Tractate Sotah, Chapter 5, Lesson 9, Rabbi Michael Abraham

[Rabbi Michael Abraham] Okay, we’re continuing. I spoke a bit about the meanings of a doubtful sotah. I want to take a kind of time-out and discuss a little the prohibition of a sotah to a kohen, but really, less the prohibition of a sotah to a kohen itself, and more how we learn it, the a fortiori argument that teaches it. I saw that this takes us a bit into the meanings of a fortiori reasoning in general, so I want to say a few words about that. And at the next stage I’ll want to move on to some continuation of the same topic, namely the relationship between a doubtful sotah and doubtful impurity. I’ve already commented on that more than once, but the Talmud here goes into things like “subject to inquiry” and all kinds of definitions that relate to doubtful sotah and doubtful impurity, so after that I want to get into those things. But that won’t be today. So I’ll start from the beginning of the passage on page 28a.

[Speaker B] I—

[Rabbi Michael Abraham] Let me share the… okay, all right? Rabbi Yehoshua said: this is how Zechariah expounded, and so on. It appears there in the Mishnah, that “she became defiled, she became defiled” means that she is forbidden to her husband and forbidden to the adulterer. The rabbis taught—the Talmud brings a baraita—three times it says in the section: “if she became defiled,” “she became defiled,” and “she became defiled.” Why do I need it? One for the husband and one for the adulterer—those are the two things that appear in the Mishnah—and here a third thing appears: and one for terumah. Meaning, after she strayed, she becomes forbidden to eat terumah. That is Rabbi Akiva’s view. “Forbidden to eat terumah” is interesting—forbidden by virtue of whom? A woman can eat terumah either because of her husband, by virtue of her husband, or because of her father, right? And if she gets divorced from her husband, say without children, then she returns to her father’s household, and if her father is a kohen then she can eat by virtue of him. So when it says here that she is forbidden regarding terumah—yes, one for terumah—it doesn’t say whether that’s by virtue of the husband, to whom she is still married until he divorces her. After he divorces her she is no longer his wife, but as long as he hasn’t divorced her, she is his wife and he is obligated to divorce her. So one possibility is to say that she is forbidden to eat terumah still before she receives the bill of divorce, which is also a novelty. After she receives the divorce document, there’s no need to innovate that; she’s no longer his wife. A second possibility is to say that she is forbidden to eat terumah because of the father, meaning even when he divorces her and she returns to her father’s household, still, even if she is the daughter of a kohen, she will not be able to eat terumah. In the Talmud itself, not in the baraita, it is not clear which of these two possibilities is meant. Rashi there writes that this means both by virtue of the husband and by virtue of the father. Meaning that she is forbidden to eat terumah both because of the husband and because of the father. It’s not entirely clear where Rashi got this from, but that is what Rashi says. So that’s the first part. Rabbi Yishmael said—so Rabbi Yishmael responds—what? Yes. “And one for terumah,” meaning that even if she is the daughter of a kohen and her husband is a kohen, she has become disqualified from terumah. Obviously—what do I care that she’s the daughter of a kohen? So Rabbi Yishmael answers Rabbi Akiva and says to him: a fortiori—if a divorced woman, who is permitted regarding terumah, is forbidden to the priesthood, then this woman, who is forbidden regarding terumah, is it not all the more so that she is forbidden to the priesthood? There is an a fortiori argument here from which he proves that a sotah is forbidden to a kohen. Yes, forbidden to marry a kohen. What’s the argument? If a divorced woman, who is permitted regarding terumah, is forbidden to the priesthood—a divorced woman is forbidden to a kohen—then a sotah, who is forbidden regarding terumah, certainly should be forbidden to a kohen. That’s the a fortiori argument. Okay? Seemingly that’s the structure of an a fortiori argument. What do you mean, what’s the connection? What’s the problem? What’s the problem? If this one is permitted regarding terumah and that one is forbidden regarding terumah, that means her condition is more severe. So if the lighter one is forbidden to a kohen, then the more severe one certainly should be forbidden to a kohen. That’s the ordinary structure of an a fortiori argument. So here, of course, Rashi already comments. Are you talking about a sotah? No, after the sotah is divorced, obviously she is forbidden to a kohen by the law of a divorcee. For example, if her husband dies. Before he divorced her. Then she is a widow, and so to an ordinary kohen she would be permitted, but since she strayed, she is forbidden. Right. So Rashi already notes here that it isn’t clear what Rabbi Yishmael wants from Rabbi Akiva. Because Rabbi Yishmael is seemingly attacking Rabbi Akiva: why are you bringing me a verse, “and she became defiled,” to forbid her to a kohen? I can forbid her by an a fortiori argument. Why do I need a verse? But Rabbi Akiva did not bring a verse for that. Rabbi Akiva was not dealing with the prohibition to a kohen. Rabbi Akiva was dealing with the prohibition to the husband, the prohibition to the adulterer, and the prohibition to eat terumah. There is no prohibition to a kohen stated here. So later on the Talmud corrects this and says that Rabbi Akiva also stated the law of prohibition to a kohen; it was simply omitted here. But that’s Rashi’s side note.

[Rabbi Michael Abraham] I want for a moment to look at this a fortiori argument that Rabbi Yishmael brings. He is basically saying that if a divorced woman, who is permitted regarding terumah, is forbidden to a kohen, then a sotah, who is forbidden regarding terumah, certainly should be forbidden to a kohen. Now if, when I speak about a sotah who is forbidden regarding terumah, I mean that a divorced woman is permitted regarding terumah and a sotah is indeed forbidden regarding terumah—okay? One could discuss from two sides what exactly this prohibition of a sotah regarding terumah is. Okay? And that’s your point. One could say that she becomes forbidden regarding terumah because her connection to her husband has been severed. Meaning, she can no longer eat terumah by virtue of him, because although they have not yet formally divorced, the very fact that he is obligated to divorce her basically means that the connection between them has been severed, and therefore she can no longer eat terumah by virtue of him. If that were the case, then of course the prohibition to eat terumah would only be by virtue of the husband, but by virtue of her father she could continue to eat terumah. Right? If we assume that the sotah is forbidden to eat terumah both by virtue of her husband and by virtue of her father, then that means that the prohibition to eat terumah—the prohibition of a sotah eating terumah—stems from her very being a sotah, not because of the connection to her husband. Rather, a sotah, because of her act, because of “she became defiled,” and someone defiled may not eat terumah if she has become defiled, and therefore it really makes no difference where the terumah access comes from, from the father or from the husband. She is simply disqualified from eating terumah by virtue of her being a sotah. Therefore there is no difference between the husband and the father. And in that sense, if I look at Rashi, Rashi says that “one for terumah” means that even if she is the daughter of a kohen and her husband is a kohen, she has become disqualified from terumah. Rashi is basically assuming here that the disqualification of a sotah from terumah is not a disqualification that stems from the undermining of her bond with her husband. I still don’t know why he assumes that, but that is what he assumes. Meaning, he assumes that her disqualification from terumah stems from her very being a sotah. She became defiled. She can no longer eat terumah. It is not because of the weakening of her bond with her husband.

[Rabbi Michael Abraham] Now of course the question is: how does Rashi know that? How does Rashi know that the disqualification from eating terumah, the disqualification of a sotah from eating terumah, is because of this sanction, because she became defiled, and not simply because she has separated from her husband just like a divorcee? Because then, of course, if that were the case, there would be no place for the a fortiori argument. Right? Because the fact that a divorcee is permitted regarding terumah—well, a sotah is also permitted regarding terumah by virtue of her father. She is forbidden regarding terumah by virtue of the husband, but that is also true of a divorcee, simply because he is no longer her husband or because the bond between them has been undermined. Therefore there is no place for such an a fortiori argument. And that itself is the compulsion in the Talmud that caused Rashi to explain that she becomes forbidden both by virtue of her husband and by virtue of her father. Right? Because if she were forbidden only by virtue of her husband, there really would be no place for the a fortiori argument. What is Rabbi Yishmael doing making an a fortiori argument? If Rabbi Yishmael makes an a fortiori argument, then clearly he understood that when the sotah is forbidden regarding terumah, that means both by virtue of her husband and by virtue of her father. In other words, she is essentially forbidden regarding terumah. That of course does not exist in the case of a divorcee. Because in fact the divorcee can eat by virtue of her father. The fact that she does not eat by virtue of her husband is not because she has been disqualified, but because he simply is no longer her husband. So there is no basis for making hierarchical comparisons between a divorcee and a sotah. Okay? So this comes out of the Talmud. But of course now one can go back and ask: how does the Talmud know this? So Rashi is compelled because Rabbi Yishmael makes an a fortiori argument. But how does Rabbi Yishmael know this? How does Rabbi Yishmael know that when a sotah is forbidden to eat terumah—and we know this from “and she became defiled,” it is learned from the verse—then clearly she is forbidden regardless of by virtue of whom? And therefore he can also make the a fortiori argument, and so on. But one could have said that she is forbidden only by virtue of her husband, in which case it is not an essential disqualification from terumah but only because he is no longer really her husband, and then there is no room to derive from a divorcee to a sotah that she is forbidden to a kohen. So how does Rabbi Yishmael himself know this? What? Exactly. So I think the simple explanation is that if we learn it from the verse—Rabbi Yishmael is not contesting Rabbi Akiva’s derivation from the verse “and she became defiled.” He only says that this verse teaches me the prohibition regarding terumah, and therefore I don’t need another verse to forbid her to a kohen. And later in the Talmud it is explained that there is another verse. Rabbi Akiva brings another verse. It doesn’t appear here, but there is in fact another verse, again “and she became defiled.” Okay, so on that Rabbi Yishmael says: there is a verse for terumah, but you can learn the prohibition to a kohen by an a fortiori argument, so you do not need another verse to forbid her to a kohen. The a fortiori argument will work. Why not? Sorry—the a fortiori argument, on the contrary, means you don’t need another verse because there is an a fortiori argument, the a fortiori argument does work, and Rabbi Akiva says it does not. What? To forbid her regarding terumah. But now the question is whether you need another verse to forbid her to a kohen. Rabbi Akiva says yes; Rabbi Yishmael argues no, there is an a fortiori argument. Okay, how does he know to make such an a fortiori argument? Because he assumes that the prohibition of a sotah regarding terumah is a prohibition both by virtue of her father and by virtue of her husband, otherwise there would be no room for this a fortiori argument. How does he assume that? Precisely because of “and she became defiled,” which he accepts from Rabbi Akiva, that the verse “and she became defiled” teaches that it forbids her regarding terumah. Now, it forbids her regarding terumah because she became defiled. Meaning, if she became defiled, that means she essentially cannot approach terumah; it is not because of the weakening of the relationship between her and her husband. A disqualification in—what? A disqualification in the object?

[Speaker B] Yes, exactly.

[Rabbi Michael Abraham] A disqualification in the object, in the person as object, in the woman as person as object. Meaning, so… therefore there is no difference between husband and father, and that is basically the a fortiori argument they make. Now if the sotah were disqualified from eating only by virtue of her husband, then there would be no a fortiori argument. Why? Because the sotah would be disqualified from eating by virtue of her husband simply because in an essential sense he is no longer her husband. So there is nothing to compare here to a divorcee. Okay, so basically the a fortiori argument requires this to be an essential disqualification, a disqualification in the object, and not a technical disqualification. Okay? So basically this means that when Rabbi Yishmael made this a fortiori argument, he assumed something about the nature of the disqualification regarding terumah. If his assumption about the nature of the disqualification regarding terumah had been different, then at the formal level I could still have made the a fortiori argument. If a divorcee, who is not disqualified regarding terumah—well, okay, who is permitted regarding terumah—is forbidden to a kohen, then a sotah, who is disqualified from her husband’s terumah, only from her husband’s terumah, let’s say for the sake of discussion, is it not all the more so that she should be forbidden to a kohen? At the technical level there is a structure of an a fortiori argument here. Meaning there is a severity in the sotah compared to the divorcee. But I would not make that a fortiori argument in such a situation, as I said earlier. Why not? Because behind the laws I assume there is an idea. The a fortiori argument is not a formal a fortiori argument between laws. The laws express an idea, and the a fortiori argument is made on the plane of the idea. Okay? It’s not that you can take four laws—here exempt, here liable, here liable, therefore here too it should be liable. It doesn’t work that way. You have to look at what lies behind the laws and see where they come from. Do they really reflect severity, or do they simply reflect some technical matter? Okay? That’s also true regarding the divorcee: the fact that she is not forbidden regarding terumah—meaning that what she cannot eat by virtue of her husband is not because she is forbidden, but because he is not her husband. She still cannot eat terumah in terms of the practical law, but that doesn’t matter, because conceptually there is no underlying idea of severity behind that. It’s not that she is forbidden to eat terumah because of some added severity.

[Rabbi Michael Abraham] So the lesson I want to draw is that when I deal with certain laws and decide whether or not to make an a fortiori argument on the basis of those laws, it is not enough to look at the laws themselves. Meaning, I need to see what idea stands behind the laws. There is an issue here of interpretation, and in a moment I’ll explain this a bit more clearly. That’s regarding the divorcee, and regarding the sotah it’s the same thing. I say: if indeed she is forbidden both by virtue of her father and by virtue of her husband, then the idea is that the sotah, in the person—in the woman as person—is forbidden regarding terumah, right? Then you can begin making an a fortiori comparison with a divorcee. But if she were forbidden only by virtue of her husband, and not by virtue of her father, then although she would have a law that is more severe than a divorcee, because the sotah is forbidden to eat terumah by virtue of her husband and the divorcee—well, he simply is not her husband, so therefore she isn’t forbidden—then seemingly there would be room to make an a fortiori argument anyway. But I would not do that. Why not? Because the prohibition of a divorcee to eat by virtue of her husband is really not that she has been disqualified from terumah; rather, for this purpose he simply is not her husband. He already has to divorce her, so he is already no longer her husband. I think I spoke at the beginning of the year about the two stages, right? Of betrothal and marriage. Did we speak about that? That betrothal and marriage are built in two stages. There is betrothal and afterward there is marriage. And I argued that with divorce too this happens in two stages. There is the dissolution of the marriage, and you still need to give a bill of divorce to dissolve the betrothal. And from the moment he sets his mind to divorce her, or from the moment he is obligated to divorce her—and in the case of a sotah he is obligated to divorce her—then again he no longer has rights to her produce, there are all kinds of practical differences from which one can see that she basically returns to the status of a betrothed woman. Meaning she ceases to be a married woman and returns to the status of a betrothed woman. And therefore it could be that this is what causes her no longer to eat terumah by virtue of her husband, because in an essential sense she is no longer married to him. She is betrothed to him, but not married to him. But then indeed that would not be a disqualification from terumah; it would simply be separation from her husband, and there would be no basis to learn from that to her prohibition to a kohen.

[Rabbi Michael Abraham] So once again you see, both on the side of the divorcee in this equation and on the side of the sotah in this equation, that the laws by themselves are not enough to generate—or not generate—an a fortiori argument. I have to think about what stands behind the laws. And this is a very important point, because for example the Briskers—in the Brisk Haggadah, it’s called MiBeit Levi, Gorlitzer, yes, yes—there, at the end of the first volume, there is… no, no, the end of the first volume is stories, which is fascinating in its own right, Brisk stories. But I mean earlier, on “Who Knows One.” On the poem “Who Knows One,” so at “thirteen” it says “thirteen attributes.” What are the thirteen attrib—

[Speaker C] Thirteen—

[Rabbi Michael Abraham] Attributes? “The Lord, the Lord, compassionate and gracious”—that’s—

[Speaker C] They say the thirteen attributes of mercy, obviously.

[Rabbi Michael Abraham] Right. But in the Brisk Haggadah, the thirteen attributes are: a fortiori reasoning, verbal analogy, deriving a principle from one verse. The thirteen hermeneutic principles by which the Torah is interpreted. It’s wonderful, this whole thing, and it’s said as if innocently. Meaning he doesn’t even… as if it’s obvious: “thirteen attributes,” and immediately below he writes: this means a fortiori reasoning. It never even enters his mind that the reference might be to “The Lord, the Lord, compassionate and gracious.” That’s not at all in the picture—there are no such attributes. What kind of attributes are those? We’re talking about a fortiori reasoning. So in his commentary there on a fortiori reasoning he brings some passage—I don’t remember the source because it’s a compilation—but he brings a passage there where Rabbi Chaim claims that a fortiori reasoning is a formal principle. It is not a principle that has logic behind it. He brings some a fortiori argument from the grasses in Midrash Rabbah at the beginning of Genesis. The grasses made an a fortiori argument from the trees: if regarding the trees it says “according to its kind,” then the grasses too should be “according to their kind.” It doesn’t matter, some strange a fortiori argument like that. And from there Rabbi Chaim wants to claim that a fortiori reasoning is a formal principle. What does it mean that it is a formal principle? Give me three laws and I can derive the fourth law. Meaning, basically the structure is like this. Maybe I’ll just do this here for a moment.

[Speaker B] It’s just technical, like, and not logic?

[Rabbi Michael Abraham] Yes. Let me do this for a second. Yes, the Briskers really love this notion that there is no logic behind things.

[Speaker B] Brisk isn’t logic?

[Rabbi Michael Abraham] No. For example, we only ask “what,” you know, not “why.” Wait—why do we—

[Speaker B] When you accept that you’re doing a commandment and you don’t think “for the sake of the commandment,” then it’s as if you didn’t fulfill it. You’re not doing it because it’s written, but because you think it’s not good.

[Rabbi Michael Abraham] If—

[Speaker B] If you don’t steal because it doesn’t fit your morality, then—

[Speaker C] It isn’t considered that you kept the commandment. Seemingly this comes from the fact that a person reasons on his own, no? That a fortiori reasoning a person can derive on his own.

[Rabbi Michael Abraham] Yes, but they still say that it is still a formal matter. You can derive it on your own—so what?

[Speaker C] This case explains why a person can derive it on his own, because it’s only formalism.

[Rabbi Michael Abraham] Ah, I thought you were saying the opposite. Wait, I’m not managing to understand how to share—screen document, okay, wait, here. Okay, can you see it? Good. I found it in the end. So look. Basically the structure of an a fortiori argument is built like this. Let’s take the a fortiori argument from the Mishnah in Bava Kamma 25, right? Rabbi Tarfon and the sages. So here we have tooth-and-foot and horn. Horn—I’m writing this in an Indian style here. Horn, and this is tooth-and-foot. Okay?

[Speaker C] Tooth—

[Rabbi Michael Abraham] And foot, tooth-and-foot, and this is… I’m sure there’s a better way to do this, but I’m not capable of it. What? Don’t worry.

[Speaker E] Fine, it’s not—

[Rabbi Michael Abraham] Don’t worry, I’m not going to write much. That’s it. Okay? We have public property, private property, tooth-and-foot and horn. All right? Now the structure is this: here exempt—tooth-and-foot in public property is exempt, right? Horn in public property is liable—actually it’s half, but let’s ignore that for the moment; I’m writing one. Okay? Liable, it doesn’t matter to me right now how much. And tooth-and-foot in private property is also liable, right? Now I ask: what is the law here? Right? That’s the question mark. Okay? Asking what the law is here. So that’s an a fortiori argument. And the a fortiori argument tells me that the law here too is one. Right? Now, one can see this in a completely formal way, and every a fortiori argument is built this way. So this is the generic structure of a fortiori reasoning. Okay? So again, there is a fortiori reasoning based on one datum—the lighter and heavier cases that appear in the Torah are based on one datum. Right? It’s an a fortiori argument: if one is liable for opening, then for digging all the more so, no? Because “if a person opens a pit” or “if a person digs a pit”—if opening an existing pit makes you liable, then if you dig the pit you certainly should be liable, right? That’s an a fortiori argument.

[Speaker F] But that’s an a fortiori argument based on one datum.

[Rabbi Michael Abraham] Right? That’s an a fortiori argument based on one datum. Okay? And from it you draw a conclusion. Talmudic a fortiori reasoning is usually based on three data points, not one. And it is always built this way. This is the generic table of an a fortiori argument. Now one could look at this and say: I have a formal rule—once you draw such a table, and you have three laws like these, one, zero, and one, then the result here is one. Okay? A fortiori. And that’s what Rabbi Chaim claims. Rabbi Chaim says: once you draw me such a table and you have these three laws, I don’t need to get into the question of the relevance of the laws, whether they are connected to each other or not connected to each other; it is a completely formal matter, there is no reasoning behind it. Okay? Today my goal is to show you that this is not true, but that is what he claims. And the claim that it is not formal basically means that just because you have a structure like this does not yet mean that you make an a fortiori argument. Here, for example, with the divorcee and the sotah, we have such a structure. Right? Divorcee, and let’s say the sotah were forbidden only by virtue of her husband, okay? So now we have here divorcee and sotah; this is the divorcee, this is the sotah, this was by virtue of the husband, let’s say forbidden by virtue of the husband, and here it would be with a kohen—whether she is forbidden to a kohen. Now their general structure is this: a divorcee does not eat by virtue of her husband, but she is not—it’s not a prohibition. A sotah is forbidden by virtue of her husband. A divorcee is forbidden to a kohen, therefore a sotah is also forbidden to a kohen. Formal matter. Okay? But I say no—what do you mean? If she is forbidden only by virtue of her husband, then this thing doesn’t interest me. This fact—that the sotah is forbidden to eat terumah by virtue of her husband—doesn’t really mean that there is a greater severity in her. Why? Because the fact that she is forbidden by virtue of her husband is not because she has been disqualified from terumah, because in relation to her father she is permitted. What I want to say is only that the connection between her and her husband has been undermined, and therefore she cannot eat terumah. So the same structure itself, the table exists, these are the data, and still I will not make an a fortiori argument. Why won’t I make an a fortiori argument? Because my claim is that one has to interpret the data and understand whether they really reflect relations of severity, and only then make the a fortiori argument. In other words, an a fortiori argument is not a formal procedure. Okay? What? Had I reasoned on my own I might have said… uncertainly, but I would have reasoned. Meaning, it is not enough to draw this structure. When I give you this structure, that still isn’t enough to derive the conclusion. Okay? There is something behind these laws, and you have to take it into account when making an a fortiori argument. And that is basically what we learn. That is what we learn from this passage: the mere fact that the laws are there does not yet mean anything. You have to think about what lies behind the law.

[Rabbi Michael Abraham] Now look at what follows, because it continues exactly in the same direction. Tosafot asks as follows. “This one, who is forbidden regarding terumah, is it not all the more so that she is forbidden to the priesthood?” Tosafot says: one cannot say, “An Israelite woman will prove it, for she is forbidden regarding terumah and permitted to the priesthood.” What are you giving me an a fortiori argument from divorcee to sotah for? An Israelite woman will prove that what? An Israelite woman is forbidden regarding terumah, right? Why? Because she is an Israelite woman. But she is permitted to the priesthood—she may marry a kohen. No, no, not that if she marries she is permitted terumah, but that she is allowed to marry a kohen. Although she is forbidden regarding terumah, she is allowed to marry a kohen. Okay, so here is a refutation of the a fortiori argument. Because you are showing me that if someone is forbidden regarding terumah, then clearly she should be forbidden to a kohen—not true. An Israelite woman proves otherwise, because she is forbidden regarding terumah and permitted to a kohen. Okay? Now of course no one would imagine that this is a real refutation except Rabbi Chaim. And Tosafot answers: this is its meaning—if a divorcee, who is permitted regarding terumah, is forbidden to the priesthood through an act that was done to her, namely divorce, then this one and so on—sorry, divorce, this one and so on—is it not all the more so that she is forbidden to the priesthood through an act of sexual misconduct that was done involving her? But an Israelite woman—what disqualifies her from terumah is not through an act. One could—

[Speaker F] Maybe say even more, that she became forbidden and wasn’t always forbidden, sort of.

[Rabbi Michael Abraham] I would say the opposite. An Israelite woman is not even “forbidden regarding terumah”; that isn’t relevant in the context of terumah. There is no prohibition upon her regarding terumah. Right? That’s the other side of the same coin. What Tosafot is basically saying—and notice the wording of Tosafot, it really is a bit formal—I would phrase it this way: what are you bringing me from an Israelite woman for? An Israelite woman is not forbidden regarding terumah. Meaning, she is permitted regarding terumah; it’s just that she has no one through whom to eat it. But she has no disqualification regarding terumah. And that is why I prefaced with the earlier analysis and said that when we compare a divorcee with a sotah, the assumption is that in the sotah there is an essential disqualification from terumah, not because the relationship with her husband has been undermined, but because there is a disqualification in the person. Meaning, she can no longer eat terumah because in her essence she has become defiled. The Israelite woman has not become defiled. She is not forbidden regarding terumah; she simply has no one through whom she can eat, neither by virtue of her father nor by virtue of her husband, because she has no husband. Okay? That’s all. So in that sense she is like a divorcee. What do you want her to prove? She proves nothing. Okay? A divorcee is allowed to eat terumah if her father is a kohen, but by virtue of her husband she cannot eat terumah. And likewise if the Israelite woman’s father were a kohen, she would be allowed to eat terumah—she just wouldn’t be an Israelite woman. Fine, but you cannot make an a fortiori argument like that. You see once again that the laws seemingly exist—you can put in an Israelite woman—but it is not relevant. Meaning, we understand that these laws do not reflect a substantive relation of severity, and therefore the Israelite woman is not a relevant parameter.

[Rabbi Michael Abraham] Now Tosafot brings another answer. Another answer: “And furthermore, if an Israelite woman marries a kohen, she will be permitted regarding terumah, which is not the case with these women, who never have permission forever.” Right? A sotah never has permission forever—and by that he means that if she later marries a kohen, okay, then she still cannot eat terumah, despite being married to a kohen. Now again, if he divorced her she cannot marry a kohen because she is divorced—but if her husband died? If her husband died, she still cannot eat terumah, and therefore clearly she also cannot marry a kohen. That’s the a fortiori argument. Okay? So I think Tosafot’s second answer is really nothing more than another formulation of the same idea. And therefore I think this again reflects that Tosafot too was a bit formalistic, because really you’re saying the same thing. You are saying the same thing through different indications. One indication is the question whether she was disqualified through an act or not through an act. The second indication is what would happen if she actually married a kohen—would that permit her? But it’s the same idea. The idea behind this is basically that in the case of the daughter of a kohen—sorry, in the case of the Israelite woman—there is no disqualification from terumah; there is only no access to the permission yet. That’s all. You can look at it from this angle or that angle, but it’s the same thing. These are not two answers of Tosafot. They are two indications of the same idea. Once Tosafot sees this as two answers, or two different indications, it seems that he too is speaking somewhat on the formal plane. Otherwise you should have said straightforwardly: what do you want? These laws do not really reflect a relation of severity. Then you’re saying the same thing.

[Rabbi Michael Abraham] Now what I basically want to say is that what we see from this brief discussion is that when we look at a set of laws, that still does not allow us to make an a fortiori argument, and it certainly does not necessarily allow us to know what the law is in the fourth, missing slot. And here I want to pause the passage for a moment and really try to talk about this issue of a fortiori reasoning. It will come up again in the next part of the passage, and I’ll need it there too. So maybe I’ll start by going back to the a fortiori table. Look. Okay, it saved it for me. Fine, here is the earlier masterpiece. Let’s look at this table for a second. Now I’ll ask you a few questions, okay? First question: actually it doesn’t say “one” here; really it says “half.” What? Innocuous? Yes, an innocuous horn, right. There the discussion is about an innocuous horn, because the whole discussion is what the law of an innocuous horn in private property will be—whether it will be half or one—so the whole discussion there is about an innocuous horn. So really it says “half,” okay? Now the question is: what should be written in the next slot? Now look—how is an a fortiori argument built? So I began by mentioning earlier that the biblical a fortiori argument is based on one datum. Meaning, if one is liable for opening, then for digging all the more so. Okay? Or, “Behold, the Israelites have not listened to me; how then should Pharaoh listen to me, seeing I am of uncircumcised lips?” If the Israelites do not obey me, then Pharaoh will obey me? Meaning, you see, this is an a fortiori argument with one premise and one conclusion. Now notice: “The Israelites did not listen to me, so how will Pharaoh listen to me?” is not a “what is true in two hundred is certainly true in one hundred”; it’s not like opening and digging. Opening and digging is that if something is included in the larger case, then certainly the smaller case is included. Meaning, when you dig it, within the act of digging there is also an act of opening. Okay? So the larger case actually contains the smaller. It’s not that digging is more severe than opening; digging actually contains opening plus something more. Okay? So it contains it in practice. That’s an a fortiori argument often considered stronger than an ordinary one; it’s an a fortiori argument of inclusion. But the a fortiori argument of “if the Israelites have not listened to me, how will Pharaoh listen to me?” is not an argument of inclusion. Rather, it is simply more likely that the Israelites would listen to me than that Pharaoh would listen to me. So if they did not listen, then Pharaoh certainly will not listen. Okay? There it is an a fortiori argument on the basis of one datum, but not an argument of inclusion. So let’s talk about that; it’s easier for me. How is that a fortiori argument built? There is actually a reasoning here. The reasoning says that the likelihood that Pharaoh will listen to me is lower than the likelihood that the Israelites will listen to me. And if the Israelites did not listen, and in Pharaoh’s case the likelihood is even smaller, then certainly he will not listen. Right? That’s how the argument is built. Meaning, there are really two premises here, not one. One premise is that the Israelites did not listen to me. That is one factual premise. A second premise is a hierarchical relation: the chance that Pharaoh will listen to me is smaller than the chance that the Israelites will listen to me. Right? So basically the a fortiori argument is built on a fact and a hierarchical relation.

[Rabbi Michael Abraham] Let’s return to the table here. What is happening in this table? In this table, in fact, the a fortiori argument is built on three data points, not on one. In this case these are laws and not facts, but that doesn’t have to be so. And from these three data points I want to infer the missing law: what is the law of horn in private property? It too is liable. In a moment we’ll talk about half or one, but it too is liable. How do I do that? The answer is that basically I do this: look at the top row. In the top row I see—yes, the row of tooth-and-foot—what do I see? That private property is more severe than public property. Right? Since tooth-and-foot is liable in private property and exempt in public property, private property is more severe than public property. I go down to the lower row, the row of horn, and I say: fine, if horn in public property is liable, and that is the lighter case, then in private property, which is more severe, certainly horn should be liable. Therefore the result is one. Okay? That is one possibility. So what exactly did I do here? I took the two data points of the top row, generated from them a hierarchical relation saying that private property is more severe than public property, and now I’m back to the biblical a fortiori argument. I have one datum, this one, and a hierarchical relation that emerged from those two. Meaning, the difference between the biblical a fortiori argument and this one is not essential. In the biblical a fortiori argument, the hierarchical relation comes from reasoning: it is more plausible that Pharaoh is less likely to listen to me than the Israelites are. Here I do not have a direct piece of reasoning; I have two halakhic data points, and from those halakhic data points I derive a relation of hierarchy: private property is more severe than public property. But after I derive from the two upper data points a relation of hierarchy, I basically have a case of one law plus a hierarchical relation, and from that I derive the result. We have returned to the case of the biblical a fortiori argument.

[Rabbi Michael Abraham] But there is another thing. In this a fortiori argument, unlike the biblical one, we have an alternative formulation. Now look at the right-hand column, okay? The right-hand column basically says that horn is more severe than tooth-and-foot. Right? Now suppose I erase the right-hand column. I now have another hierarchical relation: horn is more severe than tooth-and-foot. Now I move to the left-hand column. I have a law that tooth-and-foot is liable in private property, and I have a hierarchical relation learned from that column. I apply it here, and therefore horn too is liable in private property. Okay? Do you think these are two identical arguments?

[Speaker G] Two sources, from here and from there. What—

[Rabbi Michael Abraham] What do you mean, two sources? I have three halakhic data points. I presented here two derivations of an a fortiori argument, two formulations of the a fortiori argument. Are these formulations identical? Why not?

[Speaker G] Degrees are degrees, “song of ascents,” right?

[Rabbi Michael Abraham] But the question is: what—why is there a difference between them, if there is one? What’s the difference?

[Speaker G] Here you’re saying private property versus public property, and here it’s—

[Rabbi Michael Abraham] Meaning that the first formulation, which compares these two laws—let’s call it the “column formulation” for the sake of discussion—basically tells me that the left-hand column is more severe than the right-hand column. That private property is more severe than public property. That is the premise, and then I move to the law and apply the hierarchy. In the row formulation, notice that I look at the right-hand row and create a hierarchy between rows. So the hierarchical relation is not between domains but between damage-causers: that horn is more severe than tooth-and-foot. A completely different premise. Okay? So basically these two formulations are not two equivalent formulations of the same derivation. They are two entirely different derivations. Here, let me give you a practical difference. Didn’t we say it doesn’t say “one” here, but rather “half”? Right? Now let’s go with the column formulation. In the column formulation, what am I basically saying? I look at the top row and say private property is more severe than public property. Right? So with horn too, private property will be more severe than public property. So what will be written here? What will the law be? No—half. Half. Since all I know is that private property is more severe than public property. Horn in public property is half. So in private property it is at least half, and “it is sufficient for what is derived from the law to be like the source from which it is derived,” so I cannot say how much more than half, so it’s half. I can only obligate half. More than that I have no proof. I’m assuming the rule of sufficiency here—that’s the dispute between Rabbi Tarfon and the sages there—but for the moment I’m assuming the simple logic. So half, right?

[Rabbi Michael Abraham] Now let’s look at the a fortiori argument of the rows, not of the columns. So I say: the a fortiori argument of the rows basically means I look at the right-hand column and say that horn is more severe than tooth-and-foot, right? Now I go to the left-hand column. Right? So if tooth-and-foot, the lighter case, is liable in private property for full damages, then horn in private property, which is more severe, is liable for one. Now you see that these two arguments yield a different conclusion. The first argument yields a conclusion of half liability. The second argument yields a conclusion of one. So there you have it: these are not two equivalent arguments. They are not two formulations of the same argument. They are different arguments. Okay? Now let me show you this from another angle. Let’s suppose I found a refutation. Let’s say I found a refutation. I found a third domain: the moon. All right? On the moon, what happens? What’s the law? Fine. But on the moon something interesting happens: tooth-and-foot is liable and horn is exempt. All right? That’s the law on the moon. Now that refutes the a fortiori argument, right? The moon will prove that tooth-and-foot is liable, just like in our passage, exactly. “An Israelite woman will prove…” that in her case she is forbidden regarding terumah but permitted to a kohen. Right? Forbidden regarding terumah and permitted to a kohen. That is exactly the refutation we saw. Okay? Now this refutes the a fortiori argument. Why does it refute the a fortiori argument? Let’s try to think. Because this column basically shows me that my conclusion that horn is necessarily more severe than tooth-and-foot is not correct. Here, for example, tooth-and-foot is more severe, not horn. Right? Does this collapse the a fortiori argument?

[Speaker F] If it’s only in one domain, it’s still more severe overall, right?

[Rabbi Michael Abraham] And that knocks down the a fortiori argument from the rows. Because the a fortiori argument from the rows assumes that horn damage is necessarily more severe than tooth-and-foot damage. That’s not true. Here’s a counterexample. But the a fortiori argument from the columns doesn’t assume any relation at all between tooth-and-foot and horn. It doesn’t care about that. What it assumes is that private property is more stringent than the public domain. Right? And that remains intact. The moon proves nothing about that issue. So the a fortiori argument remains standing. I could have filled in one possibility because of the a fortiori argument from the rows and not because of the a fortiori argument from the columns. How would I refute the a fortiori argument from the rows if I added that kind of refutation? A row refutation. Here—if I added such a refutation. Say some other damaging category, a tail. Okay? And a tail has interesting laws too. What? That in the public domain one is liable for tail-damage, but if someone causes damage with a tail on private property, one is exempt. Okay? You understand that this refutes the hierarchy between the columns. Right? It means that private property is not necessarily more stringent than the public domain. But that does not touch the question of the relation between the rows. It has nothing to do with it. So in other words, whether you bring me this kind of refutation or that kind of refutation, the a fortiori argument remains as it was. It hasn’t fallen. Okay? That is basically the claim. Why? Because, as I said before, these are two different arguments. Not two different formulations of the same argument, but two different arguments. Therefore, if one falls, the other does not necessarily fall. Each thing has to be judged on its own. You have to check whether the assumption behind the a fortiori argument has fallen or not.

By the way, just a side comment. There are people who think—Adolf Schwarz, for example, a scholar of the hermeneutical principles from the rabbinical seminary in Vienna, already in the nineteenth century, I think—he wanted to argue that an a fortiori argument is deduction. Meaning, a necessary logical inference. But that’s nonsense. Obviously not. Why? Because a necessary logical inference cannot be refuted. There is no such thing as a refutation of a logical inference. You can find a mistake in a logical inference. You cannot refute a logical inference. If the logical inference is necessary, then it is necessary. You won’t find a counterexample. You won’t find a counterexample to the fact that two plus three equals five. Right? Two plus three equals five. When you can refute an argument, that means it is not a logical inference. And why? The example we saw before. What we have in an a fortiori argument is three data points. When I derive the fourth datum—say, from these three, when I derive the fourth law from those three—that is certainly not a logical inference. Why? Because it is really assuming some kind of hierarchical relation. What is it saying? Let’s take, for example, the inference from the columns. I say: from these two laws I derive that private property is more stringent than the public domain. I’ve made a wild generalization here. Because I know that it is more stringent regarding tooth-and-foot damage. Who says that this is also true regarding horn, or in every respect? I am generalizing. Right? After making that generalization, I have taken two specific laws and created from them a general rule. The general rule says that private property is always more stringent than the public domain. I made a generalization. Now I apply it to horn too, saying that with horn as well private property is more stringent than the public domain. Right?

When I bring a refutation—here, for example, I brought a refutation, right? Here, with tail-damage, private property is not more stringent than the public domain. What have I refuted? The generalization I made on the basis of this row. I am not denying the facts. The facts are facts—explicit accidental facts. They are true facts. But the generalization I made on the basis of these two laws—that in general private property is always or necessarily more stringent than the public domain—that is not true. It’s not always so. Here, in fact, is a counterexample: with tail-damage, it isn’t. Okay? So the fact that behind this inference sits a generalization—that is exactly what separates this inference from deduction. A generalization sits behind it. And the refutation always attacks the generalization. It does not attack—refutation cannot attack deduction. The refutation attacks the generalization. The generalization you made is not necessary; here is a counterexample.

By the way, this is very reminiscent of John Stuart Mill’s challenge to deduction. We are used to dividing logical inferences into three types: analogy, induction, and deduction. Deduction is an inference from the general to the particular. Socrates is a particular case within the group of all human beings. So that is an inference from the general to the particular. It is a necessary inference. Then there is an inference from the particular to the general; that is called induction. Socrates is mortal, Moshe is mortal, therefore all human beings are mortal. I take particular examples and create from them a general law. So if that was movement from the general to the particular, this is movement from the particular to the general. And analogy is from particular to particular. If Moshe is mortal, then Yankele is mortal too. Okay? That is an inference from particular to particular, not a generalization and not specification. Or if all donkeys have four legs, then all horses have four legs too. Even though both sides are general, that is still analogy. Why? Because I did not learn from the general to a particular contained within it, or to a more inclusive general category, but rather to another general category that is foreign to it. So it’s the same as going from particular to particular, except that here the “particular” is a group—that doesn’t matter. Okay, so that is analogy.

And what John Stuart Mill argues is that contrary to what we think—that in deduction the conclusion is necessary—that too is nonsense. Because let’s say: all human beings are mortal, and Socrates is a human being, therefore Socrates is mortal. But how do we know that all human beings are mortal? By induction. So in fact, at the base of deduction there is always a hidden induction. We don’t put it on the table, but it’s there. Therefore, if you find an example of someone who is not mortal—if you discover that Socrates is not mortal—there is no conceptual problem with that. It just means your assumption that all human beings are mortal was not correct. You made an unjustified generalization there. Here we have a counterexample. So in that sense, you can view the a fortiori argument as deduction—deduction, if you give up the necessity of deduction. Meaning, if you also take into account how I arrived at the major premise of the deduction. Okay? So here too it’s the same. I take two laws, create from them a generalization, and apply it to the bottom row. Then someone brings me an even lower row that shows me that this generalization is a flawed generalization. Meaning, there are examples where it does not work. And therefore you can no longer know. Okay, you can no longer know whether with horn it is so or not—and that is enough. Right? In order to refute, you do not have to prove that it is false. It is enough to raise the possibility that it is false. Because the burden of proof is on the one who wants to derive the law. All right? So I show you that it is not necessary to infer this law.

So it turns out that every refutation should have refuted only one formulation of the a fortiori argument, and then I could have rotated it and kept it in force—whether it is a column-refutation or a row-refutation. Okay? But in the Talmud we see that this is not true. Once you bring a refutation, the a fortiori argument falls. What did Tosafot say here? “The Israelite daughter will prove it.” Why will the Israelite daughter prove it? Just rotate the a fortiori argument. What’s the problem? No. Once I found a counterexample, even though it is a column-example, it refutes both formulations of the a fortiori argument. That is what we see in the Talmud. It is not yet logically clear why, but that is what we see in the Talmud. Okay? Right. So apparently what we see is that these probably are indeed two formulations of the same argument. They are not two different arguments. They are two formulations of the same argument. Why? Logic doesn’t seem to show that. We also saw with the rule of “enough” that the results of the two formulations are even different. So clearly there are two different formulations here. So why is it that when one falls, the other always falls as well? That is the puzzle here. So now I want to explain for a moment what really stands behind all this.

[Speaker E] And I’ll do that—I’ll go back for a second to… okay.

[Rabbi Michael Abraham] I’ll go back for a moment to the original table with zero and one; leave aside the rule of “enough” for a second. Okay? By the way, there are two places in all of the Talmud, I think—as far as I know—where they rotate an a fortiori argument. They rotate the a fortiori argument. Meaning, you bring some refutation or something, there’s a problem with one formulation, and you rotate it and propose the alternative formulation. One is in Niddah and one is in Bava Kamma there in the Mishnah that I mentioned earlier. And in both of those places it is an a fortiori argument that includes the rule of “enough.” Because what is written here is not one but a half. But in every a fortiori argument where what is written here is one, like here—if this and this are the same, then the diagonal is, yes, an identity matrix—then in all those places they do not rotate the a fortiori argument. That too is an interesting puzzle: why? Meaning, why is it specifically with the rule of “enough” that suddenly we notice that these are two different arguments, but in a regular a fortiori argument with refutations, for us it is the same argument? I don’t know if I’ll get to that, but it’s just a comment.

So I want to explain for a moment what actually stands behind these things. Let’s look for a second—yes, I’m leaving the refutations now. What is the… my eraser isn’t doing much. So we have—this is the structure of an a fortiori argument, right? You could have put it here. So my claim is the following. Really—and this is what emerges from the passage I showed you before, and here I return to that important point—I am basically claiming that an a fortiori argument is not based on the laws that appear in the table. Or not only on them. You need to think about what stands behind those laws and understand whether those laws really reflect relations of greater and lesser stringency or not. Only then can you carry out the a fortiori argument. Let me show you what that means, and you’ll immediately see the implications.

When I look at this table, I say to myself: I look at the top row. Let’s talk about the a fortiori argument from the columns. I look at the top row and infer that private property is more stringent than the public domain. Let’s translate that. When I say that private property is more stringent than the public domain, what that means is that private property has some feature that, in some sense, causes it to be more stringent than the public domain. Let’s call that feature alpha. Okay? So the public domain has a property at the level of alpha, and private property has a property—well, my handwriting is hieroglyphics—what, two alphas. Okay? That’s not a lambda, yes? The Holy One is a great artist, as the midrash says—not me. So private property is two alpha and the public domain is alpha, right? Then I say that if tooth-and-foot are exempt in the public domain and liable on private property, that means there is some parameter alpha present in both the public domain and private property. That creates a hierarchical relation between them. Or maybe I’ll define it the other way around—you’ll soon see why—let’s write two alpha here. Okay. Why do I say that? Look.

Tooth-and-foot have the property alpha, basically. Now, the public domain requires two alpha in order for us to impose liability there. The damager has to have special severity in order to be liable in the public domain. Only severe kinds of damage are liable in the public domain, okay? Now tooth-and-foot have a level of severity of alpha. That is not severe enough. You need a severity of two alpha in order to impose liability. Therefore this is zero. By contrast, on private property, a severity of alpha is enough to impose liability. Therefore here one is liable. You see why I marked alpha and two alpha? When private property has alpha, that means private property is more stringent, not more lenient. It means that already from alpha onward you are liable there. It is more sensitive to liability, okay? In other words, any damager with a severity level of alpha is already liable there. So it is more stringent, right? For damages also, is it common or not? Tooth-and-foot are common.

[Speaker B] What—

[Rabbi Michael Abraham] Why does it matter whether it’s common or not? So you’re asking me what alpha is; I don’t care what alpha is. I’m saying there is some alpha. If you want to say alpha is frequency, fine, I don’t care. There is some feature, alpha, which tooth-and-foot have at level one, and horn has at level two alpha, eh? Yes, exactly. So I say: horn has it at the level of two alpha; horn is more severe. And then I say this: because it is more severe, it will even be liable in the public domain. In the public domain you need a truly severe damager in order to impose liability, okay? Therefore horn will be liable in the public domain, and tooth-and-foot will not be liable, because it has only a severity level of alpha. I’ve now explained these three laws, right? I now have an explanation. And what is the result? What is the conclusion? Now I ask whether horn is liable on private property—sorry? What’s the answer? Certainly yes. Because in order to impose liability on private property, even a level of alpha is enough. So horn, which has a severity level of two alpha, certainly will be liable, right? Therefore clearly the answer is one.

But notice what this really means. It means that when I explain and carry out the a fortiori argument from the columns, I look at the top row and build a hierarchy between private property and the public domain—but implicitly I have also assumed a hierarchy between the rows. Right? You cannot say that there is a hierarchy between the columns that has nothing to do with—let’s say tooth-and-foot are actually beta. Fine? Not alpha but beta. And this is two beta; it is more severe than that, but on some other parameter, a parameter that is irrelevant. Okay? Suppose this is two beta. That won’t help, right? It won’t explain the columns in the table. Why are tooth-and-foot exempt in the public domain and liable in the damaged party’s courtyard? Beta explains nothing. You have to have the parameter alpha appear within tooth-and-foot. If that is the parameter that determines the hierarchy between the domains, it must also be a characteristic of the damaging categories. Otherwise the hierarchy between the domains will not be explained—it will not explain the data in the table. Do you understand what I’m saying? Again. If the characteristics of—suppose all of a sudden there’s a hierarchy, but—

[Speaker C] It doesn’t have to be specifically that characteristic.

[Rabbi Michael Abraham] It has to be. If it’s beta and two beta, it won’t help.

[Speaker C] If beta equals two alpha—

[Rabbi Michael Abraham] No, beta equals two alpha isn’t beta anymore; then you’re just calling it by another name. I mean beta is a different parameter, unrelated to alpha—different. That won’t help. You need the alpha-characteristics to be characteristics of the damaging categories too, not only of the domains. Otherwise the hierarchy you made between the domains won’t be enough to explain the laws in the table. In order to explain those laws, you need to tell me that tooth-and-foot have a level of alpha, and then I can understand why in the public domain they are exempt—because they don’t have enough to impose liability. Why on private property they are liable—because they do have enough to impose liability. And you have to assume some property alpha in tooth-and-foot and also in horn. Not only do you have to assume it, you also have to assume that there is a hierarchy between them. Meaning that if tooth-and-foot are alpha, right—if tooth-and-foot are alpha, then horn must be higher on the alpha scale; it must be more. Meaning if this is alpha, then this is two alpha. Do you see that the hierarchy between the domains actually projects or dictates a hierarchy between the damaging categories? It must. In other words, you cannot detach the hierarchy between the domains from the hierarchy between the damaging categories, and vice versa.

Meaning, if I now explain the a fortiori argument from the rows, then—

[Speaker C] I’m saying it’s because of an a fortiori argument, like because I’m saying that what applies in the public domain—if horn is more severe in the public domain, then automatically horn is more severe, so to speak.

[Rabbi Michael Abraham] Exactly. And the claim is that I am making—

[Speaker C] an earlier a fortiori argument already, so to speak.

[Rabbi Michael Abraham] No, not an earlier a fortiori argument. Rather: if horn is more severe than tooth-and-foot in the public domain, that means that the feature by which horn is more severe than tooth-and-foot is relevant to liability in the public domain. Because if it were more severe in a way that is not relevant to liability in the public domain, then it would be uninteresting; you would not be able to explain the laws in the table. In order to explain the laws in the table, you need to tell me that the parameter alpha that distinguishes between the two damaging categories is also relevant to the domains, at some level. And once I understand that, I look at the laws and it immediately dictates that here there is alpha and here there is two alpha. As a single assumption, it has to be that way. Okay?

Now you see that we have discovered something remarkable. We have basically discovered that once I established a hierarchy between the rows, that necessarily also implicitly assumes a hierarchy between the columns—otherwise it won’t help. And vice versa. Do you understand what that means? It means that the two formulations are the same formulation.

[Speaker E] It’s the same consideration.

[Rabbi Michael Abraham] Meaning, they are two different formulations of the same argument. I am basically saying that there is a hierarchy both among the domains and among the damaging categories, built in this way. And it doesn’t matter whether I take this column as the starting point or take this row as the starting point. In the end, if I want to explain the laws in the table, then for either formulation I must assume a hierarchy both between the rows and between the columns. If I now translate the formulations I mentioned earlier: horn is more severe than tooth-and-foot in the public domain, therefore it is also more severe on private property. I said that this is apparently a different argument from: private property is more stringent than the public domain regarding tooth-and-foot, therefore it is also more stringent regarding horn. No—it doesn’t work that way. These are not two different formulations. It is one and the same formulation. It is one and the same formulation.

The formulation basically says: please give me a model of alphas and betas and so on that characterizes the relevant levels of severity of the damaging categories and of the domains, and that will explain the three given laws. You see? These three laws—this, this, and this. Give me a model. This is the model that comes out. Once this is the model that comes out, I know that here I fill in one. Because horn has two alpha, and on private property alpha is enough to impose liability, so obviously the missing law I am looking for here is one.

[Speaker B] And that is one axis. And the second relation is the basis for the differences along the second dimension.

[Rabbi Michael Abraham] Both are true, but you need both; you can’t have one without the other.

[Speaker B] Two are given, the third is the last one. No, no, no.

[Rabbi Michael Abraham] That is exactly the point. When you establish a relation, say, in the right-hand column, then you establish a relation between horn and tooth-and-foot: horn is two alpha and tooth-and-foot is alpha. Now you want to apply that to the left-hand column, but you can’t. Why can’t you? Because you need to tell me that in the columns too alpha plays a role. Otherwise you cannot apply it. And what role does it play? Necessarily this role: this is alpha and this is two alpha. It has to be. Therefore neither of the two formulations can make do with a hierarchy on only one axis. In other words, each of the two formulations actually expresses a hierarchy on both axes—both the horizontal and the vertical.

[Speaker D] So we constructed one shared model for both?

[Rabbi Michael Abraham] Correct. There are not two arguments here; it’s one argument. Just two angles from which to look at it. It’s one argument, not two. What?

[Speaker D] What are we doing with the rows and the columns?

[Rabbi Michael Abraham] Exactly—you can’t separate one from the other. Because you need to assume that the alpha that explains the difference between the rows is also relevant to the difference between the columns. If the difference between the columns is determined by a different parameter, then we haven’t done anything; then I am not interested in the hierarchy between the rows. Okay? So it has to be that way. You cannot detach the columns from the rows.

Now look what that means. I now have a refutation, right? The moon comes back and shines upon us. Let’s say, given my handwriting, I really could be an astronaut. Right, that’s the refutation. Okay? Let’s assume it’s zero for the sake of discussion. Okay. So this is the refutation, right? Now I said: this refutation breaks the assumption about the relation between the rows, right? Because it shows me that tooth-and-foot are not necessarily more severe than horn. But the a fortiori argument from the columns can still work. Who said private property is not more stringent than the public domain? It can still be more stringent than the public domain. This refutation doesn’t break anything here, right? Well, no. Why? Because let us now look—after all, if I now want to fill in one here, I need to find a model, as an explanation, with alphas and betas for the existing laws. Now notice: the existing laws are now not three but five. Right? Those are the data. And now I ask what is written inside the missing square, the lacuna. I want to fill in the lacuna.

So I say this: I try to make an a fortiori argument. That’s how I started, right? That tooth-and-foot are alpha, horn is two alpha, the public domain is two alpha, and private property is alpha. Then I say: we need to fill in one here. But then someone says: wait, there’s the moon. The moon says that tooth-and-foot are one and horn is zero. What does that mean, in effect? That alpha alone is not enough to explain the five data points. Because if the relation were that simple between alpha and two alpha, right, then it would explain this but be contradicted by that. What does that mean? That there must be another parameter in the game. Or in other words, you also need some beta. No, you don’t need alpha plus beta; beta alone is enough. You’ll see in a moment. In principle you could do that too, but beta alone is simpler. Look. What are you saying—a parameter different from alpha? Right.

[Speaker H] It’s not two alpha, it’s different.

[Rabbi Michael Abraham] Exactly. But you don’t need to add alpha to it. Look. Okay, this is the model. Now you see this model is perfectly fine. Look. Let’s now see what happens. In principle, this explains everything, right? This model. Look, there’s no problem with two parameters. Meaning: horn has beta, therefore it is liable in the public domain, because to be liable in the public domain you need beta. Why are tooth-and-foot not liable in the public domain? Because they have no beta; they have alpha. That’s very nice, but what is needed for liability is beta. They don’t have it. Okay? What happens here? On private property tooth-and-foot are liable because they have alpha, and alpha is enough for liability. Let’s continue now to fill in the moon. In the moon case too, tooth-and-foot are liable. So the moon cannot be two alpha, right? It needs to be, say, alpha. Fine. But horn—wait, that too is alpha. Fine? Horn is not liable in the moon case. Why? Because horn has beta but not alpha, and in order to impose liability you need alpha. So you see that I have now filled in a model; I have a model that explains the five data points. And now if I ask myself what the law is here—it remains open. Not zero. No, it remains open. I cannot infer it. Why? Because both one and zero are possible. Why? If I put one here, then I can say—wait, if I put zero here, then it’s obvious, right? Because if you have beta, while private property needs alpha, then it’s zero. But I can find a model with alpha and beta that is also consistent with one. I can find one. And because of that—I’m not going to show it now, but I can find one—and because of that, it at least remains open. There’s no proof here that it is zero either, but it remains open. Once it remains open, that is a refutation.

Meaning, what the column I added on the left—the refuting column I added on the left—actually did was tell me: you will not succeed in explaining the data with a single parameter. You will need another parameter. Once you need another parameter, the whole game opens up again, because now you can arrange things so that here it will be one, and you can arrange things so that here it will be zero. It really doesn’t matter. In a refutation, what you always prove is that the more severe and the less severe—

[Speaker B] Once you have a refutation, you no longer have more severe and less severe.

[Rabbi Michael Abraham] You no longer have more severe and less severe, but then you go back—

[Speaker B] back—

[Rabbi Michael Abraham] to the simplistic formulation. But that doesn’t work. Because if you don’t have more severe and less severe in the rows, you still have more severe and less severe in the columns. That is the problem I came to solve. Therefore I’m not speaking now at all in the language of more severe and less severe. I’m saying: give me the model that explains the data. “Model” means: what is the parameter-composition of tooth-and-foot, what is the parameter-composition of horn, what is the parameter-composition of each of the three domains. That is called a model. This model explains the laws. I am basically saying: each of the damaging categories and each of the domains has features; those features are alpha and beta at different levels. A model is a list of features attached to each damaging category and each domain, by means of which I can explain the five laws. Now I can propose a model where the result here is one, and I can propose a model where the result here is zero, once there is a left-hand column. And therefore that is a refutation. If there is no left-hand column, then the model dictates that here it must be one. And that is why the a fortiori argument works.

That is how refutation works. Therefore, when I add a column-refutation—yes, when I add a column on the left—it knocks down both formulations, not only one of them. And what that really means is that I have basically proven to you that Rabbi Chaim is not right. Because I have shown you that according to Rabbi Chaim, if this were a formal matter, then one should have said that a column-refutation refutes only the hierarchy between the columns, not between the rows—or in this case, only between the rows, not between the columns. But if in the Talmud we see that every refutation, whether of a column or a row, refutes the a fortiori argument entirely, and I do not rotate the a fortiori argument, then this means that we are dealing with two different formulations of the same argument, and not with two different arguments. Therefore when one falls, the other falls too; there aren’t two, it is the same one. Okay? That is the claim.

Or in other words, we have reached the conclusion that an a fortiori argument is not a formal hermeneutical rule. It is not enough to look at the laws; I need to look at the parameters behind them. Let me show you another interesting example that explains why this is so. What happens if I have a table with two data points in it—one and one on the diagonal—but I don’t have the zero here, as one usually does in an a fortiori argument. I don’t have the zero here. Where do we find such an example? There is one in the Talmud in Berakhot 21a. The Talmud there tries to derive that one must recite a blessing over food beforehand, or over Torah afterward. So what does the Talmud say? “If Torah, which does not require a blessing afterward, requires a blessing beforehand, then food, which requires a blessing afterward, is it not logical that it should require a blessing beforehand?” So I have proven by an a fortiori argument that one must recite a blessing over food—a blessing of enjoyment. In the same way one can say the reverse: let me show you that one must bless over Torah afterward. “If food, which does not require a blessing beforehand, requires a blessing afterward, then Torah, which requires a blessing beforehand, is it not logical that it should require a blessing afterward?”

[Speaker B] Now you understand—that’s the famous a fortiori argument of the Chafetz Chaim. What?

[Rabbi Michael Abraham] If from my pocket—

[Speaker B] if I’m allowed to take from my pocket with your authorization, then you, who are allowed to take from your own pocket, certainly are allowed to take from my pocket.

[Rabbi Michael Abraham] Obviously. But I say that about myself, not about you, yes? But the idea is right.

[Speaker B] Ha.

[Speaker G] Ha ha.

[Rabbi Michael Abraham] Yes, so that is the a fortiori argument that proves it. The authors of methodological works—there are several such authors—bring the following example: let me prove to you that one must put four corners on a doorpost. “If a four-cornered garment”—that is, 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 logical that it should be obligated in fringes?” Or the opposite: I can prove to you that one must put a mezuzah on a four-cornered garment. “If a doorpost, which is exempt from fringes, is obligated in mezuzah, then a four-cornered garment, which is obligated in fringes, is it not logical that it should be obligated in mezuzah?” Now what is the problem with all these a fortiori arguments?

[Speaker G] That they’re crooked.

[Rabbi Michael Abraham] It’s easy to say they’re crooked; the question is why they’re crooked. Right, because the structure is like what is drawn here—you see the structure? There are only two data points here, not three. And you can fill this in, you can fill that in, and the two formulations are really doing opposite tricks. The first formulation basically assumes there is a zero here, and then makes an a fortiori argument and fills in one here. The second formulation assumes the opposite—after all, there is no law in the Torah here, so I assume whatever I assume. It assumes there is a zero here, and then fills in one here by force of an a fortiori argument. Okay? After all, once you have two missing data points, you can always assume one of them and infer the other, or assume the other and infer the first, right? And therefore you arrive at two opposite directions. More than that: each of those directions refutes the other direction. Because if I filled in here with an a fortiori argument that there is one, then I can no longer use an a fortiori argument to fill in one there, because that a fortiori argument is based on the assumption that here there is zero. In other words, each of the formulations refutes the opposite formulation.

Why really doesn’t it work? I’ll tell you why it doesn’t work. It doesn’t work because in a situation like this—let’s say if there were zero here and zero here, what would that mean? It would mean that there is no relevance between the two columns. The columns are not governed by the same parameter. This column is governed by parameter alpha and that one is governed by parameter beta. And this is more severe than that in terms of alpha, but not more severe than that in terms of beta. Okay? Basically, two parameters are in play here, and there is no relevance between the columns. In other words, mathematical independence—what in linear algebra is called independence between columns—this state of affairs reflects, on the logical level, a lack of relevance between the domains. And when you discover something about the public domain, you cannot infer anything from it about private property, because private property is governed by another parameter, not the parameter that governs the public domain.

Therefore what this means—and this too comes out of the analysis I gave earlier of the ordinary a fortiori argument—is that behind the laws that appear in the table it is very important to assume relevance. One must assume that the laws belong to the same set of parameters that determine relations of stringency. For example, with the doorpost and the four-cornered garment, the claim is that the obligation of fringes is governed by some parameter alpha, while the obligation of mezuzah is governed by parameter beta. So the fact that there is some reverse relation between garment and doorpost means that there is no simple hierarchical relation between them at all. In terms of alpha this is more stringent; in terms of beta that is more stringent. Fine? So the claim is that in order to infer a Jewish law conclusion from a table of laws, you actually need to assume something about the relevance of the laws—that they belong to the same semantic field, that they belong to the same space of reasoning, to the same space of parameters, to the same space of explanations. And when they do not belong to the same space of explanations, then even though the structure looks like the structure of an a fortiori argument, you will not make the a fortiori argument.

Now here is an example—let’s return to our passage. In our passage the situation is not any of the ones I described before. Here there is actually exactly such a case in our passage. Let’s look. Take the Israelite daughter. So an Israelite daughter is forbidden to eat terumah and permitted to marry a priest. Fine? Forbidden to eat terumah and permitted to marry a priest. Why shouldn’t I infer from this that a divorced woman, who is permitted regarding terumah, should be permitted to marry a priest? A fortiori, right? Here is the Israelite daughter and here is the divorced woman. Fine? Now the Israelite daughter is forbidden terumah and permitted to a priest, right? So the divorced woman, who is permitted regarding terumah—sorry, who is permitted regarding terumah—is it not logical that she should be permitted to a priest? There is your a fortiori argument. Why don’t we make it? Because the assumption is that the Israelite daughter’s prohibition regarding terumah is not a prohibition. It is not a prohibition. There is no zero written here. There is nothing written here. It is blank. So there is not a zero here; it simply remains blank. And we already saw that once this is blank and this is blank, you may not fill in one here. The a fortiori argument is no longer valid, because once this is blank, it means there is no relevance between the columns, no relation; the same set of parameters does not govern both of them. They are not relevant; you cannot put them in one table at all. That was the mistake: putting them into one table. That was incorrect. You should not have done that, because they are not relevant—they do not belong to the same field, they are not playing on the same field. Okay?

So in fact, what stands behind all the phenomena we saw when we read the Talmud—where we saw that even though the laws seemingly look like a table of a fortiori laws, we do not make an a fortiori argument, or the laws look like a refutation, but we do not treat them as a refutation—why is it not a refutation? Because the Israelite daughter is not a refutation of the a fortiori argument. What do you mean why? Because now we are already talking about an a fortiori argument with a refutation, an argument like this. Yes? So here I have the Israelite daughter. Here—this is the Israelite daughter. And here, wait. The Israelite daughter; I have the suspected adulteress and the divorced woman, right? The suspected adulteress and the divorced woman. Okay. And here: terumah and priest, right? Terumah and priest. Don’t show this page to anyone or they’ll immediately send me to a speech therapist. Okay, so: terumah and priest.

Okay, I am basically saying: terumah, divorced woman—here there is zero, yes? She is permitted regarding terumah; there is no prohibition concerning terumah. Zero means there is no prohibition, okay? Permitted to a priest, forbidden to a priest—there is a prohibition. So regarding the suspected adulteress, who is forbidden regarding terumah, is it not logical that she should be forbidden to a priest? Therefore they want to say one, right? Tosafot says: what kind of a fortiori argument is that? After all, the Israelite daughter is forbidden regarding terumah—there is a prohibition on her, right?—and nevertheless she is permitted to a priest; there is no prohibition against her marrying a priest. So that is a refutation. Why is it not a refutation? The answer is because what is written here is not one. What is written here is blank. There is no prohibition to eat terumah, and no permission either. There is nothing. She is simply not relevant to the issue. It is like asking whether virtue is triangular. Virtues—in the sense of character traits—are virtues triangular? The answer is neither yes nor no; it is not relevant. It does not belong to the semantic field of triangular or not triangular. Geometric shapes are not relevant for describing character traits, right? Same thing here. When you tell me that the Israelite daughter is forbidden regarding terumah—she is not forbidden regarding terumah. She is not relevant to the question of terumah; it does not arise in her case. Therefore it is not correct to write a zero here. Rather, one should write nothing here. In fact, it is not correct to write one here, as though there is a prohibition regarding terumah; rather, it should be left blank, meaning nothing should be written here. Once I write nothing, I say that the Israelite daughter is actually not relevant to the table—not that she is or is not a refutation. She is simply not relevant. The whole column should be erased. The entire column should be erased because it is not relevant. And now we have returned to here, and it remains an ordinary a fortiori argument as we did earlier. Okay? So what stands behind this whole matter is really the question of relevance. Well, really this is only the tip of the iceberg—you could make whole theories out of it—but we’ll stop here. That’s it. Enough.