Topics in Talmudic Logic, Lecture 6
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
- Introduction and lecture context
- The four basic tables and the meaning of a refutation
- The unified algorithm for deciding a complex table
- Defining the parameters and what counts as “good” and “not good”
- Calculating the number of parameters as a heuristic and brute-force search
- Building a graph from order relations in columns and transitive nodes
- The complexity of solving graphs and correcting mistakes
- Moving to the topic / passage in Kiddushin 5a: a fortiori reasoning, refutations, and a generalization from a prototype
- The common side as an elimination mechanism and the “miracle” of the combination
- The structure of the common-side table and its generality throughout the Talmud
- Deciding the three types of common side according to three different parameters
- Confirmation of the criterion and the claim that the parameters are necessary
- The story of the giant table, the absence of an ad hoc solution, and the zero/one mistake
- The strength of a fortiori reasoning versus a generalization from a prototype, and any slight refutation
- Relevance as selecting a table and hidden assumptions before the calculation
- An analogy to the philosophy of science: Semmelweis, Carr, theory, and facts
- The refutation of “their benefit is greater” as reality and a constraint on the solution
- Adding a document, the refutation from divorce, and a larger common side
- Conclusion and the planned continuation
Summary
General Overview
The text presents a unified algorithm for analyzing Talmudic forms of inference using data tables and graphs, so that a fortiori reasoning, a generalization from a prototype, a refutation, and the common side all come under one umbrella: a criterion for comparing two possible “fillings” of a question mark. The decision between filling in zero and filling in one is determined by topological parameters of the graph, and when there is no unequivocal preference, the result is a refutation in which “there is no result,” not “the result is zero,” because one and zero are equivalent when there is no decision. After developing the tools, the text begins to follow step by step the topic / passage in Kiddushin 5a, showing how a fortiori reasoning, refutations, a generalization from a prototype, and the common side are translated into tables and graphs and decided according to the criterion. Later on, it presents the hidden assumptions built into writing the table itself, the distinction between a halakhic / of Jewish law refutation and a factual refutation, and an explicit parallel between this logic and the construction of scientific theory and the choice of “relevant facts” in the philosophy of science.
Introduction and lecture context
The speaker opens with classroom remarks about photography and directing decisions made on the fly. He summarizes that they built a criterion involving a number of parameters, connectivity, direction changes, and number of vertices, and determined that the number of levels at which each parameter appears does not matter, under the assumption that one raises the number of levels in only one parameter out of all the parameters in play.
The four basic tables and the meaning of a refutation
The speaker reintroduces four basic tables corresponding to a fortiori reasoning, a generalization from a prototype, a refutation of a fortiori reasoning, and a refutation of a generalization from a prototype, emphasizing that under the criterion the results “come out beautifully.” He sharpens the point that a refutation means there is no result, not that the result is zero, and therefore when there is no decision, one and zero are equivalent. He adds that when there is a result, that means the inference is positive, and illustrates that a fortiori reasoning gives “an a fortiori result in reverse, strict and lenient instead of lenient and strict.”
The unified algorithm for deciding a complex table
The speaker says that there is no longer any need to identify names like a fortiori reasoning or generalization from a prototype, because as far as he is concerned there is one algorithm for all tables of every size. He describes a process in which you take a table with a question mark, build two models and graphs for the two possible fillings, calculate connectivity, direction changes, and number of vertices, and compare which is preferable on each parameter separately in the sense of minimum complexity. He says that if one filling is preferable or equal on all the parameters, it is chosen; and if preference is split between parameters, that is a refutation and there is no answer. Even an advantage of three parameters against one still counts as a refutation, because it is enough that there be no preference in one respect to undermine the assumption of a clear preference between the thing learned and the source from which it is learned.
Defining the parameters and what counts as “good” and “not good”
The speaker defines a large number of direction changes as not good, a large number of parts/elements in connectivity as not good, a large number of vertices as not good, and a larger number of parameters as less good, though each parameter is examined separately rather than as a total sum. He clarifies that by connectivity he means how many disconnected units there are, and the more disconnected units there are, the worse it is.
Calculating the number of parameters as a heuristic and brute-force search
The speaker says that calculating the number of parameters is “only a heuristic,” and that so far there is no closed algorithm for it, though someone did a master’s thesis on it, made some progress, and proved a few theorems. He notes that one can always do a “head against the wall” search that tries 2, 3, 4 parameters until it succeeds, but that is exponential.
Building a graph from order relations in columns and transitive nodes
The speaker explains that one builds a graph by determining an order relation between columns or rows—it makes no difference, because these are two formulations of the same consideration, and every table is “one argument.” He demonstrates a construction in which two vertices feed into the larger of them without unnecessary arrows because the relation is transitive. He constructs a parameter model for a basic node and explains that when moving backward, the number of parameters grows either by increasing the strength of a parameter—for example alpha, two alpha, three alpha—or by adding a new parameter—beta, gamma—under a rule that one cannot “go up in two parameters” at once.
The complexity of solving graphs and correcting mistakes
The speaker says that the graph can become complicated and have several possible solutions, and the matter becomes nontrivial, which is why an algorithm for solving such a graph is important. He notes that in the end they found places where the heuristic was mistaken and there was a simpler way, but without a closed algorithm he cannot give a general form, whereas a computerized search will always arrive at a result at the cost of time.
Moving to the topic / passage in Kiddushin 5a: a fortiori reasoning, refutations, and a generalization from a prototype
The speaker presents the Talmudic text in Kiddushin 5a step by step and translates it into tables. He brings Rav Huna’s statement that a bridal canopy effects acquisition by a fortiori reasoning, in the formulation: “If money, which does not complete, nevertheless acquires, then a bridal canopy, which does complete, should it not by law acquire?”—with money and bridal canopy against marriage and betrothal—and determines that the optimal filling comes out as one. He presents the refutation, “What about money, for one redeems consecrated property and second tithe with it,” as a fortiori table with a refutation, and the conclusion of the analysis is that filling in zero and filling in one are equivalent, so this is a refutation. He interprets “intercourse will prove it” as moving to learning bridal canopy from intercourse in the form of a generalization from a prototype, determines that there too the optimal filling is one, and then brings the refutation “What about intercourse, for it acquires in the case of a yevama,” which once again produces equivalence between the fillings and therefore no conclusion.
The common side as an elimination mechanism and the “miracle” of the combination
The speaker describes the move of “and the law returns… the common side in them,” in which money and intercourse join together to teach about the bridal canopy. He presents a schematic account in which each source is refuted by a different advantage, and then elimination cancels the alternative explanations and leaves a common parameter as the cause of the law. He illustrates this with an analogy to the natural sciences through elimination and generalization, and explains that there are two competing theories that explain the data of the teaching cases, but only one of them will allow us to infer the law for the case being learned. He ties the elimination to the criterion of simplicity by saying that a theory that grounds the law in one parameter is simpler than a theory that grounds it in two parameters.
The structure of the common-side table and its generality throughout the Talmud
The speaker argues that the common-side table has a universal structure in which there are always two unit vectors of 1,0,0 for traits that exist only in each of the teaching cases and not in the case being learned, and that a refutation of the common side is a situation where there is a characteristic present in both teaching cases but absent in the case being learned. He says that in the common side of the Sages there are not “facts” here but “laws,” and therefore the real parameters are the alpha-beta-gamma parameters that come from analyzing the table, not the halakhic columns themselves. He claims that there are three types of common side in the Talmud and no more, and that the only possible difference lies in the values of two particular cells, which are combinations of a fortiori reasoning and a generalization from a prototype.
Deciding the three types of common side according to three different parameters
The speaker analyzes the common side in Kiddushin by means of graphs for the two fillings, and arrives at the conclusion that in a common side made up of a fortiori reasoning and a generalization from a prototype, the decision favors filling in one because of an advantage in the parameter of direction changes, while the number of parameters remains equivalent because the number of levels plays no role. He continues and analyzes a common side made up of two a fortiori reasonings, showing that the decision in favor of filling in one comes from connectivity, because with filling in zero one gets a disconnected vertex that creates non-connectivity. He analyzes a common side made up of two generalizations from a prototype and shows that the decision in favor of filling in one comes from the smaller number of vertices, with equivalence in the number of parameters. He sums up that each of the three types of common side is decided by a different parameter: connectivity, direction changes, and number of vertices.
Confirmation of the criterion and the claim that the parameters are necessary
The speaker says that the topological parameters were chosen after study in books on graph theory as an attempt to characterize complexity, and then it became clear “to our astonishment” that each of the three types of common side is decided by one of the parameters. He argues that this is strong confirmation that the three parameters are necessary, because if one is missing, one of the types of common side will remain unexplained. But he emphasizes that this is not a mathematical proof, because there may be an alternative parameter that could replace one of them. He sums this up as the difference between a mathematician’s standard and a physicist’s standard, presenting it as a hypothesis that has been successfully tested.
The story of the giant table, the absence of an ad hoc solution, and the zero/one mistake
The speaker describes a case later in the passage where they reached a graph that was not decided by the criterion, and they tried looking for additional parameters but found none, until they thought of throwing the whole thing out. He says that in a large table “something like seven by ten,” it turned out that a zero had been written where there should have been a one, and after correcting it, the criterion solved the problem immediately. He presents this as strong confirmation that the method “dictates” and does not allow ad hoc tailoring.
The strength of a fortiori reasoning versus a generalization from a prototype, and any slight refutation
The speaker presents the intuition that a fortiori reasoning is stronger than a generalization from a prototype, and suggests a possible resolution according to which the number of preference criteria might express “strength,” even though every preference is decisive. He quotes the Talmud in Hullin, that a generalization from a prototype can be challenged by any slight refutation, whereas a fortiori reasoning requires a real refutation. He defines any slight refutation as a difference that exists between the teaching case and the learned case without certainty that it is relevant to the law, in contrast to a real refutation, which is a relevant difference recognized by reason.
Relevance as selecting a table and hidden assumptions before the calculation
The speaker says that the model does not decide whether a refutation is relevant or not, because that is reflected in the question of whether one adds a column to the table or not. Therefore relevance is an assumption of the calculation, not one of its results. He illustrates that one could have added irrelevant columns such as “goes into a pocket” or “can be bought at the grocery,” and the choice whether to add them is a decision based on reasoning, whereas once the table is fixed, the calculation is “mathematics” and “deduction.” He gives an example of a fortiori reasoning about a doorpost in relation to fringes and mezuzah, and explains that the table is illegitimate because it is obvious that the obligations are controlled by different parameters and therefore are not “playing on the same field.” He concludes that the table itself is the hidden assumption that prevents the inference from being necessary deduction.
An analogy to the philosophy of science: Semmelweis, Carr, theory, and facts
The speaker explicitly connects the model to the question of the relation between theory and facts, mentioning Semmelweis, who looked for causes of maternal mortality without a clear direction and therefore examined many “facts,” and the historian Carr, who argues that theory determines which facts are relevant. He says the same structure exists here: collecting halakhic / of Jewish law facts into a table requires prior intuition about what might be explained by the same parameters, because there are infinitely many possible facts, and intuition filters attempts before one arrives at the theory that emerges from the alpha-beta-gamma solution.
The refutation of “their benefit is greater” as reality and a constraint on the solution
The speaker moves ahead in the passage and brings the refutation, “What about the common side in them, for their benefit is greater,” emphasizing that the column of benefit is different because it is a matter of reality rather than a law. He argues that writing a benefit column in the table is “cheating,” and that it is more correct to treat it as a constraint on the solution: there must be some microscopic parameter that exists in both teaching cases and not in the learned case, instead of adding a column as just another law. He says that it turns out that even under a constraint one still gets a refutation, but he raises the open question whether the two approaches are mathematically equivalent, and connects this to the possibility that the dispute between Rabbi Yehuda and the Sages regarding the refutation of a stricter side could be expressed as a difference between “law” and “a constraint on solutions.”
Adding a document, the refutation from divorce, and a larger common side
The speaker describes the “a document will prove it” stage as a series of attempts: first a fortiori reasoning from document to bridal canopy, then the refutation “for it dissolves a marriage with a Jewish woman” by means of the law of divorce, and then a new combination in which money and intercourse will prove it, creating a common side between document and money-plus-intercourse. He describes a tabular expansion into a larger structure with another divorce column and another row for document, presenting it as a common side from three teaching cases to bridal canopy. He says that visually it does not matter who “goes with whom,” because mathematically it is equivalent; all that matters is filling in the laws and calculating the optimal filling. He notes that the refutation of a stricter side appears again in the form of a vector of three ones against zero in the learned case.
Conclusion and planned continuation
The speaker says he will stop here, and suggests that next time he will work through the final situation with “the biggest table,” show once how to analyze it through to the conclusion of the Talmudic passage, and then discuss the significance of the whole thing.
Full Transcript
[Speaker A] Good morning. Welcome. What? Are there new photos that I’m going to have to pose for?
[Rabbi Michael Abraham] Later on, but for now, yes. If you’re unsure, tell me and we’ll talk about it, we’ll see. Okay? Directing decisions on the fly. Okay, so practically speaking, what we did was present the criterion: number of parameters, connectivity, direction changes, and number of vertices. And the claim was that basically the number of levels at which each parameter appears doesn’t play a role. And one more assumption: that we increase the number of levels only in one parameter out of all the parameters that are in play. Now the basic structure is really a data table. We saw four of those. I’ll just do it here as a reminder. Say a table of zero, one, one, question mark—that’s a fortiori reasoning, right? And what does that mean? If here it doesn’t exist and here it does exist, then if it exists even here, it certainly exists here, right?
[Speaker A] This table, the table—
[Rabbi Michael Abraham] this one is a generalization from a prototype, right? Since here it’s the same, then here too this thing is the same. Here there is—yes, let’s do it—here there’s zero, one, one, question mark, one, zero. That’s a refutation of a fortiori reasoning, right?
[Speaker A] And this is a refutation—
[Rabbi Michael Abraham] of a generalization from a prototype. We have one, one, one, question mark, one, zero. Okay. And we saw that under this criterion everything comes out beautifully. Meaning: here the result is one, here the result is also one, here the result is zero, and here the result is zero. How do we get there?
[Speaker C] Not zero—you can’t know, because you said not zero.
[Rabbi Michael Abraham] Yes, right—that one and zero are equivalent, not that the result is zero. The result being zero, by the way, would itself mean there is an actual inference—we proved it. It would basically be a result of a fortiori reasoning, strict and lenient instead of lenient and strict. In other words, when there is a result, that means the inference is positive. A refutation means there is no result, not that the result is zero. In other words, one and zero are equivalent. So the basic algorithm basically says this: if I have a data table—and now it can also be as complex a table as you like—we’re no longer dependent on all those names like generalization from a prototype or not generalization from a prototype and exactly how and whatever size you like. In principle, let’s say here there’s one, one, zero, one, zero, zero, zero, one, zero, one, one, zero, one, one, zero, zero, one, zero, zero, one. I’m just throwing out something like that—sorry, one of them still needs to be the question mark. And I ask: what goes here? You understand that now I no longer need to look for whether this is a fortiori reasoning or a generalization from a prototype or a refutation or whatever you call it—those are names. As far as I’m concerned, the algorithm is one and the same for all of them. That’s why I say there is really a mechanism here that succeeds in bringing all these modes of inference under one umbrella, under one algorithm. Give me this thing, I’ll mark it, build the model and the graphs for filling in zero or filling in one, I’ll calculate these parameters—how many parameters there are, connectivity, direction changes, number of vertices. I’ll see which of the two fillings has an advantage. And the rule is this: if one of them has an advantage over the other in all the parameters—or is equal, either better or equal in all the parameters—then that is the preferred filling. If there is no unequivocal preference, if one is better in the first parameter and the other is better in the second, then that is a refutation. Meaning, I have no answer. When there are differences—say there’s an advantage in one parameter, in one criterion, in favor of one filling, and in the other three in favor of zero—still, that’s a refutation, because it is enough that there be no preference in one sense in order to cast doubt on the assumption that there is a clear preference between the thing learned and the source from which it is learned. Okay? Preference, we said, means minimum in each of the types—minimum. Direction changes: a large number of direction changes is not good. Connectivity: the number of parts, elements, also not good. Number of vertices, also not good. And the number of parameters? Yes, right. The larger it is, the worse it is. Right.
[Speaker C] But each one separately.
[Rabbi Michael Abraham] Not that you add them all together.
[Speaker C] And with connectivity we said: the more connections the better—that is, the fewer disconnected units.
[Rabbi Michael Abraham] Yes. The point is how many disconnected units there are. So the more disconnected units there are, the worse it is. Yes, exactly. That’s it. So I said that basically when there’s a given table, I put zero here, I put one here, analyze each of them, extract the parameters, and determine what the preferred filling is according to the question of which one satisfies these parameters—meaning which one is preferable to the other. One more thing we saw last time was the way we calculate the number of parameters, and here I said this is only a heuristic. At the moment there is no closed algorithm for this issue. I said there was someone who did a master’s thesis on this and made some progress and proved a few theorems about it, but there still isn’t a full algorithm for the matter. Of course there is the algorithm of—yes, banging your head against the wall. That always exists. You take two parameters, check whether you can do it; three parameters, four—wherever you succeed, you stop. But of course that’s something exponentially insane. In any case, what I said is this: how do we build the graph and from that calculate the criteria? So the claim is that we go, say, by the columns. In principle you can also go by the rows; it doesn’t matter. Because as we saw with a fortiori reasoning, I showed that these are not two formulations of two arguments. They are two different formulations of the same argument. And this is true for every table of every size. It doesn’t matter. It is always one argument. That’s the advantage of this way of looking at it. This perspective already draws all the conclusions we developed along the way. So I no longer need to look at the fortiori reasoning from here, the fortiori reasoning from there. A fortiori reasoning, for me, is a square table. And in this square table there is an algorithm for how to solve it. That algorithm is the a fortiori reasoning. It doesn’t matter whether it’s formulated by rows or by columns, because as we saw, it’s the same formulation. Okay, it’s the same consideration, not the same formulation. Now, how do you do this? I take the columns and establish some order relation among them. For example, if I do, say, filling in one just for the sake of the example, then if this is A, B, and C, then B is the greatest—say this is B’s side—C goes into it and A goes into it. Okay, right? Because both A and C are less than or equal to B, and there’s no relation between them. If there were a relation between them—say if here there were—well, that would require there to be three rows—but if here there were something that goes into this which goes into that, then we would need to make some kind of table like this, without this. Okay? And then that would mean this goes into that and this goes into that. Of course, if you look at the table, this also has an order relation relative to that, but the graph here reflects it through C. Okay? I don’t do it separately. I don’t put another arrow here. That arrow already tells me there is a relation between them too. It’s transitive. Okay, so now once we’ve drawn such a node—for example, this is the sample node I drew earlier. Like this. We said this is B, this is A, and this is C. Now how do you build the model for such a thing? So I start here, say, by deciding that here there is a parameter alpha. Here there has to be something greater than parameter alpha, right? If you remember, greater means weaker in the columns. In the rows, greater means stronger, right? For damaging agents, the more alpha the damaging agent has, the more liable you are for it—it’s more severe. For domains, the higher the alpha intensity of the domain, the more lenient it is. It’s harder to impose liability there. You need a very severe damaging agent in order for it to be liable there. Okay? That’s why the table was always structured so that if here it’s alpha and two alpha, then here it’s built the other way around—here it’s two alpha and here it’s alpha. Because the relation of severity here increases with the parameter, and here it decreases with the parameter. All right? And it’s always like that. So therefore if here B is alpha, then what can C be? For example, two alpha. And what will this be?
[Speaker C] Alpha and beta one more time.
[Rabbi Michael Abraham] Right. Two alpha and beta would also work, but you don’t need that. We’re looking for the simplest thing. So this is the basic node. It’s worth remembering that. When you have a node—it can be a very complicated graph with another arrow here and another arrow here and an arrow between those, all kinds of things like that—if you see this kind of node, then at this node the structure is like this, built in. Meaning: when you move backward from here, the number of parameters increases, because it becomes more lenient. But increases in what sense? Either the intensity of alpha increases, or another parameter is added to it. Those are the two ways to increase alpha. If, for example, there is another thing like this, and there’s D here, then there is no choice but to add another parameter, right? No choice. Meaning to add alpha and gamma. That’s it.
[Speaker A] Wait, and if it were in a situation like you drew before, one inside the other inside the third, then it’s the same thing? You don’t need—so that would be like three times this?
[Rabbi Michael Abraham] Same thing, right, you don’t need another parameter here. Right. And here it would be something like this.
[Speaker A] So you don’t need to change it. No, so there’s no need.
[Rabbi Michael Abraham] Then it would be—wait, not here. Let’s do it here. Here there’s three alpha. Exactly.
[Speaker A] Right. And basically this—
[Rabbi Michael Abraham] will go, yes. But if here, for example, it continues—
[Speaker A] just a—
[Rabbi Michael Abraham] riddle: what do we do here?
[Speaker C] Two beta?
[Speaker A] Then here there’ll be more.
[Rabbi Michael Abraham] Two beta is illegal. Illegal? Because you can’t go up in two parameters. So what then?
[Speaker C] Two alpha, but two—
[Speaker A] alpha plus beta.
[Rabbi Michael Abraham] Two alpha and also beta, apparently, right?
[Speaker C] But that—
[Rabbi Michael Abraham] is also not good, because there has to be an arrow to this. Two alpha and also beta would have to feed into this with an arrow. If there’s no arrow, that means some delta probably has to appear here.
[Speaker C] So we’ll add to this—it needs—
[Rabbi Michael Abraham] to add some other parameter, or add something here, it doesn’t matter. By the way, there are several solutions, but the question is: what is the minimal number of parameters, how is it distributed over the graph? There are several ways to do it. Okay? So this starts getting a bit complicated, and that’s why what I said earlier—that the algorithm for how to solve a graph like this is important—because the graph starts getting complicated, so it’s not all that simple to solve it. And the truth is that there were several things where in the end we found that we had made mistakes. We used this heuristic and then in the end saw that it could be done in a simpler way. Since we don’t have a closed algorithm, I can’t currently give a general form for doing it. Of course, again, if you’re not worried about computer time and you have a computer that can do it, then you will always get to the result—it’ll take time, but you’ll always get to the result. Okay. Now what I want to do is use these tools in order to start following the passage in Kiddushin, where my goal at this stage is at least to get as far as the common side, and there to do what I said last time we would do. Okay? To try to analyze the common side and come back and prove the criterion. I’ll show you why this criterion is the right criterion. But for that I want to start with the topic / passage in Kiddushin. So I’ll read you the Talmudic text, in Kiddushin 5a, step by step. We’ll read it and do it on the board, okay? So—a fortiori reasoning. There is a fortiori reasoning. Rav Huna said: a bridal canopy acquires by a fortiori reasoning. And there’s some discussion there, doesn’t matter, they wave away one formulation there. Rather, it challenges it like this: “If money, which does not complete, acquires, then a bridal canopy, which does complete, should it not by law acquire?” Okay? What is that? That’s a fortiori reasoning, right? So we already know how this works. I also gave all the notation there, so I’ll use that now. So we have this: we begin with a fortiori reasoning. We have bridal canopy and intercourse, okay? Bridal canopy and intercourse. No, sorry—bridal canopy and money, sorry.
[Speaker C] Bridal canopy and money.
[Rabbi Michael Abraham] And we have marriage, and we have betrothal. Okay? Let’s call it M, for whoever remembers. Now the a fortiori reasoning says: money does not effect marriage, but it does effect betrothal, right? A bridal canopy effects marriage, and the question is whether a bridal canopy effects betrothal. So Rav Huna says: a fortiori reasoning, right? We already analyzed this a fortiori reasoning, so I don’t need to go over it again. You make the table, discover that the maximal filling is one, the optimal filling is one. Okay. The Talmudic text continues: “What about money, for one redeems consecrated property and second tithe with it?”
[Speaker A] Here we add P, okay?
[Rabbi Michael Abraham] Here we add P, and redemption of consecrated property and second tithe—one and zero. What is that? That is a table of a fortiori reasoning with a refutation. That also appeared here, right? We analyzed that too, and we did the analysis. It turns out that filling in zero and filling in one here are equivalent. There is no preference for either filling, and therefore this is a refutation. Okay. Now the Talmudic text says: “Intercourse will prove it.” Meaning what? Huh? Not yet the yevama—that’s already the rejection. First of all, “intercourse will prove it” means what?
[Speaker C] Because of the yevama.
[Rabbi Michael Abraham] I don’t know what you’re saying, right? The Talmudic text doesn’t spell it out. But let’s think for ourselves what it’s saying. We basically have intercourse and bridal canopy, right? We are now learning from intercourse to bridal canopy. No money. If not money, then intercourse. In marriage it acquires, and in betrothal it also acquires, right? I know that—those are the laws. I know it. I know the laws. Here I know it’s one, and here it’s a question mark. So if that’s the case, it’s clear to me that the Talmudic text here, even though it doesn’t spell it out, is clearly making a generalization from a prototype, not a fortiori reasoning, right? Based on the shape of the table. Okay. We also analyzed this generalization from a prototype and saw that the optimal answer is one. Then the Talmudic text comes and says: “What about intercourse, for it acquires in the case of a yevama.” Again? Oy vey. Intercourse acquires in the case of a yevama, and a bridal canopy does not acquire in the case of a yevama. Again, a refutation of a generalization from a prototype. It turns out that filling in one and filling in zero are equivalent, and therefore we still have no conclusion. Now the question that interests us is that now the Talmudic text says this: “Money will prove it.” Intercourse doesn’t work; money will prove it. Wait, but with money we already got stuck—money doesn’t manage to prove it. The Talmudic text says: “And the law returns. This is not like that, and that is not like this.” Who is “this” and “that”? Intercourse and money, right? “The common side in them is that they acquire elsewhere and acquire here; so too I will bring a bridal canopy, which acquires elsewhere and should acquire here.” Okay? We’re doing the common side. Now notice how the common side actually works. We saw in one of the lessons—maybe the one before last or two before that, I don’t remember—that the common side can basically be described schematically like this. We have two teaching cases, we have the case being learned. We’re trying to learn from this because it has Z and this also has Z and this also has Z. We’re trying to learn from this. Then we say: what about this one, which has X, while that one doesn’t have X? Okay? Then it says: this one will prove it, because that one has Z and it too doesn’t have X. Then one says: no, what about that one, which has Y? Right? Then I say: and this one doesn’t have Y, okay? And that one also doesn’t have Y. Then I say: that doesn’t work—what about this one, which has Y? And then the element that—
[Speaker A] And the law returns.
[Rabbi Michael Abraham] What does that mean? “This is not like that, and that is not like this. The common side between them is that both have Z, and therefore, since this one also has Z, you can learn from both of them what you could not have learned from each one separately.” A miracle happens here. And I explained that miracle as a process of elimination, the way we do it in scientific generalization. I said: if two objects, say, fall toward the earth and I want to infer from them the law of gravity, I can say maybe this one falls because it’s black and the book maybe falls because it’s white. The common side between them is that both have mass, and therefore anything with mass falls toward the earth. That is elimination. Elimination basically says this: the parameter X, which is being proposed as an alternative—maybe that causes the law? That can’t be, because here there is no X and the law still exists. Maybe Y causes the law? That can’t be, because here too the parameter Y doesn’t exist, and yet the law still exists. That means that X and Y are not the parameters that create the law, that generate the law. Therefore there is no choice but to say that the parameter that generates the law is Z, and it exists in both of them. There is no refutation to that. Everywhere Z exists, the law exists. So if that’s what generates the law, then here in the case being learned, since Z exists, I don’t care that it lacks X and Y. Since it has Z, and Z is the relevant parameter, therefore the law exists in C as well, right? That’s basically what we said. So elimination basically shows us that X and Y are not relevant. Because if we didn’t show that, then you could say there are two theories. You could say that this causes the law—let’s call the law D, okay? And another possibility is that Z causes D, right? Those are the two possibilities, two theories. Either the existence of parameter X or parameter Y can cause the law; that would explain the law here and also the law here. This one has Y and this one has X.
[Speaker A] But it doesn’t explain the case being learned.
[Rabbi Michael Abraham] It doesn’t need to explain the case being learned. The case being learned is the result. It doesn’t need to explain the case being learned. It explains—the data are only these, that’s what I’m looking at. Here is my question mark. These I know. So it explains the data, right? Both this theory and that theory explain it. The question is which of the two is correct. Why is it important which of the two is correct? Because if this one is correct, then here it has neither X nor Y, right? Here X or Y is zero. So then D is absent. If this is the correct theory, then I don’t care that it lacks X and Y—it has Z, so therefore the law is here. So it is very important for me to know which of the two theories is correct. The process of elimination is basically built on the criterion of simplicity, where I say that this theory is simpler than that theory. It ties the law to one parameter, Z, and not to two parameters, X and Y. Now you understand that this is basically the same language as here, just in a more visual form. What I’m going to do now is translate it into this framework. We’ll do this too in exactly the same form, okay? So let’s see. I’ll leave this drawing here for now. Now look at what we’re doing. Because how does this miracle happen? When we learned from money to bridal canopy, we had a table of two by two, and then a refutation came. Three by two.
[Speaker A] After—
[Rabbi Michael Abraham] We left that aside; it doesn’t work from money, we gave up on it. Okay? We tried from money, it doesn’t work, we gave up. We went to learning only from intercourse. That also doesn’t exist. So once again this is a two-by-two table. And there’s also a refutation on it, a three-by-two one. This table is a binyan av, and this is a kal va-chomer. Okay? It could have been that both were kal va-chomers, or both were binyanei av. After we don’t succeed with either one of them separately, how can it be that the two of them together do the job? The answer is: we move to a table that contains all the parameters. All the… sorry, all the modes of acquisition. And then we basically have to make this table. Meaning: relevant. At first, in the Talmudic discussion, they tried to work only with this and only with that. If I didn’t succeed with this, I try with that. I don’t need both of them. If I had managed to work with this and there were no refutation, that would have been enough. After all, I can prove by a binyan av from intercourse to chuppah that it works. Why should I care that I couldn’t prove it from money? But I didn’t prove no. If money had proven no, that would have sabotaged the proof here. But money says: look, from money you won’t be able to draw a conclusion. I’ll draw it from intercourse. That’s why at the beginning of the common denominator we always start from just one source-teacher or just the other source-teacher. And what the common denominator does, in this language, is simply say: I’m now adding another row here, I’m making a bigger table; here I put intercourse and money. One, right? Wait, right, also wow, I need to add. Now I need to fill in the missing data, right? I took that from the tables. I also have Y as one. For intercourse this is one and here it’s zero, right? Now I… but I’m still missing these two. What do you say? So first of all let’s see. Does intercourse redeem second tithe and consecrated property? No, right? So obviously not. And does money acquire a yevamah? No. No. So I know, just from the Torah law itself, that this is zero. But in general, keep this rule in your hands: in a common denominator, the filling-in is always like this. Always. In this case I simply know the laws so I filled it in, and this is what came out, and this is always what comes out. Why? Because notice that what appears here—now I’m erasing this—notice: what I did is basically make a table that contains all the data from the two smaller tables. I didn’t add anything. It’s not that I’ve now added some extra page here that solves the problems I got stuck on before. No. Just combining the two tables together solves the problem. That’s the miracle happening here; in a moment we’ll see how it happens. Okay? But that’s basically the miracle taking place here. Now what I’m saying is: compare this to that, which is why I left the diagram here. Look. In practice what I’m saying is this: I have two source-teachers, money and intercourse, right? That’s A and B. My two source-teachers, right? I want to learn chuppah, that’s C. Right? I want to learn from the two source-teachers. Now look what characteristics there are. Money has one characteristic that also exists in… there’s one characteristic that exists in all three of them. Okay? This one, right? Z is the characteristic that exists in all three. In money, in document, and in intercourse. What? Which one? Wait, wait, in a second. Z, I’m looking here. Z, right? What about X? X exists only in A, not here and not here. Y exists only in B, not here and not here. Right? X and Y are these two. You see? Here there will always be a vector with one 1 and all the rest 0s. So that’s a representation of this thing. Okay? Now Z will be the result. Because really Y and P are not actually the parameters. X and Y are really the alphas and betas that came out of the diagram. It’s not Y and P. Remember in the class on the common denominator I explained the difference between a refutation from laws and a refutation from facts? Right? Now here what appears are laws. Acquires a yevamah, redeems second tithe, effects betrothal, completes betrothal, completes marriage. Right? We don’t say, for example, that money and intercourse involve pleasure. That’s not a halakhic parameter, just a fact. Money and intercourse involve pleasure; chuppah does not involve pleasure. Okay? Here there are no facts. Here there are only laws. So for example, Z—we’re looking for a row where all entries are 1, and you won’t find one. That will be the result. After we see that the fill-in here is 1, we’ll see that there really is something that exists in all three, and that will be our Z. But that will be a result. Why will it be a result? Because really the X, Y, and Z that appear here do not correspond to these columns. They correspond to alpha, beta, and gamma that we’ll find when we analyze the table with these columns. Because they are the factual characteristics, not the halakhic ones. Okay? So if that’s the case, we can see that this table is universal. A common denominator will always look like this. Before I go on, let’s see—this is basically the next stage in the sugya. And here the sugya claims that the result is 1. Right? We need to check whether according to our criterion we’ll succeed in proving that this is 1. But before I do that, let’s look for a moment at the different possibilities. What can vary in this table in other cases of common denominator? After all, the sugya of chuppah only serves me as a way to develop a more general theory and try to analyze every common denominator in the Talmud. What could be different? The differences can’t be here and here. That’s universal. There’s always one characteristic that exists only in one of the three, and a second characteristic that exists only in the first source-teacher and the third, and both do not exist in the thing being learned. Therefore here there will be two such unit-vectors of 1, 0, 0.
[Speaker A] And here it could be zero or one? What? Where? In that A, where you have opposite the…
[Rabbi Michael Abraham] No, that we’ll put in by hand. We’re looking for what’s there. But right now I’m speaking in terms of the data. Here there’s a question mark. The question is what could be different in the table in other sugyot. In this sugya this is the table. In other sugyot, what could there be? In what kind of common denominator could the structure vary and still look like a common denominator in the Gemara?
[Speaker C] Two kal va-chomers, or two binyanei av?
[Rabbi Michael Abraham] Right. Meaning, what could be here is either 0,1 or 1,1 or 0,0. Okay? That’s what can be here, and here it will always be one. Right, because both in kal va-chomer and in binyan av this is 1,1 and this is 1,1. Notice that this square is kal va-chomer.
[Speaker A] This square is the binyan av. These two are the refutation of the kal va-chomer.
[Rabbi Michael Abraham] These two are the refutation of the binyan av. The rest I filled in simply according to the laws I know from the Torah. By the way, if the laws had come out differently—if here, for example, there had been a 1—it wouldn’t have helped at all. If there were a 1 here, that would mean there are two characteristics here that exist in the two source-teachers and do not exist in the thing learned. That is a refutation of a common denominator. A refutation of a common denominator is when there are two source-teachers and there is a characteristic that exists in both source-teachers and not in the thing being learned. That’s how you refute a common denominator. Therefore, in the table of a common denominator there will always be only one lone 1 in this row and one lone 1 in this column. Okay? It can’t be otherwise. The only thing that can vary is these two cells. Of course you could also say zero and one, but that’s the same thing. It’s just saying this row is row one and this row is row three. It doesn’t matter; it’s the same thing. There are basically three kinds of common denominator in the Talmud, right? No more. There can’t be more. This is as broad as it gets. So let’s try to examine the black common denominator, what appears in the sugya in Kiddushin. Okay, so what we do in order to analyze it is we begin. So wait, first I’ll erase the… I’m doing the black one now. Now we make one fill-in where this is zero, and the second fill-in where it’s one. Okay? So let’s start with the blue. So with the blue, let’s make the diagram. Okay? And I’m drawing it in blue because it means this is the diagram of the blue fill-in. So we have this: we basically have A, right? Into which Y enters
[Speaker A] and P, with no relation between them, right?
[Rabbi Michael Abraham] The question is where N sits. What do you say—where, how does N enter here? If we put
[Speaker A] it in P? No.
[Rabbi Michael Abraham] N enters—N does not enter into A in this fill-in, but Y enters into N. Right? And then basically we need to write N here and Y here. Right? That’s the diagram of fill-in zero. Why does Y enter into N? Because Y is 001 and N is 011. So that means for every component this is greater than or equal to that. Right? Now if, as I said, these two both enter into A and there is a relation between them, then this will be a train. If there is no relation between them, then it will be like these two. Fine? That’s the difference.
[Speaker A] But how did you know that Y enters into N which enters into A, and not just that from Y an arrow goes into N, which is a vertex not on the new path you created?
[Rabbi Michael Abraham] What do you mean? I didn’t understand. Y enters… enters N.
[Speaker A] I mean, you could have drawn it where N now was previously Y; you could have drawn an arrow from Y to a new vertex N, which doesn’t…
[Rabbi Michael Abraham] No, that’s impossible, because N has to enter into A. You can see that N enters into A. N has to—not…
[Speaker A] Wait, wait, wait—that’s what I’m asking.
[Rabbi Michael Abraham] N does not enter into A. Wait, no, this drawing is wrong. I mixed up the fill-ins. Wait. Y enters into A, right? P also enters into A.
[Speaker A] Now N—does Y enter into N?
[Rabbi Michael Abraham] Ah, but not into A.
[Speaker A] Y does enter into N, that’s what I meant.
[Rabbi Michael Abraham] Okay, you were right. Fine? This is the graph. Okay. What about the red graph? The Potocki graph? So basically we have this: we have here A; this time A is really big, so N also enters into it, right? So this is basically Y, N, and P. Do they all enter the same A? Do you agree or not? Let’s see. No, now Y will need to enter into N, sorry. Y enters into N.
[Speaker A] Now it does. Yes, right. That’s the graph. Okay. Y
[Speaker C] enters into N, N into A; there are more connections, Y into A.
[Rabbi Michael Abraham] Okay? Now at this point let’s look for a moment at these fellows in terms of the criteria. We’ll calculate the other parameters in a moment, but in terms of connectedness the graphs are equivalent, right? They’re connected; here there’s one unit that is fully connected. Changes of direction—we’ll look in a second. Number of vertices likewise: four distinct vertices. What about changes of direction?
[Speaker A] Here there are fewer.
[Rabbi Michael Abraham] Here there are two changes of direction, right? And here there is one. So this is preferable in terms of… meaning, here there are more changes of direction, so this is preferable according to changes of direction. So if we don’t have a problem with the number of parameters, we already know that fill-in one is preferable. Right? Because it’s better in terms of changes of direction. In all the other respects they’re equivalent. What remains is just to check the number of parameters, and for that we need to go through these graphs and solve them. Suppose here we have alpha; then here it’s two alphas, here it’s alpha and beta, here it’s beta. Agreed? Right? This is an inverted node that splits into two, not two that enter into one. An inverted node basically means that here you have the intersection of these two. Fine? And you have to check that this doesn’t talk to that, right? If there were only a single alpha here, then there would have to be an arrow here. Right? Therefore it must be two alphas so that it won’t talk to that.
[Speaker A] And also
[Rabbi Michael Abraham] with beta it doesn’t talk. Right? And this also doesn’t talk to that. So this is alpha and that is beta, and everything is fine. Right? That’s the solution. What happens here?
[Speaker A] Again we have alpha,
[Rabbi Michael Abraham] two alphas, three alphas, and here alpha and beta. There are fewer parameters. Right? Now in terms of number of parameters, they’re equivalent. Right? Here, however, there’s an increase in the number of levels within the alpha parameter. But as we said before, that doesn’t matter. It doesn’t play a role, certainly not when there’s an advantage in one of these other things. I can leave open what happens when everything is equivalent and there’s only this factor. But if there is this factor, we saw that it doesn’t change the overall picture, and therefore the result is that the fill-in is fill-in one. And I remind you that we reached that result by means of the criterion of changes of direction. Let’s write on the side: what shall we call this? A common denominator of binyan av plus kal va-chomer, right? Good. Now we want to check what happens in the other two types of common denominator. Okay, so you see how this is constructed? If intercourse did not effect marriage, did not effect marriage itself, then the learning from intercourse to chuppah would have been a kal va-chomer and not a binyan av. This quartet, right? And then when I join them into a common denominator, it’s a common denominator with zero here and zero here, because it’s basically a common denominator composed of two kal va-chomers and not of a kal va-chomer and a binyan av like we did before. Okay, that’s the indication. Now let’s see what happens here. So again we have a blue fill-in and a red fill-in. Let’s start with the blue. So once again. Y enters into A and P also enters into A.
[Speaker E] And N plus another unconnected vertex, plus another vertex.
[Rabbi Michael Abraham] A vertex not connected to anything, right? You can already see what’s going to happen here. We have a problem with connectedness, right? Fine. So that’s the zero fill-in. What’s the one fill-in? So we basically have a picture like this. Right? This is A, this is Y, P, and N. In other words, what fill-in one did here was add this arrow. Right? That’s basically what happened. This arrow that wasn’t there before is now present, right? Because when there was a zero fill-in here, this did not enter into A. Now that there is a one fill-in here, it does enter. So all in all this arrow was added. By the way, it’s always like that: the difference between a zero fill-in and a one fill-in is that it simply adds some arrow somewhere. In most cases. Okay, now let’s figure out how this business works. So we have this: we have alpha; here we have two alphas; here alpha and beta. Right? This is the basic node; we already know what to do with it.
[Speaker A] What happens here? Here there is some…
[Rabbi Michael Abraham] It seems to me that A turned toward gamma, right? In principle we could have done two betas. You see here the implication. We could have done two betas here, right? That would have worked.
[Speaker C] Where? Here, yes. Two betas.
[Rabbi Michael Abraham] Right? Two betas don’t talk to alpha, and also beta. Okay, but we know we don’t go up by more than one parameter; we already did alpha and two alphas, therefore we’re forced to add gamma. Okay? What happens here? We have alpha, here we have two alphas, alpha and beta, and let’s say alpha and gamma, or beta and gamma, it doesn’t matter. No, alpha and gamma is better. Why? Within alpha. What? Right. Here we increased within alpha, right; it can’t be beta and gamma, right. It has to be alpha and gamma. And once again what do we see here? That in terms of number of parameters there is equivalence. We need three parameters both for zero fill-in and for one fill-in. Okay? In terms of connectedness, both are fully connected. In terms of changes of direction, there is one change of direction here, and here too there isn’t more than one change of direction. Right? What remains? Number of vertices is the same.
[Speaker A] Wait, what about connectedness? Sorry, connectedness. Connectedness bothers me, because this one is isolated, one stands…
[Rabbi Michael Abraham] Right, so therefore it is not connected.
[Speaker A] And therefore this thing basically means…
[Rabbi Michael Abraham] So which is preferable? The red one, right? Fill-in one. So this common denominator too—the fill-in is one, and this time it comes out because of that. A common denominator of kal va-chomer and kal va-chomer, right? You already understand what the third one will be. By the way, just in principle, a complete solution of such a thing is also supposed to give me a solution for the modes of acquisition. How do we do that? Remember? And what I found here is only how to populate these. What the chemical analysis is of these—what alpha, beta, gamma parameters there are in these and in what doses. Now how do I find these? So I know: if it has the ability to do this and this, P and A, right? That means it has alpha and beta, and this will also do alpha. So therefore it’s obvious that if this is alpha and beta… I’m only doing this for the exercise. What about H? So H, under the assumption that the fill-in is zero—I’m talking right now—H basically does only N, right? So H is basically gamma. H? Yes. Right? It does only N. So it has only gamma, nothing else. And B does A and Y. A and Y, so that means it has alpha and gamma.
[Speaker A] Right? Y… here there are two alphas.
[Rabbi Michael Abraham] What did I say? Wait. B does A and Y. So it has two alphas.
[Speaker A] Alpha and two alphas? No, just two alphas, yes.
[Rabbi Michael Abraham] Fine? So that’s how we move from the solution of the columns to the solution of the rows. I’ll explain later why that matters. But for our purposes, for the business at hand, it doesn’t matter. For our purposes we found that there is a certain number of parameters. Obviously, if I can explain all the columns with two parameters, it won’t happen that in these rows an additional parameter suddenly appears. There’s no such thing. Because otherwise there is no explanation for the columns either. In other words, I said that the consideration regarding the columns is equivalent to the consideration regarding the rows. It’s the same consideration. Therefore, if three parameters suffice for the columns, they will also suffice for the rows. That’s obvious. Okay, so that’s this exercise. Now we need to do the common denominator of the last type, which is a common denominator of two binyanei av. Right? And again we’ll do the same analysis. In zero fill-in, we have N, into which A enters, and from A come out these two fellows, right? P and Y. Agreed? And in fill-in one, we basically have—you can already see what’s happening here—A and N are one vertex. Y enters from here—sorry, P, let’s draw it this way so it will be parallel. P enters from here and Y enters from here. Right? Now you can already see that in terms of number of vertices there is an advantage to the red one. And that’s what will decide the issue here. And to be sure, we need to check the parameters. How do we do the parameters? So here we have alpha, two alphas,
[Speaker A] two alphas and beta, three alphas.
[Rabbi Michael Abraham] Right? I’m simply following the rule. From here to here I increase; I have no reason to add a parameter because I can do it by increasing the valence or the number of degrees. And from here this is the regular node that we already know. One of them adds a parameter and the other rises by one. Right? So this is the basic unit that we can already do with our eyes closed. What happens? So basically with two parameters we explain everything. And now notice: this is a stronger common denominator. A stronger common denominator than the previous one, because with two parameters we explain everything; we don’t need three like in the previous common denominator. Okay? That also makes sense, because when you have an analogy, that means they really are all similar, much closer to one another. Right? You don’t need many parameters to explain them. What happens here? So here we have alpha, here
[Speaker A] we have two alphas, and here alpha and beta.
[Rabbi Michael Abraham] It’s simply the basic node. Right? Again we have two parameters and two parameters, so the number of parameters is equivalent. And if so, that will decide a common denominator made of binyan av with binyan av. Right? Now look, I think this is very יפה—very nice—I told you this last time too, because really we should have photographed it. Well, never mind, they’ll understand from the context. Couldn’t one pass there in two domains? No, there’s no issue here; everything is connected. Yes, but here it’s fewer times, no?
[Speaker C] No, connectedness is the question of how many subgraphs you have with no connection between them. Fine? Number of parameters—number of parameters in general, or do we need to track it in terms of the…
[Rabbi Michael Abraham] Okay, sometimes the decision comes from number of parameters. For example in kal va-chomer. In kal va-chomer the decision comes from number of parameters.
[Speaker C] Isn’t there also a decision there from one of these factors?
[Rabbi Michael Abraham] No, no. Kal va-chomer is decided only by number of parameters.
[Speaker C] I’m asking
[Rabbi Michael Abraham] whether there is a place where only number of parameters determines it,
[Speaker C] or whether it will always also be expressed in one of the topological types?
[Rabbi Michael Abraham] I’m convinced there is, yes, but I’d need to reconstruct all the graphs again. We’d need to check a refutation of a kal va-chomer, we’d need to check every such thing. Because for a refutation too, the accounting matters. Because if you don’t take into account the number of parameters, it could be that you have an advantage in connectedness, but in terms of number of parameters there would be an opposite advantage, and that creates a refutation.
[Speaker C] In all the cases we checked, did it always always come together with some other topological factor?
[Rabbi Michael Abraham] Yes. Well, I’m almost convinced there is such a decision too, but right now I don’t remember it off the top of my head. In any case, this is the… this is the picture. Now I’ll tell you what’s nice here. I said this last time too, but now I think it can be understood better. When we saw that the basic algorithm built on number of parameters doesn’t work, we started looking for what does work. And then I told you that we looked a bit in books on graph theory, and flipped through them and asked ourselves what might characterize the complexity of a graph. And we came to the conclusion that it had to be numbers two, three, and four there: connectedness, changes of direction, number of vertices. Udi, you asked me last time whether perhaps one can formulate exactly how to define connectedness or how to define the number of changes of direction—whether along a maximal path or an average number of changes of direction—but on the basic level, changes of direction is the relevant parameter. I’m not committing myself that it can be—meaning that it is defined in only one way—but it is true that these three came up for us in a natural way. We flipped through, looked at what people examine there, and said: wow, it seems to us these three are the relevant ones. Why is that important? Because after we found this, we came back to the common denominator. Besides the fact that it solved the four basic graphs for us—but then we came to the common denominator, and to our amazement it turned out there are three types of common denominator. Each one is decided by one of the parameters. Meaning: a common denominator of kal va-chomer and kal va-chomer is decided by connectedness. A common denominator of kal va-chomer and binyan av is decided by changes of direction. A common denominator of two binyanei av is decided by number of vertices. Now this just came out. We didn’t tailor it ad hoc; rather, we decided that these are the three parameters, we saw that there are only three types of common denominator in the Talmud. No more, there can’t be more. And each of them is decided by one of these parameters. What does that mean? It’s basically—I don’t know if it’s a proof—but it’s a strong confirmation that these three parameters are necessary. Meaning, if any one of them were missing, you wouldn’t be able to explain one of the types of common denominator. In other words, the algorithm must include reference to all three of these parameters. It must. Right? It could be that there are additional parameters that in this context don’t matter, and in other contexts would be added. But these have to be here, that’s clear. Again, why do I say this isn’t a proof? Because I might discover some other parameter that would decide, say, this graph instead of number of vertices, okay? And it would also solve all the other problems for me, and then number of vertices would be superfluous. Therefore I can’t treat this as a proof that these are the three relevant parameters, but it is strong confirmation. Strong confirmation. Let’s say this: for a mathematician it’s not good enough; for a physicist it’s excellent. Okay? There’s no proof here, but there is good confirmation. I had an intuition, I tested it, and it passed the test beautifully. It just fell into place like a puzzle. Okay? So that means we got some confirmation, and it’s very important that the hypothesis was raised before we checked. Not that there was a table and we tried to see okay, how will we decide the table—ah, let’s try number of vertices as a criterion, and that decides the matter for us here. We didn’t proceed that way. Rather, we first defined the three parameters—two, three, and four—and discovered independently that in exactly the three graphs under discussion, the three types of common denominator, each one decides one of the types. So that means all three are required. Okay. I said something else too—there was another interesting adventure we had, and there too it was an interesting confirmation. The confirmations always come when you get stuck. Because at first we thought it was just a matter of solving the number of parameters—maybe valence, alpha-beta, how far do they go? Alpha two, alpha three. Last time we saw a contradiction between graph two and graph three; it didn’t work. We got stuck, so we started looking for topological parameters. That’s what we found here. Now what happened at a certain stage? Later in the sugya—the sugya continues—we reached a graph that we couldn’t decide with this criterion. We said: there has to be something else here. We tried, we checked, and said okay, let’s look for more topological parameters. Let’s look for something else that won’t interfere with all those we already had—that is, won’t ruin them—and will solve this problem for us, which isn’t solved by the present criterion. We didn’t find anything. Meaning, we couldn’t tailor an ad hoc solution to the problem we were stuck with. We sat over it—we were already saying, okay, apparently this whole business should be thrown in the trash, it’s worthless. After a few completely despairing days—we sat here nights—we discovered: it was already a big table, something like seven by ten or whatever. We discovered that in one place in that table we had mistakenly written a zero instead of a one. That’s all. And then after we wrote the one, this criterion solved the problem. Meaning, we got immediately to the correct answer. What was nice about that? You see that ad hoc solutions didn’t manage to rescue us. Meaning, you might have thought these were ad hoc solutions—we understood it would solve the problem, so we decided that was the criterion, and somehow it worked. But no. Here we had a problem, and we tried by hand to tailor an ad hoc solution so it would explain the—, and we couldn’t. Meaning, this technique doesn’t let you do whatever you want just to make it work. Rather, it dictates. To the extent that this thing dictates—errors of this kind are the greatest confirmations. They are the greatest confirmations of the technique, because what it really means is that the technique is not accidental. If you try to solve it ad hoc, to tailor it ad hoc, you won’t succeed. The fact is that it works, and if you deviate in some way you simply get stuck. So that means it’s good; it means this thing works. Therefore many times these impasses are actually excellent confirmation—of course under the optimistic assumption that in the end you really do find an error. And if not, then indeed you can throw the whole business in the trash, because it really doesn’t work. But it was hard to believe we’d throw it in the trash, because this technique sounds so sensible that I think it would have been very strange if we hadn’t found a solution; or to say there is no solution—that’s utterly unreasonable. Okay, now we can start discussing a few points in the meantime. For example, we have some intuition that kal va-chomer is stronger than binyan av. Right? It’s somehow directional, more decisive. Binyan av is an analogy: maybe it’s similar, maybe it isn’t. But kal va-chomer—if so here, then all the more so there. And that seems stronger than a binyan av. The question is whether within this model one can try to assess why. Okay? So indeed there are places—it’s not, I don’t think this has, we haven’t checked it fully—but there are places where you can, for example, try to suggest that if one graph has an advantage over another on two criteria, then it is more dominant, more preferable, than if it has an advantage only on one criterion. Even though according to the criterion so far, any advantage decides; it gives me the optimal solution. But if I want to explain things at a higher resolution—meaning both are good, but which is better? Which is stronger? It may be that simply the number of parameters of the advantage determines which is stronger. For example, the Gemara in Chullin says that against a binyan av one can raise even a very slight refutation, whereas against a kal va-chomer one needs a genuine refutation. What does a very slight refutation mean? A very slight refutation means something that exists in the source-teacher but not in the thing learned, but it’s not entirely clear whether it is actually relevant to the law we are trying to derive. Meaning, I don’t know, say I want to learn chuppah from money by kal va-chomer. And I say: money can be carried in a pocket, but chuppah doesn’t fit into a pocket. Okay? So maybe that’s the parameter. What is a very slight refutation? Maybe, I don’t know. Because as long as I don’t know what causes marriage and betrothal, I can’t know what is relevant and what isn’t. But there’s a feeling that it’s some kind of thing like—why is that important? What does it matter whether it fits in a pocket or not? That’s what is called a very slight refutation. There’s a difference, but you’re not sure, you can’t know, and it doesn’t seem that this difference is relevant. A genuine refutation is when there is a difference in something relevant. “Behold, the children of Israel did not listen to me, so how will Pharaoh listen to me?” So one says: obviously Pharaoh is more stubborn than the people of Israel; it’s not just some distinction. Clearly, logic says he will listen to Moses less than the people of Israel. Okay? So that’s called some kind of criterion of superiority, or a relevant refutation. I maybe—I’ll make one more comment in a moment. But just in this sense, notice here: whether the refutation is relevant or irrelevant, you won’t see that in the model. In the model you put in a refutation, there’s a column: one, zero, one, zero. What does it mean that it’s relevant or irrelevant? The decision whether it is relevant or not is expressed in the question whether to add a column or not to add a column. But once you’ve added a column, there’s no question here of how relevant it is. You decided: it’s relevant. Meaning, there are assumptions underlying the writing of the table itself—not that the table expresses them, but without them the table could not be written. Or in other words, if redeeming second tithe or acquiring a yevamah had seemed to us not to be laws connected to the question of whether kiddushin takes effect or not, then we simply couldn’t add those columns to the table; they just wouldn’t be relevant, they wouldn’t belong to this field. So the table would remain just a two-by-three table, and the two left-hand columns would not appear here. The decision to add them means that it seems to us that they are governed by the same parameters. Meaning, that this law is relevant to our discussion. Okay? Therefore relevance and irrelevance will not be a result of the calculation I do here. It is an assumption of the calculation I do here. If it is relevant, I put it into the same table. Once there is a table, I do the analysis; I have a criterion and I check. But all that is only on the assumption that it is relevant. I could have added lots of other things here. Money also can’t, I don’t know, and intercourse also can’t be used to buy groceries at the supermarket. So I could have written there “going to the supermarket.” Fine? The question is whether it is relevant or not. If it isn’t relevant, then I don’t add it.
[Speaker A] The Sages determined it, the Gemara determined what is relevant.
[Rabbi Michael Abraham] The Gemara determined it. I’m asking: how did it know? What difference does that make? You can determine it too. When the Gemara made a kal va-chomer, it decided this was relevant; when I make a kal va-chomer, I will decide this is relevant. It depends who is doing the business. He is the one who decides what is relevant and what is not. If you disagree, then you won’t accept the other person’s inference, because you think it isn’t relevant and he thinks it is. Therefore I’m saying that this thing is not a result of the model; it is an assumption of the model. And therefore when the Gemara—one second—when the Gemara tells us that against a binyan av one raises even a very slight refutation, and against a kal va-chomer one does not raise a very slight refutation, it is basically telling us: look, a kal va-chomer is a strong inference. To topple it, you need a relevant refutation. Binyan av is somewhat speculative—maybe it’s similar, maybe it’s not similar—you can’t know, and therefore even a very slight refutation. Again, not something totally absurd, but a refutation you are not convinced is really relevant—you can add it to the table. And therefore this is basically an assumption. Now I’m saying: in the table there, what do I do with the Gemara in Chullin that tells me this? Suppose I find a very slight refutation. To the table of kal va-chomer I won’t add it; it won’t refute the kal va-chomer. To the table of binyan av I will add it, and then it will refute my binyan av and not refute the kal va-chomer—but that will come out not because of the calculation. The calculation will express it, but it will come out because I simply wrote the table differently. This is very important, because it basically means there are all kinds of assumptions I haven’t put on the table, assumptions that underlie the writing of the table even before I begin the analysis and checking of the criteria. The very fact that I wrote a table here with all these things means that I assess that money, chuppah, and intercourse—marriage, betrothal, redemption, and yevamah—all play on the same field. Meaning, it is relevant. Once I have assumed that, from that point on I have an algorithm and I’m no longer interested in the contents; it’s all mathematics. Okay? It’s deduction, basically. But it is deduction only once I have already decided what my table is. The decision what my table is—what to put into the table and what not to put into the table—is a decision based on reasoning, a question of what is relevant and what is not relevant. We spoke about the obligation of a lintel in tzitzit, right? If a four-cornered garment, which is exempt from mezuzah, is obligated in tzitzit, then a lintel, which is obligated in mezuzah, all the more so should be obligated in tzitzit. Why does this kal va-chomer sound strange? Because it is obvious to us that the obligation of tzitzit and the obligation of mezuzah are not caused by the same parameters. These are not relevant axes. You cannot write one table with those two columns side by side. Because when I write those two columns in the same table, I am basically assuming that the solutions, the alphas and betas I find there, will explain both the tzitzit and the lintel and the mezuzah and everything. But if I think they won’t be the same alphas and betas—that these are two things that don’t speak to each other—then you’re not allowed to write this table at all. That won’t emerge from the table; rather, that’s the assumption because of which I write the table. This is extremely important, because for a long time after we found this idea and it really explained all these inferences in a very nice way, we said: wait a second, so basically we have here an inference that is completely mathematical. Give me the data and I’ll tell you the result. So why isn’t it deduction? Why isn’t it a necessary inference like deduction? What’s the difference? How can there be refutations here? What is going on? The answer is that it is not deduction because there are hidden assumptions here that we did not take into account. The hidden assumption is the very writing of the table. Writing the table is itself a kind of assumption. It is an assumption that says these columns and rows are parts of one table, that they play on the same field, and that the same set of parameters operating among them will explain everything together. And maybe the connectedness of the graph somewhere shows us that we made a mistake here. Meaning, the parameters that explain this part and the parameters that explain that part do not speak to each other. We should never have written those columns together. Okay? Here that really comes out as a result of the calculation. But many times it is an assumption on which the calculation is built. Yes.
[Speaker F] So basically what we once said about theory and facts? Why? Because even the question of which facts to choose in order to build a theory from them is based on
[Rabbi Michael Abraham] Basically, what we talked about regarding Semmelweis and Carr. Right, so that’s exactly the next point I wanted to make. Yes. Really, if you remember, we talked about the non-trivial relationship between theory and facts, and I said there that Hempel already talks about this. At the Open University there’s also a description of this in the introduction to the natural sciences, philosophy of the natural sciences, philosophy of science. So they say there, period: Semmelweis wanted to check why there was mortality among women giving birth in his ward, mortality higher than in the second ward. Now he didn’t have the faintest clue where to look. So he started making, basically, he made these kinds of tables. He basically said: let’s check the direction the priest walks in both tables and see whether the mortality lines up with that parameter. Is there a similar criterion? Does it face east? What is the floor made of? All kinds of things, how many windows are there, I don’t know, all kinds of things, what do they wear, what clothes do the doctors wear, how many male doctors and how many women? I’m just throwing things out. All kinds of completely bizarre things. Why? Because he didn’t have the slightest idea what actually causes mortality among women giving birth. Puerperal fever was an unexplained disease at that time, and therefore they didn’t know in what direction to look. Like I said regarding the historian Carr, I said it’s similar: when you’re looking for… a theory that explains, a theory that explains, say, victory in battle, then you need to collect facts, but you don’t know which facts to collect as long as you don’t have the theory, because the theory determines which facts are relevant. Say if it was a morale advantage that decided the battle, then you look for facts indicating what the morale was in this camp and in that camp. But if it was superiority in weaponry, then you need to check what the armaments of the two sides were. And if you have no idea what causes such a thing, and it’s not weaponry and not… you know nothing, then what facts will you collect? There are infinitely many facts. You have to assume something about the theory in order to begin collecting the facts. That’s exactly what happens here. I said that what happens here is exactly what happens in science. Same thing. There is absolutely no difference between what we do in Jewish law and what we do in science. This is the logic of scientific thinking exactly like midrashic thinking. It’s the same thing. Why? Because here too, basically, what am I saying? I’m asking which facts are relevant. I collect facts: what redeems consecrated property and second tithe? What effects acquisition in a yevama? Those are facts. I gather facts from the Torah: does money do it, does intercourse do it, does chuppah do it. I collect facts in order to build a theory from them. Now which facts should I collect? I’m looking for relevant facts, but as long as I don’t have a theory, how do I know which fact is relevant? I don’t know. So it turns out, as I said there and as is true here too, although we still don’t have a theory, we have some kind of sense of smell that tells us what not. What cannot be, what is unlikely to be. And we play with the intuitions; sometimes we’ll make mistakes. We play, we collect facts. We managed to explain things very well. Didn’t manage? We switch things around a bit. But there are things we won’t even try at all, because if we had no intuition, there would be infinitely many attempts to make here, and we would never manage to arrive at the right solution. You can’t get through infinitely many wrong attempts. You do elimination; you wave aside a great many attempts that you don’t even think are worthwhile. You’re left with a few that are possible, and among them you play around. The intuition operates in us before we know the theory itself. It helps us filter which facts to take. And that is exactly what happens here. We derive the theory after we’ve analyzed the table. The theory is the alpha, beta, gamma — what is found in each thing. But the construction of the table itself we build on some intuition as to what the theory will be. What can be explained on the basis of those parameters and what cannot. Exactly what happens in philosophy of science. It’s exactly the same thing.
[Speaker A] Can a situation arise where you don’t include the right parameter in the theory?
[Rabbi Michael Abraham] Of course that can happen. Then you made a mistake.
[Speaker A] And then you’re basically searching in the dark.
[Rabbi Michael Abraham] Right. Obviously, there’s no guarantee; that’s why this isn’t straightforward, that’s why this isn’t deduction. That’s exactly the point.
[Speaker A] Just for example, the column of redemption compares something that in principle maybe shouldn’t be there, because here you have marriage and money in a yevama, which is fairly similar to those things. One could have said that this redemption column is a bit unrelated.
[Rabbi Michael Abraham] Right, the Sages thought it was,
[Speaker A] because of the connection of
[Rabbi Michael Abraham] the legal effect — either it’s legal effect, or perhaps it really belongs to holiness. After all, kiddushin is like “holiness has spread over all of it,” there on page 6, there in that discussion, you see that it’s basically like consecrated property. So then what creates consecration can perhaps also redeem consecrated property. I don’t know, you can think of why that is, but the fact is that the intuition of the Sages said, before they had a theory, this seems relevant to us. So they added it to the table. True, not always is that clear, and sometimes we also may not understand even after we’ve seen that the Sages did it. Fine. It could be that the Sages also made a mistake, I don’t know, but right now I’m trying to trace what they did. Okay, now the next stage in the passage, let’s move a little further, the next stage in the passage: we finished the common denominator. So the Talmud says: what is common to them is that their benefit is great. Chet one one, pasik chet. A big H. Benefit. Benefit — in English, benefit. So we’ve basically added a parameter here. Now notice, I’m already giving some hint of what’s coming, but before I give the hint of what’s coming, basically, how is such a refutation built? We can already see — we also saw it in the visual form of the common denominator. I said: how can there be a refutation of a common denominator? The advantage of this one and the advantage of that one is not a refutation; that’s the meaning that there is always a common denominator in such a case. When is there a refutation? When both source cases have an advantage that the target case does not, right? A stringency that the target case does not have. That’s exactly what happens here. You see? There is some feature that exists in both source cases and not in the target case. So of course that is what a refutation of a common denominator will look like, in every common-denominator passage basically. Of course, if I return to the passage, then here it was zero. I just took a time-out to show the other two types. This is the refutation. Now in principle we’re supposed to analyze this thing to see that here already filling in one is no better than filling in zero. We’ll do the calculation, and we’ll discover that filling in one and filling in zero are not preferable in this situation, and that basically means that the a fortiori argument, the common denominator, has fallen. The next step in the passage — I’m not going to do it explicitly on the board, it’s not all that important, I just want to show the principle. The Talmud says, yes: what is common to them is that their benefit is great. I needed one more comment. This column is different from all the other columns, did you notice?
[Speaker A] Which one? The H column.
[Rabbi Michael Abraham] Right, it’s not a law. It’s a fact of reality. It’s not a law, it’s reality. You remember that in the common denominator that made a difference, because a refutation based on a stricter side can work only if here you are raising a legal refutation. In this case, for example, a stricter-side refutation won’t apply, because the refutation here is a factual refutation, not a legal one. Right? So it’s very important to notice that. More than that, I’m actually cheating when I write the table in this way. Because now what I’m going to do is build the table, I’m going to build the — sorry — the graphs, I’ll analyze them, and I’ll discover that indeed zero and one are equivalent. Zero has no advantage over one.
[Speaker C] Would you also expect there to be a special parameter for H?
[Rabbi Michael Abraham] Exactly. Fine, more than that, I don’t need to write H at all. It’s a mistake to write H in the table. H is a constraint on the solution. Let’s say I erase H; I’m basically saying this — that’s really the correct way to do it. The question whether it’s equivalent or not equivalent is a mathematical question that at the moment we don’t have a clear answer to. Can you write H here and analyze it as though it were a law, and will that always give the same answer as the real analysis, which in another moment I’ll show, or not? I don’t know. But it’s clear that the real analysis is not to write the H column here, but to solve the issue under the constraint that there must be one microscopic parameter that appears in A and in B but not in H. That constraint will be different from the previous solution of the common denominator. Meaning, nothing is added to the table. We previously did an analysis of the common denominator; we found an advantage of filling in one over filling in zero. Now I say, okay, now I’m looking for a solution under a constraint — like alphas and betas; by solution I mean theory, yes? Which alphas and betas populate the columns and the rows. And now I’m looking for such a solution where there will be a gamma parameter that exists in A and in B but not in H, whether in filling in zero or in filling in one. That is a constraint on the solution. That is the right way to solve this issue. It turns out that even that way it comes out as a refutation. Meaning, filling in zero and filling in one are the same thing, and that is the correct way to do it. At first we did it as though it were simply one more law, and that also comes out okay. The question is whether these two ways are equivalent, because this is easier to do — that is, to add another column here — easier than finding a solution under a constraint. To find a solution under a constraint, there is no simple way to do it; you have to think it through logically. There’s no algorithm I can think of for doing that.
[Speaker A] Why assume that it can’t be done and that it isn’t technical?
[Rabbi Michael Abraham] The question is whether it’s equivalent. I have no idea; one would have to prove mathematically that it’s equivalent. I don’t know. There may be situations where it would count as a refutation if these are laws, but it would not count as a refutation if it is a constraint on the solutions. For example, one who raises a stricter-side refutation — ostensibly I would expect that according to his view, if you solve this when it’s a law, then he would expect there to be equivalence between filling in one and filling in zero. That’s a refutation. Right? But if you solve it in terms of reality, meaning as a constraint on the solutions, then it won’t be a refutation. Right? According to the view of Rabbi Yehuda, who raises a stricter-side refutation; according to the Sages, no. Okay? So that’s why I don’t know. One has to examine the mathematics of how this business works, whether there is equivalence or there isn’t equivalence. I assume assumptions will come into it there — meaning, under one set of conditions there is equivalence, under another set of conditions there won’t be equivalence. Okay? Whether that equivalence can be proven and is always mathematically true, I don’t know; those are somewhat complicated questions. But in any case, that’s the right way to proceed. H is really — H is really alpha, it’s not H. It is: take the alphas and betas you found in the solution and make sure that you have one of them — alpha, beta, or gamma, it doesn’t matter which — that exists in A and in B but does not exist in H. If you don’t have one, you need to add one. That’s a constraint. Okay? And therefore it can change the result even though the table is the same table, because you are looking for a solution under a constraint. Okay? That’s basically what you’re supposed to do here. And then the Talmud says, after we did the common denominator, the Talmud says: a document will prove it. What does that mean, “a document will prove it”? What do we do with the document?
[Speaker C] We added the document.
[Rabbi Michael Abraham] In principle in the Talmud it’s open-ended, but in principle, pedagogically, it seems to me one should correctly read the Talmud like this: we make a new a fortiori argument from a document. To chuppah. Two by two. That’s it. First of all let’s see — if it works from a document, then no problem. Throw this whole business in the trash and learn it from a document, because this need not bother us. But then we say: what is distinctive about a document? It dissolves marriage with a Jewish woman; here you have divorce. A document also effects divorce. Chuppah does not. Right? So there is a refutation. And once we refuted the document, then we say: the common denominator. Money and intercourse will prove it. Now there is a common denominator between document
[Speaker A] and money and intercourse together.
[Rabbi Michael Abraham] Meaning now we add the document here, and of course acquisition with a Jewish woman, or divorce, is another column, okay? And we say that here there is the H column, here there is another divorce column, and here there is another row. You can already see where this is going — by this stage the table is already four by six. And we make a common denominator from three source cases, one, two, three, to chuppah. But we make a common denominator from three source cases in this way. It is a common denominator of mamon and of money and intercourse on one side, and on the other side there is an a fortiori argument from document. Okay? And those two together make a common denominator for chuppah. That’s the visual form. In the form of the table, nobody cares how you describe it visually and who goes with whom. It doesn’t matter. We have a table, we fill in all the laws, and we calculate what the optimal filling is, one or zero, and it comes out right.
[Speaker A] And then again a refutation.
[Rabbi Michael Abraham] A stricter-side refutation; once again there is a refutation. A refutation that exists in money, in intercourse, and in document, but does not exist in chuppah. That’s a vector of three ones and one zero.
[Speaker C] Is there a reason they didn’t try to make a common denominator between document and money alone, or document—
[Rabbi Michael Abraham] I think not. In any case, according to how we’re describing it here, it changes nothing at all. It just—
[Speaker C] It can’t make a difference. I mean, because the move each time, the move is that it didn’t work formally.
[Rabbi Michael Abraham] They worked with logic. The logic took them down this path, but mathematically it is completely equivalent. It changes nothing.
[Speaker C] Like, maybe I’m identifying a pattern that isn’t exact, but they did try the simplest argument they could at first, and then they assembled it further.
[Rabbi Michael Abraham] Obviously, but I’m saying, but that—
[Speaker C] Why didn’t they give up on the simple arguments?
[Rabbi Michael Abraham] Because they didn’t know how to do it this way. Okay. No, but they do know — after all, they know what an a fortiori argument is, they know what a binyan av is, and they try to build from that.
[Speaker C] No, I mean, but they know that a binyan av from two sources and not from three sources is simpler. Why did they— because they tried it.
[Rabbi Michael Abraham] Ah, you mean why they didn’t do document and one of those two alone. Yes. I thought you meant the larger common denominator with document and money on one side and intercourse on the other. Ah no, there’s no point in that. Then you’d have to check whether there is any common denominator here at all. It may be that there is no common denominator here, because you have to check these columns: do these columns contain a one while all the rest are zeros? Because if both contain a one then that refutes
[Speaker C] the common denominator; there won’t be a common denominator here.
[Rabbi Michael Abraham] So maybe they already identified that, and then maybe — one has to check there what, fine, one has to check what they did there. And maybe it’s just shorthand, I don’t know. Okay. Okay, I think we’ll stop here. What we’ve basically reached is more or less the end of the passage. Maybe next time I’ll do one more thing — the final state, meaning the largest table, we’ll see once how to do it, and we’ll reach the conclusion of the passage, and then we’ll see what this thing means; we’ll talk a bit about the significance of it.