2019-04-22 – Between Midrash and Logic – Lesson 15

Table of Contents

• The development of the table in the Kiddushin passage and the presentation of the refutations
• Benefit as a theoretical feature rather than a halakhic column
• A refutation as a column versus a refutation as a constraint, and continuing to build the table
• The question of mathematical equivalence between the two representations
• The distinction between empirical and theoretical, and the parallel to Bava Kamma
• The passage in Makkot: binyan av, common denominator, and the refutation of a stricter side
• The interpretations of the medieval authorities (Rishonim) in the Ritva regarding the stricter side
• A proposed simple explanation: the stricter side as a result of a gap between the halakhic and the microscopic
• Returning to the Kiddushin passage and continuing the work with constraints

Summary

General Overview

The text summarizes a line of study in which a logical “table” was built for the Kiddushin passage, and the question of the correct way to fill it in was repeatedly examined by means of the model’s criteria. But it argues that the representation used until now was naive, because it mapped the passage in a way that blurred the meaning of the refutations. The central claim is that “benefit” is not another halakhic outcome of the same kind as marriage, betrothal, redemption, and acquisition through a yevama, but rather a theoretical, microscopic characteristic of the actions themselves. Therefore, it is more correct to represent it as a constraint on the solutions rather than as an additional column that artificially inflates the table. On the basis of this distinction, the text also discusses the refutation of the “stricter side” in the Makkot passage, together with the interpretations of the medieval authorities (Rishonim), and proposes an explanation according to which the difference between halakhic-macroscopic refutations and theoretical-microscopic refutations changes both the way the refutation operates and the possibility of seeing equivalence between different models.

The development of the table in the Kiddushin passage and the presentation of the refutations

The text describes how a table was built containing rows of halakhic outcomes such as marriage, betrothal, redemption, and acquisition through a yevama, and columns of actions such as money, intercourse, and chuppah. At each stage, the question was examined whether one gets a “proof” for filling in a one, or a “refutation” that makes filling in a one and filling in a zero equivalent. The text reconstructs the study of the common denominator from money and intercourse to chuppah, the binyan av from intercourse, the refutations of the type “intercourse acquires in the case of a yevama and chuppah does not,” and the return to the law of the common denominator. The text presents the way two standard refutations appeared in the table as symmetric patterns of a “unique property” of money versus a “unique property” of intercourse.

Benefit as a theoretical feature rather than a halakhic column

The text argues that presenting the refutation, “What is unique about money and intercourse? In both of them there is benefit, whereas in chuppah there is not,” as another column in the table is an incorrect presentation, because benefit is not a law or halakhic result but a characteristic of the actions. The text explains that when one identifies a property that exists in money and intercourse but not in chuppah, one may think of “benefit” as a feature that explains the difference, and therefore it should not be placed in the same row as marriage and betrothal, but should rather be treated as microscopic. The text states that the correct representation is to delete the benefit column and leave the halakhic table as it is, while solving the problem under a constraint according to which there must be a microscopic parameter shared by money and intercourse that is not present in chuppah.

A refutation as a column versus a refutation as a constraint, and continuing to build the table

The text distinguishes between two forms of refutation of a common denominator: a refutation that adds another halakhic result and therefore adds a column to the table, and a refutation that does not enlarge the table but instead adds a constraint on the space of solutions, so that only solutions with a suitable microscopic parameter will be considered valid. The text argues that even without a benefit column, one can still reach the point where filling in a one and filling in a zero are equivalent by means of the constraint, and that this better reflects the way of thinking. The text describes how, later in the passage, when a document is added and then a bill of divorce and a Hebrew maidservant, columns are added only for new halakhic outcomes, while the condition of benefit is preserved as a changing constraint—for example, that in money and intercourse there is benefit, while in chuppah and in a document there is no benefit.

The question of mathematical equivalence between the two representations

The text raises the question whether one can always translate a microscopic refutation into a column-type refutation in a mathematically equivalent way, so that the result will be identical, and admits that there is no general proof of this. The text describes that the “incorrect” representation did work over the course of the passage that was studied, but it argues that this may be because if benefit really were a halakhic outcome, one could construct the passage in a similar way. The text seeks a logic that represents the way of thinking and not only formal equivalence, and therefore emphasizes that the equivalence is not trivial.

The distinction between empirical and theoretical, and the parallel to Bava Kamma

The text proposes a conceptual framework of “macroscopic” versus “microscopic,” analogous to empirical versus theoretical, in which the halakhic outcomes are the observable phenomena, from which one infers abstract features such as alpha, beta, and gamma. The text points out that at the beginning of Bava Kamma, most of the arguments are theoretical characteristics of the primary categories of damages, such as “its initial formation was for damage” and “another force is involved in it,” and therefore the work there is, from the outset, under theoretical constraints rather than by adding columns of outcomes. The text uses this comparison to illustrate that theoretical refutations do not necessarily behave like halakhic refutations.

The passage in Makkot: binyan av, common denominator, and the refutation of a stricter side

The text brings the passage in Makkot concerning Rabbi Yehuda’s view that one is flogged for a prohibition that involves no action, through a derivation from one who brings forth a bad name and from conspiring witnesses, the refutations on each one individually, and the return to the law of the common denominator. The text presents the Talmudic refutation, “What is common to them? They both involve a fine,” and its rejection according to Rabbi Yehuda, who does not hold like Rabbi Akiva that conspiring witnesses are a fine, and then the refutation, “for they have a stricter side,” together with the position, “Rabbi Yehuda does not refute on the basis of a stricter side.” The text formulates the basic difficulty: the refutation of a stricter side seems as though it would cancel every common denominator in the whole Torah, because every teacher will always have a stricter side; otherwise, one could learn from it alone.

The interpretations of the medieval authorities (Rishonim) in the Ritva regarding the stricter side

The text cites Tosafot as saying that the refutation of a stricter side is stated only when the stringencies are “very unusual, with nothing else like them in the entire Torah,” such as receiving lashes without warning in the case of conspiring witnesses, and receiving lashes and paying in the case of one who brings forth a bad name, and it expands with examples from other places. The text cites Nachmanides as saying that the refutation of a stricter side is made when both of the teaching cases have stringencies that the learned case does not have, and the learned case has no stringency that the teaching cases do not have, and presents this as a formal criterion of offsetting stringencies. The text cites “my teacher, the Ri”a” as saying that the refutation of a stricter side applies when the stringencies relate mainly to the very same area from which one wants to learn, and here that area is lashes. The text cites an interpretation in the name of Rabbi Meir Ashkenazi, which turns “the stricter side” into an ordinary refutation based on the possibility that conspiring witnesses might receive lashes and pay in a pathological case, and notes that the wording “stricter side” does not fit this interpretation. It concludes with the Ritva’s comments on the connection between Rabbi Yehuda’s position and the rule of “any amount at all” in refuting a common denominator.

A proposed simple explanation: the stricter side as a result of a gap between the halakhic and the microscopic

The text proposes an explanation according to which the refutation of a stricter side rests on the fact that the stringencies brought there are halakhic properties. Therefore, it is possible that two different halakhic properties derive from one and the same microscopic characteristic, so that the alternative, “either X or Y causes the law,” is not necessarily more complicated than the alternative of the common denominator. The text argues that in such a situation there is no clear advantage to the theory of Z as a single cause, because one may claim that behind X and Y there is a shared “alpha” that explains both, and then one gets equivalence at the level of simplicity, which undermines the proof of the common denominator. The text states that the refutation of a stricter side arises, if at all, when the refutations are macroscopic-halakhic, whereas when the refutations are theoretical-microscopic there is no room for a stricter side in this sense. Therefore, the distinction between the two types of refutations is essential and not merely a matter of wording.

Returning to the Kiddushin passage and continuing the work with constraints

The text ties the conclusion about the stricter side back to the starting point and argues that it demonstrates why it is not always correct to assume equivalence between “adding a column” and “placing a constraint on solutions,” as was done earlier with benefit. The text states that the continuation of the analysis of the passage will proceed as in the previous lecture, but with benefit treated as a theoretical refutation and as a constraint rather than as a column, and that in this way it will be seen that the argument still works. The text emphasizes that the method does not require one actually to identify alpha, beta, and gamma, but only to show the existence of microscopic characteristics within the structure that explains the halakhic data, and that an actual identification of those characteristics will be discussed only later.

Basically, what we’ve done until now is this: we looked at the progression of the passage in Kiddushin, we tracked how the table develops, how the table keeps expanding over the course of the passage, and each time we tried to follow the model’s criteria and see whether we get the right result. Meaning: is there a proof here that the filling is one, or is there a refutation here—meaning that fillings of one and zero are equivalent? And what I want to do now is really finish the fundamental move by looking at some other aspect of the passage, because I cheated a little. I sort of staged the passage a bit—I cheated. I cheated on purpose in order to show how this model works in the normal way, but in fact, if you look carefully at the meaning of what’s going on in the passage, what we did isn’t really the passage. It’s a naive picture of it. Because if you remember, we had there a common denominator from money and intercourse to chuppah. From money and intercourse we had money—I’m marking this only symbolically—and intercourse, which together teach me chuppah. If money and intercourse include marriage and include—sorry—intercourse works, money works from marriage to betrothal. Money does not include marriage, but it does include betrothal, so chuppah, which includes marriage, all the more so should include betrothal. That’s the derivation from money. Then we say: money redeems second tithe. Fine? So then we say, okay, intercourse will prove it. Intercourse effects both marriage and betrothal; that is a paradigm case. Intercourse is a paradigm case, so chuppah too, which effects marriage, should also effect betrothal. And then we say: intercourse acquires a yevamah and chuppah does not. So there too there is a refutation. Then we say: the argument returns, the common denominator. And then what is the next stage? The next stage is: what is unique to intercourse and money? That in both of them there is benefit, whereas in chuppah there is not. This refutation is a problematic refutation—not problematic exactly, but I didn’t present it correctly. I presented that refutation as though there is basically another column in the table here. We really had a table of—let’s draw it the way it appeared for us. The table looked like this: marriage, betrothal, redemption, and yevamah, and here you have money, chuppah, and intercourse. Zero, one, one, one, question mark, one, and here the two refutations. The two refutations always look the same. Here you have one and two zeros, and here you have one and two zeros. These two refutations—the P is the unique property of money and the Y is the unique property of intercourse. That’s what the table looked like. Now, when I say that there is a refutation from “what is unique to money and intercourse? That in both of them there is benefit,” then how did I present it? I presented it like this. There is a column here called benefit. Benefit exists in money and intercourse but not in chuppah. And why is that a refutation? It’s a refutation because it tells me that there is some shared property of money and intercourse that is not found in chuppah. And that is exactly the property that produces this whole matter. And if so, maybe that is also the property responsible for effecting betrothal. And if so, these two will succeed in effecting betrothal, but chuppah will not succeed in effecting betrothal. That’s really how the refutation worked. But the problem here is that benefit is something fundamentally different from all of these. All of these are laws, halakhic outcomes. Marriage, betrothal, redemption, and acquisition of a yevamah. Here, though, this is a characteristic. In fact I’d say even more than that: when we tried the two fillings, zero or one, and we opened everything up and it came out that here there is, I don’t know, one, zero. Zero, one, two, whatever. And here there is one, zero, one, whatever. Okay, what does that actually mean? When I looked here and said to myself, wait a second, there is some characteristic alpha that exists in money and intercourse but does not exist in chuppah—it is probably benefit. Right? Because what is the property that exists in money and intercourse and not in chuppah? If I need to identify it now, then I’ll say: it’s simply benefit. That’s the benefit. Okay, so therefore, therefore, it really isn’t correct to place benefit here. You could say—it would be more correct to say—that benefit may be a microscopic characteristic and not a criterion. Exactly. That is the point where I cheated. Meaning, benefit—I treated it as though it were just another halakhic outcome. There’s betrothal, there’s redemption, there’s yevamah, there’s benefit. Another halakhic outcome that money succeeds in producing, intercourse succeeds in producing, and chuppah does not. But that’s not true; it is not an outcome that money and intercourse produce. It is a property that money and intercourse have and chuppah does not. Really, benefit should have been some constraint on the solution. I should have erased that column, remained with this table, such that the refutation really says: what is unique to money and intercourse? That there is benefit in them. What does that actually mean? Solve the problem without this, but choose only a solution that has some shared property of them and of B that does not exist in H. Only from among those solutions—and now find the minimum. And I claim that that is a more accurate representation of the refutation. If we had presented the refutation that way as well, it still would have worked out, even though the table is the same table—notice. And if I present the table without the column, in principle what is the correct filling here? Zero. One. Common denominator. Right? A derivation by common denominator proves that it is one. Now I claim—correctly—that if I add here another column with a refutation, that really will change the situation, and now the filling one and the filling zero will be equivalent. That is a refutation. But I’m saying: the second option too—that is, solve the table as it is, but choose only the solutions that have this property, where one of the microscopic parameters is in M and B but not in H. I want only such solutions. And from among them we take the—so as to refine the passage down to the things that are more relevant and ignore things that were refuted and so on. Meaning we’re really ignoring things that aren’t relevant from our point of view. Not that they aren’t relevant—rather, they aren’t correct. Any solution that does not show me a shared property that M and B have and H does not have is simply not a correct solution. Because after all, I already know—I bring some prior knowledge with me; I’m not learning it from within these properties. All these models concern the halakhic properties. I apply my algorithm to them and learn from the halakhic properties something about the abstract characteristics present in all the actions and all the outcomes. Right? I don’t know what these alpha, beta, gamma things are, but I know there are some characteristics alpha, beta, gamma that appear in this way in the actions and in another way in the outcomes. And where do I learn that from? Not from my knowing anything here—I know nothing here. I don’t need to know what B is, or H, or M. I just do a logical analysis of the table and that’s what comes out. But if I have some prior knowledge that I bring from home, then I say: I actually do know these things, I know what they are. I know what money, intercourse, and chuppah are. Exactly. So I know something about the solution. I actually know that the solution must include some parameter that appears in M and B but not in H. If so, then I throw out all the formal solutions I got that are not of that type. And only from among the solutions that satisfy this constraint—this is really solving the problem under a constraint, right? Under a constraint that there is a microscopic parameter that exists in this and this but not in that. After I solve the problem under the constraint, I choose the simplest solution for zero and for one, for a zero filling and for a one filling. And my claim is that if this really is a refutation, then there is no need to add that column. It is enough to introduce the constraint. The constraint too will cancel out the one here. It too will create the refutation. Meaning that without the constraint, the correct solution is one. But if I impose the constraint, I can see that this constraint will dictate that the one-filling and the zero-filling are equivalent. Meaning that a refutation of a common denominator can be of two types. Either I find an additional halakhic property that money and intercourse succeed in producing and chuppah does not succeed in producing—a halakhic outcome that chuppah does not succeed in producing—or I simply look at the things I’m talking about, I now ignore the letters, and I want to see what M, H, and B really are, and I see that there is in fact something shared by M and B that H does not have. That too is a way to refute a common denominator. If I see that M and B share something that H does not have, then I say: if so, then the common denominator has been refuted. And then I do it in such a way that the table does not actually grow, does not expand; rather, there is always a constraint that the solutions I consider are only solutions that introduce a parameter into M and B that does not exist in H. I only consider those, and I take the simplest among them. Now when we continue onward and add the document, and after that the bill of divorce, and the Hebrew maidservant, and everything that appeared there later on, then every time there is an additional halakhic outcome I will add another column and another column and another column, but all the while I need to preserve one thing: that in money and intercourse there is benefit, and in chuppah there is no benefit. And after I add the document, I know that in the document too there is no benefit. So the constraint will be that in money and intercourse there is benefit, and in chuppah and the document there is no benefit. And I’ll continue the whole process I did last time, but I’ll do it with the constraint. Then the tables will always have one less column. The benefit column will be deleted, because it will appear as a constraint and not as a column in the table. I don’t understand why a theoretical example couldn’t be like that. Correct, it could be that it represents the thought process correctly. Logic tolerates everything, but I’m looking for a logic that represents how I think. And it seems to me that it isn’t correct to represent how I think in this way. Now there is an interesting question whether there is a theorem saying that I can always translate one into the other. That wherever I have a microscopic refutation, I can always present it as a column-refutation, and the result will be the same. There might be such a theorem; I don’t know, we haven’t proved it. There could be such a theorem, and then that would mean: true, it doesn’t represent how I think, but it is mathematically equivalent. So I can still do it the way I did until now. In the meantime it worked for us. But I don’t know if it is a general theorem. Because we saw the whole passage, we went through it when I presented it in the wrong way, and it worked. It worked because of course it had to work, because if benefit had been a halakhic refutation rather than a microscopic refutation, the passage could have been built in the same way and everything would have been fine. So it’s okay that it worked. I’m just saying that the question is whether it will also work in the second way, which is the really correct way. Now I want to show you why this is not trivial, or why one should not assume so simply that these two forms are equivalent. Because you can always say, look, true, on the conceptual level there is a microscopic refutation—let’s call it now a microscopic refutation. Meaning, not a refutation that adds a column; that would be the macroscopic one, the properties that I can see. Say, in a scientific context, I look at phenomena: which bodies are attracted, which bodies fall, which bodies behave in this way—behave, that’s the phenomenon, that’s the macroscopic level, that’s what I see before my eyes. And from that I infer some abstract properties of those bodies, like mass, charge, things like that. I call them microscopic—maybe that’s not the most precise term—but let’s call it theoretical, perhaps, not microscopic. Fine. There is the empirical and there is the theoretical. And this refutation is a theoretical refutation, because it is a refutation that is really telling me: I see nothing that refutes this common denominator just by looking at Jewish law. But by looking at money and intercourse, at the meaning of those concepts, I can refute it. So I am really understanding what is inside those actions not through their halakhic consequences, but simply by looking at them themselves. If you remember the passage at the beginning of Bava Kamma, that whole passage is built on such characteristics. The passage at the beginning of Bava Kamma, when it talks about the four primary categories of damages, says: a pit, from the start of its formation, is for damage; here these are characteristics—what are they? Fire is like this; from the start of its formation it is for damage; its way is to go and damage; another force is involved in it; its intent is to damage—things of that type. These are all characteristics in alpha-beta terms. It isn’t right to mark it in this way. It is all in terms of what the theoretical characteristics are. And then the table is two by two. And everything will supposedly work through microscopic constraints, or theoretical constraints, exactly. Now I want to show you why this distinction is important. This distinction between theoretical refutations and—let’s call them—halakhic refutations. This is a passage in Makkot with Ritva on it, and let’s learn it for a moment in order to see what. There’s a parallel passage also in Ketubot 32a—not parallel in the sense of the same topic, but with the same structure. It’s called the refutation of a severe aspect. Let’s look at the passage in Makkot. “It follows that Rabbi Yehuda holds that one receives lashes for a prohibition that involves no action. From where does he derive it?” From where does he know that one receives lashes for a prohibition that involves no action? “Ulla said: he derives it from one who brings out a bad name. Just as one who brings out a bad name is a prohibition involving no action, and one receives lashes for it, so too any prohibition involving no action receives lashes.” So what is this? This is basically a paradigm case or an analogy, yes? Then they say: what is unique to one who brings out a bad name? That he is lashed and pays. One who brings out a bad name has a unique property, that he is lashed and pays, so you cannot derive from him to every prohibition involving no action. Rather, Resh Lakish said: he derives it from conspiring witnesses. Just as conspiring witnesses are a prohibition involving no action, and one receives lashes for them, so too any prohibition involving no action receives lashes. The Gemara refutes this: what is unique to conspiring witnesses? That they do not require prior warning. So they too have some unique property: they do not require prior warning. So the Gemara says: one who brings out a bad name will prove it. And the argument returns: this is not like that, and that is not like this; the common denominator between them is that they are prohibitions involving no action and one receives lashes for them; so too any prohibition involving no action receives lashes. The Gemara says—so let’s write on the board where we are working. We have two teachers, one of them is conspiring witnesses and one of them is one who brings out a bad name. Fine? And here put—doesn’t matter—some other prohibition involving no action, okay, it doesn’t matter exactly which one. There it is “you shall not leave any of it until morning.” What? Where? In the Mishnah in Makkot. Why? I’m not saying this for no reason. In the passage it doesn’t matter, some prohibition, okay? So this is notar. Fine. Good. So now we basically have two teachers here, each one of which has a unique property, and therefore each one separately cannot do the job, right? Conspiring witnesses do not need prior warning—there is no prior warning—and this one is lashed and pays. Right, lashed and pays—that is the property. Okay, and in notar there are neither of those two properties. Fine? And therefore we need to construct a common denominator. Now what does the Gemara say? “What is unique to the common denominator among them? That they are both fines.” Meaning both conspiring witnesses—it’s a tannaitic dispute, not important right now—and one who brings out a bad name are fines. And notar has nothing to do with a fine, and therefore you cannot derive from them. Maybe because they are fines, they are more severe. So this is a refutation like we saw here; it is basically a refutation like this: a characteristic, that it is a fine—it is a halakhic characteristic. Okay. Now the Gemara says: Rabbi Yehuda does not hold like Rabbi Akiva, for he holds that conspiring witnesses are monetary liability, not a fine. Rather, what is unique to their common denominator? That they have a severe aspect. Now that is already a real blow. Meaning, how do we refute the two teachers? That in both of them there is a severe aspect. Rabbi Yehuda does not refute by a severe aspect. Okay, but there is a very strange refutation being raised here in the Gemara: a severe aspect. What is the severe aspect? In one who brings out a bad name there is a severe aspect because he is lashed and pays. In conspiring witnesses there is a severe aspect because they are without prior warning, right? So since both of them have a severe aspect, you cannot derive from them to notar. And that of course raises the question: then no common denominator in the Torah can ever stand, right? Because every common denominator in the Torah has precisely this structure: for each of the two teachers, each one has a severe aspect. It is always like that. If it did not have a severe aspect, we would derive from it alone. The question is whether a severe aspect is really an advantage, because not every advantage is supposed to count as severe. It has to be, otherwise you derive from it alone. After all, how does a common denominator work? You derive from conspiring witnesses, and then you find in conspiring witnesses a severe aspect and therefore you cannot derive from it, right? Then you say, okay, so let’s derive from one who brings out a bad name. Then they say no, one who brings out a bad name also has a severe aspect, so you cannot derive from him. Then you say: the argument returns, let’s derive from both of them. So always, by definition—in fact from the very schema of a common denominator—both teachers will each have a severe aspect. That is true by definition, because if one of them did not have a severe aspect, we would derive from it alone. Okay? So in fact, if I accept the possibility of a refutation by severe aspect—which is what the Rabbis here are using to refute—then in practice there is no common denominator in the Torah. None. Every common denominator can be refuted in this way. Right? So the medieval authorities are already troubled by this, Tosafot in Ketubot too, but here Ritva brings several approaches, so let’s read him. Ritva says as follows. Maybe I’ll mention this before entering Ritva. Look. Let’s present it another way. On what was the principle of common denominator based? The principle of common denominator—we asked this very question, in one form or another, already when we first discussed common denominator. And what did I say there? That in a common denominator there are two ways to explain the—say, let’s call “lashed and pays” property X, and “without prior warning” property Y. So in notar there is neither X nor Y. Right? Then I say: okay, so probably the property Z, which exists here, and here, and here—what is that property? That it is a prohibition involving no action. Right? So if Z is the cause, then we can derive from both of them to here as well. Then I asked there: wait a second, but who says so? Maybe Z is not the cause; maybe either X or Y is the cause. And here, after all, there is neither X nor Y, and therefore the law does not apply. We brought a simpler explanation. Exactly. Meaning, I had an alternative theory. One theory says Z causes the law. A second theory says no, either X or Y causes the law. And where is the practical difference between these two theories? In the question whether notar receives lashes or does not receive lashes. Right? If Z causes the law, then notar has that too. But if X or Y causes the law, notar has neither X nor Y. So for notar there will be no lashes. Right? So why did we say that in fact it is better to take the common denominator? Because it is simpler. Occam’s razor, right? We said that the simpler explanation is always preferable. So the explanation that one parameter governs the law is simpler than the explanation that either one parameter or a second parameter governs the law. There is some almost metaphysical assumption here, that the Holy One, blessed be He, created Jewish law in such a way—or let’s formulate it differently—that every halakhic result has only one reason. That is really the assumption of common denominator. It is unitary. Yes. Fine? The assumption of common denominator really says that every halakhic result has one reason. Meaning, it cannot be that there is a halakhic result that has two different reasons, each of which can bring it about. Because if that were so, it would be impossible to derive by common denominator. And common denominator is one of the interpretive principles of the Torah—it is not just in one place; it is a general principle. So there is some assumption here about how Jewish law as a whole is structured. When I find two things in Jewish law that have the same law, that means that there is something shared that causes it. For example, for a pit one is liable to pay, and for fire one is liable to pay. What does that mean? It means that payment probably does not depend on “from the start of its formation it is for damage,” because that is only a property of a pit and not of fire. Nor is it connected to “another force is involved in it.” “Another force is involved in it” belongs entirely to “its way is to go and damage”—that is fire. So that does not exist in a pit. Rather, it probably depends on something shared by both of them. What is that shared thing? “It is your property and its safeguarding is upon you,” right? The Mishnah at the beginning of Bava Kamma. Okay? So the assumption is that there is always one reason that causes the law, and not alternatively one of two other reasons. Sometimes, by the way, that one reason may itself be a combination, as in the Mishnah in Bava Kamma: “it is your property and its safeguarding is upon you.” So that can be treated as two conditions joined together. But two things that are both this and this, for me, count as one. It doesn’t matter—that is one reason. There are not two alternatives here. There is one possibility that is a conjunction of two conditions. Okay? So that is the simplest theory and therefore the correct one. Can I ask? Yes. Instead of saying it is either X or Y, could we say that X is composite, or I don’t know, that X and Y inherit from something above them, that there is something common over them? In a moment—you’re anticipating me. I think you absolutely can say that, but hold on a second. Where does the refutation by severe aspect stand? The refutation by severe aspect really undermines this whole logic of common denominator. Why? Because the refutation by severe aspect is basically saying to me: if here there is X and here there is Y—these are the severe aspects—and here there is neither of them, then I knock down the common denominator. What is it really saying? That of a paradigm case. A paradigm case from two verses. Common denominator. Yes. Meaning, it says that actually it is not true; there can be two different reasons for the same halakhic outcome, and I do not necessarily choose the simplest result. The less simple result is also a legitimate alternative. That is really the meaning of a refutation by severe aspect. Which is a very strange thing. So what then? Then there is no common denominator in the Torah? What kind of thing is this, a refutation by severe aspect? So here the medieval authorities—look at Ritva: “What is unique to the common denominator among them? That they have a severe aspect.” Tosafot asked: if we refute by severe aspect even though the severe aspects are not similar to each other, why does he add this? Because if the severe aspects are similar to one another, then of course we refute. Right? If here there were X and there too there were X, that would certainly be a refutation. That is like the refutation of benefit, right? What is unique to money and intercourse? That there is benefit in them. Because that is the same property. If they are different properties, like redemption and yevamah, then on the contrary, that strengthens the generalization. But if it is the same very property, that is a refutation of common denominator, because maybe that is the property that causes the law. Okay? So therefore he emphasizes here: if we refute by severe aspect even though the severe aspects are not similar to each other, then you have abolished every common denominator in the world. That is what I asked earlier. And they answered: we only refute in this way when the severe aspects are very unusual, with nothing like them anywhere in the Torah. Meaning, the claim is really the following—an interesting claim. What does “very unusual” mean? How do you know whether the refutation is extremely unusual or only somewhat unusual? Usually when we refute, we say: oh, this is different from that. Why should I care whether they are unusual or not unusual? You can compare any refutation to another refutation and ask whether it is unusual by some standard, no? How will you compare refutations? With the foot, “its way is to go and damage,” and a pit, “from the start of its formation it is for damage.” Which is more unusual than which? Huh? You can compare refutations according to the way they work in the tables, if it is unusual relative to the way a refutation usually works. No, no—it is not a refutation that works differently. He says if it is very unusual. The table looks the same; it is just that the alpha or alpha-property hidden behind the refutation is extremely strange. So what counts as extremely strange? Especially when we know that the Gemara in Chullin 15 says that a common denominator can be refuted by even the slightest refutation. Meaning, not only a very unusual refutation—any refutation. What difference does it make? As long as you propose an alternative that is possible, you can no longer make the proof of common denominator. Right? That is really the idea behind it. So what does “very unusual refutation” mean? So look, he adds something else: severe aspects that are very unusual, with nothing like them anywhere in the Torah. For conspiring witnesses—meaning, usually a normal refutation says: there is something in the teacher that is not in the learner. But that something that exists in the teacher could also exist elsewhere. What matters to me is just the relation between the learner and the teacher—that’s all. Do you understand? But what Tosafot is really claiming here is that that is true for a normal refutation. But if there is a refutation that singles out only this teacher—not only in comparison to the learner is it special, but in general, in the rest of the Torah it does not appear; it appears only in this teacher, and similarly in the second teacher—if those are two unusual refutations of that sort, then we refute by severe aspect. According to the Rabbis. Rabbi Yehuda does not refute even there. But according to the Rabbis, one does refute by severe aspect. And how does he spell it out? He says as follows: for conspiring witnesses receive lashes without prior warning, and one who brings out a bad name is lashed and pays. Meaning, receiving lashes without prior warning is a unique property; it exists nowhere except with conspiring witnesses. Fine? And one who brings out a bad name—that too is half-testimony. What? And also half-testimony. What do you mean? That’s a property of the action itself. But not receiving lashes without prior warning. No, but in terms of—ah, look at the receiving lashes without prior warning. And one who brings out a bad name is lashed and pays, which is also a unique property. Okay? And since it is a unique property that does not exist anywhere else in the Torah, here yes, we do refute by severe aspect—so says Tosafot. And in the chapter “These Young Women,” where they refute in this way the common denominator of one who injures his fellow and conspiring witnesses—one who injures his fellow is liable for five payments. Yes, there too there is a severe-aspect refutation; and one who injures his fellow—what is the refutation? That he is liable for five payments. Again, that is truly unique: only one who injures is liable for five payments. No other damager is liable for five payments. Okay? And in tractate Sotah regarding one who immersed that day and earthenware vessels, where one who immersed that day is a primary source of impurity and an earthenware vessel imparts impurity from its airspace, and this is somewhat similar to what we say in Sanhedrin: what is unique to a prince and a deaf person? That they are unusual. Yes: “You shall not curse a deaf person,” and “a prince,” and “a father,” and there they derive by common denominator. So that is the first direction of Tosafot. The direction—still, we need to understand: why does it matter if it is extremely unusual or not? It is still a more complex theory. Why not adopt the simpler theory? The logic of common denominator applies even to very unusual refutations. Why does that undermine the logic of common denominator? More than that: this explanation does not really work very well. Tosafot here leaves out part of the discussions. In “one who injures his fellow” the refutation is that he is liable for five payments. I don’t recall right now what—conspiring witnesses here, the refutation is that they did not receive prior warning, so he omits that. But in short, some of these characteristics—for example, that it is a primary source of impurity—there are other primary sources of impurity too; is a primary source of impurity some unique thing in the Torah? There is something here that is not entirely clear. Tosafot says this because he was pressed, but the criterion here is a somewhat problematic one. Then he says as follows: “And our Rabbi, Ramban, of blessed memory, explained that we only refute by severe aspect where both teachers have severe aspects that the learner lacks, which of course is always true in every common denominator, and the learner has no severe aspect that the teachers do not have, as in our case.” Meaning, Ramban says there are offsetting refutations. Meaning, where do we refute by severe aspect? In a place where in both teachers, each one has a unique severe aspect, and the learner has no unique severe aspect not found in the teachers. But in a place where the learner too has a unique severe aspect not found in the teachers—say the learner would have, I don’t know, A, a thing that does not exist in the two teachers—then there will be no refutation by severe aspect. Why? Because the learner too has a severe aspect, not only the two teachers. So the two teachers are not unique in having a severe aspect; the learner too has a severe aspect. What really stands behind this? The claim behind it is that the learner has some unique property that perhaps is what causes the law not to apply to it. The absence of that property is the reason why the law applies in the two teachers. And with the learner, which has that property, call it A, where here there is A and here non-A and here too non-A. So Ramban says: if here there is a unique property, this one has a unique property X, this one has a unique property Y, and this one has a unique property A, then Ramban says that in such a case there is no refutation by severe aspect. There, there’s the problem: what is unique to one who brings out a bad name? That it had prior permissibility; it had a previous state of fitness, and nevertheless. Fine, one moment. I agree that this is a bit shaky, but first I want to explain it. So here too there is a unique property, and then what? Then we do not refute by severe aspect. Why? Because this too has a severe aspect. You can always flip it around. You can almost always find something about anything. But it probably has to be that this property must be a relevant property. So what makes it relevant? Notar is written with a nun. Exactly—that’s my point. Anything can be made into a relevant criterion, anything can be… That’s why I say: but that is always true when you make an analogy. You don’t make an analogy between a prohibition involving no action and impurity in the public domain. You have some idea in your head of what the connection is. Exactly. So Ramban says the same thing here. But notice that it is really strange. Ramban takes the refutation by severe aspect as something entirely formal. I would say: these two have a severe aspect and this one does not have a severe aspect—and this one, sorry, does not have a severe aspect—therefore we refute by severe aspect. Just a second. But if this one too has a severe aspect, then you can derive as usual. Which is very strange, obviously. I would say the opposite: maybe precisely the fact that these two do not have A is the shared property, and therefore that is the refutation. True, that is not so simple, because the fact that they don’t have A is a leniency in them, not a severity in them. After all, the A has to be the severe aspect. And the absence of A is an absence of stringency. So true, that is not a standard refutation; you cannot say that because these two do not have A, therefore they have this law. So Ramban requires that it be a severe aspect. But still, what is going on here? Meaning, if this really is more severe than those two, then you’d want to make a kal va-chomer, say. If those two, which do not have A, nevertheless receive lashes, then this one, which has A, all the more so receives lashes. That is basically a kal va-chomer, right? Then you say: what are you talking about? What is unique to one who brings out a bad name? It has X. So I answer: conspiring witnesses will prove otherwise. Then I say: what is unique to conspiring witnesses? They have Y. The argument returns, the argument returns—but still, both of them lack A. Fine, but the lack of A is a leniency. We could treat it, as we said earlier, by removing a column. Maybe you can look at it as a characteristic and not as another factual column, in the end, in the whole severe-aspect matter. Wait, I’ll get to that in a moment; that’s where I’m heading. Just one more second. Fine, so that is really what Ramban says: he truly takes the very existence of a severe aspect as constituting a refutation. Now, I think that in a certain sense Ramban is slightly pulling the ground out from under the logic of common denominator, because we understood earlier that the logic of common denominator is: let’s choose the simplest theory, right? But choosing the simplest theory—even in a place where there is no A—the simplest theory is still that Z is the cause and not X or Y. Ramban says: that doesn’t interest me; it’s not the simplest, and still I don’t choose it. Meaning, something here becomes entirely formal. If both of them have a severe aspect and it too has a severe aspect, then it’s fine; and if not, then not. Why not? If what you want is the simplest theory, that is true whether there is such an A or not. It is simpler for everything to depend on Z than for it to depend on either X or Y. So there is something here that alters the logic of common denominator somewhat. So that is Ramban. “And my teacher, the Re’ah, of blessed memory, would explain that we only refute by severe aspect when the severe aspects are in the very matter we are trying to learn from them. As here, where we are trying to learn regarding lashes. And one who brings out a bad name is severe in that he is lashed and pays, and conspiring witnesses are severe in that they are lashed without prior warning.” When the refutation pertains to the very law that you are trying to derive—you are trying to derive here about lashes. And precisely in the area of lashes, one who brings out a bad name has a unique property, and conspiring witnesses have another unique property; they are highly exceptional precisely in the matter of lashes. So the Re’ah says: if that is the case, you cannot derive in the matter of lashes from these two to notar, because you see that at least in the context of lashes each of them is highly exceptional. In such a situation we refute by severe aspect. But if they had some other halakhic property, another halakhic severity—still a severity—but it did not concern lashes, rather something else, then we would not refute by severe aspect, and we would indeed use a common denominator. Why is lack of prior warning connected to lashes? Because prior warning is a law within lashes. And one is liable for lashes even without—it doesn’t get lashes without prior warning. Meaning, liable to lashes without prior warning—that’s the… Can’t you define a criterion of relevance? I don’t see how. That is exactly the point, and I’ll talk about it later. That is precisely the background within which this whole model operates. We haven’t turned these matters into mathematics—despite the illusion. “And I saw from Rabbi Meir of Ashkenaz, of blessed memory”—I assume that means Maharam of Rothenburg, as distinct from Rema—“who wrote that a severe matter means that sometimes, with conspiring witnesses, there can be the same severity as one who brings out a bad name, namely being lashed and paying.” Meaning, conspiring witnesses can sometimes be lashed and pay. How? Such as if they testified concerning a person that he had brought out a bad name. The conspiring witnesses who testified about a person that he had brought out a bad name made him liable for lashes and payment. Now they were proven false. Since they were proven false, they themselves receive lashes and pay. So it turns out that conspiring witnesses too can be lashed and pay, and therefore this is really a refutation. What is he really saying? That this isn’t a refutation by severe aspect at all; it is just a standard refutation. There is a shared property of one who brings out a bad name and conspiring witnesses—what is it? That both are lashed and pay, that’s all. And that does not exist in notar. So this is just a standard refutation. Yes, that does not exist in notar, so it is just a standard refutation. Obviously that is not the meaning of the Gemara, yes? The pressure simply pushes the medieval authorities to suggest all kinds of explanations. “And Rabbi Yehuda does not consider this a refutation, since most cases of conspiring witnesses are not like that.” Usually conspiring witnesses do not receive lashes and pay; that is only a pathological case, yes, a practical difference for the betrothal of a woman, as they say. “And the language of ‘severe aspect’ does not fit this explanation, nor is this found to fit nicely in those places where they raise ‘severe aspect.’” There are other places in the Talmud where they ask from severe aspect. Therefore he says: that is certainly not the correct explanation. Good. His final remark is itself interesting. “And Rabbi Yehuda does not refute by severe aspect—apparently he does not hold by that statement in Chullin 115,” what I mentioned earlier, “that every common denominator can be refuted by any slight refutation, for the refutation of severe aspect is a form of slight refutation, as Rashi explained there regarding the matter of prince and deaf person in tractate Sanhedrin.” Meaning, Rabbi Yehuda simply does not accept the Gemara’s statement in Chullin that a common denominator can be refuted by any slight refutation, because Ritva here is really saying that a refutation by severe aspect is just a particular kind of slight refutation, since it isn’t really a strong refutation. But still, there is some property in the teachers that is not in the learner, and that is enough to refute a common denominator. And we already discussed that a common denominator is a weak analogy, and therefore it suffices to refute it with even the slightest refutation—but perhaps I’ll come back to that. In any event, up to here these are the words of the medieval authorities. In my humble opinion, there is a much simpler explanation here—almost trivial. And that explanation is what you suggested earlier, and what you suggested afterward in slightly different language. Look: the properties we are talking about here—are they properties of this type or properties of that type? No, these are Y-P properties. Right? These are halakhic properties. This one is lashed and pays, and this one is lashed without prior warning, right? Now the whole claim of common denominator is that I prefer the simplest theory. So I say there are two possibilities. Either—let us ignore the A now, yes, here is Ramban’s proposal—there are two possible theories. Either X or Y causes the law, and then of course in notar it will not apply, or Z causes the law, the common denominator in both of them is Z. And Z also exists in notar, and therefore notar too will have the law. Since Z is a simpler proposal than either X or Y, I prefer it. But the Rabbis say: what are you talking about? Maybe X and Y, after all, are both halakhic properties. Maybe they are both caused by the same microscopic characteristic. Maybe the reason one who brings out a bad name is lashed and pays is that he has property alpha. And conspiring witnesses, who are lashed without prior warning, that too is because they have property alpha. Fine? Only, besides alpha, they have something else too, and therefore there it is not “lashed and pays,” but the halakhic expression is “no prior warning.” Fine? But still it stems from the same microscopic characteristic. So you have no proof that the common denominator is a simpler theory. Because even the theory that appears more complex—either X or Y—may in fact not be complex at all, and the burden of proof rests on the one asserting the common denominator. I say: you have brought me no proof. You cannot prove to me that the filling here is one. Because I have a possible alternative in which the zero-filling is no less simple. Where will there be a practical difference? Look through the Talmud in all the passages where they refute by severe aspect, and you will always see that the characteristic is a halakhic characteristic. Always. If the characteristic were a microscopic one—for example, “what is unique to one who brings out a bad name? That he”—I don’t know—“went and abused…” that’s too severe because of some practical reason. Meaning, how he abuses his spouse, I don’t know, or something like that. Fine. Or the severity of conspiring witnesses is that they use a court to do something terrible. Doesn’t matter. Some sort of reasoning that is not a halakhic result but characteristics of the thing itself. There will never be a refutation by severe aspect there. There cannot be a refutation by severe aspect. You can’t compare between things. Exactly. Because there we are already speaking about the microscopic properties, so you cannot say that both of them derive from—as you wanted to say earlier—from some third property. That is no longer plausible; I see two different things. But if you bring me two halakhic characteristics, halakhic characteristics like the ones we mention here can conceal behind them an entire microscopic world. You do not know what alphas and betas and gammas are functioning there. It is entirely possible that X and Y are both affected by the same microscopic characteristic. And then what happens? You actually have two alternative simple theories. One simple theory is that the characteristic causing lashes is Z, and that does not exist in notar. A second theory, no less simple, is that the characteristic causing lashes is alpha, and alpha is absent in notar. Right? And since that is so, then it will not receive lashes. Okay? So it comes out that whether it receives lashes or does not receive lashes, the theories are equally simple. You cannot say that the fact that it receives lashes leads me to a simpler worldview. And therefore, a refutation by severe aspect, in my view, is a rule: a refutation by severe aspect exists only where—if it exists at all—it exists only where the refutations we are talking about are halakhic refutations. But where the refutations are theoretical refutations, there will never be a severe-aspect refutation. And that is exactly why I’m bringing this example, this consideration, because it shows you what I said earlier. When I talked about benefit, if you remember, I said we presented benefit as another column, and then I checked whether it works with tzedakah and so on, and I said: benefit is not another column; it is a theoretical refutation, not a macroscopic refutation. And here we already see that there is a difference. A macroscopic refutation and a theoretical refutation are not supposed to work in necessarily the same way. According to the Rabbis against Rabbi Yehuda, one refutes by severe aspect when the refutations—when the properties, sorry—are macroscopic, halakhic. But when the properties are theoretical, one does not refute by severe aspect. Meaning, this does not give—both are refutations, but they are refutations with different properties. One of them is now the subject of a tannaitic dispute, and the other would not be the subject of a tannaitic dispute. So here is an example of why it is important to distinguish between a halakhic refutation and a theoretical refutation. What would a diagram of such a thing look like, say, if you put in arrows? Or if you use a table? A diagram of parameters. Meaning, if we have alpha and alpha, then everything will be connected in all directions and we won’t be able to prove anything. Say in… No, no, no, no. That won’t be expressed in the diagram at all. I’ll do it in a moment. Because if we have alpha and alpha, we’ll have to connect them somehow. No. We’ll take care of that. The diagram is the diagram determined by the table. That is the diagram, and H does not appear here. Benefit does not appear. That is the diagram. The solutions I’ll look for for this diagram will be only solutions where there is a parameter that exists in A and B but not in H. I won’t relate, in the severe parameter—I won’t relate in the diagram? No, not in the diagram. In the solution of the diagram I’ll relate to it. We’ll do it and then we’ll see. Now look, it’s entirely possible that some of this is also present in the intuition of the medieval authorities that Ritva cites here. Because when, for example, what he writes there—what Tosafot says—that the severe aspects are very unusual, not appearing anywhere else in the Torah. And I asked: why does that mean that there one can refute by severe aspect? I would say as follows. Why indeed does Rabbi Yehuda not refute by severe aspect? What is problematic about a refutation by severe aspect? The problem is that when you have different halakhic properties, why assume they have the same cause? That is somewhat speculative. And still it is simpler to say that Z is the cause and X and Y—if that is intuitive then maybe yes, but assuming we have no simple intuition about it, we are making some move here. Why would the simple intuition—why does this question arise in the first place? If we have no intuition to assume that they share some common place. No. It doesn’t matter. X and Y have no common place. Yes, I understand. If we have no intuition to assume it, why do we say they do have something in common? How can we even begin a severe-aspect refutation without intuition? Why not? You have X and Y; they are halakhic properties. You have no idea what causes them. After all, even here when you want the analysis, you have alpha, beta, gamma. Which one is alpha, which is beta, which is gamma—you don’t know. I had an intuition that it was… No, you know nothing. You don’t know at all who alpha, beta, and gamma are. How do you know what they are? We’ll talk about that later. For now it is entirely formal work. You have laws here, and from them you extract some abstract theoretical structure with alphas and betas and gammas. That’s all. So I say: fine, such a structure could theoretically also lie behind X and Y with the same alpha. You don’t know. If you have intuition, great, then perhaps it won’t work. But in any case, where you don’t know—then Tosafot says this: since usually this whole business is dubious, refutation by severe aspect—because here you have two different halakhic properties and you want to tell me that both come out of a shared characteristic, a shared theoretical characteristic—we do not say such a thing unless we have some indication of it. What would count as such an indication? That these two characteristics appear only here, in the whole Torah. Then—why assume there are two different microscopic parameters, each of which is unique in the entire Torah? I say: if I see something here that is unique in the whole Torah, that somewhat strengthens the possibility, at least, that maybe this really is the same thing. And therefore Tosafot says that here one refutes by severe aspect. I need some additional indication telling me that behind the halakhic properties sits the same theoretical characteristic. So if indeed both are extremely exceptional, look: when we see two very strong exceptionalities, especially if we add what the Re’ah writes, that the refutation is in the very matter we are trying to learn from, then what does that really mean? That these very unusual exceptionalities both concern lashes. So what does that mean? Then it greatly strengthens the possibility that maybe they are really caused by the same thing. And this is really the same property. Therefore there one refutes by severe aspect. Because in such situations, behind X and Y—different properties, this one is lashed without prior warning and that one is lashed and pays—different halakhic characteristics, but since they are so exceptional and run against all the Torah’s rules, and both are connected to the laws of lashes, according to the Re’ah if I combine him here, then that gives me some indication or legitimacy to say that if so, then perhaps there is indeed one theoretical characteristic behind both of these characteristics. And then that is no less simple than the proposal of Z. On the other hand, when you say “severe aspect,” the phrase itself sounds as if it places each one in a separate corner and doesn’t connect them. Therefore Tosafot says: but if it is a severity… It’s just that intuitively, when I hear “severe aspect,” I don’t think they’re talking about something that connects them, but specifically something that separates them. On the contrary. Obviously. But what are we saying? You said there is something connecting them, namely alpha. נכון. I’m saying that when there is an exceptional severe aspect, a severe aspect that is very, very strange, then I say: fine, something must be causing it. Now why assume there are two very, very strange things? Severity and leniency are equally strange. There are many things in the Torah that are more severe than this one or more severe than that one, so there is no problem as long as it does not suddenly become some kind of anomaly. Only when there is a very special anomaly do I say: okay, that at least strengthens—makes more plausible—the possibility that there is some shared characteristic here. I can no longer dismiss it. After all, to throw out the common denominator, it is enough to say that I can no longer rule this out completely, reject it out of hand. I am not claiming that this is in fact the case. I am only saying that here it already becomes legitimate to propose this too as an alternative. And that is a refutation by severe aspect. Okay? And then what this means is—and I see I’m going to do this next time already—this example has shown us that it is very important to notice whether we have a refutation whose meaning is the addition of a halakhic column, or whether the refutation is some constraint on the solutions. Fine? And what we are now going to do—up to here I was just finishing this example, now I return to the passage—we will analyze the continuation of the passage, what I did in the previous class, and now we’ll analyze it in these terms. Meaning, we will treat benefit as a theoretical refutation, as it really is, and not as a halakhic refutation. And we will see that it will work that way too. What? To build some methodology that explains which potential data can—what sort of dominant and recessive gene they have that can appear in the phenotypic display in order to create the interpretations of alpha, beta, and gamma and identify them. To identify them, yes, identify them. I’ll talk about that at the end. I’ll talk about it at the end. I don’t have a lot to say about it, but maybe I can show through examples how one can work with it a little. We’ll speak about it at the end. Okay? There’s a sign-in sheet here for anyone who wants. And at the end of this passage, with what alpha? What? In this passage, with what alpha? We don’t identify the alpha at all. Never, really, the whole way through. For me it is enough to say that there is such an alpha. That is the whole strength of this method: I do not assume anything speculative about who the microscopic characteristics are. I build a model that says there are microscopic characteristics alpha, beta, gamma. Alpha exists in this and this and this and not in that. Beta exists with intensity two in this one, with intensity—and so on—and that explains the whole matter. And I do not need to identify alpha, beta, and gamma at all. Fine? For the inference, I do not need identification. If I want to understand the passage by identifying them, then I say—I’ll talk about that at the end.