually, but in this Mishnah there’s a point I’ll need later. “How is it with an ox that causes damage on the property of the injured party?” So what is the law of an ox that gored on the property of the injured party? “If it gored, pushed, bit, crouched, kicked in the public domain, he pays half-damages.” All these are subcategories of horn. He pays half-damages. “On the property of the injured party, Rabbi Tarfon says full damages, and the Sages say half-damages.” So there is a dispute about the law of horn in the courtyard of the injured party. This is an a fortiori inference — let’s write the table here. Public domain and the property of the injured party; tooth-and-foot and horn. Okay? The data are basically: here there is zero, horn in the public domain is liable for half; tooth-and-foot on the property of the injured party is fully liable. And the question is what the law is for horn in the courtyard of the injured party. That is basically the problem we’re dealing with. Until now I ignored the fact that here there is a half. Actually I brought this a fortiori inference as an example and wrote a one here just in order to understand how an a fortiori inference works, but the truth is that here it says half. Meaning, the meaning of this half is simple: horn — we pay horn damages — if my ox gored my fellow’s ox in the public domain, I have to pay half the damage if this is the first goring; after it becomes forewarned it turns into full damages. I’m speaking, of course, about a harmless horn. So that’s the table. Now this is a table we haven’t encountered until now. Because until now we encountered only tables of zero and one. And here for the first time we see an inference — in this case an a fortiori inference — built over a field of three values: zero, half, and one, and not over a binary field of zero or one. And the question is what to fill in here. So if here it said one, then it would be a simple a fortiori inference and the result would be one, and we already analyzed that and saw why it is one, and also, to remind you perhaps, why rotating the a fortiori inference also changes nothing and nullifies no refutation — all of that we saw with one. That’s something that can apparently be explained pretty easily with the model. Even if there is a half, according to the dynamics of things, let’s see. I assume this has to be — and that’s the dispute of Rabbi Tarfon and the Rabbis; now we’re learning it, let’s see. So the Talmudic text says as follows: Rabbi Tarfon says full damages and the Sages say half-damages. Okay? If it is in the courtyard of the injured party. So Rabbi Tarfon says that in fact we have to fill in full damages here, and the Sages say half. So it’s a tannaitic dispute about what should be filled in here. And what is the dispute? So the Mishnah says as follows: Rabbi Tarfon said to them — now Rabbi Tarfon comes to attack the position that says half, in order to show that it is one. “Now if in a place where it was lenient regarding tooth and foot in the public domain, where one is exempt, it was stringent regarding them on the property of the injured party, to pay full damages — in a place where it was stringent regarding horn in the public domain, to pay half-damages — is it not all the more so that we should be stringent regarding it on the property of the injured party, to pay full damages?” So he says an a fortiori inference; he phrased it in a certain way. That phrasing can be interpreted in several ways; there’s no point going into the details. The result is one. In our language, how would we explain that? We would explain it — after all, we saw that there are in fact two ways to formulate such an a fortiori inference. You can formulate such an a fortiori inference by comparing columns or by comparing rows, right? Comparing columns is built as follows. I’m looking at a comparison of rows. I look at the right-hand column, the public domain. In the right-hand column I see that horn is more severe than tooth-and-foot, right? Within the comparison in the right-hand column you see that: horn is more severe than tooth-and-foot. Now I move to the left-hand column and I assume that here too horn should be more severe than tooth-and-foot. So if tooth-and-foot is liable for one, then horn too is liable for one, right? And that’s if we do a comparison — right? Less one, half.

What? One and a half.

No, it’s less one; we’re in the principle of “it is enough.” So in such a “it is enough,” enough. So this comparison essentially gives us one. And if we go like this, no? If we go like this, then we’re now looking at the top row and you see that left is more than right, right? No — what does left more than right mean? That for every missing cell it will need to be at least what is on the right, and “at least” is always what is on the left, not more. As we did here. Why didn’t we put two here? Why one? We saw that half is more than zero, right? Suppose here there is one, then here there would have to be two, three. We put one, because the assumption is that you have no proof to do more than what is here. It’s at least that. Everything beyond that — what you said earlier, actually — everything beyond that needs proof. It could be correct; that doesn’t contradict. Bring me proof. If you bring me proof that there’s a two here, I’ll put two, no problem. As long as there’s no proof, I put one. That’s the minimum. Okay? So if I now make this comparison, then in this comparison I look at the top row. In the top row the property of the injured party is more severe than the public domain, right? We see that from tooth-and-foot. Now we go down to the bottom row and say, well, so here too the property of the injured party should be more severe than the public domain, right? And if in the public domain horn is liable for half, then on the property of the injured party it is also liable for at least half. But we said “at least half” means half, right? So that means that this a fortiori inference gives me a filling of half, while that a fortiori inference gives me a filling of one. Therefore this case, apparently in this case we have a situation we haven’t encountered until now, because this cell and this cell are not symmetrical. This matrix, this table, is not symmetrical, so the result seemingly depends on the direction of the inference. Okay? Until now what we saw was that if the matrix is symmetrical — here there is one and here there is one — then the result does not depend on the direction of the inference. What does that mean? That even if we put a refutation here or put a refutation here, if the refutation refutes the inference of the columns, it will also refute the inference of the rows and vice versa. We saw that when the basic matrix is symmetrical, but here it is not symmetrical. So what would we expect to happen in the Mishnah? That Rabbi Tarfon and the Sages would actually argue about the direction of the a fortiori inference. The Sages, who say it is half, presumably learned this a fortiori inference; and Rabbi Tarfon, who says it is full, presumably learned this one. Now what actually happens in the Mishnah is not that. The Mishnah tries to go in this direction but then backs off. Its conclusion is not that. So after Rabbi Tarfon says, brings his a fortiori inference for full damages — and from the structure of the Mishnah, leave the wording aside, the wording is very convoluted — from the structure of the Mishnah it’s quite clear that at first he brought this a fortiori inference, this one specifically. And the result is one. In the wording you could understand otherwise too, but from the structure of the Mishnah it’s clear, because afterward that’s exactly what he does with a change. And then the Sages say to him: not at all, “It is enough for what is derived from the law to be like that from which it is derived.” That’s how they answer him, right? They said to him, “It is enough for what is derived from the law to be like that from which it is derived: just as in the public domain it is half-damages, so too on the property of the injured party it is half-damages.” So it’s clear we are talking about this a fortiori inference, and therefore the Sages say to him: “It is enough.” So what does Rabbi Tarfon do? He says to them, “I too will not derive horn from horn.” What does “horn from horn” mean? This a fortiori inference, because in this a fortiori inference we learn this empty cell from here, which is exactly horn on the property of the injured party from horn in the public domain. That is learning horn from horn, right? So I won’t derive horn from horn; that proves that earlier he did derive horn from horn. “I will derive horn from foot.” What does that mean, horn from foot? In fact, this a fortiori inference. A brilliant move. Meaning, I didn’t try this one; they told him, “What are you talking about? That’s enough,” so he says to them, okay, then I’ll do this one; to that you certainly have no answer. Here it’s definitely one and not half, right? And the Sages don’t even blink: “Now if in a place where it was lenient regarding tooth and foot in the public domain, it was stringent regarding horn,” etc. They said to him, “It is enough for what is derived from the law to be like that from which it is derived: just as in the public domain it is half-damages, so too on the property of the injured party it is half-damages.” What is this? Strange. You said in this a fortiori inference — how can you apply “it is enough”? After all, from here we proved. From here we proved that horn is more severe than tooth-and-foot. Now we move to this domain, to the property of the injured party. Here too horn is supposed to be more severe than tooth-and-foot, so how does it come out as half? Half is less than tooth-and-foot. That contradicts what we proved from this column. What is this “it is enough”? This “it is enough” contradicts what came out of this column. And the Sages say “it is enough.” Now Rabbi Tarfon too is a bit problematic, because when he opens the discussion, he opens it specifically with this a fortiori inference. Why? Go straight for the winning card, start here. It seems that Rabbi Tarfon says that in this a fortiori inference too the result is one. Meaning, in the simple reading of the Mishnah — that’s what most of the medieval authorities say — in the simple reading of the Mishnah what comes out is, in fact it appears in the Talmudic text, because where the a fortiori inference is not symmetrical it applies “it is enough.” But from the simple reading of the Mishnah it comes out that in the end, after all the maneuvers and reversals they try to make here, both Rabbi Tarfon and the Sages do not distinguish between the directions of the a fortiori inference. For Rabbi Tarfon the result is always one, and for the Sages the result is always half. It doesn’t matter whether you did this a fortiori inference or this a fortiori inference. Meaning, I told you that in the entire Talmud there are two examples where the Talmudic text itself or the Mishnah rotates the a fortiori inference. There are only two examples of rotating an a fortiori inference: one in tractate Niddah and one here, as far as I know at least, and in both these examples it’s a case of “it is enough,” a non-symmetrical case where there is one here and half here. All the other examples are symmetrical — one and one — there are no rotations of an a fortiori inference, and that’s precisely one of the confirmations I brought for the model I dealt with last time. But here, where it is asymmetrical, there they really do always try the rotation. But after they try the rotation they back off. They try to rotate, but in the end they say no, it doesn’t depend on the rotation. Rabbi Tarfon, who says full, says full in both directions, and the Sages, who say half, also say it in both directions. And Rabbi Tarfon too, according to the words of the Sages, said to them — that’s what Tosafot says there.

Yes. Fine, there are many more nuances here in the medieval authorities; I’m not going into all the details right now. But broadly speaking, what comes out from the simple reading of the Mishnah in the end is that, surprisingly, even in the non-symmetrical case, where we have one and half, there is an attempt to use the rotation of an a fortiori inference, but in the end neither Rabbi Tarfon nor the Sages rotate. Yes.

Is there an a fortiori inference regarding muktzeh from kindling on a Jewish holiday or on the Sabbath?

That appears in this Talmudic passage here later, immediately after this Mishnah. And then — do they also say “it is enough” there?

No, no. There, I talked about that at the very beginning — that is an a fortiori inference that comes out of one datum, not out of three. An a fortiori inference of reasoning. It seems to me that maybe this discussion can be read on two levels. On the first level it is indeed binary, it is indeed with our older perspective, that to be lenient means exemption — that’s zero — and to be stringent means not exempt, meaning there is liability. The man does not go home clean. Because that’s how you can see it at the start, when he opens with the rotation: “Now if in a place where it was lenient regarding tooth and foot in the public domain, where one is exempt…” from there you understand that the meaning of being lenient is zero, complete exemption. “It was stringent on the property of the injured party; in a place where it was stringent regarding horn in the public domain to pay half-damages…” You see that the word “to be stringent” — here they aren’t distinguishing between paying full damages and paying half-damages. Stringency means he is obligated in something, that is, not exempt. So in fact there is here first of all a primary discussion of the old type. And now on that they say “it is enough” regarding quantifying the penalty, not regarding exemption. But why, in the quantification, do they arrive at one?

What?

There are two levels here. Come hear this, but still how does that ultimately explain the result? So why do they arrive at one or at half? But you lost the first one. The first is only about liability — so then how does he arrive at the conclusion of full damages? And also why do the Sages reject him? So how do you arrive at one or half? Maybe the Sages don’t accept the…

Maybe Rabbi Tarfon really does learn like that, that there is zero, one, and half. And the Sages say no, there is liable/exempt, and afterward we’ll talk about the strength of the liability. They don’t accept the afterward, and there they make the claim of “it is enough,” and the claim of “it is enough” is always in the bottom line, between horn and horn. Why?

That doesn’t help me divide it, because in the end I still have to understand the second stage. So why is it really like that? I don’t mind dividing it. By the way, from the sugya it also apparently comes out that it has to be broken into two stages. For that you need to see the Talmudic discussion. But that still doesn’t explain the result. How do we arrive at one or half? And that comes out of the a fortiori inference; it’s not that they bring it from somewhere else. The a fortiori inference proves liability, but how much liability — that they bring from a verse or something like that.

If that were so, you’d be right, but no: this a fortiori inference also leads them to the halakhic result. It’s as though you look at horn in the public… What are the sides here? Public domain… public domain, let’s say. It’s a little complicated to explain. It’s like a legal precedent that was ruled, and now they have to be consistent with that precedent. And that precedent says “it is enough.” In other words, the precedent says that horn doesn’t pay more than half. And they stick with that; they don’t want to create…

Why? But maybe in the public domain it won’t pay half, and on the property of the injured party it can pay full damages? You have to understand that in the end this result of half contradicts the assumption of the a fortiori inference. That’s a very problematic result. Because the assumption of the a fortiori inference is that horn is more severe than tooth-and-foot.

No, there is liability there. In the place of the question mark there is liability, and in that sense the a fortiori inference works.

Wait, but now I’m talking about the results, not about liability. In terms of results, is this more severe than that? Is horn more severe than tooth-and-foot?

In the sense that he pays.

Yes. Then why not here too? He also pays.

No, but why not here? In terms of the amount. Now I have a reverse a fortiori inference to prove that you’re not right regarding the amount of payment, on the second stage. Here, I’ll show you here. If here this is more severe than that, then it cannot be that here it will be more lenient than that. Fine. Here we have tooth-and-foot, here horn; these are the causes. And the characteristics are public domain and the property of the injured party. What causes liability is not the domain. The domain gives conditions for the liability, but the cause of liability is the horn or the tooth-and-foot. Therefore the fundamental discussion is what I do with horn versus horn, not with horn versus tooth-and-foot. That’s a different kind of thing altogether.

Why? I don’t see it.

And therefore they go back…

The Sages stick to the original a fortiori inference in horn, from public domain to the property of the injured party.

No, they don’t stick to that a fortiori inference. They also say regarding the a fortiori inference that you’re saying, Rabbi Tarfon: we will apply “it is enough.” They don’t say that this a fortiori inference isn’t correct. They say the same words, and it’s as though they ignore your second attempt altogether. They say “it is enough.”

Exactly. So that I understand. I agree that Rabbi Tarfon made his second a fortiori inference, they simply ignore it — I explained that to you.

It doesn’t seem that way from the wording of the Mishnah.

What?

It doesn’t seem it’s exactly the same quotation. They also ignore…

Why ignore it? Word for word. You can prove that horn is more severe than tooth-and-foot, so here too you prove that horn is more severe than tooth-and-foot. What — how can you ignore that? It’s written word for word; that means it’s not…

It doesn’t have to be exactly the same thought. The simple reading of the Mishnah doesn’t look like they’re ignoring it. It looks like they’re saying “it is enough” also about the second formulation. The second formulation doesn’t prevent saying “it is enough.”

But the words are not correct.

Why not correct?

Because in the first case he says… in the first case they said, “It is enough for what is derived from the law to be like that from which it is derived: just as in the public domain it is half-damages, so too on the property of the injured party it is half-damages.” Right.

And here it should have been something else.

No, it couldn’t be something else, because here it was one. You have to apply “it is enough” from here. The wording of “it is enough” must be like that. In truth, they are ignoring the second a fortiori inference.

No, they’re not ignoring it; they’re saying that it too is exposed to “it is enough.” But say, “just as in the public domain…” You started the a fortiori inference from here, because here it is one and not half. You can’t apply “it is enough” from here.

But they say that “it is enough” regarding the second formulation. They don’t say the second formulation is wrong. There is no invalidation there at all of the second part — so too on the property of the injured party.

What?

Yes. In any case, there is here — you are trying to present the possibility of seeing this in rows and columns. In pure logic it is hard to say something against that; as though here, you can rotate it 45 degrees. But in any case, from what we might call a linguistic standpoint and from the standpoint of human perception, the case is damage of the type horn that happened in the context, in the setting, of the public domain or private property. You don’t say: I had here damage of the type private property that happened in the context of a strike by a horn. Obviously not, so what?

So there is a certain asymmetry here.

Why? Obviously there is.

And therefore all the time “it is enough” works in this. I’m trying to make this a fortiori inference. This a fortiori inference basically says that the damager of horn is more severe than the damager of tooth-and-foot. Is it legitimate to infer that conclusion from this column?

Yes.

Legitimate, right? So if I know that horn is more severe than tooth-and-foot, now I move to the property of the injured party. Here too horn should be more severe than tooth-and-foot.

Who said so? It’s true that one becomes liable for tooth-and-foot — in that sense, yes — but in the sense of not being exempt. It can’t be that he’s exempt, as we once learned.

No, but when we’re talking about the size of the fine, the size of the compensation, why?

I can tell you why, but that’s already an explanation. Who said that if in one place it is more severe, then in the second place too it must be more severe? Who said that?

That is a basic assumption of an a fortiori inference. And that is a foundational assumption of an a fortiori inference. If the property of the injured party is more severe than the public domain with regard to the damager tooth-and-foot, who said that the property of the injured party is more severe than the public domain also regarding the damager of horn? Maybe there it flips. You can throw out every a fortiori inference like that if you make such claims.

What do you mean? Regarding private property…

Regarding private property it works. I’m talking in rows. Ordinary wind, unusual wind — you also have to protect what belongs to you in your own yard. Yes? Now in the public domain, the public don’t have to guard themselves from your private ox; you have to guard your private ox. What I’m trying to say is, who said that if something is more severe in the public domain, on private property who said it also has to be more severe on private property? You are raising a refutation. But the Talmudic text doesn’t raise a refutation; the Talmudic text brings these two formulations. If the Talmudic text had brought your refutation — and by the way this refutation may perhaps arise in Tosafot later — then we’d discuss it as a refutation. But the Talmudic text doesn’t mention any refutation. For it, this is a perfectly fine a fortiori table; no additional datum is needed. And from here one comes out for Rabbi Tarfon and half for the Sages. Not because there is some hidden refutation that was not said; at least there is no hint of that in the Talmudic text or the Mishnah. “It is enough for what is derived from the law to be like that from which it is derived” means not more than that — at least half and not more than that?

No, yes — exactly not more than that. So at least half on the side of the a fortiori inference, and “it is enough” says okay, but if you don’t have proof for more, don’t go beyond what is enough.

But didn’t “it is enough” propose something like the minimum of what is in the table or something?

No, it doesn’t seem that way. Why not? You know what “it is enough” is? At least the way it’s usually understood — and it also makes good sense — “it is enough” is perfectly reasonable. The whole law here is very reasonable. Before I get to the a fortiori inference. First of all there’s a law here that tooth-and-foot are exempt. Why is he exempt? Why isn’t he liable for something? There that’s your domain to walk in. Because — because — because that’s the way of animals to walk, and you put your things there, so the responsibility is on you. Okay? We need the animals to walk, so the Torah said: since we need that, you should guard your food from the animals. And therefore — but on the property of the injured party, I don’t have to guard against someone else’s animals coming to me. It’s like Shmuel said earlier. So you’re really telling me that there is a refutation here. But the Talmudic text doesn’t bring a refutation to the second formulation of the a fortiori inference. On the side of the refutation — the other side — there is a reasoning in the matter, but the Talmudic text doesn’t present it as a refutation. Otherwise the Sages should have attacked Rabbi Tarfon and said to him: there is a refutation to this second a fortiori inference you are bringing us. They don’t say that. They say the a fortiori inference is perfectly fine from their point of view; only on it too we’ll say “it is enough.”

Maybe “it is enough” is a formal legal rule, not a substantive one, but in the background stands the substantive discussion.

That’s already — I don’t know — philosophy of law. Fine, I’m not responsible for that, and you’re not responsible for that. Okay. Fine, so that is the course of the Mishnah, and what comes out is this. Only in this case and in the parallel in Niddah does the Talmudic text or the Mishnah try to rotate the a fortiori inference. It’s very understandable why that appears only here, because here there really is some asymmetry between the direction of the rows and the direction of the columns. So why this attempt appears here is totally clear. But for some reason it turns out that in the end both sides do not agree to this attempt. Both sides say there is no difference between the different directions of the a fortiori inference. If it’s one, it’s one; if it’s half, it’s half. Meaning even in this case, in the end, there is no difference between the directions of the a fortiori inference. So we need to understand that too.

But Rabbi Tarfon does say there is a difference between the two directions.

No, no, it seems not. From the structure of the Mishnah it seems not. Because the Mishnah starts with this a fortiori inference. Rabbi Tarfon, in order to justify his determination that it is one, starts specifically with this a fortiori inference. And then when they tell him “it is enough,” he redirects the a fortiori inference. Meaning, from his point of view that one too led to one. So at least the simple reading of the Mishnah, although there are medieval authorities from whom it appears — meaning — that this is only a methodological presentation: he tried to propose it, the Sages answered, and rightly so he accepted the answer and then attacked from the other side, and that is actually his a fortiori inference. Most of the medieval authorities do not learn that way, and the simple reading of the Mishnah also does not seem that way. So that is regarding the two sides of the a fortiori inference. Let’s see — you know what, here I’ll start from here instead of Tosafot. Fine. So how do we actually analyze such a thing? One thing is clear: in a case of this kind, where we have three values — zero, half, or one — in the hidden cell too we need to check three values and not two. We now actually need to compare three possible fillings: either a filling of zero, or a filling of half, or a filling of one. Right? If we follow the path I laid out until now, then basically we have this table. And these diagrams. What?

What? Sorry, one second. You’re saying if from the outset you’re willing to accept three values and check which of them is better. So suppose, you say, this has one that is better than that. In other words, let’s put it this way: how much better than that do you want to say? Suppose — no no. Suppose, suppose that from this model it came out that if we divide the points, then zero will get four and half will get three and one will get three points on the “goodness” scale. Fine? So then you’ll say it’s zero, even though it makes more sense that it’s half in such a case. Meaning, if you — suppose — suppose you assign a score to what fits better here. I don’t know if you assign scores, but suppose the criterion of preference also, you say, has to be not binary but something more complex: by how much is this preferable to that?

No, no, it’s not mathematics. It also depends on the question. But I’m saying suppose I get, say, a score of who fits better here. Then I’ll come to the conclusion that zero fits here four.

How will you get there? You know, until now what we’ve done is compare models. Either this model is better or it’s worse. Zero versus half — which is better? If it’s simpler, then it’s better. If not, then not. So far we don’t even have any tool that says zero is better than half and better than one. It will say that it is the correct filling.

Not necessarily.

Why?

Because although maybe it didn’t happen here, but theoretically suppose I have — again, I’m saying theoretically — suppose they tell you with a die that there is a forty percent chance that zero gets zero, a thirty percent chance that you get half, and a thirty percent chance that you get one. Will you go by the expected value of that? The expected value of that is zero?

Of course not. No, I’m talking about expectation; I’m not talking here about expectation. Why is expectation the criterion? The criterion is which filling is more sensible, simpler, among the three. I’m not comparing one against the other two; I compare each one against each of the others. Again, maybe one could build on this a more complicated model, but I’m going with the simple one. So if you get to a situation where every filling has some kind of advantage, then do you need to say zero?

Apparently yes. But I won’t say zero; I’ll say it remains open. But “the burden of proof rests on the one seeking to extract from another,” and that will be zero. Then you’ll say zero and not half and not one. Obviously. But that is also true with zero and one — if there is an advantage to zero over one. So here what was true earlier about one is now also true about half. Good. So how do we check that? Let’s begin with the first method, the obvious one. So I say let’s start with filling in one. Okay? In filling in one the diagram is roughly on A and B. So in filling in one we essentially have a diagram like this. Right? That’s in filling in one because the values here are higher than the values here in all the rows. Okay? And then we’re usually accustomed to doing here alpha and two alpha, and that’s the model of filling in one. What happens with filling in zero? Wait — alpha and two alpha? Why not?

Wait, how are you dividing A and B?

Ah — how to divide A and B is an interesting question; I’ll get to it in a moment. But I’m doing this in stages. Not to draw?

To draw — I’m done drawing. The question is how to build the model for A and B. Excellent question; I’ll get to it. But for now I’m proceeding as though this isn’t a different problem from what we’ve seen until now. Here this is what comes out. Now if the filling is zero, then there are no constraints here. In filling zero, then we basically have independence, right? No connection — the columns are not dependent. And if we have half, then it’s the same diagram. Right? That’s also for filling half and also for filling one. And this is for filling zero. Okay, which is preferable? Obviously this is alpha and beta, so what is preferable is this. It is preferable both in dimension and in connectedness, right? That’s a strong preference.

But there’s no preference between one and half?

Right. Between one and half we have no preference.

Right, and that’s connected to what he said earlier, that this only comes to teach me liability — that the half is only what he wanted to say earlier, to separate it into two stages, that the a fortiori inference really only teaches me liability.

Right. Now after you’ve gone down to the microscopic level, you can say that. So now how do we explain the opinion of the Sages, why it comes out half? But your model still isn’t complete; now you need to check how A and B fit. Wait — and they fit exactly as though there were a one written here. For the moment I’m assuming that. Meaning, for example, you see — let’s fill it in. A, A contains only B, right? So A has alpha; in this case A has alpha, and B contains A. Filling in one-and-a-half contains both A and B. So that means it has two alpha. Fine? As though there were a one written here and not a half. And the assumption is that in fact some kind of assumption similar to what Ido said earlier: that what I’m discussing here is whether it contains liability or not. How much liability it contains — perhaps that is a unique feature of the public domain or something like that — but I want to know whether it contains liability or doesn’t contain liability. When you rotated the a fortiori inference, did you reverse the diagrams?

What?

No, they won’t be the same diagrams, and I’ll do that in a minute. Not exactly. I’m saying that essentially only zero is what contradicts the a fortiori inference. There is no significance between one and one-and-a-half. Right? Really — exactly. And our conclusion is something very interesting. No, that is exactly the point. If I really assume that in this model things are handled as I handled everything until now, then in fact I get these simple diagrams. Obviously this is the preferred thing and not zero. Now we only need to decide why it’s half and not one. Suppose we’re now talking about the opinion of the Sages. The Sages hold that it is half. So here precisely the principle of “it is enough” enters. If half and one are equivalent fillings — in the case of two, when two fillings are equivalent, that’s a refutation. And you don’t know what to do, zero or one, so it remains open. But here there are two equivalent fillings out of three. Meaning, this one is rejected, and one of these two remains. That’s not a situation of refutation; that’s a situation of “it is enough.” Meaning, we choose the minimal one, as people do — that’s the logic of “it is enough.” We have either one or half; both stand in the same relation vis-à-vis zero, and therefore we say “it is enough,” so half. That’s what the Sages say. And who is “what is derived from the law” and who is “that from which it is derived” here?

What?

Who is “what is derived from the law” and who is “that from which it is derived” here? Because it’s one versus half?

No, you don’t see “what is derived from the law” and “that from which it is derived” when you look at the table. When I do this analysis there is no “what is derived from the law.” It’s as though we use the analysis of “it is enough,” but not really.

No, I’m saying this is the microscopic reflection of what is called the principle of “it is enough” on such a table. This is its mathematical expression, or the expression of the microscopic model. “What is derived from the law” is the question-mark cell, and “that from which it is derived” is the known half. Fine, obviously, but he says in that language here. “What is derived from the law” is basically — you are comparing alpha to two alpha. Everything that alpha knows how to do, A so to speak, two alpha certainly knows how to do. Fine? That is essentially the learning. Okay.

The diagram just says whether there is an a fortiori inference or not, that’s all, nothing else.

What?

In this particular case. But that’s how it comes out; it’s not necessary. In this case it really came out that the diagram does not distinguish between one and half. One and half are in the same status. Obviously both are better than zero. So the conclusion is not relevant to the a fortiori inference at all.

No, it’s always binary, it doesn’t matter.

No no — you’re reaching a conclusion that it’s always so. Who said it’s always so? Slowly. Here that’s how it came out, and that’s correct. I didn’t understand — I understood it differently. I understood that if I make a refutation on the a fortiori inference then it doesn’t matter what will be there. Who spoke about a refutation? Wait. When I say zero, I’m saying there is no a fortiori inference. There is no a fortiori inference here? Yes. Filling in zero in the diagram, or a refutation, is as though there is another column. The whole idea revolves around the question whether you introduced some logical rule that created the half. It is what created the measure. If there is no logic, this whole story is nothing, it interests no one, and the results have no significance. But again, in this specific case, I won’t infer from here the general conclusion that there will never be a difference between one and half. Why? There could be many tables. In this specific case you are right. And therefore I say that what Ido said earlier — to divide the a fortiori inference into two stages — that really is what comes out here. But it just happens to come out in this case. I’m not sure that every such a fortiori inference can always be analyzed in two stages and in fact ignore the quantitative differences. You’re left with the name. In this case that’s really what comes out. That’s all. Since there are no other cases in the Talmud, I have no way even to check it. Meaning, I can raise hypotheses, but I have nothing against which to check them, no other cases. There are only two cases, and both are of this type. There isn’t anything else. Okay, so that’s — that basically the… yes.

So you explained why it is half, because “it is enough” says take the minimum.

Right.

But that doesn’t explain what you wanted at first.

No no, for now I only explained the Sages. Okay? And then what comes out from here? Notice now, what basically comes out from here is that it doesn’t matter at all whether we reverse the a fortiori inference or not. Again, according to the Sages, “it is enough” exists in both cases. Because when I make microscopic analysis, what difference does it make whether I reverse it or not? I’m proposing a model for this table. This is the model and these are the data. I’m not here making this a fortiori inference or that a fortiori inference, exactly as in the symmetrical case. Therefore according to the Sages there is “it is enough” whether you raise this a fortiori inference or raise that one, because on the microscopic plane what you are comparing is this diagram with this diagram. It doesn’t matter from where you look. It’s a mode of looking. If you reverse this diagram, won’t you get the same thing?

Absolutely not. Here, both in half and in one, the left-hand column is more than the right-hand one; it is greater than or equal to the right-hand one. Right. But if you reverse the a fortiori inference and put half, half becomes independent; half becomes worse than one. No — again. I’ll get to the unequal diagram. For the moment the assumption is that topology is checked on columns. In terms of dimension, it won’t matter; number of parameters, it won’t matter. If topology is checked on columns, then you didn’t reverse it.

No, the criterion is always that topology is checked on the columns. That’s the model. I’ll get to that nuance in a minute. I’m just trying to build this in stages.

Yes.

But if we say “it is enough for what is derived from the law to be like that from which it is derived,” that is, horn on the property of the injured party to be like that from which it is derived, namely tooth-and-foot on the property of the injured party — that is, we learned it along the left-hand column — then you would indeed infer that the conclusion is half.

Exactly! And the Sages do not do that. Why not? That’s the question. The answer is because it doesn’t matter at all how you look at it. The microscopic model says that “it is enough” is a statement along a row or a column; it is not about the whole. That is the statement of “it is enough.” That’s what the Sages are saying: not so. It is the passage from cell to cell, a comparison between two cells. And that is exactly what the Sages say — no, not like that. That is exactly what this model explains. Because Rabbi Tarfon is trying to rotate the a fortiori inference, and the Sages, in a kind of autistic way, repeat again: “It is enough for what is derived from the law to be like that from which it is derived.” “It is enough for what is derived from the law to be like that from which it is derived.” You’re stuck on the earlier a fortiori inference. Who is “what is derived from the law” and who is “that from which it is derived”? That’s the question. Explain to me verbally who is “what is derived from the law.” Exactly that’s the question. So obviously something here is not understood. After all, if you come from here the result ought to be one and not half. My answer is that the concept of “it is enough” ultimately rests on this consideration. And in this consideration it doesn’t matter how you formulate it. So here there is no “what is derived from the law,” here there is no “that from which it is derived”; here there is a filling of half, here a filling of one. You take the minimum. You take the minimum. That is “it is enough.” Because the minimum is always what you take. Why are you comparing it at all?

To half and not to one — I compare half to one.

I don’t understand. From the standpoint of the injured party, “it is enough” that he should receive the maximum.

What do you mean? That’s nice. The damager gets special data…

What are you talking about? What are you talking about? “It is enough” is what the Sages said — half and not one. What maximum? What are you talking about? Does it come to strengthen the law or weaken it? Exactly what I just heard a moment ago. That’s true from a verbal standpoint. Who is “what is derived from the law”? That’s exactly the point. What does that expression mean, and who is “that from which it is derived”? What does the expression mean? These are not fillings. If you want to speak on the verbal plane, “that from which it is derived” and “what is derived from the law” are both there, because otherwise you can’t apply “it is enough.” Here there is no half; only here there is half. And why do the Sages refer to both of them even when Rabbi Tarfon is speaking? Exactly, that’s what I’m asking — why? And the answer is this. From their perspective, when you make such an a fortiori inference — sorry — this a fortiori inference is no different from that a fortiori inference, just as in the symmetrical case. Why, in the symmetrical case, when I put a refutation here, did it also refute that a fortiori inference? After all, it really showed that there is a problem in the hierarchy between the rows or between the columns. Why did it refute it? The claim is that behind such an a fortiori inference there really sits a microscopic model; but if so, then it also sits behind this. It is one a fortiori inference, not two. And the claim is that here too, despite the asymmetry, it is one a fortiori inference and not two as it is drawn here. And once this diagram leads you to the result of half, it really doesn’t matter how you formulate it, whether you formulate it this way or that way. The results — the microscopic explanation for this set of laws, these four laws — dictate a result of half. That’s what the Sages say. That is exactly this autism — that it doesn’t matter to them if he rotates the a fortiori inference. Why? Because rotating the a fortiori inference is not changing the argument; it is the very same argument. It’s another formulation of the very same argument. I hear it, but I can’t internalize it. Because it seems that the Sages simply took a position in favor of the damager, to obligate him in the minimum. That’s how they decided. They suddenly got empathy for the injured party…

“It is enough” is on the lower amount. What are you talking about? I’m now offering you an explanation of what the logic is that keeps him at the lowest possible amount. What is preferable? Do the Sages have favoritism toward damagers? Sorry, that’s a result. I’m now showing you a reasonable explanation of the Sages’ way of thinking. Now if it’s reasonable, why assume it’s arbitrary? If it’s not reasonable, then let’s argue about it. Leave the conclusions aside. I think it’s a reasonable explanation. The Sages continue to speak this way; this is exactly what we did with a symmetrical a fortiori inference. In a symmetrical a fortiori inference the claim was that in the simple verbal expression of the a fortiori inference, seemingly there are two arguments here that are completely different. Completely different. And what is refuted — this doesn’t refute that and vice versa. It turns out, after we go down to the levels, to the complexes, the microscopic parameters, that it is actually one argument, not two. One cannot exist without the other. So what, are you reversing the a fortiori inference? Isn’t the topology the same? Won’t you get the same…

I — in just a second I’ll reverse it; I’m getting there. But you…

So — so it can’t be. In a second. What I’m taking us back to is simply that it’s still not clear to me. Meaning, according to the model you built, the least understood thing is tooth-and-foot in the public domain. Meaning, if you say A is more severe than B, where B is alpha and A is two alpha, and little A contains one alpha and little B contains two alpha, then why doesn’t A — that is, if it has one alpha — give half in B? In other words, according to this, tooth-and-foot in the public domain should give half-damage.

So I said: that is the assumption here in this treatment — and in a second I will indeed do this half — for the moment the assumption is this: that the a fortiori inference here basically says, as Ido said earlier, there is some factor here that is sufficient to impose liability. Now why here is it half? I don’t know. Maybe in the public domain there is something that, once there is already liability, lowers it to half. It does not take part in the logic of the matter, but it characterizes the public domain.

A statement that in fact the liability is of a different kind — meaning the liability of horn is because of a parameter of…

It’s ordinary horn liability, but for some reason in the public domain ordinary horn liabilities are half. I don’t know. But it would already be preferable to learn one, so to speak: if I give names to the parameters, I’ll say that horn is a fine and tooth-and-foot is really restitution of the damage, then you can’t learn one from the other, because what works for horn won’t necessarily work…

No, the assumption — the fact is that we do rule here a fine too, so all the same they do make this a fortiori inference. Meaning, the assumption is that even in a fine there is some obligating reason; they don’t just impose fines for no reason. It doesn’t fit with the model the rabbi presented, that it’s as though in two stages — because if it’s a fine, then seemingly from the outset there should be no liability, and only afterward…

That’s what I said, that’s what I said. Therefore even if in this fine it said one, if it’s a fine, then de facto it is already half, even if one is written here. Because in fact it is not required by the core law; by core law one owes nothing, only the fine imposes one, so it only says that it’s a bit more. It doesn’t say exactly how much; the quantity here is hard to quantify.

A technical question: in the microscopic model, does the arrow go from the lenient to the severe or from the severe to the lenient?

No, in terms of the results — the question is what you define, in terms of results or in terms of actions. In terms of results, the higher result — the higher column — is the column that is easier to impose. The standards, the criterion required to impose it, the threshold is lower. Okay? Therefore B, for it alpha is enough; for A two alpha are needed in order to obligate in A. Okay? In actions it’s the opposite. In actions it’s the power of the action. Here it’s what is required to impose the result. We’ll see that in a moment. Okay, now really, as several of you already rightly said, the business isn’t entirely complete. There are still one or two problems here intertwined. One problem is that up to now I haven’t related to half; I related to half as though it were one. There is some kind of liability here, basically somewhat like what Ido suggested earlier. There is some kind of liability here, without getting for now into the question whether it is half or one, at least in the first logical part of the argument. And the second question is that when we now compare rows to columns and make the same diagrams but on rows, it is not certain that the result we get will be the same. Okay? So let’s try to do that. Up to now I tried to offer some explanation for the Sages; afterward we’ll come back and see how far it can really be relied on. But now I move to Rabbi Tarfon. To move to Rabbi Tarfon: Rabbi Tarfon, after all, does not accept “it is enough.” So if he does not accept “it is enough,” that means that when there is half and one, you cannot decide that it is half. The Sages, after all, say half and one are equivalent and both are better than zero. So zero drops out, I have two possibilities left, either half or one, and I take the minimal one — “it is enough for what is derived from the law to be like that from which it is derived.” So they take the half. So according to the Sages it is a resolved problem. But according to Rabbi Tarfon there is no “it is enough.” If there is no “it is enough,” that means the problem is unresolved; half and one are equivalent. So what is the result? Why is there no “it is enough”? He doesn’t say “it is enough” from the one.

What?

No, because beyond one it isn’t even called “it is enough”; everyone agrees on that. There won’t be anything beyond one here. Half — the question is why…

There won’t be. Where will you stop it? After all, even one who learns an a fortiori inference with “it is enough” — where will he stop? When there is half versus one, he has somewhere to stop — he’ll stop at one. When he comes to learn from the a fortiori inference, the maximum is one. So he says exactly: I’m deriving half; the maximum is one. Okay, so he does apply “it is enough,” but “it is enough” is not to this one. He has no “it is enough”; he fills in one here. One is not “it is enough.” One versus two is “it is enough,” but here one is the top. So not “it is enough”?

“It is enough” means less. “It is enough” means not more, not more. “It is enough” is the minimum. Therefore I say it is against the two — but two isn’t an alternative at all, because why get to two? Why stop at two? That’s what I asked you earlier: what does “it is enough for what is derived from the law to be like that from which it is derived” mean? Not to be more than that from which it is derived.

Right, minimum means this and not more than this. So if he stands at one, that’s not more than one; it’s one. He says: he himself says, what is the minimum of the maximum.

No. Where there are two possible outcomes before us, half or one, and I proved that it is more than half or at least half, there I call this principle “it is enough.” “It is enough” tells you: don’t go to one, stay at half, because half is the minimum. Where I don’t have two possible results, there is only one possible result, only one — because where am I going to stop? At two? At ten? At a thousand? No result is on the table at all. That’s not called “it is enough.” Everyone agrees to that. Meaning, there is no more than one here. It’s not only “don’t pay more”; rather, exactly — there is no such option. “It is enough” excludes an option that could otherwise have existed. But I understood that Rabbi Tarfon has “it is enough,” because otherwise he wouldn’t say, “I will derive horn from horn.”

No, no. That’s Tosafot. Look at the first words: “I will not derive horn from horn” — he said that according to their words; but according to his own view, he does not accept “it is enough.”

So you’re saying that it’s as though he just said it?

No, he wants to persuade the Sages. You have “it is enough”? Fine, then I’ll attack you from here. I myself don’t accept “it is enough” at all. But wait — if he attacks them from here, that also contradicts the “it is enough” here, because if he said that this a fortiori inference is such-and-such — as though if I make an a fortiori inference this way and that way, it’s very similar anyway — and then if he makes this move, that also contradicts the conclusion.

That’s what the Sages tell him. They say to him, what do you want? This a fortiori inference is identical to that a fortiori inference. “It is enough,” I say to you, applies here too. Okay? Fine. So really, what do we do here? Here it’s a bit more complicated. Let’s begin with filling in half. I’ll do now — maybe before that, once according to Rabbi Tarfon you can’t decide by columns, let’s try to decide by rows. You determine the big A and B and the…

I’m now making a diagram on the rows, not on the columns. Because in the columns it’s an unresolved problem, meaning, it can’t be decided: half and one are the same thing, right? So according to Rabbi Tarfon the problem remains open. Let’s try — maybe it can be decided from the rows. Fine? So what happens here now? If we have half, then in the expression — right? This is half. So little A and little B are independent, sorry, little A and little B. And I’m now doing it between the rows, right? So they are independent. Which ones are you looking at right now?

What? From top to bottom?

This row against this row. So here there are two halves. This half is greater than zero, but this half is less than one. So they are independent; it’s like one and zero. So here there is… Here it is independent. So if so, what now do we need to fill in? Apparently what would we have done? Alpha and beta — that’s what we always do. But now I want to continue and also check how the half comes out here. Now a more complete model. Fine? Now how the half comes out here is complicated. You already did the half.

What?

The half.

No — how do I now need to explain not only A and B, but also big A and big B, so as to see how here we get half, here zero, here one, and here also half. What? How does this miracle happen? How can the model explain it? So I had two possibilities. One possibility was to assume that if in the action there is only one characteristic out of the two needed in order to apply the liability, that would be half. Fine? For example, if A requires both alpha and beta in order to incur liability, and little A has only alpha, then the liability would be half. And whoever doesn’t even have alpha would be zero. That doesn’t work; I simply checked. What is needed is that A should require two alpha, and little A should have only alpha. It has to be inside the parameter. And that is very reasonable, right? If A requires some parameter at intensity two, and action little A has that parameter, then it succeeds in operating, but at a weaker intensity — it has it only at intensity one. So the result will be half. Now what we need to do is build a solution — and this is no longer simple. If we put alpha here and beta here as we were used to doing, do the calculation yourselves and you’ll see there is no way to find a solution. You won’t succeed in finding a solution that explains why here and here there is half and here there is zero and here there is one. Therefore the solution that comes out is two alpha here and alpha-and-beta here. And in the domains of damager and public, right? One second — there is a microscopic parameter here that is in A, yes? That is in A, not that A should require. Of course. Yes. Little A. Okay, that’s the solution. Now we can check and see that it works. Look — I didn’t find anything simpler. It has to be this. And then it means as follows. When A is — this has two alpha, right? Then big A it doesn’t manage to apply. Why? Because beta is missing. Right? B, by contrast, it does manage to apply at intensity one; it has everything required to apply B. Right? And what about this? This has alpha and beta. Filling 1.5, sorry, half, contains both A and B — then in big A, which requires two alpha and one beta in order to incur liability, it will only be half. Beta it has completely; it has half of the alpha. So the result will be half. And regarding B, which requires two alpha, and this has alpha and beta, that too will be half. Therefore half and half. Again. The claim is the following. I am now saying little A is two alpha, little B is alpha and beta — which of course is no problem to do, right? In principle I would have had to put alpha and beta here, try and see that there is no solution. I can’t produce a solution. Then start complicating it: say this is two alpha, this is alpha and beta. The main thing is that this should be independent. Since this should be independent, you need to define it as two alpha, because here there was alpha and here alpha and beta, so of course there should have been half here. Okay, so this is two alpha and this is alpha and beta. Now I assume that, and let’s see that I succeed. So I say this: if I adopt a solution for big A, that is, requiring — for big A, which is the public domain, yes? — in order to impose liability in the public domain, two alpha and one beta are needed. Let’s see that this works. Little A has alpha with intensity two, but no beta at all. And beta is also required. So it doesn’t succeed in imposing the liability, right? By contrast in B, what is required is only two alpha. And A has two alpha. So it succeeds in imposing B. By contrast, here in horn we have alpha and beta, both at intensity one. So in the public domain, big A, we need two alpha and beta. Beta is no problem, but alpha we have only half. Therefore this is half. Right? Here what is required is two alpha, but we have alpha only at intensity one. Right — we also have beta, but that’s irrelevant; it’s not relevant here. Therefore here too it is half. So this really is a solution that explains all the data in this table. Right? In fact, one should know that if I now wanted to take the table of the columns — just one second — the table of the columns I did earlier and really take into account the actions and the half, this is what would come out. Because the argument is the same argument; it doesn’t matter from which side we look. What explains these four data is this model. It doesn’t matter — the topology here doesn’t change anything. Okay? So it doesn’t really matter that I moved here to rows. If I had done it on the columns, but yes taken into account what is obtained in little A and little B and why here it is half and here it is one, this is what I would have gotten. Because the argument is one argument. Again, if I had done this argument on the columns. Okay? Let’s do that. We have one and half, and zero and half. Okay? In terms of the columns I know that here we have B, here A, and an arrow like this. Right? In filling half. Right, because these — half is less than half and zero is less than half, less than or equal. Okay? So that is the expression. Now I need to fill this in. So what did I do last time? Alpha, two alpha. Right? But if I now checked what you asked for, and rightly so — check what happens with little A and little B and why exactly here it is half and not one — then I would have had to go back through the process; I wouldn’t have found a solution. I would have had to arrive at the conclusion that A…

…and two alpha and beta, and B is two alpha — you see that the arrow exists. Right, two alpha and beta as against two alpha, that explains this model. Then I would find for the actions two alpha and beta. This model is a correct model both for the actions and for the results; it makes no difference at all from where you look, the argument is still one argument. Therefore it isn’t — wait, what are you saying? That if you always take the microscopic parameters in the actions and their results in the domains, then the model won’t change even if you zoom?

Maybe. No matter how you draw it, you arrive at the same letters. Where will the difference indeed be?

In the form of the graph.

Exactly. Fine. The difference will be — meaning, the model won’t change. What will indeed differ is the shape of the graph. And we said that topology also has an influence. Therefore the shape of the graph can indeed have an influence, and therefore here, rotating the perspective can indeed change the result. Now what happens here? Let’s see. Fine? So now I return to the analysis of the rows. Leave the columns aside now, only as a remark. Now let’s go back to the rows. So we did filling half. Fine? Now let’s do filling one. Okay? So in filling one we have A going to B, right? Okay. Now again, in principle we can put alpha here and we are used to putting alpha and two alpha here. And again, I’m already telling you now: that won’t work. When you try to explain the half and everything, it won’t work. What does work? I’ll already give the result. This is alpha; this is alpha and beta. Notice what happened here — something a bit opposite from what we are used to. Notice: the stronger one is here at the tip of the arrow, not here. Why? Because it’s an action, because these are actions. Right. In actions, the stronger the action, the higher the intensity of alpha, the more results it contains, the stronger it is. In results, the weaker it is, the more ones there will be in it, right? Meaning, the direction is different. Therefore the rules of the diagrams here have to be reversed. When we have an arrow like this, we fill it in as though in results there were an arrow like this. Okay? And now we say that on the property of the injured party, that is B, there is alpha, and A is two alpha and beta. Let’s see that this works. We are talking about filling one. Let’s see that this works. So we have for A — it has alpha, right? B requires alpha, so it succeeds in extending B. But A also requires beta, and it has no beta at all, so that is zero. Right? By contrast B — B has alpha and beta. So public domain A requires two alpha and beta, so that is half. But this requires one alpha, therefore that is one, because it has alpha. The fact that it has beta doesn’t matter; it has alpha. Okay? This is filling one. You just have to notice that the order of the diagram rules gets reversed here.

What? It’s just notation — what difference does it make how you draw it?

It doesn’t matter. It only means that if you want simple mathematical rules for how to solve this, that’s all. No, but if all the arrows are reversed you’ll get the same result.

Right, you can mark the arrows here, but then what will be reversed is not the model but the direction of the arrow, and then that matters. So the lower one will be — the tip of the arrow goes toward the lower one; something here has to reverse in that way. Okay. Now regarding zero. But what does that show? When you rotated it to work with results, you didn’t show that it rotated, because the actions remained actions and the results remained results.

Exactly, exactly — and therefore the move is in fact correct. And therefore the idea remains correct all the time. It doesn’t matter from which direction you look; it’s the same argument. The rotation I’m doing here from rows to columns is not an alternative formulation of an a fortiori inference; it is another model for the a fortiori inference. But here, according to this diagram, we see that one and half are not the same thing.

What?

Because one and half are not the same thing — something here…

Okay, one second — right, exactly that is where I’m heading. Okay? Maybe — why don’t you do a real rotation? That is, when you have an a fortiori inference with others, say canopy and money, where those are actions, and betrothal and redemption and things like that, where it is clear that these are results. But here you could relate just as well to public domain and the property of the injured party as actions, and to liability for horn and liability for tooth-and-foot as results.

The logic says not that way, because you ask yourself: will a high index lead to higher results or to lower results? Then that will reverse severity and leniency instead of leniency and severity. Right, right, exactly. You didn’t use this as an a fortiori inference; you simply completed the previous model and gave it either this way or that way — what difference does it make, it comes out the same. What he’s saying, if I understand him, is that canopy is indeed an action, and the result is betrothal. But here, all in all, for an event, a damage event in the world, there are two parameters: what happened and where it happened. Why do you say that what happened — no, it operates differently. There is an asymmetry between them.

There is an asymmetry between the actions and the domains. Because the stronger the actions, the more liable you are; but the stronger the domains, the more exempt you are. That’s your definition.

Fine, but the definition is asymmetrical. However you define it, it is asymmetrical. There is a difference. By the way, the rule-makers speak of what is called an a fortiori inference of places and an a fortiori inference of domains or of places, and people don’t understand what they mean. I think they mean this. When the a fortiori inference is based on the parameter that is the passive parameter, meaning the one that should be high and when that parameter is high there will be more zeros in the column — in this language that is called domains. There we were discussing whether it is easy to apply betrothal — no, easy to apply the status change, exactly. And what is money? The question is what intensity it has so as to make it easy to apply the status change; that was the subject there. Here the question is whether it is easy to create liability or hard to create liability. Right, in places. Fine. So the third thing that comes out here in filling zero — to make it short, filling zero comes out as two disconnected things. Not connected. And the model again is two-dimensional; right now it doesn’t matter exactly how you make it, it’s exactly the same thing. What is the result in the end? Notice: the result is one. Right? Because all are two-dimensional. This is connected, and those two are not connected. Right? So one is the preferable one — that’s Rabbi Tarfon. And then it doesn’t matter from which side you look. The big question is why the Rabbis don’t accept that. Because with the Rabbis we kind of fudged it. Meaning, with the Rabbis I didn’t explain how it fits with the rows and why here it is half and not one. Here I have a few suggestions, but since we have only one case in the Talmud, I have no other places to check it. There’s one more case, but it’s exactly the same thing. In a table like this. Just replace the names, so it’s not — the logic will be exactly the same logic. So I have no way to say. Either the Sages say that one direction is enough for us in order to decide. Rabbi Tarfon says one direction is enough in order to prove that the result is one. And the Sages require that it be correct in both directions. So it’s enough that in the direction of the results it doesn’t work. That’s the simplest possibility. Or indeed, along the lines we said earlier, the Sages are talking about the very fact of liability and not about its intensity. They ignore the difference between half and one. But these are all suggestions that I have no way to check. We don’t have more than two cases in the Talmud, and in both of them it is the same.