2019-04-22 – Between Midrash and Logic – Lesson 10

Table of Contents

• [0:01] Opening: review of a previous a fortiori inference
• [1:19] Differences in ordinary a fortiori inferences in the Talmud
• [2:34] Converting a quantitative a fortiori inference into a logical intuition
• [6:29] Refuting a fortiori inferences of rows
• [9:07] The grades example: Yossi, history, and physics
• [11:28] Two independent arguments within an a fortiori inference
• [19:46] The connection between ability in physics and history
• [23:30] The thieves example: a fortiori inferences of pockets
• [28:31] A hierarchy of alpha and beta and their effect
• [29:53] Asymmetry between proof and refutation
• [31:16] Microscopic parameters that affect the comparison
• [33:47] The frog analogy and generalization
• [37:11] Logical tables and binyan av
• [44:05] Binyan av from two verses and its structure
• [51:04] Common denominator, refutation, and their impact
• [52:35] The table technique and looking ahead
• [54:53] The connection of money and betrothal in the parameters
• [57:16] Redeeming second tithe — the theoretical characteristic
• [59:05] Aristotelian logic — examples and simplification

Summary

General overview.

The text draws a distinction between several types of kal va-homer and develops a tabular description showing how such an inference works, how it is refuted, and why a single refutation in the Talmud collapses the entire kal va-homer, even though on the face of it there seem to be two independent formulations in terms of rows and columns. The text argues that kal va-homer inferences are not merely a formal rule, because beyond the “hard” data in the table there are implicit assumptions of relevance and a hidden theory of “microscopic parameters” that explains the data and links the different formulations. The text suggests that understanding this theoretical layer would make it possible to deal systematically with complex tables, determine algorithmically how to fill in the missing cell, and also uncover the conceptual “why” behind legal rulings, not just “what” the conclusion is.

Types of kal va-homer and the quantitative structure

The text distinguishes between a kal va-homer of “from two hundred to one hundred,” which looks like a deduction that cannot be refuted though in fact even it can be challenged, and an intuitive kal va-homer based on one datum plus a hierarchy supplied by reasoning, such as “if her father had spit in her face, would she not be humiliated for seven days,” which assumes by logical intuition that the Holy One, blessed be He, is more stringent than her father. The text defines the ordinary kal va-homer found throughout the Talmud as a quantitative kal va-homer based on three data points in a fixed tabular structure of two “rows/columns” and a “question mark,” where two data points are used to extract a hierarchy and the third is the anchor point from which one proceeds to the conclusion. The text explains that a quantitative kal va-homer can be converted into an intuitive step by first extracting from the data a relation of greater and lesser stringency between A and B, and then applying it to the third datum.

Two formulations, rows and columns, and refutation

The text presents two formulations of a quantitative kal va-homer: a “rows” formulation, in which one extracts a hierarchy between B and A and then infers from it regarding the second row, and a “columns” formulation, in which one extracts a hierarchy between Y and X and then infers from it regarding the second column. The text states that at first glance the two formulations are not equivalent, because a refutation of a row-based kal va-homer is carried out by means of “an additional row” that undermines the generalization produced about B>A, while a refutation of a column-based kal va-homer is carried out by means of “an additional column” that undermines the generalization produced about Y>X. The text concludes that according to the tabular logic, one refutation ought to topple only one formulation and leave the other intact, so apparently two refutations should be needed to bring down the whole kal va-homer; but in practice, in the Talmud, a single refutation brings down the entire kal va-homer.

The grades example and the gap between refuting rows and refuting columns

The text illustrates this with grades in history and physics, showing how one inference can be formulated as a hierarchy between subjects, while another can be formulated as a hierarchy between students. The text shows that a refutation of the “third person” type can reverse the hierarchy between the subjects without touching the hierarchy between the people; conversely, a refutation of the “third subject” type can undermine the hierarchy between the people without touching the relation between the subjects. The text uses this to sharpen the point that the tabular square seemingly contains two different arguments, and then asks why the Talmud does not simply “rotate” the kal va-homer into the second formulation after a refutation.

The exceptional cases in Bava Kamma and Niddah, dayyo, and the attempt to “rotate”

The text argues that throughout the Talmud there is no example in which two refutations must be brought in order to collapse a kal va-homer; one refutation always brings it down, except for two places in tractate Bava Kamma and tractate Niddah where they try to “rotate” the kal va-homer. The text states that in both of those places the table is not a regular one but includes a “half,” and the rotation is done against the rule of dayyo rather than against a refutation. The text shows that in a “half” table, the row formulation leads to a result of “half,” while the column formulation leads to “one,” and that this gap resembles a refutation in that it distinguishes between the formulations and demonstrates that they are not equivalent.

Kal va-homer from two data points, symmetry, and the blessing over Torah and Grace after Meals

The text brings a rule cited by “the rule-formulators” that one does not make a kal va-homer based on only two data points, because the absence of a datum is symmetrical and makes it equally possible to fill in the opposite way. The text notes that there are examples in the Talmud where they nevertheless try to do this, such as with the blessing over Torah study and Grace after Meals, where the Talmud attempts a kal va-homer on the basis of “it is not written,” and then rejects it. The text emphasizes that the difficulty is not only logical but essential, because without a third datum there is no indication of relevance between the contexts.

Relevance, “microscopic” parameters, and the claim that kal va-homer is not formal

The text proposes that the need for three data points stems from relevance: the third datum provides an indication that the hierarchy learned on one side may also be relevant to the other side. The text gives an example involving “Esther, who hates jazz, likes reading literary fiction” to show that even an argument that fits the structure of a kal va-homer may still be problematic if the hierarchy is not relevant or if different parameters explain the data. The text formulates the point as follows: a kal va-homer assumes a hidden generalization — the relation of greater stringency learned in one row is not only with respect to X, but reflects a more general feature that is supposed to affect Y as well — and the refutation reveals that the stringency exists only in one aspect, while another aspect may reverse the hierarchy. The text defines refutation as showing that both “one” and “zero” are equally plausible, not as proving that the correct result is “zero.”

Analogy, binyan av, and the hidden generalization

The text connects the discussion to a previous lecture on induction and deduction and presents an analogy of “frog A is green,” from which one infers something about frog B, arguing that behind the analogy there sits a theoretical generalization according to which “being a frog” is the parameter relevant to being green. The text describes how this can be represented as a 2×2 table rather than as a 2×1 vector, because the missing row is precisely the theory assumption. The text argues that behind inferences such as kal va-homer and binyan av there is always a theoretical layer that is not exposed, and that sometimes the conclusion appears “intuitive” only because the relevance assumptions have already been absorbed.

Tabular language, algorithmization, and variations of the interpretive rules

The text proposes that using tables makes it possible to generalize, classify, and combine data situations, and to ask, for each pattern, whether the data allow one to infer that the missing cell should be filled with “one,” “zero,” or “unknown.” The text describes how certain tables represent a kal va-homer with a refutation, in which the result remains open, and how one can chain moves like “let this prove it” and create larger 3×3 tables. The text argues that if one develops an orderly method, it should be possible to formulate an algorithm that takes the accumulated data and returns the status of the missing cell without having to follow the “linear form” of the Talmudic passage.

Binyan av from two verses, the common denominator, and refutation of the common denominator

The text identifies a complex table as a representation of binyan av mi-shnei ketuvim and explains the structure in which two source cases (A and B) try to teach C; each has its own unique refutation, and in the end one learns from “the common denominator.” The text argues that there are three types of binyan av from two verses: both are binyan av, both are kal va-homer, or one is of one type and one of the other; and that the Sages call all of them by the same name, even though they may have different logical properties. The text explains that a refutation of the common denominator arises when one points to an additional feature shared by both source cases that does not exist in the target case, in which event it is no longer clear that the selected common denominator is the determining factor.

The planned continuation: solving tables through microscopic parameters and conceptual meaning

The text postpones to a future continuation the presentation of the orderly technique that will again make use of microscopic parameters to explain the data and decide the missing cell. The text promises two “gains”: the ability to determine in a rigorous way how the cell in the table should be filled once all the data are laid out, and the ability to understand the theory behind Jewish law, not only the result. The text illustrates this with the idea that one may identify a theoretical parameter that explains why money and intercourse can effect betrothal, and why the bridal canopy can effect marriage, so that the model is not merely formal but a tool for uncovering reason and principle; simple problems do not need it, but large tables do.

Okay, let’s begin. Last time we started discussing an a fortiori argument. We saw that there are several kinds of a fortiori arguments. There’s an a fortiori from the rule of “from two hundred, one hundred” which is, supposedly, a deduction that can’t be refuted, and I said in parentheses that even that isn’t precise; you can also refute that kind of a fortiori. There’s an a fortiori that’s based on reasoning. Those are all the a fortiori arguments that appear in the Bible. Those are arguments that start from one given fact and infer from it a second fact, where the relation of stringency between the contexts comes from reason. “If her father had but spit in her face, would she not be shamed seven days?” So that means that the Holy One, blessed be He, is more stringent than her father. How do we know that? By reasoning. So one datum is enough here. I know that logic tells me that the Holy One, blessed be He, is more stringent than her father. I need one datum: what happens with her father? She is shamed seven days, and from reasoning I say: then with the Holy One, blessed be He, at least seven days, or perhaps more than that. In the ordinary a fortiori arguments in the Talmud, that’s not how it works. Those are what I called a measured a fortiori. That’s an a fortiori based on three data points, not one, and the structure is always like this. Always like this. There’s something like this here. Okay? Here there’s this, here there’s this, and here there’s a question mark. Okay? I have some action that fails to achieve result A. I have an action that succeeds in achieving—this same action succeeds in achieving—result B. Okay? A second action succeeds in achieving result A, so all the more so it will also succeed in achieving result B. Now X, Y, A, and B change each time depending on the passage, but it’s always the same structure. The only differences in the context of an a fortiori can be here or here, where it might say here “half,” or something incomplete, and then the question of “dayyo” starts to arise. Right? Like with tooth and foot, and horn in the public domain and the damaged party’s courtyard. But for now I’m leaving dayyo aside. So that’s the ordinary structure of an a fortiori argument. We said that an a fortiori of this type can be formulated in two ways—or, before the two forms, how does such an a fortiori actually work? I can take one step and turn it into a reason-based a fortiori, an argument of the previous type. How? I take this row, and I infer from it that B is more stringent than A. Once I’ve inferred that, I can already erase the whole upper part and look only at this. I have the datum that Y includes A, so if B is more stringent, then certainly here too there will be a one. Y also includes B. Okay? So what’s really the difference between the last two a fortiori types, those last two categories? It’s the question of where the hierarchy between B and A comes from. If it comes from reason, then one datum is enough for us. Reason tells me that the second datum also has to be at least like it. If it doesn’t come from reason, then I need two data points from which I derive the hierarchical reasoning that B is greater than A, and now I can throw them away and use the third datum. Okay? The same thing, of course, also with the columns. I can take this column and derive from it a hierarchy between X and Y, and then use only that datum and infer the conclusion. In both of these formulations, which are different, two of the data points serve us in deriving a hierarchical reasoning—who is stronger than whom, or who is more stringent than whom—and the third datum is the footing I plant my foot on when I move forward with the a fortiori. Okay? So really the difference between the second type of a fortiori, the reason-based one, and the measured a fortiori, is the question of where the reasoning comes from. Does the hierarchy come from reason, or does the hierarchy come from two halakhic data points? That’s all. That’s why you need two more data points in the third type. We also saw—and I already mentioned this last time—we also saw that there really are two different formulations for a measured a fortiori. You can take these two data points, derive this hierarchy from them, and then do the a fortiori on these two cells. Or you can take these two data points, derive this hierarchy, and then do the a fortiori on this datum and infer the conclusion from it. These are two formulations that can appear in every a fortiori by the very… We said they were equivalent. Are they equivalent? We talked about that too, right? They aren’t equivalent. Meaning, these aren’t just two different formulations of the same argument. At least that’s how it appears at first glance—we’ll soon see that I’m not sure about that—but at first glance it seems there is no connection at all. Meaning, when I make this a fortiori, yes? A row-based a fortiori. Then I take the first row and infer from it that B is greater than A. Okay? Now I go down to the lower row. If Y contains A, then certainly it will also contain B, which is more stringent—if that is more… more easily applicable than A. Applicable with an h. Right? So that’s a row-based a fortiori. When I refute this a fortiori, what am I supposed to do? I’m supposed to find some Z here, an additional row. A row-based a fortiori is refuted with an additional row. Okay? And that row’s structure will basically be this. Let’s write another row here. Its structure will be this. Why is that a refutation? Because it basically proves that the hierarchy we extracted from the first row is not necessary. Here, look, it doesn’t hold. It isn’t general. Okay? There may be some aspect by virtue of which B really is more stringent than A, but there may be another aspect that says A is more stringent than B. And now I don’t know which of those two aspects is relevant to Y. So therefore I can’t infer my conclusion. That’s a refutation of the row-based a fortiori. What does a refutation of the column-based a fortiori look like? Adding another column. C. Where here there is a one and here there is a zero. Okay? What I’m finding, basically, is another result that X succeeds in applying and Y does not succeed in applying. What does that refute? It refutes this. Because if from this row it looks like Y is more stringent than X, or stronger than X, then in this row we see that there is at least one aspect in which X is stronger than Y. And since I don’t know which of the strengths is relevant to column B, I can’t infer my conclusion. I’m already formulating it now in a somewhat more careful way. I’ll sharpen that further later. So what is the meaning of this picture? The meaning of this picture is that a refutation of this type refutes the column-based a fortiori. A refutation of this type refutes the row-based a fortiori. Which means that in order to refute an a fortiori, we should really always have to present two refutations. One isn’t enough. That is, if we find some a fortiori, yes? This square, in some Talmudic discussion, and we want to refute it. The Gemara says: what about X, since it contains C while Y does not? Sorry—and Y does not contain C. Okay? That refutes nothing. What it has refuted is only that now I can’t make the column-based a fortiori. But in the row-based a fortiori there is no problem continuing to make it, even after I presented this refutation. This refutation refuted only one of the formulations and not the other. Why? Because it undermined the relation between Y and X. It says nothing about the relation between B and A. Right? What I see here is that X and Y are not related the way I thought. But what does that have to do with the relation between B and A? I learn nothing from here about the relation between B and A. So that assumption remains intact. And I said last time that refutations always attack the side of the hierarchy, or the assumption of hierarchy. Meaning, this assumption or this one. Right? Everything else besides that is deduction; there’s nothing there to attack. Meaning, refutations of all kinds always attack the assumption of hierarchy. But since there are two completely different, entirely independent hierarchy assumptions here, then when I attacked one of them I brought down the row formulation. But the column formulation still remains in force. And vice versa. Okay? Let’s take an example. Right? Yossi got 70 in history and 80 in physics. Okay? Let’s go with something counterintuitive. 70 in history and 80 in physics. Okay? Now Yankele got 80 in history. How much will Yankele get in physics? At least 80, right? Now I bring a refutation. Look, there’s Shimon, who got 80 in history and 70 in physics. So you can’t infer the conclusion you inferred before. There’s an even better refutation: physics and history aren’t connected. No no, we’ll get to that in a second. But first there’s Shimon—I’m talking first about data, not about assumptions. Let’s look at data, let’s collect grades; that’s hard data. Look, we have Shimon. What does that mean? It means that my assumption that physics is easier than history is not necessarily correct, okay? But it says nothing about the relative intelligence of Reuven and Yankele, or whoever the first one was—Yossi. Yossi and Yankele, right? There’s no connection there. Levi proves nothing about them. This one—Levi, Shimon—proves nothing about them. Let’s make it Reuven, Shimon, Levi, whatever. Okay? The question is whether Reuven—one formulation of the a fortiori says: look, Reuven is smarter than Shimon. And if Shimon got 80, then Reuven will get at least 80 in physics. Okay? That’s one formulation. How do you refute that? You find a subject in which Reuven got less than Shimon; he’s not smarter than Shimon. Okay? But there’s another formulation. I say: physics is easier than history, and the proof is that Reuven got 80 in physics and 70 in history. So if that’s the case, then Shimon, who got 80 in history, will certainly get at least 80 in physics. How do I refute that argument? I refute that argument with a third person, not a third subject. Right? With a third person and not a third subject. I say there is a third person who reverses the hierarchy between the subjects. So that refutes the assumption that physics is easier than history. But of course that doesn’t touch the question of who is smarter—Reuven or Shimon. Yossi proves nothing about that. Okay? So that basically means that an a fortiori argument hides behind it two independent arguments, completely different. Now if it really hides behind it two independent arguments, then something here is broken. Because in the Gemara, whenever a refutation is presented, the a fortiori falls. Nobody says, wait, wait, wait—I have another formulation. That was this refutation. It refuted the row-based a fortiori, but I still have the column formulation, or vice versa. What’s the problem? We need one more refutation to definitively close down this a fortiori? No. The whole debate is always whether there is a refutation or there isn’t. If there is a refutation, the a fortiori has fallen. Why has it fallen? Why does a refutation like this also throw out the row-based a fortiori? What’s the connection? And throughout the entire Talmud, throughout the entire Talmud, there is no exception. There is never a case where you need to bring two refutations in order to throw out an a fortiori. Whenever a refutation is presented, the a fortiori falls. There are only two places where there is a deviation, where they really do try what’s called to “rotate” the a fortiori. Meaning: you knocked out my row-based a fortiori, so now I’ll formulate for you the column-based a fortiori. Those two places are in Bava Kamma and Niddah. And in both those places we’re dealing with a table that isn’t like this. Not full? Rather half. Yes. A table like this. And only—in short—the rotation is not done in response to a refutation, but in response to dayyo. But if you use dayyo, then here if you go with the row-based a fortiori, the row-based a fortiori says B is more stringent than A, so if Y produces half in A it will certainly produce half in B. Right? Not more than half, because of dayyo, but it will produce half in B. Right? That’s if I went with the rows. But if I went with the columns—if I went with the columns—then what I get here is that Y is more stringent than X, right? So if X produces one in B, then what will Y produce in B? One. Right? The column-based a fortiori gives me one; the row-based a fortiori gives me half. That is basically like a refutation. Dayyo is a logical creature that resembles a refutation. It too distinguishes between the row formulation and the column formulation. And when the row formulation falls, the column formulation will still remain. You don’t arrive at the same result, and that itself is proof that we are dealing here with two different arguments. It looks as if it uses both. Meaning one and a half. No no, one in blue and half in black. That’s not—okay? So the fact that two different conclusions are reached is also an indication that we are dealing with two different arguments. Just like the previous indication—that a refutation knocks down one and doesn’t touch the other, and vice versa. So that means they aren’t equivalent. They aren’t two formulations of the same argument, but two different arguments. In every such square, you can present two arguments to prove that a one should sit here. But the problem is that the Gemara doesn’t acknowledge that. When the Gemara raises a refutation and knocks out one of the arguments, the a fortiori has fallen. It doesn’t rotate. Only in the case of dayyo is there an attempt to rotate, and that attempt fails. Even there, in the end, they don’t rotate. The Gemara says the rotation won’t help you. But there at least some attempt was made. In the other places, where the table is a normal table without halves, where there is no dayyo, there isn’t even an attempt. Once someone raises a refutation, the a fortiori has fallen. So it also fell. The question is why. What do you mean, why? After all, ostensibly it seems that these are two different formulations of the same argument. If so, once I knock out one, the other is just equivalent, so it falls too. But when you look at the logic of it, we’re dealing here with two different arguments, not two formulations of the same argument. That’s another hierarchical datum. What do you mean? You need the hierarchy of the properties, say, in order to invalidate the a fortiori. Why? You don’t need anything. Look. If you now prove from the row-based a fortiori that B is more stringent than A, that says nothing about the question of what—even if X were more stringent than Y. Maybe the Gemara really assumes that X is not more stringent than Y. No, and it doesn’t need to assume that—that’s exactly the point. Because whether X is more stringent than Y or X is not more stringent than Y, you can still infer from this row that B is more stringent than A. So that means the Gemara isn’t assuming any hierarchy between X and Y. No, I’m saying maybe it assumes there is no hierarchy, as you say—no hierarchy good enough for an a fortiori—and that’s why it doesn’t rotate it. No, why? What do you mean? Why shouldn’t there be such a hierarchy? The Gemara would have rotated it. What do you mean? After all, there’s a one here and a zero there, so look—you see there is a hierarchy. What does that mean? These are facts, not assumptions. Forget your arrow; there’s a one written here. There is a fact here that this is more stringent. How is that different from the other direction? There is something here that seems problematic. Okay, this is a first indication of what I’ll elaborate more on later: that an a fortiori argument is not a formal rule. Meaning, you don’t take three data points like these and derive a fourth datum from them by virtue of the table’s structure alone. Behind the data there are still some assumptions, and as we will see, those assumptions tie together the two formulations, the columns and the rows, and in fact this is one argument and not two. We’ll see that later, but I want to sharpen this point more. Look, and this is where I come to the relevance that was mentioned earlier. There is a rule among the writers of rules that you do not make an a fortiori argument based on two data points. An a fortiori like this, an a fortiori like this—these are the only two data points—so I say, okay, here it’s unknown and here it’s one, so this is more stringent than that, therefore, therefore I’ll also fill in a one here. You don’t do that. There has to be a datum here that is zero. Meaning, the absence of a datum cannot serve as the basis for an a fortiori. And the reason is of course very simple, because the absence of a datum means that to exactly the same extent I could have filled it in the other way. If there’s nothing here and there’s a one here, then this is more stringent, so I’ll fill in a one here. Right? It’s completely symmetrical. Now, if I filled in a one here, then by doing that I’ve broken the a fortiori in the opposite direction; I’m sawing off the branch I’m sitting on. Okay? Therefore, therefore, you don’t make an a fortiori on the basis of two data points. What lies behind the fact that you don’t make an a fortiori on the basis of two data points? By the way, there are examples in the Talmud where they do do it, even though the writers of rules say not to. What kind of example might that be? For example, the blessing over Torah study and the grace after meals. In blessings? In the blessing over Torah study we recite a blessing beforehand, and it doesn’t say what happens afterward. And in grace after meals it says afterward, and we don’t know what happens beforehand on a Torah level. Right? So we make an a fortiori: if the blessing over Torah study, where we do not bless afterward, we do bless beforehand, then grace after meals, where we do bless afterward, all the more so should we bless beforehand. But by the same token, of course, one could make the reverse a fortiori. If grace after meals, where we bless afterward, we do not bless beforehand, then the blessing over Torah study, where we do not bless afterward, certainly we should not bless beforehand. But we do bless beforehand. So something here doesn’t work. Rather, okay, now we have to see what the point there is, why the Gemara nevertheless tries to make a sort of a fortiori like that. It rejects it in the end, but it does try to make such an a fortiori. But what is more important to me is not to understand that passage, but to understand what the problem really is. Why do you need the third datum at the substantive level, not just because of the logic? At the substantive level, the reason you need the third datum is a reason of relevance. At least that’s what I think. What does that mean? If there is no third datum, then maybe there is no hierarchy at all between these two data points; they may just be two unrelated data points. Like you said earlier: what connection is there between ability in physics and ability in history? These are two different kinds of skill. Okay? If you show me that one person failed in physics and succeeded in history, and the second succeeded in history, then I understand that there may be at least some connection, unless it is refuted. It’s not a necessary connection, but it could be. But if you tell me, look, this one succeeded in physics and the other succeeded in history, and I have no idea what he did in history and what the other did in physics, then I have no indication at all for the assumption that there is a connection between the skills for physics and the skills for history. What’s the connection? Only if there is a shared cell that gives me some indication of the shared nature of the hierarchy, of the fact that the hierarchy on one side could at least be relevant to the other side. Of course it doesn’t have to be, but it at least could be. But if there is no such cell, then I have no indication. Let me give you another example. Let’s make an a fortiori with three data points that suffers from the same problem. Okay? An a fortiori like this—how? I say as follows: if Esther, who hates jazz, likes to read fine literature, then Rachel, who likes jazz, will certainly like fine literature. All the more so, that’s the argument, surely that is the law. What’s wrong with that? Jazz. No, jazz is wonderful. What’s wrong with the argument? What’s wrong with the argument? Maybe it’s exactly the opposite? Meaning, no, it can’t be the opposite. Such an a fortiori is a closed a fortiori; it has three data points. What’s not okay with this a fortiori? Ostensibly it meets all the standards of an a fortiori. There is a connection. Meaning, what? There is a substantive problem here and not a logical one. Why? The hierarchy isn’t correct. Why? It’s not more. What is it not more than? I’m raising data points. Esther likes fine literature, likes jazz—no. Fine literature she likes—yes. Rachel likes jazz—yes. I’m asking about fine literature. What’s the problem? The data are data that hold. So these are data that need another assumption. What? The placement where you put them—why do you put them specifically there? What do you mean specifically there? I’m building a table. Is it forbidden to build a table? I’m writing the data I have in a table. Esther, Rachel, jazz, fine literature. You already assumed the hierarchy in advance. Why? It could be that fine literature is the earlier one, that whoever likes fine literature doesn’t like jazz, and then… Ah, you see, that’s not right. What do you mean I assumed it? Everywhere I assume it. Every a fortiori I assume it. In tooth and foot and horn, where here there is a zero and here a one, you assume that this is more stringent, right? That’s the rule of a fortiori, no? How do you know this is more stringent? No, I don’t know. What are you talking about? If I knew, I wouldn’t need the two data points. That’s why I introduced beforehand that there is a second kind of a fortiori based on one datum where the hierarchy comes from reasoning. But in an a fortiori of three data points, a measured a fortiori, the hierarchy is learned from two data points. And therefore you need three data points there—two of them to teach the hierarchy, and one further datum where you begin the argument. So that means I didn’t derive the hierarchy from reasoning; I derive it from the data. That’s exactly the difference between a measured a fortiori and a reason-based a fortiori. So what’s wrong here? In principle, nothing is wrong. It’s a good a fortiori. Right? Like an a fortiori brought by the writers of rules, those who deal with the hermeneutical rules—I don’t remember whether I saw this in Rabbi Ostrovski or in Rabbi Hirschensohn, I don’t remember anymore. They’re all from the old Mizrachi crowd, those who deal with the rules; that’s an interesting phenomenon in itself. In any case, he says there as follows: if a four-cornered garment, which is exempt from mezuzah, is obligated in tzitzit, then a doorpost, which is obligated in mezuzah, surely should be obligated in tzitzit. That resembles the a fortiori of the thieves: my fellow’s pocket, where my pocket is permitted to me and his is forbidden to me; my fellow’s pocket, where his is permitted to him, surely I am forbidden with respect to it. Right, exactly the same idea. Right? So what’s wrong with this a fortiori? Ostensibly it’s a wonderful a fortiori. What’s wrong with it? What’s wrong with it is something not in the structure. In the substance. Exactly. Meaning, I’m trying to show through this that we assume additional assumptions beyond the data. Notice, this is not a reason-based a fortiori. In a reason-based a fortiori, the assumption is on the table. I say: the Holy One, blessed be He, is more stringent than her father, so if her father confines her for seven days, then the Holy One, blessed be He, certainly confines her for seven days. On the table there is some assumption that the Holy One, blessed be He, is more stringent than her father. So there there is no problem. There it’s clear to me that there is some additional assumption. Here, ostensibly, there is no assumption. I take two halakhic data points—I have no proof who is more stringent than whom. I take two halakhic data points, from this I derive that this is more stringent than that, and then I learn over here. Ostensibly I’m not mixing in reasoning here, but it is mixed in here. The reasoning is mixed in here in the sense that the stringency I learned between these two with respect to X is relevant also with respect to Y. The fact that I learned that B is smarter than A in physics still doesn’t mean that those are the same skills required to be smarter in history. But that doesn’t appear in the logic of the a fortiori. That’s an assumption that has to be exposed. It’s sitting there behind the scenes. Okay? That’s what you said about not being an equal basis. There? Also. No, there on the contrary—there there is no assumption at all. What assumption? Only afterward in the application, when you apply it, then you take only those things that seem relevant to you to apply. But the very comparison between two things in a verbal analogy is a textual datum. When we compare two animals, the connection between them is that they are both animals. But that’s not verbal analogy. You mean paradigm construction. Ah, paradigm construction certainly. All the logical rules are like that. I’m getting to that. So basically we need to understand—the point here is meant to sharpen for us the following fact: when we assume a relation of stringency—not assume, infer a relation of stringency—between A and B with respect to X, as I said last time, we make some sort of generalization before we apply it to Y. We say that this relation is more general; it’s not specific only to X. When I say that Reuven is smarter than Shimon, it’s not only his skills in physics, but he has some sort of intelligence in a broader sense, more than Shimon. That’s the assumption. The refutation may refute that, but that is the assumption. Therefore, I want to learn also with respect to history. So notice: these two data points are not enough; there is some additional generalization here. Because from here I infer that B is more stringent than A with respect to this parameter. He is smarter—Reuven is smarter than Shimon with respect to history, but with respect to physics there may be completely different skills involved. Okay? I assume not when I make the a fortiori. And that’s why I make the a fortiori. When the refutation comes—and the refutation here, in this case, shows me that it isn’t true that Reuven is more stringent than Shimon; look, in geography for example he’s less good than Shimon—what does that mean? It means there are skills for geography, there are skills for physics, and when I want to infer conclusions with respect to history, the question is what it resembles. What are the skills required to succeed in history? Are they similar to the skills of physics, or similar to the skills of geography? That will be very important. Because there is no simple, general hierarchy between B and A. This datum points to a hierarchy, and that will never be refuted. This datum always points to a hierarchy in which B is more than A. What the refutation does is not to say that B is not more stringent than A—that’s why I formulated it carefully earlier—but rather that it is more stringent than A only in one aspect, whereas there can be aspects in which A is more stringent than B. And the big question is which of these aspects is relevant to the row I’m interested in, to history. Are the skills of geography or the skills of physics—which of them is closer to the skills required to succeed in history? Which means that behind the a fortiori there is not just a formal mechanism—one bigger than zero, I go down here, automatically there is a one written here. There is some implicit assumption here that no one ever says, but it’s there. And that assumption says that the relation of stringency I obtained in this row is also the relation that affects this row; it is the relation that matters for this row. And if that isn’t true, the whole a fortiori isn’t true. Sometimes I understand by reasoning that it isn’t true, so I won’t make an a fortiori at all. Sometimes what will help me realize it is the refutation. The refutation raises for me here that it isn’t true. So the refutation reveals to me that this hierarchy relation—for example the hierarchy relation I learned here—is one certain hierarchy relation, but there is another aspect that has the opposite hierarchy relation. And the question is: which of the two relations is relevant here? So the refutation shows me that the stringency here is a stringency only from one aspect, not the general stringency. So it’s not that we refute the fact that Y is more stringent than X—Y is more stringent than X, stronger than X. The fact that there is a one here and a zero there—there is no arguing with that. It may just be that it is more stringent than X in one aspect and in another aspect it isn’t. Say, aspect alpha is more stringent than X, and aspect beta—on the contrary, X is more stringent than Y. So here there is alpha, here there is beta. For me the important question is which one, alpha or beta, affects this column. If alpha affects this column, then it is correct to fill in a one here. If beta affects this column, then it is correct to fill in a one here, but here there will be a zero. Okay? Otherwise, yes. So then I also don’t understand how one can make a refutation to an a fortiori. What do you mean? Because now I really—now I brought case Z to prove, not to prove that B is not greater than A, but to prove that sometimes B is not greater than A. But I assumed in advance that X and Y are connected. This Z that I brought, I can say it’s not connected. No, but my assumption when I bring a refutation is that the refutation is also connected. The same assumption that you assumed—that it’s connected. No more, the same thing. I have no indication at all which is more connected to what. So on X’s side there is an a fortiori, on Z’s side there isn’t an a fortiori, and therefore I don’t know. So one has to remember that the result of a refutation is never proof that here one should write zero. The result of a refutation—an indication that this is one. Maybe it really is one; I don’t know. But you cannot infer from here that it is one. You have to remember: there is an asymmetry between proof and refutation. Proof has to show that one is the correct answer. For a refutation it is enough to show that one and zero are equally plausible. It doesn’t have to prove that zero is the correct result. It is enough to prove that zero and one are equally plausible in order to say that the a fortiori has fallen. And the question remains open. After a refutation the question remains open; it is not that the question is closed in the opposite direction. That is the meaning of a refutation, and it will be very important later, so it’s important to remember that. So what does that mean? Just one more second. Here again is an indication of what I said earlier: that behind these dry data there sit some parameters—let’s now call them microscopic or theoretical parameters—that do not appear in the table. That’s what we earlier called the skills for physics, or a person’s level of talent for physics and his level of talent for geography or history or whatever it is. These things do not appear here; they are behind the table. This table teaches us something about them. If one person succeeded more than the other in physics, that means he has some talent—we’ll call it alpha—at a higher level than the other. But if he succeeded less in geography, that means there is a parameter beta whose value in one person is lower than in the other. And now, when I’m interested in history, the question is: what are the skills of history? Is it alpha, is it beta, is it a combination of the two, is it neither of them, maybe it’s a third thing? You can’t know. If I manage to show that what determines geography is alpha, then the a fortiori remains in force. But if I don’t know, then I don’t know. And for the Sages this is a first principle, and today we are decoding that first principle? When you analyze a rabbinic a fortiori, you’re decoding it. When you make an a fortiori of your own, then you’re doing what they did. Depends what you’re doing. Fine. But understand: an a fortiori argument—people make a fortiori arguments in every field of human thought and activity. It is not something special only to Jewish law. Therefore there has to be some universal logic here. It isn’t something that… what? Aharon Barak invented it. Everybody uses a fortiori; he didn’t invent it either. But there are—people have already written books about it. Chaim Perelman, I already mentioned him, on legal rhetoric—he brings there some German, I think, who counted thirteen legal inference rules. Interesting. Thirteen inference rules. Some of them are paradigm construction, from one text and from two texts, and a fortiori. He doesn’t call it that, no matter—he calls it analogy and induction and a fortiori. And another part are all kinds of other rules; by chance he got thirteen, doesn’t matter. It’s obvious. But a fortiori and paradigm constructions are done everywhere; there’s nobody who doesn’t do them. Okay? Yes. But in terms of the meaning of an a fortiori. You said in the previous lessons that there is something that is supposedly disproven and something that was never proven. An a fortiori is something that… it isn’t certain. Not certain. An a fortiori will never in life be certain? Certainly not. Any addition of data is by definition uncertain. But still people rely on it? They rely on it, even though it isn’t certain. It is intuitive enough for us. Yes. When you board a plane, you board it and risk your life, right? That’s not certain either. That is a scientific result; science is never certain. Science is like here: it’s a hypothesis you assume is reasonable and rely on, but it is not certain. Science is not mathematics. So how is this a fortiori different now from the explanation in the previous lesson of induction and deduction? No, it isn’t different. It’s the same thing. The explanation here is a line of reasoning that exists in every induction. I’m about to show that, exactly. I’m about to show that. In the previous lesson I talked about how we make an analogy. When we say: the frog is green, this is also a frog, so this too is green. We make an analogy between frogs. What is such an analogy based on? Such an analogy is based on an implicit generalization, right? Because I’m basically saying that all frogs are green, not only this specific frog. And once I say that all frogs are green, in particular the second frog is green. Right? What is that generalization? That generalization is finding the microscopic parameter. Because that basically says that the parameter that matters for knowing whether something is green or not is its being a frog. Right? Because look, from a simple perspective, when we want to build a table for an analogy, how do we make a table for an analogy? Frog A is green; wait—green, frog A, frog B. Frog A is green, therefore frog B is also green. Ostensibly that is the table of paradigm construction, right? That’s a table of two by one, a vector of length two, right? But that’s not right. Why not put here, I don’t know, a race car? You assume there is something shared by these two. I would say, if you want to write it explicitly, then I would say: frog. A is a frog, right? B is also a frog. So if A is green, then B will also be green. And that is a table of two by two, not of two by one. And it is always like that. That is actually the expression of the generalization. That missing row is actually the expression because what is my comparison between A and B, that they are both green—what is it based on? On the fact that they are both frogs. I’m really assuming that all frogs are green. So I can write it in the form of a two-by-two table. Okay? But behind every inference from datum to datum sits a theory. The theory that says this is a property of frogs; it is not some property of this particular frog, but rather a property of a frog as such. And that is a generalization, because it basically says that its being a frog is the relevant parameter for its being green. Just as talent alpha is the relevant parameter for success in physics or mathematics or history or whatever it may be. Okay? So behind these kinds of considerations there always sits a theoretical layer that is not exposed to our eyes. And we—well, we basically assume it, while we ourselves often aren’t even aware of its existence. And this is a large part of the problems regarding the logical rules, as we will see later. Once we understand that behind these inferences of a fortiori, paradigm construction, and so on, there sits a theoretical layer of such parameters, we will suddenly understand many things that you don’t understand without that. Such as: why one does not rotate an a fortiori anywhere. The Sages were right not to rotate an a fortiori anywhere. Okay, so that is a second indication of the importance of the theoretical layer. Give me a third indication. Only because we have a theoretical layer can we even think of making a certain a fortiori. Okay, so yes. Look, until now we talked about a fortiori and paradigm construction, right? Basically, if we look at tables, then we’ve come out… the first form was an a fortiori: zero, one, one, and a question mark, right? Once I adopt a certain form for presenting an inference or a problem, the advantage of that, as we know from logic, is that now it can be generalized and all the possible types can be checked, classified, combined, and so on. So now we already have a language; a language is beginning to form for us. This is usually the first step in building a logic. So look: now I can just ask about inferences of other types, without naming them. What about these data, for example? Or I don’t know, or this? These are the data I have. Can I infer some conclusion about the missing cell or not? Right, each of these would be some kind of rule. A hermeneutical rule. We can call it this or that, it doesn’t matter. Now, we already know the name of what happens here. Say I have a structure like this. What should the answer here be? We already know. The answer is: unknown, either zero or one. Why? Because there is an a fortiori here that has a refutation against it. An a fortiori after there is a refutation against it—our situation is that we don’t know whether the result is zero or one; the question remains open. Right? So this table, for example, we already know what rule it represents. It represents a compound a fortiori rule with a refutation against it. Okay? The same thing, a table like this. Without this. Right? That too is the same thing. It’s an a fortiori with a refutation against it, and we also already know what the result here will be: zero or one, we don’t know, it’s open. But now I can ask myself more—I can go wild as I like. What about… what about something like this? X, circle—start playing around here. Whatever, doesn’t matter. What about this? We don’t have names for this table, but in principle this too presents some state of data, and I can always ask myself whether from these eight data points I can infer something about the ninth datum. That is a legitimate question. Right? For example, if we made a refutation to an a fortiori, at the next stage the Gemara says, okay, this one will prove it, and brings another row. Now we have a three-by-three table. An a fortiori, and a refutation, and “this will prove it.” Okay? Sometimes another row like this will come here, and we have to ask what it does. So all these things have verbal descriptions—it’s just that when we look at the verbal description it’s very hard to follow, very hard to know whether the inference is correct or not, whether they handled it correctly or not; you constantly have to activate your intuition. And the more complicated it gets, the more the intuition slips away from us. But if we manage to develop a systematic language with an orderly method for how to handle every table, then there is no need for all those intuitions. We can make an algorithm that will tell us: given this data, in a given n-by-m table, with such and such data, what is the answer in the missing cell. I can tell you: it’s one, zero, or unknown. There is a rigid algorithm. Okay? It doesn’t matter at all. Therefore this is the great advantage of this form of presentation. Once you find a form of presentation, that is the first step toward making something logical. You define how to present it. Now you can make variations, present other things, and ask about them. You can combine things. Say you have two a fortiori arguments. Two a fortiori arguments—here too there is one, one, zero, question mark, and here too, with different entries. Meaning, as if the rows and columns—the columns can be the same, say, and the rows different, or something like that, or at least one of them different. Now the question is how to combine the two together, and whether together I can reach a better answer than either one alone. All in all I build the complete table and ask myself what to fill into the cell. Okay? What is this table, for example? What is this table? Paradigm construction, right, what we did before with the frogs. Frog A is green, frog B—sorry, A is a frog, B is a frog, A is green, is B also green? The answer is one. Again, under the assumption—and again there is an assumption—that being a frog is relevant to the property of being green. The same assumption as the skills in physics and history. Okay? But with the assumption—the assumption is already basically hidden by the fact that I presented all these data on one table. The moment I present all these data on one table, I have basically already assumed implicitly a connection between them, that they are governed by the same microscopic parameters, the same alphas and betas. Okay? So here of course I am already after that stage. So here we have a name for this thing. Of course I can also ask what happens with zero and zero; I also have the tendency that here too there should be one. That too is a kind of paradigm construction. I simply compare hierarchy relations. Just as here one corresponds to one, here too one will correspond to zero. I can make such an analogy, right? That too is a kind of paradigm construction, but slightly different. So the Sages would call both this and this paradigm construction, but they can have different logical properties. And we will see that when we draw the table, when we solve the table. Okay? So often something is called by the same name, but in fact it has slightly different properties because it is really a somewhat different variation. I’ll now show you a slightly more complicated example. Let’s see how much I need. Here I have A, B, and C. Here I have one, one, question mark. Here I have one, one; and here I have one, and here I have one. That’s the table. What do you say this is? The longest thing in the table inside this. Come on, what do you say? Without the last column if you want. We’ll see afterward whether we need the last column or not. This is paradigm construction from two texts. Why? Look at something very simple. How is paradigm construction from two texts built? In paradigm construction from two texts, we have here two teachers, A and B, and a thing learned called C. Okay? How does it begin? It begins: I try to learn C from A. Right? I say, if A, in which there is no X and there is Y, then C, which has X, surely has Y too—a fortiori. Right? Then I say no: what about C, since it has Z and here there is no Z? Okay? So I say, B will prove it. Why? Because B has Z, right? And it has Y. And here—and wait. If it has Z, then it is even more stringent. That’s backward. There should be Z here, and maybe there should be Z-prime here, right? So now I say, say this has X and doesn’t have Z, right? It doesn’t have Z. So now let’s learn from it. Then we find something else that C has and B doesn’t have. So in the bottom line, what’s happening here? I’ll erase all these confusing things. In the bottom line, what we have here is basically the following. A, B, and C have some common property—let’s call it Z: Z and Z. There is something that A has and C doesn’t—that is the refutation. There is something that B has and C doesn’t, right? That is the refutation of this. Now I say as follows: when I try to learn from here, I say, what about this one, since it has Y-minus, let’s say. Then I learn from here and I say, what about this one, since it has X-minus, okay? Then I say: let them both come. Together they do manage to teach. Their common side is that both have Z; this one too has Z. That is basically the structure of a common denominator. So look what this means in terms of the table: that A has something by which it is more stringent—wait, how does it go there? B and C are here. B is more stringent than A in a certain respect—let’s call it zero and zero for the start, okay? B is more stringent than A in a certain respect, right? Therefore I want to learn by a fortiori from A that here there should be a one. Then I say no, but there is a refutation here, and this is the refutation of this a fortiori, right? So I say, let’s learn from C as well in the same way by a fortiori. It says no, C also has a refutation; this is that refutation, right? So this column is a refutation of this, and this column is a refutation of this. And this has some common denominator. Or you could say it this way: there is a common denominator to all of them, and that is what ultimately teaches me. Although we’ll later see that maybe we don’t need that at all. So basically what is here is a common denominator. Again, look—let’s leave the last column aside for the moment because it is more confusing. Look, it’s like this. Look for a moment at this two-by-two table. This is an a fortiori from A to B—this is how they start, right? From A to C, sorry, from A to C. This is this cell. I also have an a fortiori relation from B to C, right? Each one individually attempts an a fortiori, but there is a refutation against each such a fortiori individually, right? So against this a fortiori there is a refutation—where is it located? Here. Right? This property serves as a refutation against this a fortiori. And against this a fortiori there is also a refutation; this is that property, right? Therefore in terms of the number of properties, there is X, Y, Z, I don’t know, T, whatever, okay? By definition, in paradigm construction there will always be four properties and three things being compared. There will be an a fortiori from A to C, an a fortiori from B to C, and a refutation against each of the a fortiori arguments, right? Now, paradigm construction from two texts can also start not with an a fortiori but with a paradigm construction from one text. Right? I can start by comparing A to C because they are similar, not because this is more stringent; I start with paradigm construction. It’s the same thing. Then I say no, they are not the same—what about that one, since it has such-and-such. Then I move to the second and prove from the second either by a fortiori or by paradigm construction, no matter. Which means that here there can be either zero or one. There are really three types of paradigm construction from two texts. There is paradigm construction from two texts built on two texts that teach by paradigm construction, one built on two texts that teach by a fortiori, and one built on one text by paradigm construction and one text by a fortiori. So here there is zero and here there is one, right? Obviously when here there is zero and here there is one, it’s the same thing; just flip the rows and columns, it doesn’t matter. So there are really three different tables, all of which the Sages call paradigm construction from two texts. We’ll see later that they have different properties. Even though in all of them the correct answer is indeed one. You can learn that the result is one; that is the correct answer. And it is not trivial that this is the correct answer; on the face of it, it is refutable. The correct answer is one. We will see that this is true. Okay? So here there is a larger table that has a name—paradigm construction from two texts—and this is its representation. Now, of course there can also be more. Now, how do you refute paradigm construction from two texts? There, what should be here? You want the common denominator itself to be more lenient in some respect? Not the common denominator, only A and B, without C. You can show that A is something like this? No no, that is a text—he says a refutation on, sorry, a refutation on zero zero one. He says, their common denominator is that both require the altar, and then you say: what about them, since their blood was offered in, I don’t know what. So when I look at A and B, they have some common property that does not exist in C. So when you learn from each one individually, you have a refutation that singles it out. Fine, the second one shows that this refutation is not relevant. But the second one has a refutation that singles it out, so the first one shows that that refutation is not relevant. So what is relevant? The common denominator to both of them, Z. But wait a second—there is something else they share too, also A-prime. And this one doesn’t have A-prime. So who says Z is the determining factor? Maybe A-prime is the determining factor, and then you can’t learn what you want to learn. Okay? That is a refutation, that is a refutation. And basically what we now have to show with our technique is that when there is such a table, the correct filling here is no longer one. That also works with zero one zero, no? What? That also works with zero one zero. No no, that’s not… that means C has… both of them have A-prime. If C has A, then C is more stringent than both of them. So if they have A-prime, then certainly C, which has A, will only be better than them; it can’t be worse than them. Okay? Therefore a refutation on the common denominator is simply another column. So here we already have a name for this table too: a refutation on the common denominator, a common denominator with a refutation. Okay? And so on. You can reach tables of one hundred by five hundred and twenty-four. It doesn’t matter; the words aren’t necessary here. That’s the advantage of mathematical symbolism. Okay? The words aren’t necessary here. Now what we have to see is what one does with this. How do you handle this kind of table, and how do you really show that the result is one or zero? Why is this a refutation? Why does this prove? What is the technique behind it? In order to understand the technique behind it, I’ll have to do that next time, I’ll have to do it next time. We’ll need once again to resort to the microscopic parameters. Because now again we’ll have to see—remember the alphas and betas with the links to physics and history? Here of course it’s much more complicated, but still if there is an inference here, that means there is some theoretical system behind this table. That means there are some microscopic parameters that explain all the data, except for these—these are what are known to us. After we find them, we’ll ask ourselves what they say about this cell, and thus we’ll know whether to fill it with zero or one. Okay? So basically the microscopic parameters are the root of the solution to all these problems and the root of the systematic treatment of this problem. Now the nice point—and we’ll get to this later—is that when I show you how to do this in an orderly way, we’ll gain two different benefits from it. First, give me a complicated table however you like, and I’ll tell you what should be written here, one or zero, it doesn’t matter. There’s no need to go through anything, not to understand anything, not even to follow the stages in the Gemara. Give me all the data after the Gemara has already brought all the refutations and all the rest. Write for me in a table all the data the Gemara has raised. I don’t care what the initial assumption was and what was a refutation and what they did afterward; the linear shape in which the sugya works is not important. In general, give me all the outcomes, all the data that accumulated by the end, write it in a table, and I’ll tell you the answer. Okay? That’s one point. The second point is that we will manage to understand what theory stands behind the Jewish law. Meaning, for example, if there is intercourse, canopy, and money here, and I ask myself whether this effects betrothal, whether it redeems second tithe, whether it effects marriage, all sorts of things like that—not only will I be able to show whether intercourse effects betrothal or something like that, but I’ll also show you why it effects betrothal. Because it has some certain parameter, alpha, beta, or some combination of them, that is what matters in order to effect betrothal. So this tool is not only a tool that helps us understand how to fill the empty cell. This tool also helps us understand things that no one even tries to understand. The question is why money really effects betrothal. Why? What is there in money, or what is there supposedly… in betrothal that exists in money, and therefore money effects betrothal? No one even asks himself that question. From this model it comes out immediately. The answer comes out mathematically. Again, an unidentified answer. Some parameter emerges—we’ll call it alpha—which exists in money, in canopy, and in intercourse, but not in other actions. And that parameter is what matters in order to effect betrothal. To identify alpha, of course, doesn’t come out from here. Who is this alpha? That we will have to think about. We will have to ask ourselves what is common to money, canopy, and intercourse. We know the concepts, so let’s see what they have in common, and from this we can infer conclusions about what is required in order to effect betrothal. We are basically asking some kind of rationale of the verse, or some kind of question of what the idea is that stands behind the law. Okay? So this instrument is not merely a formal instrument; on the contrary, it helps us move one layer deeper in the question of why Jewish law is really the way it is. What stands behind it? Yes. What I was trying—what I understood from what you said is that basically we are trying to create an a fortiori on a gap between two variables, and afterward we refute it by means of a third variable. Then we introduce a fourth variable and discover that the whole reason we initially thought it was more stringent than the other variable is not the reason. We had a vague intuition, and now we’re clarifying it. Correct. Once we introduced this variable, it became partially clarified. It may be that alpha still matters, but beta is also needed. It is more stringent, but not for the right reason, and then also not—in fact your true parameter is another parameter altogether. Once you say it, then we already know the point, the deep point of the whole… we already formulated it. Why do I need to formulate it this way? When they tell me the common denominator among them is, I don’t know what, that they redeem second tithe, or that they… No—the common denominator among them that they redeem second tithe is a property. But I’m asking what is in them? Why do they both redeem second tithe? But that is our Z. It is the… it is never written in the Gemara? No. It is never written in the Gemara. Why? It just isn’t written. The common denominator among them is that both are his property, his property. I gave you a bad example. But that they redeem second tithe. Redeeming second tithe is a halakhic property. I’m asking why it redeems second tithe. Don’t tell me a law; tell me what its theoretical characteristic is by virtue of which it redeems second tithe. That’s the question I’m asking. To go from the laws to the theory that grounds those laws. What does the table… you’re saying, say, second tithe is a property and it’s not the point. Fine. Now I want to ask what in second tithe is the point, and in what sense the table advanced me. I know before and after the table that second tithe is the point. No, you don’t know. Because now from this table you know that, say, A has alpha and beta and lacks gamma; C has another combination, and so on. That gives you some indication to understand who alpha is, who beta is, and who gamma is. You know that it exists in this, this, and this, and it also exists in this and this but not in this and this. So that gives you some indication to understand who this alpha is. Maybe it’s, for example, benefit? Money and intercourse—say if we discover a property that exists in money and intercourse but not in canopy. Right? For example, betrothal. Betrothal is effected by money and intercourse, right? Canopy does not effect betrothal. But marriage—canopy does effect that. So what does this language mean, basically? That in order to effect betrothal you need to transfer benefit to the woman, in microscopic language—either intercourse or money. But why should canopy do that? In canopy there is no benefit. Ah, but canopy effects marriage and money does not effect marriage. So we see that it is more stringent than money. More stringent than money in another parameter, the one that matters for marriage. But in terms of benefit, which is what matters for betrothal, it is not more stringent. What does canopy have that intercourse and money do not? Some kind of bringing into one’s domain. And that is really the meaning of marriage. Therefore canopy does it and money and intercourse do not. All this could also have been said without arranging it in a table. Obviously that’s simple. But if you have a table of ten by seven, let’s see you say that without a table. Obviously, in simple problems it’s always like that. When you take Aristotelian logic—if you have a very simple problem, you don’t need any logic. Everyone knows how it works. Why do you need Aristotelian logic to understand that if all human beings are mortal and Socrates is a human being, then Socrates is mortal? But let’s see you analyze an argument that says: all human beings are not green, some of the green things have wings, all winged things are not frogs, and some frogs are human beings. Now the question is whether human beings have wings. Understand.