Halakhic Positivism, Lecture 8
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
- [??:??] The connection between constraint and the Talmudic text (NONE)
Summary
General Overview
The text presents a formal way to analyze Talmudic inferences like kal va-chomer, binyan av, refutations, and tzad ha-shaveh by means of success/failure tables and their translation into models with parameters and directed graphs. The demonstration is built on the passage in tractate Kiddushin 5a, on the question of whether a wedding canopy effects acquisition, moving from Rav Huna’s kal va-chomer, through refutations and bringing intercourse as proof, to the common denominator between money and intercourse and the refutation that their benefit is greater. Later, a general method is proposed for determining preference between a “fill with one” and a “fill with zero” according to four indices in the graph, and the philosophical significance of the method is described as an algorithm that mechanizes the move from data to theory, similar to scientific generalization and legal reasoning.
The Purpose of the Lecture and the Methodological Framework
The speaker defines his goal as showing how we derive inferences of kal va-chomer, binyan av, refutations, tzad ha-shaveh, and the like, and how to analyze a passage systematically using tables. He uses a table in which each column represents an action or domain, such as betrothal, marriage, redemption, and the values one and zero indicate success or failure in performing the action corresponding to the column. He presents two main filling options for a question mark in the table, “fill with one” versus “fill with zero,” and translates each table into a parametric model in which strengths like alpha, beta, and gamma explain the pattern of success.
The Wedding Canopy Effects Acquisition by Kal Va-chomer: Money versus Canopy
The speaker presents Rav Huna’s kal va-chomer: “If money, which does not complete, acquires, then a wedding canopy, which does complete, is it not all the more so that it should acquire,” and maps it onto a table of money and canopy against betrothal and marriage. He shows that in the filling where the canopy acquires betrothal, a simpler model emerges with a single parameter, whereas in the filling where the canopy does not acquire betrothal, a model with two parameters is required. He concludes that the one-parameter model is preferable because it is simpler, and formulates a systematic rule using arrow diagrams: when there is an arrow, what is at one end must contain the requirements of the other end, and the arrow marks inclusion.
The Refutation from Money: Sacred Property and Second Tithe Can Be Redeemed with It
The Talmudic text challenges the kal va-chomer from money: “What is unique about money? It is that sacred property and second tithe can be redeemed with it,” and the speaker adds a redemption column to the table and explains how a refutation arises when a dimension is added in which money has an ability that the canopy does not. He demonstrates that in the graph/parametric analysis, two models emerge with the same essential complexity, so there is no clear preference for filling with one over filling with zero. He defines a refutation as meaning that the method fails to decide in favor of one filling, because each side gains an advantage on a different index.
Intercourse Will Prove It: Moving to Binyan Av and the Refutation from a Levirate Widow
The Talmudic text brings “intercourse will prove it,” and the speaker explains that he is temporarily focusing on intercourse as teaching about the canopy and ignoring money, as part of the dynamic of tzad ha-shaveh, where each teaching source is considered separately and then challenged. He defines the move as a binyan av in the sense of analogy, and shows that when one tries to decide between filling with one and filling with zero, one gets stuck because a difference in “resolution” alone, like alpha versus two alpha, is not enough for a decision. The Talmudic text challenges intercourse: “What is unique about intercourse? It is that it acquires in the case of a levirate widow,” and the speaker shows that in this case it is clearly a refutation, because the structures obtained for the two fillings are equivalent in terms of complexity.
The Common Denominator: “Money Will Prove It, and the Argument Returns”
The Talmudic text combines the two teaching sources into a tzad ha-shaveh argument: “This is not like that, and that is not like this… the common denominator between them… so too I will bring the canopy,” and the speaker notes that the wording “it acquires elsewhere and it acquires here” sounds strange, because money acquires in the domain of redemption and intercourse in the domain of a levirate widow. He builds full diagrams for the two fillings and shows that at this stage both sides require a model with two parameters, so there is still no decision based only on the number of parameters. He concludes that additional criteria are needed to determine preference beyond the dimension of the model.
Graph Criteria for Preference: Connectivity, Number of Vertices, Direction Changes, and Dimension
The speaker defines three parameters from graph theory for evaluating complexity: the connectivity of the graph, the number of vertices, and an index of direction changes along a maximal path, and adds as a fourth parameter the dimension of the model, that is, the number of parameters like alpha, beta, and gamma. He argues that surprisingly, each of the three possible kinds of tzad ha-shaveh is decided by a different graph index, according to the composition of the inferences, whether two kal va-chomer arguments, two binyan av arguments, or a combination of kal va-chomer and binyan av. He formulates a rule of decision: if filling with one is preferable or equal on all four indices relative to the other filling, it is decided in its favor, and if one index favors one filling and another index favors the other, a refutation results.
The Mechanization of Inference and the Intuition in Choosing the Table
The speaker argues that once the table and the facts entered into it are chosen, the rest of the process is a closed algorithm that a computer could perform, so the conclusion “necessarily comes out of the data.” He explains that the non-deductive point lies in the very choice of which domains and facts to place together in the same table, because that is a hidden assumption that they belong to the same conceptual framework. He compares this to scientific generalization using the example of a falling book and a bookstand, and describes how enlarging the table with more examples leaves, in the end, the “common denominator” as an explanatory characteristic like mass.
Philosophy of Science: The Context of Discovery and the Context of Justification
The speaker presents Hans Reichenbach’s distinction between the context of discovery and the context of justification, and argues that contrary to the view that discovery is inspiration, the method offers an algorithm that produces discovery of a theory out of data. He clarifies that the algorithm does not identify what alpha actually is in practice, but rather supplies a formal theory of parameters and their strengths, and that the initial decision of which facts to collect and enter into the table relies on prior intuition, similar to the historical examples mentioned in an earlier lecture.
Legal Implication and an Idea for Predictive Application
The speaker proposes in principle that the method could be used to predict decisions, such as a judge’s expected sentence, on the basis of previous patterns, assuming some degree of consistency. He argues that the method can expose parameters of which the judge himself may not be aware, so consistency may still be preserved as long as those parameters are in fact operating. He adds that one could in principle also enter variables like mood or external circumstances, if they can be collected, and the prediction would be better than a shot in the dark, even if not absolute.
The Refutation of the Tzad Ha-shaveh: “For Their Benefit Is Greater” and the Distinction Between Types of Parameters
The Talmudic text concludes with a refutation of the tzad ha-shaveh: “What is unique about their common denominator? Their benefit is greater,” and the speaker interprets this as meaning that money and intercourse involve benefit, whereas the canopy does not. He explains that this is a case of refutation because there is a shared parameter in both teaching sources that does not exist in what is being taught. He adds a methodological reservation, according to which “benefit” is not of the same type as the halakhic columns in the table, such as betrothal, marriage, redemption, and levirate widow, and therefore it is better to represent it as a constraint on the parametric model rather than as a new column, so that the existence of a shared gamma in the two teaching sources is imposed and we then see that the preference between the fillings is canceled in accordance with the refutation.
Full Transcript
[Rabbi Michael Abraham] You could stop with the formal matters, but still I want to sort of complete the picture, to see how this whole thing works. So whoever… fine, can you see the board there? Okay, I’ll just briefly remind you: basically I’m trying to show how we derive inferences of kal va-chomer, binyan av, refutations of them, tzad ha-shaveh, and the like. We saw how kal va-chomer works. Basically the claim is determined according to… I’ll do the review while starting the passage in Kiddushin. I’m just going to follow the passage in Kiddushin, and I’ll try to show how to analyze it, with a few skips so this won’t get too long, but more or less following the passage.
[Speaker C] Where is this in Kiddushin?
[Rabbi Michael Abraham] Kiddushin 5a. There the Talmudic text discusses the question: does a wedding canopy acquire? We know that a woman is acquired in three ways; there’s a discussion there at the beginning: money, document, and intercourse. The canopy is done by… meaning, marriage is effected by intercourse and canopy. And now the question is, what happens if we try to do the betrothal by means of the canopy? In other words, can betrothal take effect through a canopy? So the Talmudic text begins with a kal va-chomer. Rav Huna said: a wedding canopy acquires by kal va-chomer. I’m skipping a few things, but in the end Rav Huna’s kal va-chomer is: “If money, which does not complete, acquires, then a wedding canopy, which does complete, is it not all the more so that it should acquire?”
[Speaker A] So that’s the first datum. Let’s start with that. So basically we have—let’s make a legend here.
[Rabbi Michael Abraham] Yes: money, canopy, document, intercourse. Fine, that’ll be on the side and we’ll remember it. Fine. Now we have like this: “If money…” Let’s put here money and canopy. Okay. Now money acquires… let’s say this is betrothal and this is marriage. So marriage and betrothal—how did I mark this? K. Ah, yes, okay. So money, which does not effect marriage, does effect betrothal; canopy, which does effect marriage, here there’s a question mark whether it effects betrothal. Right? That’s basically the kal va-chomer; that’s exactly the structure…
[Speaker C] Maybe we know that.
[Rabbi Michael Abraham] What? Ah, marriage and betrothal are… ah, betrothal is K, you see that right away. Wait, okay. Fine. So these are actions and this… these are rows and these are columns; that’s also important to remember. Meaning, these are the rows, and at the top and bottom these are the columns.
[Speaker A] Is there enough room in the columns now? What? Better to number them, then it can get bigger…
[Rabbi Michael Abraham] Ah, okay. You’re right, you see it from farther away than I do.
[Speaker A] Wait, let’s try to do it this way to save columns.
[Rabbi Michael Abraham] We said money and canopy, marriage and betrothal, zero one, one question mark. Okay, that’s the initial datum. Now we say like this: what do we need to do? We need to fill this in with either a one or a zero. Now I analyze these two fillings. So regarding kal va-chomer we already did this and saw that basically, suppose this is a fill with one. If it’s a fill with one, then money has a value of alpha; this it can’t do because this requires two alpha. Someone who has alpha can’t do that, but alpha is enough for this, so money does it. Right? Now the canopy has two alpha. How do I know? Because it manages to do this too, and this after all requires two alpha. If it has two alpha, then of course it also manages to do this. So that we solved quite easily. Okay, what happens in the fill with zero? In the fill with zero, the situation is that we have—say this is alpha—then this it can’t do because this requires beta.
[Speaker D] Alpha is enough for this, and this is beta.
[Rabbi Michael Abraham] Right? So it can do this and can’t do this, and these are two columns in the tables. Fine? So those are the two models, and then I said the blue model is simpler because it has only one parameter, while the red has two parameters, so we prefer the blue one, and that’s how we proved that this is one. Fine? Now at the end of last time I tried to show how to do this systematically, because when the refutation comes in and more remote things come in, things you won’t be able to do by ear like this, then what I said was the following. I basically take I and N. I take I and N and ask what the relation between them is. Here I’m speaking about the fill with one, because that’s the blue. The diagram represents the blue. So in the fill with one, I know that this is included in this, since in every cell what’s on the left is greater than or equal to it. Right? So I mark it like this. N is included in I. Now the rule is that I start with I, I call it alpha. You have to start with something. Now the question is: what can I say about N? And I know that it’s two alpha. Why? Because I know that whatever can do this can also do this. Right? That’s really the point. So if whatever does this also does this, that means this has stronger requirements than that. Therefore the rule is always: if there is an arrow—now let’s forget the explanations, now we already know the rule—once there is an arrow, what’s here has to contain what’s there. Okay? What happens in the red? In the red case, the diagram goes like this: I have I and I have N, right? But there’s no arrow between them. There’s no relation between them, neither is contained in the other, right? Therefore basically this is alpha and this is beta, or the reverse, it doesn’t matter. Okay? So that’s how I can take a table systematically and extract a model from it. After I’ve extracted the model, I see here there is one parameter and here there are two parameters. For now I’m ignoring the fact that here there are two degrees of that parameter; for me that doesn’t matter, and in terms of the conclusion it also won’t matter. Okay, now the next stage in the Talmudic text: the Talmudic text brings a refutation. What does it say? “What is unique about money? It is that sacred property and second tithe can be redeemed with it.” So we draw here the logic—and once again I switched to English. Sacred property and second tithe can be redeemed with it, so money has some ability to do something that a canopy cannot do. Right? And now again I need to do exactly the same exercise. How do I know that this is a refutation? So let’s see.
[Speaker A] Sorry, sorry, can I ask a methodological question? What does the one mean?
[Rabbi Michael Abraham] The one means it succeeds. That it succeeds. We entered the data, yes.
[Speaker A] I don’t understand how you can add a column when the one isn’t even talking about marriage at all. Why?
[Rabbi Michael Abraham] It succeeds in redeeming second tithe.
[Speaker A] What does “succeeds” mean?
[Rabbi Michael Abraham] It does it; it succeeds in doing the action described here. The action described here is redemption, the action described here is betrothal, and this is marriage.
[Speaker A] Fine, I’m talking about the one, the meaning of the one. Before you added the P column, the meaning of one was succeeds in effecting marriage.
[Rabbi Michael Abraham] No, succeeds in doing something. Marriage or betrothal. Succeeds in doing something. Here too, what is this one? It succeeds in effecting marriage. And this one succeeds in effecting betrothal. Marriage? So the one here isn’t universal. One means it succeeds in doing the action. The action is defined here: marriage, betrothal, redemption. One means succeeds, zero means does not succeed. And last time I said that the assumption when we put everything in one table is that apparently success in this and success in that belong to the same conceptual framework. And that is of course an assumption we can never be certain about, but I talked about the intuitions that exist even before doing the calculation and so on. Okay. And now we return to exactly the—again I’ll erase these alphas and let’s write them again for the new table this time. So I begin with the fill with one, but now I’m not doing it by ear anymore because by ear it’ll be harder to do. It’s still possible, but harder, and I’ll just show you that here you can do it much more easily. So we have like this: I have I and N, I and N and P. So like this, I and N. I’m talking about the fill with one. So I—if N is included in I, that still remains. Right? N is included in I. But what about P? P too is included in I, but it has no relation to N. Right? You can see that P basically has the same relation to this, and this also has the same relation, but between those two, between these two, there is no relation. So between those two there is no arrow. That’s important. Okay?
[Speaker E] How do we know the arrow goes from P to I and not from I to P?
[Rabbi Michael Abraham] Because P is included in I, it’s contained in I; I is the larger one. The arrow always goes from the smaller to the larger. Now what happens in the red? In the red, basically, you can see that I and P become the same thing. Basically I have P in here, and this stays the same. No arrows. Right? I and P are the same column. Now how do I build the model? So here it’s clear, we already know: this is alpha and this is beta. Right? There’s no relation between them. Both I and P are beta. In other words, what does it mean that they are both beta? It means that what’s needed in order to do this is also needed in order to do this. Whoever has beta will succeed in doing this, and will also succeed in doing this, because it’s the same thing, at least according to the data here. Fine? That’s basically what it means.
[Speaker C] I don’t understand why P is included in I.
[Rabbi Michael Abraham] I understand if N and I are the same thing. If you put in the red filling, here it’s one zero and here it’s one zero, it’s the same thing, it’s the same column. When there’s an arrow, one is contained in the other. When there’s no arrow, there’s no relation between the two. When they’re identical, it’s simply the same point.
[Speaker A] Yes, but the arrow there is one-directional. P and I? P and I aren’t one-directional, they’re the same thing. P is included…
[Rabbi Michael Abraham] No, but below, below. In the red. No, no, below below.
[Speaker A] P to I isn’t one-directional, they’re the same thing. P goes into…
[Rabbi Michael Abraham] No, in the blue that’s the fill with one. The red symbolizes the fill with zero. The blue is the diagram for fill with one. In the fill with one, here it’s one zero and here it’s one zero, so it’s included in it. And this also is included in it. But between these two there is no relation, which is exactly… ah, in the blue P— I is stronger than P. Exactly. That’s exactly the difference. Right? Now what does that mean? Let’s fill it in. So here we already did alpha and beta, we know that. How do you do it? So again I start—maybe let’s do it so it’ll match—let’s call this alpha and this beta.
[Speaker A] Huh? Alpha and beta, here alpha and here beta—you did that in blue. Ah, that’s beta.
[Rabbi Michael Abraham] It doesn’t matter to me right now, I just want to do between these two. The previous part isn’t important to me. So here I also call this alpha. What do you do here? P is two alpha and this is alpha and also beta. The reverse. It has to be.
[Speaker A] In a structure like this it’s always the reverse. Alpha and beta are in I, and this is alpha and this is beta. Because it’s contained.
[Rabbi Michael Abraham] It’s contained. Both this is contained in that and this is contained in that. There’s no difference. I could have put two alpha here and beta there. There’s no difference. I could also have said alpha and beta here and two alpha there.
[Speaker A] It doesn’t matter.
[Rabbi Michael Abraham] No, I’m only doing it this way so beta will always appear in relation to N, because I want you to see the difference between the diagrams. Do you understand why it’s like this? It has to be contained in that. So between this and that it’s alpha and two alpha just like before. But here it has to be something that on the one hand swallows this, contains it, but has no relation to that. Right? So I have to make alpha here that is smaller than two alpha. It can’t be two alpha and it can’t be three alpha. If this were two alpha, what would happen? Then there would be an arrow like this. Right? But there isn’t. So therefore here it has to be alpha and beta, and here it’s two alpha. A structure like this always gets filled in this way. Now of course you can switch this with that, it doesn’t matter, but what do I always see here? That in both cases we have a model with two parameters, alpha and beta, and here too alpha and beta, so what is a refutation? It means there is no preference for fill with one over fill with zero, and vice versa. Right? Again we remember that although here we rise to two alpha, there is something slightly more complex here from the standpoint of the calculation. Yes, but as I said, we do not relate to the level of detail inside each parameter. Later I can prove that, but right now I’m just saying it. Okay. Now we move on. And the Talmudic text says: “Intercourse will prove it.” “Intercourse will prove it”—I’ll write it here already on this table so we won’t have to do it again. Okay. What does “intercourse will prove it” mean? Intercourse effects marriage, effects betrothal, and of course you can’t redeem second tithe with it. They tried once, but you can’t. Okay. So what does that mean? Now we have a new table. Three by three. But as I said last time, this is on the way to the tzad ha-shaveh. For now we’re still… “Intercourse will prove it.” Meaning we start from intercourse. After that they say intercourse doesn’t work because there’s a refutation against it, then we go back and do both together. But right now we’re talking only about intercourse. Before, I spoke only about this, I made a refutation, and now I’m speaking only about this. You see that I’m building the table but for now I’m analyzing only one part.
[Speaker A] Is that called binyan av?
[Rabbi Michael Abraham] What? This is binyan av. Three sides—is that binyan av? Yes, because binyan av is basically analogy. It’s the same thing. It’s not kal va-chomer, it’s stronger. Binyan av is the same thing. Okay, so once again back to our matter. So what—let’s do the calculation for a moment. So now we have the blue diagram. So like this: in the blue, of course I is here, P enters it, right? P is one and here all three are one. And N enters it too. Right? But between P and N there is still independence, because P is one here and N is two ones here.
[Speaker A] Nothing changed from before.
[Rabbi Michael Abraham] What? Nothing changed. Yes, let’s see, that’s the blue. Okay. Now in… ah, wait, what, I’m doing it on—
[Speaker A] The big table?
[Rabbi Michael Abraham] No, it’s the lower table. I’m doing only the… only the… only the I and N. The two of them here. So I’m saying like this: in fill with one, basically I and N are the same thing. Yes. Right? So that’s—
[Speaker A] N.
[Rabbi Michael Abraham] I and N are the same thing, they’re both here. That’s the fill with one. In the fill with zero, then we have like a kal va-chomer, right? In the fill with zero basically we have I, N like this, the picture—the arrow—
[Speaker A] Flips. Okay? How are you doing a kal va-chomer like this in the lower table?
[Rabbi Michael Abraham] No, I’m not doing a kal va-chomer. A kal va-chomer could work like this if I were doing that, but it’s not a kal va-chomer. That’s why I’m going like this; that’s how this binyan av works. This is just the binyan av that doesn’t work, meaning under the assumption that the filling is zero.
[Speaker A] But in the blue, how are you ignoring N? How?
[Rabbi Michael Abraham] No, because I’m doing only a binyan av. “Intercourse will prove it” regarding canopy. I’m forgetting about N. I wrote it here, but later I’ll come back to it.
[Speaker A] But the issue of N was already mentioned.
[Rabbi Michael Abraham] It was mentioned, so what? I want to prove canopy. In tzad ha-shaveh it always works this way. I start learning from this, then I leave it. It was mentioned, but I left it. I learn from this; each one gets a refutation. You remember, we wrote this in one of the previous lectures, and then I go to both together. That’s why I’m doing it in the same table, because it’s for later. Really I should have made a little two-by-two table here. Okay? So what happens here? These are the two tables. What is the decoding? Here we know this is alpha in two alpha, exactly like—sorry, the reverse, exactly like… no… and the one above is alpha. Right? And why is blue preferable? Oh, now here apparently we’re stuck, right? Because in both cases we have only one parameter, and the whole difference is only a difference in resolution, meaning that there are several levels of alpha. So it would be very tempting to say that blue is preferable because where everything else is equal, where everything else is equal, then it does play a role. Only if it’s not equal, then it doesn’t matter. Again, exactly—before I didn’t do it because here too, you see, everything was equal, so I left it written that way. Everything was equal and still there was a difference at the level that here you have alpha and two alpha and here you have only one alpha. And nevertheless I said that it’s equivalent. Right? So that means—I’m just doing the calculation, trying to build the model out of the data. And the data say that basically this doesn’t play a role. Which means that here I’m stuck. I’m stuck because my model won’t explain to me why binyan av works. Why?
[Speaker A] But maybe it does play a role that there isn’t an additional parameter, that there’s only one parameter?
[Rabbi Michael Abraham] Yes, but then if there is—
[Speaker A] Also beta then yes.
[Rabbi Michael Abraham] But then it becomes terribly ad hoc. Why? What difference does it make? After all, here in both cases there is beta, only here alpha is alpha and two alpha and here it’s only one alpha. So what I said here, that there is no beta at all, still the relation between them is the same.
[Speaker A] Yes, the beta—you don’t know what its strength is in each one of them.
[Rabbi Michael Abraham] Fine, but it exists in both. No, look, its strength is the same, that’s what I’m saying; there isn’t two-beta here, it’s only one beta. Therefore from the logic of it, it doesn’t seem that… this distinction is relevant. Later you’ll see that I can prove that, that it isn’t relevant. And therefore for now I’m stuck; I’ll come back to this in a moment, there is an explanation, but I’m building it in stages. So at this stage I’m basically saying I’m stuck, because from there it’s proven that the values, the number of values of alpha, don’t play a role. But here, if I adopt that they don’t play a role, then it’s not clear to me why the blue.
[Speaker A] So in any case, binyan av is logical, not something added to the models. I mean, I’m blond and I wear glasses. He’s blond, so does that mean he wears glasses? What kind of logic is that?
[Rabbi Michael Abraham] Again, and let’s look here… what’s the connection? Why? What kind of logic? If you didn’t succeed in chemistry but you did succeed in literature, and the other one succeeded in chemistry, so did he succeed more in literature? What does literature have to do with chemistry?
[Speaker A] Fine, you said that when there is here—
[Rabbi Michael Abraham] Here too I’ll say the same thing: there is some assumption that it belongs to the same field. That’s exactly the point. The moment I put it in the same table, I implicitly assumed that it really isn’t blond and glasses, but that there is a connection between the things. Otherwise it really isn’t relevant. Okay, so now once we understand that, let’s move on. I’ll come back to this point so you can later see how it works in the expanded model. What does the Talmudic text do now? “What is unique about intercourse? It acquires in the case of a levirate widow.” Okay? We have another English word here, levirate widow. Okay. So now I move over here, and here I already showed the comparison to the right-hand side; you saw that there’s some contradiction with the number of values of alpha. What does that mean? So for now notice that I’m referring only to… by the way, money does not acquire in the case of a levirate widow, so I can already fill that in here; that’s a fact. Fine? But that’s not important to me for now because I’m only—I’m writing it in one table, but think as if the table is really these four and these two. That is basically binyan av with a refutation next to it. I’m writing it here so I won’t have to write it again when I use all of them. Okay? What happens now? So in the binyan av model, again, I’m looking—notice—at a two-by-three table, these two and a two-by-two. Fine? Think of it that way. Now, if the filling is one, then I have here I into which Y and N enter.
[Speaker D] Wait, this is one and one and this is one.
[Rabbi Michael Abraham] Sorry. Y and N together. Okay, that’s in the fill with one. And in the fill with zero, fill with zero, then I have I and Y together and that enters into N. Right? Now here you can see very clearly, you don’t even need to make the model, it’s obvious that it’s the same thing. Therefore this is a refutation. And this does fit, right? Alpha and two alpha, and here too—
[Speaker D] You have beta and two beta.
[Rabbi Michael Abraham] Yes, it doesn’t matter, but it’s the same… but it’s the same complexity. Fine? Therefore here it’s clear that this is a refutation and everything is fine. Now the Talmudic text says, all right, so we tried the kal va-chomer, we refuted it. This I filled in from my own knowledge. I know that intercourse doesn’t acquire in the field of redemption, so I already filled that in, although in fact the Talmudic text didn’t use that. It wasn’t needed yet, but that’s the fact, it’s true. And by the way it’s always like that. It’s always like that. In the entire Talmud you won’t find a counterexample. A refutation always branches out from the table to a tzad ha-shaveh; here it will be zero. Always.
[Speaker A] Otherwise you won’t find a tzad ha-shaveh.
[Rabbi Michael Abraham] And also above and to the left. Here we made a binyan av, a refutation on it, and here I filled in my own information, yes, that money does not acquire in the case of a levirate widow. And that’s it. What happens now? So now the Talmudic text doesn’t add another column, right? Because it basically says: “Money will prove it, and the argument returns. This is not like that, and that is not like this; the common denominator between them”—in intercourse and money, yes—“is that they acquire elsewhere and they acquire here.” “Acquire here” means one, yes. “So too I will bring the canopy, which acquires elsewhere and acquires here.”
[Speaker E] Meaning, there’s another column, “acquires elsewhere”?
[Rabbi Michael Abraham] No, “acquires elsewhere” means that elsewhere, in other places, it acquires; here, it acquires; in both cases it acquires. Now on the face of it, as I said in the lecture where we discussed tzad ha-shaveh, that’s very strange. Because this acquires in redemption and this acquires in the case of a levirate widow—who says that’s the same thing? If you remember the alpha and beta I did there in the tzad ha-shaveh, do you see this diagram? We take two teaching sources, yes, which teach about this. This one has a special property and this one has a special property. Now here we’re seeing it from another angle, but it’s basically the same thing. Okay? Basically the question is how this works. So look how it works. I want to draw the tables for a moment—the diagram, sorry. So let’s start with the blue. It’s already starting to get more complicated. You can see that here it’s already harder to arrive at the model without doing it systematically.
[Speaker A] The number one next to the question mark will also be blue? What?
[Rabbi Michael Abraham] The number one next to the question mark—shouldn’t it be written in blue? I wrote it in blue.
[Speaker A] Ah, it looks a little black.
[Rabbi Michael Abraham] So now I’ll start with fill with one, okay? So in fill with one I have I, it’s the biggest, yes, it’s all one. L enters into it, you see? L enters into it. But note, okay, let’s start for a second: L enters into it. Now Y also enters into it, but it enters through N. Meaning it’s true that in the end it reaches this, and that’s an important point: don’t draw a direct line from here to here because it doesn’t matter; this enters into that and that enters into this, so it’s transitive. But P is not the same. P enters from the side. P does not enter via N. Ah, right. P enters from the side, that’s the diagram. Okay? Now what is the diagram of fill with zero? So like this: I have I. P and Y enter; P and Y enter from two sides, right? Y enters from here, P enters from here, and N—Y enters into N. Y enters into N and that’s it. Wait, Y enters into N but N doesn’t enter anywhere. So Y enters into N, so it’s something like this, right?
[Speaker A] So that’s I and that’s N, yes. Fine?
[Rabbi Michael Abraham] Now let’s fill it in. So we have like this: this is alpha, this is two alpha, three alpha. This is alpha and beta too. Okay? You see? This can’t go into that and this can’t go into that, so I’m set, right? Now what happens here? Again, this is alpha, let’s say this is two alpha. This is alpha and beta too, right?
[Speaker A] That’s—
[Rabbi Michael Abraham] That’s alpha and beta too, and here, wait, the reverse.
[Speaker A] You need to write from what alpha plus beta.
[Rabbi Michael Abraham] Yes, like this. Right, you’re right. No, that’s not enough—you’re not right.
[Speaker A] You need to write alpha plus beta at the top, beta on the right side, and two alpha below.
[Rabbi Michael Abraham] This is alpha and beta too—
[Speaker A] And two alpha below.
[Rabbi Michael Abraham] This is two alpha, but then we have to see what happens with N. Beta. Beta. So wait, ah, okay, fine. Okay? Now when I look at this thing, again there is a difference at the level of resolution. Here it gets up to three alpha, but in both cases it’s a two-parameter model. Now again the question is—we’re stuck, right? Once again we’re stuck, because this was supposed to work, the tzad ha-shaveh. So what does that mean? It basically means, including the problem we were left with earlier, that we probably need to introduce additional criteria into the question of preference. Meaning: to determine which filling is preferable, it’s not enough just to determine the parameters in each model. Now when we thought about it, we said that the logic suggests doing it according to the complexity of the graph. The more complex the graph is, the worse it is. What is the complexity of a graph? So we went to graph theory, books on graph theory, and in graph theory we found three parameters that seemed relevant to us—parameters that are actually used, meaning we didn’t invent them. Three parameters that are used. I’ll write them on the side because something very surprising and interesting came out here. One parameter is connectivity. Connectivity means the graph is all connected to everything. For example, if there were no arrow here at all, then it wouldn’t be connected. There would be one unit here and another unit here with no connection between them; they don’t talk to one another. Here both graphs are completely connected.
[Speaker A] But what’s more connected is more—
[Rabbi Michael Abraham] Is it more complex or simpler? Connected is simpler. Right. The number of vertices in the graph—if it’s fewer, that’s preferable, because that’s simpler. And the third thing is direction changes, the directionality of the graph, right?
[Speaker E] That’s what overlaps here, yes.
[Rabbi Michael Abraham] Direction changes. Direction changes basically mean that, say, in this case direction changes are necessary. Why? I go here, you see, I flipped direction, meaning the arrow flips—the head meets a head or a tail. Right? And here too it flips, so that’s two direction changes, right? And in this one there is one direction change at most. The largest path you can find on the graph, with the greatest number of direction changes—that’s the index of direction changes of the graph. Now here it’s one and here it’s two, so that means this one is preferable. But an important point to note: we did this before we saw that. Meaning, we looked for what might characterize the graphs and said these three seemed right. Look, this is something very surprising. I said there are three types of tzad ha-shaveh, right? Where both of these are one and both of these are zero, and where this is one and this is one. Each one is decided by a different one. One is decided by connectivity, one by the number of vertices, one by direction changes. And I think that very strongly confirms that these really are the three relevant criteria for determining preference—which, again, I say we determined in advance.
[Speaker A] Isn’t there some explanation of this in the language of the Talmudic text?
[Rabbi Michael Abraham] No, the Talmudic text doesn’t think in these terms; this whole framework is from a different dimension. No, I didn’t understand, again.
[Speaker A] I understand the first two.
[Rabbi Michael Abraham] Why the third? Direction changes mean there is no transitivity, the relation between the things is not simple. If this is bigger than that and that is bigger than that, then this won’t be bigger than that. Why? It’s not logical; something here is more complex, there is no simple relation between the components of the graph. All of these, by the way, are parameters that appear in topology and graph theory; it has nothing to do with us. We just took them because these seemed to us the three relevant parameters for determining the complexity of a graph. Okay, so let me explain them again.
[Speaker C] No, that’s not his problem; the problem is how you move from complexity to the three… I didn’t understand. What again? The interesting point, I didn’t understand.
[Rabbi Michael Abraham] So I’m saying: I determine, I check—I want to compare the graph of fill with one to the graph of fill with zero. I compare them in terms of connectivity, in terms of the number of vertices, in terms of direction changes. So here connectivity is full; this is fully connected and this is fully connected too, right? The number of vertices is also four here and four here. Right? Remember earlier there were vertices that merged; A and N were together, so that reduced the number of vertices—we’ll come back to that. Okay? Now what remains is direction changes, which is exactly what happens here. Look, the maximal path with direction changes is from here to here. How do I do that? I go like this—boom—here I meet head to tail, so that’s one direction change, right? It’s like the same direction, so there’s no direction change, but here there is, right? So here there is one direction change. Continue here, here too there is a direction change—it’s tail מול tail, so that’s two direction changes.
[Speaker E] I could have said exactly the opposite, that there—
[Speaker A] There’s a greater path length there, so it’s significantly more complex. Length of the path… the path length…
[Rabbi Michael Abraham] The longer the path length, the simpler it is, because that means there is a simple relation among all the components.
[Speaker A] This connects to that and this connects—
[Rabbi Michael Abraham] To that, and everything is simple.
[Speaker C] Now you said something interesting. What’s interesting?
[Rabbi Michael Abraham] That now—
[Speaker E] You’re telling me there are several levels: three alpha, two alpha, alpha. We would have thought it was simpler—
[Rabbi Michael Abraham] Beyond the greatness of having three such things that didn’t have that kind of relation between them, that’s more—
[Speaker A] Complicated in Talmudic decision-making.
[Rabbi Michael Abraham] If there were really three at all, you’d be right, but that’s the number of vertices. But if there are already three, then if there’s a simple relation among them, that’s better. If transitivity holds, it’s simpler, more intuitive. When there’s no transitivity, we always get stuck. Okay, now what’s interesting? What I said before. Look, we decided that by changes of direction. So next to changes of direction I write down: this is called an a fortiori inference.
[Speaker A] And all the more so—a diagonal a fortiori inference plus an analogy from a common source.
[Rabbi Michael Abraham] Meaning, when I have a common side where one side of it is an a fortiori inference and the other side is an analogy from a common source, that gets decided by changes of direction, right? We remember that. Now let’s look at the second example: here we have two a fortiori inferences. I changed this from zero to one. Also because there is—I’m thinking now in the topic, I’m thinking how you solve the problem when you have a common side where both sides are a fortiori inferences, not when one is an a fortiori inference and one is an analogy from a common source. Let’s see what comes out.
[Speaker F] Now let’s relate to the data.
[Rabbi Michael Abraham] There are three. Either one-one, or zero-zero, or one-zero.
[Speaker C] Why are you saying that? Why are you saying that?
[Rabbi Michael Abraham] There aren’t any more, there aren’t any more common-side cases. I explained this in one of the earlier classes, because a common side always starts either with an a fortiori inference and then a refutation, and then they bring another a fortiori inference, and that too gets a refutation, and then the law returns. Second possibility: you start with an a fortiori inference, a refutation, and then bring an analogy from a common source, which is what we had in our passage. Third possibility: an analogy from a common source, a refutation, a second analogy from a common source, a refutation, and then the law returns. That’s it, there’s no more. What else could there be? Either analogy from a common source plus analogy from a common source, or a fortiori inference plus analogy from a common source, or analogy from a common source plus a fortiori inference, or a fortiori inference plus a fortiori inference. That’s all, there can’t be more.
[Speaker C] Each one of them—you said analogy from a common source and a fortiori inference and analogy from a common source.
[Rabbi Michael Abraham] No, no. I said either an a fortiori inference, or two analogies from a common source, or two a fortiori inferences. What can make up a common side? Two simple inferences that together make a common side. So I’m saying, let’s now see what happens in a situation like this, where here it says one. Okay, so I start with filling in one, and I say this: I have M, I, and N together. Right? I and N together. Fine, you can already see what’s going to happen here. Here it’ll be the number of vertices, because I and N already took one vertex from us, right? So I and N together—here, X, Y, and P go into M, right? We already know the solution to that: that’s two alpha, that’s alpha and beta. Right? The schemes more or less repeat themselves. What’s the ending with filling in zero? I have M, and into it goes I, right?
[Speaker A] And P goes into M, and Y goes into M.
[Rabbi Michael Abraham] Right, this is Y and this is P. Agreed? Look what happens here, because Y is one, it goes into this, by way of this, into this of course, and this one goes in from the other side. There’s no connection between them. Right? That’s exactly what I drew here—there’s no connection between them. Now let’s do the calculation. So that’s alpha, two alpha, three alpha, two alpha, and also beta. You can already see the pattern: when there’s something here, then this will go up by one and beta will be added, or zero, it doesn’t matter; here beta gets added and here it drops, it doesn’t matter. Okay, now let’s do the calculation—what are the priorities here? In terms of number of times, they’re equivalent, because both are alpha and beta. Right? In terms of connectedness, same thing, everything is connected. What’s left? The number of vertices. Here there are only three, here there are four. What do we say about changes of direction? Directions. Changes of direction are the same. Here there is one change of direction and here there is one change of direction, no more. The maximum number of changes of direction is the same. So that means that the number of vertices decides it—it decides analogy from a common source plus analogy from a common source.
[Speaker A] Rabbi? Yes, two very short questions. This whole consideration you’re making—is it after you failed to decide only by the number of parameters? Yes. Second thing, no—
[Rabbi Michael Abraham] No, no, it all gets combined together. I combine everything together at the end. It’s not one by one. Taken together, that’s the stronger side.
[Speaker A] Is there no situation where the order there gets reversed between the parameters? Is it always only that one is better and the rest are equal? Couldn’t it be that—
[Rabbi Michael Abraham] No, that—we can, I’m not going into all the details here, we can prove that you only have to raise one parameter, to increase its quantity.
[Speaker A] No, no, I wanted to talk about between connectedness, number of vertices, and changes of direction—couldn’t there be a case where the connectedness—
[Rabbi Michael Abraham] Yes, there could be such a case, I’ll talk about that too—
[Speaker A] Where the order gets reversed and then there’s no decision either.
[Rabbi Michael Abraham] Right, very nice. How did you get that one is an a fortiori inference—
[Speaker A] And number-wise analogy from a common source and analogy from a common source?
[Rabbi Michael Abraham] Yes, so this is the table of analogy from a common source and analogy from a common source. We decide it with the criterion of the number of vertices. The a fortiori inference and analogy from a common source we decided by changes of direction. What’s left for us now? Zero-zero, right? That’s the third possibility. Let’s go back to that: that’s a fortiori inference and a fortiori inference. Yes, a fortiori inference and a fortiori inference.
[Speaker A] Y inside I, P inside I—but let’s see how in filling in one.
[Rabbi Michael Abraham] In filling in one we have I, right, everything is inside I. Right, Y goes into I and there’s no connection between them.
[Speaker F] Y, P, and X. Maybe that means that the assumption of taking these four means there’s no connection between them at all?
[Rabbi Michael Abraham] So I said, that’s what we assume when we made the table—that’s a judgment call. About that—this isn’t part of the model. From the outset you assume there’s some relation among them, because otherwise you wouldn’t put them in the same table. That’s exactly the non-deductive part of this model. It’s N, not I.
[Speaker C] Right?
[Speaker A] What? Ah yes, sorry.
[Rabbi Michael Abraham] Okay, what happens—
[Speaker A] In the filling where P and Y are inside I?
[Rabbi Michael Abraham] P and Y go into I, and N in T is connected.
[Speaker A] Right. There—there’s no connectedness. That’s the change, yes.
[Rabbi Michael Abraham] Right? That’s the diagram. Okay? Now, you can already see, it can be the same thing; the only difference is in connectedness. Now, this is pretty amazing.
[Speaker A] Why are there two changes of direction at the top? No, only one. You don’t have vertices in the middle.
[Rabbi Michael Abraham] You don’t have a path along a path. You can’t get from here to here, only straight, not through this. The only path you have is like this. So there’s only one change of direction. Draw me a path with two changes of direction.
[Speaker A] No, but from Y to N there’s a change of direction, and from Y to P there’s—
[Rabbi Michael Abraham] No, no, no. A path that has the greatest number of changes of direction along it—that’s the measure you have here. Because counting changes of direction one by one doesn’t tell you how complicated it is. So now what happens here again is the same thing once more: if you want, here you have alpha, two alpha, alpha and beta, and here too. Right?
[Speaker A] And here there’s alpha, two—
[Rabbi Michael Abraham] Alpha, alpha and beta, alpha and gamma.
[Speaker A] Right? Alpha plus gamma?
[Rabbi Michael Abraham] Yes. Okay. And then that basically means we have—what happens here now? Notice. The number of vertices is the same, four and four. Changes of direction are the same, one and one. The number of parameters in the model—three: alpha, beta, gamma. Same thing. The only thing that decides it is the connectedness. Right? And therefore this is where a fortiori inference plus a fortiori inference comes in. Now that’s very interesting, because it means we only have three—we only have three common-side forms in the whole Talmud, there can’t be more. Okay? Each one of them requires a parameter. I showed that only this parameter decides it. That’s a proof that these three parameters are relevant in determining priority. And now I say this: now I make the generalization, and I say, my criterion for determining priority—and it will explain all the things we ran into before as well, refutation, a fortiori inference, and analogy from a common source—the criterion is this: we take four indexes. One index is connectedness, a second index is the number of—
[Speaker A] Vertices, a third index is changes of direction, a fourth index—
[Rabbi Michael Abraham] The dimension of the model, how many parameters the model needs. Okay. Now it works like this.
[Speaker D] If filling in one is preferable in terms of all the indexes, then that’s prioritization, meaning, then it decides it. Why is the number of parameters one of the indexes, or you don’t assign priorities within the indexes? No. Okay.
[Rabbi Michael Abraham] Now I’m saying: if filling in one is preferable in terms of all the indexes, then that’s prioritization, meaning, then it decides it. If there is some index in favor of filling in one and an index in favor of filling in zero, and I don’t care how much—because the logic, notice, the logic of a refutation is a logic that says it’s enough that there’s one side on which the other is stronger; it could be, right? Or you can’t know, because you don’t know which kind of superiority is the determining one. Right? So the criterion is very simple: I have four indexes. If there is a clear priority in terms of all the indexes—greater than or equal in favor of one of the fillings—that is the correct filling.
[Speaker C] The parameters are also an index? Yes, that’s the dimension. That’s the dimension, but it’s not what decides it. Why? No, it’s one of the four.
[Rabbi Michael Abraham] Now I’m saying, here now we have—
[Speaker C] Dimension, alpha and beta and gamma, another dimension.
[Rabbi Michael Abraham] Now I’m saying: these four, I need to check in every pair of tables. From filling in one and filling in zero, I check connectedness, number of vertices, changes of direction, dimension. Now I check what came out with filling in one, what came out with filling in zero. If there is priority for one of the fillings in all the parameters, then it is—
[Speaker A] The decisive one, then it’s decisive prioritization.
[Rabbi Michael Abraham] If each of them has priority in at least one index, and I don’t care how many indexes in each direction—if the priority isn’t univocal across all the parameters, it could be that in three parameters it’s the same and in one it’s better. Greater than or equal—that counts as preferable for me. Okay? By contrast, if there is one parameter with respect to which filling in one is preferable, and another parameter with respect to which filling in zero is preferable, that’s a refutation. Now once you assume this, it explains everything. Now everything exists in the Talmud, everything, everything we checked, all the arguments of every type and kind in all the tables—it all works, always. When you check it, it always works. And it also works in places where I told you that before I gave the lecture on this, two days earlier my son brought something from the yeshiva, some Tosafot that they couldn’t work out there in the yeshiva; everyone was racking their brains, that Tosafot made no sense—there’s an a fortiori inference, there’s a refutation, there’s this—it didn’t work out for them. I made a three-by-three table; it wasn’t even complicated. A three-by-three immediately gives you the result. Not only does it immediately give you the result, you can even understand a little why it’s more reasonable, because you look at the decoding, at the parameters. Here there’s some index that isn’t there—you can see why it makes more sense. But the question is how the Amoraim and so on—
[Speaker A] After all, they didn’t study modern graph theory.
[Rabbi Michael Abraham] No. It’s known: the Amoraim formulated the logic of the simple methods of inference—an a fortiori inference, an analogy from a common source, a refutation of each of them, and combinations. And by combinations I mean an a fortiori inference and an analogy from a common source together making a common side, or two analogies from a common source together making a common side, a refutation of the common side. They always build it out of the basic building blocks, and then you can track it more easily. The problem is that then you’re very limited, because if there’s a table that you can’t build in the form of an a fortiori inference and then an analogy from a common source—after all, you can make here whatever table you want. You can make here a table: zero, zero, zero, one, zero—whatever you want, literally whatever table you want—and I’ll tell you what the result will be. I don’t have to do the accounting: wait, this is an a fortiori inference and it has a refutation, then comes an analogy from a common source, that has a refutation, and a common side is made, and there’s a refutation to the common side—that requires you to keep the logic of the inference in your head. And that’s what the Sages did. But they did it in the simple cases, the relatively simple ones—not all that simple; I said the passage in tractate Kiddushin is really not simple. But this tool tells you: take any table of any size you want, of any kind; you don’t need to build it out of methods of inference, and everything comes out the same way. You don’t need to think and strain your mind over anything. Do your calculation, build the table, you know exactly what the result will be, you can give it to a computer. And that’s exactly what I told you bothered me: if you can give it to a computer, then why isn’t this deduction? So it’s basically deduction: you have the data, the conclusion necessarily follows from the data, so why isn’t it deduction? The answer is because we decided to put everything into the table. Yes, it’s that linkage, that’s the deduction—that’s the hidden assumption people don’t notice. And I brought the case of the doorpost and the fringes and all those things where it’s clear to us that it isn’t correct. It’s just that that’s where it enters. Meaning from there onward it’s really a closed algorithm that a computer can execute. Now I remind you that I showed last time that an inference or a scientific generalization is like this too—it’s just a common side. Right? What is a scientific generalization? I said: I put down a book and such-and-such, I let go of the book and it falls. What does the book have? It’s made of paper. And here, I let go of this and it also falls. What does this have? It’s made of wood. I said yes, but the returning law will prove it—book and lectern—what? The common side between them is that they have mass and they fall; so too anything that has mass falls. I look at what they share. Now it could be that I need more, because maybe they also share something else—they’re both inanimate. So I’ll take something else, I’ll take a pencil. Fine, but after I do this enough times—and therefore I enlarge the table, I take another pencil and another thing—but in the end, after I take enough data, what remains for me is the common side: having mass. That’s basically this alpha, which says as if they all have alpha or something like that, roughly. It needs to be more complicated, but roughly. Do you understand? So in fact legal inference, scientific inference, inference in our everyday life—they all work with this logic, and this is basically the logic of non-deductive inference. And it’s a fully ordered logic; you can mechanize it. In fact, I once looked into this with a friend of mine, but you still have to think through a few interesting points there. Basically, what are we doing here? We’re trying to identify the parameters that govern the data, the problem—to find the theory. I talked about the fact that what we’re really doing here is: I have data and I don’t know what it is in levirate marriage, in redemption, in money, in canopy-marriage, that succeeds or doesn’t succeed in doing this. This table basically tells me, helps me build the theory that explains the data. If you remember with gravitation: something falls, so I write one; it doesn’t fall, I write zero; and if I do the analysis, I’ll discover that there’s a parameter M that appears in all of them, and that’s really mass. They all have mass, right? This is basically the way to find the theory underlying the data. Except that this theory is a formal theory, because I don’t identify who alpha is, who beta is, who gamma is. I say there is some alpha, some beta, some gamma; it appears here with this strength and there with that strength, and that I can know from the data. And then I say that this thing basically mechanizes the way I get from data to theory. In philosophy of science they distinguish between the context of discovery and the context of justification. Hans Reichenbach wrote that he was the first to make this distinction: as for how one makes a scientific discovery, there is the context of discovery. The context of discovery means: a scientist thinks of a certain theory. We have no way of knowing how he arrived at that theory. It’s a guess—his grandmother appeared to him in a dream, it doesn’t matter, it’s not interesting anyway. He himself might not know either, right. Let’s say the Holy One, blessed be He, gave it to him. Maybe, yes. Not maybe—definitely. I don’t know if definitely, maybe. In any case, the context that matters for us is what he calls the context of justification. After you have a theory, you test it experimentally and see whether it works or not. I don’t care where you got the theory from, that’s not interesting. And traditionally in philosophy of science people are used to thinking that the context of discovery is inspiration. It’s not something we know how to say anything about. We can only—the context of justification, sorry—we can only test whether it works or not; that’s in our hands. What I’m trying to propose here is an algorithm that creates discovery. That’s how I discover the theory. Not discovery—discovery. Discovery is not inspiration. Discovery is not inspiration. There is an algorithm that gives you discovery.
[Speaker A] Provided that you choose the A and the B, and that’s the discovery. So what is the intuition that tells you what to put into the table?
[Rabbi Michael Abraham] Do you remember that in the previous class I talked about this—with the historian Carr and with Semmelweis, the examples I gave last time—that this is true in any case. Meaning, you need to decide which facts to collect in order to build a theory. Remember Napoleon’s victory? It’s exactly the same thing. To decide which facts are relevant for building the theory before you know the theory. So how do you know which facts are relevant? I have some intuition. That’s the intuition with which I build the table. To build the table is to say which facts are relevant to the theory and which are not. Exactly. And afterward from that I extract the—you see that this is basically the logic of science, how science works. But what’s nice here is that it’s a logic that explains discovery in science, not justification. Nobody—people think that’s inspiration, but it isn’t. Meaning, at bottom the building blocks the Sages built in the Talmud, with—
[Speaker A] The core of inspiration is what you put into the table.
[Rabbi Michael Abraham] Right. Building the table. But beyond that there’s really a system for how you get to the theory, which I think is very interesting. And the Sages found these building blocks. By the way, in Latin logic, I once saw in some article, there are also thirteen interpretive principles. Not exactly like ours. A fortiori inference and analogies from a common source appear there, but the others are not like ours. But I’m saying, there are some building blocks here—I’m talking only about the logical building blocks, not general-and-particular and the like. That’s a different game. Therefore this theory here connects only a fortiori inferences, analogies from a common source, refutations of them, common side, and all generalizations. That’s it. Don’t bring in here general and particular, don’t bring in here a matter that left the general category not to teach about itself but about the whole category, two verses that contradict one another—they don’t enter here. These are not the logical interpretive principles. I’m talking about logical inferences, as in science, as in law. In this case, this is the logic that stands behind them. Now, for example, one implication that I once tried to examine with a friend—with Zohar—I once tried to see whether we could actually make a start-up out of this. Because here, I’m giving you the idea, you do it. Because basically now, let’s say a lawyer wants to know—he’s appearing before a judge and he wants to know what punishment his client is likely to get. Now he already has data on what the judge did in previous cases. Say, if someone drove without a license he got such-and-such; if someone drove through a red light he got such-and-such; if someone drove without a license and through a red light he got this. Someone who drove like this was acquitted; someone else got such-and-such a fine; that one went to prison. We have a collection of things here—put them into a table. Now a new case comes. A person drove, as the comedians say, against the direction of traffic, through a red light, without—ran into a kiosk. Fine? So now the question is what’s likely to happen to him.
[Speaker A] No, but to do that you have to assume that the judge is consistent. Yes, that’s it. Okay.
[Rabbi Michael Abraham] Therefore no—but the nice point here, the nice point here, is that a judge is consistent as long as he isn’t aware of it. And what I’m doing here is something the judge himself doesn’t know about himself. Because I’m uncovering within the judge certain parameters that, from his perspective, affect sentencing. Now he himself isn’t aware of that. In that sense I think there’s a good chance he is consistent.
[Speaker A] Yes, but there could be other parameters—for example his mood in the morning, his wife gave him—
[Rabbi Michael Abraham] That too is another parameter, put that in too. Put that in too.
[Speaker A] You don’t know what they put into him—
[Rabbi Michael Abraham] In the morning.
[Speaker A] But you don’t know—
[Rabbi Michael Abraham] You don’t know what. No, don’t know. No—what he ate for breakfast, morning and noon. Fine. In principle I can collect it. It’s always like that, and that’s why there will be refutations. If it doesn’t work—but it will give you a prediction that isn’t a shot in the dark. A better prediction than a shot in the dark. True, it’s not absolute.
[Speaker C] That’s exactly what you were talking about with more severe—
[Speaker A] Less—
[Speaker C] And more severe, that also goes into this table, that the refutation in the Talmud is this is more severe and this— You talked about more severe and less severe. I don’t remember what. There was a refutation, they didn’t understand it. I think the Rosh—the Rosh among the great ones there—tried to solve it, but he didn’t understand. Meaning this whole issue of additional parameters—
[Rabbi Michael Abraham] No, that’s something else that I’ll get to; I’ll comment on it next time. I’m not going to do mathematics anymore, but I’ll comment on it next time, a bit formally. Why? This. Wait, sorry, I haven’t cheated yet.
[Speaker A] Now we’ve moved on.
[Rabbi Michael Abraham] In the next stage the Talmud says: a refutation. A refutation of the common side: what is there about their common side? Their benefit is greater. What does “their benefit is greater” mean? With money and intercourse there is pleasure; with the marriage canopy, not really, right?
[Speaker A] There are more parameters.
[Rabbi Michael Abraham] So therefore, therefore you actually see what a refutation is. It’s a refutation because it refutes both sides: a parameter that exists in both sides and not in the thing being learned. Remember the A, B, and C? So in A and B there is a shared parameter that doesn’t exist in C. That is exactly a refutation—it succeeds in refuting a common side. Now do the calculation, do the calculation, and you’ll see that this really is a refutation,
[Speaker A] Because it’s exactly equal to red. You don’t need to do the calculation; if it’s equal to red that means red is possible. You don’t need this whole graph here.
[Rabbi Michael Abraham] Right, right, if H now drops in the number of points. No, no, no, without getting into that.
[Speaker A] The column of A—let’s not—
[Rabbi Michael Abraham] Forget—in the table, it is identical to the column of H.
[Speaker A] Which means that if H exists, you can’t say that A is not like H by definition, even before all the graphs.
[Rabbi Michael Abraham] That’s an interesting question. I need to think whether that really always works, but I can do it in my calculation and say that the moment—when I put in filling zero here, then this and this are the same thing. So A and H would be one circle, right? And that would compensate for the advantage the other side had before, because that’s an advantage of filling zero. Okay, so therefore this—now that was interesting, I need—
[Speaker A] To think about it.
[Rabbi Michael Abraham] I can almost prove it to you. I’m not sure at all; I have a few thoughts about it, but maybe—I need to think about it. In any case, the point is that they won’t add more parameters, because the number of parameters needed to explain A will explain that as well, but then it goes back to my consideration that A and H are simply the same vertex.
[Speaker A] What I’m doing here, I’m already cheating—
[Rabbi Michael Abraham] But that’s the point I want. Why am I cheating? Because H is not of the same type as these parameters. Those are halakhic / of Jewish law parameters, right? It effects marriage, it effects betrothal, redemption of a yevama. This is a factual parameter: there is pleasure or there isn’t pleasure. It’s not what it succeeds in doing or doesn’t succeed in doing. Remember what I remarked about the medieval authorities (Rishonim)? Right. Therefore it really isn’t correct to do it this way. Okay, in the table the Talmud speaks this way—it doesn’t do it—but
[Speaker A] I leave it as—
[Rabbi Michael Abraham] An option, that it is a refutation from the Talmud’s standpoint. I accept that it’s a refutation, it’s a good refutation, but its analysis should not be done this way within these tools.
[Speaker C] Wait, but what about H? You just told us—pleasure, pleasure, pleasure.
[Rabbi Michael Abraham] These have pleasure—money and intercourse. Fine, so what now? Now—fine. So I’ll already tell you now, basically, how you do it. Can we have a few more minutes? I’ll tell you now how you do it, and then next time I’ll open up the whole next possibility. Basically what I’m claiming is this: the moment I say that these two have pleasure, pleasure is one of the alpha beta gamma, not one of the I, M, P, Y. Pleasure means that in money and intercourse there is something relevant to causing or not causing things—a parameter. So basically I take this table and solve it with the two fillings, but there is a constraint. The constraint is that in M and in B there must be some shared gamma. That’s it. Now continue, continue solving: filling in one, filling in zero, do that and you’ll see that now a priority of one over zero will no longer come out. That’s a refutation. Now we haven’t proven a theorem that this always works—that is, that every table, every column you add to the table, if you do it in the form of a constraint, will always come out the same way—because the Talmud doesn’t make that distinction. In our analysis it ought to make a distinction, but I have a suspicion that it should work. I don’t have a proof, I don’t know, I need to think about it. But in all the cases we checked, it works. Okay.