Topics in Talmudic Logic, Lecture 5
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Table of Contents
- The purpose of non-deductive logic and the justification of synthetic thinking
- A parametric theory behind kal va-chomer
- Tzad ha-shaveh as scientific generalization and dependence on law-generating properties
- The plan for building an arsenal of inferences
- The Talmudic topic of chuppah in Kiddushin as a demonstrative framework
- The course of the inferences in the topic: kal va-chomer, binyan av, tzad ha-shaveh, and refutations
- Tables, fillings, and parametric models
- A refutation as a demand for equivalence between fillings, and the debate over “valence”
- Using graphic diagrams to build models and criteria of preference
- Three topological parameters and a criterion of preference without tradeoffs
- Resolving the contradiction between the refutation of kal va-chomer and binyan av through topology
- An additional constraint: increasing levels in only one parameter
- Preparation for what follows: the three kinds of tzad ha-shaveh and deciding among them by the topological parameters
Summary
General Overview
The text builds a systematic logic for non-deductive inferences in the writings of the Sages, and assumes that behind those inferences there stands a theory of abstract parameters that generate the laws, rather than the laws themselves. It demonstrates this through kal va-chomer, binyan av, and tzad ha-shaveh, and shows how refutations do not prove the opposite, but rather cancel the preference between competing models. It then proposes a formal framework of tables, graphic diagrams, and criteria of preference among models, in order to explain the flow of the Talmudic discussions consistently and non-arbitrarily.
The purpose of non-deductive logic and the justification of synthetic thinking
The text opens by arguing that non-deductive arguments require a systematic logic that will justify synthetic thinking and turn it from something arbitrary into a systematic structure, similar to ordinary logic. It sets a philosophical goal of building a framework that does not depend on the private content of the examples, but on a formal schema that determines how conclusions are to be inferred.
A parametric theory behind kal va-chomer
The text argues that the structure of kal va-chomer and the way the Sages relate to it teach that the inference rests on a theory: a set of abstract parameters that make up the concepts under discussion. It presents a model in which torts and domains have a set of features (alpha, beta, and so on) that determine liability or exemption. In torts, more alpha expresses stringency, while in domains more alpha expresses leniency, and therefore the relation between rows and columns is reversed.
Tzad ha-shaveh as scientific generalization and dependence on law-generating properties
The text analyzes the logic of tzad ha-shaveh and concludes that it requires assuming a theory at the base of the inference, because the inference operates on properties that generate the laws, not on the laws themselves. It identifies tzad ha-shaveh with scientific generalization through the example of things falling to the earth, where one strips away irrelevant features (color, shape) and remains with the essential feature (mass), and shows how a case like a balloon requires sharpening the rule rather than abandoning the logic of elimination.
The plan for building an arsenal of inferences
The text declares that it is returning from kal va-chomer to a systematic construction of tools: analysis of kal va-chomer, binyan av, refutations of each of them, and then tzad ha-shaveh in three forms. It defines this as the first part and the main body of the work, after which the continuation is presented as straightforward.
The Talmudic topic of chuppah in Kiddushin as a demonstrative framework
The text chooses the topic of chuppah in tractate Kiddushin in order to develop the framework through a progression that becomes more and more complex. It explains that the question is whether chuppah acquires, meaning whether it effects kiddushin, within a two-stage structure: kiddushin as acquisition and marriage as completion, where marriage is not presented as having essential halakhic significance like kiddushin.
The course of the inferences in the topic: kal va-chomer, binyan av, tzad ha-shaveh, and refutations
The text presents the course of the Talmudic discussion as a chain of stages: Rav Huna learns that chuppah acquires by kal va-chomer: “Just as money, which does not complete, acquires, so chuppah, which does complete, should certainly acquire.” And that kal va-chomer is refuted by the claim: “What is unique about money is that sacred property and second tithe may be redeemed with it.” It brings “intercourse will prove it” as a route of binyan av, and that too is refuted by “What is unique about intercourse is that it acquires in the case of a yevamah.” It describes the move to “money will prove it, and the law returns… the common side between them…” and the refutation of tzad ha-shaveh, “What is common to them is that their benefit is great,” followed by the opening of a “new game” with a document, its refutation by “What is unique about a document is that it effects release with regard to a Jewish woman,” and a return to a broader tzad ha-shaveh from money, intercourse, and document. It concludes with the move in which the refutation “What is common to them is that they can operate against her will” is raised, and Rav Huna rescues the derivation by arguing, “At least with money, in matters of marriage, we do not find against her will,” because a unique property that is not shared by all the source cases does not topple the inference.
Tables, fillings, and parametric models
The text translates the inferences into tables of rows (modes of acquisition: money, chuppah, document, intercourse) against columns (results/laws: marriage, betrothal, redemption, acquisition in the case of a yevamah, great benefit, divorce, against her will). It describes filling-zero and filling-one as alternatives, fills the tables with zeros and ones, and builds for each filling a parametric model (such as alpha and two alpha, or alpha and beta) that explains the data. It prefers simpler models according to the principle of simplicity, and interprets refutation as a situation where there is no preference between fillings, not as proof of negation.
A refutation as a demand for equivalence between fillings, and the debate over “valence”
The text concludes from the refutation of kal va-chomer that the “number of levels” in which a parameter appears (such as alpha versus two alpha) is not supposed to be decisive, because a refutation requires equivalence between filling-zero and filling-one. It presents a difficulty vis-à-vis binyan av, where preference seems to depend specifically on valence, and formulates the tension as a contradiction proving that the definition of preference among models is still missing another component.
Using graphic diagrams to build models and criteria of preference
The text develops a graphic representation in which each column in the table is a vertex, and there are arrows when one vector “fits into” another, meaning when it is less than or equal in every component. It uses diagrams to force a parametric structure such as alpha, two alpha, or alpha and beta in branching situations, and emphasizes that stringency is qualitative rather than quantitative when there is no order relation between vertices.
Three topological parameters and a criterion of preference without tradeoffs
The text imports from graph theory three features: connectedness, number of vertices, and number of direction changes along a maximal path in the graph. It interprets direction changes as damage to the hierarchy that makes kal va-chomer and binyan av possible, and links this to the example of mezuzah and tzitzit, where there is no shared world of parameters. It defines a criterion of preference in which a diagram is preferable only if it is preferable or equal on all criteria, and if there is reciprocal preference in different directions, the result is a refutation, with no tradeoffs between considerations.
Resolving the contradiction between the refutation of kal va-chomer and binyan av through topology
The text presents how combining the topological criteria resolves the contradiction between the stage of refutation and the stage of binyan av without needing valence as a decisive parameter. It shows that where a refutation is required, equivalence emerges because one side has an advantage in connectedness or vertices while the other has a corresponding advantage, whereas in binyan av one filling gets a consistent preference because of an advantage in the number of vertices without parallel disadvantages.
An additional constraint: increasing levels in only one parameter
The text adds a heuristic assumption according to which, when building models, one rises in levels (alpha, two alpha, three alpha) only in one parameter, while other parameters remain in a single version, and if qualitative diversity is needed one adds a new parameter (gamma) instead of “two beta.” It justifies this by saying that a single quantitative hierarchy should be preserved along a connected graph, whereas different qualitative colors (beta, gamma) mark different branches with no order relation between them. It demonstrates this with two extreme graphs in which alpha determines the quantitative scale and the other parameters “color” different paths.
Preparation for what follows: the three kinds of tzad ha-shaveh and deciding among them by the topological parameters
The text states that there are three inferences of tzad ha-shaveh, and argues that it turns out each of the three topological parameters determines preference in a different kind of tzad ha-shaveh. It presents this as evidence that these three features must be included in the systematic logic of inferences, and closes with the promise that next time it will enter into tzad ha-shaveh and show how its three types receive preferences exactly according to those criteria.
Full Transcript
[Rabbi Michael Abraham] Okay, we’re in the topic of logic. I started with non-deductive arguments, I explained their significance and why it’s important to try to build some kind of systematic logic that deals with them, why this matters on the philosophical level—that is, to give some kind of justification for what I called synthetic thinking, to turn it into something that isn’t arbitrary, not a shot in the dark, but something built in a systematic way like ordinary logic. Then I began, I tried to show regarding kal va-chomer that the structure of the inference, and also the Sages’ attitude toward that inference—that they never just turn the kal va-chomer around anywhere even though there are refutations—means that at the base of the inference there is a theory. And theory here means, like in chemical analysis, some collection of parameters, some collection of characteristics, that build the concepts involved in the discussion. If we were talking about the ox that gores, tooth, and foot—those are the damaging agents—or the injured party’s courtyard and the public domain—those are the domains. The claim was that every damaging agent and every domain has some set of abstract properties—we marked them with Greek letters, alpha, beta, and so on—and those are really what are responsible for whether you are liable or exempt and so on. The damaging agents have properties that indicate their severity, or how much that damaging force really deserves liability; and with the domains, which were the columns in the example I gave, the parameters basically describe what has to be present in order to obligate you. That is, the bigger alpha is in the domains, that’s a leniency, right? Because it means that in order to obligate, you need something very, very severe to obligate—in other words, that’s a lighter domain. And the more alpha there is, five alpha, ten, I don’t know, the more units of alpha there are in the damaging agent, then it’s more severe. In other words, the relation works in the opposite direction between the rows and the columns. Therefore in kal va-chomer, for example, the structure of the kal va-chomer was alpha, two alpha, but here it was two alpha and alpha. The first one was specifically two alpha and the second was alpha, and that means this is more severe than that. Because from the standpoint of the columns, the smaller alpha is, the more severe it is. More severe means any damaging agent, even if it has only alpha, is liable, therefore that is a stricter domain. The damaging agent, the more alpha it has, the more severe it is. Fine. So it basically works in the opposite way.
[Rabbi Michael Abraham] What I then did was try to show from another angle why we need to assume that behind the inferences there stands a theory, and I did that through an analysis of the common side. I tried to show the logic of the common side, still not yet with parameters exactly like I did with kal va-chomer—we’ll do that today, I hope. And I tried to show there that you have to assume there is some theory at the basis of the inference. That is, the inference doesn’t operate on the laws involved in it, but on some properties that generate those laws. And if you adopt that conception—that is, if you adopt that there is a theory behind the things—then suddenly you understand the refutation called “a stricter side,” and the whole course of the Talmud in Ketubot and in Makkot that we saw last time. And along the way I also showed that the common side is really just a kind of scientific generalization. Just as we make a generalization from certain cases in science, we make a generalization from certain cases in Jewish law. I said that if there are these two things—this falls to the earth and this falls to the earth—and I want to know whether this too will fall, then I want to learn from that, so I say: what about this one, since it’s made of plastic? Then I say, that one will prove it. No, that one is also made of plastic. I don’t know—what about this one, since it’s green? Fine, I say, that one will prove it. Then I say: what about this one, since it’s round? I say, that one will prove it, since it’s rectangular like that one, box-shaped, like this. And the law returns: this is not like that and that is not like this, and the common side between the two is that it doesn’t matter what their color is and it doesn’t matter what their shape is, none of that matters—what matters? Their common side, that both of them have mass. So because of that, anything that has mass will also fall to the earth. Therefore scientific generalization is nothing other than the common side. Scientific generalization means finding a common side to all the examples we saw, stripping away from them the features—the unequal features, the irrelevant ones—remaining with the essential feature; that’s the elimination we do. And the claim is that that feature is responsible for the phenomenon we’re talking about, in this case gravitational attraction. I used the common side…
[Speaker C] Here you gave one parameter—for example, I take
[Rabbi Michael Abraham] a balloon, and you say
[Speaker C] that this balloon should also fall, but it flies upward. Okay. So in some sense that refutes your whole rule.
[Rabbi Michael Abraham] Right, right, we need to understand exactly what’s going on there. Does the balloon really fly upward? The answer is no, it doesn’t fly upward. It goes upward only because there’s air here. If there were a vacuum here, it wouldn’t go upward. And therefore, okay, it’s more complicated than what I described here, it continues further, but it’s all according to the same logic. You can continue the elimination further as well; it will continue in the same form. Of course I’m using basic generalizations only in order to demonstrate the logic.
[Rabbi Michael Abraham] What I want to do now is go back from kal va-chomer and start building this logical arsenal, these logical tools, in a systematic way. So just as we did with kal va-chomer, afterward I’ll do the same for binyan av, then I’ll examine the refutation of kal va-chomer, the refutation of binyan av, and after that I’ll do the common side, of which there are three kinds. That will be more or less the first part. And that’s the main work. From there on, it’s already just straightforward.
[Rabbi Michael Abraham] Good. So I want to start with a short reminder of what we did with kal va-chomer, just so it’ll be in the background. And I’ll begin by simply following a passage in tractate Kiddushin, the topic of chuppah. One of the more complicated sugyot in this sense in the Talmud, and therefore it’s useful to demonstrate things through it. There it develops slowly and becomes more and more complex, and I’ll try through that to show how this whole business works—to develop the logical framework along the sugya itself. So, the sugya begins with the discussion whether chuppah acquires. “Acquires” means effects kiddushin. In the creation of the marital unit there are two stages: there is kiddushin and there is marriage. Okay? What is called acquisition in this context is kiddushin. Marriage doesn’t have essential halakhic significance. So the sugya in Kiddushin wants to know whether chuppah can effect kiddushin. That is what is called “whether chuppah acquires.”
[Rabbi Michael Abraham] So first I’ll go over the sugya from above so you can see the structure from a bird’s-eye view, and afterward we’ll break it down into its components and go through it stage by stage. Rav Huna said: chuppah acquires by kal va-chomer. I’m skipping a little… Rather, the refutation is this: “Just as money, which does not complete, acquires, so chuppah, which does complete, should certainly acquire.” There’s something at the beginning that gets rejected, so I’m skipping it, it’s not important. But the formulation of Rav Huna’s kal va-chomer is this: “Just as money, which does not complete, acquires, so chuppah, which does complete, should certainly acquire.” “Completes” means effects marriage. Marriage is the completion, kiddushin is the beginning, that’s the acquisition, and after that comes marriage, which is the completion. Okay? So money does not effect marriage; it effects only kiddushin. So if money, which does not effect marriage, does effect kiddushin, then chuppah, which does effect marriage—
[Speaker C] Marriage
[Rabbi Michael Abraham] is effected by chuppah—certainly it should effect kiddushin, kal va-chomer. So that’s the kal va-chomer the Gemara brings. Now they bring a refutation against the kal va-chomer: “What is unique about money is that sacred property and second tithe may be redeemed with it.” Money has certain capacities that chuppah does not. For example, with chuppah you can’t redeem second tithe, but with money you can. Okay? Or sacred property. So because of that, apparently there is something in money that isn’t in chuppah, and then you can no longer learn from money to chuppah. Okay? That’s a refutation of the kal va-chomer.
[Rabbi Michael Abraham] Suppose the two damaging agents I’m talking about—like I said earlier—are money and chuppah, these are the two modes of acquisition I’m discussing, and the results are marriage and kiddushin. I’m already sketching the table; we’ll get to that later. So now they have basically added another column, which is redemption of second tithe. There is effecting kiddushin, effecting marriage, redeeming sacred property and second tithe. It’s like a column-refutation. Okay? So we dropped the kal va-chomer. Then the Gemara says: intercourse will prove it. What does that mean, intercourse will prove it? Intercourse acquires. And what does intercourse do?
[Speaker B] Both marriage and kiddushin, right?
[Rabbi Michael Abraham] So how does it prove regarding chuppah? By binyan av, not by kal va-chomer, right? Meaning to learn from money to chuppah. Intercourse accomplishes both kiddushin and marriage. Chuppah effects marriage. It’s reasonable—we make an analogy—that it should also effect kiddushin. Okay? Everything here is either yes or no. This is non-deductive logic. Everything is either yes or no. And then the Gemara says: “What is unique about intercourse is that it acquires in the case of a yevamah.” What does that mean? We took out—we added another column, right? We have intercourse and chuppah, marriage and kiddushin, and now acquisition in the case of a yevamah. A third column, right? Again, a column-refutation. We refute binyan av in the same way we refute kal va-chomer. So binyan av also falls.
[Rabbi Michael Abraham] Then the Gemara says the next stage, the fifth stage: we had kal va-chomer, refutation of kal va-chomer, binyan av, and refutation of binyan av, right? Fifth stage: “Money will prove it, and the law returns. This is not like that and that is not like this. The common side between them is that they acquire elsewhere and acquire here; so too I will bring chuppah, which acquires elsewhere and acquires here.”
[Speaker C] We learn
[Rabbi Michael Abraham] this by the common side, from intercourse and money together. Meaning, intercourse by itself didn’t manage to do the job, money by itself didn’t manage to do the job, but money and intercourse together will manage to teach about chuppah. That’s the miracle of the common side that I talked about last time. So at this point we already have a more complex derivation, with a comparison among three modes of acquisition, right? Until now we always spoke about two. We compared intercourse to chuppah, and chuppah to intercourse, and chuppah to money. Now we’re taking chuppah in light of intercourse and money together. We already have a table with three entries, yes? Three modes of acquisition.
[Rabbi Michael Abraham] Stage six: a refutation of the common side. “What is common to them is that their benefit is great.” This isn’t the refutation of a “stricter side” like I discussed in the previous lecture; this is a classic refutation of the common side. If I find something that is distinctive of both source-cases in the common side, that’s a regular refutation and the common side falls. Because if both source-cases have a special property and the target case doesn’t, it may be that this is the property that is really responsible for the law, rather than the property you originally thought. And because of that, you can’t learn from the source-cases to the target case.
[Rabbi Michael Abraham] Then the Gemara says: a document will prove it. A new game. We erased chuppah, intercourse, everything—we erased everything. The common side fell, we didn’t manage to do anything. Document, a new game. What happens with a document?
[Speaker C] A document acquires. Moving on.
[Rabbi Michael Abraham] A document doesn’t effect marriage, but it does effect betrothal, right? So it’s like money. So let’s make a kal va-chomer from document to chuppah, the way we did from money to chuppah, right? The Gemara says: “What is unique about a document is that it releases in the case of a Jewish woman.” A document effects a bill of divorce. Fine?
[Speaker C] Another column-refutation.
[Rabbi Michael Abraham] Yes, exactly. A document effects release through a bill of divorce, and therefore you can’t learn from document to chuppah. Chuppah doesn’t effect divorce. Then the Gemara says: money and intercourse will prove it, and the law returns. This is a common side from three source-cases. Money and intercourse are one side together, and document is on the other side. Okay? And document on the other side. So basically this is now the common side. So at this point we really have four modes of acquisition, right? Intercourse, document, money, and chuppah. And now we want to derive chuppah from the other three source-cases. This is a broader common side than a common side from two source-cases, or from three source-cases, or from two where one of them is doubled. And this is a common side between a common side and a document, which is a kal va-chomer—between the common side and kal va-chomer. Okay? This is a larger common side.
[Rabbi Michael Abraham] Then the Gemara says there is a refutation: “What is common to them is that they can operate against her will.” All three can operate against her will. So the whole thing falls. The Gemara says: and Rav Huna says, “At least with money, in matters of marriage, we do not find against her will.” Money doesn’t have the feature of against her will, so there is no refutation here. Why, if money doesn’t have against-her-will, does that save the derivation? After all, document and intercourse do have it. This is basically just another unique property that belongs only to one of the sides in the common side, and that never causes a problem, right? Because we know it is not the crucial property, since money doesn’t have it and nevertheless it acquires, right? Therefore we basically saved the common side. According to Rav Huna that is stage eleven. Okay? After that there is even some continuation, so this whole business is fairly broad. Meaning, there is a kal va-chomer that gets refuted, a binyan av that gets refuted, the combination of the two creates a common side, that too gets refuted. A kal va-chomer gets refuted; the kal va-chomer and the common side together make some large common side, all of it gets refuted, and then they change something else and rescue that common side back again. Okay? That is basically the structure of the sugya.
[Rabbi Michael Abraham] Now I want simply to go through the eleven stages of the sugya one by one and analyze them exactly with the same method I used for kal va-chomer, and I’ll try to show you how one analyzes every inference at whatever level of complexity it may have. It doesn’t matter. Here we’ll get up to a certain level of complexity, but you can go as far as you like. On what page? Thirty? Page 5? Yes. So I’ll start from the beginning. It begins with a kal va-chomer, and we already did that, so no problem. I’ll write some notation first—two readings of the text and one translation, okay? This is marriage, this is M, okay? In English, marriage. Betrothal, in that same English, is A. P is redemption, redemption of sacred property and second tithe.
[Speaker D] That’s acquisition in the case of a yevamah—wow, yes, yevamah in English.
[Rabbi Michael Abraham] Benefit—their benefit is great. G is divorce, yes, “releases in the case of a Jewish woman.” And K is of course against her will, in English. Okay? Those are the results, those will be the columns. What will the rows be? The rows and columns, yes, Baal HaTurim. And the rows—this works with lowercase letters.
[Speaker C] If this is money, H is chuppah,
[Rabbi Michael Abraham] S is document,
[Speaker C] B is intercourse, and W is document.
[Rabbi Michael Abraham] Specifically here for some reason we used the actual English terminology. Okay? Okay. Lowercase w, yes, in lowercase letters. Okay. So this is the notation, and what I want now is to go through the stages of the sugya slowly. So the first stage of the sugya is the first kal va-chomer. The first kal va-chomer we already did, so I can be brief. Here we have marriage, betrothal, money, and chuppah. Okay? The filling of kal va-chomer we already know. A question mark. Right?
[Speaker E] What
[Rabbi Michael Abraham] we do is fill it with zero and one, check the—build the model—and we basically get that the preferred filling is one. Okay? The filling is one. So look, what I’m going to do is do the kal va-chomer again, to show how this works. So I’ll do this: in filling zero, I have M and A, zero one, zero one; this is M and this is H. That’s how I filled it here, okay? And in filling one, it’s one zero, one one. M, H, M and A. Okay? Those are the two tables, and I need to build a model for each of them. We already did this, so I can already know that this is alpha and this is two alpha, and this is two alpha and this is alpha, right? If this is alpha, then it can’t effect marriage. Money isn’t strong enough to effect marriage, because for marriage you need the power of two alpha, and money has only the power of alpha. Okay? But for betrothal—
[Rabbi Michael Abraham] and for betrothal, yes, alpha is enough, and therefore money manages to do it. With chuppah it has the power of two alpha, because in fact it does effect marriage—but in the filling, sorry, in the filling, sorry, no, that’s below, in the lower table. In filling zero, here alpha and two alpha, then of course it also manages to effect betrothal, for which alpha is enough, right?
[Speaker C] What’s the filling here?
[Rabbi Michael Abraham] So we already did that too. Say this is alpha and this is beta and this is alpha, this is beta and this is alpha, right? Why is this the correct answer? Simple. Because here there is one parameter, true, but in two levels, alpha and two alpha, and here there are two parameters. This is the simpler model, so I prefer filling one.
[Rabbi Michael Abraham] Now I want to explain how we find—well, before I explain that actually, now I want to move on to the next stage. We do a refutation. Okay? So stage two: we have a refutation. Basically it’s the same table, I’m just adding a column.
[Speaker C] Redemption.
[Rabbi Michael Abraham] Right. Redemption—second tithe and sacred property.
[Speaker C] Zero,
[Rabbi Michael Abraham] one, one, question mark, zero, one, right? Money effects redemption, chuppah doesn’t effect redemption. Okay? Now once again, we repeat the exercise. Filling zero. Last time I didn’t do this and now I am.
[Speaker C] Alpha and two alpha won’t help here.
[Rabbi Michael Abraham] Okay? These are the two columns. Now I need to find the model for these two tables. This is already a bit more complicated. The upper one is simple. Why is the upper one simple? Because basically we have here two identical columns.
[Speaker C] This
[Rabbi Michael Abraham] is basically exactly the same thing as this, right? It will have exactly the same structure as this, the same properties. So clearly the solution is alpha beta, beta alpha and alpha, right? By definition that will be an explanation. What happens here? Here it’s already more complicated.
[Speaker C] Here it doesn’t
[Rabbi Michael Abraham] let you do alpha and two alpha. So what we can do is exactly the same thing: we do alpha, two alpha, and this is two alpha alpha, and now this thing is identical to that, so here too two alpha.
[Speaker C] Why? Because P is like L.
[Rabbi Michael Abraham] P is exactly like L, right?
[Speaker C] Then it doesn’t work. Ah, right. Right, now it doesn’t work.
[Rabbi Michael Abraham] Something here doesn’t—exactly—
[Speaker C] That’s no good anymore.
[Rabbi Michael Abraham] Here it has to be beta, which is independent of alpha, say, or something like that. So I’ll formulate it—okay, here you can see it’s already starting to get complicated visually. So I’ll show you how to do this systematically, okay? How do you do it systematically?
[Rabbi Michael Abraham] The point is this: what we need to do is build a tree—this comes from graph theory. We build a tree where every point on the tree is a column in the table. We begin by establishing an order relation between columns. That is, say between these two columns for example: this thing contains this thing, is greater than or equal to this thing in every entry. Say if there were columns of length four or five or six, it doesn’t matter: if it holds that in every entry this is either greater than or equal to that, then from my point of view—it fits into A; that’s how I mark it. Do you understand what I’m saying?
[Speaker G] Not really.
[Rabbi Michael Abraham] Again. I take—I make a diagram here. I call this P, and this I call A, and this I call N. Okay? Now I mark what the relations are between the columns. So look: P fits into A, agreed? A is greater than or equal to P, right? So that means P fits into A. That’s how I mark it, notation. Okay? You’ll soon see why this is useful.
[Speaker C] N fits
[Rabbi Michael Abraham] into A.
[Speaker C] Wait, A certainly
[Rabbi Michael Abraham] fits into A, it’s there. But N—N also fits into A, right? And the relation between P and N? There is no relation. So they stay like that; there’s no arrow between them. In fact two points with no connection are two points with no arrows between them. If there is a connection, one contains the other, then the arrow goes from the smaller to the larger. Okay? The smaller fits into the larger—remember it through that metaphor.
[Rabbi Michael Abraham] Now, the way to solve it, to find the model, is very simple. This is a heuristic. There was someone who did a master’s degree on this and he proved a few theorems about it, but we still don’t have a closed algorithm that takes a given table and extracts from it the minimal model. But heuristically you do it this way. Suppose we start with A as alpha—I’ll mark it in another color—suppose A is alpha, then P has to be bigger than alpha, right? So P is two alpha. And I remind you that the less severe one has more alphas, because I’m talking about the columns. Okay? So if P fits into A, then P will have greater power than A. Greater power means less severe, in the columns. Okay? So if this is alpha, this is two alpha. What will this be? There’s also—
[Speaker C] Two? No. Alpha…
[Rabbi Michael Abraham] It can’t be beta, because there is an arrow between them. Alpha and beta don’t talk to each other. It can’t be two alpha, because then there would also have to be an arrow here.
[Speaker C] So we have to say beta on top of alpha, and then that’s two beta?
[Speaker G] No,
[Rabbi Michael Abraham] not two beta, rather this will be alpha and beta. Right? Alpha and beta is stronger than alpha, just as two alpha is stronger than alpha. So it fits into this, but it doesn’t talk to this. You see? Between two alpha and alpha-plus-beta there is no simple relation. If it were alpha-one, for example, then there would be a simple relation, because alpha-plus-beta is stronger than that. But between these two there is no simple relation. So whenever you see a branching point like this—and this is a very basic node in these diagrams—the way to start is to mark some alpha here, it doesn’t matter which parameter, and go backward, build it in such a way that each time you move to something greater, but you take care that it maintains the correct relations with the other points, or lack of relations. Okay? So at a point where, say, if there were here for example another—if there were a table like this, for example—what would we say? Three alpha, right? That also maintains the relation between this and that? The answer is yes, right? This and that don’t talk to each other. This isn’t more severe than that and that isn’t more severe than this. They don’t talk to each other. But if there were an arrow here, if there were an arrow here for example, then what would you say?
[Speaker C] Then it’s no longer alpha.
[Rabbi Michael Abraham] It’s not beta.
[Speaker C] We’d have to say that it’s also beta.
[Rabbi Michael Abraham] Three alpha and also beta, right? Because it has to be both more severe than this, fit into this, and fit into this. Okay. Therefore you can go backward from the lenient to the severe. These are basically steps of a fortiori that we’re constantly making here. Okay? We’re simply building a fortiori. Just as a fortiori leads us from alpha to two alpha, from the severe to the lenient, here too we are simply building the whole diagram through steps of a fortiori.
[Rabbi Michael Abraham] Okay. So now if we return to the refutation, to the a fortiori, then this is the diagram of this, right? What is the diagram of this? I did it in the wrong color. Just a second, fine. For the sake of future generations, I’ll do it in the same color as the table. This is A, this is P, and this is L. Okay? So this fits into that and that fits into this, and now I build the model like this. Say alpha, two alpha, alpha plus—
[Speaker C] Beta, alpha beta.
[Rabbi Michael Abraham] Okay. What does beta mean? A parameter that plays on the field. You won’t manage to explain all this data using one parameter.
[Speaker C] But I see that A is a vector that is a composition of B plus N, so how can A not have beta?
[Rabbi Michael Abraham] No, no, that’s exactly it—there is no plus relation here. There are no plus relations here.
[Speaker C] That doesn’t mean one-one?
[Rabbi Michael Abraham] No. It only means more severe, but apparently it also has another component. The severity is not quantitative; it is qualitative. If it were quantitative, then there would be a relation among all the—there would also be an arrow here.
[Speaker C] No, P and N don’t need an arrow.
[Rabbi Michael Abraham] Exactly. Therefore that’s what proves that you can’t write the severity of this one with a single parameter. Understand? In this case, for filling one, you can. That’s what I did here. Meaning, what do you mean “you can”? It means you have alpha and beta, and here it’s two alpha and alpha. So the relation between these two is the same parameter. But you won’t be able to write this here also as two alpha or three alpha or however much you want, even though that would explain this arrow. But it wouldn’t explain the lack of connection here. Fine?
[Rabbi Michael Abraham] Now if I make the corresponding diagram here, then let’s do it in exactly the same way. We basically have A, P and A, and here we have N. Right? It’s the same thing. We mark P and A in the same way; there is no arrow between them. Right? No arrow between them because there is no relation. Those are two independent things. So how do we build the model? Nothing. Say this is alpha, then this is beta. Okay?
[Speaker E] There’s nothing between them.
[Rabbi Michael Abraham] Right? There’s nothing, so they don’t talk to each other. This has an alpha component, this has a beta component. They operate on different planes. There is nothing between them. Okay? What this one manages to do, that one doesn’t manage to do, and vice versa. Are you with me? Okay? Chuppah manages to do this but not that. Marriage manages to do that but not this. So now I ask myself which model is preferable. The red one.
[Speaker C] There’s no nature to this—it’s like two parameters. The red one. Why? Because here there’s alpha, and with beta there’s a connection between the parameters.
[Rabbi Michael Abraham] No, there’s no connection. Beta is independent of alpha.
[Speaker C] No, beta, okay.
[Rabbi Michael Abraham] There is a connection between the vertices, not between the parameters. The parameters are independent.
[Speaker H] Actually the blue one, because it really is much simpler.
[Rabbi Michael Abraham] Only two parameters, no connection. The blue one has two parameters—before connection. The connection we haven’t yet—at this point the diagram serves me only to build the model. I’m not looking—we’ll get to properties of the diagram in a moment.
[Speaker C] It won’t get you anywhere.
[Rabbi Michael Abraham] Seemingly that’s simpler.
[Speaker C] But it won’t get you anywhere.
[Rabbi Michael Abraham] No connection to what?
[Speaker C] It brings you.
[Rabbi Michael Abraham] P and A have a parameter. You’re speaking in words I don’t understand. There’s mathematics. Meaning, this thing has two parameters. This thing also has two parameters. That’s all. So I manage to explain both models with two parameters, both this table and that one. But here one parameter appears in two levels. Right. And here it doesn’t. So seemingly I would expect that this thing is simpler. No—the reverse. This is more complex, this is simpler. The upper one is simpler. Right? This is simpler.
[Speaker C] No, certainly. After all, the fact that here there’s…
[Rabbi Michael Abraham] Again, you’re not speaking the right language. No words. There’s mathematics. I built the model: here there is alpha and here there is beta, that’s what I built. Now I can erase it because I found the answer. Okay? I’ve forgotten the diagrams, okay? I only used them to build the model. So let’s write: N is beta, and alpha, and this is alpha, and this is beta. Right? That’s the property. I complete this, I extract those from the diagram, and this I complete already. Something simple. Now this—P is two alpha, A is alpha, and this is beta.
[Speaker C] Now this: two alpha and alpha, and alpha and beta. Right? That’s more complicated.
[Rabbi Michael Abraham] Now you understand. Now leave that aside; we’ve forgotten about it. Okay? This part is more complicated. We shouldn’t let it confuse us. That’s it — it only served me in order to solve the… to find the model. Okay? Now I ask myself: which model is simpler? The upper one. Seemingly, the upper one. Because in the upper one there are only alpha and beta, and here there are alpha, two alphas, and beta. So seemingly it’s simpler. But we’ll also see later that the number of levels of alpha probably doesn’t play a role. Or if it does, then it’s minor. Right. Now look, an important point: if I were to assume that it does play a role here, then it would come out that this model is the preferable model. Right? And that’s filling in zero. But a refutation doesn’t mean that filling in zero is preferable. A refutation means there is no preference for filling in zero over filling in one, right? We didn’t manage to prove it. So already from here you can conclude that apparently the number of levels in which one of the parameters appears — meaning, on how many levels it appears — doesn’t play a role. What matters is only the number of parameters. Later on maybe I’ll comment on this; it could be that it plays some minor role, but it’s not a significant role. As far as we’re concerned, these two things are supposed to be equivalent. That’s an indication that the number of levels in which each parameter appears isn’t important.
[Speaker C] And why are they supposed
[Rabbi Michael Abraham] to be equivalent? Because that’s what refutation means. A refutation means that you can’t decide whether to fill in zero or fill in one. A refutation is not a proof that the opposite is true. When you prove something by an a fortiori argument, you prove that a wedding canopy acquires. In a refutation, you have not proved that a wedding canopy does not acquire; rather, you refuted the proof that a wedding canopy acquires. So that means filling in zero and filling in one are supposed to come out equivalent. That’s the logical meaning of a refutation. I’m saying — without getting into the question of what really happens with the wedding canopy and all that — exactly what Aristotle did for logic: I’m trying to examine the schema without assuming anything about the substantive components that appear through it. And in the schema, a refutation has to give me equivalent results for filling in zero and filling in one. Okay? And therefore the first conclusion we can already draw is that apparently the number of levels at which a certain parameter appears doesn’t play a role, certainly not a significant one. Okay. We move on. That was about the a fortiori argument and the refutation of it. Now the next step — that was number two. Number three is an inductive paradigm. Okay. Now we move to step three. Step three — so we erased that. There’s a refutation; we didn’t succeed in learning wedding canopy from money. The next step: we build a table. Wedding canopy here, and we learn it from intercourse, right? From intercourse. Now this is marriage and betrothal. Intercourse does this one, this and this. The wedding canopy does this, and the question is what happens here, right? Let’s go back again over the whole story.
[Speaker C] Zero and one. Yes.
[Rabbi Michael Abraham] We have H, B.
[Speaker C] Zero and one, N, I. Wedding canopy effects betrothal. Okay?
[Rabbi Michael Abraham] Now what am I doing? Again the… Here it’s actually fairly easy; I don’t even need the diagrams. We’ll do it just for practice.
[Speaker C] We already solved this, right?
[Rabbi Michael Abraham] This is exactly that. Just switch the role of the rows and the columns — the two rows and the two columns, right? So basically we already know that this will be alpha, two alpha, two alpha, alpha. Exactly like the table of the a fortiori argument, right? But let’s do it just for sport. So basically we have I and N. Sorry. Wait — you’re not doing the… no… the upper one. We have I and we have N, where I goes into N, right? And then we know that this is alpha and this is two alpha, right?
[Speaker C] So now we have two alpha and alpha.
[Rabbi Michael Abraham] If we fill in here, then of course this is two alpha and this is alpha. Right? You see that what we got is exactly this. Where? Exactly this, but reversed. Reverse the two columns and the two rows; it’s the same thing. What we’re doing below.
[Speaker C] Why didn’t you do it… what? It’s really…
[Rabbi Michael Abraham] Because I always want the thing I want to learn to be here. That’s the convention, but it’s just that; it doesn’t matter. This diagram is much simpler, right? It’s two identical points, I and… yes, this is alpha. This is alpha, this is alpha, this is alpha, this is alpha, this is alpha. Okay? Now notice: the inductive paradigm is supposed to be — we’re now in a tangle. I’m taking you through the stages, where we got stuck each time at the stage where we got stuck. Because what is the result supposed to be in an inductive paradigm?
[Speaker C] It has secondary — this, this
[Rabbi Michael Abraham] It’s supposed to be a preference for filling in one, right? After all, the inductive paradigm is a proof that here we should fill in one; meaning this is supposed to be preferable. Right. Now why is this preferable to that? Numerically. Only numerically, right? Because here alpha appears on two levels and here on one level. And I remind you that here we saw that what we may call the valence — we call it the level of values, the quantity of values of alpha that appears in the matter — doesn’t matter. Right? So we have a contradiction between the refutation of the a fortiori argument and the inductive paradigm.
[Speaker C] So in the inductive paradigm it does matter.
[Rabbi Michael Abraham] We’re in trouble. For the moment, in trouble. In fact, we did this in the reverse order, not in the order of the Talmudic passage. We solved the a fortiori argument, we solved the inductive paradigm, and we reached the conclusion that this really is preferable — sorry, that this is preferable because there is valence here and valence also plays a role. But then we got stuck when we got here. Okay? And then we didn’t know what to do; we’ll soon see. At the stage when we thought this, at the stage when we thought valence played a role, it was still fairly clear to us that valence, even if it played a role, would probably play some kind of minor role. It wouldn’t play a significant role. So maybe that’s why, for example, an inductive paradigm is considered weaker for us than an a fortiori argument? Just by simple intuition: an inference of inductive paradigm is weaker than an inference of a fortiori argument. Why? So I’m saying, translated into our terms: because in an a fortiori argument the preference is significant — here it’s two parameters versus one. Here the preference is only on the level of valence, and in both cases it’s one parameter. It’s just that one is two-valued and one is one-valued. Okay? So it’s more minor. That’s what we thought at first. Again, I’m saying this is about to collapse. Because of this. Here we see that this thing doesn’t play a role — alpha versus two alpha. So for now we’re left with a puzzle. We don’t know what to do. Okay? But let’s continue as if we know what to do, and then we’ll solve the…
[Speaker C] How did you do three before two? I don’t understand. How did you do three before two? You should have done it in serial order, three and then two. Why?
[Rabbi Michael Abraham] I didn’t understand. The order of the Talmudic passage is one, two, three. But the logical order is before the refutations. First I do the a fortiori argument, then the inductive paradigm, then the refutation of the a fortiori argument, then the refutation of the inductive paradigm. Yes, obviously. Okay? That’s the order it appears in the Talmud.
[Speaker C] Here, look — three, wait, this is four.
[Rabbi Michael Abraham] The refutation, the refutation of the inductive paradigm, tells us that it acquires in a yevama.
[Speaker C] Acquires in a yevama — we have Y,
[Speaker E] I, N, P, H.
[Speaker C] This
[Rabbi Michael Abraham] is a sign that this is one.
[Speaker C] This acquires in a yevama — one, one.
[Rabbi Michael Abraham] And the wedding canopy does not acquire in a yevama. Right? There is no wedding canopy. Now again the two fillings — it does not acquire. Okay. Now we have to solve this in order to see that this really is a refutation. So let’s do our diagrams. Let’s start with this. What do we have here? This is a diagram we already know, right? It’s like the refutation of an a fortiori argument, just reversed. Okay? So we take — meaning it’s not that, not a fortiori and not a refutation of an a fortiori argument. On the contrary, it’s an a fortiori argument. So we have Y and I going into N, right? That’s the diagram. Y and I are the same thing, going into N, so we know this is alpha and two alpha. Okay?
[Speaker C] What does “goes into” mean again?
[Rabbi Michael Abraham] That it’s bigger.
[Speaker C] Equal or bigger?
[Rabbi Michael Abraham] Equal or bigger. Meaning Y and I are smaller, in each component separately, than this column, than this vector. Okay? If in each of them it is bigger or equal, then that means it contains it. There is an order relation between these vectors. Now, okay, so basically we’ve already solved this, so this is alpha, this is alpha,
[Speaker C] two alpha.
[Rabbi Michael Abraham] Wait, no, this is two alpha, sorry. This is alpha; B is two alpha, alpha; and H is alpha, right? Let’s now solve this. So now we have I — it’s actually the same thing, right? You can already see that it comes out immediately. Y goes into I and I, right? Exactly the same diagram. We have no problem at all. So this is two alpha and this is alpha, right? So I and N: alpha, alpha, two alpha, two alpha, alpha. Right? Here we got it smooth, right? They’re exactly identical. And this is a classic refutation. A classic refutation says there is no preference at all for this filling over that filling, right? Everything is fine; it’s exactly the same kind of model.
[Speaker C] Now we are
[Rabbi Michael Abraham] only in trouble with these two, with two and three. Why? Something is wrong with our algorithm. Because we expect that here there will be a preference for red over blue, and here there will be equivalence between red and blue. Right? A refutation means equivalence. Okay? But in both cases the difference is only at the level of the number of values that alpha gets. So you have to decide one way or the other. This is proof that apparently this model is not complete. That is, in the definition of the preference of models, we need to add something else. And then we started thinking what to add. And here something very nice happened. We started thinking, and that’s exactly what you were saying in your comments while I was sketching these models — we began to look at diagrams, and also more complicated diagrams, we’ll encounter them shortly. And we began to think that perhaps the structure of the diagram could also be an indication of simplicity. Some diagrams are simpler than others. Now graph theory deals with this kind of characterization of diagrams, these tree-like diagrams. No — actually they’re not trees, because there are also closed paths here. And there they distinguish among three main parameters. So we went to graph theory and asked ourselves: what are the characteristics attached to a given graph? What distinguishes a graph? We found three parameters. One parameter is connectivity. For example, this diagram, this one — it has no connectivity, right? There’s one here and one here and they’re not connected to each other. That’s a diagram that isn’t simple. Not simple in the sense that there are things here that have no relation to each other; you can’t infer from one to the other. Is it not a function? What? I wouldn’t call it a function, but there’s no relation between them; they belong to different conceptual worlds. So connectivity sounds reasonable as a parameter of the simplicity of a diagram. Okay? Another parameter is the number of vertices in the tree — or not in the tree, in the graph. Okay? Because, for example, we saw that in the last one — the lower left red one, or also this blue one above it — there are two vertices that are identical. In the tree there are really only two vertices, not three. Right? That’s an advantage. It’s simpler. A tree with three vertices or four vertices is more complicated than a tree with two vertices. Okay? The number of vertices is part of the complexity of the graph. And the third parameter is the number of direction changes. When we travel along a path in the graph — we’re talking now about more complicated graphs, which we’ll meet shortly — let’s say there’s some I here that goes into N that goes into P and this also goes into Y and this goes into K and this goes into here. Okay? A diagram like that. Or if you want even something else here and B. Okay? Whatever you like; it doesn’t matter. So what is the number of direction changes? When I choose two points, say this one and this one, and I ask myself what the path between them is — then there can be, say, this path, there can be this path, right? How many direction changes do I go through along the path? So every time the head of an arrow meets another head, or the tail of a tail meets a tail, that’s a direction change. Right? And if here, for example, I go along this path, then from here to here there’s no direction change, right? The arrowhead meets a tail. But here there is a direction change. Okay? So from here to here there is one direction change. But if I go like this… the maximum. We choose the path along which there is the maximal number of direction changes. Okay? That’s the third parameter. Now why direction changes — this is maybe the less intuitive part — why does this topological parameter, this is topology of networks or of graphs, why does this parameter represent complexity? In our context it’s very clear why. Because after all, an a fortiori argument, or an inductive paradigm, basically says that there is something shared by the source and the target, and therefore you can learn this from that. Right? If this is more severe than that and this is more severe than that, then this too is more severe than that. Now when there are direction changes, that doesn’t hold. This is more severe than that, this is more severe than that, but here it could be that this won’t be more severe than that. The arrow changes direction, so when you make a full turn you encounter a direction change. That means the hierarchy, the a fortiori argument or the inductive paradigm — the directionality in the graph is not preserved. It’s a more complicated graph, or a weaker inference. Okay? You can’t learn from one to the other if there isn’t a simple hierarchical relation between them. In an a fortiori argument — you may remember that I mentioned this a fortiori argument about the obligation of the doorpost in a mezuzah. Remember that matter? “If a four-cornered garment, which is exempt from mezuzah, is obligated in fringes, then a doorpost, which is obligated in mezuzah, is it not all the more so obligated in fringes?” And therefore every doorpost must have fringes. Where does that even begin?
[Speaker C] Okay? Why? Because there’s no relation between the parameters.
[Rabbi Michael Abraham] What do you mean there’s no relation? Seemingly it’s an a fortiori argument like any other a fortiori argument. Except what? Our intuition says — and by the way it’s not only intuition, but never mind — our intuition says that here we are comparing two things that don’t operate on the same…
[Speaker C] They
[Rabbi Michael Abraham] are not characterized by the same parameters. Let’s speak in that language. Meaning, the properties of one — the properties relevant to obligating something in fringes are not relevant, they’re completely different from the properties relevant to obligating something in mezuzah. There’s no relation between these two things, so you can’t learn from one to the other. Okay? Basically that means they won’t preserve hierarchy. If this is more severe than that in the sense of the alpha axis, that still doesn’t mean there’s some beta axis in terms of which it is less severe than it. Right? There’s no simple hierarchy. You can’t learn from the hierarchy between these two here to their hierarchy here. You can’t make an a fortiori argument or an inductive paradigm. Therefore the number of direction changes is a parameter that clearly says this is a weaker inference, that from our standpoint the graph is less preferable. So into the preference criterion we now add another requirement, and I define the preference criterion as follows:
[Speaker C] Fewer vertices dried out and less directionality. Until
[Rabbi Michael Abraham] you photographed the degrees of freedom, the intermediate degrees? The preference criterion asks when one diagram is preferable to another.
[Speaker C] Is this already worth photographing?
[Rabbi Michael Abraham] In a moment, I’ll just write it, although there were already things I erased, so we’ll see. The preference criterion goes like this: number of parameters — I’m giving up for the moment on valence; in a moment I’ll show you we no longer need it — connectivity, direction changes, and number of vertices. Okay, these three are the topological parameters, the properties of the graph. Here this is only the model that I built by means of the graph, so how many parameters there are.
[Speaker C] Direction changes and number of vertices — is that the same thing? No.
[Rabbi Michael Abraham] Once the vertices meet, they don’t merely meet; they become identified. This graph, for example, has two identical vertices, right?
[Speaker C] A and N, Y goes into A and N.
[Rabbi Michael Abraham] A and N are the same vertex. So how many vertices are there in this graph? Draw a circle around each vertex. One. Two. Even though there are three columns, there are three columns but only two vertices. So that’s the number of vertices. It has nothing to do with direction changes. Direction changes are when the head of an arrow meets another head or a tail meets a tail. Now what I need to do: if there is a diagram that is preferable in the number of parameters but inferior in terms of connectivity, or also in terms of direction changes — how do I decide? What do you suggest? What seems reasonable to you?
[Speaker D] Parameters before topology. What? Parameters take precedence over topology — that’s how it sounds.
[Rabbi Michael Abraham] What do you say? Agree? Intuitively I would say — in the end that’s also what worked — but intuitively I would say no. I would say: if the preference is in all the parameters, then the preferred graph is the one selected. If there is one that isn’t, then that’s already a refutation. Right? A refutation — after all, in order to show a refutation there are no offsets. Suppose in an a fortiori argument I showed that the wedding canopy is preferable to money, right? Because it effects marriage. Okay? And suppose I found a refutation. Okay, so let’s see whether the preference overcomes the refutation and maybe we can still make the a fortiori argument? Right? No, we don’t do that. Why not? Because once there is one side for stringency and one side for leniency, you can’t infer. You can’t know what is more and what is less. The same idea, I say, applies here as well. I say that if there are four criteria of preference, okay? If in all of them one diagram is better than or equal to the second, then it is preferable. If there is one yes and three no, one yes and two no, whatever it is — you cannot infer a conclusion. I think that would also have been my initial intuition. It also said that; that’s what we thought from the outset. Okay? That’s much more sensible than starting to think what is more important. What is more important depends on the context; you can’t draw conclusions. And I remind you that we are trying to build something universal here, that won’t depend on the content of the graph — whether we’re talking about a wedding canopy or money or betrothal or tort law. I want to infer conclusions from the structure alone, exactly as Aristotle did with logic, okay? So I’m looking for something that seems to me not dependent at all on the contents that populate this formalization. In other words, the formalization should give me the answer. That’s what I’m looking for. Okay? Therefore I say: this is the preference criterion. A diagram is preferable to another if it is preferable in all the criteria. Preferable means either better or equal, okay? If there is preference to one side and preference to the other side, I don’t care in how many parameters this way or that way — that is a refutation. Meaning you cannot prefer one diagram over the other.
[Speaker C] Is there another field where they do this counting of direction changes in graphs? Topology. Not in some kind of problem you happen to know?
[Rabbi Michael Abraham] It’s a property of graphs. I don’t know exactly what they do with it there, but they do things with it there, yes. I’m simply saying: we opened books on graph theory, looked for the parameters that characterize a graph. We found that it’s mainly these three. We said okay, these three are interesting; let’s use them. Now it turns out this does two things. And that’s the wonder of it — it really strengthened the… because once you guess something and in the end discover that it works, you get positive feedback on what you guessed. We didn’t build everything ad hoc. We assumed it sounded reasonable that this should be so, and suddenly we discovered — amazing. First, it solves for us the contradiction between two and three; you’ll see in a moment. Okay? This criterion gives you the right answer for all the graphs. There is no longer any contradiction between two and three; I’ll show you in a moment. Second — and this is even more amazing — I told you there are three types of “common denominator” inference in Jewish law. A common denominator built from two a fortiori arguments, a common denominator built from two inductive paradigms, and a common denominator built from an a fortiori argument and an inductive paradigm. And of course the reverse — an a fortiori argument and an inductive paradigm — is the same thing, just swap their roles. Okay? Three. Now it turns out that in order to prove preference in each of the three types of common denominator that exist in the Talmud, one is decided by connectivity, one by direction changes, and one by number of vertices. That is clear proof that these three are the determining parameters — at least these; maybe there are more. But it proves that these three certainly have to be included. Because that is exactly what creates the preference in each of the types — we’ll see this shortly. I’m just showing you why there is confirmation for our intuitive assumptions here. Okay? We got confirmation after we tested them. All right? Now let’s try to test this. First of all, let’s see that it solves our problem here. Now I completely ignore valence; I’m not interested now in the number of levels of alpha in which it appears. Okay? So I say like this. Here we already know, right? The two diagrams are… wait, let’s do it systematically. Take a picture, but I’m the director. A and M — that’s with filling in zero, right? With filling in one, now you can photograph a full board. With filling in one it’s N going into A. Okay? M goes into A. Let’s make the diagrams here. Here we have P and I versus N, right? And here we have P, I, N, both going into I. Okay?
[Speaker C] Right,
[Rabbi Michael Abraham] we did this earlier; I’m just summarizing here so that… okay? Good. Now three. So the blue is like an a fortiori argument: I goes into N, and the red is simply a point.
[Speaker C] Okay?
[Rabbi Michael Abraham] Here we have Y and I going into N, and the red is
[Speaker C] Y goes into I and into N.
[Rabbi Michael Abraham] Okay? These are the four diagrams, right? Now, this is alpha beta; alpha two alpha. Here alpha beta. Here alpha two alpha alpha and beta. Here alpha and alpha two alpha. And here alpha two alpha, alpha two alpha. Okay? That’s all the… photograph this now, this masterpiece — it contains everything. Yes.
[Speaker C] Okay, do you also want to stand next to the board? Smile.
[Rabbi Michael Abraham] Thank you. All right? Okay. Now this really contains everything; you don’t need all the previous ones, you need this. Now look. Let’s start checking. This thing — so here we have alpha and beta, and here there is… this is preferable, right? In terms of connectivity, this is also preferable. Right? Because here there are two parts and here there is one part. There are no direction changes here at all, and the number of vertices is the same. So this is preferable. Right? So it fits the criterion. Next. This — they are equivalent. I said I’m ignoring the valence, that there are two alphas here. Okay? So the parameters are the same parameters. Right? Now in terms of connectivity this is preferable. Why?
[Speaker C] What is that black thing next to alpha below?
[Rabbi Michael Abraham] No, no, no — below, what is it? Alpha and also beta. It’s
[Speaker C] Beta is a bit…
[Rabbi Michael Abraham] Do you want to correct the photograph?
[Speaker C] Photograph it again, so it looks like a W.
[Rabbi Michael Abraham] “On the mount, the Lord shall be seen.” There. Okay. Now look here. So in terms of number of parameters, this is equivalent. Right? In terms of connectivity this is preferable. In terms of number of vertices, equivalent. In terms of direction changes, this is preferable.
[Speaker C] Why? Here it’s…
[Rabbi Michael Abraham] Because here there is a direction change.
[Speaker C] No, in terms of number of vertices, why is it equivalent?
[Rabbi Michael Abraham] Sorry, in terms of number of vertices this is preferable. This is preferable, right? In terms of direction changes, this is also preferable. In terms of connectivity, this is preferable. What does that mean? Refutation. Right, a tie. Exactly right — it’s a refutation. Notice, I arrived at this being a refutation by ignoring the fact that here there are two alphas and here there aren’t two alphas. I ignore the valence. That did the job for me. Okay? Now we had a contradiction with three; let’s see how three works. So we have it like this: in terms of number… in terms of number of parameters, equivalent. Right? In terms of connectivity, equivalent. In terms of direction changes, equivalent. In terms of number of vertices, this is preferable. Right? And therefore the result is one. The contradiction between two and three has disappeared. Right? I can prove both this and this with the same preference criterion. Of course, I’m assuming in the background that the number of… levels at which a parameter appears doesn’t matter. What happens in the fourth case? So we have it like this: the number of vertices is the same; everything is the same. So here there’s nothing to do at all; it’s obviously equivalent. Right? In both cases it is one parameter; in both cases even the same level of valence, connectivity, direction changes, number — it’s the same graph, simply. So here there is no… it’s just the same thing. In other words, this criterion does the job. For all the presentations I’ve done up to now, and without getting tangled in the contradiction between two and three that I had before. Okay?
[Speaker C] So basically you cleaned it up. Two and three are no more. What? Two and three disappear?
[Rabbi Michael Abraham] Not that they disappear, but there’s no contradiction between them. They fit together very well. I can show the preference of the blue here, of the red here, and the absence of preference here with the same criterion. There’s no need to assume something about valence, which got us into the previous contradiction. Okay? Now I need one more assumption for what follows. In all of them the red is preferable?
[Speaker C] What? In all of them the red is preferable.
[Rabbi Michael Abraham] Well, if there is a preference, then it’s for the red or they’re equivalent. Because I chose filling in one to be red. So now I’ll do one more thing whose explanation really lies ahead. But I need it only so that the picture will be complete, and from here on we simply work — this is the algorithm. From here on we simply work.
[Speaker C] The definition of direction changes — is it between every vertex and vertex?
[Rabbi Michael Abraham] No, the maximum. The maximal path that exists along the graph. All right? The path with the largest number of direction changes that you can find in the graph. One more point — okay, now I erase this.
[Speaker B] How did we say connectivity comes into play?
[Rabbi Michael Abraham] Because what is preferable — if it isn’t connected, then it’s less good. More connected, yes, obviously. When there’s a relation between the things, then it’s more reasonable to infer from one to the other, right? If there’s no relation between the things, it’s like the mezuzah and the fringes. They don’t speak to one another; they don’t belong to the same world. So how can you infer from this to that?
[Speaker G] So in more complicated models, the more and more connections I find, I have one more reason to think it’s more preferable.
[Rabbi Michael Abraham] Right, exactly. If they don’t preserve hierarchy — direction changes, things like that — then exactly, that will offset it. Okay? Now, in order to complete the algorithm, there is one more point, whose explanation will come next time or later on. And that is: when I build the model in a given graph, then as I told you earlier, how do I do it? The assumption is that I do not enlarge more than one parameter. That is, I go up to alpha, two alpha, three alpha, four alpha — beta will remain only beta. If I need to add something else, it will be gamma. It won’t be two beta. Only one parameter rises in levels. Okay? This is something I need to assume. I’m saying, first of all, without it you can’t explain the Talmudic passage. But there’s also a rationale to it. I’ll explain what the rationale is. After all, if I want there to be a relation between the parameters, and therefore I can learn from this to that — from wedding canopy to money, and intercourse and document to wedding canopy — I basically want there to be a relation between them, right? But that’s what I described up to now. But I also want the hierarchical relation to be preserved. When the hierarchical relation is preserved, what does that mean? It means that basically there is one basic scale along which they all line up. Right? If this is preferable to that and it is preferable to that, they are preferable in the same sense. Therefore, when I want to make a preference between two things, it will always be a quantitative preference; it will always be in the same parameter. Meaning: alpha, two alpha, three alpha, five alpha — but beta will always remain one. I’ll show you two examples that may illustrate this best. Look, I’m erasing these two now; I’m leaving only the criterion.
[Speaker C] Is there a relation between these things and operations research — the traveling salesman problem and the simplex and all those things?
[Rabbi Michael Abraham] Traveling salesman is also a network, but I don’t know exactly what — it depends what you do with it, I don’t know. Obviously, in traveling salesman you need to get from site to site, there are bonds and so on, but it’s also a graph.
[Speaker C] It seems to me this reminds me of the free…
[Rabbi Michael Abraham] It’s also a graph, but the question is what you do with it. I don’t know.
[Speaker C] They want the optimum flow.
[Rabbi Michael Abraham] Yes, no, I know, but you’re asking how you do it — whether you use topological properties of the graph — I don’t know, I’m not familiar with treatments of that kind in the traveling salesman problem. Anyway, look, I’ll give you an example.
[Speaker C] After all, what basically — I have a bit of a problem with the notebook,
[Rabbi Michael Abraham] What bothers you about… this constraint that I’ve now put on the table, that we increase the strength of the parameters only in one of all the parameters. What bothers you is that it will actually force me to add parameters artificially. Because if I’ve raised alpha to two alpha or three alpha, beta is already fixed. So if I need to create preference in beta, I have to introduce beta and also gamma. I can’t rise to two beta when the structure of the graph itself doesn’t require it. But I have to do it — and that’s very bothersome. So I’ll show you two extreme examples so you’ll see why it still makes sense, okay? The more one waves one’s hands, that’s the weak point, you understand? I mean, when I had a study partner and we argued passionately, I would say, no hands — put your hands behind your back and explain it to me like that. Okay, usually when you can’t explain, you start waving your hands; that’s a weak point. But I still think, it seems to me, that in the end it is convincing. So look, I’ll show you the following two graphs. Something with lots of hierarchies, yes? I’m taking something extreme. Something like this, to here and so on. Okay? I continue.
[Speaker C] Each one contains the other?
[Rabbi Michael Abraham] Yes. Suppose I got some graph like this. So what do I do? Notice what happens now. After all, I can’t increase in two parameters, right? What I would otherwise have done is say: this is alpha, two alpha, three alpha, four alpha, five alpha — and also on this somehow with beta, two beta, three beta, and that’s it. And now I need to introduce a lot of parameters, right? So no, not a lot — I need to introduce one more. Now look how it works. This is alpha and also beta. Okay, this is two alpha and also beta; this is three alpha and also beta; four alpha and also beta; and this is four alpha and also beta and also gamma. Now this is alpha and also gamma; three alpha… wait, five alpha?
[Speaker C] Doesn’t matter, wait,
[Rabbi Michael Abraham] three alpha and also gamma, two alpha and also gamma, yes, it gets smaller — three, two alpha, yes, alpha and also gamma, and if you want put gamma here too, it doesn’t matter. Okay? Look, you understand that this meets the criteria, right? I rose only in alpha; I’m rising to higher levels, right? In beta and gamma, no. But notice, this forced me not to introduce too many parameters. It forced me to introduce one parameter that accompanies the rise here, which is beta, and one parameter that accompanies the rise here, which is gamma. What does that actually mean? It basically means that there are quantitative levels here indicated by alpha, two alpha, three alpha, four alpha, such that each of these paths colors that quantity with a different quality. Okay? This colors the rise — this is actually gamma — and this colors the rise with beta. Therefore, if I have two paths, this will be described by two scales, beta and gamma, while the hierarchical relation between them is determined by one parameter, the parameter alpha. Now you understand that this makes sense. It makes sense because I want these two graphs to belong to the same sack, so I’m basically saying that the hierarchy between these two is related to the hierarchy between these two; it’s on the same scale. Why is there no relation between this and this and between this and this? Because this is colored in the color beta and this is colored in the color gamma, but the quantity — the quantitative dimension — behaves in the same way. In other words, alpha is the same quantitative parameter in both things; it’s just colored differently.
[Speaker C] Right, but gamma and beta have no relation to each other.
[Rabbi Michael Abraham] Right, therefore there are no arrows here. If there were an arrow from here to here, or from here to here, or from here to here, I don’t know, the whole business would change. But I’m taking a simple example just so you can see the meaning of the constraint I’ve added blindly… now. The meaning of the constraint is basically this: the quantitative scale is one within the graph, and what determines it is some one parameter we choose — alpha, it doesn’t matter. All the rest can color that quantity in the color of another quality. If you need several qualities, introduce parameters, but the number of parameters is not the same as the number of levels; it is the same as the number of qualities. The levels of alpha already handle that — both the levels here and the levels here, even though there is no relation between them. Do you understand? In other words, there is logic to this assumption. We found that you have to assume it; later we’ll see that without it this won’t work. In only one place, by the way; everything else works even without it. But there is logic to this assumption, it isn’t just something ad hoc and that’s it. I’ll bring you…
[Speaker C] Why are you unifying between the two wings? There’s no relation at all.
[Rabbi Michael Abraham] No, there is a relation. The relation is from here. This is the graph I got; a table came out and gave me this. It can happen; there’s no problem. There can also be a table; I can describe this table to you, I can draw this table for you here if you want.
[Speaker C] One wing is mezuzah and this is fringes, B is fringes. No. What no?
[Rabbi Michael Abraham] In mezuzah and fringes there was no relation at all. There was a line here, a line here, and they wouldn’t connect.
[Speaker C] So they…
[Rabbi Michael Abraham] No, but here it does connect — that’s exactly the point. In mezuzah and fringes they don’t connect, therefore I say — that’s exactly the point — I’m talking about one graph. One graph, so there is a relation between the things. If there is a relation between the things, then the quantitative hierarchy is determined by one universal parameter. On top of that you can color it in different colors; every channel that creates a hierarchy has its own color. Look, here I’ll do something similar in reverse.
[Speaker C] And that meeting at the top — is there a relation between C, B, and A…?
[Rabbi Michael Abraham] The arrows are simply in the opposite direction, yes? Okay? A graph like this. What do we do with a graph like this?
[Speaker C] The zero is the one in the middle. The quantities change. Three alpha and beta, two alpha and beta, alpha
[Rabbi Michael Abraham] and beta, and beta alone, and this is alpha.
[Speaker C] And this is alpha and also gamma.
[Rabbi Michael Abraham] And this is two alpha and also gamma; it has to be, because alpha has already risen so I can’t change gamma. Three alpha and also gamma, and so on. Okay? Exactly the same thing as you see here, you also see here. Only the hierarchy is in the opposite direction, and that means that with one parameter I can build the quantitative hierarchy, but I need to color it in different colors. The direction is simply a direction derived from the directions of the arrows, but that’s all. Okay? This thing… what?
[Speaker F] Why… how is this separated from that branch up there? What? The three alpha?
[Rabbi Michael Abraham] No, no, that’s just… these are two different drawings; there’s no connection. Yes, yes, obviously. Wait, so maybe I’ll do here… I said, come on, photograph. Hope you’re not forgetting us, David. Okay. Wait…
[Speaker C] I wrote down all this here and…
[Rabbi Michael Abraham] Okay. So these are basically the criteria for how to build. All these things are ultimately just criteria for how I find the number of parameters for item one of the preference criterion. Right? Two, three, and four are simply according to the shape of the graph. Okay? But as for how to determine the number of parameters, I need all the heuristic I mentioned before: the basic node is a node like this, where two arrows go into one — alpha, two alpha, alpha and also beta. That is the basic node.
[Speaker C] How do you define connectivity in graphs?
[Rabbi Michael Abraham] This is fully connected. There aren’t two parts here that don’t speak to each other. Each of them is fully connected. If this thing weren’t here, if this thing weren’t here, then there would be no connection between them at all, and then you’d have to build the hierarchies accordingly; but then there really wouldn’t be connectivity. Here there is connectivity. As long as there is one path that gets from this to that, they are connected. Okay? There has to be no way at all of getting there in order for the graph not to be connected.
[Speaker C] It’s simply a completely separate branch. Yes? Okay? The direction changes are identical in the two graphs.
[Rabbi Michael Abraham] In this case, yes. Yes. There is one direction change.
[Speaker C] Only at the heads of alpha.
[Rabbi Michael Abraham] All right? Yes, but these are just examples — of course not… I’m only trying to show you why this constraint that says we increase the number of values only in one parameter makes sense. In other words, correct, it came out for us as the result of a constraint, unlike the criterion there, which I told you came directly from intuition and turned out to work. This one didn’t. Meaning, here we did it because we got stuck in one of the diagrams that we couldn’t find.
[Speaker C] In both of them the number of vertices is one. Why?
[Rabbi Michael Abraham] No, the number of vertices — every circle is a vertex.
[Speaker C] No, a vertex is when there is a direction change.
[Rabbi Michael Abraham] No, no. A vertex is a circle. How many points are there in the graph? And there are bonds and sites. So the arrows are the links, and the circles are the vertices. Okay? Okay, that’s the criterion. Next time we’ll get into the common side, and I’ll try to show you how, in the common side, the three types of the common side receive priorities exactly according to the three topological criteria. Okay? Each one requires a different one, and that’s evidence that you need all three. Good. See you for now. Excellent, next time we’ll get into the common side, and I’ll try to show you how, in the common side, the three types of the common side receive priorities exactly according to the three topological criteria. Okay? Each one requires a different one, and that’s evidence that you need all three. Good. A little more confidently.