Lecture dated 19 Tammuz 5777
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- Tanna d’vei Rav: generalization and specification, and eight specifications
- Tanna d’vei Rabbi Yishmael: generalization, specification, and generalization around each specification
- Choosing one feature in the Talmudic wording and the hierarchical structure in the table
- Rashi’s difficulty and a methodological solution: carrying features “in the bag” through the stages
- The relationship between exposition by generalization-specification-generalization and the common denominator, and the claim that “there is no combining”
- Rav Acha’s refutation and his proof from chargol: its meaning within the new reading
- “Resonance”: extracting relevant aspects instead of combining results
- The role of “chagav,” the problem of “tzartzur,” and adding “its name is chagav” to Jewish law
Summary
Overview
In this Thursday lecture, the 19th of Tammuz 5770, Rabbi Michael Abraham presents two methods of exposition in the topic of the kosher signs of locusts: Tanna d’vei Rav sees in the verses a structure of generalization and specification, in which even “according to its kind” counts as an inclusive specification, while Tanna d’vei Rabbi Yishmael sees “according to its kind / according to their kinds” as language of generalization and therefore expounds generalization-specification-generalization around each specification. The Rabbi continues the exposition of d’vei Rabbi Yishmael through a table of signs and group diagrams in order to explain why the Talmud formulates the results through certain features such as “has no humpback,” and he proposes a methodological reading according to which the sugya does not use a classic common-denominator argument on the results of the expositions, but rather a process of neutralizing features and extracting “relevant aspects.” He explains the role of “chagav” as an excluding term that defines the field of discussion and excludes the “tzartzur,” and shows how the halakhic conclusion emerges: the four signs together with “its name is chagav,” as indeed was ruled.
Tanna d’vei Rav: generalization and specification, and eight specifications
The Rabbi states that according to Tanna d’vei Rav, the structure in the verses is a general category followed by a list of specifications, where “according to its kind” is not a generalization but an inclusive specification that adds another similar species. He describes that the verses “the arbeh according to its kind… the salam according to their kinds… the chargol according to their kinds… the chagav according to their kinds” create eight specifications, because each named item and its accompanying “according to its kind” are both counted as specifications. He explains that “according to its kind” is a general term and not an individual, but according to Rav it is still a specification that includes one more specific case, such as the “vineyard bird,” which is learned from “arbeh according to its kind.”
Tanna d’vei Rabbi Yishmael: generalization, specification, and generalization around each specification
The Rabbi explains that Tanna d’vei Rabbi Yishmael defines “according to its kind / according to their kinds” as “general forms of generalization,” and “arbeh, chagav, salam” as “specific forms of specification,” so that each specification stands between a first generalization and a second generalization, creating a structure of generalization-specification-generalization. He describes the exegetical sandwich in which the main general category opens, the specification stands in the middle, and “according to its kind” closes as an inclusive generalization. He continues from the point where the exposition begins: “Arbeh means govai; ‘according to its kind’ comes to include tzipporat keramim,” and defines the features of the arbeh through the table: it has no humpback, no tail, and a short head.
Choosing one feature in the Talmudic wording and the hierarchical structure in the table
The Rabbi presents a hierarchical diagram within the “set of four” signs from the Mishnah and explains that “short head” includes all the items, “no tail” is more restrictive, and “no humpback” is still more internal. He asks why the Talmud summarizes the result of the exposition around arbeh specifically through the feature “has no humpback,” rather than as a broader combination of all the signs, and answers that in terms of “the first one is decisive” as opposed to “the last one is decisive,” this teaches how the Talmud operates the rule of generalization-specification-generalization. He sharpens the point that a precise halakhic formulation should really have required wording that combines all the relevant conditions and not only “has no humpback,” because there could be creatures without a humpback that are not within the field of discussion at all if they do not satisfy the set of four signs.
Rashi’s difficulty and a methodological solution: carrying features “in the bag” through the stages
The Rabbi raises the claim that Rashi explains the sugya consistently according to the approach of “after the exclusion,” and shows that this seems difficult, because according to that, the result of the exposition of arbeh should have been broader and not narrowed to “has no humpback.” He resolves this by saying that the Talmud does not intend an essential narrowing, but rather a stage-by-stage formulation in which additional features remain in the background, and the discussion proceeds by neutralizing one parameter after another. He demonstrates this from the continuation of the Talmud: after including from salam “one that comes and has a humpback,” the humpback is neutralized as a parameter, and then the Talmud explicitly moves to the next parameter, “tail,” and learns from chargol that also “has a tail” is permitted, so it is clear that the features did not disappear; they simply had not yet become the focus.
The relationship between exposition by generalization-specification-generalization and the common denominator, and the claim that “there is no combining”
The Rabbi mentions the claim from the previous lecture that the sugya appears at first glance to be the only example of a combination between expositions of the generalization-specification type and logical expositions of the “prototype” and a fortiori type, and he presents the only principled possibility for such a combination: to begin with generalization and specification and continue with the common denominator. He argues that here too there is no true common denominator operating on the results of generalization-specification-generalization, and he brings signs of mismatch from within the sugya itself, foremost among them the point that it makes no sense for a common-denominator argument to seek to erase a feature shared by all three source cases, such as “short head.” He points out that the Talmud proceeds as though the common denominator works, and only afterward does Rav Acha’s difficulty appear, and he emphasizes how implausible it is that Rav Acha is “proving” that one can refute a common denominator, when the Talmud is full of such refutations.
Rav Acha’s refutation and his proof from chargol: its meaning within the new reading
The Rabbi quotes Rav Acha’s refutation, “What about these, whose heads are not long,” and presents the continuation of his claim, “And if you say… then include chargol as well… and let it come from arbeh and salam,” as a structure trying to justify a refutation of what appears to be a common-denominator argument. He explains that according to his reading, Rav Acha is not responding to a regular common denominator, but to a process that is not really a common denominator at all, but an attempt to extract relevant aspects from several expositions of generalization-specification-generalization, and therefore there is room to ask whether there is even such a thing as a “refutation” of what seems to be generalization-specification-generalization. He uses this to argue that the term “binyan av” in section 6 is not a formal exegetical rule but an everyday usage meaning learning from two cases.
“Resonance”: extracting relevant aspects instead of combining results
The Rabbi proposes that the individual expositions around arbeh, salam, and chargol do not stand as separate rulings whose results are then combined; rather, they are stages that neutralize parameters and reveal which “two aspects” remain as the basis for the general exposition of generalization-specification-generalization. He stresses that the process is not a “combining” that collapses all differences into the result of the “set of four” alone, because in generalization-specification-generalization, according to “the last one is decisive,” the result remains within the framework of two aspects and does not immediately collapse into a broad generality. He formulates this as a renewed activation of generalization-specification-generalization on the aspects extracted from the expositions, not as the operation of a common denominator on halakhic results.
The role of “chagav,” the problem of “tzartzur,” and adding “its name is chagav” to Jewish law
The Rabbi explains that “chagav” is not a specification like arbeh / salam / chargol, but a category-name that defines the field of discussion and excludes, and therefore it operates “with a different logic” within the list. He describes the Talmud’s question about the “tzartzur,” which has four legs, four wings, jumping legs, and wings that cover most of its body, and the answer, “The verse says ‘chagav’—its name must be chagav,” which teaches that the name-feature is decisive and excludes the tzartzur even if it has the four signs. He states that from here a structure emerges in which the discussion takes place within the world of “those that are called chagav,” and therefore the practical result is written as “anyone who has the set of four and whose name is chagav,” and he notes that this is how Rabbi Yose in the Mishnah, the Shulchan Arukh, Maimonides, and the Tur all rule in practice.
Full Transcript
[Rabbi Michael Abraham] Day
[Speaker B] Thursday, the 19th
[Rabbi Michael Abraham] of Tammuz, 5770, a lecture by Rabbi Michael Abraham.
[Speaker B] “According to its kind,” so
[Rabbi Michael Abraham] that “according to its kind” includes tzipporat keramim; arbeh and tzipporat keramim, those are the two things that get included. After that, from chargol it’s tisofya, from salam yokhna, and so on; each one, its “according to its kind,” includes additional specifics. That is basically Tanna d’vei Rav. Maybe let’s go back again to the structure. Tanna d’vei Rav see the structure here as kind of a generalization and specification—not generalization-specification-generalization, but generalization and specification. And then they basically say: look, the verses appear in front of you at the top of the source sheet. When it says there, “Of these you may eat: the arbeh according to its kind, and the salam according to their kinds, and the chargol according to their kinds, and the chagav according to their kinds,” those are four specifications, and the “according to its kind” of each one is also a specification. It simply comes to include one more specification, so you have eight specifications here. There are eight specifications here preceded by a general category, right? “That goes on four, that has jumping legs above its feet with which to leap on the ground” — that’s the general category, and after it comes a list of eight specifications, because each “according to its kind” is also a kind of specification.
[Speaker C] “According to its kind” is what you’d call kind of a general term, not an individual.
[Rabbi Michael Abraham] According to Rav—according to Rav—it comes to include one more case; it’s not a general category. It’s a specification that comes to include something else, like a kind of arbeh. There’s the arbeh, and “according to its kind” includes one more of that type, which is tzipporat keramim. Fine, and so on. But basically, as the continuation of the sugya explains—just so we keep the structure in our heads—according to Tanna d’vei Rav, the structure here is basically a structure of generalization and specification, where the specifications are a list of eight specifications: the named item and its “according to its kind.” And what about Tanna d’vei Rabbi Yishmael? Here they learn it through the exposition of generalization-specification-generalization, because they understand that “according to its kind” and “according to their kinds” are general expressions. It doesn’t come to include one extra isolated case. And since they’re general expressions, what we really have here is a three-part structure around each specific item. There’s the general category at the beginning, the general generality. “Arbeh according to its kind” is generalization-specification-generalization. Then salam according to its kind and chargol according to its kind. Each such specific item is basically in a sandwich between two general categories: the main general category at the beginning, and its “according to its kind” on the other side. So the structure here is basically generalization-specification-generalization. And Tanna d’vei Rabbi Yishmael really do expound it that way, and not as just generalization and specification like Tanna d’vei Rav.
Last time we started the exposition of Tanna d’vei Rabbi Yishmael, and I want to continue from that point, so let’s just recall where we were. Basically, we were at the beginning. Look in your Talmudic text, section three. Section two is the exposition of Tanna d’vei Rav; section three is the exposition of d’vei Rabbi Yishmael. So d’vei Rabbi Yishmael taught: these are general forms of generalization, and these are specific forms of specification. Arbeh—in other words, they claim that “according to its kind / according to their kinds” are generalizations, while arbeh, chagav, salam, and so on are specifications. That’s what they mean by “these are specific forms of specification and these are general forms of generalization.” As opposed to Tanna d’vei Rav, who also treat the “according to its kind” as specifications, d’vei Rabbi Yishmael say no—these are the specifics: arbeh, chargol, salam, and so on; and these are the general categories: “according to its kind,” “according to their kinds,” and so on. Therefore what we have here is basically a structure of generalization-specification-generalization.
So the exposition begins like this: “Arbeh” means govai; “according to its kind”—I’m in section three, yes?—“Arbeh” means govai; “according to its kind” comes to include tzipporat keramim. Meaning: anything similar to the arbeh. The features of the arbeh can be seen in the table: it has no humpback, no tail, and a short head. Fine? Those are the features of the arbeh, and “according to its kind” says: in generalization-specification-generalization, anything that has those three features. Now in section four, the Talmud continues: “I only have one that comes and has no humpback. From where do I know one that comes and does have a humpback? Scripture says: salam.” So what did we actually learn from arbeh? Look at the opening phrase: “I only have one that comes and has no humpback.” We already learned that. Any winged swarming creature that has no humpback may be eaten. Where did we learn that from? From the exposition in the previous section, right? From arbeh. In other words, the result of the generalization-specification-generalization around arbeh is all those that do not have a humpback.
Now if you look at the table, arbeh has other features too: no humpback, no tail, and a short head. Why did the Talmud choose specifically the first feature of arbeh, that it has no humpback? So that’s what I spoke about last time, and basically the claim was this: you can see from looking at the table that there’s a certain hierarchical structure of the vertices here. So there’s a set of four—the four signs in the Mishnah, with the jumping legs, the hoof-like feet, the short head, no tail, and no humpback. That’s the structure. How do I know? Just look at the table and you’ll see. “Short head” is the group that contains all the items, right? And the set of four is of course the overall domain. We’re dealing only with things that have the four signs that appear in the Mishnah. Right?
[Speaker B] Within that domain, we take the short-headed creatures.
[Rabbi Michael Abraham] That’s the next subgroup down, and it basically contains all the items. Now inside that are the tailless ones. Right? Why? Because the tailless ones are arbeh and salam. Both are included in the short-head group, right? What? What about chagav? Chagav is a general term. I said—it’s not really a specific species of creature at all.
[Speaker C] So the set of four is actually a set of three?
[Rabbi Michael Abraham] No, no—the set of four means the four signs: it has legs, wings, coverings, jumping legs, and so on. The set of four signs that characterize all these items.
[Speaker B] Jumping legs and so on, yes.
[Rabbi Michael Abraham] So obviously “no tail” is the next stage, and “no humpback” is a feature only of the arbeh, which is included within the tailless ones, right? Because arbeh is one of the tailless ones. On the other hand, if you look at the table—if we assume that the relations in the table more or less cover everything we have in reality—then the structure is like this. Once the structure is like this, if the result is all those that have no humpback, what is it saying? Right? “I only have those that have no humpback,” the beginning of section four. “I only have those that have no humpback.” I asked why the exposition from arbeh picks specifically that feature, “no humpback.” It has other features too. And the answer is that “no humpback” is the innermost feature. It’s an inner feature, and then what? Then if we expound—remember, we checked two possibilities: the first one is decisive, or the last one is decisive. Remember?
[Speaker B] We have generalization-specification-generalization. So if we say the last one is decisive, then that means we have—
[Rabbi Michael Abraham] Specification and generalization broadens on one side, while the first general category limits it to two aspects.
[Speaker B] And if we say the first one is decisive, then it’s the opposite.
[Rabbi Michael Abraham] We begin with generalization and specification, and then we have, say, five aspects in this case. Five aspects meaning there’s no inclusion at all—
[Speaker C] just the items written in the verse. Fine? Not even the minimal inclusion.
[Rabbi Michael Abraham] And the second generalization gives us the minimal inclusion, which means four aspects. Exactly the number of aspects. So apparently if that’s the result, then apparently they expounded it here according to “the first one is decisive.” Fine? It simply gave us a way to see how the Talmud is working. I already said this; it was at the end of the previous lecture. Yes.
[Speaker D] So what about the five levels? If you had said “those with tails,” then you also could have made the diagram.
[Rabbi Michael Abraham] Those with tails—you have the chartzol; look at the table.
[Speaker D] It also has all the signs, and it’s inside the humpbacked ones and inside the short-headed ones. Right. So I asked why not—why did they choose humpback specifically? Because arbeh is written first?
[Rabbi Michael Abraham] No, not because it’s written first. So why did they do it with humpback? Because right now I’m expounding arbeh, so the determining features are the features of arbeh.
[Speaker D] So why not just arbeh?
[Rabbi Michael Abraham] Yes, right now we’re doing generalization-specification-generalization around arbeh. Just arbeh. The order doesn’t matter. Fine? But still, that’s a good point—you have to notice this. I decided that the relevant features are no tail and short head. Why not has tail and long head, with the negatives flipped? Because I’m taking the arbeh around which I’m doing the exposition, and the features of the arbeh are the features I draw in the diagram. And the features of the arbeh are no tail, no humpback, short head, and the set of four. And in fact you’ll see in the next stage, when we expound salam for example, then the feature will be “has a humpback.” Or when we expound chargol, then the features will be “has a humpback” and “has a tail.” In other words, it all depends on the item sitting in the middle of the sandwich of generalization-specification-generalization. That item determines the relevant aspects.
[Speaker B] The Talmud expounds in this order because that’s how it appears in the verse. Meaning, it’s not accidental that the Talmud took it this way—it just works out.
[Rabbi Michael Abraham] Not true—but that doesn’t matter. The order won’t change the result here. We’ll see later how it continues, and then you’ll see—I think the order doesn’t change the result here. Fine. Now, I said that if indeed we’re going according to the approach that the first generalization is decisive, then it’s very simple: what does the Talmud say? “I only have humpback”—the meaning is this. But notice that it says humpback—what does that mean? Suppose we stopped here. Suppose that’s all the verse said—only arbeh. Fine? The later items weren’t written. What would you write in your Shulchan Arukh? Remember we said there’s a question here whether we’re looking from the perspective of writing the Shulchan Arukh, or from the perspective of each item that comes before us and we have to decide whether it’s kosher or not—or how you write the list of items, the Shulchan Arukh of kosher winged swarming creatures. So how do you write it? What would you write? You could write “no humpback,” but that’s not entirely precise. Because there may be something here that also has no humpback, but it’s not in the field of discussion at all—it doesn’t have the set of four, it’s not short-headed, so I’m not including it. I don’t know—I’m not familiar with this zoology—but there may be some winged swarming creature with no humpback, only it’s not part of the discussion because it doesn’t satisfy the set of four or anything else. Therefore, if someone wanted to write the Shulchan Arukh more precisely, he should have written: anyone who has the set of four, a short head, no tail, and no humpback. Fine? On the principled level, that’s the more precise formulation. It’s true that if the structure is this structure—I just don’t know what creatures are actually out there—but if the structure is this structure, then it doesn’t matter; it would be enough to write “no humpback,” because that already implies it, since everyone with no humpback is anyway inside—
[Speaker C] Like the kosher signs in animals. What? In kosher signs of animals there are ruminants that aren’t kosher, and split-hoofed animals that aren’t kosher, and all kinds of other combinations.
[Rabbi Michael Abraham] Exactly. There’s something from that group that sits here. I draw it this way because for me the set of four defines the discussion. So within the set of four, apparently the order is—maybe the order is like this; that’s the first proposal. The problem is that Rashi, for example, in this sugya consistently explains it according to the approach of “the last one after exclusion.” It’s not the first one after exclusion, but the last one after exclusion. And with “the last one after exclusion,” this comes out problematic, because according to that, what should have come out? Specification and generalization, and similarity in one aspect—what does that mean in terms of the diagram? The whole set of four, right? The big circle. Everyone inside the big circle is similar in one aspect to the arbeh. Okay? It doesn’t matter whether it has a long head, short head, no tail, has a tail—that’s not important. As long as it’s in the set of four, it has one aspect of similarity to the arbeh, and therefore the result should have been the set of four—which is also the result of this Mishnah. But that’s still before the other items, which we still have to go through. Yes. But we still have the first general category. The first general category limits it and says: not one aspect, but two aspects, right?
[Speaker B] And in “the last one after exclusion” we begin from here—that’s one aspect—
[Rabbi Michael Abraham] one aspect, and the first general category limits it more: no, I want similarity in two aspects, not one, and then the result should have been everyone with a short head, one circle inward. Okay? That’s what should have happened. But that’s not what is written here. Here it says everyone with no humpback—that’s the result of the exposition around arbeh. So how can Rashi explain here that it’s “the last one after exclusion”? That’s what I said quickly at the end last time, and I want to expand on it now because it’s a very important point. It seems to me this is one of the mysteries of the sugya, and I think we can explain it if we understand this properly from the outset.
Basically, the assumption I made here is an assumption that came out of the table. I looked overall at the three creatures that appear in the table—salam, arbeh, and chargol—and from that I tried to draw the diagram. That’s speculative, and who knows what actually happens in reality—how these features relate together in real creatures. It seems to me that the most general structure, the most general structure, is this. If these three features—yes, this is—
[Speaker C] Wait, I’m sorry, I didn’t follow: short head, no tail, and no humpback. Right?
[Rabbi Michael Abraham] The three specific creatures written in the verse don’t necessarily represent everything that exists. And notice: when we draw the diagram, we should draw this diagram according to reality, not according to the verse. I only used the table because I don’t know zoology, so I made a first attempt through the table. But basically what the Sages ought to have done is go to a zoologist—or whatever—do research, and ask themselves what the relationship is between these features. From the word arbeh they say that the relevant features are these four: the set of four, short head, no humpback, and no tail. That comes from the item written in the verse. Fine? But what the relationship between the features is—that’s a factual question, a scientific question, and it has to be checked. I don’t know—if there are creatures that have no tail but do have a humpback, then that means one isn’t included within the other, and vice versa. You have to examine the pairwise relationship between every two such features.
[Speaker B] Why did you put everything inside the set of four? Why isn’t it—?
[Rabbi Michael Abraham] Because the whole discussion is apparently taking place within the set of four. In principle you could make it even more general, three-dimensional. Yes, exactly. But I’m simplifying, because it’s hard to represent graphically. The Talmud too assumes that the entire discussion takes place within the set of four, because as I explained last time, the set of four is a feature shared by all the items in the verse, so that’s probably the domain under discussion. There’s no argument about that. The whole question is whether there’s some additional limitation or not.
Now when we look at this, and let’s go with Rashi now—what is the result here according to the approach that the last one is decisive? We said “the last one is decisive” means two aspects, right? “The first one is decisive” means one aspect; “the last one is decisive” means two aspects. Two aspects means that this is the result: all three of these circles. Everything inside one of these three circles is similar in at least two aspects. If it’s here, then it’s in three aspects. If it’s here, then even four aspects. But two aspects exist in everything that’s in these three circles. Two aspects relative to what? Relative to what’s in the middle—to the arbeh. The arbeh is the focal point of the diagram; that’s where the arbeh sits. It has all the features. And around it there are different circles of generalization. The radius of generalization here is this. That’s basically the radius of generalization if we’re talking about two aspects.
[Speaker C] Why is it specifically two aspects if it’s found in one of the outer circles?
[Rabbi Michael Abraham] Here? Yes. No, no—here? Yes, here. Yes, then it’s similar in the set of four and also in no humpback. So that’s two aspects. The set of four always gives you one aspect, and you want one more.
[Speaker D] And what about the white area there?
[Rabbi Michael Abraham] There it’s similar in only one aspect. Let’s see—we need to go a little further—
[Speaker B] A little, against the light.
[Speaker C] A little. What’s outside those three and inside the set of four—that’s one aspect.
[Rabbi Michael Abraham] Exactly. And therefore, because there’s the first general category, we require two aspects. The first general category excludes all the surrounding white space. Okay. Now what’s the result of that? How do I write it in terms of intension and extension, in terms of scope and content? So I could write it as the set of four and short head—that would indicate this circle, right? This circle is the set of four and no humpback. And this circle is the set of four and—
[Speaker B] no tail.
[Rabbi Michael Abraham] Okay? A union of these three, right? The union of these three intersections basically gives the result. Okay, so that’s the way set theory would write this result. Now our question is: why does Rashi write—or really, why does the Talmud; according to Rashi, why does the Talmud summarize the exposition of arbeh by “no humpback” here? It chooses this intersection specifically. Why not this one or this one? In principle all three are relevant: anyone similar in two aspects. Anyone similar in two aspects is anyone in each of these three groups. Why choose specifically “no humpback”?
So let’s see. I think the reason is that, in truth, just as before we saw that when it says “no humpback” here as an inner element, we really mean no humpback and short head and no tail and everything else. It’s called “no humpback” because that’s the innermost group. Now something similar happens here too, as we’ll see. It’s just a methodological construction of the exposition. Basically what the Talmud meant to say was all of these. The Talmud meant to say: I only have all of these. And now let’s see what the second one does with this—salam, I think. No, chargol—right? The second one is—no, salam. Fine? The second one is from salam.
Now how do we do the exposition from salam? Look in section four: “I only have one that comes and has no humpback. From where do I know one that comes and has a humpback? Scripture says: salam”—that is nifol—“according to their kinds,” to include the oshkhaf. Fine? So salam includes for me the “has a humpback” case. Or in other words, it neutralizes the humpback parameter. Right? Let’s look at the table. Look at your table. Salam differs from arbeh only in humpback, right? What does that actually mean? It means that if we now repeat the process we did on arbeh, we’re now doing another exposition on salam. What changes? Nothing changes except that “no humpback” becomes “has a humpback.” That’s all, right? Everything else stays the same. And the three features of salam are: has a humpback, short head, no tail, all within the set of four. So it will be exactly the same diagram, right? There’s no point even doing it again if we continue with “the last one is decisive” like Rashi. So we just continue exactly the same. By the way, if these had been concentric diagrams, there could have been a change. Because if we took that picture, then as Yehudit said earlier, from the fact that “no tail” could be located here, “has tail” might be over there. “Has tail” is the dual group; it could contain different properties and not be contained by them. But here, since we didn’t assume anything about the relation between the groups, it’s completely universal, so it doesn’t matter whether it says “no humpback” or “has a humpback.”
[Speaker B] Why don’t we insert salam into this exposition? What? Why don’t we take—try to place salam on this graph and build it?
[Rabbi Michael Abraham] Why would we do that? Because if salam is here—
[Speaker B] Okay, salam—
[Rabbi Michael Abraham] has no humpback, short head, and no tail. Now inside this fish—this, only this, not here.
[Speaker B] Fine, only if you take two properties similar to it, then what do you get?
[Rabbi Michael Abraham] You mean two properties that—no, you can’t, because you’re now doing a new exposition on salam.
[Speaker B] But I’m asking: let’s do a new exposition only on this graph.
[Rabbi Michael Abraham] But when you do it on this graph, you have to take the properties of salam. Fine.
[Speaker B] Those aren’t the properties of salam. This graph doesn’t describe the properties of salam. The properties of salam are “has a humpback.” But what do you mean the graph doesn’t describe it? This graph describes the whole world. No, no. The whole world is mapped in a very particular language—the language of the arbeh. That’s exactly the difference.
[Rabbi Michael Abraham] The specific item is very important here. The specific item determines the language in which you map the world.
[Speaker B] Wait, but this graph describes all the relations between has and doesn’t have a humpback, among all of them. Now I’m claiming there are eight cells here, right? Which presents the whole—yes.
[Rabbi Michael Abraham] It presents the whole world in a division made from the perspective of the arbeh.
[Speaker B] Fine, I want to think as if I were an arbeh. And if I apply salam to this graph—I draw it and take two properties from salam—then I’ll basically get… is it supposed to reach the same result?
[Rabbi Michael Abraham] What’s similar to salam in two properties? It’s a little hard to see from here; it would come out twisted. You’d have to check in which properties it’s similar and in which not. Leave it—let me do it more simply. You take salam and draw the exact same picture. After all, it’s the same picture. So instead of “no tail” I write “has a humpback”—that’s the whole difference. It’s the same thing, right? Once I write the same thing, I go straight here, according to “the last one is decisive,” and the only difference in language is that it will say “has a humpback.” Right? That’s all. Okay. Fine? So now notice. I’ll erase this for a moment.
[Speaker C] Doesn’t that solve the fact that humpback just isn’t relevant?
[Rabbi Michael Abraham] Exactly—that’s exactly what it means. Exactly. So now we got this from arbeh. Fine? And this from salam. So we have the set of four and short head—that’s the same thing.
[Speaker B] Here we have the set of four and—
[Rabbi Michael Abraham] humpback, and here the set of four and no tail. Fine? Those are the results we got from salam. And now what are we supposed to do with two expositions of generalization-specification-generalization? We now have two results. What are we supposed to do with them? Simply unite them, right? Meaning, if the Torah tells me you may eat this and you may eat that, then everything I may eat is the union of the two results. What comes out of the union of the two results? Well, obviously this is that group, and this is that group. Those two unite, and what do you get? Only the set of four, right? So basically what comes out is the set of four, the set of four and short head, and the set of four and no tail. Right? That’s basically what comes out of the unions among these three groups. Okay? That’s basically what comes out.
[Speaker B] Unions?
[Speaker C] Yes, of three groups.
[Rabbi Michael Abraham] What comes out is the union of all these results. I’m taking the union of these three—that’s the result of arbeh; the union of these three—that’s the result of salam.
[Speaker B] What suddenly? You made a union with the set of four, so it stays the whole set of four.
[Rabbi Michael Abraham] Right, so I wrote “the set of four.”
[Speaker B] So between those two—
[Rabbi Michael Abraham] it’s just the set of four.
[Speaker B] What suddenly? You took the union of the set of four with the other two, so it came out the set of four.
[Rabbi Michael Abraham] Ah—but what really happens if you continue this further is that the union of these three just gives you the set of four; the other two become irrelevant. Right? So basically we’re not talking here about a simple union. And this is, once again, the sugya. So what is being done here? The Talmud says nothing. It phrases it in a very general way. It says: I only have arbeh that has no humpback. From salam we learn that even one with a humpback is included, and therefore we removed humpback. How did we remove humpback? We removed everything. Only the set of four remains, and that’s it. We already reached the result of the Mishnah. We already reached the Mishnah. The Mishnah indeed says only the set of four. But that’s not right; it doesn’t continue that way. We still need to go further: salam, chargol, and chagav. So how does it continue? Therefore already here we see that something is happening that you don’t notice at first glance, and it’s connected to what we talked about last time.
I said last time that this sugya is the only example in rabbinic literature—or at least as far as we checked—of a combination between expositions from the family of generalization and specification and expositions from the logical family, matters like prototype reasoning and a fortiori reasoning. Fine? We said that theoretically such a combination can’t even exist except in one kind of case only: only if we begin with generalization and specification and continue with the common denominator, and not the other way around. Because if the common denominator comes first, you can’t then continue with generalization and specification. Generalization and specification is a rule that works on a text, not on halakhic results. Therefore the only possibility one could even imagine is something that begins with generalization and specification and afterward brings in the common denominator. The medieval authorities (Rishonim), at least some of them, seem to learn our sugya that way.
Later in the sugya, look in section six: “I only have chagav and no humpback, chagav and with a humpback, chagav and no tail, chagav and with a tail, chagav and its head is not long. From where do I know chagav and its head is long?” That’s just the continuation of the exposition. You said, after all, that you derive it by a binyan av from the three of them. “Binyan av from the three of them” means the common denominator, of course. “The case of arbeh is not like the case of chargol, and the case of chargol is not like the case of arbeh; the case of both of them is not like the case of salam, and the case of salam is not like the case of both of them. The common denominator among them is that they have four legs,” and so on. So here, ostensibly, they derive a common denominator from the results of an exposition that is generalization-specification-generalization—the only place in rabbinic literature where you have such a thing. And now I’m going to prove to you that here too, there isn’t. It’s not true. Fine?
So first of all, you can already see here what exactly they did in this combination. This combination is just a common denominator, a binyan av from the two of them, right? We take the generalization-specification-generalization that came out of salam, and the generalization-specification-generalization that came out of arbeh. Now those are the results—what do we do with them? We simply make a common-denominator argument. Remember how a common denominator works? A common denominator basically says: if I have two source cases—let’s try to draw it. If I have two source cases and I want to learn to a third case: in this source case there is, say, property X and not property Y. In this source case there is Y and not X. Fine? And both of them also have Z. And this one also has Z. And this one also has neither X nor Y. Fine? So we can’t now learn from this one alone to that one—why not? Because this one has X and that one doesn’t, so there’s a refutation. We can’t learn from this one alone to that one because this one has Y and that one doesn’t. But the Sages say: yes, but the common denominator of both is that both have Z, and this one also has Z, so we learn from both of them.
We explained how this common denominator works, not through the algorithmic tables but through simple logic. The simple logic basically says that from the fact that the law we’re learning exists both here and here, it’s clear that this law is not caused by property X, because even in the not-X case here, the law still exists. Likewise it’s clear that it is not caused by property Y, because here there is no property Y and still the law exists. So we neutralized the relevance of properties X and Y, and the claim is that property Z is what causes the law—and property Z exists here too. Therefore this case too has the law. That’s how a common denominator works.
What happens if we have only one of them? No Y. And has a humpback and no humpback. Fine? Same thing, right? It’s just a union. I’m only saying that humpback is not a relevant parameter; only the other shared characteristics are the relevant parameters. Humpback is not a relevant parameter. So in fact the same idea of the common denominator appears here too. It’s basically a union of relevant groups, right? When you unite groups, that’s the set-theoretic expression for proving that a certain parameter is not relevant. To prove that it’s not relevant means that the whole world, regardless of whether it has or doesn’t have a humpback, has this property. So it’s the same thing. To say that a property is not relevant is basically to unite the group that has the property with the group that doesn’t have the property. Fine? So therefore, on the face of it, what is being done here is some kind of common denominator.
But if it really were a common denominator, then the result would be only this. And that’s the real union of the groups here. The union of these two with this gives this. This contains them both. Is the common denominator a union? Yes, the common denominator is a kind—
[Speaker C] of union.
[Speaker B] You see, if we formulate
[Rabbi Michael Abraham] it in the language of
[Speaker C] the common denominator:
[Speaker B] “No tail” is also not a relevant
[Rabbi Michael Abraham] property, right? Because this one doesn’t have the property of “no tail” and still it has the law—our law is that it’s permitted to eat, right? So “no tail” is not a relevant property. “Short head” also isn’t a relevant property, right? So therefore we erase everything. Here you don’t need that. There’s a bit of union of if and only if. In a moment we’ll see—you’re right in your point about short head; we’ll get to that in a second, because the Talmud changes later. But at least for now I’m trying to show that from the flow of the sugya, even though the structure looks very much like a common denominator, it probably isn’t a common denominator. It’s something else. And I’ll prove that to you more sharply.
[Speaker C] One second, wait—before the diagrams and all that, I’m trying to read the Talmudic text the way they taught me in school. I can’t manage it. No—what does it say there in section 6? Basically the goal of section 6 is to check whether the issue of a short head is a relevant criterion, right? And there’s some kind of jump here and they reach the conclusion that the relevant criterion is the four parameters, and therefore a short head is… So let’s read. “The arbeh is not like the chargol.” Meaning, there’s some difference in the parameters between them. “The chargol is not like…” which is supposed to rule out the relevance of something. I’m jumping to the table. Short head is what’s under discussion right now. No, no—so its hump and its tail are…
[Rabbi Michael Abraham] What? The Gemara itself asks that. Rav Acha asks the question you’re asking.
[Speaker C] But before Rav Acha—the initial assumption of section 6. How does that…
[Rabbi Michael Abraham] That’s my proof that this is not a common-denominator derivation. I’ll get to that in a moment. I’ll get to it. You’re right.
[Speaker C] The arbeh and the chargol that appear here at the beginning in “the arbeh is not like…” are identical in terms of… you’re right. They’re all identical.
[Rabbi Michael Abraham] There’s complete overlap on that common denominator. What do all three have? That they have a short head. That’s really what you’re asking, bottom line.
[Speaker C] Before that, before that—what are we learning from the sentence “the arbeh is not like the chargol”?
[Rabbi Michael Abraham] “The chargol is not like…”?
[Speaker C] That the hump and the tail are irrelevant.
[Rabbi Michael Abraham] That what? That the hump and the tail are…
[Speaker C] Irrelevant. Right? Because the arbeh and the chargol don’t have a hump. Right. And both of them…
[Rabbi Michael Abraham] No, the arbeh has no hump; the chargol does have a hump. So that tells you the hump is irrelevant. Same with the tail. The arbeh has no tail; the chargol has a tail. So tail is irrelevant too.
[Speaker C] In one stroke they knock out the hump criterion and the tail criterion—both are irrelevant.
[Rabbi Michael Abraham] And still what remains is that all of them have a short head, and that’s exactly Rav Acha’s question.
[Speaker C] So wait—then what is the Gemara coming to teach us in the next sentence? “And neither of them is like the sela’am.”
[Rabbi Michael Abraham] So think—maybe you need…
[Speaker C] Both of them—he’s a hybrid, he has a hump.
[Rabbi Michael Abraham] Why? Because maybe… maybe only something in which both features are either absent or present. And then the sela’am shows you that that’s not true.
[Speaker C] That even if one exists… here’s an example of one like that, and still…
[Rabbi Michael Abraham] We’ll get into that in a bit more detail in a moment. But… but in the Gemara right now, exactly—come look…
[Speaker B] at this table—that if both are yes, both are no, that’s something that… that recurs in… generally in this topic and elsewhere. I didn’t understand. What I just suggested, for example. That… there are two things here. Not necessarily unique—something we’ve learned and used elsewhere. Why not? What? That there are two things here.
[Rabbi Michael Abraham] Could be, I’m not ruling it out, it could be. Fins and scales. Maybe only that… or that it doesn’t have a feature… its feature is that the tail and the hump are in the same state. Either both exist or both don’t. That’s the relevant feature.
[Speaker C] The table proves that “short” doesn’t have to be one of them. Exactly. So let’s see.
[Rabbi Michael Abraham] It could be. You can’t rule it out. So look—let’s keep reading this Gemara for a moment and you’ll see… you’ll see what I mean. So in section… I’ll do it briefly now and afterward I’ll explain in more detail, just so you can see the overall picture, because the picture is a bit complicated. So we said that in section 4 we included something that has a hump, right? The hump was neutralized. And in the meantime, what did we use? The arbeh and the sela’am. Right? And we only neutralized the hump. Now the Gemara says—but notice what else arbeh and sela’am have in common: they have no tail, right? Neither has a tail. You can see it in the table, look at the table. With arbeh and sela’am we neutralized the hump, because one has a hump and the other doesn’t, but neither has a tail. So now let’s read section 5. “I only know one that comes and has no hump; one that comes and has a hump…” wait… “one that comes and has no tail…” wait, section 5. Section 5. “I only know one that comes and has no tail; from where do I know one that comes and has a tail? Scripture says: chargol.” Yes. 5. Five. What are you reading? Five. Ah, yes. “I only know one that comes and has no hump; one that comes and has a hump”—that’s what we learned until now. Where does “one that comes and has no tail” come from? Because that’s common to both of them: both the arbeh and the sela’am, as we said, had no tail.
[Speaker C] We didn’t say that…
[Rabbi Michael Abraham] We didn’t say it, but…
[Speaker C] We’ve been operating all along within the group of those lacking a tail, right?
[Rabbi Michael Abraham] Exactly, so that was taken for granted. And now we’re going to see that this too is irrelevant—that having no tail is irrelevant. So notice what I said to you earlier about Rashi. When Rashi says “no hump,” I asked: why only the feature “no hump”? So that’s the result from arbeh. Why did he choose specifically “no hump”? So I already said that it’s only a methodological formulation; basically… what are we left with? Nothing? The whole fourfold set? No, we’re left with a short head. Rashi didn’t say that at all, and the Gemara didn’t say it either, because it was obvious—not because they didn’t relate to that group. They did relate to it. That group is there. When they mentioned only the feature of “no hump,” they didn’t mean only “no hump”; they meant these three groups. The Gemara’s methodological form for ruling out the relevance of the parameters is to begin by saying “no hump,” prove from sela’am that hump is irrelevant and erase it; then we’re left with short head, with the four, and no tail. Now in section 5: from where do I know “has a tail”? Scripture says: chargol. Chargol throws out the tail—tail is irrelevant too—and we’re left with the fourfold set and short head. That proves that Rashi has no difficulty at all. I asked earlier—the question we started with was why the Gemara chooses specifically the feature of “no hump” for arbeh, when it has three features. And we saw that in fact the result is three groups, so why did they choose only that group? According to the first version, that’s actually clear, because arbeh really comes out as the inner one, right? Only that one comes out; you don’t get three groups, you get one. So that would seem to be proof specifically for the first version. But Rashi explains our Gemara according to the later version, and I argued that he has no difficulty at all. Why? Because what the Gemara really means to say is: I only know the fourfold set and short head, or no hump, or no tail. It says this briefly because it does it in methodological stages, so it says: I only know these three. One that comes and has a hump, while the other two remain the same, so I’m not repeating them—how do I know it? That’s “the sela’am after its kind.” So we erased hump. What are we left with? We’re still left with those other two. Here’s the proof: because the Gemara now says, I only know no tail—what about has a tail? It’s talking about the tail now. It didn’t ignore it as if it weren’t there, as if it wasn’t in play—it was there. The Gemara is carrying the feature of “no tail” with it already from the first stage.
[Speaker B] Maybe if that’s so, then also—I’m looking here at the table—arbeh and chargol alone would have been enough for me, and the result would have been exactly the same.
[Rabbi Michael Abraham] Wait—the Gemara itself asks that too. In a moment you’ll see, the Gemara itself asks it. I’m doing this deliberately step by step because I want to show you that I’m right here, that this is not a common-denominator derivation.
[Speaker D] That’s a proof. So then it’s essentially the same thing, and it wouldn’t have been between arbeh and chargol because maybe between arbeh and chargol only what is similar works, so hump and tail yes, but what isn’t similar, no.
[Rabbi Michael Abraham] No, obviously—but what about arbeh and sela’am?
[Speaker D] You said what? No, arbeh and chargol.
[Rabbi Michael Abraham] No, arbeh and chargol—maybe the similarity between them requires similarity regarding tail and hump.
[Speaker D] If you have one like this and one like that, one plus and one minus, then it’s not like that.
[Rabbi Michael Abraham] That’s what they asked earlier.
[Speaker B] Why? Obviously, if I take arbeh and chargol, then obviously hump isn’t the feature.
[Rabbi Michael Abraham] No, but maybe the hump-and-tail feature has to be the same—either minus-minus or plus-plus; it can’t be minus-plus or plus-minus. Maybe, yes, maybe. The third option—the one they asked earlier.
[Speaker B] No, no, yes, yes…
[Rabbi Michael Abraham] Exactly. That’s why it’s a third option—the third option is split. Okay? So now let’s continue. So first of all I showed you that Rashi has no difficulty at all; the Gemara itself works this way. When the Gemara says “I only know one that has no hump; one that has a hump,” it doesn’t mean only “no hump”; it means those three things. We neutralize the “no hump” with the sela’am, and then indeed we come back to “no tail,” and it’s still with us. The fact that they didn’t say it is because in the meantime it wasn’t yet relevant, but it was with us all along in the backpack behind us. It’s not that they weren’t dealing with it till now. Therefore Rashi is right, and the Gemara here follows the later version and not the first version, even though at first glance it looks like the first version. It’s not true—it’s the later version. All right? And I think Rashi derives it exactly from here—from this he learned that the sugya here follows the later version.
[Speaker B] Where is it written that the features are according to the first version as opposed to the later version? Is that stated explicitly somewhere in the Gemara? What? That we go by the number of features—two, four, five, six?
[Rabbi Michael Abraham] The Gemara in Eruvin. That’s why we started with the sugya in Eruvin.
[Speaker B] But does the Gemara say it explicitly? Six features?
[Rabbi Michael Abraham] Yes, yes, yes. It says it explicitly. We saw it in the sugya in Eruvin. That’s why I think that’s the foundational sugya for this whole issue of common-denominator derivation.
[Speaker C] So now the Gemara continues—
[Rabbi Michael Abraham] So now: “And from where do I know one whose body is… and whose head is long?” And this is absolutely astonishing, because the three items that appear in the table all have short heads. So what does the Gemara do?
[Speaker C] It wants to prove that a short head is irrelevant.
[Rabbi Michael Abraham] It wants to prove that short head is irrelevant too. From where? From these three, by a common denominator. There’s no such thing as a common denominator here—it’s obvious…
[Speaker C] that it’s not a common denominator.
[Rabbi Michael Abraham] But in a moment I’ll show you what it is. But it’s obvious that it’s not a common denominator. Now true, Rav Acha immediately asks this right afterward, okay—but what kind of initial assumption is this, like you already asked? What sort of thing is this? What, the Gemara doesn’t know how to make a common-denominator derivation? It’s absolutely absurd, and against all the logic of such a derivation. If you refute it at some marginal point, obviously it doesn’t work that way. It’s exactly the opposite. Right, exactly. So now the Gemara makes a common denominator from all three.
[Speaker C] Meaning, formally it’s obvious that what should have been proved here is a new case that hasn’t appeared until now. Right—it needs a new equation.
[Rabbi Michael Abraham] Someone with a long head.
[Speaker C] It can’t be that with the three elements that have participated in the game until now, you can suddenly prove…
[Rabbi Michael Abraham] something different about a long head. You need someone with a long head. Now the Gemara—and then the Gemara concludes, in section 6, middle of section 6: “The common denominator among them is that it has four legs, four wings, jointed legs, and its wings cover most of its body”—that’s what I call the fourfold set—“and a short head”? That’s it? No—the Gemara says no: “The Torah revealed the case of short head, [to teach] any creature that has four legs,” etc. Then the Gemara asks—notice: at this stage the Gemara has no problem with it at all; Rav Acha’s question hasn’t even arisen yet. The Gemara keeps discussing it calmly: yes, fine, we proved long head too.
[Speaker C] It seems they assumed all the students fell asleep at that point. Could be.
[Rabbi Michael Abraham] “But what about the tzarzor—doesn’t it have four legs?” Wait, what are they asking—what about the cricket? Then they answer: no, no, it needs to be “whose name is chagav.” Yes—“Scripture says ‘chagav,’ whose name is chagav.” Notice: chagav plays a logical role opposite to the first three; it excludes, it doesn’t include. It excludes the tzarzor because its name is not chagav. The others include; it excludes.
[Speaker C] The discussion left the issue of short head versus long head and moved to dealing with the necessity of the four signs.
[Rabbi Michael Abraham] Correct. We already proved long head; now we have some problem with the tzarzor, so chagav solves that and everything is fine. And then suddenly Rav Acha comes in, in section 7: “Rav Acha objected: what is unique to these is that their heads are not long.” Boom—the wheel turns over. How can you make a common denominator from three things that share a common feature and then try to erase that very feature? That can’t be. Erasing means, as we said, showing that this feature is irrelevant—but this is exactly proof of the opposite: that this is the relevant feature. If I had to prove something from this table, I would say: a short head is required. Not only do I not know otherwise—no, a short head is required.
[Speaker C] Short head—that’s what you see in the common denominator, and in all four things also a short head.
[Rabbi Michael Abraham] So suddenly now Rav Acha—now they remember Rav Acha’s difficulty. And the Gemara says—just notice Rav Acha’s wording: “Rav Acha objected: what is unique to these is that their heads are not long.” “And if you would say: what is unique to these is that their heads are not long”—what kind of common denominator is that? And then look at Rav Acha’s extremely strange continuation—this is still part of the objection. “And if you say that since they are alike in the four signs, we derive from them and do not raise an objection”—if you tell me that you can’t object to a common denominator on the basis that all three source cases share the same feature, then I’ll prove to you that you can. Now look at the proof. “If so, chargol too, which is like them, should not have been written, and it should have been derived from arbeh and sela’am.” Notice in the table: suppose we have arbeh and sela’am, okay? And we don’t have chargol. Fine? And we don’t have chargol. So what do we do?
[Speaker C] What—you have a problem with the tail?
[Rabbi Michael Abraham] Correct. We have a problem with the tail. Right. We have a common denominator that neutralizes the hump—the hump is irrelevant. We have a problem with the tail, so what would we say? We would object: what is unique to these two is that they have no tail. From that Rav Acha proves that you can object to a common denominator. Because if you couldn’t object to a common denominator, then chargol would turn out to be superfluous, because I would derive it from arbeh and sela’am. What would you say as an objection—that they happen to have no tail? But after all, you don’t object to a common denominator. Rather, the fact that chargol was written means you can’t derive it from arbeh and sela’am, which means that this common denominator can indeed be objected to. Very nice. Meaning, he proves that you can object to a common denominator on the basis of a shared feature. An objection? A proof of that? The Talmud is full of that. You don’t need any proof for such a thing. The whole idea of a common denominator is that you can make that kind of objection. That’s the whole idea of a common denominator. Do I need proof that you can object to a common denominator if the source cases share a feature that the derived case lacks?
[Speaker B] That’s totally absurd. The later authorities don’t answer the question.
[Rabbi Michael Abraham] No, this whole thing doesn’t even get off the ground. Now, either it was obvious to them—what I’m about to say in a moment—maybe, I don’t know, but that’s how it seems to me. Not really obvious; I would expect someone to comment on it. I didn’t find anyone who discusses this whole business…
[Speaker C] a kind of common denominator that isn’t a common denominator.
[Rabbi Michael Abraham] Externally, maybe for the medieval authorities (Rishonim) what I’m about to explain was obvious. Maybe. I don’t know. I searched and didn’t find any discussion.
[Speaker C] Externally, in terms of the key words, the contrast, the “what is unique to those,” everything simply looks like that. Until you read it deeply.
[Rabbi Michael Abraham] And that’s my proof. Now look. There are several things here. What’s the proof? First, this common denominator doesn’t even begin. The common denominator is that they have a short head—
[Speaker C] and here you’d have to erase “short head.”
[Rabbi Michael Abraham] Second, why do they wait with the objection until after chagav? What kind of initial assumption is that? And even if there is such an initial assumption, why not attack it immediately and throw it out completely? Why drag this law along until after chagav in the middle?
[Speaker B] Why is the objection itself not understandable?
[Rabbi Michael Abraham] And this is the objection. And third, the objection itself—Abaye finds himself needing to defend it, as if to prove to us that you can object to a common denominator in this way. What, we don’t know that? The Talmud is full of objections to common denominators. What needs proving here? If you have such a case, it isn’t a common denominator—it’s not even an objection; it’s just not a common denominator at all. It isn’t an objection in the first place. What on earth is going on here? You can’t read it this way. Therefore, in my opinion—and now I’m returning to stage four—this whole Gemara has to be read differently, in my view. And what this Gemara is really doing is saying the following: there is no such thing as combining a common-denominator derivation with a general-particular-general structure. As I said in the previous lesson, no—even here it doesn’t happen. It doesn’t happen. What they’re doing here is not a common denominator, not an av-category derivation. That’s a mistake. It’s phrased like an av-category derivation, but that’s not what they’re doing here. So what are they doing?
[Speaker C] But they do use the term av-category derivation.
[Rabbi Michael Abraham] No—look, av-category derivation here is in its everyday sense: let’s learn from these two. Not in the technical sense of one of the hermeneutic rules. So what are they doing? The point is this. Look: we have four items, right? Around each one of them we expound a general-particular-general pattern. What do we do with the results? We have four results. What do we do with them?
[Speaker C] Why four? Three.
[Rabbi Michael Abraham] What? Three, sorry. Three results. What do we do with the results? One could have said: we take the union. That’s what we did up till now. But we already saw that the Gemara doesn’t take a union. And the continuation isn’t a common denominator either—union is a common denominator, right? And the continuation isn’t that either. The Gemara does not take a union of results. This is a very important point. There is no union among the results of general-particular-general. Why? Because there is no such thing as layering a common-denominator derivation on top of the results of general-particular-general. Union is a common denominator. It doesn’t work; you don’t combine them. Even the only possibility that might in principle have been considered—to apply a common denominator on top of general-particular-general—even that the Gemara refuses to do. So what does it do? It says: fine, somehow we still need to decide. So what do we do with this? In my opinion, what the Gemara does—we called this operation resonance, reverberation. What we’re really doing is this: we think there is one single structure here of general-particular-general. It’s one structure. It’s general-particular-general. Inside it there is a list of items. In the end we need to identify what the relevant aspects are for the general exposition. We are not making three private expositions. The private expositions are stages along the way through which we identify what the relevant aspects are. The structure of general-particular-general needs three aspects in principle, as we also saw in Eruvin. Meaning, we really need to find the two relevant aspects; in a moment we’ll see who the third aspect is. The third aspect is already “whose name is chagav”—I’m telling you that now already. But aside from that, there are two more aspects. And all we’re doing here is not a common denominator. We’re identifying: what are the aspects? We have different aspects; in each of the items there are different features. How do we make the exposition over the whole set? How do you do that? We make small general-particular-general expositions around each item, but the result is not a halakhic result. They are tools by which we neutralize aspects. We say: these aspects are irrelevant, those aspects are also irrelevant, and who remains in the end? That will be the relevant aspects for the exposition. Now I return to stage four. Look at stage four. Why is the result of the union of these two not simply this? Because we are not taking a union. What are we doing instead? We have to remain in a situation in which we have two aspects, because this is a general-particular-general exposition, right? And in a general-particular-general, when it comes specifically, the result is two aspects. Therefore specifically these two are the result, and not that. Because we are not taking a union between groups. We are identifying what the two aspects of the general exposition are.
[Speaker B] The general exposition in this case is, for the moment, only on those two items.
[Rabbi Michael Abraham] We’re proceeding step by step, after all.
[Speaker F] So we made an exposition on…
[Rabbi Michael Abraham] This, we gave a lecture on it; now we’re looking at these two together. Let’s say only those two are written. Let’s say only those two are written in the verse. One second. Let’s say only those two are written in the verse. We now have to identify who the two sides are that come out of the exposition, because the result of a general-particular-general exposition is specifically two sides at once. And once again, evidence for Rashi that here we’re going with “at once.” How do you find the two sides here? What? How do they emerge from them? So it comes out that for inclusion, with a short head and two sides, or inclusion and between them a tail of two sides. Those are two sides. It’s either-or. Right. Here, like in this kind of table, what is the result of two sides? Either this or that. You have to write it in the form of either-or, and that’s two sides. This is called two sides in a diagram like this, and there are also three such ones, but we neutralized one of them. Okay? So therefore this is really the result of an exposition; it’s not the result of a common denominator. It’s the result of a general-particular exposition on these two items together. That’s something else. You do a general-particular-general exposition on this, a general-particular-general exposition on that, take the results, and from them produce the result of an exposition as though these two items were written together, and now let’s see what the relevant sides are. Meaning, this is not a common denominator; it’s an attempt to identify sides. And now, wait a second, is this really the whole space that resembles both this and this in two sides? What do I mean? If I want to try to formulate what the rabbi just said in scientific-technical language, then this union, these two things, is the space that resembles both the locust and the bald locust in two sides—or am I mistaken? Or either the locust or the bald locust. Yes, yes, but you have to explicitly include both the hump-backed one and the non-hump-backed one; I’m ignoring that because it’s not relevant. No, I’m saying: similar in two sides to both of them. No, both-and? No, there’s no such thing, because one has a hump and the other doesn’t. And no one will resemble both with respect to hump-ness. But “in two sides” doesn’t mean similar in all sides in the world; it means in two sides. Two sides means reproduction, it’s… so you can also take the absence of a hump and absence of a tail; it too is similar in two sides. So why is the hump not relevant? I’m saying: similar in two sides to both of them. Exactly. Meaning there’s some more complex process here. I don’t know—again, general statements of the sort you’re asking about have to be proven. I don’t know whether I can state some general principle: take the properties of these two items, how do we directly derive the result? That’s exactly what I’m saying—no, that which is similar in two sides to both of them, that’s not the same… To both of them together? Similar in two sides also to this—no, to whom? To the locust and the bald locust? To both this and that. So that’s short head and tail, and no tail. But then you need someone who has… ah, reproduction and short head, so that’s similar… reproduction and short head is similar to both of them, and reproduction and tail is similar to both of them? Yes, in principle yes. Rabbi, how is this different from the common denominator? What we wanted to learn before. In the end we still—the rabbi said the two statements written below, and we make an “or” between them, a union, same thing. No, it’s not the same thing. The union between these two is not reproduction, absolutely not. Absolutely not. Ah, making a union between both of them each time without reproduction. What I marked here? No, just without reproduction—that’s the difference? Exactly. Okay? So that’s why I’m saying it has the appearance of a common denominator, but in my opinion it’s really a process of eliminating sides. We’re doing some process between the two expositions of general-particular-general in order to create a final or comprehensive general-particular-general exposition. But the final result is a general-particular-general exposition, not a common denominator. And therefore the result here is not reproduction. That’s why Rav Acha comes and says—Rav Acha here, sorry, before Rav Acha—what do these people do with the… with the common denominator in section six? We asked: what kind of common denominator is that? The answer is: it’s not a common denominator at all. It’s not a common denominator; it’s a general-particular-general exposition. Now look how this fits in. It’s simple; it’s a general-particular-general exposition. How do we do general-particular-general? After all, in a general-particular-general exposition we always expand beyond the properties found in the source cases, right? It resembles them only in two sides, not in three, not in all sides, right? Without a short head… we include the long-headed one. We include the long-headed one not because it is a common denominator to all the source cases. That would be nonsense; the common denominator to all the source cases is that they have a short head. Rather, after we make a common denominator among all the source cases, we identified what the relevant sides are in the passage. And which sides did we identify in the end? What? So in the end, what are the sides? So here we have… short head and reproduction, and no tail and reproduction—that comes out from the bald locust and the locust. Now the grasshopper tells you: it has a tail. Reproduction and short head. So reproduction and short head is basically what’s written here, right? Short head is included within the reproduction set. Reproduction and short head. Okay, but that’s not all. We asked why the hagav comes in in the middle. Why not attack this common denominator directly? This now explains what the Talmud explains when it gets to section six. “How do we know the one that comes with a head—how do we know? Is this a paradigm?” Yes, common in two properties. What? How does he explain it? Wait, I’m right here. General-particular-general by definition is something that doesn’t need to be completely similar to the source case. I always give up some of the source case’s properties. The question is how many properties; that depends on whether the first general term is decisive or the last general term is decisive, it depends how many points of similarity I require. But I always give up some of the properties, and that is exactly the claim here. That short head is common to the three source cases—so how can the result be long head? The answer is because we showed that short head is not a relevant parameter, because the answer is that we have two sides. Two sides—that whole circle—and therefore it also includes long head. Yes, two sides. Reproduction. Not yet—sorry, I moved too fast there. Before the two sides, we showed that short head is not a relevant parameter, and therefore the result is the set of four. Now we need to fit this into our pattern. After all, if the last general term is decisive, we go with Rashi’s approach that the last general term is decisive. “The last general term is decisive” means that general-particular-general gives two sides. But here all we have altogether is two sides, so really the result should have been this. They only wanted to expand to long head, so they bring the hagav in in the middle. Now look: this again explains every line in the Talmud if you read it correctly. Look. Now hold on, there’s something here that bothers me a bit; I think maybe this can be explained differently. Section six in a certain sense repeats the steps that were in three, four, and five. Right. But it doesn’t do it in exactly the same way. Because if I formulate what happens in three and four—where we’re talking about the locust and the bald locust—in the terminology, in the style of section six, I would say it like this: the locust is not like the bald locust, because the locust has no hump, unlike the bald locust which has a hump; and the bald locust is not like the locust, because it has a hump unlike the locust which does not. That proves that the hump is not relevant. The formulation is different. Wait, and then I get to the logical step done in step five, and I would say this: and the two of them—the locust and the bald locust, one and three—which have no tail, are not like the grasshopper, which has a tail; and the grasshopper, which has a tail, is not like them; from here, the tail is not a relevant property. Right. That is basically what the Talmud did in stages three, four, and five. Suddenly they get to section six and choose to lay out the space differently. No, no, they’re not laying out the space differently. Instead of reconstructing this process… No, but they’re not reconstructing it—they distinguish using one and two, the locust and the hagav, and that is exactly the point I’m making. Because the first three sections—three, four, and five—do a kind of private general-particular-general around the locust, around the grasshopper, around the bald locust. Section six performs a completely different logical operation. It does resonance. It doesn’t do general-particular. It now takes the results of the three private expositions and extracts from them what the relevant sides for the general exposition will be. That was not done in the previous stages. In the previous stages they did three private expositions, and therefore it is explicitly formulated as general-particular-general. Allow me to suggest another possibility? Okay. Come—when we explained earlier here “the two of them are not like the bald locust,” it was in order to prove that these two properties together are not both required. Let’s assume that’s not how the Talmud understands it; let’s assume there was no initial assumption like that at all. Okay. If so, then what the Talmud is doing in section six is saying this: the locust is not like the grasshopper, and the grasshopper is not like the locust, and by that I prove that neither the hump nor the tail is relevant. Because one has plus-plus and the other has… Ah, then why do I need the bald locust? Seemingly, why did they bring me a third example if by the locust and the grasshopper alone I can prove the irrelevance of the hump? So if the answer is not that exclusion, that either-or, then evidently the bald locust proves something new. And the something new is perhaps the irrelevance of short head, and the result is long head. Maybe that’s how the Talmud means it? No, you’re right, but that’s section eight. Rav Achai says exactly that in section eight, but Rav Achai disputes this common denominator. He rejects it. You’re right, that’s Rav Hai’s reading. Okay? We’ll get to him in a moment. In the end we asked: now I understand why there was an initial assumption here that we make a common denominator and learn a property that isn’t present in all the source cases, and that this is the common denominator. But we still haven’t explained how in a diagram like this the result comes out to be all of this. Two sides is really only this. On the other hand, I remind you that we had a few more difficulties in the course of the Talmudic discussion. One difficulty was: why did they bring the hagav in in the middle? Why not attack the common denominator directly with Rav Hai’s attack? The second difficulty was: why does Rav Hai need proof that one can refute a common denominator? The answer is: Rav Hai knows what he’s talking about; he knows that standing opposite him is not a common denominator. Standing opposite him is a process of resonance, resonance. He knows that we’re searching here, and he argues that even this thing can be attacked using the tools of common denominator—but that he has to prove. How do you know? After all, concerning general-particular-general we said there is no refutation. There is no refutation against general-particular-general. So if here you’re doing a general-particular-general from two sides, like a common denominator, but it’s general-particular-general—who says it can be refuted? That isn’t true; that’s exactly what I’m claiming here. After all, what is a refutation? The whole idea of general-particular-general is to extend to things that are not completely similar to the source case. So a refutation would always be a refutation. The whole idea of general-particular-general is that it is not exposed to refutational attacks. So Rav Hai is right—he needs to bring proof that one can attack such a common-denominator process by means of refutation. If it were an ordinary common denominator, no proof would be needed; even Rav Hai would not bring a proof. But a common denominator like this absolutely does need proof. More than needing proof—in fact it isn’t even understandable even after the proof. Why really? If there is a general-particular-general, then by definition we expand more than what is found in the source case, so why on earth can you refute it? Even after the proof, I still don’t understand what Rav Hai means. We began with the question: why on earth does Rav Hai need to bring proof? What he says is trivial. The conclusion is: not true—even after the proof, more explanation is needed as to what he means. I’ll come back to him in another moment. But let’s continue with this difficulty: what about the hagav at the beginning, that they bring the hagav in in the middle. So what about the cricket? At the end of section six the Talmud asks—we said the fourfold set is the key one. The Talmud asks: but isn’t there this cricket, which has four legs and four wings and jumping legs and so on—should it be permitted? Talmud: a cricket? Who said it isn’t permitted? All those that have the fourfold set are permitted, no? Who said the cricket is not permitted? Tradition maybe, or something—but it must have some property; not just tradition, because otherwise they’ll tell you it’s a scriptural decree and you don’t need to explain it with the tools of exposition. If we return it to the tools of exposition, then that means I can also show by means of exposition that the cricket is not—it’s not some novelty that comes alongside the exposition in parallel. What is the property? The cricket, as I at least understand it, has the four common characteristics, but it is not a hagav. It is not called hagav. And in this table it doesn’t appear? In the Talmud itself no; in the Torah the cricket is not mentioned. No, but hump, hagav, short head—so what? I have no idea. I have no idea what the internal properties are, but we already saw that all those properties are not relevant, so it doesn’t matter. Okay? If the cricket had been written in the Torah, I would need to know what its properties are. But we proved that all those properties are not relevant. So all I need to know about the cricket is only the following: that it has these four properties and that it is not called hagav. Hagav is written in the Torah. In “back of hagav”? Is “cricket” what’s written? It’s a shame to get into that now. The nicest thing in learning Talmud is that you need to know when to leave the page alone. Truthfully it would have been better to know; what can I do, I don’t know. So the Talmud rejects the matter of the cricket. How do we know that the cricket is nevertheless forbidden to eat despite having the four characteristics? Because it is not called hagav. What does that really mean? Notice. It means that either there is a set of five and not four. Right, there is a set of five and not four, and in the language of a Venn diagram that means there is another circle around this. This whole discussion takes place within the world of those that are called hagav. And therefore even one that fulfills the conditions of the fourfold set—the cricket is here somewhere outside. Yes, the cricket is here. Okay? It fulfills the conditions of the fourfold set, the quartet, so to speak, but it is not called hagav. Okay? We say: no—so it is erased. What does it mean that it is erased? That the whole discussion takes place within the world of hagavs. Therefore now notice what the resulting diagram is, as if by magic: three circles one inside the other. And what is the result of general-particular-general in such a situation? General-particular specifically. Two sides, simply this. Yes, you can do it differently—you can avoid making hagav have four properties, and instead of the fourfold set write there that there is a set of five, we have five properties, and there you go. No, but the Torah says hagav. There is something special about hagav. The Torah writes hagav. The Talmud’s assumption is that when the Torah writes a name that is not a proper name—we said from the beginning that hagav is unlike locust, grasshopper, and bald locust, because it is a category. It is not a particular creature. Here the Talmud teaches us something new. This name, this particular that is actually a category in the list of particulars, operates with a different logic. The items work with sides, all that we’ve done until now, the Venn diagram. When something is written that is a category, that means it bounds the sides. It means that the entire discussion takes place only inside the world of hagavs. It does not function as an additional particular in the structure of general-particular-general. One second—the proof is based on the fact that we have ambiguity and lack of knowledge as to whether hagavs have a long head or a short head. And that remains blank. I don’t know, but it doesn’t matter. We don’t know. And therefore you say the only thing hagavs have is these four properties. No, hagavs have short head; there are some with long head, there are hagavs of this sort and hagavs of that sort. Hagav includes the bald locust and the grasshopper. Meaning, being long-headed or short-headed is not a relevant criterion for being a hagav. Right, only the four properties are relevant to hagav-ness. And not even those. And not even those. Here’s the cricket—sorry, the cricket here—there are hagavs that don’t, there are hagavs that don’t have the property of the fourfold set. There are; I infer from the Talmud that there are. Seemingly one could say: hagav, period; hagav is four properties. You see that that’s not so. First of all, from the fact that the Torah also says hagav you learn that you are wrong. And that is exactly the point. That’s why I’m saying that with hagav it’s a circle like this. And now look, we asked one more thing, just one second—we asked: how does the result of the general-particular-general exposition, after the resonance, after everything, come out? We have two properties: the fourfold set and short head. If the last general term is decisive, we include two sides. Two sides should have been only this. Not true—two sides inside the hagav in the middle. You have to explain what you’re talking about before you derive net profit from it. So they insert the hagav here, and now you understand that the result really is this and not this. And therefore notice: you include long head. How do you include long head in this common denominator? One second. How do you include long head in this common denominator? Because this common denominator is not a common denominator; it is general-particular-general. It is general-particular-general, and this is its diagram. Short head and the fourfold set—those are the two sides that the resonance process left us from among the first three characteristics. Hagav—the Torah itself added it. When you draw this diagram and apply the rule that the last general term is decisive in general-particular-general, the result is the fourfold set, and that is what is written in the Mishnah. What is written in the Mishnah is the fourfold set, and Rabbi Yosei added that its name is hagav. Meaning: the whole fourfold set is only the fourfold set that is found within the field of discussion of hagavs. It may be—it’s not just that it may be, we know there is—the cricket also has the fourfold set, but its name is not hagav. So therefore this diagram is not a diagram of zoology. If you asked a zoologist, this would not be the correct diagram. This diagram—the diagram would look like this. Okay? Wait just a second, I want to finish explaining. If we asked a zoologist, the diagram would look like this. But since the Torah wrote hagav, the Torah is saying: leave that alone, ignore it. Now deal only with those that are called hagavs. Within the world of hagavs there are some that have the sign of the fourfold set and some that do not. Zoologically, etymologically as well. Okay. So now this description—notice—this description is not the result of zoological research; it is the result of what is written in the Torah. And in the Torah it says hagav, so it cuts off for us what the zoologist would have added here. Now we have a structure like this; general-particular-general in such a structure yields the signs of the fourfold set. But as I already explained, notice: when the fourfold set appears within the hagavs, how is the result that comes out in the Shulchan Arukh actually written? Anyone that has the fourfold set and whose name is hagav. Because otherwise the cricket would also be included. Therefore “anyone that has the fourfold set,” says Rabbi Yosei in the Mishnah—look in your section one—Rabbi Yosei in the Mishnah says, “Rabbi Yosei says: and its name is hagav,” and everyone rules this way in Jewish law—the Shulchan Arukh, Maimonides, the Tur—that its name must also be hagav. And that is the result. And now it is understood what the common denominator in section six is. The common denominator is not a common denominator. After we extracted these two sides, we identified the relevant sides. Now we perform a general exposition of general-particular-general on the sides we found. This is not a common denominator. It is an extraction of sides. And now I take the whole verse; there is a list here of a quartet of particulars. So what will the diagram of the sides relevant to the quartet of particulars be? This. That is the change in the resonance process. And now you apply a general general-particular-general exposition to it. And that is what gives the Mishnah, and it gives exactly the Mishnah. Just one more second—I only have to draw the picture that Rav Hai wrote in the middle. What Rav Hai disputes, I think at least—so what is Rav Hai attacking? Rav Hai argues that we do not continue the sides. We do four—expositions of general-particular-general, each one separately, really three—and it is not correct at the end to take the two principles we received and perform on them some renewed comprehensive general-particular-general exposition. He argues that we remain with the private expositions and somehow need to connect them, but when you connect them, then indeed you connect them like a common denominator. That can even be refuted if the sides are not relevant. That is what Rav Acha argues; he knows that the discussion here is general-particular-general, not common denominator. He only disagrees with the first stage in the Talmud, which makes some kind of general common denominator after we finished the private common sides. He stops at the first stage. We do four little general-particular-generals around each item, and that’s it—we don’t continue. That is what Rav Acha disputes. What would happen if we stopped with the hagav being inside the four sides, and then the cricket comes out between the hagav and the fourfold set, and again the relevant circle is the circle of the fourfold set plus hagav? Yes, it changes nothing, because in the end this is not zoology. What is written in the Torah—let’s say that were the situation: hagav, cricket, fourfold set—sorry, hagav and this is fourfold set, and then the cricket comes out in the middle—but the Torah says only one whose name is hagav. That means all of this is irrelevant even before we did the general-particular. We erased it; it does not appear in the diagram, because our entire discussion is only inside the hagavs. The most you’re saying is that this circle sticks to that one, and then it makes no difference. No no, then it’s not good. I thought about it incorrectly, because if that counts, then why don’t you count the fourfold set as four? What? If you count two when the hagav sticks to the fourfold set—because the fourfold set is truly common to all of them; hagav is not common to all of them. If the hagav and the cricket… Fine. By the way, I think next time we’re supposed to finish. In the next lesson we’ll finish.