Topics in Talmudic Logic, Lecture 7

Table of Contents

• Criteria for comparing table fillings
• A theory underlying midrashic inferences and microscopic parameters
• Evidence for the existence of a theory: a fortiori reasoning and the refutation of a stricter side
• Bava Kamma versus the huppah passage: the language of factual properties and the language of halakhic properties
• The parallel between halakhic inference and scientific inference, and the example of mass
• The limits of the algorithm and the shortcuts of intuition
• Predictions and testing: science versus Jewish law
• Identifying parameters and reconstructing the rationale of the verse
• Factual refutations as constraints on the model’s solution
• The huppah table example and an illustration of the complexity of the solution
• Multiple minimal solutions and the difficulty of arriving at meaning
• A natural conceptual basis versus an unnatural one, and a qualification of the optimism

Summary

General Overview

The text presents an algorithmic method for analyzing non-deductive inferences באמצעות tables, diagrams, and topological criteria for deciding between competing fillings, and grounds the claim that beneath midrashic inferences there sits a theory of microscopic “factual” parameters that explain halakhic properties. It explicitly compares model-building in Jewish law to theory-building in science, demonstrates this through passages in Kiddushin and Bava Kamma, and shows how certain refutations function as the imposition of a constraint on the model’s solution. Alongside the optimism that one can reconstruct the “rationale of the verse” from the model, the text ends cautiously: sometimes there are multiple equivalent minimal solutions, and therefore the conceptual identification of the parameters and the meta-halakhic meaning depend on choosing a natural conceptual “basis,” which is not guaranteed.

Criteria for Comparing Table Fillings

The decision between a zero-filling and a one-filling is made according to four criteria: the number of parameters required for the explanation, the number of direction changes in the diagrams, the number of vertices in the diagrams, and the number of connected components in the graph. The comparison is made across the two diagrams together, and its purpose is to choose the preferable filling when two fillings are possible.

A Theory Underlying Midrashic Inferences and Microscopic Parameters

The text states that at the foundation of the inferences there is a theory of microscopic parameters explaining things like “damaging agents and domains” or “actions and results,” such as huppah, intercourse, a document, money, as against marriage, betrothal, redemption, second tithe, divorce, and a yevamah. It argues that the halakhic characteristics are what appear in the table as results (“effects betrothal”), whereas the factual characteristics are the table’s solution itself—that is, the alpha-beta elements that explain the halakhic force of the actions.

Evidence for the Existence of a Theory: A Fortiori Reasoning and the Refutation of a Stricter Side

The text argues that the a fortiori reasoning in the literature of the Sages is not “circular,” and the natural explanation for that is a microscopic solution underlying it, such that an a fortiori inference of rows and of columns is the same hierarchy within the same parameter. It adds that the refutation of a stricter side in a common-denominator argument clarifies the distinction between a refutation from laws and a refutation from parameters: a refutation from laws is not accepted by all the Tannaim, and there is a tannaitic dispute over whether it overturns the common denominator, because the unique halakhic properties may rest on the same “factual characteristic” shared by the two source cases.

Bava Kamma versus the Huppah Passage: The Language of Factual Properties and the Language of Halakhic Properties

The text presents the beginning of Bava Kamma as a case in which the Talmudic text itself speaks explicitly in the language of factual properties, such as “its way is to go and cause damage,” “another force is involved in it,” and “the beginning of its making was for damage.” It presents the huppah passage in Kiddushin 5a as a case in which the Talmudic text speaks mainly in halakhic properties (“effects betrothal,” “effects redemption”), and from analyzing the table one tries to extract alphas, betas, and gammas as a factual model underlying the laws.

The Parallel between Halakhic Inference and Scientific Inference, and the Example of Mass

The text argues that there is no fundamental difference between halakhic inferences and scientific inferences: in both there are observed phenomena and a theoretical model that explains them through microscopic parameters. It illustrates this with an analogy to gravitation and mass, and develops a physical remark about the difference between inertial mass in Newton’s second law and gravitational mass in the law of gravitation, and how the question of why they are the same quantity led to general relativity. It argues that one could in principle build an enormous “table of phenomena” in physics and extract alpha-beta parameters from it without identifying what they are, and even discover parameters that seem different but are always identical, so that one of them can be eliminated.

The Limits of the Algorithm and the Shortcuts of Intuition

The text explains that the algorithm becomes difficult when one reaches large tables, because the difficulty rises exponentially. Therefore both the Sages and physicists do not work directly with giant tables, but use shortcuts of understanding and intuition to identify relevant and irrelevant properties. It presents the “law of conservation” according to which, if you don’t understand, you have to work harder, and suggests the suspicion that if all the data were available, one could in principle reach all the correct conclusions even without understanding, by means of a giant table that would split into independent subgraphs sorting different domains.

Predictions and Testing: Science versus Jewish Law

The text suggests that just as in science a model can be tested through predictions, so too in Jewish law one could in principle hide a column in the table, such as divorce, infer from the model what should appear there, and then check it against the Torah. It emphasizes that in Jewish law this is generally not done because the data are already before us, but the logic itself is identical.

Identifying Parameters and Reconstructing the Rationale of the Verse

The text describes two ways of using the algorithm: a formal use when there is no intuition for identifying parameters, and a reverse use in which, after finding a model, one tries to identify what the alpha-beta elements are and expound the “rationale of the verse.” It gives hypothetical examples such as identifying “benefit” as common to money and intercourse but not to huppah and a document, and identifying “connection” as common to huppah and intercourse but not to money and a document, and suggests that this could explain asymmetries, such as why what effects betrothal does not effect divorce.

Factual Refutations as Constraints on the Model’s Solution

The text distinguishes between a halakhic refutation and a factual refutation such as “their benefit is greater,” and states that a factual refutation is not just another ordinary column but a constraint on the model: there must be one microscopic parameter present in money and intercourse and not present in huppah. It demonstrates how one solves a one-filling and a zero-filling under such a constraint, and how the constraint mainly changes the number of parameters, and therefore can turn an inference into a “refutation” by offsetting the topological advantage through a disadvantage in parameters.

The Huppah Table Example and an Illustration of the Complexity of the Solution

The text presents a four-by-seven table summarizing the huppah passage, with columns such as divorce, against her will, yevamah, benefit, and redemption/betrothal/marriage, and rows of money, huppah, intercourse, and document. It describes Rav Huna’s line of argument: an a fortiori inference from money, a rejection based on the redemption of consecrated property and second tithe, “intercourse will prove it” and a rejection based on yevamah, the common denominator of money and intercourse and a rejection from “their benefit is greater,” “a document will prove it” and a rejection from divorce, and then the refutation “they can exist against her will” and the response “we do not find money in marriage.” It emphasizes that the solution is not trivial even in a relatively small table, notes the possibility of error in the drawing or the result, and argues in principle that the preferable filling should emerge according to the criteria.

Multiple Minimal Solutions and the Difficulty of Arriving at Meaning

The text states that simple graphs can have several equivalent minimal solutions in terms of the number of parameters, and therefore there is no formal preference between them, even though the possible meaning of alpha and beta changes dramatically. It illustrates this with a basic structure in which a solution can be represented as alpha, two alphas, alpha and beta, or as alpha, two alphas, alpha and beta together, and shows that in the passage of redemption/betrothal/marriage two equivalent solutions would lead to different interpretations of “benefit” and “connection.”

A Natural Conceptual Basis versus an Unnatural One, and a Qualification of the Optimism

The text argues that the question of what counts as “simple” or “complex” depends on the system of concepts, and that a composite such as “alpha and beta together” can be defined as a single basic parameter without being any less simple. It explains that if one lands on an unnatural basis, one cannot intuitively identify the parameters, and therefore reconstructing the rationale of the verse from the table becomes problematic. It concludes that the initial optimism about extracting meta-halakhic meaning depends on the chance of reaching the natural solution. It states that in “reverse thinking,” where one identifies parameters in advance and uses them as a shortcut, one arrives more directly at the natural solution.

Okay, we’re in the middle of a discussion about non-deductive inferences, and last time we talked about—well, I tried, I mean not tried but expanded—the criterion, the criterion for deciding or preferring between two fillings of the table, not only with additional topological parameters. Meaning, when we compare a zero-filling with a one-filling of the two tables, we ask which of the two is preferable, and then we have to compare them according to four criteria. One criterion is how many parameters are needed in order to explain it—the model with the smallest number of parameters. The second criterion is the number of direction changes in the two diagrams. The third is the number of vertices in the two diagrams. And the fourth is the degree of connectedness—how many independent parts there are in the graph. We saw that, in terms of the overall move, beyond developing the technique itself, I tried to convince you that it has a justification. The first attempt at persuasion was to show that at the foundation of the interpretive inferences there is actually a theory. By “theory” I mean some kind of model of microscopic parameters that explains the damagers and the domains, if we’re talking about that, or the actions and the outcomes—like chuppah, intercourse, document, money, and so on. Those are the actions. And the outcomes are marriage, betrothal, redemption, second tithe, divorce, the yevamah, and so on. So I tried to show that at the foundation of the inferences there is a theory. I showed this mainly on the basis of two things. One thing was that I tried to show that a kal va-chomer is never reversed anywhere. In the literature of the Sages they do not reverse a kal va-chomer, even though on the face of it reversing it turns it into a completely different argument, such that an objection to one direction would not knock out the other direction. The only way—I don’t know if it’s the only way, but the natural, obvious way—to explain this is to make a microscopic solution that underlies the kal va-chomer. And then I showed that, in fact, the kal va-chomer of the rows and of the columns is the same kal va-chomer, the same hierarchy in terms of the same microscopic parameter. So that was the first proof. The second proof was the objection of “a stricter side” in a common denominator, where I tried to show that when the objection is made from laws, as opposed to an objection made from parameters, an objection made from laws is not accepted by all the Tannaim. Meaning, there is a tannaitic dispute over whether one makes an objection of a “stricter side.” When I say that the two source-cases each have some stricter side—not the same stricter side, but a different stricter side. This one has its own unique strict law and that one has its own unique strict law. There is a Tanna who says that such a thing knocks out the common denominator. And the medieval authorities (Rishonim) struggle with what that means, because if that knocks out the common denominator then there is no common denominator in the Torah. It is always like that, every time; each of the two source-cases always has some stricter side, and that’s exactly why we arrive at a common denominator. Therefore it is built into the structure of the common denominator. So I explained that if the objection were about a property of the two source-cases, then it really would not knock out the common denominator. But if these are different halakhic properties, special to each of the two source-cases, then there is an opinion that says this does knock it out. And why? Because underlying the special properties, it could be that the same factual characteristic is sitting there. Now, in the language of microscopic parameters, that factual characteristic is our alpha-beta. Because it is really telling me what there is in chuppah, or in intercourse, or in money—what the factual property is, what is there that causes it to be able to effect betrothal, effect marriage, or not effect it. There is something responsible for the capacities of each of these actions, for the legal force each of these actions has. Therefore I call these factual characteristics. The halakhic characteristics are what is written in the table. In the table it says that this effects betrothal, so that is the halakhic characteristic. The factual characteristic is the solution to the table. That is the alphas and betas that I derive from it. Why? What is the idea behind this? Because the claim is that behind all the halakhic characteristics there stands a theory. And that theory is a theory connected to factuality. Now in the context of the beginning of Bava Kamma, which we discussed in previous classes, there it’s out on the table. The Talmud there talks about the primary categories of damages—pit, fire, ox, tooth and foot damage, horn, and so on. And what does it say about each of them? What are the properties? “Its way is to go and damage,” “another force is involved in it,” “its making begins as damage,” all sorts of things like that. What are those? Those are properties. Factual properties. Those are the alpha-betas. The Talmud there speaks in the language of alpha-beta; I don’t need to use laws in order to try to reconstruct from them what the factual properties are. There the Talmud itself tells us the factual properties. Okay? By contrast, in contexts like chuppah—in the topic of chuppah in tractate Kiddushin on page 5—the Talmud speaks about halakhic properties. This one effects betrothal, this one effects redemption, this one does not effect redemption—that’s all halakhic properties. And from my analysis of the table, I try to extract what the theory is, meaning in terms of factual properties, the alpha-betas-gammas that stand behind the laws. Okay? So where can you ask about a stricter side? When the objection is from halakhic properties. Because if the objection is from halakhic properties, then I say, suppose there are two source-cases A and B, and the case to be learned is C. Source-case A has a special property X, B has a special property Y. If X and Y are alpha-beta properties, factual properties, that does not knock out the common denominator, because that is always what happens in a common denominator. Each one has a unique property, but together they teach that those are not the relevant properties—not alpha and not beta, but the common denominator. But if these are halakhic properties, then essentially it could be that the alpha responsible for property X in source-case A is the same alpha responsible for property Y in source-case B. And it is the same alpha. And if the two source-cases have the same strict property, then that knocks out the stricter side. So I’m saying that what I said two classes ago—I don’t remember exactly when it was—when I tried to show the necessity that there have to be factual parameters at the base of the inferences, now I’m closing the circle and saying that when we analyze the table and extract from it the model, the alphas, betas, gammas, and so on, we are essentially finding the theory that underlies the laws. Just as in scientific inference we are actually finding the relevant parameters responsible for the phenomena. The phenomena in the scientific context are parallel to the laws in the halakhic context. Yes, in the scientific context you say certain things do not fall to the earth, other things do fall to the earth. I ask: those are the facts, which are parallel to the laws—yes, those are the phenomena that I need to explain. Now I ask what causes this? What theory stands behind it? I say perhaps mass. Mass is really the alpha. Everything that has alpha, that has mass, will fall to the earth. Then I have found a theory that explains the phenomenon of gravitation. That theory says there is a microscopic parameter that I call mass, and this parameter is responsible for falling to the earth—which are the phenomena that I observe. The same in Jewish law. I observe halakhic phenomena, laws that the Torah establishes, and I try to explain them in terms of a theoretical model. Then I find all sorts of parametric characteristics, alpha-beta sorts of things, parallel to mass, and they explain the halakhic phenomena. Okay? So this goes entirely in parallel, exactly the same thing; there is no difference between halakhic inferences and scientific inferences. It works in the same way. There is only one point where you found that there it was mass, where alpha was mass. Here I don’t identify the alphas. I’ll get to that perhaps today already. I’ll get to it. Right, that is indeed a difference. Let me maybe give you an interesting example—it’s just a physics note, but it’s interesting logically too. When I taught physics in high school, our textbook was Sears and Zemansky. I don’t know, in the younger generation I think they use other books already. The book was Sears and Zemansky. In Sears and Zemansky, in the mechanics book, there is a small short chapter that talks about inertial mass and gravitational mass. A chapter everyone skips in high school. The nicest chapter in the book, the only chapter that is really conceptual in the book, and everyone skips it because it doesn’t help solve any problems. It’s no use for the matriculation exam. When I taught, there was a year when I made the mistake of going to teach in a high school, and there I tried to teach them this chapter. In any case, what that chapter basically says is this: there are two contexts in physics in which mass appears. Two different contexts. One context is in the law of gravitation. When there are two bodies with mass, say m1 and m2, then the force they exert on each other is proportional to the product of the masses divided by the square of the distance. Okay? So that is one appearance of mass in the physical world. The second appearance is in Newton’s law of inertia, or in Newton’s second law, okay? Newton’s second law says that force equals mass times acceleration. Now, someone who finishes high school physics—if you ask him—has never once thought about the fact that this question, why the same quantity appears both here and here, is really not at all self-evident. Why in Newton’s second law, F=ma, does m appear, and in the law of gravitation m also appears—the same quantity? Why is it the same quantity? These are two completely different appearances. It’s not at all… think for example of Coulomb’s law, right, the electric force between two electric charges, where you have the same form of law. Between two charges, say two charged bodies, one with charge q1 and the other with charge q2. What force do they exert on each other? Coulomb force—the product of the charges divided by the square of the distance. Okay? Exactly like the product of the masses divided by… What does that mean? That mass here functions like a kind of charge. It is a gravitational charge, meaning it is the property of a body that causes it to be attracted to another body, just as electric charge causes electrical attraction; mass is the charge that causes gravitational attraction. F=ma is not connected at all to attractions and gravitation. The F that appears there is not specifically the gravitational force. It can be Coulomb force too; it can be any force whatever. So why does m appear there? What does it have to do with m? Einstein essentially was—I don’t know if he was the first—but he was the one who made use of this. General relativity came out of this simple question. How can it be that inertial mass, the mass in the law of inertia, in Newton’s second law, and gravitational mass, the charge in the law of gravitation, of gravitational force, are the same quantity? It’s m. Why? These are two totally different things. Now in high school nobody thinks about this. We study: yes, this is m and that is m, all fine. Nobody thinks to ask, wait a second, why does the same m appear here and here? And then Einstein says space is curved and gravitation and force and the curvature of space are the same thing, and so on—in short, general relativity comes out of that. Now this is really, if you were—if we were looking at it in this language, to go back to your remark—if we were looking at it in this language, then basically I would have had to do this. I would have had to make a table of phenomena. All the bodies in the world—what physical properties do they have? This one is attracted by electric force, this one is attracted by gravitational force. This one accelerates in this way and that one does this and this one behaves under friction in that way and when you heat it it becomes liquid. Let’s say I collect all the physical phenomena. And all the physical bodies—those are the phenomena. And now I want to build physics. How do I build physics? I make a table. I could in principle do exactly what we did here. I would do what was done here and I would say there must be some parameters, whatever they are, alpha beta gamma delta, such that some combination of them causes you to be attracted by gravitational force. Another combination causes attraction by Coulomb force. Another combination causes you to melt at such-and-such a temperature. Okay? And so on. All the physical properties are essentially a property—or all the physical phenomena are the result of properties of bodies. Certain properties of bodies. Okay? Now those properties are the parameters. And from those phenomena I could build all of physics without identifying who alpha is, who beta is, who gamma is, and who delta is. Rather I would say there are alpha beta gamma delta, some kind of fundamental charges or basic properties of different bodies or of different materials if you like, it doesn’t matter, without identifying them, and I could derive all of physics without identifying anything. Then I wouldn’t know that my alpha is really mass; I wouldn’t even call it m, because it wouldn’t be mass. I’d call it alpha. And strangeness or whatever, all the other properties people extract today in particles, I’d call beta gamma delta, whatever. And from that I would derive all the properties. In principle one can develop all of physics this way. There is no obstacle to doing that. Exactly the same thing. And then I wouldn’t need to identify the parameters at all. By the way, in that case I would seemingly discover that there are two different parameters, alpha and beta, which for some reason are always equal. That is gravitational mass and inertial mass. They would appear as two different parameters, but somehow it would turn out that whoever has parameter alpha also has parameter beta. And then that would really mean I don’t need both of them. I could call them by the same name and erase one of them from the picture, because they are really identical. Okay? That is really what I would discover there without identifying who alpha is and who beta is. I would understand that there are two parameters here playing two roles. They are always equal. Then I would erase one of them simply because—and since I don’t identify either one of them—then I would erase one of them. Okay? What happens in physics? Why don’t they use this in physics? First of all because this technique is a hard technique when we begin to get tables of, I don’t know, twenty by thirty. It becomes very hard; the ability to solve the table grows exponentially. What do you do in such a case? By the way, the Sages also did not work with tables, and physicists also do not work with tables. Why not? Because we find shortcuts. We basically say: we already know more or less what is plausible, which relevant properties are at play here and which are not. I talked about Semmelweis, right? And about Carr the historian, which is exactly the attempt to identify, even before you understand the whole picture, what might be relevant and what might not. That is what we do, both in physics and in Jewish law. Then we don’t need the tables—or at least not completely. We can shorten the path by understanding that there are certain relevant and irrelevant properties here, and we know: this is the ox’s way, this is derivative damage, this has another force involved, and I understand that it makes sense that this would matter. But how does it work? It works in the opposite direction from the table. Because here I start from the logic of the thing. I start from making use of some understanding of what really makes a property matter, and that lets me shorten the route. I can say, okay, then I already know that mass is what will cause it, and not something else, so I already know there is a parameter here that is mass even without doing a table analysis, and then I can make much simpler tables or really skip the need to solve tables at all. Because understanding the rationale of the verse—the understanding—can shorten the way for me. That is exactly what Semmelweis did. Because if he had to test everything without understanding anything, he would have had to test an infinite number of things; he would never finish. You do elimination by assuming what might be relevant and what might not; you shorten the route, you drastically reduce the table, and then you can work. If it’s small enough, you don’t even need a table; you do it intuitively. That is kal va-chomer, binyan av, common denominator. These are things for which you do not need a table. The Sages did this because they understood the logic, so you don’t need a table. It works with a table, but you don’t need it—when the tables are small, you don’t need the tables. In other words, when we know how to identify parameters, that makes the way easier. How did we know how to identify properties—I mean alpha-beta parameters? How can you identify that a parameter is relevant to your discussion? You can—you have an intuition. Like the way we identify that mass is relevant to gravitation. Otherwise we could say there is some alpha here, I have no idea what it is, but there is some property of bodies that causes them to be attracted, and then I’d see that this property exists in this body and that one, but light doesn’t have that property because light isn’t attracted. Then maybe I could work backward and say, ah—what is the property that every body in the world has but light doesn’t have? Or sound waves don’t have? That’s mass. Right. So I would go—one second—I would identify the properties by seeing which bodies have those parameters and which do not, and that would give me a guess as to what the property is. Think for example—let’s go back a second to betrothal—the same thing happens here; it is exactly the same thing, you need to understand, exactly the same thing. This is the logic of science, of Jewish law, of law in general, of everything. This is the basic logic. There are shortcuts. You can take shortcuts. But at the principled level, one could go with this technique and solve the whole universe, find all the theories of the whole universe in all fields this way. And if I gathered, of course, all the facts, and I were equipped with all the facts, and I knew which of them were relevant and which were not—of course that is a very theoretical statement—but in principle it would really give me the full picture. By the way, even if I didn’t know what was theoretical, there would emerge subgraphs that were not connected to each other, and I would understand that the properties here are not relevant to that subgraph, and I would discover what is relevant and what is not through analysis from the collection of properties. But of course that’s for messianic times; there is no computer that can do such a thing, not even come close to such a thing. But in principle there is an algorithm that does it. Meaning, one could build all of physics, all of law, all of Jewish law in such a way, algorithmically from the data. How will you know that the parameters you collected are relevant? I don’t know. I conjecture. I have intuition. Take Semmelweis, for example. How did he—well, no, Semmelweis was easier. Because after Semmelweis did it, he had predictions; he could check. He had predictions of what would now happen. If I’m right, I can put it to an experimental test as is done in science. In Jewish law, for example, this could be relevant if I simply leave one of the columns out of the table entirely and analyze what happens here in the table, and then I’ll try to see what I predict will happen in that column. Which of them will effect divorce and which will not effect divorce. Suppose I don’t look, I close my eyes, I don’t see what the Torah says about divorce; I make an analysis of the parameters, and then I discover whether chuppah, money, document, intercourse—which of them effects divorce and which does not. And now I go to the Torah and check myself. Then it would be like making a prediction and testing it in an experiment, whereas we usually don’t do this because we already have the data in the Torah, so we don’t ignore something and test it, but in principle one could do it exactly like in science. In science we don’t have the data until we do the experiment. Therefore we make some prediction and then do the experiment to check whether the prediction works or doesn’t work. And in principle it is the same logic. The point I want to sharpen here is that identifying the parameters, on the one hand, is not needed. Meaning, if you go with the tables without shortcuts and without anything, you don’t need to identify the parameters. In all the work I did up to now, I never told you what alpha is, what beta is, what gamma is; I simply solved the model and everything was fine. I can explain the inference even without that. Right? If I want to shorten the route because my table is too large and this algorithm has trouble solving it, then I shorten the route, and then I do need to identify parameters. I need to say, okay, there are some parameters here—for example, in intercourse and money there is something in common, and in document and chuppah there isn’t. What is it? Pleasure. Right? In intercourse and money there is pleasure; in chuppah and document there is no pleasure, they’re just formal acts, I get no pleasure from them, right? So for example I can assume that the pleasure parameter is relevant here. Now I ask myself why it would be relevant. I check which halakhic phenomena I can produce by money and intercourse but not by chuppah and document. Those are the phenomena in which pleasure is apparently important. Maybe not only pleasure, but pleasure is needed too. Okay? So I have a way to go back and forth. Meaning, to go from the table to the parameters without identifying them, but I can also try to identify relevant parameters and save myself the trouble of solving the table. More than that: it could be that I don’t know, I have no intuition about which parameters are operating. I solve it straightforwardly from the table. I find the model and I know what there is in chuppah, what there is in intercourse, what there is in money, what there is in document—which alphas and betas and so on there are in each one. And now I go backward. I found that there is a parameter in intercourse and money that is not in chuppah and document. So I know that alpha is pleasure. Right? What do I mean “know”? I conjecture. It could be that I am wrong, maybe there is something else in common, but it is not a bad guess. Okay? Then I ask myself, suppose there is something in chuppah and intercourse but not in money and document. What could that be? Chuppah and intercourse. Marriage perhaps? No—what is their property? A factual property, alpha-beta. I would say some kind of union between husband and wife, right? Chuppah is standing together in one domain. Intercourse is a physical union. Document and money do not do that. Right? So if I discovered a parameter that exists in intercourse and chuppah but not in money and document, I would say, ah, this parameter is probably some kind of union, some sort of bond, couplehood, something in that direction. Again, I do not always know how to carry this all the way through, but it already gives me a clue as to what it is. And then understand that this method basically allows me to derive the rationale of the verse. One second. This method allows me to derive the rationale of the verse because now I take the laws, solve the model, extract what parameters exist in money, intercourse, document, and chuppah, and from this I can discover what really characterizes actions that effect betrothal, or divorce, or redemption, or marriage, or whatever. And that is deriving the rationale of the verse, because I am basically saying: why does this thing effect betrothal? Because it has pleasure. So I understand: pleasure can effect betrothal. If I did this without the table, that would be deriving the rationale of the verse. Right? But if I do it with the table, I have logical support for the matter. I can test myself. It’s not a shot in the dark. If I thought that deriving the rationale of the verse, for example, was problematic because maybe I’m wrong, then there would be room to say that if I do it in this way perhaps it is permissible to derive the rationale of the verse. Because here I have a logical way to test whether I’m mistaken or not. There it is just speculation; you raise a hypothesis and have no idea whether it helps or doesn’t help. Here I can propose predictions and put them to the test. So perhaps here it would be permissible to derive the rationale of the verse. And these things may have many implications. So now notice—I’ll summarize—what I’m saying is this: I essentially have two ways in which I can derive benefit from what we learned, from this algorithm, from this algorithmic method. One possibility is to make inferences where we have no intuition, and then I don’t know how to characterize the microscopic parameters, so I use formal alphas, betas, gammas—I don’t know what those alphas are, but I can know what the correct filling is. That is what we have done until now. Basically I used this in order to mechanize the inference, meaning to make an algorithm that performs the inference. The opposite use of it says this: I made the inference, explained it, found a model—now I take the model and ask myself: what is this model? Who is alpha? Who is beta? Who is gamma? And I begin to see, wait, if alpha exists in chuppah and intercourse, then alpha is probably union. If beta exists in money and intercourse, then beta is probably pleasure. Then suddenly I can understand the rationale of the verse, the meaning—why did the Torah determine that these things effect betrothal and those do not? Because from the Torah’s perspective, apparently pleasure is required to effect betrothal, or pleasure also is required to effect betrothal. Then I can go backward—not to explain the inference, I already explained the inference—but to use the model that came out at the base of the inference, from within the inference, in order to understand the theory that stands behind the law, to derive the rationale of the verse. Then I have a way to understand what betrothal really is, what marriage really is, how one determines that something effects betrothal and that something does not effect betrothal. It could be, by the way, that this really is how the Sages determined it—they understood by reasoning what might effect betrothal. Behind the expositions and verbal analogies they bring, there still stands some kind of understanding in general. Then I say: I can reconstruct the line of thought the Sages went through. They probably looked for something that contains pleasure, so that is money and intercourse, okay? Or they looked for something that creates union, so that is intercourse and chuppah, okay? Or things of that sort. Say, something that has both union and pleasure together would do both. Something that has only union but no pleasure, namely chuppah, will not do betrothal, only marriage. Something that has pleasure but no union, namely money, will do betrothal, because for betrothal you do not need union; pleasure is enough. Okay? The discussion here about union—it sounds more like the beginning of a shared life; that too, spreading a cloth over her and intercourse. No matter, fine, I’m not going into those resolutions right now, exactly how to formulate it. I’m only trying to show the scheme, meaning how I can use this in order to decode what really the theory is that stands behind the halakhic rules. Why is a woman acquired in three ways, and why does she acquire herself only through the husband’s intercourse and through a bill and a document? What is the asymmetry? Why does what effects betrothal not effect divorce? So I have actions, but I want to know the theory—what really stands behind this? Or the same in damages. In damages, as I said, the Sages immediately put the alpha-beta on the table because they understood what made sense, so I did not need this whole route. But in principle I could take the rabbinic results, the Talmudic law—what tooth and foot do, what horn does, how much one is liable for, where one is liable and where exempt, and so on—and from that derive various parameters, and then suddenly I would discover that there is a parameter that characterizes tooth and foot but does not characterize horn. I would say that is probably “its way is to go and damage,” or “the normal course of its movement,” or something like that. Or “there is benefit in its damage,” which would be tooth but not foot. Or in horn, “its intent is to damage,” and that would be my alpha and beta. That is how I would identify who alpha and beta are, and I could reconstruct Bava Kamma without studying it. I could write the first topic of Bava Kamma based only on the laws; if you just gave me the laws, I would derive “its way is to go and damage,” “its making begins as damage,” “another force is involved in it,” “its intent is to damage,” unusual, not unusual. All those things I would derive by identifying parameters. I would call it alpha, beta, gamma, delta without understanding what it was, but afterward when I asked myself what it was I would search: what exists in horn but not in the others? That is probably intent to damage, or unusualness. What exists in pit? That is probably “its making begins as damage.” What exists in fire? That is probably “another force is involved in it.” And so on. Understood? So in different topics we sometimes start from the understanding and build the inference; sometimes we start from the inference and can build from it the understanding. Usually one does not do that, but I am saying that this algorithm opens before us the possibility of doing such a thing. We can generate understanding even where it is not present—in the Sages, or perhaps even the Sages did not know it, not only that they didn’t write it. We can reconstruct understanding where it is absent. Okay? Meaning, this thing is a very powerful tool. It is not only an algorithm that helps me decide whether the filling is one or zero; afterward I can use the model I found in order to decode the entire theoretical infrastructure of the topic. That is a very, very significant thing. Okay, now what I want to do is really move one more little step, one more little step beyond what I did last time. I hope today we’ll finish this matter. So I want to do the following. We talked about the… And also a question. If you take small diagrams of four by four or five—you can create a situation where from such a diagram you derive the conclusion, you don’t know what it is—what do you mean the conclusion? The parameters, the alpha-beta—and then afterward you assemble them from the whole diagram into another parameter, meaning… You’re talking about pleasure and money. Okay, you reached the conclusion that pleasure and money really fit for you. That money contains pleasure. Fine. Now you can take the conclusion—okay, I got money. I can take this parameter and transfer it—meaning, I already know these two, so I can give them up. What, in order to simplify the problem? Yes, simplify the problem. Sure, sure. That’s what I said. For example, the very fact that we put something into the table and not something else—that already simplifies the problem. In principle we could take all the laws in the Torah, all the kinds of damagers there are, all the kinds of liabilities there are, build one giant table of I don’t know, a million by a billion, okay? All the possible kinds of damagers you can imagine, in all the possible domains, where liable and where exempt. Fine? Now I would discover that these graphs would split into subgraphs that are unrelated to each other. And in each such subgraph I would say: there is public domain, private domain, horn, tooth and foot, and ox. That’s it. Meaning, spaceships and all that would not come in there; that would be a different wing of Jewish law. Now I understand this intuitively, so from the outset I build only a little table of three by four that includes only these things. At the principled level, understanding always saves work. The famous conservation law. Meaning, if you do not understand, you have to work harder. What is optimistic here is that I think—at least it’s a suspicion, I don’t have proof—that you can understand nothing and still reach all the correct conclusions. If you took all the data and all the… if all the data in the Torah were available and you could extract all of them, in principle, right, you would build a giant table. Fine? You would extract all of Jewish law from that table without understanding anything. And that table would already give you which parameters are relevant to this and which are not relevant to that. And you would discover there are many subparts. You could write all of Maimonides. There are the laws of monetary damages and there are the laws of the Sabbath. In the table they would all appear together. You would only discover that the laws of monetary damages have nothing to do with the laws of the Sabbath. It’s a different connection. Then you would say, ah, so that gives you fourteen different subgraphs—those are the fourteen books of Maimonides. Of course I’m being wildly optimistic. There would emerge the fourteen different books of Maimonides. Within each such book there is a set of laws—say, in the Book of Times. So there are the laws of Sabbath, the laws of holidays, the service of the Day of Atonement, Passover, and so on. Subgraphs would also emerge inside that table. In principle, I suspect, one could reach all this in a fully mathematical way. You are assuming that everything is built on reasoning, not on textuality, on comparisons. The textual aspects don’t interest me because the textual aspects reflect reasoning. So you couldn’t write all of Maimonides. No, no, it doesn’t matter. The textual… even textual differences, even textual conclusions—behind them there is ultimately an explanation. Why does the text say one has to compare this to that? Because apparently there is a similarity between this and that. Tradition tells us to make a verbal analogy. But why did the Holy One, blessed be He, make a verbal analogy between these two? I don’t know, but there is some logic behind it, right? Why did the Holy One, blessed be He, say that a slave and a woman require a verbal analogy? Apparently there is something similar between a slave and a woman. I may not know how to identify what it is, but there is something similar there. That’s all; that’s enough for me. I don’t need to identify what it is. Alpha, I’ll call it—what difference does it make. I don’t have to identify it; that’s exactly the point. So I’m saying it doesn’t matter that my trigger is textual. In the end, even a textual trigger reflects something that really is similar; otherwise why would the Torah compare this to that? It only tells me about it through some textual form. It doesn’t matter. But ultimately there is some similarity between these two things. Yes, otherwise why would there be a similarity between a woman and a slave? There has to be something similar between them. Okay? So the point is as follows. I’ll now do one thing just to show you how far this goes in the topic of chuppah. The table that emerges in the end—in fact here I’m really not going to go through—look, maybe, you know what, I’ll just remind you. We are searching whether chuppah effects betrothal, right? Rav Huna argues that chuppah effects betrothal. So he begins with a kal va-chomer from money. Money, which does not effect marriage, does effect betrothal, so chuppah, which effects marriage, certainly effects betrothal. Then they reject: what about money, since one redeems consecrated property and second tithe with it? Intercourse will prove otherwise. What about intercourse, since it acquires a yevamah? The common denominator of money and intercourse. They reject the common denominator because their pleasure is great. Money and intercourse, remember? Their pleasure is great. Document will prove otherwise. What about document, since it releases a Jewish woman—it effects divorce. The common denominator is composed of document and money on one side, and money and intercourse on the other. And from both those arrows… we learn about chuppah. They make an objection to the common denominator: what about the common denominator among them, since they can apply against her will. All three of these source-cases have some situation in which they can work even against her will. With money this does not happen. Money does not work against his will. Then they say—there is a rejection of that—“with money in the realm of marriage we do not find this,” never mind, some claim that money doesn’t work. So now I want to take this—that is the conclusion of the topic. And therefore Rav Huna says that with money in the realm of marriage we do not find this, and therefore from his perspective they really do succeed in revalidating the basic kal va-chomer of chuppah. Meaning, chuppah really does effect betrothal according to Rav Huna. We do not rule that way, but that is how Rav Huna learns. Now I’ll show you—just so you get an impression of how far these things go—I’ll show you a table. The table that comes out here in the end is the following. It’s a table of four by seven. For the moment I’ll leave it open because I need the clock, sorry. Okay. Now even background music. I wrote you a table while I write. Think of it as elevator music on the phone while you’re waiting for the relevant clerk to answer. Okay. So this is divorce, this is against her will, yevamah, pleasure, all the stages we went through on the way. Now I fill it in. There are several tables of classes. Okay, this is the table. Now. What is the A? Pleasure. That’s pleasure in English. Fine, so this is the table; it’s really four by seven. And it summarizes the whole topic. So really let’s just go back for one word—what is K? Against his will. Her will—also in English. And what is G? G is divorce, A is pleasure, Y is yevamah, P is redemption, E is betrothal, and N is marriage. And the parameters are? If it is money, that’s M. H is chuppah, B is intercourse, and S is document. Good. Okay. Fine, so now this is really the general table. Now just so you get an impression and see that it is not trivial to solve, and we are talking about four by seven—this is a relatively small table. But by the way, this is one of the most complex topics, I think, if not the most complex topic in the entire Talmud. So you really don’t get to very very high levels of complexity in the Talmud at the principled level, only because we know how to shorten the route since we understand what is relevant and what is not. If we didn’t understand, one could put the entire Babylonian Talmud into one giant table—that would be the Talmud. I would hand you a table here—that’s the Babylonian Talmud, that’s it. Meaning, one could now take this table and from it derive the whole Talmud, all of Maimonides, all of… of course purely hypothetical. Okay, so now we’re talking about a filling—zero is green and one is red. These are the two fillings we make, these are the fillings. Okay. Now look what comes out—I’ll just draw it instead of giving you separate puzzles here. This is G, this is K, this is A, this is Y, this is M, this is P, this is H. I’ll try to connect them all. And this contains the… wait, wait. This is the table. Now look, I’ll just explain for a moment what I did here. What we are doing is basically taking these columns, and now we’ll talk about filling one, right? This is the red one. Yes. Filling one. So obviously A is the fellow that everyone enters into, right? Am I right? Why doesn’t H enter into it? H—why doesn’t H enter? A… who needs to enter into what? Into A. Into A yes, it should enter. I have some mistake here; this is filling zero. H is filling zero. Does N enter A? Is N supposed to enter? Yes. N enters A. S also enters A. Fine, I think a mistake fell here in the book’s drawing, we’ll see in a minute. In any case, let’s see how this thing works. We have A, which of course is the biggest one, it’s one one one one, okay? Into A enters N, everyone enters it in principle. N enters it, and H also enters it, right? Of course, everyone enters it. But Y, what about Y? Yes. Y enters through N, right? And also through H, right? And also through K, right? Y also enters through K. It enters through all of them. That is why you see that all the routes from Y to A are either through H or through N or through K. Okay? And so on. And if you go column by column and draw its relation to the other columns, you’ll get this table. Okay? All in the same direction. What? All in the same direction? I draw it in a direction, meaning this is the direction I always draw it. Only if there is no choice—if there are direction changes, that’s another matter. What I mean is that like this it’s the same direction. We’ll see in a second. No, here there is a change of direction, we’ll see in a second. The table in filling zero, we have it like this: we have A—wait, why did a mistake fall in my drawing. A—what really changes? N doesn’t enter, right? Into A. Only N. So let’s stop all of them. K here, P G here, only G changes for us. P here, and H here. Okay? Now let’s see. Does P enter H? P into H? Yes, right? Good. Yes yes. Does H enter A? H? Yes, right? Does P enter N? No. No. So P does… okay, so earlier too it wouldn’t have entered. Does Y enter H? Yes. Yes. Y into H? No, why? Y enters H. Does Y enter N? Yes. Yes. Okay? Does N enter A? One one yes. What? N into A? No. No, right? That arrow was cancelled. Right? N remains isolated. That arrow was cancelled. And does G enter K? Yes. G into K yes. G enters K and K enters A and Y enters K. Only arrows entering A can really change; all the rest are the same, only A changed, right? So G enters, does K enter A? Yes. Yes of course. Okay. So that’s all, only this arrow was cancelled. And Y enters K. No. What? Y enters K. Y… yes. Y enters K. Also… everything is the same. The only change that can exist from filling zero to filling one is only in A itself, right? Because all the relations among all the others remain the same. So only in all the things that enter A do I need to check whether they still exist—meaning whether this exists, whether this exists, and whether this exists. All the rest can really be copied. Okay? Now what we basically see here is that N remains outside. Not outside, it’s simply not connected to A. Exactly. But there is no problem of connectedness here because it is connected through Y, doesn’t matter. So now what we have here are really two graphs, and I need to build their solution. And their solution, you’ll see, is not simple at all. Say I begin here with alpha; here there are two alphas, and here say three alphas. Okay? Now here there is alpha and beta too, right? We talked about the basic triangle where this goes to two alphas and this goes to alpha and beta. Here there is… what will be here? Two alpha and beta too. Yes. You have to check very carefully whether it is consistent with this. Consistent with this, right? Because since this enters this, if there were no arrow here I wouldn’t be allowed to put the beta here. Right? And these and these of course don’t speak to one another, fine. Right? That works. What happens here? Here alpha and gamma too, say, right? I can’t do this with betas. It won’t help, right? Wait, how does Y enter H? Yes, I could have done it with beta, say if I wrote here two alpha and beta. That would be fine, but it wouldn’t be fine with this, right? Because two alpha and beta is bigger than alpha and beta, there should be an arrow here. Okay, and therefore there has to be some new gamma here. But there you have two alphas; Y enters our H. Where? Y enters H. Yes, Y and H, Y enters H. Of course. Ah, okay, so therefore you must have something like this here, right? But then it’s no good because then it should enter this, right? So you have to make here two alpha and here one alpha. Am I right? But then it doesn’t—it doesn’t fit from Y, Y and X are the same thing. Ah, they don’t speak to one another, so right, I need two alphas. Two alphas—that must be. No, fine, now it’s okay. No, but Y and H. It can’t enter here, right? And it also can’t enter here. No, but Y and H also can’t enter. Y and H? Doesn’t work out. Ah, Y and H, okay, right. Correct. No, and it also can’t be two alpha and beta because then it doesn’t speak with this. So it would enter it, right? So here alpha and beta—no, it has to be two in order to enter this. This is alpha and beta, here it is alpha—no, H must be two alpha, H must be two alpha. Two alpha, am I right? But then you need to note that it can’t enter here, so this I have to add three alpha, and here three alpha, here four alpha. First of all, right? Once there are three alpha here and here two alpha and gamma, they don’t speak to one another. It is lower in alpha and higher in gamma. Okay, I’m trying to show you how this is done. The solution itself isn’t important right now, but you see you have to play around, and you can miss things here. So now of course this has to be built in such a way that it enters this, so let’s say this is three alpha and also gamma and beta. It has to be at least that. Now I want to raise it more. Do the three alphas work? No, that won’t work with this, right? Because it doesn’t enter, doesn’t work. Yes, it enters. So I’ll do four alpha here—no, if I do four alpha here that also won’t be good, it will enter. Why not? Five alpha will also enter. Because four alpha isn’t good because it will enter. I need the opposite, I need less. I need two alpha here, and two alpha won’t help me. Alpha—I need to check also that there is no arrow here, you see? That there is no arrow here. Okay, in short, I managed to convince you that this isn’t simple, right? One has to find these two parameters. Now it turns out—never mind, we did it here, checked carefully, though we may have made a mistake because we don’t have an algorithm. Meaning, we can never know, we have no proof that we found the minimal model; maybe we missed some possibility and there is a model. In any case, what came out for us is that in both these diagrams there are four parameters here—you cannot manage with fewer than four parameters, alpha beta gamma delta. And therefore? What? Yes, therefore no implementation. We would have had to add another parameter here, if I am right and we didn’t miss something. But in the end filling one comes out preferable because of direction changes, that’s what it says to me. Yes, as if it has to be, because everything else is exactly the same. What? Connectedness is the same, you only have gamma… No, and here actually it doesn’t come out that way. I need to check it. Everything is connected. The connectedness is the same and the number of vertices is also the same. The question is whether it is true in terms of direction changes; actually it seems to me here there are more. Look. Here there is one direction change, two, three. Three direction changes, right? Here there is one, two, three, four. Yes. You went back, you’re at the point… No, I went like this, right? So that’s one. Here. Two, here three, right? Okay, so he wants to get there, doesn’t want to get to G? Because there you got to G. No, I went from P to H. I’m looking for the path with the largest number of direction changes. So the path from P to H that goes like this is the path with the largest number of direction changes. And the fact that N is not connected in green? Yes. Isn’t that more problematic? Because there is—there are more disconnected things there, less connectedness. No, because connectedness is according to subparts of the graph that don’t speak to one another. Here everything is connected. A path… wait, N is entirely by itself. The path with the fewest direction changes that does not repeat itself, that doesn’t cross itself, that’s what you mean? What? Crossing is fine, but not overlapping. What does not overlapping mean? Not to go over the same path, the same bond, the same site again. Okay. Fine, in short, I’m saying one has to work out here. What came out in the book—and again I say, everything may contain a mistake—but what came out in the book is that in the end this is preferable in terms of direction changes; filling one is preferable in terms of direction changes. And if filling one is preferable in terms of direction changes, then that means filling one is the better filling. And therefore this inference is a valid inference. Wait, your criterion is the fewest direction changes? Right. That is why I’m saying it doesn’t work here. I’m saying one has to check both the drawing and the solution. But I’m saying that in principle it should have come out here that filling one is the preferred filling. Fine? I only brought this not because—we didn’t go through the whole way. Only so you can see that it’s not all that trivial. In a table like this, in a diagram like this of three or four sites, you can understand the solution fairly easily. But all in all it can be very nontrivial, and here we are speaking about four by seven. That’s nothing. In Maimonides it isn’t that simple. Exactly. “Not that simple” is an understatement I haven’t heard in a long time. Okay, so just to get the impression—what I now want to do further is to explain what I noted in the previous class. I have… we’ll work like this, it’s always… this is the common denominator. This is the common denominator, okay? Right? From money and intercourse to learn about chuppah, where I have an objection in favor of money, I have an objection in favor of intercourse, right? And I have an objection to the common denominator. An objection to both of them. Okay? This objection, by the way, is it a halakhic objection or a factual objection? Factual. Pleasure is factual. And therefore in principle I didn’t write—it’s true. If this objection looked like a halakhic objection, and a halakhic objection—for example there is room to say that result one here is the correct one because this is an objection of a stricter side. That means that each one of them has a stricter side, right? Each one has a stricter side—sorry, that means… wait… no, sorry, no. If these two properties were halakhic, then you could say there is an objection of a stricter side, and then the filling of one here would not be preferable to the filling of zero. Here I’m talking one stage further. I made an objection to the common denominator, brought another parameter. I made an objection to the common denominator. Now here, if it were a law, I would show that there is a law that money and intercourse do, but chuppah does not do. That would be an ordinary objection, in the way one writes it. But here it is pleasure. And pleasure is different from yevamah, redemption, betrothal, and marriage. Why? Because pleasure is a microscopic parameter. And therefore what really needs to be done is something else. This is the correct table. Exactly like the common denominator. This is the correct table exactly like the common denominator, but now I need to solve filling one and filling zero under a constraint. And the constraint is that there be one microscopic parameter that exists in M and B but not in H. Right? This is the common denominator before the objection. This is the common denominator; I solved it and found that one is preferable to zero. That we did in the previous class. Okay? Now there is an objection: what about money and intercourse, since they involve pleasure? Great pleasure too. Okay. Exactly. Now I wrote this as a column but that is not correct. This is a factual objection, not a halakhic objection. Therefore how should one take this objection into account? Now I do the same exercise I did with the common denominator. Only now when I search for the models for filling one and filling zero, I search under a constraint. Only a model where these two have the same parameter that this one does not have—only that is admissible. Under that constraint I look for the simplest model for filling one, the simplest model. Exactly. That model—we’ll call it, say, that parameter gamma. That is really the claim. Let’s draw everything, just so gamma is always there. Exactly. Then I search for solutions given that here and here there is gamma and here there isn’t. Aside from that, do what you want. This of course changes the result. Of course it won’t change either the connectedness or this, so clearly what will change here is the number of parameters. Because this is an objection. An objection means that the inference itself gives me one, but after the objection the result should be one or zero as equally weighted, right? Now clearly what this constraint will do is basically change the number of parameters, because that is the only thing that changes here. The graphs remain the same, it only changes the number of parameters. So if this common denominator was decided by, I don’t know, direction changes—I no longer remember because there are three common denominators, each one was decided by a different topological parameter. Say this one was decided by direction changes. Then what the constraint will do is give the other side a model with more parameters. And then it will balance the advantage of direction changes with a disadvantage in the number of parameters, and then it becomes… That is exactly what happens here. Okay? That is exactly what happens here. Let’s just do this one thing as an example. And alpha and beta with the base, with gamma here? What? Yes. Let’s try the example. I’m currently drawing diagrams of a common denominator. Fine? Common denominator without objection, because it is the same diagram, the same table. So I begin with filling one. So I have Y, N, and A. Here I have P. Right? Let’s see. A is one one one. N enters it, and Y enters through N to A, right? And P enters A independently. That is the diagram. Okay? And H—you couldn’t connect it here as a row? No. Why not? Because H is not an action. H is a property. Intercourse and money have pleasure. Chuppah does not have pleasure. That is a property. That is the alpha that will be here and here but not here. Fine? Now filling zero gives me the following diagram. Y, N, and A and P. Let’s check. Of course again Y enters N as usual and does not enter P—that’s always the case. What changed is only entries to A. We talked about that. So the only thing that changes is that N does not enter A, right? N does not enter A, aside from that it is the same thing. Okay? So for now the red is preferable. Wait. Now how did we decide this? Clearly there are direction changes here, right? This is two direction changes and this is one. Okay? So this diagram is decided by direction changes. Apparently the number of parameters is equal in both. Let’s do the calculation. We already did the calculation, doesn’t matter. The number of parameters came out equal in both, and the number of direction changes is decisive. What will happen now when we impose a constraint? When we impose a constraint, then basically in M and B there is another parameter, namely H, right? In B and Y? M and B. M and B. Because money and intercourse now have pleasure. Money and what did we say? Intercourse. We basically want this: we want money and intercourse to have pleasure. Now notice, this diagram is according to columns, not according to rows, right? What appears here are the columns, not the rows, but I think we can already guess how this needs to be built. It needs to be built so that the parameter will be here and here. How do I know? Because if it is here and here then it will be in M, because M has to contain it, and it will be in B because B has to contain it. So basically I’m saying that Y and P have to share the same parameter. Gamma, gamma, and here too gamma and gamma, and it must not be in A, in H. Meaning in M and B yes, and in H it must not. And one of them is contained within N. So this parameter is supposed, as it were, also to be in… Fine, but my requirement is a requirement on these. Addition of parameters, not on the vertices. In the end I want to get to the point where this gamma is in M and B but not in H. I’m saying, how can that happen? M, okay, okay. Fine? Yes. Because H cannot manage to contain either Y or P because it won’t have gamma. Right? That is exactly the point. If it had gamma it would do both of them. Okay? So through the columns I simply reconstruct how I’ll arrive at the correct result in the rows. Fine? So that is basically my constraint. In the end—you see, in principle one does this with equations. Meaning, you write alpha theta, say, alpha and also gamma and also theta, and here this is alpha. Wait, I got to… I’m doing H. I don’t know, I’m solving. Alpha—no, this is gamma. Right. So what came out here? I’m doing this—ignore the… I started with filling one. Okay? What did I really do? I said I want there to be gamma in Y and P, fine? Aside from that I now want to find a solution, provided that here and here there is gamma, and I need to check that gamma does not occur in H. Fine? After I find the solutions—short version, when you substitute everything in, there are all sorts of constraints on the variables theta and omega. Fine? And after deriving it, you get that theta is different from omega and both are nonzero, and therefore omega equals gamma and theta equals alpha. Meaning omega equals gamma and theta equals alpha. That is the resulting solution—sorry, omega equals beta. You have to add another parameter. Here is the parameter added because of the constraint. Omega equals beta and theta equals alpha. What, what? Then this basically means that here we have two alpha. Ah, okay. Theta too, two alpha. And this is beta. Fine? You see that this solution gives us gamma here and gamma here, right? And here we do not have gamma—not in A and not in M. Why does M need beta? What? Why does M need beta? There has to be beta there. If there were no beta then gamma and alpha would mean there would be a line from this to that. There would be no point calling it… as if the need for omega and theta was because… These are temporary variables and then in general… Yes, I simply solve the equations to find what thetas and what minimal theta and omega will satisfy the properties. Because there really is no difference between beta and omega, between us. You just added here… Yes, right, right. I could have written omega there, it doesn’t matter. Okay, so I mean I need a new parameter—that’s the point. Okay? Now look, and this can be done in the same way. Now look how I verify that I really solved the problem. I basically already verified it when I did the equations, but I’ll just show you. Look, now I want to build the model for this. It really matches the table. Let’s see. What about M? M succeeds in doing P and A, right? P and A means A has gamma alpha beta, right? It succeeds in doing P and A. Okay. H—you see how one goes from the solution to the columns; this is how I go to the rows, how I find the solution for the rows. Now H—H in filling one succeeds in doing N and A. That means it has two alpha. It does not have gamma, and that is exactly what I needed. I don’t care about beta, but gamma must not be there. That is a constraint. And B contains everything except P. So that means it probably has two alpha and gamma, right? Two alpha and gamma. Okay? You see the condition is met that here there is gamma and here there is gamma and in H there is no gamma. That was of course a guess because I went through the columns and tried to guess what would give me the results, because my constraint is on the rows. To constrain them. But I went through the columns and understand that if this is one then it controls that. And if this is… one then it controls this. So what I set in these two, I basically set these two, and here it has zero, so clearly here it will not have gamma because it will fail to contain either this or this. Okay? And therefore what happened here basically is that the number of direction changes was in favor of this. But since this is an objection, clearly the number of parameters here will now be smaller than here. Exactly, this could be done with fewer parameters. Okay? Let’s see. Without beta. So here there is gamma, two gamma, I don’t care, and here too there is gamma, and in N there is no gamma, as I expected. Now let’s see what comes out for us in the columns—if this really holds, then money contains P and A. P and A means two gamma, right? Chuppah does only marriage, so alpha. And intercourse does everything except P, so it has alpha and gamma. Okay? So again we see that the constraint is met: here and here there is gamma, and in chuppah there is no gamma. So that is fine, it is under the constraint. But within this constraint we got that the number of parameters here is smaller than here, while the number of direction changes here is larger than here, and therefore this is an objection. Meaning, what the constraint did was force me to add a parameter in the model, and that is how it refuted the inference. It is exactly the same diagram. Now in principle how should one continue the topic? Maybe one can take a picture. How should one continue the topic? You see, we are in the columns. The next thing in the column is divorce. Fine? Divorce is a column. But I always have to remember that there is one column missing here, the column of pleasure. And therefore all the solutions from here onward I always do under the condition that intercourse and money share a parameter and chuppah does not have it. All the solutions from here onward are to be done under that constraint, even though it came up at the beginning. So we enlarge the table and enlarge the table, and in the end we get a table of four by six, not four by seven, because pleasure will not be there. But that four by six has to be solved under a constraint. The constraint is that there be a pleasure parameter in money and intercourse but not in chuppah, and that it also appear there. What? That gamma appear, exactly. Fine? And when you reconstruct everything, it works. Okay, what I still want to do now is first, the point with which I opened earlier, I want to say a few more words about that, and with that we’ll really finish this topic. There are some more cases here, dayo and things like that, but that really starts getting us more tangled in the calculations. Let’s take the basic structure. Say we have something like this. Good, such a table. So we said that I can solve this in the following way: alpha, two alpha, alpha and beta. Right? But of course I can also solve it this way: alpha, two alpha, alpha and beta. That too is a solution. There is no preference for one over the other, right? Both solve the table, both are equally simple, and therefore there is no preference for one over the other. Why is this important? Because sometimes when we want to find the meaning of these parameters, as I discussed at the beginning, there will be a difference. Because now we will search—say this is betrothal and this is marriage and this is, I don’t know, something else. Let’s take a concrete example—why? An objection to the premise of a kal va-chomer, fine? We did an objection to the premise of a kal va-chomer. So we have P, N, and A. Yes, we have money and chuppah, marriage, betrothal, and redemption. Okay? Zero, one, one, question mark, one and zero. Okay? When we make the diagram of this, then again, in filling one, we get exactly such a diagram, right? This is A, and N, and P. Right? This is in filling one. In filling zero we basically get P and N and N disjoint. Okay? With no connection. Right, with no connection. Okay. Now let’s look at this. When we look at this, we can basically make the two solutions I made there. Two alpha, alpha, alpha and beta, and alpha and beta, alpha and two alpha, right? Or the green solution or the red solution. Both are possible. Where will the difference be? Look. In the red solution, P and A have the same single property, only with a difference of intensity. When I look at redemption and betrothal, I say both have the same property. So maybe it really is something that contains pleasure—what contains pleasure redeems. Only perhaps more pleasure is needed to redeem than to effect betrothal. Right? Then I say—and for marriage? For marriage perhaps one also needs some kind of union. Therefore one also needs beta, not only pleasure. Okay? So the red solution is very intuitive. I know how to identify who alpha is and who beta is. Now the green solution does not change the truth. Fine, in truth what is needed for redemption? Only pleasure. Only pleasure at a higher intensity. And for betrothal, a lower intensity of pleasure is enough. Okay? But now pleasure is alpha and beta. Pleasure alone—and this is marked as alpha and beta. There is no choice but to say that in this case alpha and beta together are actually the minimal thing—they are pleasure. And two alpha is the more complex thing. You know, after all, that we can construct concepts in all sorts of ways. Something that is both triangular and wet. Okay? So is that a basic property or a combination of properties? It depends on your conceptual world. You can say that being triangular and wet together is the basic property. And if there is something that is only triangular—ah, that is complex. It is composed of the property of being triangular and wet after removing from it—which is really a construction—after removing the wetness. Do you understand what I’m saying? This depends basically on the basis. The basis in which we lay out our space of properties can be a basis that is natural. In this case the green solution—the red solution—is the natural basis. But there can be an unnatural basis. And our need, in order to reach meta-halakhic answers, is only if by chance we landed on the natural basis. If we did not land on the natural basis, we will have a very hard time finding the real meaning of the matter. And then if they gave us the green solution and I asked you—forget it, you don’t understand, there’s no logic. I don’t know what betrothal is, what redemption is, what marriage is, I know nothing. I just want, from this pattern alone, to understand what alpha is and what beta is—you won’t manage. You will not be able to find it. Why? Because alpha and beta together are pleasure. And alpha alone is pleasure plus exclusivity, plus union. Okay? Because the basic concept is pleasure plus union; that is the basic concept. And if you remove the union from it, only pleasure remains. Then it turns out that pleasure is actually the complex concept. So if you take the complex concept as basic and subtract something from it, then you perform an operation on the thing in order to get the result. Do you understand? Look, in philosophy there is a discussion about this with what I said earlier about triangular and wet. After all, we have concepts that are compound concepts. Look for example at damage payments in monetary damages. A famous conceptual inquiry among the later authorities (Acharonim): why does one have to pay when his property causes damage? Is it because of negligence in guarding it, or because of responsibility? Responsibility—my property damaged, I’m responsible and I have to pay. Regardless, as it were, of what the guarding does. So now I can say this: I have the possibility that it is negligence, alpha. I have the possibility that it is responsibility—my responsibility for my property, right? But why not say that both are required together in order to obligate? And of course that is also true—you need both in order to obligate. Then I say you need both alpha and beta, right? I can treat it differently. “You need both” is my alpha—call it gamma so we don’t get confused, fine? Gamma means both negligence in guarding and responsibility for my property. When I want to build a thesis that is only responsibility, that will be a compound thesis. Take the basic thesis: responsibility plus negligence in guarding, and operate on it, meaning remove the negligence and then only the responsibility remains. Or remove the responsibility and only the negligence remains. So it depends very much on what is considered a basic concept in your eyes. If you had a person whose mind, for example, is built differently from ours—he, from his perspective, responsibility always comes with negligence in guarding. In his world it never appears separately. On the contrary, when he sees something that is only negligence without responsibility, he doesn’t recognize such a thing; that is something complicated, probably a composition of things. Understood? It depends very much on what seems simple to us or what seems basic to us and what seems complex to us; it’s a matter of definition. And therefore the same thing here. That is why the system is universal, but the interpretation is dependent on the conceptual system. And in our conceptual system, we will need to land on the natural solution in order to be able to give it meaning. Fine? If we do not land on the natural solution, then very high creativity is required to understand that it is not natural and to try to reconstruct from it the meaning in our own terms. And that is a problem, in the sense I said earlier. What? Doesn’t Occam work here? No, when they are equally simple. It’s simpler to say that… No, alpha and beta together are not less simple than alpha alone. Because alpha and beta together are a single parameter. Alpha or beta is more complicated. Alpha and beta together are no more complicated than… define alpha and beta together as gamma. Now what is needed is one condition, gamma. That’s it. What’s the problem? I can always define something that way, right? Suppose I say: to merit the World to Come, one must be generous. Now what is generous? If you want to break it down into pieces, it could be that generosity is a combination of several properties: that you have money, and that you are kind-hearted, and that you have empathy for the other. Together that creates generosity. So is “generous” a basic concept or a compound concept? It could be that someone says: in my world, generous is the most basic thing there is. I cannot decompose it. Give me only the property of having money—that is a compound property, having money. That is generosity without the other two things. Do you understand what I’m saying? So many times the question what seems simple to me and what seems complex to me depends on my conceptual world. And that is not an objective thing. What is objective here is the inference, or the quantity of parameters needed. You will not get below two parameters; no explanation will give you fewer than two parameters. But how to place the parameters on each column—you can do that in many ways, and not all of those ways will fit the natural interpretation. Many times you will see, in most of these cases, that when you try now to build the model and identify the parameters, you won’t succeed. We tried. Check at every stage in the topic, you’ll see. You won’t find what chuppah and money have in common that intercourse doesn’t have, say, if there is such a parameter. I don’t know, I can’t understand what they have in common. But it could be that in fact there is nothing that money and chuppah share without intercourse. That is a composition. It is some complex thing that exists in money and intercourse, and in chuppah there is half of it—except that the half itself is the complex thing in this language. Do you understand what I’m saying? Good—lukewarm response. Why does that make it harder to infer what the parameter is? No, it makes it harder to identify the parameter. I don’t know what the parameter is. Think—how would you identify the parameters in the green one? Go identify them for me. Very simple, right? Three columns. Identify for me the parameters of the green one. What parameter is shared by marriage and betrothal and is not in redemption? It is the… sorry, the opposite. Which one is in redemption and not in marriage and betrothal? Pleasure—there is also pleasure in betrothal because money also gives pleasure. Money also effects betrothal. You won’t be able to identify it; it is a composition that exists here and doesn’t exist there. Define that composition as parameter gamma—a composition of alpha and beta and delta, all that for me is gamma—and that gamma, that composition, exists here and here and not there. Do you see what I mean? If you look at simple parameters, you won’t find a simple parameter that exists here and here and not there. But if you look at this simple thing as a composition, you can decompose it and see that it exists here and here and not there. It depends on your conceptual world. So the conclusion is that from this whole table you basically can’t understand anything from what you got? Right. You won’t be able to decode the rationale of the verse. What I described at the beginning of the class was too optimistic—more optimistic than it should have been. Meaning, that is assuming we reached a model that fits the natural language, our natural way of thinking, our natural conceptual world. But the chance of landing specifically there is small, because there are many equivalent models and the solutions can be very numerous, especially in complex tables. So in backward reasoning we are likely to be able to build it? Right, in backward reasoning you go straight from the par… right, you will arrive directly at the natural solution. Exactly. Fine? Now let’s start returning the notebooks. Okay, fine, that’s it.