A Lesson from 5777

Table of Contents

• Learning from what was itself learned, and the assumption of composition in Hazal versus the formal model
• The exceptional status of sacrificial matters in Zevahim 50 and the a priori difficulty
• The source of the principle in sacrificial matters, Rashi, and the structure of the Talmudic discussion
• The dispute between Maimonides and Nachmanides and the status of Torah-level law versus rabbinic enactment
• Hypotheses to explain the exception: “the verse repeated it to make it indispensable,” value structure, and topology
• Mapping the combinations in Zevahim: four hermeneutic methods, twelve combinations, and what is relevant to the model
• Building tables for a kal va-homer from a kal va-homer, and the question of sub-tables
• The Talmud’s example: “Something learned by kal va-homer — can it itself be learned by kal va-homer?”
• A kal va-homer from a kal va-homer in the model: the 3×3 result favors one-one
• A possible translation of a double kal va-homer into a single kal va-homer, and the lack of other examples
• Binyan av from a kal va-homer: a quick determination
• Kal va-homer from a binyan av: no determination
• Binyan av from a binyan av versus the Talmud, and the gap in sacrificial matters
• A clarification of “binyan av,” mah matzinu, and even a minimal refutation
• A double refutation in a kal va-homer: not just rejection, but proof of zero

Summary

General Overview

The text presents a methodological and logical doubt about whether, in a formal model of tables, one can justify composing valid inferences on top of one another, as Hazal assume with regard to learning from what was itself learned — in particular, a kal va-homer from a kal va-homer, and a binyan av from a kal va-homer and vice versa. It ties the examination to the topic in Zevahim 50, which deals with sacrificial matters as an exceptional domain in which not all such compositions are accepted, and raises a principled difficulty: how can something that is logically correct throughout Jewish law fail to work in sacrificial matters? It then examines the combinations relevant to the model — kal va-homer and binyan av — through diagrams of 3×3 tables, points out where the model decides and where it remains undecided, and concludes by showing that a double refutation — row and column together — not only knocks out a kal va-homer but can force a zero entry.

Learning from what was itself learned, and the assumption of composition in Hazal versus the formal model

Hazal assume that if a kal va-homer is a valid argument and a binyan av is a valid argument, then you can compose them one on top of the other and derive a kal va-homer from a kal va-homer, a binyan av from a kal va-homer, a kal va-homer from a binyan av, and so on. The text says that in the model one would need to prove a general theorem according to which, in a table without two missing cells, if one cell can be filled by means of a valid argument and, once it is filled, the second can also be filled validly, then the double filling is valid. It says this is not self-evident and needs to be checked. The text clarifies that the question is not about the validity of the logic as such, but about whether Hazal’s assumptions fit the formal model that was presented earlier.

The exceptional status of sacrificial matters in Zevahim 50 and the a priori difficulty

The text points to the topic in tractate Zevahim, in the chapter “Eizehu Mekoman,” on page 50, where sacrificial matters are an exception to the assumption of learning from what was itself learned, whereas the basic assumption is that in the rest of the Torah these compositions do work. The Talmud discusses many such compositions on their own terms and clarifies which combinations work in sacrificial matters and which do not, going through possibilities such as a gezerah shavah from a kal va-homer, a hekesh from a binyan av, a binyan av from a kal va-homer, and more. The text asks how it can be that if a kal va-homer from a kal va-homer is a valid inference just like a single kal va-homer, then in sacrificial matters it would not work. It suggests that the problem is not that the logic “doesn’t work,” but that perhaps the system accepts the inference only when there are explicit and well-established verses.

The source of the principle in sacrificial matters, Rashi, and the structure of the Talmudic discussion

The text says it does not recall a source in the Talmud for a general principle that “in sacrificial matters one does not learn from what was itself learned,” but rather that each composition is discussed separately, and the ruling is learned from the arguments and the data themselves. It cites Rashi on “just as it is removed,” with the wording: “If so, from slaughtered sacrificial offerings he should have learned from it that in sacrificial matters one does not learn from what was itself learned, unless the verse explicitly reveals it.” It remains unsure whether Rashi is citing an established datum or deriving a principle from it. The text presents an explanation attributed to Rashi: “That is to say, anything learned through one of the thirteen hermeneutic methods by which the Torah is interpreted is not considered, in sacrificial matters, as something explicitly written in the Torah, and we cannot learn from that learned matter another new derivation. Therefore Scripture was particular to say ‘it shall be removed’ explicitly, so that what we can learn is a new derivation from this. But if we had learned this through one of the thirteen hermeneutic methods, then we could not have learned from it a further derivation.” But even so, the question remains: what is the source, and what is the principled reason for this exception?

The dispute between Maimonides and Nachmanides and the status of Torah-level law versus rabbinic enactment

The text rejects linking this directly to the dispute between Maimonides and Nachmanides, and argues that an inference built on top of another inference is “certainly correct throughout Jewish law according to all views.” The only question is whether the result is considered Torah-level law or rabbinic. The text says that even if a conclusion from a kal va-homer is regarded, according to Maimonides, as rabbinic, one can still build on it another kal va-homer as an operative halakhic conclusion. It focuses the exception on sacrificial matters, not on the dispute over normative classification.

Hypotheses to explain the exception: “the verse repeated it to make it indispensable,” value structure, and topology

The text raises an ungrounded hypothesis that in sacrificial matters the value structure plays a different role, and connects this to the notion that “the verse repeated it to make it indispensable,” where even an explicit verse is not enough to make something indispensable and an additional verse is required — so that each verse teaches, schematically, “half the force of obligation.” It considers the possibility that the model’s assumptions about strengths and obligations are not justified in sacrificial matters, or that “topological” elements such as a change of direction behave differently there. But it notes that if change of direction did not matter in sacrificial matters, that should affect individual inferences too, and not only learning from what was itself learned. The text ends this part by saying it has no clear answer and leaves it as a question mark.

Mapping the combinations in Zevahim: four hermeneutic methods, twelve combinations, and what is relevant to the model

The text says that in Zevahim the discussion concerns compositions of four hermeneutic methods: hekesh, gezerah shavah, binyan av, and kal va-homer, which together create twelve unordered pairings times two. It says that in its model hekesh and gezerah shavah are not relevant because they are “textual methods,” so what remains to examine are the four combinations involving kal va-homer and binyan av: kal va-homer from kal va-homer, kal va-homer from binyan av, binyan av from kal va-homer, and binyan av from binyan av. It reports the “data” from the Talmud: a kal va-homer from a kal va-homer is valid, a binyan av from a kal va-homer is valid, a kal va-homer from a binyan av remains open, and a binyan av from a binyan av remains open.

Building tables for a kal va-homer from a kal va-homer, and the question of sub-tables

The text explains that a kal va-homer requires a structure of “zero, zero, and two ones on the diagonal,” and therefore, in order to carry out a kal va-homer from a kal va-homer, one has to expand to a 3×3 table in a way that allows a second kal va-homer on the basis of the first result. It raises a methodological doubt: are we allowed to rely on sub-tables as decisive evidence inside a larger table, or can the additional data change the “optimal model” and make a local conclusion invalid? It suggests that instead of filling things in stage by stage, the model fills two cells together from the full set of scriptural data, and in that way avoids the difficulty that a “Jewish law” conclusion that was learned is not a verse and therefore cannot serve as the basis for another inference.

The Talmud’s example: “Something learned by kal va-homer — can it itself be learned by kal va-homer?”

The text notes that the only example it found of a kal va-homer from a kal va-homer is the Talmud’s own discussion of whether a kal va-homer from a kal va-homer is valid, where the Talmud tries to derive the permissibility by means of a kal va-homer based on another kal va-homer, and rejects it as a methodological circularity. It cites the Talmudic wording: “And just as a gezerah shavah, which is not learned through hekesh … is learned by kal va-homer … then a kal va-homer that is learned from hekesh — is it not logical that it too should be learned by kal va-homer? And this is a kal va-homer son of a kal va-homer.” Then comes the rejection: “It is the grandson of a kal va-homer,” and the Talmud replaces it with a different structure so as to remain a “son of a kal va-homer” and not a “grandson.” The text stresses that the rejection is not because of a lack of logical validity, but because you cannot prove a doubtful instrument by means of itself.

A kal va-homer from a kal va-homer in the model: the 3×3 result favors one-one

The text combines the relevant Talmudic tables — whether according to Rav Pappa or according to the dissenting view — and shows that in a 3×3 structure one gets constraints under which one cell must be zero and another must be one in order to prevent refutations of the first and second kal va-homer. It checks four combinations for filling the two missing cells and concludes that the filling “one-one” is the simplest, because it avoids dependencies, avoids changing direction, and reduces dimension, so that one parameter can explain the structure. It presents the model’s answer to the difficulty that “one does not derive a kal va-homer from a legal conclusion” by saying that the model does not use an intermediate result as a basis; rather, it infers two conclusions simultaneously from the total set of verses.

A possible translation of a double kal va-homer into a single kal va-homer, and the lack of other examples

The text comments that sometimes one can take “the two sides of the cells” and make a single kal va-homer instead of a double one, and wonders whether this is why there are almost no examples of a kal va-homer from a kal va-homer besides the discussion of the tool itself. It suggests the possibility that the Talmud first went through the double route in order to ensure that, given additional data, there is no refutation — and only afterward can one “go back” to the smaller kal va-homer.

Binyan av from a kal va-homer: a quick determination

The text describes how a binyan av from a kal va-homer does not require a 3×3 table but a smaller structure in which a kal va-homer fills one cell, and from there one performs a binyan av to fill another. It says that here, “even בלי doing the calculation” — that is, even without doing the full calculation — it is clear that the one-one filling is the best, because an alternative filling would create a refutation against the kal va-homer, whereas one-one coheres and creates an optimal structure with fewer points.

Kal va-homer from a binyan av: no determination

The text explains that a kal va-homer from a binyan av requires a 3×3 table with a diagonal, because one must begin from a binyan av that fills one cell and then use it for a further kal va-homer structure. It says that in this case, “a kal va-homer from a binyan av is not necessarily valid,” and describes an outcome in which x is determined as one but y remains open, so that the first binyan av is valid but the move from it to the kal va-homer is not determined. It says it did not find a simple example of this.

Binyan av from a binyan av versus the Talmud, and the gap in sacrificial matters

The text says that according to its considerations in the model, a binyan av from a binyan av comes out valid, whereas in the Talmud, in sacrificial matters, the question remains open, and it does not know how to resolve that. It stresses that in the rest of the Torah a binyan av from a binyan av “certainly works,” so fitting the model to that is not a problem. The problem is only why it would not work in sacrificial matters, and which of the model’s assumptions would need to change in order to explain that.

A clarification of “binyan av,” mah matzinu, and even a minimal refutation

The text clarifies that it is using binyan av in the sense of an analogy of the type discussed in the model, and not the common denominator form of a different type. It notes that in mah matzinu its refutation is “even a minimal refutation,” and that in a binyan av from one verse it is not clear whether even a minimal refutation is enough, because that is a tannaitic dispute, whereas in a binyan av from two verses even a minimal refutation is enough.

A double refutation in a kal va-homer: not just rejection, but proof of zero

The text returns to the structure in which a kal va-homer is understood as having “two formulations,” one in terms of rows and one in terms of columns, though in fact they are the same inference. Therefore, a row-refutation or a column-refutation on its own “knocks out the kal va-homer.” It then checks what happens when you place two refutations together, both on the rows and on the columns, and shows that in the table calculation the preferred filling is “zero-zero,” so that the conclusion is that the central cell must be zero. It formulates this as an “anti-kal va-homer,” in which a double refutation is not an open situation but a positive proof that the correct value is zero.

Okay, now we want to fill in the second slot as well, also מתוך this model, an analogical prototype or something like that. So actually this is a problem, a problem of a new type. The Sages’ assumption is that if there is an argument that is valid in itself, then you can build it on top of another argument. Meaning, if an a fortiori argument is a valid argument and an analogical prototype is a valid argument, then you can derive an analogical prototype from an a fortiori argument, or an a fortiori argument from an analogical prototype, or an a fortiori argument from an a fortiori argument, and so on. That is the explicit composition of this thing among the Sages. As for our model, there is room to examine this. I haven’t really checked it fully; that is to say, this actually requires proving a theorem. To prove a theorem that in every such table—or theorems, I don’t know—that in every such table where there aren’t two missing slots, if one of them can be filled by means of a valid argument, and given that the first one is filled, then filling the second is also valid, then you can fill them both. And that is not self-evident; it’s something that needs to be checked. What, is this only at the level of an axiom? Not an axiom; it could be—but if it doesn’t work out then it becomes an axiom, but if it does work out then… No, this is what the Sages assume, but the question is whether the model sustains it. The question is whether the model I presented before sustains it. That is, one has to check whether indeed in this model, every time I make an a fortiori argument from an a fortiori argument, or an a fortiori argument from an analogical prototype, or something like that, the result will also always be valid. That’s not obvious. The Sages assume it on the intuitive level; the question is whether the formal model upholds that assumption. Fine, that’s something that requires examination. We’ll see later that it’s not entirely simple. I mean, I’ll show where there is an explicit discussion of deriving from something already derived; it’s a topic in tractate Zevachim, in the chapter “Which place is theirs,” page 50. And there the topic is specifically sacrificial matters, because in sacrificial matters there is an exception to this assumption. In sacrificial matters, we are not always willing to accept a derivation of something derived from something already derived. Of course, the basic assumption there in the passage is that generally in Jewish law you can always derive from something already derived, but sacrificial matters are exceptional. There are certain structures that in sacrificial matters won’t work. Not all of them. And the Talmud there starts discussing: so what does work and what doesn’t—an verbal analogy from an a fortiori argument, an juxtaposition from an analogical prototype, an analogical prototype from an a fortiori argument—and there are structures that do work and structures that don’t, and the Talmud goes through almost all the possibilities of composing inferences or arguments. But as I said before, from the Talmud’s discussion it is clear that the simple assumption is that in the rest of the Torah everything works. What they discuss there is which of these compositions will also work in sacrificial matters, but the plain assumption is that it always works.

Now even before actually checking these structures, I asked myself the question: what could the explanation even be? Meaning, if indeed an a fortiori argument from an a fortiori argument is a valid inference, valid in the sense we’re talking about here, exactly like a single a fortiori argument, then how can it be that in sacrificial matters it won’t work? Either way, if really—like if someone tells me that in this area logic doesn’t work—there’s no such thing. Logic always works; that’s what logic means. What is true, however, is that the a fortiori argument in principle works only on things that are clear and explicit in the verses. That’s the first assumption. We started with things such that if an a fortiori argument from an a fortiori argument didn’t work, it couldn’t work anywhere, not only in sacrificial matters, and therefore every such thing is a novelty. It’s not a question of whether the logic is okay; the question is whether I accept it at all. The logic is okay, but I don’t accept it. But I’m saying again: I’ll also make an a fortiori argument from an a fortiori argument out of verses. The way I present it in my model, we basically make a table with two empty slots, and we won’t relate to this as filling one by means of an a fortiori argument and then, on that basis, filling the second. No. I’ll fill them both together. But isn’t it still the first one? What? You’re not doing the first one, so you fill both step by step. No, I’m not doing the first one. What—I don’t know what that means. Okay, we’ll see. But I don’t remember, although there’s a dispute here between Nachmanides and Maimonides. No, I don’t think it’s connected to their dispute. It may be connected to their dispute because an a fortiori argument from an a fortiori argument, or an inference on top of an inference, is certainly true throughout all of Jewish law according to all views. Whether that comes out as Torah-level according to Nachmanides, or whether Torah-level or not is the dispute between Maimonides and Nachmanides, but that’s true even for a single inference. To make rabbinic things I don’t need all these structures. What do you mean, I want to make—if it’s not Torah-level. What do you mean? I want to make—if it’s not Torah-level, then again it’s rabbinic rules, I can fold it. No, this is an exposition; these aren’t just decrees, these are things that come out of expositions. Right, so according to Maimonides we don’t care. According to Maimonides we don’t care and according to Nachmanides we do? No, what do you mean according to Maimonides we don’t care? Even according to Maimonides, if the Talmud didn’t say that it is Torah-level, then the whole logical structure doesn’t matter. No, that’s not important. When we reach a Jewish legal conclusion by means of an a fortiori argument, whether you regard that conclusion as rabbinic legislation or as Torah-level, it is still the Jewish legal conclusion, and on that basis you can make another a fortiori argument. That’s obvious. There is no dispute about that. The question is what happens in sacrificial matters. In sacrificial matters this is exceptional, but in the rest of Jewish law you can. So that means it’s not connected to the dispute with Maimonides? Maybe in the rest of Jewish law this is a novelty, and in sacrificial matters they didn’t innovate it? Okay, that’s fine, but still, what does it mean that it’s a novelty? After all, if it really comes out of this same analysis of tables, then it’s just a necessary conclusion from the analysis of the individual inferences. Who says that the fact that in sacrificial matters they don’t do it comes from verses? As though it’s some kind of scriptural decree? Okay. Until… No, we’ll see, it’s more complicated. These matters in sacrificial law—there is no basic source; at least I don’t remember the Talmud bringing a source for the principle itself that sacrificial matters are exceptional. Not that sacrificial matters are exceptional, but that we do not derive something already derived from something already derived in sacrificial matters. There is that distinction. No, there isn’t. What there is, is a discussion of each composition separately. An analogical prototype from an analogical prototype, an a fortiori argument from a verbal analogy, a verbal analogy from an analogical prototype. Each such composition is discussed separately in the Talmud, and the interesting point is that they derive it through these very arguments themselves. We’ll see an example of that today. He cites here in Rashi something—“as he takes it off,” this is learned from this and this—from which we learned that it was repeated in order to teach that in sacrificial matters one does not derive from something already derived. I don’t remember this Rashi exactly. Well, that needs checking, because in the Talmud, it seems to me, there is no source given for the fact that one does not derive from something already derived; rather, each composition is discussed on its own merits and we make various considerations to see whether it works in sacrificial matters or not. You can call that a source. Because if the biblical data contradict the fact that we derive from something already derived, then you can in effect say that those biblical data themselves innovate that in sacrificial matters there is no deriving from something already derived. Even though elsewhere in the Torah, even if we had such data, we would not infer from that a rule that one does not derive from something already derived; rather we would say: this is an a fortiori argument with a refutation, and therefore we don’t derive it. That’s all. There is a biblical datum that contradicts it. Do you see? To infer from that some rule specifically that one does not derive from something already derived in sacrificial matters—it seems there must be a tradition, or it comes from some conception of the area of sacrificial law. I really don’t know. I had various suspicions about where this comes from. Maybe because there are endless refutations in sacrificial matters? What do you mean? Because every time the Rabbi finds… because it’s so important, then… Is it impossible always to find refutations? I don’t know. In any case, the Talmud doesn’t tie it to that. The Talmud simply brings—we’ll now see one discussion as an example—the Talmud computes it, proves it from data. It shows that you can’t do it. Yes. Does that happen specifically in sacrificial matters? Yes. Now, no: the Talmud brings it from topics of sacrificial law, but why the generalization is that from here emerges a rule that in sacrificial matters you can’t derive from something already derived—that I really don’t know. They had some tradition, or some conception of the realm of sacrificial law; I don’t know exactly what. And in sacrificial matters there is also the issue that “the verse repeated it to make it indispensable,” as I said earlier, which I somewhat suspect may be where this comes from. I think it may be connected to that, but I’m not sure. It’s a hypothesis. “The verse repeated it to make it indispensable.” How does that connect to relying on your intuition if it’s not really deductive? Then don’t trust your intuition in sacrificial matters. No, “the verse repeated it to make it indispensable” isn’t connected to intuition. “The verse repeated it to make it indispensable” has an explicit verse. And despite there being an explicit verse, it still is not indispensable. You need another verse for it to be indispensable. Why? Because what… one verse isn’t enough? It seems that… Why? Cast doubt on what seems simple to you in sacrificial matters. On that very general level maybe, but then why not cast doubt on a single a fortiori argument? Why specifically on deriving from something already derived? More than that: deriving from something already derived is not necessarily weaker than a single derivation. And still, deriving from something already derived, yes, and deriving not? Why? “As he takes it off”—there is a verse: “as he removes it from the ox of the peace-offering sacrifice, the priest shall burn them on the altar of the burnt-offering.” Rashi says: if so, from the slaughtering of sacred offerings he should have learned from it that in sacrificial matters one does not derive from something already derived, had the verse not explicitly revealed it there. No, fine, he brings it as a datum; he doesn’t derive it. He brings it as a datum from the Talmud; he doesn’t derive it from here. No, that’s not how I understand him. But again, you can take every computation the Talmud makes as a source. The Talmud shows you that there is some law which, if we derived from something already derived, that law would not be correct, and the verse says that it is correct. So that refutes the… But usually when we encounter such a thing in other contexts, we say: fine, then there’s a refutation of this derivation. As happens many times: we derive something by an a fortiori argument and suddenly find an explicit verse that the result of the a fortiori argument is not correct. Fine, then the verse innovated that it’s not correct. We do not derive from this some principle that one may not use an a fortiori argument. “Meaning to say that anything learned through one of the thirteen principles by which the Torah is interpreted is not considered in sacrificial matters as something explicit in the Torah, and we cannot derive from that already-derived matter yet another new derivation; and therefore Scripture was careful to say ‘remove it’ explicitly, so that what we can learn from it is a new derivation; but if we had learned it from one of the thirteen principles, then we could not have learned from it another derivation.” Fine, but still the question is whether that is the source. Here he explains Rashi, but you’ll need to narrow down a lot of interpretations here. The Talmud shows that deriving from something already derived in sacrificial matters doesn’t succeed. Yes. But still, deriving from something already derived in general is permitted? Yes. How can that be? What do you mean, how can that be? How can it be that it shows it can’t be? It shows it through data. We’ll see an example today. We’ll see an example today. But this is logic. Meaning, if it’s logic, then it shouldn’t work in other things either. Wait, wait—that’s exactly the problem. That’s why I started, and didn’t yet manage to get there—the first question is an a priori question: before I check where the Talmud derives these principles from, how can such a thing be? Either way, if indeed this conclusion really does follow from the data—that is, it is logically valid—then what does it mean that in sacrificial matters it is not true? If it follows from the data, it follows. Nobody would tell me that in sacrificial matters you can’t, I don’t know, use deduction. What do you mean? Deduction is simply true; it’s not some kind of… What? Also in tractates about purity? In purity too they use the thirteen principles, a fortiori arguments, a lot. “Why is this case different”? No, no, they use everything. At one point I studied Zevachim; in purity there are many too. By the way, there is a Hazon Ish, in Hazon Ish on hands. There the Hazon Ish tries to argue that maybe there is no deriving from something already derived in other areas too, not only in sacrificial matters—but that is all rabbinic, hands and mikveh. In tractate Yadayim? Yes. Let’s see what can… No, but maybe he is speaking about some derivation, I don’t know exactly in what context it comes up there.

In any case, as I said, the a priori question is a very interesting one, and I have no clear answer to it. I can raise various hypotheses, but I have no clear answer. One possibility, for example, is that maybe in sacrificial matters, I don’t know, maybe valuation plays a role, even though until now we assumed it doesn’t play a role. After all, all kinds of assumptions entered here, and those assumptions may not be justified in sacrificial matters. And here I thought it connects to “the verse repeated it to make it indispensable.” “The verse repeated it to make it indispensable” means that each verse teaches you not the full force of the obligation, right? In order to tell you that the sprinkling of the blood is indispensable, you need two verses. If there were only one verse, I would think that you have to do it, but it isn’t indispensable; the second verse says that it is also indispensable. What does that mean? That each verse essentially teaches only—let’s call it schematically—half the force of the obligation. So this actually means that valuation works differently here than in other areas. When you say “obligated,” you’ve really written here half, not one. Do you understand? Then perhaps that changes various assumptions, but I haven’t checked it systematically. There could be various hypotheses here. Of course, all the topological elements—someone might say, I don’t know, maybe in sacrificial matters the issue of reversing direction doesn’t play a role. I have no idea. Only then, of course, we would have to check all the specific considerations in which a change of direction was decisive. Meaning, an inference of an a fortiori argument, for example—or not an a fortiori argument, but a refutation of an a fortiori argument—seems to me to have been determined by a change of direction. Now if change of direction doesn’t play a role in sacrificial matters, then that should show itself not only in deriving from something already derived, but also in the individual inferences. I don’t know of such a thing, meaning it apparently doesn’t happen—at least as far as I know—only in deriving from something already derived is sacrificial law exceptional. In regular derivations, sacrificial law is like the rest of Jewish law, which greatly limits the possibilities of how to explain this within the model. So I really don’t know; I’m simply leaving it with a question mark.

Now, in order to examine the matter a bit more systematically, we actually have to go through all the combinations, because as I said before, the Talmud does not state it as a sweeping principle that in sacrificial matters there is no deriving from something already derived. On the contrary: many combinations are possible even in sacrificial matters. There are certain combinations—and that’s one of the proofs that there is no general source for it—there are certain combinations that do not work in sacrificial matters, but there are many that do, and the Talmud discusses each combination separately, whether it works or does not work. So what I really should have done in order to reach a conclusion about why exactly there is a difference between sacrificial matters and the other areas, is to go through systematically and see what characterizes the considerations that are valid also in sacrificial matters, and what characterizes the considerations that are not valid. I didn’t find a clear answer to that, and therefore I’m saying: I don’t know. I can only say that there is no frontal contradiction to our model in that passage there. There are things that I’ll explain in a moment.

So, the compositions the Talmud deals with in Zevachim are not all thirteen principles one on top of another, but only—only, I think—four: juxtaposition, verbal analogy, analogical prototype, and a fortiori argument. That is, compositions of four of these. That means sixteen combinations, right? Actually fewer—twelve. Because the combinations of an a fortiori argument from an a fortiori argument you must not count twice. If you reverse it, it will still be an a fortiori argument from an a fortiori argument, right? So in effect there are twelve combinations, and with each of those twelve combinations the Talmud discusses whether it works or does not work. I’ll take four combinations that are relevant to us, because verbal analogy is not found in this model, so I won’t be able to examine it within the model; it’s not relevant. And juxtaposition isn’t either. Those are textual principles, juxtaposition and verbal analogy. So what remains for us is a fortiori argument and analogical prototype. Now “common denominator” does not appear there in the Talmud, only an analogical prototype from one verse. So in effect we have four combinations to check within our model. There is an a fortiori argument from an a fortiori argument, an a fortiori argument from an analogical prototype, an analogical prototype from an a fortiori argument, and an analogical prototype from an analogical prototype, right? Those are the four possibilities we need to check. And the data are these: an analogical prototype from an analogical prototype remained unresolved in the Talmud—whether you do that in sacrificial matters or not. It’s somewhat scattered there, so I’m collecting it. An analogical prototype from an a fortiori argument, in the conclusion, is valid. An a fortiori argument from an analogical prototype is also an unresolved problem. And an a fortiori argument from an a fortiori argument is valid. Those are the data first of all—that’s what comes out of the Talmud. Now what I want to do is check within our model how this business works, and see what characterizes the problems regarding which the Talmud says they were resolved, and what characterizes the problems that remained open. And again I say: that’s what I wanted to see. I didn’t find anything unequivocal. Therefore, what I’ll do is simply use this as one more example of a more complex application, an application of the model.

All right. Now in order to check this, we actually need to draw a table of an a fortiori argument from an a fortiori argument. So we can try and ask ourselves how such a table could even be structured. So look: we saw that an a fortiori argument is this table. What is an a fortiori argument from an a fortiori argument? When I fill this one in through a primary a fortiori argument, that is the simple a fortiori argument. But I need a few more empty slots such that on the basis of this one I’ll try to fill them in by means of another a fortiori argument. Now look, an addition like this or that will never do it. Right? From below. It cannot be. Because whatever we fill in here, we won’t succeed in creating here a structure of an a fortiori argument out of these four. Whatever we fill in here. A structure of an a fortiori argument requires zero, zero, and two ones diagonally. That won’t happen here. If I fill one here and one here, fine? So that means I can’t have a structure of an a fortiori argument, and the same here too. Therefore it’s clear that it has to be something like this. An analogical prototype, yes. What? An analogical prototype in the other corner. Wait till I get there. It has to be something like this. Fine?

Now when I looked for an example to try and examine, because in fact it’s not entirely clear how to fill in the rest of the data. Some of it I can try to conjecture. If here it says one, right? As a result of the first a fortiori argument, and I want to use it in order to make another a fortiori argument, then I assume this is a possible structure, for example, right? After I fill in one here from that a fortiori argument, then the structure of the quartet found here will create another a fortiori argument and I will have to fill in one here as well. Right? This is a structure of an a fortiori argument from an a fortiori argument—or the reverse, of course, if we put zero here and a question mark here; that’s not important. Okay? The question is, of course—so this is basically given, it has to be something like this, or with the question mark and the zero reversed. But what goes in these two? What would you… give me the eraser. I’m not erasing. No, I’m not erasing anything. After all, this is some kind of datum. What is the law? What is the law of c and b? If in big c and little a there is an inference, then you can make the a fortiori argument here from above. You don’t need the a fortiori argument derived from the already-derived. From both of them? No, but maybe there is some refutation here? No, that there is a refutation. There may be data. Once there are more data, then you already need to be careful. One of the interesting questions in this area—and again this is also a matter for theorems that need to be checked, which we haven’t checked systematically—is whether sub-sub-tables that yield a valid inference, if I don’t see some refutation around them, does that mean that this will also be the solution to the full table? I don’t know. So far it came out that yes, but I’m not sure it always works. Maybe it’s the proof of a theorem. Maybe separate the tables, make a new table. What do you mean? Remove this one; here there’s a zero in little c and big b. But I can’t ignore the data. I don’t close my eyes. Meaning, if I have more data, I need to take them into account. Now here it remains open, the question is what to fill in here, but in order to know what to fill in here, it may be, for example, dependent on what gets filled in here. Let me give you an example. For example, if here there is zero. Suppose here there is zero, right? Then notice that if here there is one—sorry. That assumes there is relevance. What? That assumes that there is… Yes, yes, I’m assuming there is relevance between the rows and columns. There is no necessity that there be relevance. Fine, true, but then indeed I’m not sure you’ll have an a fortiori argument from an a fortiori argument. The moment you combine the tables, that means it is on a common axis. So if it’s relevant to b and a, and a is relevant to b—but there is no transitivity in relations of relevance. So I don’t know, maybe there is such a case, but then we’ll need to see it. In principle, okay, maybe. Perhaps, maybe. But then we won’t make an a fortiori argument from an a fortiori argument. Why? From little b to little c you can make it. What? From little b to little c you can make it because there is relevance, but from a to c you can’t. Interesting question. I’m not sure that’s correct, because if it won’t be relevant, then I think there will be some refutation here at the second stage. Because the reason you arrived at the one here is that you ignored a parameter found here, and it can actually affect it, and it affects both here and there. Fine, I don’t know. Interesting question.

So now I say as follows: if indeed, yes, for example—not like that; let’s put it this way—if for example there is one here, not here, right? Say I fill in only the… just for the example. Notice what happens. This thing constitutes a refutation of the first a fortiori argument. Do you see? This a fortiori argument—I want to fill in a one here, but here there is a column refutation. It doesn’t work. Therefore, it is actually fairly clear that here there needs to be zero. Yes, but that forces… So that forces that there is a refutation of the a fortiori argument, because otherwise—what I asked earlier—that this comes out… No, but you already assumed that you can take a sub-table and derive from it an a fortiori argument. Who says that in the full data table this will also come out as one? It’s not at all clear to me; it needs checking. You want to make an a fortiori argument from these two pairs of slots, make from them a quartet of an a fortiori argument. Why not? Because you have more data here. Maybe those data will refute it? Maybe, taking the rest of the data into account, the optimal model won’t be with a filling of one? I don’t know. I’m saying there should be a theorem for this. Maybe there is one; I’m not sure. Meanwhile I’m saying, so far it has fit for us, but I don’t know if it is a general theorem. So if here there is one then there would be a refutation, and therefore it’s reasonable that here there should be zero. What happens here? Here it’s not clear, right? Here it could be… So obviously, if we filled it that way, then this will be zero. Now the question is what goes here. It’s not entirely clear. If here there is zero, that’s not terrible, right? No problem; it doesn’t contradict anything. It could still be that both are one. And even if here there is one, it could still be that both are one. Right? It doesn’t contradict anything. Okay? No—if here there is zero, sorry, if here there is zero, that would refute the second a fortiori argument. Because after we fill in the one here, then here is one and here is one and this is zero, and I would want to fill in one here, right? But if here is zero, that means c is more lenient than b, so that would refute the second a fortiori argument. Therefore here it must be one. Okay? What is the refutation of the second a fortiori argument? I didn’t understand. What is the refutation of this a fortiori argument? I didn’t say there is one; it needs checking. You don’t always see a refutation with your eyes. Now this isn’t visible. No, I’m saying: it needs checking. You may be right, but it requires checking. You can’t just immediately make this a fortiori argument, because once there are more data, you have to take them into account as well. Maybe it comes out the same way, okay. No, but if it comes out the same way then you have a problem. Why? Because then you haven’t shown a model in which one needs to derive something already derived from something already derived in sacrificial matters, because look, you can derive it without that too. Therefore the suggestion… only if there is some refutation. Right, and if you show me a refutation, then you’ve missed the point. But if you haven’t shown me a refutation, then the meaning is that you haven’t shown me a case where one needs to derive something already derived from something already derived in sacrificial matters, because in any case, or in the cases you’ve shown me, you don’t need the double a fortiori argument; a single a fortiori argument is enough. And therefore my suggestion for the corners is: neither zero nor one, but z and w. And z and w solve the problem for you: you can derive neither this nor that, and you can’t make… Interesting, I hadn’t thought of that.

The example I found is not sacrificial law. The example the Talmud itself uses—I’ll show you in a moment—is an example where everything is filled in. Everything is filled in, and still the Talmud needs the double a fortiori argument, to make the a fortiori argument from an a fortiori argument. That’s really an interesting question; I hadn’t thought of it—why didn’t they just do this directly? Meaning, it could be that you can always say that the Talmud worried that maybe given the additional data it doesn’t come out right—check it. Once you’ve done that, you can already go back and make a small direct a fortiori argument, but only after you’ve checked. Do you understand? Meaning, it could be that after I prove that this is a valid schema and it really does give one, we can infer the conclusion backward and say okay, then in every structure of an a fortiori argument from an a fortiori argument you can go directly to these two and make a single a fortiori argument. The nice point here is that when I was looking for an a fortiori argument from an a fortiori argument to sink my teeth into—some concrete example—the Talmud doesn’t bring one. The only example I found is an a fortiori argument about this very principle itself. Meaning, the Talmud discusses whether an a fortiori argument from an a fortiori argument teaches in sacrificial matters, and it learns this by means of an a fortiori argument from an a fortiori argument. That is the example, the example I found. And perhaps the explanation is what you said, because in most cases where there is a situation of an a fortiori argument from an a fortiori argument—or in all cases, actually, where there is such a situation—the Talmud simply makes the a fortiori argument directly on those two sides of the slots and does not need to make an a fortiori argument from an a fortiori argument. So this only answers the question: then what is the whole discussion in the Talmud? What is the whole discussion? That’s an interesting remark; I hadn’t thought of it. If nevertheless I make the table larger in such a way that there is a refutation on this and not on that. Yes, but in a three-by-three it probably doesn’t work, because as I said, zero here is mandatory and one here is also mandatory. Unless there can be another case. There could be a case where here there isn’t, say, zero—and right, if here there is one, for example. So if here there is one, then there is a refutation of this a fortiori argument, but it could be that now, when you look at the whole table, the optimal filling will still be one here. Even though with respect to the small a fortiori argument there is a refutation here. No, no, this is not a common denominator. Two empty slots; a common denominator is one empty slot. That’s why I said it’s different from the expansions we make in the Kiddushin passage. In the Kiddushin passage we take more and more data on the same empty slot. We say, what does it mean that we add more data? Here, no—we also add another empty slot. It’s something else. So it could be that indeed when I make this single a fortiori argument there will be a refutation here, but when I also add these data, then despite the refutation here, the filling here will be one. Theoretically, but all that of course has to be checked, okay?

So now look at the amusing case in the Talmud, on your page: “A matter learned by an a fortiori argument—can it itself teach by an a fortiori argument?” So we begin with an a fortiori argument from an a fortiori argument. We said there are four combinations that need checking. “A matter learned by an a fortiori argument—can it teach by an a fortiori argument?” The Talmud says: an a fortiori argument—meaning, we learn this itself by an a fortiori argument. “And just as a verbal analogy, which is not learned by juxtaposition according to Rabbi Yohanan”—we won’t go into all the details of the passage; it isn’t important—verbal analogy is not learned by juxtaposition. What does “not learned by juxtaposition” mean? A structure of juxtaposition followed by verbal analogy does not work in sacrificial matters. “Yet it teaches by an a fortiori argument,” as we said. Verbal analogy does indeed teach by an a fortiori argument. “Then an a fortiori argument that is learned from juxtaposition, as taught in the school of Rabbi Ishmael, is it not all the more so that it should teach by an a fortiori argument?” And that is an a fortiori argument from an a fortiori argument, because it is an a fortiori argument whose use is itself based on an a fortiori argument.

Now look at table one—what you have, the lower table—that is basically the representation of that argument. Do you see? “A verbal analogy is not learned from juxtaposition”—that’s the zero at the top right. But it teaches by an a fortiori argument. So an a fortiori argument learned from juxtaposition certainly should teach by an a fortiori argument. Okay? That’s the first table. Now we continue. “It is the grandson of an a fortiori argument, isn’t it?” It’s not an a fortiori argument from an a fortiori argument, it’s the grandson of an a fortiori argument. What does that mean? It’s an a fortiori argument that is itself based on an a fortiori argument from an a fortiori argument. It’s not based on an a fortiori argument; it’s based on an a fortiori argument from an a fortiori argument. Why? I’ll explain in a moment. But that’s what the Talmud says. So what? What’s the problem with that? If you can do two, you can do three, four, five. No, because this is exactly what we are discussing here. We are discussing whether one can make an a fortiori argument from an a fortiori argument. So you want to learn that by means of an inference that is itself an a fortiori argument from an a fortiori argument? That can’t be. Therefore the Talmud says this can’t be; it is the grandson of an a fortiori argument, and it says this won’t work. So then what? The Talmud says: rather, an a fortiori argument. “And just as juxtaposition, which is not learned from juxtaposition, according to Rava and Ravina, nevertheless teaches by an a fortiori argument, as taught in the school of Rabbi Ishmael, then an a fortiori argument learned from juxtaposition, as taught in the school of Rabbi Ishmael, is it not all the more so that it should teach by an a fortiori argument?” And that is an a fortiori argument from an a fortiori argument. Not a grandson, but a son. Fine? So that’s the second inference: if you look, juxtaposition is not learned from juxtaposition and teaches by an a fortiori argument—exactly the same table as above and the same columns. What changes is that my teaching principle is not verbal analogy but juxtaposition. That’s all; apart from that it’s the same thing. Okay? The whole basic structure is the same. Very interesting. What? It’s very interesting. Because juxtaposition is not perceived as one of the thirteen principles by which the Torah is interpreted. Yes, true. A question that does have what to rely on. Sefer HaKeritut writes that it is like something written explicitly in the Torah, connected to what you noted before, and therefore one can derive an a fortiori argument from it. Sorry? Yes. What? I don’t understand the story of the grandson of the a fortiori argument. You don’t need to say that; it’s enough to say that he’s a son, and that’s what we’re discussing. What do you mean? The problem is whether it is permitted to derive an a fortiori argument from an a fortiori argument, right? The Talmud says “the son of an a fortiori argument.” So if it brings such a thing, that’s an example of an a fortiori argument from an a fortiori argument, right? So you are asking about that thing itself—how can you derive it? What do the words “an a fortiori argument” mean? “An a fortiori argument from an a fortiori argument” means… You don’t need to bring the grandson. What? Some words are missing here. That is why they rejected it; they said that the grandson cannot be learned. But also the son—the question is whether you can, whether you are permitted to derive the son of an a fortiori argument. What is an a fortiori argument from an a fortiori argument? What do the words “an a fortiori argument” mean? The grandson of an a fortiori argument from an a fortiori argument, which itself comes out of an a fortiori argument. That’s okay? That’s a derivative of an a fortiori argument. That’s okay? Yes. I don’t understand what Yossi says there on his original card. “An a fortiori argument from an a fortiori argument” means to derive an a fortiori argument from an a fortiori argument, which is what we wanted. No—the right to use this a fortiori argument I learn from one a fortiori argument, and not from an a fortiori argument from an a fortiori argument. Okay. Fine.

The first “a fortiori argument” in the phrase “an a fortiori argument from an a fortiori argument” is the same a fortiori argument that we want… it is the thing under discussion, that’s how I understand it, yes. Although here it’s not completely clear, but that’s how I understand it. “From an a fortiori argument” means from the a fortiori argument of this table. Yes. How do I know that the thirteen principles were also taught in order to derive things that are not laws or rulings? Here I’m really learning rules of the game; I’m not learning laws. True, the Talmud assumes that that too is possible. “If her father had but spit in her face, would she not be shamed seven days?” That too is not some law; it’s a mode of conduct. Yet they learn it by an a fortiori argument. So by means of derivations like these I can arrive at, I don’t know how many principles—not thirteen, maybe thirty-two principles? No, but these are principles; what’s the problem? These are regular principles from which Jewish law is learned; they’re simply being applied also to domains that are not legal. Not additional principles, but those same principles themselves, only here they are not being used in the practical legal domain but in the meta-legal one. Fine. They are also used in the area of aggadic literature, so one can use them in the Talmud’s meta-law. Between Rabbi Yosei HaGelili—there all those principles appear, and more principles too.

All right. So now, since we are looking for an example of an a fortiori argument from an a fortiori argument, right? I told you the only example I found is this one. That is, how do they learn that one may use an a fortiori argument from an a fortiori argument? It was rejected precisely because of the circularity here. But let’s take it. It was rejected not because it is not logically valid, but because we cannot learn from a tool whose own validity is in doubt. But it is still a good example for checking the structure of an a fortiori argument from an a fortiori argument. So I take it. That is, I take the option rejected here. I didn’t find another option, okay? Or another example. But still, it is an example. What? Just continue, continue. Now why is there really an a fortiori argument from an a fortiori argument here? Because the law that verbal analogy teaches—look at the first table in inference one, “verbal analogy teaches by an a fortiori argument.” Do you see that first table? Do you see it? Top left, right? Correct? That law itself is learned by an a fortiori argument. The reverse. That law, that verbal analogy teaches by an a fortiori argument—the top-left law in the upper table—itself is learned by an a fortiori argument. Therefore what we have here is actually an a fortiori argument from an a fortiori argument. Where does that appear? It is another Talmud passage on page 50b, the next passage on your page: “A matter learned from a verbal analogy—can it teach by an a fortiori argument?” Yes, you see, that is exactly the top-left datum in the upper table, right? “A verbal analogy teaches by an a fortiori argument.” Where do we learn that from? That’s this Talmud passage, fine? On page 50b. “A matter learned from a verbal analogy—can it teach by an a fortiori argument?” The Talmud says: by an a fortiori argument. We learn this itself by an a fortiori argument, and therefore there is here both a son and a grandson of an a fortiori argument. “And just as juxtaposition, which does not teach by juxtaposition—whether according to Rav Hamnuna or according to Rav Avin—teaches by an a fortiori argument, as taught in the school of Rabbi Ishmael, then a verbal analogy, which teaches by juxtaposition according to Rav Pappa, is it not all the more so that it should teach by an a fortiori argument?” Pay attention: this is your inference three. Just so it will be easier to see it before your eyes. In inference three, you see here the right column is “teaches by juxtaposition,” not “learned from juxtaposition” as above, but “teaches by juxtaposition.” What does “teaches by juxtaposition” mean? That it comes before juxtaposition, right? We see that “verbal analogy teaches by juxtaposition” means that the verbal analogy comes first and on top of it we build the juxtaposition. Above, the right column is “learned from juxtaposition,” meaning the juxtaposition is first and not second. Fine? So the upper table basically learns verbal analogy from juxtaposition. So notice what we actually have here: that verbal analogy teaches by an a fortiori argument is itself learned by an a fortiori argument from juxtaposition. And now they use yet another a fortiori argument in one table in order to learn from verbal analogy to the a fortiori argument itself, that it too teaches by an a fortiori argument. That is called an a fortiori argument from an a fortiori argument. Again, the Talmud rejected it, but only on methodological grounds, not on logical grounds. That is, this is an a fortiori argument from an a fortiori argument that is fine. Since the content here is the very question whether one may learn an a fortiori argument from an a fortiori argument, then it isn’t legitimate to make this argument. But if the contents were different, this would be the structure of an a fortiori argument from an a fortiori argument. So let’s ignore for a moment what these contents say. Let’s take the structure. This is basically the example we were looking for, right? Afterwards the Talmud says—again, table three is built on the upper-left column of table one, right? It leads to the upper-left column of table one. Now the Talmud continues: “But according to the one who does not accept Rav Pappa, what is there to say?” Who is Rav Pappa? Rav Pappa is the one who says that verbal analogy teaches by juxtaposition. Yes, table three, lower-right datum: you see “verbal analogy teaches by juxtaposition.” That is only Rav Pappa’s opinion. But one who puts zero there does not accept Rav Pappa’s opinion and will not be able to make the a fortiori argument. So the Talmud brings another a fortiori argument, and that is basically the a fortiori argument of inference four. “And just as juxtaposition, which does not teach by juxtaposition, teaches by an a fortiori argument”—you see, juxtaposition doubled, right? “It teaches by juxtaposition, so it teaches by an a fortiori argument; it does not teach by juxtaposition, yet it teaches by an a fortiori argument. Then a verbal analogy, which teaches through another verbal analogy according to Rami bar Hama, is it not all the more so that it should teach by an a fortiori argument?” Now notice carefully—exactly—I could have written the right column as “teaches by juxtaposition,” but that would not be correct. The right column is “doubled.” Juxtaposition doubled on top of juxtaposition, and verbal analogy doubled on top of verbal analogy. Therefore the datum, the relevant parameter here, the relevant definition of the column, is “doubled.” It is not “teaches by juxtaposition” or “learned from juxtaposition.” Fine? It’s a somewhat different angle. Fine. But this too is some kind of a fortiori argument. So we have two options, basically.

Now what does this actually mean? That if we go with Rav Pappa, we have to merge table one and table three. Leave the contents aside for the moment; that’s not interesting. According to Rav Pappa, we merge table one and three. According to the one who disagrees with Rav Pappa, we merge table one with four, right? Let’s see what comes out here. Table one with three—the union is as follows. What? In the last table on the left, you should have written “learned from juxtaposition”? Learned from juxtaposition, yes, that’s a mistake. B. Yes, that’s a mistake. Learned from juxtaposition. The moment you say “learned,” then it’s obvious what it is. Learned from juxtaposition. So we have juxtaposition, verbal analogy, and a fortiori argument. Notice: the first a fortiori argument is verbal analogy from juxtaposition. Right? That’s table three: verbal analogy from juxtaposition. But table three gives us only four of the data; we will need to complete two more. If you look in the Talmud, you can complete those two as well, simply by drawing those additional data from the Talmud too. So I already did that; I’ll tell you in a moment. The second a fortiori argument is to learn a fortiori argument from verbal analogy. Okay? Now the filling works like this: zero, one, one, one, x, y, zero, zero, one. Notice, as we said, here there is zero and here there is one. That has to be so, we said before, right? Because otherwise, if here were zero, it would be a refutation of the second a fortiori argument; if here were one, it would be a refutation of the first a fortiori argument. Okay? So indeed that is what comes out here, and this is the structure of an a fortiori argument from an a fortiori argument.

Good. So now what exactly are we doing here to solve the problem? We are basically checking four kinds of fillings. And note: this is what I said before. I don’t first fill this and then, in light of that, fill this, because I have no way to fill this so long as that isn’t filled. I need to check both together. I basically have four combinations: zero-zero, zero-one. The left one is x and the right one is y. Here the Talmud also suggests using that same second a fortiori argument that I described, doesn’t it? No. The a fortiori argument you suggested is to take these two and these two. Right, and that’s what it does. No, that’s not what it does. What does it do in the end? “Rather, an a fortiori argument: and just as juxtaposition…” Is the conclusion of the Talmud? “And just as juxtaposition, which is not learned by juxtaposition…” Is that not this zero on the left? No, no, not at all. Look—that’s why I broke it into tables, precisely to sharpen it. The first juxtaposition is juxtaposition three. Right? After I finish juxtaposition three and find that verbal analogy teaches by an a fortiori argument, I go up to juxtaposition one. Excellent, but now the Talmud rejects that. No, it doesn’t reject it. What do you mean reject? It says “and that is,” and in the middle it moves to another inference. It’s not simple. Inference four with one. I’ll do that in a moment. No, it’s table two, inference two. What? No, no, no—inference two is indeed it. Inference two, right. Now inference two is exactly the same table I suggested taking directly from above. No—juxtaposition learned from juxtaposition. Right. The Talmud really says: here, you can do it directly. But it doesn’t say that. It rejects it because it is the grandson of an a fortiori argument, and you cannot learn from that whether to use the grandson of an a fortiori argument. It doesn’t say: this whole structure of yours is unnecessary, let’s just make a direct a fortiori argument. Rather it says: look, here we can’t use it, but I have another idea, I have a new a fortiori argument. It doesn’t say that your entire structure is unnecessary and can always in fact be translated into a single a fortiori argument. Interesting question why I didn’t notice that that is indeed the option that comes up there at the end of the Talmud. But the Talmud doesn’t reject it that way. And if there are more cases? I don’t know. That needs checking. Since I don’t know of any cases, I can’t say. Fine. It may well be no accident that I don’t know of cases—precisely because of your consideration. That they always translated it directly into a single a fortiori argument, and that’s it. They never made an a fortiori argument from an a fortiori argument.

Okay. So basically what we need to do now is examine each filling. So let’s begin with zero-zero. Once there is zero-zero here, then clearly yes—we call them a, b, and c. a, b, c. What is big c? Okay. So with zero-zero, what does the diagram look like? Three independent circles. No arrows, right? We see that none of them depends on the other; each one has opposite zero and one. What happens in zero-one? Here there is zero and here there is one. So if there is zero here… zero and one, okay. Then notice that c enters b, right? So c enters b, and what about a? It also enters where? Okay. Now one-zero: here between these two I have independence, between these two also independence, but this enters that. c enters a. And in b—zero-zero, doesn’t c need to enter a? What? In zero-zero, doesn’t c need to enter a? In zero-zero? c enters a. Wait, let’s go through them one by one. c enters a regardless. Right, c enters a in any case, and b stays down here. Okay. And in one-one that means c enters a enters b. c enters a enters b. Right? What is the best filling? What? Obviously it’s one-one. You see that immediately, right? Because here there is a problem of connectedness, here there is a problem of connectedness, here there is a problem of reversal of direction. Right? Here there is nothing—neither reversal of direction nor connectedness. And the number of points in all of them is the same, so clearly this is the simplest. Also in terms of dimension, by the way—in terms of dimension, here it would be alpha, two alphas, three alphas. Meaning, one parameter can explain everything. Here we would need two parameters, both here and here. So both the dimension and something else. In other words, an a fortiori argument from an a fortiori argument is a strong argument, and we prove that in fact the best filling is one-one. What does that mean? It means that if I now return to your question—you asked: this filling I made is no longer a datum written in the Torah. How can I use it as the basis for a new a fortiori argument? After all, an a fortiori argument is only something based on biblical data. And this is not a biblical datum. The answer is that I am not doing it that way. I take x and y together, and I rely on the rest of the data, all of which are biblical, and I fill in both of them. So I have no problem; that’s fine. I simply take these biblical data and derive from them two conclusions. I’m allowed to derive two conclusions. And that’s it. So that’s why it’s okay.

Yes, but it’s strange to call it an a fortiori argument that learns from an a fortiori argument. What? It doesn’t really fit to call it an a fortiori argument that learns from an a fortiori argument. Why? Because if it learned from an a fortiori argument, then you have one x at the beginning in x. I do have one in x, that’s true. No—one-one is really on y. What? Then the question is only about y. No, because I begin with x first, fill it in as one, and then move to y. In my model I fill them together. Intuition works step by step. But then they have the same status. x and y have the same status. Exactly, the same status. Because this is built as an a fortiori argument from an a fortiori argument. In my model I fill them together—what difference does it make? The mathematics does not follow the intuition; it only has to fit it. I’m not doing a simulation of the stages through which intuition works. On the contrary, I am trying to go in a completely different direction and show that it fits. That’s all. So that’s exactly the point. In fact, here it is much clearer why I am permitted to do this, because in the intuitive picture his question is a good question. There is even a tannaitic dispute, but one of the tannaim says that one does not derive an a fortiori argument from a halakhah given to Moses at Sinai. Why not? Because it is not a verse. Meaning only things written in a verse can serve as the basis for an a fortiori argument. So how can something that emerges by an a fortiori argument be a datum that itself serves as a basis for an a fortiori argument? The answer I’m suggesting here is that this datum is not the basis of that. The basis is these five, or these seven. And from these seven—which are biblical data, perfectly straightforward biblical data—I derive those two conclusions. So that’s fine. Okay.

Now here, a contradiction. Yes. No, not a contradiction. Leave contradiction aside. Because what? If there is a very strong contradiction between them. Maybe there is a refutation? How is there a refutation? Because there is a refutation that catches here, but it catches here. There is a refutation that goes here—one-zero, right?—to refute this a fortiori argument. Now you tell me: if here there is one, then there is a refutation of this a fortiori argument. No, once again, you are thinking in terms of templates. I told you I’m not sure one is allowed to think in terms of templates. It has to be. No—if here there is zero, then there is a refutation of this a fortiori argument. Meaning that necessarily, if you refuted… this a fortiori argument, then the first one would also do so. No, obviously, but I’m asking the reverse. You are proving that there cannot be a refutation such that you draw a sub-table of the larger table. But I’m asking whether one is allowed to think at all in terms of sub-tables, and whether that is equivalent to the insight of the smaller table. Always, always, always when we added dimensions, then even then basically—why? Wait, wait—I added a dimension, and now what… Certainly that is exactly what we did. Every larger table we made, we checked everything anew. I did not assume the inference of the three-by-three table once it suddenly became three-by-four. When it became three-by-four, that was a new problem; I solve it again. And this is a very interesting question. The question is whether there is such a theorem for every… whether I can draw a conclusion from a sub-table, look around it and see there is no refutation against it, and assume it is valid, or whether I must check it on the full table. Does it come out the same or not? I don’t know. Do you understand? Okay, so this is… this is the proof that an a fortiori argument from an a fortiori argument works. As I said before, the Talmud too indeed concludes that an a fortiori argument from an a fortiori argument is okay even in sacrificial matters. So on this issue there is no problem, because it really is a strong argument, and there is no reason it should not work in sacrificial matters as well.

What happens according to the one who disagrees with Rav Pappa? Let’s write it briefly. The one who disagrees with Rav Pappa joins table four with table one, on your page. Right? Table four with table one. Now something interesting comes out here. Notice. Here a is doubled. Right? It is not learned from something; it is doubled. It teaches by an a fortiori argument and is learned from juxtaposition. Here too it should be “from juxtaposition.” Now here there is zero—wait, here there is juxtaposition, verbal analogy, a fortiori argument, zero, one, zero, one, x, zero. And here—what is here? That’s the only difference. What is here? Look at the table and you’ll see it. In “not learned by an a fortiori argument.” In y, right? Why? Because an a fortiori argument that teaches by an a fortiori argument and a doubled a fortiori argument are the same thing. Whatever we say about this, we’ll say about that. It’s just that here there is a table with a constraint. There are indeed three empty slots here, but there are only two variables that I need to search for. So on the principled level that’s not important, right? I still check four fillings. Whatever I decide about y, I will fill into those two slots each time. Okay, apart from that it’s exactly the same thing. Now what comes out is very similar. So in zero-zero once again we get three separate circles, and again. We get three separate circles. In one-zero, one-zero, we get a entering b, with c, and so forth. In zero-one, here c enters a and b. And here again a chain comes out: a enters b and c enters a. A chain. And again, of course, the one-one filling is the best. So whether according to Rav Pappa or not according to Rav Pappa, the conclusion is that indeed the filling is one-one. Each time, notice, what I determined in y, I am basically saying that… I am filling it into both slots each time, and that’s how I do the calculation. Yes. Other than that it’s exactly the same.

Now notice another interesting point. Basically this is… okay. Another point: what happens if I now do something like this? After all, this too can be a structure of an a fortiori argument from an a fortiori argument. Right? We fill in one here, one here, right? Now here there will be zero. And one-one and zero will also give me one here, by an a fortiori argument. Right? So in fact, this too is some kind of table of an a fortiori argument from an a fortiori argument, but now again the question is what I fill in on the two sides. And here, in short—I won’t now go into all the details—what comes out is that indeed the sub-tables work. That is, every… notice what happens here. What happens here is that once here there is one and here there is zero, the first a fortiori argument does not work. It has a row refutation against it. Right? The first a fortiori argument, which says this is more stringent than that, is refuted by these two data. Okay, so it can’t work; here there must be zero. And after that an a fortiori argument. And here, when I fill in one here, then if I fill in one here, that too is a refutation of that, and therefore here it must be one. It’s symmetrical to what we had before. Yes. Symmetrical to what we had before, and then of course it comes out, it comes out the same. And indeed one can check—I tried to check, again without proving a theorem—what happens if I put one here or zero here; it really doesn’t work. The general structure too doesn’t work. You can see it—wait, it’s not completely symmetrical. What? Why? Also in zero-one-one, that’s the right column. One-one x y—that’s the middle column, and zero-zero-one is the column. It’s rotated. Yes. So here too it comes out the same way. Again I say: from the cases I checked, whenever there is a sub-table that yields a certain result, but elsewhere in the table there is a refutation against it, the overall calculation too says that this weighting is not correct. Meaning, at least according to the data we have so far, you can certainly look at sub-tables—what you suggested before—and work with them. You don’t always need to do the general calculation. What is the general calculation? When in the sub-tables it doesn’t go through, then the general calculation means that all of them are equivalent? No. The general calculation means doing the calculation on a three-by-three table. For example this one. Meaning that all four of these are equivalent? What do you mean equivalent? That they have the same number… No, not all four—at least two of them. For example, it came out for me that if the refutation is on the second a fortiori argument, say here there is zero, then there would be a refutation of the second a fortiori argument, right? From here to here. But the first would be okay. What came out there was that indeed this filling is one, but in the one-filling y could be either zero or one, and that remained open. Meaning it came out that one-zero and one-one are equivalent fillings, and they are the best. So I proved that x is one and y remains open given that x is one. And that also comes out from the general calculation. But again, I don’t have an overall theorem, so I don’t… okay.

An analogical prototype from an a fortiori argument. As for an a fortiori argument from an a fortiori argument. An analogical prototype from an a fortiori argument doesn’t require a three-by-three. This is an analogical prototype from an a fortiori argument. Right? Here the business is simpler; it isn’t like with the a fortiori argument, where I need to add another row, because here if I fill in one by an a fortiori argument, now from one-one I make an analogical prototype; I want to fill this too with one. Okay? So an analogical prototype from an a fortiori argument is basically this table. Notice that if I fill in one here and zero here, this is a table of a refutation—a refutation of an a fortiori argument. Right? So here, in fact, even without doing the calculation we already know that the one-one filling will be the best. There is no doubt about it. You can see it if you want. One-one means that these two actually merge completely, one-one, right? And this enters both of them. The best possible. Meaning it is both connected and has one less point, just two points instead of three. So clearly that is the best filling. There is no question here. But here there really is a problem, and that is why I bring it, because regarding an analogical prototype from an a fortiori argument, the Talmud remained—no, sorry—the Talmud concludes that it is valid. Fine. But an a fortiori argument from an analogical prototype. An a fortiori argument from an analogical prototype—how is that structured? I begin with an analogical prototype. One, one, one, x—that’s the analogical prototype; I want to fill in one here. And now I use that one in order to make an a fortiori argument. Right? So how do I do it? Here we do need to get to a three-by-three table. An analogical prototype from an a fortiori argument is a two-by-three table, as I did earlier. But an a fortiori argument from an analogical prototype is a three-by-three table. It has to be diagonal, for exactly the same reason. So if this is one, I try once again y zero. Fine, like before, only the difference is that here there is one and not zero. Apart from that it is the same as before. Okay? So this is basically an a fortiori argument from an analogical prototype. It turns out that what comes out there is that… And what about the corner? What? What happens in the corner? The same thing. Here there will be zero, right? Because if there were one—here actually there could also be one, I think, because if I make an analogical prototype here, then it could also be one-one. Here it could be either zero or one; I don’t think it changes anything. And here… here too it doesn’t change, I think. And if there is one there, maybe you can derive it from itself by an analogical prototype? What again? If in the bottom-right corner there is one… Here? Maybe one could derive it from itself by an analogical prototype? Derive it from itself? From above? Like what you did before. Unless once again the overall calculation can disrupt it, and you need to check it using all the data, as I said before. Fine? Now perhaps that’s why the cases where they might have done this were always cases of zero and not… perhaps. Again, I don’t know. Since I have no cases, I can’t check it. But what comes out here is that an a fortiori argument from an analogical prototype is not necessarily valid. And indeed it also comes out that this too is an unresolved issue in the margins. Is there an example of this? What? Something simple? I didn’t find one. I didn’t find one. The only thing that is problematic is an analogical prototype from an analogical prototype. Wait, sorry, does it not matter that you derive from above? What do you mean it doesn’t matter? Everything changes. No, it changes, but there is no a priori dependence for me; it could be zero and it could be one, and you have to check each separately. Yes. Meaning, I can’t know in advance that in every case there will be zero here, the way I knew in an a fortiori argument from an a fortiori argument. And what happens in this case? And this is not necessarily valid. What? And this is not necessarily valid. What happens in this case? It comes out that the diagram of one-one is equivalent to another diagram. It has an advantage and a disadvantage relative to another diagram. So one comes out one, x comes out one, but zero remains open. The first analogical prototype is fine, but deriving an a fortiori argument from it remains open. Wait, could there be a case where… after all there are four diagrams. Could there be a case where two diagrams are equivalent, and one of the other two is better than the second? Yes, but then it is decided. The two equivalent ones must always also be the best; otherwise the problem is certainly decided. Otherwise the problem is decided. The only problem is an analogical prototype from an analogical prototype. What? If the better ones are also equivalent, then there is nothing to do. That’s called a refutation; it’s an open case. When the two best are equivalent, that means there is no decision. Fine? An analogical prototype from an analogical prototype actually comes out valid according to my consideration, while in the Talmud it remained open. That is the point where I don’t… I don’t know what to do with it. Other than to assume—but in any event we know that in sacrificial matters, whereas in the rest of the Torah an analogical prototype from an analogical prototype certainly works. So it’s fine that it comes out well in the model, right? The problem is why not in sacrificial matters. So we need to give up something, but it isn’t clear to me exactly what: that in sacrificial matters somehow one may not take valuation into account, or changes of direction, or I don’t know exactly what. I don’t have enough examples to try to do a systematic study of it, so I can’t. You don’t know what, and if so, you also don’t know why. Right. A concise description.

All right. Wait, and what about the a fortiori argument from an analogical prototype that we discussed? What? Is that only in sacrificial matters? What? Is that only in sacrificial matters or always? Yes. Wait, and what is an analogical prototype? What? What is an analogical prototype? Analogy. There are several kinds of analogical prototype. So we saw here at the beginning… when speaking of an analogical prototype that is an analogy. Zero-zero-one-one or one-one-one-one. I’m not speaking about a primary analogical prototype. I’m speaking about an analogical prototype that is… What? A common denominator? We once learned here that it says in the Torah “witnesses”… witnesses means two. And about witnesses—about that type I’m not speaking to you; I’m speaking of “if it was found among them” and “for generations.” You’re speaking about “why is this case different.” Yes. Now, the refutation of “why is this case different” is any refutation whatsoever. What? The refutation of “why is this case different” is any refutation whatsoever. Not clear. From two texts. In an analogical prototype from one verse it isn’t clear whether any refutation suffices or not; that is a tannaitic dispute. But from two texts any refutation whatsoever suffices.

The last point I want to check here is really not a question of deriving from something already derived, but it resembles it in some sense. We said that in the structure of an a fortiori argument there are two inferences. Yes, this is an a fortiori argument. There is the inference of the rows, and there is the inference of the columns. And all along we saw that this is not correct; these are two formulations of the same inference. Therefore, contrary to what we might perhaps have expected intuitively, it is enough to refute only a row—or sorry—it is enough to refute only a row or only a column and that will knock out the a fortiori argument. But it is just interesting to ask what happens if we make a refutation both on the rows and on the columns. Which is what, according to intuition, should have knocked out the a fortiori argument. Right? Notice—what did I ask at the very beginning? I said that basically the two formulations, rows and columns, are two different a fortiori arguments. So I refute the a fortiori argument of the rows with a column refutation. Right? Because it shows that this is not more stringent than that. And the a fortiori argument of the columns I refute with a row refutation. Right? Because I show that this is not more stringent than that. Okay? But all along we saw that this is not correct. Such a refutation refutes both a fortiori arguments, and such a refutation also refutes both a fortiori arguments. Now the question is just an interesting one: what happens if I place both refutations? I have two refutations. Basically what does that mean? Here too I don’t know what there is. This isn’t a question mark; I don’t know what there is. There could be several possibilities. Fine? Meaning, this is basically what interests me. But when I have two refutations, I still always need to ask the question: what is written here? This is another biblical datum that I need to find. But since I’m asking a general question, I’ll again mark this as x and this as y, and I’ll see for which y’s what happens to x. Fine? That will basically tell me what to do in all cases.

Now when you do the calculation of this table, it turns out as follows: zero-zero gives that b and c and a and b are dependent. One-zero gives c entering b and a by itself. Zero-one gives b entering c and a by itself. And one-one gives three separate ones, a, b, c. What is the significance of this? That zero-zero is the preferred one. What? Exactly. That zero-zero is basically preferable. Because notice: all the tables are not connected. All the tables. So the fact that this isn’t connected is not a disadvantage, because those aren’t connected either, and this isn’t connected, and that one is even less connected. We’d need to finish it off with alphas and betas; this is only the beginning. What? This is only the beginning of the work; afterward you have to finish it with alphas and betas, that’s not… Okay. It may be that if we do it with alphas and betas, we’ll reach the conclusion that it’s not… you won’t reach any conclusion. Here there are two parameters, here there are two parameters, here at least two parameters—it’s alpha and two alphas and this is beta. Here too the same thing. And here it is even three parameters. So it would still come out preferable. But this model is surely going to fit with the atoms of here and here, I’m not… I understand. If you put two parameters here, it will definitely fit on top of c; this is alpha, yes, this is alpha and this is beta; that’s the solution here, right? Now in none of them will you find fewer than two parameters. They are not connected. So there is no problem. In terms of dimension it certainly won’t work. Only connectedness doesn’t work either, so only the number of points is decisive, and therefore this is the best.

And what does that mean? Contrary to an ordinary refutation, which tells us that… what this means is that contrary to an ordinary refutation—if you place only a column or only a row—it leaves two possibilities open, either filling zero or filling one, and that remains open. That is called a state of refutation. Here, this is a proof that the filling is zero. This is a proof by a fortiori argument. Exactly. This is an anti–a fortiori argument. It is basically a proof that the correct filling is zero. It is not a refutation. Okay? Why? But I think you shouldn’t have done x and y; you should have taken x as x and y once fill in one and see what happens. It’s the same thing; I’m doing that. No, once fill in zero and see what happens, and once one. Here: when y is zero. When y is zero, that’s these two, right? Let’s see—then which is the preferred filling, zero or one? This one, right? When y is zero, then the preferred filling is this. So in any event, yes. Right, I should have, but I’m only showing the problem here in full. I’m only showing an interesting point here that basically closes the circle for us: all along we saw that the two inferences of an a fortiori argument—the rows and the columns—are not two different inferences, but one inference. And therefore either a column refutation or a row refutation refutes both. So what does the refutation do that, intuitively, should have refuted the a fortiori argument? A refutation that is both row and column. The answer is that it proves that the filling is zero; that is stronger than a refutation. A refutation only says that filling zero and filling one are equivalent. A double refutation says that the filling must be zero; it is a proof. It is a proof that the filling is zero. Fine?

And just a theoretical question: how would you derive one from one? If there were a refutation that reversed the a fortiori argument, build me a one here. If y is zero, show me how you reverse the a fortiori argument from below to arrive at this in b. No, you can’t. Why? Because if you come from below—if y is one—reverse the a fortiori argument from below so that you arrive at this in b. The reverse, from the more stringent to the more lenient, yields a filling of zero. Huh? If you didn’t praise a zero-filling that way, why would you praise it if I concluded a filling of one? Okay, try it at home. Let’s go to the other side. If you didn’t praise…