Halakhic Positivism, Lesson 5

Table of Contents

• Positivism, deduction, and information-adding inferences
• The hermeneutic principles as rules of non-deductive inference
• A fortiori reasoning, refutation, and the claim that a fortiori is not deduction
• A fortiori of the “included in two hundred is one hundred” type, and the tension with “we do not derive punishments from logical inference”
• Mathematical models and the caution required in moving from logic to life
• Talmudic a fortiori as an argument built from three data points, with an illustration from Bava Kamma
• Asymmetry in the table, two directions of inference, and the rule of “it is enough”
• Refutation as raising doubt, not as proving the opposite
• Reversing an a fortiori argument, and the question why the Talmud almost never does this

Summary

General Overview

The claim is that halakhic / of Jewish law judgment cannot be translated into a deductive logical argument, because deduction does not add information, whereas induction and analogy do add information but are not necessary and therefore are not certain. Despite the criticism that non-deductive inferences are subjective and hard to validate, the position presented here is that halakhic / of Jewish law thinking is highly systematic and can even be formalized, and that it resembles forms of inference used in everyday life, science, and law. What is unique about Jewish law is that it contains an explicit set of inference rules for information-adding inferences, mainly from the logical hermeneutic principles such as a fortiori reasoning and binyan av, together with a system of refutations and combinations among them, even though there is no algorithm that guarantees a certain transition from individual laws to a comprehensive “theory.”

Positivism, deduction, and information-adding inferences

The claim is that deductive inference from the general to the particular does not generate new information, because the conclusion is already contained in the premises. The argument is that inferences that add information, like induction and analogy, are not necessary and therefore cannot be certain, and the same is true of the scientific move from facts to theory. The question that arises is how logical halakhic / of Jewish law thinking really is, and how subjective it may be, given the possibility that different people will draw different analogies without any deductively valid way to settle them. The goal is to show that there is a systematic character to halakhic / of Jewish law thinking, to the point that it may be formalized, while emphasizing that there is nothing uniquely halakhic / of Jewish law about this; it is true in every context of everyday, scientific, or legal inference.

The hermeneutic principles as rules of non-deductive inference

The position is that Jewish law contains a set of inference rules that try to describe non-deductive inferences that add information, especially a fortiori reasoning, binyan av from one verse, and binyan av from two verses, with the possibility of also including two verses that contradict one another, though that is not discussed here. A possibility is presented of combining these rules with one another to create complex inferences, and they are presented as building blocks from which one can produce almost any inference, scientific, legal, or halakhic / of Jewish law. There is also a link to the Hanukkah lecture on preparing the lamps, where the conclusion was that there are no algorithms that lead from the question to the solution or from the laws to a theory, and that the insight resembles a “find” that comes when one’s mind is elsewhere. Still, one can try to show how this “falling into place of reality” occurs within a non-positivist mode of thought.

A fortiori reasoning, refutation, and the claim that a fortiori is not deduction

The opening point is that a fortiori reasoning sounds like the most logical of the hermeneutic principles, and a biblical example is brought: “Behold, the children of Israel have not listened to me; so how then shall Pharaoh listen to me, and I am of uncircumcised lips,” as an a fortiori argument based on a hierarchy between the children of Israel and Pharaoh. The claim is that even if such an inference looks deductive, it depends on the assumption that the hierarchy of likelihood that Pharaoh will listen as against Israel is a given. Any argument can be presented as deduction if one is allowed to choose premises at will, as in the analogy of “a chair with four legs,” which can be presented as deduction only by adding a general premise that was never actually observed. Rabbi Adolph Schwartz’s claim is cited, that a fortiori reasoning is deduction, and it is said that this is mistaken, because an a fortiori argument can be subject to a refutation, whereas a mathematical proof is not subject to refutation, only to the discovery of an error in the proof. Refutation is defined as a counterexample that undermines the inference and shows that the conclusion is not necessary, and therefore the very possibility of refutation proves that the inference is not certain.

A fortiori of the “included in two hundred is one hundred” type, and the tension with “we do not derive punishments from logical inference”

A type of a fortiori argument is brought that appears more deductive, such as “If a man opens a pit, or if a man digs a pit,” where digging includes opening and more besides; therefore, if one is liable for opening, then one is liable for digging “all the more so,” not because of greater severity but because of inclusion. The example is brought from the Kesef Mishneh regarding “one who passes some of his children to Molech, but not all of his children to Molech,” about which it is said that this is an a fortiori argument of the “included in two hundred is one hundred” type, because passing all of one’s children necessarily includes passing some of them. The Maharsha in the second edition on Bava Kamma is cited, explaining that the Mekhilta learns from here that “monetary penalties are not derived from logical inference,” whereas the Babylonian Talmud seems to disagree and hold that in monetary matters punishments may indeed be derived from logical inference. The Maharsha resolves this by saying that an a fortiori argument of the “included in two hundred is one hundred” type is deduction, and therefore there is no concern for error or refutation. A criticism of the Maharsha is presented, based on the example of “one who passes all his children to Molech,” which is not punished even though this is an a fortiori of the “included in two hundred is one hundred” type. The Kesef Mishneh answers that “we do not derive punishments from logical inference” also stems from the possibility that the more severe act warrants a more severe punishment, and if the lesser punishment is applied, the more severe case will wrongly escape with too light a punishment, though the Holy One, blessed be He, will settle accounts with him. This answer is presented as a refutation of the conclusion of the a fortiori argument, not as a challenge to the very claim of greater severity. From here the conclusion is that even an a fortiori argument that looks mathematical is not simple deduction, because life includes considerations and model-based assumptions that accompany the application.

Mathematical models and the caution required in moving from logic to life

The position is that mathematics and logic are consistent and captivating, but the problem arises in the transition from the model to reality, where assumptions become embedded that may fail. The example is brought of vector addition in physics: two forces of five newtons, one northward and one eastward, do not yield ten but rather five square root of two, not because “five plus five equals ten” has broken down, but because the assumption that a straightforward arithmetic model describes the situation is a mistaken physical assumption. From this it is argued that even an a fortiori argument of the “included in two hundred is one hundred” type is not pure logical deduction, and certainly not an a fortiori based on severity, like Pharaoh and the children of Israel.

Talmudic a fortiori as an argument built from three data points, with an illustration from Bava Kamma

It is argued that biblical a fortiori reasoning is built on one datum from which a conclusion is inferred, whereas Talmudic a fortiori reasoning is usually built on three data points. The example from Bava Kamma is brought: tooth and foot in the public domain are exempt, tooth and foot in the injured party’s courtyard are liable, horn in the public domain is liable, and from this it is inferred that horn in the injured party’s courtyard is liable. It is explained that the a fortiori argument derives from two data points a general hierarchical relation, such as that the injured party’s courtyard is “more conducive to liability” than the public domain, and then applies that to the third datum in order to infer the conclusion, similar to the directly assumed hierarchy in the case of Pharaoh and Israel. An alternative formulation is also suggested, deriving the hierarchy between the damaging agents themselves, horn as opposed to tooth and foot, but it is said that the formulation in the Talmud is not simple, and the Mishnah introduces further complexity.

Asymmetry in the table, two directions of inference, and the rule of “it is enough”

The claim is presented that asymmetry may appear in the data table, such as in the case of horn, where the liability in the public domain is only half. Then the two ways of deriving the hierarchy lead to different results regarding horn in the injured party’s courtyard. It is argued that one inference may yield a result of full liability and another may yield half liability, and this is connected to the dispute between Rabbi Tarfon and the Sages over “it is enough for what comes from logical inference to be like the case from which it was derived,” with the Jewish law following the Sages. The claim is that this is not merely a different formulation, but two different inferences that generate different results in an asymmetrical situation, and therefore different refutations may undermine one inference but not the other.

Refutation as raising doubt, not as proving the opposite

Refutation is defined as presenting a counterexample that undermines the generalization underlying the a fortiori argument and leaves a question mark, rather than as proof that the opposite conclusion is correct. It is argued that there is an asymmetry between inference and refutation: the inference has to show that the result is liability, while the refutation only has to show that it is not certain that the liability follows. It is explained that a refutation can undermine the hidden inductive generalization, for example the claim that horn is more severe than tooth and foot in every context, by presenting a context in which the relation is reversed. Then it is no longer clear whether the injured party’s courtyard resembles the original context or the opposing one.

Reversing an a fortiori argument, and the question why the Talmud almost never does this

It is argued that if the two directions of the a fortiori argument are indeed different inferences, one would expect that when a refutation strikes one direction, the Talmud would “reverse” the a fortiori argument and use the other direction. But in practice this almost never happens. It is argued that such a reversal appears mainly in only two places, in Niddah and in Bava Kamma, and in both cases when the table is asymmetrical and the rule of “it is enough” is the focus of the dispute. From this it emerges that the Talmud seems to treat the two directions as two formulations of the same claim in ordinary cases, even though on the face of it they can be seen as different inferences, and the question of how to understand this remains open at the end.

We’ll begin. We’re in the middle of discussing positivism, and I tried to sketch, briefly at least, what the claim is. The claim is that halakhic / of Jewish law judgment cannot be translated into a deductive logical argument. It’s better to move in that direction, because I’m going to write on the board. So halakhic / of Jewish law judgment cannot be translated into a logical deductive rule of argument, and I tried to show different kinds of arguments—deduction, induction, and analogy—and point to the differences between them. The claim was that deduction, moving from the general to the particular, is an argument that doesn’t add information. All human beings are mortal, Socrates is a human being, therefore Socrates is mortal—the conclusion contains no information beyond what was already in the premises. The fact that Socrates is mortal is already included in the premise that all human beings are mortal. Arguments that do add information—that is, analogy and induction—are not necessary arguments. In other words, they are not pure logic. And therefore, for example, the stages in the scientific process, the stages in which we move from facts to theory, scientific induction—those are the stages in which we accumulate information. Those stages are always uncertain; they cannot be certain. Once we have a theory, we can derive various consequences from it deductively. So basically, as I presented things at the beginning of this series, this raises the question of how logical halakhic / of Jewish law thinking really is, how rational it is—that is, to what extent it is not something subjective. There are all sorts of claims against analogies and inductions, that this is really a subjective matter: everyone will make different analogies, and we have no real way to ground them or validate them, unlike deduction. What I want to do starting today—and I think this will take us a bit more than maybe two sessions—is to try nevertheless to show the systematic nature of halakhic / of Jewish law thinking. And the claim is that even though this is not deductive logic, there is completely systematic thinking here. I can even formalize it, that is, turn it into a formal structure, and what I’m going to try to do is show a bit how this whole thing works, while constantly noting that there is actually nothing unique here to halakhic / of Jewish law thinking. This is true of all our thinking in every everyday context, in scientific contexts, in legal contexts, in every context whatsoever. And I’ll try to show how ordinary inference works, as distinct from logical inference. What is unique to Jewish law—aside from the fact that in these classes we try to touch on Judaism—is that it’s good to approach this from the halakhic / of Jewish law side and not from other sides, although I’ll say once again: this is not unique specifically to halakhic / of Jewish law thinking. But what is unique to Jewish law in this matter, in my opinion, is that it got there a bit earlier than other fields did: in Jewish law there is a set of rules of inference that try to describe non-deductive inferences, inferences that add information. I’m talking about at least some of the interpretive principles by which Torah is expounded: a fortiori reasoning, deriving a general principle from one verse, and deriving a general principle from two verses. It seems to me—maybe “two verses that contradict one another” could also be brought in here—but these are the principles that one might call the logical principles. When I take these three principles—an a fortiori argument, deriving a general principle from one verse, deriving a general principle from two verses—and all the various refutations that can be raised against each of these inferences, and all the combinations you can make, because you can combine them with each other, deriving a general principle with a fortiori, two verses, deriving a general principle from two verses with a fortiori, you can combine more and more—the inference can be as complex as you like. These are basically building blocks from which you can generate almost any inference, scientific, legal, and halakhic / of Jewish law. And so it seems to me it would be interesting to follow these rules of inference and try to see how these things work. And within that I’ll also try to show what I spoke about in the last class, which was actually a Hanukkah class about preparing the lamps, and I ended there by saying that you have to prepare the lamps, and in the end the flame rises on its own. That is, we basically don’t have algorithms for getting from the laws to the theory of the laws, from the difficulty to the solution. It’s something like what the Talmud says: a lost item comes to a person when his mind is elsewhere. And I’ll try to show here how nevertheless these “findings,” how one can show the falling of these findings upon us, which is really to show non-positivist thinking. Good. I’ll start with the principle that perhaps sounds the most logical, and that is a fortiori reasoning. Thirteen—first of all maybe one sentence about the principles. The thirteen principles of Rabbi Yishmael—we spoke about this once, but just briefly—the thirteen principles of Rabbi Yishmael are actually a relatively late formulation. The period of Rabbi Yishmael and Rabbi Akiva is already after the destruction of the Temple, and the assumption of the medieval authorities (Rishonim) is that these interpretive principles are a law given to Moses at Sinai. And that itself already tells us that this is not just mathematics, because for mathematics I don’t need authorization from Sinai in order to apply mathematical rules. In other words, things that are true in and of themselves do not need some innovation of the Torah to permit their use. So once we accept that this is a law given to Moses at Sinai, that means that what we have here is not trivial. That is, there is something here that we might also have thought not to use, and the Torah says yes, you may use it—or the tradition tells us you may use it. And among these principles, which developed over the generations—and I already spoke about that in the past—in Rabbi Yishmael’s formulation, in the thirteen principles of Rabbi Yishmael, as I said, four of them are principles that one can call logical principles. The others are textual principles, and there is some logic behind those too, but that’s for another time. The four logical principles, which are not tied to the text but to the content—and this is somewhat connected to the recent posts, maybe I’ll bring that in—and it has to do with content, are: a fortiori reasoning, deriving a general principle from one verse, which is analogy, deriving a general principle from two verses, which is the common denominator, and as I said maybe also two verses that contradict one another. But two verses that contradict one another is not exactly—I won’t deal with that; it’s a topic in its own right. Maybe I’ll speak about it; we’ll devote a separate meeting to it. So I want to show these three logical principles. As I said, we’ll start with a fortiori reasoning. It sounds like the most logical principle, so much so that someone named Rabbi Adolf Schwartz, from the rabbinical seminary in Vienna, wrote several books on the principles by which Torah is expounded—a book on each principle. There’s one on a fortiori reasoning, one on verbal analogy, I think also one on general and particular. Two of them were translated into Hebrew; the others are in German. And in the book on a fortiori reasoning he wanted to argue that a fortiori reasoning is deduction. That a fortiori is a necessary argument, basically a logical syllogism. And that’s a mistake; it’s not true. And the clear indication of that is that an a fortiori argument can be refuted. I’ve never heard of a refutation of a mathematical proof. There is no refutation. You can find an error in a proof, fine—if someone made a mistake, you can find the mistake in the proof. But once there is a proof, you won’t find a refutation. There is no refutation of a mathematical proof. Once there is a refutation, that means the inference itself is apparently not necessary, not certain, and it needs to be examined. If there is a refutation, we’ll give it up; if there isn’t, then we’ll have to understand why and adopt it. So a fortiori reasoning cannot be a syllogism; it cannot be deduction. What led him to say such a thing? What? What is a refutation? A refutation is a counterexample. Usually, most refutations are counterexamples. Suppose, I don’t know, you make an a fortiori argument: “If horn damage, unlike tooth and foot damage, where one is exempt in the public domain yet liable in the damaged party’s courtyard—if in the case of horn damage, where one is liable in the public domain, is it not all the more so that one should be liable in the damaged party’s courtyard?” That’s an a fortiori argument in the second chapter of tractate Bava Kamma. And then you say: “What about horn damage, which is lenient in a context of exemption?” You see that horn damage also has a leniency, not only a stringency. There’s a counterexample here showing that you can’t so simply derive that a fortiori argument, okay? So there are refutations to a fortiori arguments, there are refutations to deriving a general principle from one verse, there are refutations to deriving a general principle from two verses—we’ll see all of this. I’m putting the refutations into the pool, meaning: we basically have three ways of inferring, and for each of them there are several refutations, and those are the six or eight or however many basic building blocks there are. And afterward there are all the combinations; you can combine them with each other and see what comes out. We’ll do that, so I’ll show it. So I’m starting with a fortiori reasoning. The simple a fortiori argument is sometimes brought by the Sages in midrash, where they say there are ten a fortiori arguments in the heavier and lighter cases that appear in Scripture itself. One of them, for example: “Behold, the children of Israel did not listen to me; so how shall Pharaoh listen to me, and I am of uncircumcised lips?” In other words, there is some a fortiori argument here saying that if the children of Israel did not want to listen to Moses our teacher, and if the children of Israel do not listen to my voice, then will Pharaoh listen to my voice? All the more so, he will not listen. In this case, maybe it is more “stringent and lenient” than “lenient and stringent,” but it doesn’t matter; it’s the same logic. And what does that really assume in the background? It basically says that it is clear a priori that the chance that Pharaoh will listen to me is lower than the chance that the children of Israel will listen to me. That is clear a priori. Now, if the children of Israel do not listen to me—that is given—and since it is clear that Pharaoh will listen to me even less, then clearly Pharaoh too will not listen to me. Okay? Now, this sounds a bit like a deductive inference, like a necessary logical inference, right? But it isn’t. At least not entirely. Why is it not entirely? Because the assumption that Pharaoh listens less to Moses than the people of Israel do is an assumption that you have assumed. Anything I can present as deduction, if I’m allowed to choose the premises as I please. So let’s do an analogy. For example: this chair has four legs; that too is a chair, therefore it too has four legs. That is an analogy, not a deduction, right? I’m learning from one chair to another, not from the general to the particular. I can immediately present it as a deduction: this chair has four legs, all chairs have four legs, that thing is a chair, therefore it has four legs. Why is that not deduction? Because the premise that all chairs have four legs is my premise. I didn’t see that; that’s my conclusion. Okay? Once I’m allowed to add conclusions, I can present any argument in deductive form. That’s no great wisdom. The question is what the really solid premises are from which I start, without all the intermediate steps of inference. Okay? So this basically means that we inserted here some assumption that Pharaoh will listen less to Moses than the people of Israel will. Once you have that assumption, then it is a deduction, obviously. If the people of Israel don’t listen, and Pharaoh is even less likely to listen than the people of Israel, then clearly Pharaoh too won’t listen. Again, when I spoke about likelihood, then again it’s not exactly clear, but the probability that he’ll listen is lower. The chance—the chance, yes. It doesn’t matter; I wanted to present it in a form of certainty, not probability, just for simplicity. It’s not important. But there are a fortiori arguments—and I once spoke about this too—that are better candidates for being deductions. An a fortiori argument called “included in two hundred is one hundred,” and I also spoke about that already. For example, the a fortiori argument brought in the Mekhilta or in Tosafot at the beginning of Bava Kamma: “If a man opens a pit” or “if a man digs a pit.” So the Talmud says: if one is liable for opening, then for digging all the more so. Why? Because when a person digs a pit—he digs the pit in the public domain, we’re talking about damage caused by a pit in the public domain—if you dig a pit in the public domain, then in particular you have also opened it, because you removed the upper part of the pit. Opening an existing pit is an act included within digging a pit, right? Digging the pit means creating the pit and also opening its upper part; I basically did everything. So it includes within it the component of opening an existing pit. So if I am liable for opening an existing pit, then certainly I am liable for digging. Why am I liable? Not because digging is more severe than opening, but because digging includes opening plus something else. So at the very least I should be liable as one who opened it. In other words, I’m not liable as a digger; I’m liable as one who opened it. The fact that I did something additional should not exempt me. Another example: in one of the early classes—maybe Rabbi Steinberg was here, I remember I had an email exchange with him afterward about this matter—I brought the Kesef Mishneh. In one of the first classes I spoke about these things, and I brought that Kesef Mishneh explaining, regarding passing one’s children through to Molekh, that the Talmud says: “from among his children to Molekh”—but not all his children to Molekh. So the Kesef Mishneh asks there: what do you mean? It’s an a fortiori argument. If “from among his children” counts as passing them through, then “all his children” certainly counts as passing them through. And more than it being an a fortiori argument, it is one of “included in two hundred is one hundred,” because when you passed all your children through to Molekh, then in particular you also passed some of them through, because you also passed the others. So you should at least be liable on the basis of “passing some of his children through,” not on the basis of “passing all of his children through.” In other words, an a fortiori argument of this sort is not one in which A is more severe than B, but rather one in which A includes B plus something else. Now between Pharaoh and the people of Israel, that is not such an a fortiori argument, right? “If the children of Israel did not listen to me, how will Pharaoh listen to me?”—that’s not that kind of a fortiori argument. Here it really is a matter of greater stringency, meaning Pharaoh is less likely to listen to me than the people of Israel are. So that’s the ordinary a fortiori argument of stringency. But the a fortiori argument of the pit, or of “passing from among his children through to Molekh,” is an a fortiori that seemingly is pure deduction. That is, you cannot say: one who passes some of his children through is liable, but one who passes all his children through is exempt, because one who passes all his children through is in particular also one who passes some of his children through. And with Pharaoh there is also the well-known difficulty of shortness of spirit. In hard labor. What? And in Pharaoh’s a fortiori argument. Yes, that the children of Israel had shortness of spirit, so there was a reason why they did not listen, right. And also the digging—sorry, Rabbi, this is the second time you’re saying this, the second time, I’ve been restraining myself, but okay, I want to give you a simple example. Okay. I dig in the public domain, everyone sees the dirt coming out, everyone knows there’s a pit here. Okay. In a pit that has metal panels closed over it, no one knows what’s underneath, and one day someone comes at night and removes it, no one saw the dirt coming out, uncovering a hole is much more severe than digging a hole. So now let’s ask what happens if I dug the pit, removed all the dirt from the public domain—out of the public domain? No, no, that was already two years ago. I dug the pit, removed all the dirt from the public domain, and left the pit there. That is certainly more severe than—or includes within it—the opening. I didn’t understand, say it again. I dug a pit, removed the dirt entirely from the public do—after two years someone fell into the pit. No. Why? Was he here two years ago? He’s an immigrant; he arrived here two months ago. He didn’t see anything. You’re giving an example in which it might be so, but that example concerns uncovering; you are surprising me with a new reality. No, I understand, but I’m explaining again: that’s not correct. Because when digging you can’t—yes, I am surprising you. The fact that if I dug it two years ago you’ll be surprised no less than by a pit like that. No, I know this street as covered. The first time I’m in a place I look at it; when I’m here every day—no, but I can also cover it and then uncover it. What difference does that make? So that’s uncovering. Then yes, for uncovering you’re liable, but not for digging. No, but digging and uncovering—I can separate them and I can do them together; what difference does it make? Do them together—that’s what I’m telling you, that new people here are the surprised ones. Okay, fine, I don’t think you’re right, but it doesn’t matter because it’s just an example. The principle of “included in two hundred is one hundred” does exist. “From among his children” and “all his children” is an a fortiori argument of “included in two hundred is one hundred,” so let’s make do with that. In any case, the point is that with such a fortiori arguments, seemingly this really is deduction. That is, because we saw when I spoke about deduction that I said the power of deduction comes from the fact that it does not innovate anything, that the conclusion is basically included within the premise, right? With the hot-air balloon and all those things. Now here too it’s the same. The conclusion that one who passes all his children through to Molekh is liable is included in the premise that one who passes some of his children through to Molekh is liable. Because passing all his children through is a particular case of passing some of his children through. You can pass some of your children through, and you can pass all your children through—which is also some of your children plus something else. It’s a particular case. So this really is deduction in every sense, right? And therefore there is in fact a Maharsha in the second edition on Bava Kamma, where the Maharsha claims that that Mekhilta which Tosafot brings there on page 2 says that from there we learn that we do not impose monetary penalties through logical derivation. Why did the Torah write both opening and digging? After all, if one is liable for opening, then for digging all the more so. In order to teach you that we do not impose monetary penalties through logical derivation. I thought that was about maternal half-sister: if you’re liable for the father’s sister, the mother’s sister is the same thing, right? “From his father and from his mother” is a particular case of “from his father or from his mother.” Yes, same thing. Yes, there are several examples here. So the Maharsha says—the Maharsha brings there that Tosafot brought the Mekhilta, but in the Talmud on page 49 in Bava Kamma the Talmud derives something else from it. And it sounds like the Babylonian Talmud disputes the Mekhilta, and he claims that indeed one does impose legal sanctions by logical derivation in monetary matters. Now one can understand imposing legal sanctions in monetary matters, because money is not a penalty; money is compensation. Now one can ask what that has to do with imposing legal sanctions by logical derivation. You’re paying damages—pay him. It’s not a punishment. As for punishment, we don’t punish by logical derivation; but if we learned it through an a fortiori argument, well, didn’t you cause damage? Then pay the person whom you damaged; compensate him. But the Maharsha doesn’t explain it that way. The Maharsha—there really is evidence in the Talmud that damages are nevertheless called a punishment for this purpose. The Maharsha claims it is because this is an a fortiori argument of “included in two hundred is one hundred.” The Babylonian Talmud holds that in this kind of a fortiori argument we do impose sanctions by logical derivation, unlike an ordinary a fortiori argument. And why? Because the reason we do not impose sanctions by logical derivation is that maybe you made a mistake, maybe there is a refutation. But an a fortiori argument of “included in two hundred is one hundred” cannot be mistaken; it is logical deduction. Okay? So here there is no concern at all, and certainly we would impose sanctions as well. The practical implication is that we would impose sanctions even not in monetary cases. That in an a fortiori argument of “included in two hundred is one hundred” we would impose sanctions even outside monetary law. And by the way, that’s difficult from the case of one’s sister for this Maharsha, because there it isn’t a punishment, it’s not monetary. In any case, that is what the Maharsha claims. But as I already mentioned earlier, the Maharsha is apparently not right, because—as the Kesef Mishneh brought—the a fortiori argument of “from among his children, but not all his children” is also an a fortiori argument of “included in two hundred is one hundred,” and there not only do we not impose sanctions—we do not impose sanctions. In the final conclusion. Not only is there another source needed in order to punish; there is no such source, and in fact we do not punish. One who passes all his children through to Molekh is not punished, only one who passes some of his children through. Now that is an a fortiori argument against which there is no refutation, as the Maharsha says, so what does that mean? If for some of his children you should be punished, then for all his children certainly you should be punished. According to the Maharsha there should have been no need for any source; clearly you should punish even one who passes all his children through. So the Kesef Mishneh asks this there and answers it. And several other—several answers. One of the answers he brings there, two I think—one of the answers he brings there is that the rule that we do not impose sanctions by logical derivation is because it may be that for the more severe act you are liable to a greater punishment than for the less severe act. And if you give him the punishment for the less severe act, then you have exempted him with too light a punishment. Leave it to the Holy One, blessed be He; He’ll settle the account with him. And that treats deduction like mathematics. We know, for example, that if there is a religious court in which everyone convicts a person, then he is acquitted. As opposed to seventy out of seventy-one. Why? There’s a reason for it, fine, there are explanations and derivations. But seemingly you could say: what, by the same sort of a fortiori argument? You ask again: why don’t we make an a fortiori argument? If twenty-one convicted him—twenty-one against two means he is liable—then twenty-three, why not? Fine, but there we know there is no a fortiori argument from the acquitting side, not from the convicting side. Because when there are two who acquit, then he is liable; when there are zero who acquit, then he is—then he would be exempt. Seemingly that is like “some of his children” and “all his children.” So I’m saying: if you look from the side of the number convicting, but not from the side of the number acquitting. Because from the side of the number acquitting there is a problem: if no one acquits, that means the deliberation there was probably biased. Good. All I’m saying is that even when something looks like mathematical deduction, there can be reasons. Okay, so I completely agree, and that is exactly my claim against the Maharsha. I completely agree. That is, what I’m bringing now is, for example, one of those reasons—the Kesef Mishneh’s point. What? That the punishment for the more severe act should be harsher than the punishment for the less severe one. So clearly he is liable at least to the punishment for the lighter offense also for the heavier offense, but it may be that this is not enough—you need a higher punishment. And because of that, we do not give the lighter punishment, so as not to exempt him with a light punishment, and the Holy One, blessed be He, will settle the account with him. And that itself is a reason that is actually a refutation of the a fortiori argument. This needs to be understood. It’s not just some side consideration. That reasoning is a refutation of the a fortiori argument, because the a fortiori argument says you need to punish, and this reasoning shows that you do not necessarily need to punish. In other words, this itself is reasoning that refutes the a fortiori argument. Yes, meaning in the context of the conclusion you wanted to derive from the lighter case. It does not undermine the fact that the severe case is more severe than the lighter one. A normal refutation usually undermines the idea that the severe case is more severe than the lighter one. Here certainly the severe case is more severe than the lighter one; there is no challenge to that. But the conclusion you want to reach—that if it is more severe, then obviously one should also punish for it—that is what is being refuted. And this exists in the laws of prohibition and permission as well as in monetary law: this person is liable for money. At the end of the day, you can say maybe he owes more. Fine, but that same person is liable. In money we say “included in two hundred is one hundred”; that’s exactly the phrase. Correct. “Included in two hundred is one hundred,” absolutely. Therefore indeed in monetary law it is much more severe. Correct. In any case, the point I want to make is that even an a fortiori argument of “included in two hundred is one hundred” is not simple logical deduction. There are always reasons, like you said before, there are always reflections, and this connects to what I already spoke about more than once: whenever you press a mathematical model onto life—and we spoke about this with quantity and quality also in the session before last—whenever you press a mathematical or logical model onto life, you have to be very careful. Because mathematics and logic look wonderful to us: consistent, valid, certain. But in life it doesn’t work that way. Life is more complicated, and very often the translation from mathematics to life assumes all sorts of assumptions that may be the problem. Not the mathematics. The mathematics is fine. But the move from mathematics to the world or to life can always be problematic. I gave the example of vectors and forces, that it’s not as though this breaks arithmetic, if you remember. I say: there is a force of five newtons northward and a force of five newtons eastward. What is the total force, the resultant force acting on the body? Five plus five is ten, right? But no: when there are two such forces, the result is five times the square root of two, seven point something. So did we break the law that five plus five equals ten? No. We broke the assumption that the law five plus five equals ten correctly describes this reality. In other words, that assumption is an assumption in physics, not in mathematics: that this mathematical model is suitable for describing this reality. Okay? Therefore the mathematical model is wonderful, always good, and that is its danger, because it captivates us. In other words, reality is a model for mathematics, as mathematicians say—but never mind, in the language ordinary people use, they say it the other way around. And therefore in the transition from theory to model there may be some problem. And so even an a fortiori argument of “included in two hundred is one hundred,” it is not correct to say about it that it is logical deduction, and all the more so not an a fortiori argument like Pharaoh and the people of Israel. Now all the a fortiori arguments I’ve described until now are a fortiori arguments that appear in Scripture, and what characterizes them is that this is an a fortiori argument that starts from one premise and infers a conclusion, yes? The people of Israel do not listen to me, conclusion: Pharaoh too will not listen to me. There is one premise, no more than that, one datum at the base of the a fortiori argument. A Talmudic a fortiori argument, or at least most Talmudic a fortiori arguments, is an a fortiori argument built on three facts, not one. And I gave the example earlier of horn damage in the damaged party’s courtyard. That is, if tooth and foot damage in the public domain are exempt, and in the damaged party’s courtyard are liable, then horn damage, where in the public domain one is liable—is it not all the more so that in the damaged party’s courtyard one should be liable? So such an a fortiori argument assumes three data points, not one. One datum is that tooth and foot damage in the public domain are exempt. A second datum is that tooth and foot damage in the damaged party’s courtyard are liable. A third datum is that horn damage in the public domain is liable. And from that I infer that horn damage in the damaged party’s courtyard is certainly liable. In other words, here I infer—I’ll show it on the board, so the details don’t matter at the moment—the conclusion emerges from three data points, not from one. So this is a different kind of a fortiori argument from the scriptural a fortiori arguments. So let’s mark it on the board, because I’ll want to use it. But with Pharaoh there are two, no? Why one? The people of Israel do not listen to me, conclusion: Pharaoh too will not listen to me. But he also says Pharaoh listens less than the people of Israel. That I’m adding already; that’s me adding. The data given are one. The given data are one. Okay, a methodological given. Look at what we have in the a fortiori argument of Bava Kamma. It will accompany us further on, so pay attention. Suppose I’m speaking about horn damage and about tooth and foot damage, and I’m speaking about the public domain and the damaged party’s courtyard. All right? Those are domains and types of damage. Fine? Now, I have three data points. Tooth and foot in the public domain are exempt—I’ll mark that as zero. Tooth and foot in the damaged party’s courtyard are liable. Horn damage in the public domain is liable. And I want to know what the law is here. Okay? Now I say: there are three data points, and the data basically say that tooth and foot in the damaged party’s courtyard are liable, in the public domain exempt; horn damage is liable in the public domain. From this I infer—how does the inference go? So I say this: “If tooth and foot, for which one is exempt in the public domain, are liable in the damaged party’s courtyard,” what does that mean? Because it is more severe, because it is more severe: the damaged party’s courtyard is easier for imposing liability than the public domain. “Severe” and “lenient” are a bit complicated here, but it is easier to impose liability in the damaged party’s courtyard than in the public domain. So if that is so, horn damage, for which one is liable even in the public domain—“Behold, the children of Israel did not listen to me; so how shall Pharaoh listen to me,” right? In other words, if for horn damage one is liable even in the public domain, which is the harder case in which to impose liability, then in the damaged party’s courtyard one certainly should be liable. Okay? That’s how the a fortiori argument is built. So basically what this a fortiori argument does is take three data points, take two of them—these two—and derive from them an intermediate conclusion, which often people don’t say aloud, but it is sitting there in the background: that the damaged party’s courtyard is an easier place for imposing liability than the public domain. And now I erase this row. Now I have a normal scriptural a fortiori argument, right? Because what does the scriptural a fortiori argument say? If in the damaged party’s courtyard it is easier to impose liability than in the public domain, and in the public domain horn damage is liable, then “how shall Pharaoh listen to me?”—that is to say, horn damage—then clearly horn damage will also be liable in the damaged party’s courtyard. In other words—just a second—in other words, what I’m doing here is taking two out of the three data points, extracting from those two the principle that in Pharaoh and the people of Israel was known to me a priori, and from there I raise it into something like a scriptural a fortiori argument. Right? I have one datum with this kind of hierarchical relation—which is more severe than which—and I infer the conclusion. So basically behind the Talmudic a fortiori argument sits the same logic as the scriptural a fortiori argument, but there is an added step here. The added step is to go from these two data points to the general hierarchical relation. Once I have the general hierarchical relation, I apply it to this datum and infer the conclusion, and therefore it is clear that the result here is one. Okay? And that is exactly what the Torah laid out when it says “Behold, the children of Israel did not listen to me”—that is one datum, the children of Israel are not listening to me—“so how shall Pharaoh listen to me?” So Pharaoh certainly won’t listen to me. Why? Because the assumption—the hierarchy between Pharaoh and the people of Israel—is given to us directly. I do not infer it from two data points. I do not make an inductive step in the background; I simply begin with that assumption and then infer the conclusion. Here in this case I’m doing the same thing, only here there is another induction in the middle that goes from these examples and creates a general relation between the damaged party’s courtyard and the public domain, and then I apply it here. Why must one say it that way? You could say the assumption is: if tooth and foot are exempt in the public domain and liable in the damaged party’s courtyard, that means horn damage is more severe than tooth and foot damage, and then you can say okay, now I know that horn damage is more severe, so if tooth and foot are liable in—the damaged party’s domain, all the more so horn damage will be liable. Okay, I fully agree. That’s my next step, and in fact let’s now look whether I explained well how this a fortiori inference is constructed. So basically what we find here is this relation, right? Namely that the damaged party’s courtyard is stronger, or easier for imposing liability, than the public domain. What is larger here as compared to the numerical value here. The numerical value here is larger than here. In explanatory language that means it is easier to impose liability. Okay. But in fact the comment is correct. I can also formulate it differently. I can also say that I’ll take these two data points and learn from them that horn damage is easier to impose liability for, or that horn damage is a more problematic kind of damage than tooth and foot damage. Then I say: if tooth and foot, which are the less problematic type of damage, are liable in the damaged party’s courtyard, then horn damage, which is the more problematic kind of damage, will certainly be liable in the damaged party’s courtyard. Right? But that’s not how it is formulated in the Talmud. What? But that’s not how it’s formulated in the—maybe. The Talmud’s wording is not entirely simple in terms of what it means. It’s not entirely simple what it means, and the proof is that the Mishnah there—it’s actually a Mishnah—the Mishnah there, in the dispute of the Sages, which I didn’t bring here from Bava Kamma, the Mishnah itself reverses the direction. Because actually I cheated here a little, because what is really written here is half. Horn damage in the damaged party’s courtyard is liable for half. Okay? Why does that matter? Because now look. Could it be only half? No, it depends, depends on which picture you go with. Now look. If we—top to bottom, horn damage in the public domain is half? Horn damage in the public domain is liable for half; you didn’t say in the damaged party’s courtyard. No, in the public domain. Now so what do we really have here—look how it works. If this is the table, you suddenly see that there’s a difference between the two inferences. Why? Because if I go by the rows, then I say this: from this row it emerges that the damaged party’s courtyard is more severe than the public domain, right? So I apply that also to this row. Then here too the damaged party’s courtyard is more severe than the public domain. What will come out here? Half. Half, right? At least half and above—but “it is enough for what is derived by logic to be like the source case,” and that is the issue there in the Mishnah in Bava Kamma, so the result is half. Okay? But if I go by the columns, then what happens in the columns? Horn damage is more severe than tooth and foot damage—we see that from the public domain, right? Now here too horn damage should be more severe than tooth and foot damage. If tooth and foot is one, then horn damage should be one. No, more than one. At least one. At least one. And “it is enough for what is derived by logic to be like the source case.” So once there is “at least,” we take the minimum, because we have no proof for anything more than that. So this basically means that—You turned reasoning into mathematics. And originally, that first row that you described as something added in the Talmud as compared to Scripture is a row that comes to demonstrate something that it is very natural to think: that if a person was damaged in the public domain, the degree of his own responsibility—here, here it is very natural to think this way: if you damage intentionally, you are more liable than with tooth and foot, which is the normal course. What do you mean tooth and foot? I walk in my normal way. The Talmud itself brings those rationales; you don’t need to guess them. But the Talmud makes an a fortiori argument. We’ll still get to the rationales in the background. Look. So what I really want to say is this. I’ll mark this not in blue but in red. And this inference gives me the result one, right? This inference that says this is greater than that—sorry, this inference gives me the result one. And this inference gives me the result half, right? This inference that says the damaged party’s courtyard is more severe than the public domain. Okay? What does that actually mean? It means that if I translate the Talmudic a fortiori argument into a scriptural a fortiori argument, then it’s actually not so clear what the correct translation is. There are two translations. Now one might have thought that these two translations are two different formulations of the same argument. But as we just saw—the zero gives you half after all; if the zero gives you one then the half certainly gives you—By law it would have been one. Why? You reversed it. You did—I say that if the one—I don’t know if zero gives one. I say that one is more severe than zero; I don’t know by how much and I don’t know how, that’s what I can know. So here too it has to be more severe than that. This is half, so it is more severe than half. How much? Epsilon. But the first one doesn’t only give you the “more severe than,” it quantifies for you by how much it is more severe. You can’t take quantities. In an a fortiori argument you never take quantities. And you should have put one and a half here. Nobody ever put one and a half here. You take the minimum. Why the minimum? One and a half—make it quantitative: from zero to one, from half to one and a half. The minimum means “more severe than.” That is the Talmud’s assumption. Try to think about this in everyday life; maybe I’ll get to that later. We make considerations like this all the time in everyday life too. Ninety. Now I ask what the results in physics will be. The one who got seventy got eighty in physics. Let’s ask what the other one will get in physics. So if I make the a fortiori argument, I can draw the same table here. If I make the a fortiori argument, I say he’ll probably get more than eighty. I don’t think it’s correct to say he’ll get a hundred. The gap doesn’t have to be identical. But I can indeed estimate—again, not for sure, but estimate—that if he is better in mathematics, he is probably also better in physics. By how much? Now I can’t know, so I say the minimum. That actually proves—you could seemingly, according to the logic, arrive at the conclusion that he’ll get less than ninety. Because if the other one got seventy and you see that the physics test is probably a bit easier, then if he got ninety here he surely won’t get less than ninety in physics. So again, you reversed the a fortiori argument. That is exactly what you did. You took the blue argument and arrived at ninety. And I took the red argument and got to “more than eighty.” That is exactly what you did here. That’s why I say these things appear everywhere, not only in the halakhic / of Jewish law context; it’s also in life. We take the psychometric exam—this whole process of admitting students to university on the basis of the psychometric exam is built on just such an a fortiori argument, right? Because what does it say? I rank everyone according to the psychometric axis—let’s say that’s the tooth and foot—and now I say: whoever was better on the psychometric exam will also be better in literature, better in physics, better in mathematics. It doesn’t always work, but apparently for now we have nothing better. I’m not belittling it. By the way, there is lots of criticism. It’s very easy to criticize; much harder to propose better alternatives. That is why I say we have to get used to this—and I’m noting it because it’s important for us too. Clearly these inferences are not necessary. But on the other hand, the criticism that says that because it is not necessary, it is subjective and worthless—I don’t think that criticism is correct. And I want to dwell on exactly that seam, because that is precisely positivist thinking. It’s a tool. It’s a tool that isn’t certain. Correct. Correct. And to be precise, it’s a tool that I do not disparage. It gives certain possibilities. You have to be careful with it—respect it, but also suspect it. Now there’s the pocket a fortiori argument. What? The pocket a fortiori argument. What? That you are forbidden to put your hand in my pocket and allowed to put your hand in yours. Right. My own pocket—of course I’m allowed. All the more so I’m allowed. That’s like, you know, obligating a doorpost in fringes. “If a four-cornered garment, which is exempt from mezuzah, is obligated in fringes, then a doorpost, which is obligated in mezuzah, is it not all the more so that it should be obligated in fringes?” By the way, there are such a fortiori arguments in the Talmud too. Regarding the blessing over Torah and the blessing after meals, before and after, there really is such an a fortiori argument. One of the problems with such an a fortiori argument is that it is based on two data points, not on three. Look. There he already notes this. It’s like the a fortiori argument between a woman and a man: if a woman is obligated in something, then a man certainly is, since he is obligated in more commandments—regarding shaving and so on. You see, like Shapira. No, nothing is necessary, but here it sounds absurd because it is not necessary. For clearly there is a difference between a man and a woman. No, clearly. I’m saying no a fortiori argument is necessary. That’s not the criticism. But here the criticism was that you arrive at absurdities, like a doorpost not being obligated in fringes. No, that’s humor, and we have to understand why. What really is wrong with this a fortiori argument? Seemingly this is an a fortiori argument like those that appear in the Talmud. What’s wrong with this a fortiori argument? One of the problematic things in this a fortiori argument is that there are two data points here. Just look at these two data points. Suppose this zero weren’t written here. Then I’m basically saying I can now put zero here and the result will be one. By the same token, I can put zero here and the result here will be one. It becomes a question of which direction to take the a fortiori argument. Sometimes the Talmud in tractate Berakhot wrestles with this there, with the blessing after meals before and after, and therefore this is a problem. You need three data points for an a fortiori argument. Two aren’t enough. Once there are three data points, that also makes it more—I’ll explain later also why the third datum is important. It’s not just so that we can’t reverse it; it is an indication of something much more substantial: that there is some linkage between these axes. If it is on the diagonal, then these two axes may not be speaking to each other at all. Good, I’ll get to that later. Yes. In the Talmud there, just as a clarification question: when the Talmud says “is it not right that he should be liable,” what does it mean—half or one? That is the dispute between Rabbi Tarfon and the Sages. The question is whether we say “it is enough for what is derived by logic to be like the source case” or not. The Rabbis say half because they say that principle; Rabbi Tarfon does not accept it, and therefore he says one. And there is something concealed here, and it is orthogonal to that question. Is that principle pointing to one or to half? No—the claim is that the “it is enough” being discussed there is in a situation where the inference by rows gives one, the inference by columns gives half, and then basically you have two possibilities. Either the conclusion is one or the conclusion is half. And since you have no proof, take half. That is the principle there. The ordinary “it is enough” is where both places come out one, and then I say why don’t we infer that it should be two, as you pointed out. In other words, if zero gives me one—And then basically you have two possibilities: either the conclusion is one, or the conclusion is half, and since you have no proof, take half. That is the principle. The ordinary “it is enough” is where in both places it comes out one, and then he says: why don’t we infer two? As you pointed out, if zero gives me one, then from one should I infer two here? No—that’s where the ordinary “it is enough” applies. About that “it is enough,” it seems to me Rabbi Tarfon does not disagree. Rabbi Tarfon disagrees about “it is enough” only in the sense that when there are two rows, two directions of a fortiori argument, that give different results. By the way, there are only two such cases in the Talmud, and in both of them there is a dispute about “it is enough.” One is in tractate Niddah and one is in Bava Kamma, and in both of them there really is a dispute about that principle. What? Rabbi Tarfon disagrees? No, no. In any case, what is the conclusion that emerges from here? The conclusion that emerges from here is that although on the face of it one might have thought that these are two different formulations of the same argument, the same inference—no, they are two different inferences. And the fact is they yield different results: this gives result one and this gives result half. So that means the inference is a different inference, and it is not merely a different way of phrasing the same inference. Let’s see this in another way. Let’s now speak about a refutation, since someone asked earlier what a refutation is. So let’s take—the halakhic / of Jewish law conclusion here, because both exist, is that I can only impose liability for half. That is the “it is enough” according to the Sages, who apply it; according to Rabbi Tarfon, who does not apply it, it is one. Fine? The law follows the Sages. In any case, suppose I have a refutation—I don’t know—about the moon. I found on the moon that there horn damage is exempt and tooth and foot damage are liable. Suppose I found that law somewhere in the Torah. That refutes the a fortiori argument. Now we can understand why it refutes the a fortiori argument. Why? Because the a fortiori argument—let’s talk about the blue a fortiori argument—the blue a fortiori argument basically assumed that horn damage is more severe than tooth and foot damage. And then there is a counterexample that on the moon horn damage is not more severe than tooth and foot damage. So what is the question? The question is whether the damaged party’s courtyard is more like the moon or more like the public domain. And therefore you cannot know whether here it should be liable or exempt. That’s how the refutation works. Basically, this refutation undermines the relation between horn damage and tooth and foot damage. It says: who told you that horn damage is more severe than tooth and foot damage? The induction you are making is not correct. It is not certain that these two data points can teach you that horn damage is more severe than tooth and foot damage in every context. You understand that there is some generalization here. You take two data points that are specific to the public domain, and now I say: ah, from here I learn that horn damage is more severe than tooth and foot damage in every domain whatsoever, and specifically in the damaged party’s courtyard. That is the induction that always lies at the basis of deduction, as we saw many times. Now here I am refuting the induction side. I am basically saying: from here you learned that horn damage is more severe than tooth and foot damage in every area? Not true. Here, see, there is a counterexample. So apparently there are contexts in which yes and contexts in which no. Once there are contexts in which yes and contexts in which no, now I do not know whether the damaged party’s courtyard belongs to this category or that category. So I cannot infer the conclusion. What characterizes a refutation, by the way, is that the refutation does not have to prove that the result is zero; it only has to show that the result is not certainly one. In other words, there is an asymmetry between a refutation and an inference. The inference has to prove that the result is one. The refutation has to show that it is not certain that the result is one. It does not have to—If you prove that the result is zero, then the refutation is not correct? The refutation here reaches the assumption that—Yes, but I’m saying in terms of the data, the refutation does not have to put a zero here. The refutation only has to leave the question mark here. Suppose it also puts the red here. Wait, that’s our next point. Okay? So with a refutation there is a difference between refutation and inference, and that is an important point; you have to remember it. In an inference, its role is to prove something. A refutation does not prove the opposite; it leaves doubt. Yes, exactly. And if there were a proof that there is a zero here, would it still be an a fortiori argument? It wouldn’t be “light to heavy,” it would be “heavy to light,” but it would still be an inference that proves for me what the law is here. As far as I’m concerned that too is an a fortiori form. A refutation means leaving the question mark; that is called a refutation. So if you’re holding it, can you not derive a law from that—is that considered a doubt in monetary law? I don’t believe anyone addressed it that way; I’m not familiar with anyone who treated it as a doubt. Because doubt has two kinds of doubt. There is doubt—you know, like equilibrium in physics. There is neutral equilibrium, meaning you can put a ball here and it won’t roll either right or left. And you can put it on top of a mountain here, and again it won’t move right or left, but that is not stable equilibrium; it is unstable equilibrium. And there is equilibrium like this where here, even if it wanted to, it wouldn’t—anyway, there is a doubt that leaves you in the middle because there is force to this side and force to that side, and there is a doubt that leaves you in the middle because you simply don’t know anything. A doubt of that kind you cannot seize upon. You need to seize when there is a direct monetary stake, meaning when there is some side of the doubt in your favor. In any case, this refutation refutes that blue sign, the relation between horn damage and tooth and foot damage. And now the really interesting question is: what does that do to the red a fortiori argument? Nothing, right? For the red a fortiori argument, what’s the problem? What am I saying? The damaged party’s courtyard is more severe than the public domain, therefore here too the damaged party’s courtyard is more severe than the public domain. What—what do you see here? I see that the moon is not more severe than the public domain. Why should that interest me? Who was talking about the moon? No, but the moon here is more severe than the public domain, so the moon itself sort of breaks the direct ordering relation between the domains. Correct. No, not between the domains. Between these domains, but between these it remains. Maybe it refutes the very claim that there is an ordering among the domains. Why? Why? Here—you have a case where it doesn’t work. What does that have to do with it? It says nothing about my data. In the blue a fortiori argument—we’ll come back to this—but in the blue a fortiori argument the moon directly refutes the assumption that horn damage is more severe than tooth and foot damage. Here it’s something very hypothetical. In other words, you say I could have made the same a fortiori argument over here, and then I would have gotten one, or half, doesn’t matter. That’s what I’m saying. I’m saying that if this reality exists, maybe it shows that one cannot create an ordering from greater to lesser among the domains. A fact is a fact. You’re not going to overturn all the a fortiori arguments in the Talmud because once one a fortiori argument failed somewhere. That’s like refuting an a fortiori argument because of an entirely different datum. What does that have to do with it? There is a relation between these two domains. I didn’t say every domain and every domain. Is there a relation between these two? There is. I don’t care that there is another place where it wouldn’t work. So here you see there is a relation and two different data points. Yes. Isn’t that enough to establish a relation? Why? Look, without that it was enough. So why with that is it not enough? One datum is enough to establish a relation between this and that. So seemingly this refutation refutes the blue a fortiori argument, but does not refute the red a fortiori argument. How do you refute the red a fortiori argument? With a refutation like this. I find another one—I add, I don’t know, say here. Fine? Suppose pit damage in the public domain is liable, and in the damaged party’s courtyard it would be exempt—just for the sake of example. Then of course that would refute the red a fortiori argument, but remain indifferent to the blue one. Okay. Basically, what this means is that these two formulations are not two formulations of the same line of reasoning, but two different arguments, each of which has a different result when there is asymmetry in the table, and a different possibility of refutation: a refutation that refutes one does not refute the other. The big problem is that if you go through the entire Talmud, you won’t find them doing this. I would have expected that when there is an a fortiori argument in the Talmud and a refutation is raised against it, then they would rotate the a fortiori argument. In other words, suppose there is such an a fortiori argument, fine, someone brings a refutation from the moon—okay, then I’ll derive the red a fortiori argument, not the blue one. And where do we find such a rotation of the a fortiori argument? Only in two places, as I said earlier: in Niddah and in Bava Kamma. And in both those places it is always when there is an asymmetric table. There is one and half here. In both cases there is a table of one and half, and that is the “it is enough” of Rabbi Tarfon and the Rabbis. In every other ordinary a fortiori argument, once a refutation is raised, the a fortiori argument falls. Unless you answered the refutation somehow. This goes back to what that gentleman asked here. What it does show, supposedly, is that the example of the moon can show that the assumption you make horizontally does not necessarily lead to the same conclusion. I’m saying it doesn’t work. Because after all, that same relation that exists for tooth and foot in the two domains—that’s what I asked. And I’m saying that it refutes the relation between the public domain and the moon, but why do you infer from that that I’m forbidden to make such a relation between the public domain and the damaged party’s courtyard? You see that a relation between zero and one in domains for tooth and foot does not necessarily bring the same conclusion in other domains. According to that logic, you could knock down all the a fortiori arguments in the Talmud. Here, you see an a fortiori argument doesn’t work. Here, look, there’s a moon; the a fortiori argument doesn’t work. So let’s knock down all the a fortiori arguments in the Talmud. It’s that same specific relation. Not a specific relation. It’s a relation to the moon, and this is a relation to the damaged party’s courtyard. There is no refutation connecting those. But that’s something else. But I didn’t understand. It never says that on the moon it is one or something. He took it as a datum. That’s called a refutation. Yes, fine, I took the example. Good, in any case, I’ll come back to this. I’ll return to it after I show why it is correct. Because on the face of it, it’s not so simple to say this. So what this basically should have led me to expect is that in every place where the Talmud brings an a fortiori argument and presents a refutation against it, the Talmud should rotate the a fortiori argument and say: fine. By the way, that happens in the Mishnah in Bava Kamma, for those who remember there with Rabbi Tarfon and the Rabbis, where Rabbi Tarfon makes an a fortiori argument leading to the conclusion that horn damage is liable for full damages in the damaged party’s courtyard, and then the Rabbis rotate the a fortiori argument on him and say: there, you see, it’s half. A kind of hocus-pocus. So they do rotate it there. But in principle they do this rotation only where there really is a difference in outcomes because of asymmetry in the table. But in every other table, if these were ordinary tables where here it says one and not half, so then the result would be one in both directions, they never rotate it—even though there is a refutation, even though the text notes both directions of the a fortiori argument. Where? Where? Only where they rotate it. When they rotate it, then yes, it formulates precisely the opposite formulation; that is called rotating it. But elsewhere, no. You say the a fortiori argument in one formulation, a refutation is raised, the a fortiori argument falls. That’s it. Either it falls or they repair it, but they don’t say: no, that refutation doesn’t bother me at all, I’ll just rotate it and that’s it. So that means that the Talmud apparently understands that these two directions, the blue and the red—and from here on we’ll deal with a symmetric table, let’s leave aside the half, maybe I’ll return to it later—these two formulations are nevertheless two formulations of the same claim, of the same argument. Even though they look like two different arguments, at least that is what emerges from the Talmud. And the question is how to understand that conclusion. So I don’t know if—maybe we’ll stop here, because.