Halakhic Positivism, Lesson 7
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Table of Contents
- A fortiori reasoning, fillings, and the simplicity of the model
- Parameters versus laws in the example of tooth and foot and horn
- The rationale of the verse and the conception behind the laws
- Common denominator, refutation, and the refutation of a stricter side
- The distinction between halakhic characteristics and factual characteristics
- A chemical analysis of Jewish law as a model of inference
- Relevance, tables, and the example of the doorpost and tzitzit
- Not deduction, positivism, and the limits of an algorithm
- Philosophy of science: Carr, Bacon, Semmelweis, and Kahneman
- Algorithmizing inference and reducing intuition
- The common denominator as scientific generalization and refutation as an attack on the table
- Extracting the model from a table by means of order relations between columns
- The topic in Kiddushin: “Canopy effects acquisition” as an expanding table of a fortiori reasoning, common denominator, and refutations
- Rav Huna’s answer and clarification of “doorpost and mezuzah” within the topic
Summary
General overview
The lecturer summarizes a formal model for analyzing a fortiori reasoning, common denominator, and refutations באמצעות tables of actions and domains/areas of authority, and argues that behind halakhic laws there stand theoretical “parameters” such as alpha and beta that generate the outcomes. He prefers simpler models that require fewer parameters, and explains that refutations mainly undermine the assumption that the cases belong to the same “family” and therefore deserve to be placed in the same table. He compares this move to the logic of science and to a process of elimination and “chemical analysis” of phenomena in order to reconstruct hidden components, while emphasizing that there always remains an intuitive component at the stage of choosing the data and the table.
A fortiori reasoning, fillings, and the simplicity of the model
The claim is that a fortiori reasoning is sometimes built as a table of two actions and two domains in which one value is missing, and that the missing value can be filled in two ways and we can examine which filling produces a simpler explanation. In the blue filling there is a single parameter, alpha, where a strength of two alphas is enough for the two results and a strength of one alpha is enough only for one. In the red filling there is a dependence that requires two separate parameters, alpha and beta, so that one case requires beta and another requires alpha, and therefore the blue solution is preferred as simpler.
Parameters versus laws in the example of tooth and foot and horn
Tooth and foot and horn are defined as damagers, and the public domain and the injured party’s courtyard are defined as domains, and the table describes who is exempt and who is liable where. The alphas and betas are not the damagers and domains themselves, but rather internal traits of them that have not yet been identified, and the structure of those traits is what explains the halakhic outcomes. A suggestion is raised to interpret a parameter like alpha as a trait such as “damage in its ordinary manner,” which helps explain why tooth and foot are exempt in the public domain and liable in the injured party’s courtyard, but the essential identification of alpha and beta remains an interpretive question.
The rationale of the verse and the conception behind the laws
The lecturer connects the model to the idea of the rationale of the verse and cites Maimonides in Guide for the Perplexed about those who find it difficult to give reasons for commandments because that makes them seem “human” rather than “divine.” He argues that the laws are not merely “scriptural decrees for no reason,” but express a conception that is not written explicitly, and the way to uncover it is to infer from the laws to the characteristics that generate them. He emphasizes that the model is incomplete because there is no direct way to identify alpha and beta, but one can retrospectively examine characteristics that are stronger in horn than in tooth and foot and test their meaning by reasoning.
Common denominator, refutation, and the refutation of a stricter side
The common denominator is described as a move from several attempts at a fortiori reasoning that collapse because of refutations, to looking at a broader table in which the combination of two sources together does create a valid derivation without adding another verse. The refutation of a stricter side is explained as a case where one proposes an alternative explanation for why the law in the two teaching cases derives not from the shared trait Z but from an additional trait W that exists in both of them and is absent from the case being learned, and therefore that mere alternative possibility is enough to make the derivation non-necessary. The difficulty is that in the Talmud in tractate Makkot 4 there appears a refutation, “What is unique about the two teaching cases is that they have a stricter side,” and Tosafot, the Ritva, and the other medieval authorities struggle with this because apparently it undermines the entire logic of the common denominator.
The distinction between halakhic characteristics and factual characteristics
The proposed explanation is that the refutation of a stricter side is possible when X and Y are halakhic laws, because two different laws can reflect the same shared microscopic parameter that does not exist in the learned case, and then this is a refutation like W. By contrast, when X, Y, and Z are factual characteristics there is no room for a refutation of a stricter side, and the lecturer connects this to characteristics in Bava Kamma such as “its initial formation was for damage,” “its way is to go and cause damage,” and “another force is involved in it” as factual characteristics, as opposed to characteristics such as “exempt in the public domain,” “exempt for vessels,” and “exempt for concealed items” as halakhic characteristics. He clarifies that by “factual” he means a fact that is legally relevant, and the law determines which facts count as relevant, but the question is whether two different laws necessarily express two different rationales, and he argues that not necessarily.
A chemical analysis of Jewish law as a model of inference
The lecturer compares the process to chemical analysis, in which hidden components are identified based on observable reactions in experiments, and “components” are built out of the phenomena by way of elimination. So too in Jewish law, one looks at halakhic phenomena (zeros and ones in a table) and infers from them theoretical components (alpha, beta) that explain the patterns. The thesis is that there is no direct access to the “theory of the Torah,” and therefore the analysis proceeds from the laws back to the characteristics.
Relevance, tables, and the example of the doorpost and tzitzit
An example of a fortiori reasoning is brought: “If a doorpost, which is exempt from tzitzit, is obligated in mezuzah, then a four-cornered garment, which is obligated in tzitzit, all the more so should be obligated in mezuzah,” and this is rejected intuitively because there is no connection between what obligates mezuzah and what obligates tzitzit. The conclusion is that behind every a fortiori reasoning there sits an additional assumption of belonging to the same semantic field, and that the construction of the table itself expresses the assumption that all the items are “from the same family.” The claim is that the problem with the doorpost argument is not an internal refutation of the structure but the very act of placing the items into the same table, as is also emphasized by the question why not insert “pig is forbidden to eat” into the table of damage liabilities.
Not deduction, positivism, and the limits of an algorithm
The lecturer presents the topic as part of a series on positivism and emphasizes that halakhic inferences are not necessary deductions, even if one can mechanize the calculation within a given table. The difference between “mathematics” and halakhic inference lies at the stage of determining the table, meaning in the decision of what to include and what to leave out, which is a stage laden with assumptions and intuitions. He argues that the Sages assumed a connection when they performed a fortiori reasoning or a common denominator, and if it seems to us like the doorpost and tzitzit, then we need to clarify what assumption the Sages were making.
Philosophy of science: Carr, Bacon, Semmelweis, and Kahneman
A criticism by Carr is brought against the Baconian description of collecting facts from which one extracts a theory, because there are infinitely many facts and the question of which facts to collect already depends on a hidden theoretical hypothesis. The example of Semmelweis and puerperal fever illustrates a situation in which, without a theory about microorganisms and cleanliness, it is not clear that handwashing is relevant, and it looks like an arbitrary act. The lecturer describes a constant movement between facts and theory and between theory and facts, and connects this to unconscious versus conscious thinking in Daniel Kahneman, in which initial intuitions guide the choice even before an explicit theory is formulated.
Algorithmizing inference and reducing intuition
It is argued that one cannot get rid of initial intuitions entirely, but one can reduce their scope by turning the stages of inference within the table into mechanical and algorithmic steps. The Sages carried out complex moves using intuition alone, whereas the proposed method tries to leave intuition only at the stage of constructing the table and afterward continue with calculation. Even attempts of big data and data mining to “expand” data collection do not eliminate the need for assumptions of relevance, but at most reduce the space for error while continuing to depend on intuition and framing.
The common denominator as scientific generalization and refutation as an attack on the table
The lecturer describes scientific generalization as a common denominator: examining different cases with mutual refutations and inferring a shared trait that explains the result, like moving from different observations to formulating a general law. “Diversity of evidence” is explained as adding rows/columns that increase the strength of the generalization, but there always remains a possibility of error, and therefore this is “probability” rather than mathematical proof. A refutation is explained as the claim that the table itself is mistaken because the items do not belong to the same domain, not as a disproof of a deductive move within a given framework.
Extracting the model from a table by means of order relations between columns
A method is proposed for building a graph of inclusion/strength relations between columns or rows: when in every row the values in one column are greater than or equal to the values in another column, a dependence is formed that makes it possible to assign alpha to the weaker column and two alphas or alpha+beta to the stronger column, with preference for the simpler solution. The absence of an arrow between columns requires a separate parameter like beta, because the model needs to explain both dependence and absence of dependence. It is argued that from here one can return to the table and complete parameters on the other side as well, but in complicated graphs filling in the parameters becomes a non-trivial mathematical problem.
The topic in Kiddushin: “Canopy effects acquisition” as an expanding table of a fortiori reasoning, common denominator, and refutations
The lecturer follows a topic in Kiddushin in which Rav Huna says, “Canopy effects acquisition by a fortiori reasoning,” from money, which does not complete yet does effect acquisition, to canopy, which does complete, that it should certainly effect acquisition. The Talmud refutes this: “What is unique about money is that sacred items and second tithe can be redeemed with it,” and proposes, “Intercourse will prove it,” but that is refuted: “What is unique about intercourse is that it effects acquisition in the case of a yevamah,” and then comes “Money will prove it, and the rule returns,” as a structure of a common denominator. The common denominator is refuted with “What is common to them is that their benefit is great,” and then “A document will prove it,” in which there is no pleasure, but that is refuted with “What is unique about a document is that it dissolves a marriage for a Jewish woman,” and then a higher-order common denominator is built between money and intercourse on one side and document on the other, and that is refuted with “What is common to them is that they exist against her will.”
Rav Huna’s answer and clarification of “doorpost and mezuzah” within the topic
Rav Huna answers, “We do not find money acting against her will in matters of marriage,” and disconnects the example of acting against her will by money in the case of a Hebrew maidservant from the domain of marriage, so that including it in the table of marriage is invalid. The overall picture is presented as a process in which the Talmud adds columns of teaching cases and refutations, and at times builds a “common denominator squared,” in which one of the components is itself a common denominator, something that makes it hard for human intuition to follow without a computational mechanism. The lecturer concludes with a personal example about an a fortiori reasoning in Tosafot where a three-by-three table confused “the whole yeshiva,” whereas drawing the table and calculating gave a clear result, and from that he emphasizes the advantage of the computational method when intuition breaks down.
Full Transcript
[Rabbi Michael Abraham] Let’s summarize for a moment where we’re standing. I spoke a bit about a fortiori reasoning, and I showed roughly
[Speaker C] briefly, just to get into it, that the claim
[Rabbi Michael Abraham] was that a fortiori reasoning is usually built in this kind of way: we have two actions and two domains, zero one one, and here there’s a question mark. And my claim was that we basically make two kinds of fillings, either zero or one, when in one filling I can show that this has an alpha and two alphas, this requires two alphas and this requires one alpha. This model explains the table in one filling, right? If this has a strength of two alphas, then it will manage to do this, it will manage to obligate in the injured party’s courtyard and obligate in the public domain. It will manage to do both because it has enough power. And this one has only the power of alpha, so for this it isn’t enough, but for this it is enough. So the blue filling is its model. And in the red filling we see that there is a dependence, right? One zero and zero one. So it’s clear that if here there is alpha and here there is beta, here there will be beta and here there will be alpha. It looks like that’s the solution for the zero filling. You see that this has the power of alpha, so it doesn’t manage to do this because here you need the power of beta. And this one needs alpha, so it does it. Here beta, so it does this because here beta is enough, and this it doesn’t do because alpha is needed. Okay, basically there’s some sort of dependence here, and therefore the conclusion was that the blue filling is preferable. Because in the blue filling the solution is a simpler solution. We always choose the simpler solution. In the blue solution we have only one parameter, only alpha. In the red filling the assumption is that we need to resort to two separate parameters. That’s what we had regarding a fortiori reasoning. Now last time I took one more step, because my goal was to show why this model is so important, these parameters, as I call them. Maybe I’ll just clarify, I’ll go back for a moment to clarify what the relation was between the parameters and the variables in the table. Here, let’s say, we’re talking about tooth and foot and horn. Tooth and foot and horn are two damagers. The injured party’s courtyard and the public domain—or sorry, the public domain and the injured party’s courtyard—are domains. And the table shows me what does what. Meaning, if tooth and foot are exempt in the public domain and liable in the injured party’s courtyard, and horn is liable in the public domain, then therefore it is liable also in the injured party’s courtyard. What are these alphas and betas? Meaning, these are the players on the board, the domains and the damagers. These alphas and betas are characteristics of the domains and the damagers. I haven’t identified what those characteristics are, but I showed that there are such characteristics, and that the structure of the characteristics in horn and in tooth and foot is what generates the halakhic outcomes. That’s why I call it a model. Meaning, inside horn, tooth and foot, the injured party’s courtyard and the public domain, they have certain features, and those features basically determine why horn manages to do this and not that, and why tooth and foot manage to do this and not that. Because tooth and foot have such-and-such a feature. Let’s say that it’s damage in its ordinary manner. Let’s say damage in its ordinary manner would be, yes, let’s say if we say
[Speaker D] that tooth and foot are A, then damage in its ordinary manner, let’s say, is alpha.
[Rabbi Michael Abraham] And therefore if something causes damage in its ordinary manner, it is less liable, it is not liable in the public domain. Okay? This is something not in its ordinary manner or something like that. This is an attempt to locate the meaning of these parameters. But basically the underlying assumption is that at the foundation of every law and every set of laws we’re looking at there stand certain characteristics. Theoretical characteristics. And this connects a bit to—once we talked about—really when there was no board, in this class when there was no board, I talked about the rationale of the verse. And Maimonides there, I think I mentioned him, Maimonides in Guide for the Perplexed says that there are those for whom it is easy—rather, difficult—to give reasons for commandments, because then it turns the commandments into something human. If they have no reason, then apparently it’s something divine. So I’m really continuing that line of thought here and saying: these are not just scriptural decrees for no reason. Rather, behind the Torah’s determination that tooth and foot are exempt in the public domain and liable in the injured party’s courtyard, and horn is liable in the public domain, there stands some conception. It wasn’t just determined for nothing. I don’t know what that conception is; it isn’t written in the Torah. This method helps me try to understand what the conception is from the laws. I look at the laws and see what really stands behind them. Now this isn’t complete because I’m not identifying what alpha is and what beta is. I have no direct way to identify them. Later I can look and see whether there is a certain characteristic that exists more strongly in horn than in tooth and foot, and I can try to see what its meaning is on the basis of reasoning. But that is already a question of interpretation. Why am I saying this? Because in the previous class I tried to show the importance of this distinction between the laws and the characteristics that generate the laws. And what we saw last time was that there is—we saw the inference of the common denominator, and I said that we can really see it here in the table. The inference of the common denominator—let’s look here for a moment, let’s look at these four as the sources for the time being. Okay? So this is an inference of the common denominator. What does an inference of the common denominator mean? Let’s look at these four for a moment. In these four there is basically a fortiori reasoning, right? This is the structure of a fortiori reasoning. We already know that it gets filled in somehow. Okay? Now we say there is a refutation of the a fortiori reasoning. A refutation will always look like this. Okay, the a fortiori reasoning has fallen. Now we’ll try from another direction. Now we do a fortiori reasoning from pit. Okay? A fortiori like this. We’ll see that this too has a refutation, this refutation, so this too falls. And then we say, “and the rule returns,” without adding anything. Let’s not add another verse, but simply draw the larger table now. We’re no longer looking only at a three-by-two table, but at a three-by-four table. And if you look at the whole table with both of them together, it does work. That is basically what the common denominator does. Okay? And I tried to show that the common denominator too is really based on some microscopic parameters. There too there is an explanation sitting behind it; it’s not just some kind of hocus-pocus. Now I also tried to show, through this claim that there is an explanation, the meaning of the refutation of a stricter side. We saw in the Talmud in tractate Makkot that there are situations in which the Talmud refutes a common denominator and says: what is unique about the two teaching cases is that they have a stricter side. Now Tosafot already asks, and the Ritva and everyone—all the medieval authorities—so if that’s the case, then you’ve knocked down the entire logic of the common denominator.
[Speaker D] Where exactly is that?
[Rabbi Michael Abraham] In Makkot 4. So the medieval authorities ask, basically, what is the logic of the common denominator? We have two teaching cases from which we want to learn about C. Now here there is some characteristic that isn’t here, and here there is a characteristic that isn’t here, and there is also a characteristic common to all of them. Okay? So we basically say: I try to learn from B to C. I say no—what is unique about B is that it has X; can you say that about C, which does not have X? I try to learn from A. Then no—what is unique about A is that it has Y; can you say that about C, which does not have Y? Okay? Now I say: “and the rule returns”; the common denominator is that both of them have Z, and it too has Z, and therefore one can learn from both of them to Z. Now if we make a refutation of a stricter side, then you can’t do this. Because what we’re saying is—what is unique about A if—meaning, how does one refute a common denominator? How do you really refute a common denominator? When you add W here, and here there is no W. Right? And we say: what is unique about the two teaching cases is that they have W. Why is that a refutation? Because it could be that the law that exists in the two teaching cases does not derive from Z, in which case it would also exist here because here there is Z, but rather derives from W. And W is not here. So that is a refutation; you can’t know. We said that for a refutation it is enough to show that it is not necessary. You don’t need to prove that it is not true; rather, since there is an alternative explanation, that alone is enough for it to be a refutation. Now if there is no W, then there is a common denominator, right? If there is no W, then everything is fine. But suddenly it turns out that in several places the Talmud says—at least according to one tanna—that there is a refutation from a structure like this: what is unique about A and B is that they have a stricter side. Meaning, A has Y and B has X. Now if that’s the case, then every common denominator in the Torah collapses. That is always the structure of a common denominator. And therefore the medieval authorities struggle over how to understand this thing called a refutation of a stricter side while still preserving the logic of the common denominator. How can that be? And they give various explanations. I proposed the explanation that is based on the microscopic parameters, and what I basically wanted to claim is that in a place where X and Y are laws—meaning, what is unique about A is that it has the law Y, and what is unique about B is that it has the law X—then one can show a refutation of a stricter side. And why? Because it could be that the two laws X and Y reflect the same parameter.
[Speaker C] That it
[Rabbi Michael Abraham] does not exist here, except that on the halakhic level it appears here in one form and here in another form, and then this really is a refutation, because when I show that there is the same characteristic in the two teaching cases and it is not in the learned case, that’s
[Speaker E] like this, exactly, it’s like this.
[Rabbi Michael Abraham] By contrast, if the X and the Y and the Z are factual parameters and not halakhic ones, then there is no such thing; there is no refutation of a stricter side. And that is indeed what Tosafot will find—a refutation of a stricter side in a place where there are factual characteristics, and they learn Nazir—but in Bava Kamma, for example, the characteristics are factual. “Its initial formation was for damage,” “its way is to go and cause damage,” “another force is involved in it”—all these are factual characteristics. Right? Someone who is exempt in the public domain, exempt for vessels, exempt for concealed items—those are halakhic characteristics. But the characteristics of “its way is to go and cause damage,” and so on, are factual characteristics. Okay. So I brought this—I jumped a bit ahead of the order in which I’ll get to it in a moment—but I jumped straight to it in the previous class in order to show why this distinction between the laws and the characteristics that generate them is important. Because if you think about it, this thing is a kind of chemical analysis. When we take a certain substance and want to see what components it has, we subject it to various experiments in the lab. We pour a certain substance into it and see whether it—I don’t know—explodes, heats up, changes color, does various things. Now I say: if two different substances react the same way to some third thing, then apparently they have something in common, alpha. So I write alpha in both of them. By contrast, to something else this one reacts this way and that one reacts differently. So I say, ah, apparently this one also has beta and that one doesn’t have beta. And then I build the chemical components that are inside the substance out of the phenomena. I look at phenomena—that’s what I see in the lab—and from the phenomena, by way of elimination, I try to identify the components inside the substance. That is exactly what we are doing here. In other words, this is a chemical analysis of Jewish law. I’m trying to show, through the laws, what the chemical components are that generate the laws. Now I have no way to know directly what the Torah’s theory is; it doesn’t tell me. So what do I do? I look through the laws and say: ah, if this law exists in A but not in B, then apparently in A there is some chemical component that B doesn’t have. And if in A and in C—if C also does it—then C also has that component. And if both of them do some third thing, then they have yet another component that B doesn’t have, and so on. That is basically what we’re doing here. One has to understand the basic claim. Only a moment ago someone here asked me to wait? I don’t remember anymore. Okay.
[Speaker C] And with tooth and foot it makes sense that there would be a difference between the public domain and private property. But I don’t remember right now, though there are many a fortiori reasonings in the Talmud that are very formalistic, where you ask: okay, here you have alpha and beta, but what’s the connection between this and that? And they say it anyway.
[Rabbi Michael Abraham] The point here is—that’s an excellent question, and to answer it I need a bit of time, but I’ll do it for a few minutes just because it really is a good question.
[Speaker D] I think I didn’t understand the question; I don’t know if everyone did.
[Rabbi Michael Abraham] Ah no, the question is like this. Look, the question is like this. I talked about the a fortiori reasoning from the doorpost to the four-cornered garment. Remember? We said: if a doorpost, which is exempt from tzitzit, is obligated in mezuzah, then a four-cornered garment, which is obligated in tzitzit, all the more so should be obligated in mezuzah. Meaning, why is this a fortiori reasoning not valid? Apparently it is built exactly like every a fortiori reasoning in the Torah. Because it’s obvious to us that there is no connection. There is no connection between the chemical components that obligate mezuzah and the chemical components that obligate tzitzit. Now where is that obvious to us from? I don’t know—some sort of feeling, intuition, common sense, I don’t know exactly what—and that’s the feeling. What does that mean? It means that behind a fortiori reasoning or a paradigm case or a common denominator and all these things there sits one more assumption. That’s basically what you said. There sits some assumption that this belongs to the same semantic field, the same context. Meaning, that they are characterized by the same parameters, from the same family, playing on the same field. Then I can compare them. Now I do that when I write the table, even before the analysis. The moment I wrote a table of data, or this table, I thereby said that all these guys are from the same family. Meaning, that I can do the analysis on them. Because the problem—you won’t find a refutation of the a fortiori reasoning from the doorpost to the four-cornered garment. Once I wrote this table, the a fortiori reasoning would be perfectly fine. The problem was in the fact that I wrote the table. Because when I write them in one table, I’m basically saying that all of them belong to one conceptual framework, and then I need to start discussing. Therefore, contrary to what it might seem, and it took me time to understand this, but contrary to what it might seem, there is real mathematics here. I mean, it looks like something—and in a moment you’ll see. I can let a computer calculate it, it will do a fortiori reasoning, paradigm case, refutations, common denominator, whatever you want, at any level of complexity, and it will always produce the correct result, at least in all the cases. It will produce the correct result. Meaning, it becomes real mathematics, like deduction. Why is this not mathematics? Because these things are not deduction; that is, halakhic inferences—about this we already spoke. This is really our subject. I spoke about positivism. This part is the last part of the series on positivism, and what I’m really trying to show is the logic of non-deductive thinking, of thinking that is not necessary, not logically valid. So where exactly is it not logic, if I have an algorithm that says: give me the data and I’ll tell you the result? There’s no cleverness here, it can’t make a mistake, it always works—so what is the difference between that and logic? The difference is in building the table. Here we have to make all kinds of assumptions when we build the table. The question is whom to include in the table. Why not also put in here, I don’t know, the fact that pig is forbidden to eat? Pig is also forbidden to eat. Tooth and foot are permitted to eat, horn is permitted to eat, pit is also permitted to eat—not tasty, but permitted. Pig is forbidden to eat. Why don’t we put that in here? Because it’s obvious to me that the prohibition of eating is not from the family of liabilities for damages. Meaning, it is not controlled by the same components, the same characteristics. So here I’m already introducing certain assumptions. Many times I don’t even notice it, but I’m introducing certain assumptions by the very fact that I put everything into one table.
[Speaker E] Or what I put into the table and what I didn’t put into the table.
[Rabbi Michael Abraham] Obviously. Yes. So I’m saying: the fact that I built a table—what I put into the table and what I didn’t put into the table—rightly so, is not always obvious. I agree, it’s not always obvious.
[Speaker C] Again, it seems to me that I still have to insist that there are examples in the Talmud that are more like the doorpost and tzitzit.
[Rabbi Michael Abraham] What I’m claiming is that if you find such examples in the Talmud, apparently you’re mistaken. Meaning, the Talmud assumed there was a connection between the things. On some of the places I tried to examine, by the way, I succeeded in showing that there is a connection. Meaning, one can take Rabbi Chiyya’s first a fortiori reasoning—we learned this not long ago. Rabbi Chiyya’s first a fortiori reasoning: that one’s own admission should not be stronger than the testimony of witnesses by a fortiori reasoning. In Bava Metzia there, where they say that there is an obligation in the case of partial admission: if someone is sued for one hundred and admits to fifty, then he has to swear regarding the other fifty. What happens if witnesses come and testify about the fifty, that he owes fifty? Rabbi Chiyya says that there too he must swear about the rest. Why? Because there is a fortiori reasoning. One’s own admission is not as strong as witnesses. If it obligates an oath, then witnesses certainly obligate an oath. Now here this is a fortiori reasoning that doesn’t even get started. They ask there: what’s the connection at all? Who was talking about the strength of the evidence? What does strength have to do with it? First of all, it’s not even clear that this really is the relation, that one’s own admission really is weaker than witnesses. That’s not true; one’s own admission is like one hundred witnesses. But let’s say that is true too—so what difference does it make how strong your evidence is for the first fifty with respect to the obligation of an oath about the second fifty? What does that have to do with anything? What kind of a fortiori reasoning is this? Now, this is an example of a fortiori reasoning like that—perhaps Rabbi Chaim speaks exactly about it, I don’t remember who speaks exactly about it and brings the example. I once brought the a fortiori reasoning of the snakes at the beginning, when I said that Rabbi Chaim says that a fortiori reasoning is—from the Brisk Haggadah, where they bring there—didn’t I bring it? You know that in Chad Gadya, there is there “who knows thirteen, thirteen who knows.” What does “who knows” mean? Measures. Which measures?
[Speaker B] A fortiori reasoning?
[Rabbi Michael Abraham] Exactly, the hermeneutical rules. In the Haggadahs of the usual kind, or the Hasidic Haggadahs, you’ll find “The Lord, the Lord, compassionate and gracious, abundant in kindness”—in the Brisk Haggadah, what are the thirteen measures? A fortiori reasoning, paradigm case, gezerah shavah, a fortiori reasoning, a paradigm case from two verses—those are the thirteen measures. Now if you look there in the Brisk Haggadah, you’ll see that it brings some piece from Rabbi Chaim, I think it’s called from Beit HaLevi, maybe Gerlitz. So he brings there from Rabbi Chaim that a fortiori reasoning is a formal matter; it is not our ordinary logic. It belongs to that camp that Maimonides talks about, those who say that something logical is human, so there can’t be logical things in the Torah. So he says: a fortiori reasoning seems logical to you? Mistake, mistake. It’s something formal. He brings some a fortiori reasoning of snakes from Bereishit Rabbah at the beginning, that snakes themselves made a fortiori reasoning, never mind, and that it’s not logical. And Rabbi Chiyya’s first a fortiori reasoning too—I saw someone there bringing exactly the same thing: a fortiori reasoning is not a matter of logic. And in both places I can show why it is logical. So in short, my assumption is that when the Sages put things into one box, one table, and made a fortiori reasoning or a paradigm case or a common denominator, they assumed that it was the same kind of thing. If to us it looks different, we need to think about why the Sages assumed that, but they certainly assumed it, because they don’t just do formal things like the doorpost and tzitzit.
[Speaker B] The distinction between a factual stricter side and a legal one is a bit difficult for me. Meaning, since behind everything there seemingly stands only some legal rationale. Let’s say with fire, whose way is to go and cause damage, then there is more reason to obligate him for the damage. So similarly if there is an exemption, then perhaps there is some rationale not to obligate him for the damage. But the rationale behind both things is basically a legal rationale. Right. Otherwise it wouldn’t belong here.
[Rabbi Michael Abraham] That’s exactly what I’m claiming.
[Speaker B] Ah, okay. Fine. But no, here you wanted to resolve that difficulty by distinguishing that in one place we’re talking only about a factual stricter side and in the other—but that distinction, the question is whether that distinction is correct. No, clearly it is.
[Rabbi Michael Abraham] Precisely because of what you’re saying. Precisely because of what you’re saying. Because what am I saying? I’m saying this: if these two things are factual characteristics, let’s call them alpha and beta, then you can’t refute with a refutation of a stricter side. That is the ordinary common denominator. What I’m claiming is that in a place where these characteristics are halakhic characteristics, true, behind every halakhic characteristic there sits a legal or factual characteristic, but it could be that it’s the same characteristic. The fact that the laws are different, X and Y, doesn’t mean—because it could be that the same alpha itself, in pit, produces Y and in fire produces X. And because of that, despite the fact that there are two different characteristics here, since what matters is the legal issue and not the halakhic issue, that is exactly the point.
[Speaker B] So the factual—but I’m saying, the legal issue stands behind both the halakhic and the factual. After all, the legal issue is a reason to obligate or exempt with respect to damages. Okay, yes. Different reasons. But there can be different reasons to exempt from damage liability.
[Rabbi Michael Abraham] One time they exempted you because it isn’t yours, another time they exempted you because you weren’t negligent. Right, okay.
[Speaker B] That means these are different reasons. But always—the “side” is always the legal side. Obviously, but I said, it seems to me the distinction between factual and halakhic—
[Rabbi Michael Abraham] When I speak of factual, I mean what you mean by legal. Meaning, factual means: what legally relevant fact caused us to exempt him? What is there in tooth and foot that exempts him in the public domain? It’s a legal rationale. I’m saying: the fact that it’s normal for his animal to walk there and the damage is common—that’s a legal rationale. I call it a factual characteristic because it depends on a fact, but of course the law determines whether that fact is a relevant fact. And the legal rationale says that if it is normal for a person to walk with his animal in the public domain and the damage is common, then the owner does not need to guard himself. And because of that there is a legal rationale to exempt him. But there are many legal rationales that can exempt. And the question is whether behind two different laws there necessarily stand two different legal rationales. I claim not. Or at least Rabbi Yehuda, who makes a refutation of a stricter side, claims not.
[Speaker F] And that somewhat contradicts what you said about the pig. No, you don’t bring a pig into the table because—even though it’s forbidden to eat—it isn’t relevant. Now you’re saying maybe there is some relevant side. We understand what a relevant side is that stands behind the—
[Rabbi Michael Abraham] But we do nevertheless have intuition about what does and does not fit.
[Speaker F] But why do you bring it into the table when we’re talking about—
[Rabbi Michael Abraham] I’ll add two more sentences from that lecture I said I wouldn’t give. Look, in the philosophy of science there is—maybe I once talked about this, I already don’t remember, after all the years we’ve been here—there is a very interesting phenomenon. There is a British historian named Carr, and he wrote a book called What Is History? in Hebrew too, I don’t remember, it was translated from English. What Is History? And there he says that there is a problem with the Baconian description of historical research. Francis Bacon built the logic of modern science, what he called inductive logic, in the sixteenth century. And he said that basically we do an elimination of facts. We look at the facts—this is what we’re talking about here. We collect facts. Suppose we want to know why Blücher defeated Napoleon at the Battle of Waterloo. Okay? So I collect facts about his soldiers, his soldiers, his tactics, morale, things like that, and through this I understand why someone won. That’s how a military historian works, in this case. Meaning, he tries through the facts to explain the—sorry, through the microscopic data, those alpha-betas, to explain the fact: he won. That is the fact; that is basically the one or zero in the table. Now Carr argues that in fact this is a naive conception of the work. That’s how Francis Bacon understood scientific research. You collect facts, and from within the facts you extract the theory. He says that can’t be right. Why? Don’t confuse us with facts. What causes victory in wars? We are now coming for the first time to investigate this issue. Yes? We are military scholars, military researchers. We want to examine what causes victory in wars. And now the research begins. We have no information whatsoever. So we approach various battles and examine who won, the facts, and try through that to understand what causes military victory. Now there are infinitely many facts.
[Speaker D] So the question is which facts you choose? Which facts am I supposed to choose?
[Rabbi Michael Abraham] Is the color of the pants a relevant fact? Is the height of the mother of the third soldier in the fourth battalion a relevant fact? What counts as a fact now? Remember, we have no idea what causes anything; we have no idea what causes victory in battle, military victory. So we have no way of knowing what a relevant fact is. So what do we do? Fine, let’s collect all the facts. If we collect all the facts, we’ll never finish. There are infinitely many facts. So how do we know which facts to collect? What Carr is really arguing—and by the way, they were arguing this in the natural sciences in parallel with him, these guys didn’t know about each other, Hempel made the same claim in the natural sciences—the claim is basically that when you approach something, supposedly Francis Bacon taught us that theory is built on facts. But he didn’t notice that facts are also built on theory. Because when you decide which facts to collect, you’re implicitly assuming some sort of theory. You’re saying to yourself which facts are relevant. Okay? And then you say, ah, these are relevant facts, let’s collect them. Fine, but if you already know the theory, then what do you need the facts for? You already know. It’s not like that. It’s subtler. You have a hypothesis about what it might be. You don’t know. But you know what it isn’t. Meaning, you can guess. And you don’t know the importance of every single thing, you don’t know the details. So I know which facts I don’t need to collect. Sometimes, by the way, I’m wrong. As you said, not necessarily. But I have some initial intuition; before I’m equipped with a theory, I have an initial intuition. So I collect the facts that seem relevant to me. From those, I really try to see which facts, in all places, cause victory in battle. Then I go back and construct the theory I had an intuition about at the start in a more explicit way. After that, by the way, the process continues. Then I collect more facts from more battles, and I try to see if it works. If it doesn’t work, the theory has to be corrected. So this race is constantly going from facts to theory, from theory to facts, and it’s a mistake to think that we start with facts and arrive at theory. And the same thing appears in the Open University book on philosophy of science. It mentions a Hungarian Jewish doctor, Semmelweis, who discovered childbed fever. There was a high mortality rate among women giving birth. He was head of a department in a hospital, and in his department many mothers were dying. In the department next to his, there wasn’t, or there was much less. And they tried to understand what caused it. They didn’t yet know about microorganisms, they knew nothing. They had no clue what could cause such a thing. He starts looking. He says maybe it’s the route the priest walks through the ward, where the door is located, in the east or in the west, where the sun is, the age of the students, I don’t know, all kinds of things—you’ll see there, there’s a description in that book, wonderful things. Because he has no idea what causes it, so what are you going to look for? You don’t know which facts to look for. It’s exactly the same phenomenon as Carr’s. Meaning, if you don’t know what the theory is, you don’t know which facts to look at. And that’s what is called collecting facts and then discovering what the theory is. Now at some point it was really almost a miracle—they decided to test having the students wash their hands. And they had no idea that handwashing was relevant, that cleanliness was relevant to health. That still wasn’t known. And it turned out that this eliminated the effect. Then it turned out that the students in his department came after performing dissections, after pathology. They had a pathology course and then came to work in the ward, and they didn’t wash their hands. They didn’t wash their hands, and the mothers died. In the other department there were no students, or they came from another course, I don’t know exactly what. Right? Now you understand that if you don’t know that cleanliness is relevant to health, then from his point of view washing hands is like saying Psalms in the morning. It’s exactly the same thing, forgive the comparison. It seems totally unrelated. What does it have to do with anything? You could test whether everyone stands on one leg every morning, or every evening three times. How is that connected at all to the matter? Meaning, if you don’t know which facts are relevant, or what affects the phenomenon you’re looking at, you won’t be able to collect the facts. But on the other hand, if you know what’s relevant, then you already know the answer. So what do you do? That’s exactly what’s happening here. Thank you, Rabbi. Now let me just—now I’m going back to the answer, the answer to your question. When I build a table like this or a table like that, I have an intuition about what is connected to what, what cannot be connected to what and what can. I don’t come as a blank slate. You know what is connected; you don’t know exactly how the connection works.
[Speaker D] I still don’t know, but I think… Right, right. Okay, maybe even in the world of the natural sciences nobody can
[Speaker F] explain it before they discovered the theory. After all, we have some sort of intuition that guides us in what to look for. If there weren’t such an intuition, today we would still be
[Rabbi Michael Abraham] with the science of
[Speaker F] Aristotle, you have to understand,
[Rabbi Michael Abraham] because we would keep going and searching for facts all the time, and to this very day we still wouldn’t have finished finding the facts needed to discover the correct theories as against Aristotle. Rather, even before we know the theory, we have some intuition that we don’t know how to explain, about what might be relevant and what might not. Then we collect facts and test them. Maybe we’ll make a mistake; nothing is certain. We test it. It didn’t work. But many times it does work, and it creates a kind of unconscious elimination. We’re not aware of it, but we make lots of eliminations without being aware of it. It’s like what Kahneman calls, right? System 1 and System 2. There’s Daniel Kahneman—this quick, unconscious thinking versus conscious thinking. And many times the unconscious thinking works better than the conscious thinking. We have some abilities within us that we activate unconsciously, and very often they save us much more than our conscious thought does. That’s exactly what happens here. When we build the table, we build it out of those intuitions. We still don’t really know why these are relevant and the pig is not relevant, but we have a feeling that it’s not the same kind of thing. Like with the doorpost and the fringes—just now all of you laughed, right? With the doorpost and the fringes. Can any of you tell me what obligates a four-cornered garment in fringes? Or what obligates a doorpost in a mezuzah? I can’t tell you such a thing. But on the other hand, I also laugh when someone makes an a fortiori argument from a doorpost to fringes. Why? Why? Because we have some intuition that says this isn’t the same thing, even though we don’t really know why here fringes are required and here a mezuzah is required; we don’t know how to explain it. But somehow we have this intuition that says they’re not from the same family, there’s nothing to learn from one to the other, they don’t belong in the same table. That’s exactly what happens here. We build the table, and after we’ve built the table, from that point on we’re scientists. Everything is explained, we have a technique for how to do it, we extract the microscopic parameters—but in the background we have an intuition that we don’t know how to explain. Exactly the same thing happens here too.
[Speaker D] I wanted to say first of all regarding the relevance of the parameters—the unconscious elimination you’re talking about—it seems to me that science in recent years, through big data and data mining, is trying to get rid of that and is looking in places that aren’t relevant.
[Rabbi Michael Abraham] It will never succeed.
[Speaker D] But it expands it tremendously; that’s exactly the point.
[Rabbi Michael Abraham] It expands it tremendously. It expands it, but it will never manage to get rid of it. This process itself is an expansion. Because basically, what did the Sages do? The Sages didn’t make these tables with all the analyses. I said, I’ll show you—for every table like this, give it to me and I’ll do it with mathematics, no need to strain your brain at all, I’ll tell you what the answer is. Any ten-by-eight table, huge—you have no chance of doing it intuitively. Is there an algorithm? Yes, there’s an algorithm, and I can show you how to do it. Now the Sages didn’t know that. How did they do it? They did it with intuition. With the initial intuition they got all the way to the end. What am I doing here? I’m narrowing it down, exactly what you said we do. I’m saying okay, I won’t be able to get rid of part of the intuition; I’ll have to use a certain kind of intuition here in order to build the table. But from that point on I can already do it by calculation. So I reduce more and more the place where our non-recursive thinking operates, our intuitive thinking, in favor of—and this is what mathematics does, if you remember, it all connects here—if you remember the example of convex shapes. There I was trying to show exactly this: basically, we don’t get rid of the intuitive assumptions that we can’t justify; we sweep them under the rug. And we focus only on the part from that point onward that is mathematics, which is exactly what I’m doing here. It’s exactly the same thing. You can’t get rid of initial intuitions; without them you can’t move at all. What you can do is mechanize as many parts of the algorithm as possible, meaning turn them into something mechanical.
[Speaker D] So there’s no chance that robots will take over.
[Rabbi Michael Abraham] Take over, maybe yes, I don’t know, but think like us—I don’t think there’s any chance, it seems to me not.
[Speaker D] But I wanted one more small thing. About whether this is relevant: there’s some Talmudic passage that seems to me to have a bit of trouble with this theory, the one about Hillel the Elder and the sons of Beteira, the eve of
[Rabbi Michael Abraham] Passover that falls on the Sabbath,
[Speaker D] there’s something there—it looks like a doorpost and a mezuzah.
[Rabbi Michael Abraham] Fine, but I’m saying that’s one of those questions. So every a fortiori argument like that—you have to go in and see why it’s not like a doorpost and a mezuzah, but that’s not important to me for the principal point. Okay, now let’s really try to move toward the theory. Understand that basically what perhaps I’ll do is add one more sentence. What I’m doing here is really the logic of scientific thinking, or legal thinking, or everyday thinking—it’s all the same. It’s the same logic. The logic we’re now going to learn, or are now learning, is that same logic, but instead of Francis Bacon’s logic, I’m offering you now an algorithmic logic. Completely algorithmic. Meaning, I’ll give you an example. When we want to infer the law of gravity, okay? So what do we say? I take this thing and let go of it and it falls downward. I say wow, good, so now I ask whether if I let go of this, it too will fall downward. So I say yes: if this falls downward, then this too falls downward. But this one has spaces in the middle. Right? And that one doesn’t have spaces in the middle. Remember, I still don’t know anything, so I don’t know what matters and what doesn’t. I don’t know that every mass falls downward. I’m now looking for the law. Okay? But what about this one, which has holes? And I take it, drop it, and I say this too falls downward, while that one has no holes. Right, but what about this one, which is made of plastic? And that one isn’t made of plastic, so maybe it won’t fall? So I say, that will prove it. And the argument comes back around. This is not like that, for this is made of plastic and that is not. This is not like that, for this has holes and that doesn’t have holes. The common denominator in both is that both have mass. So I too will say that everything that has mass will fall toward the earth. Scientific generalization is a common denominator. It is nothing but a common denominator. And one of the rules that guides the scientific researcher along the way is what’s called diversity of evidence. Diversity of evidence means, say I want to know the color of ravens. That’s the example philosophers of science always give: all ravens are black. I think Hempel was the first one who drilled this into our heads. So I want to prove that all ravens are black. So I go outside and I look and I see that all the ravens I see really are black. Great, excellent. Then someone says all the ravens you saw were Israeli ravens, ravens from the Land of Israel. Maybe in Australia ravens are pink? Fine, so I go to Australia to look. Maybe it’s only Australia and the Land of Israel, but in, I don’t know, Indonesia or the United States they aren’t? So I go and look there too. Now if I saw five ravens on five continents, that’s better than five ravens from the same place. Why? Why is that a better-grounded generalization? Because of the common denominator. Because on each of the five continents there is a refutation: what about that one, since it’s Australian? What about that one, since it’s American? So then the argument comes back around. This is not like that, and that is not like that; each has a unique characteristic. Right, that’s the point. This one has X and that one has Y, but clearly neither X nor Y is responsible for this—for their color. Therefore apparently Z is responsible, what they all share. So scientific generalization is basically a common denominator. That’s all. And the more diversity there is in the evidence, the more columns there will be in the table, or rows in the table, and the more rows there are in the table, the stronger the conclusion is. Okay? But just a second, that’s still only plausibility.
[Speaker C] It’s just more plausibility, not—I don’t know—
[Rabbi Michael Abraham] Plausibility, reasonableness, or whatever. It’s not probability; I’d call it plausibility.
[Speaker C] In practice it’s not—
[Rabbi Michael Abraham] Plausibility, I’d call it plausibility.
[Speaker C] It’s rational, but it also proves. Right, yes.
[Rabbi Michael Abraham] So I’m saying, of course you can make mistakes. Science is not mathematics. And therefore science is subject to refutations. Exactly as in Jewish law, a halakhic inference is subject to refutations. And why? Because in truth it’s not mathematics. In the end, what does a refutation say? A refutation says that these two things that we placed in the same spot don’t belong in the same spot. Think about it. I made a refutation here. There’s a C here. Why is that a refutation? It’s a refutation because it shows me that these two fellows do not sit in the same table. There is no simple relationship between them. It is not correct to place them in the same table. So what did the refutation refute? It refuted the table, not my inference. My inference is correct. The refutation shows that the table should not have been built, because these two do not belong to the same field, to the same area of discussion. That’s what a refutation does. So in the scientific world it’s exactly the same thing. Meaning, if I found something with mass that didn’t fall to earth, then they would tell me: fine, then the fact that you grouped them all together—your Z, right?—that they all have mass is not the relevant thing; there’s some W, there’s something else. This whole business is simple: it’s exactly the logic of science, the logic of law, the logic of Jewish law, the logic of everything that is not mathematics.
[Speaker D] What about all the combinations of characteristics? Can’t they have an effect?
[Rabbi Michael Abraham] Yes, sure they can. We’ll get there in a moment. We’ll get there in a moment, but for now an a fortiori argument is a simple thing. We’ll get to more complicated things, combinations. Okay. Was there someone else? Okay. So what I now basically want to do is move forward and show you the following. How do we, say we have a certain table—how can we extract from it what the explanation is, what the model is, what the theory is? Right? So as I told you last time, when I have a table with a question mark, I fill it once with a one, another table with a zero, and I explain each table separately, and I see which explanation is simpler. That determines which filling is simpler and which I choose, one or zero. Fine? So at the moment, what I need to do is this: given a table that has a certain filling, let’s say a one, I want to know how I find these alphas and betas—how do I find the model that explains the table. And what I do is this. Suppose here there was—right? Suppose here there was a one below. Suppose here there was a one. What I basically want to say is—I take the… ignore this for a moment, for now I’m looking only at this table. Okay? Now I look at A and B as two vertices. A and B as two vertices—I look at the relation between them. This is greater than that, right? Let’s mark it like this. What does greater mean? That in every row you check, what’s written here is greater than or equal to what’s written there. The opposite? There’s a simple relation between them. The opposite?
[Speaker C] No.
[Rabbi Michael Abraham] It doesn’t matter right now whether it goes from big to small or from small to big, but there is a simple relation between them, okay? By contrast, if here it says zero—if here it says zero—what happens in these four? Different. A, B, and there is no arrow between them, right? There’s no connection between them. Fine? Now the point is this. If there is such a relation between them, say if this is alpha, then this really has to be something that contains alpha. It can be either two alphas, or alpha and beta. Fine? Something such that whoever does this also does that. Right? So if I find this ordering, then I now have a way to find the model immediately. I say, I assume this is alpha, as an initial assumption. Now I ask, okay, what will this be? This will be either two alphas, or alpha plus beta. Now of course this is simpler, so I prefer it: this is two alphas. Fine? So then, notice, this is my model. B is alpha, A is two alphas. From here I go back to the table and say: little a here doesn’t do alpha and it does do alpha, so little a is alpha. Once I already know these, then I have no problem explaining these too, right? And this one does both of them, therefore it is two alphas. Meaning, I start with the columns—you can also do it with the rows, it doesn’t matter. I start with the columns, I make some kind of graph like this, build the parameters from it, and then go back to these, although you don’t really need to do that, but in principle I go back to these and I can also find these. Okay? What happens here? I start from the fact that B is alpha, but this has no connection to it, so this is beta. Right? There is no relation between them. Therefore this is beta. Good. What would happen, for example, with a table like this? With A, B, and C. Right? A three-by-three table. Say look, for example, at this. Suppose I fill this with a one. Fine? Now look at this and this. You see that this is contained within that, right? Compare those two columns. This is one, and all these are smaller than it, right? And this too is contained within that. Right? So the diagram is exactly the same. Okay? How do I build this diagram? Suppose this is alpha, then this, we said, is two alphas, and here what? Two alphas or alpha plus beta. It can’t be two alphas, because then there would be an arrow between them. It can’t be three alphas, it can’t be anything like that—there has to be something extra here. Therefore this is alpha and beta. It has to be. Okay? And so on. Now once we understand this, we can do any table we want. But there’s another point here. For the moment I’m trying to show from… Why can’t this be alpha and beta? Why can’t it be only alpha?
[Speaker C] Because if it
[Rabbi Michael Abraham] were only alpha, then there would be an arrow
[Speaker C] like this.
[Rabbi Michael Abraham] Sorry, like this. And there isn’t an arrow.
[Speaker C] Ah, okay.
[Rabbi Michael Abraham] Meaning, the absence of an arrow is also an indication. You have to explain every arrow and explain every absence of an arrow. Meaning, the model has to fit completely. Every relation between every two columns has to be described in the model. Therefore it’s convenient to do this by drawing it, because looking at the table itself is complicated. I’m not going to do too much more of this; I can already feel you’re getting a bit tired of the matter. But I want to show the rationale, the logic of the matter, the essence of the matter. And what I’m basically claiming is that once I have a table, I can draw some sort of chart like this of relationships of
[Speaker C] dependence, yes, dependence relations between the columns.
[Rabbi Michael Abraham] Once I have the dependence relations between the columns, I can fill it in and find the model. That’s how I find the model in every table. Okay? This is of course heuristic, because with complicated graphs it’s not so simple exactly how to populate them. That’s a nontrivial mathematical problem. One of our students worked on it—on how to fill these in—and found some theorems about these things, about how to populate this structure, but he didn’t cover the whole—he didn’t solve the problem completely. Meaning, it’s a nontrivial mathematical question, topology and things like that. In any event, now there is also an additional point, and here, so as not to exhaust you, I’ll skip a little. When you begin to check, what we did was go through a passage in tractate Kiddushin. The passage in Kiddushin tries to learn whether the wedding canopy effects acquisition. Acquisition here means effecting betrothal. The canopy effects marriage; the question is whether the canopy can effect betrothal. The Talmudic passage—this is one of the most complicated passages in these matters, because it keeps becoming more and more complex, and therefore we thought to follow it from beginning to end and in that way build our model. Okay? So basically the passage works like this. I’ll read it briefly so you can see what it’s about. Rav Huna said: the canopy effects acquisition by an a fortiori inference. Meaning, we have an a fortiori inference that says that the canopy succeeds in effecting betrothal. You see, it’s like tooth, foot, and horn in the damaged party’s courtyard or in the public domain. I ask whether canopy, money, and intercourse—those are the categories of damages—effect marriage, effect kiddushin. Fine? What do they do? Those are the domains. It says: the canopy effects acquisition by an a fortiori inference. But the challenge is as follows: if money, which does not complete the process, nevertheless acquires, then isn’t it logical that the canopy, which does complete the process, should acquire? Money cannot effect marriage, and yet it effects betrothal, so the canopy, which effects marriage, certainly should effect betrothal. Right? A fortiori. The Talmud says: refutation—what about money, for by means of it one can redeem consecrated items and second tithe. Money can redeem—well, give me your consecrated item and your second tithe in exchange for a canopy, and I won’t thank you very much for the deal. So by means of a canopy you can’t redeem consecrated items and second tithe. Okay, so the Talmud says: intercourse will prove it. What does that mean, intercourse will prove it?
[Speaker D] What kind of refutation is that? That’s a doorpost-type refutation. Why? What’s the connection? What is there between a canopy… what are you saying, that with money I can buy Bamba snacks and with a canopy I can’t? It’s the same thing. But it’s not relevant.
[Rabbi Michael Abraham] Why? It’s very relevant. With money—when you do kiddushin and you want to do it with money—that means money has purchasing power. Kiddushin is a purchase, like the field of Ephron. And if you show that in matters of purchase money has an advantage over the canopy, then maybe in kiddushin too it’s like that. We see there’s a connection between kiddushin and purchase—it’s “taking, taking” from the field of Ephron. Fine. So the Talmud says: intercourse will prove it. What does that mean? We’re working on it—here, we made this a fortiori argument, right? We found a refutation. Fine, so intercourse will prove it. We’ll make from here—but this time not an a fortiori argument, rather there’s a one here, this is a binyan av, okay? Some sort of common denominator version of an a fortiori argument—whoops, I got tangled. Intercourse will prove what? That intercourse effects both kiddushin and marriage. So it’s one-one: it effects both kiddushin and marriage, and therefore the canopy, which effects marriage, also effects betrothal. Then the Talmud says: what about intercourse, for it acquires in the case of a yevamah? Intercourse can acquire a yevamah, can perform levirate marriage. Levirate marriage is not done with a canopy. A canopy is marriage with a woman, but levirate marriage is only through intercourse. So that means we’ve refuted it. So now we have this refutation on intercourse. You see? This is this refutation. Before there was this refutation, now there’s this refutation. Fine? Basically, learning from intercourse also doesn’t work. So then the Talmud says: money will prove it, and the argument comes back around. Actually until now we tried from each one separately, and now we say okay, let’s take them both together—and notice, without any addition. We remain with the same table; we don’t add another column. Until now the whole process kept adding columns. We started with this table, added this column, and then a refutation. So we went back to this table, added this column, and a refutation. Right? Each time something else was added. But the common denominator is the miracle we talked about, that nothing is added at all; you simply look at the whole table instead of just part of it. And when you look at the whole table, suddenly it does work. Okay? And that’s what the Talmud says: this is not like that and that is not like this; the common denominator between them is that they acquire elsewhere and they acquire here, so I too will bring in the canopy, which acquires elsewhere and acquires here. Then the Talmud says: what about the common denominator between them, for their benefit is great. What is that—“for their benefit is great”? What does that mean? Let’s now look here. You see that this is tooth and foot and horn and intercourse? It’s all the same because it’s the same schema. The A-B-C doesn’t matter at all; it doesn’t matter what you fill in here. “Their benefit is great” means the following, always. Why? How do you refute a common denominator? There’s something that these two have and this one doesn’t. Remember the W? That both teaching cases have and this one doesn’t have? Therefore it will always be a column like this, where these two have it—that’s benefit. These two have it and this one doesn’t have the benefit, right? Intercourse and money have benefit, but in a canopy there is no benefit. So the Talmud says: a document will prove it. What happens with a document? It has no benefit, right? That’s the whole point. What are we doing? We’re adding another teaching case. A document will prove it, because it has no benefit, right? And it effects marriage and also effects kiddushin—which is basically a binyan av, if it’s not an a fortiori inference—and likewise these two don’t apply to it, no matter. Fine? So necessarily here it’s zero and zero. Therefore, therefore, a document will prove it. And now notice where we are: a refuting column has already been added for the common denominator. Now we’re making a common denominator squared. Meaning, we’re making a common denominator between these two—the entire common denominator here—and this teaching case, and together they form some larger common denominator, where one side of the larger common denominator is itself already a common denominator. Okay? You understand that we lose our intuition this way. If we don’t have a computational mechanism, it will be difficult to track what works and what doesn’t. Then the Talmud says: what about a document, for it dissolves marriage with a Jewish woman? Here there is another table. A document can effect divorce, right? Intercourse cannot, canopy cannot, money cannot, and a document can. So here there’s yet another table: zero zero zero one, another column, right? So here you understand what’s happening. Now there’s the thing that refutes the common denominator, and there’s the divorce feature that refutes the document. And now we make a common denominator, but of course this is now a common denominator of a higher order. So the Talmud says yes, that’s exactly what it does. And now understand: the Talmud did all this without tables and without charts and without anything, meaning it’s an intuition that is hard to follow. Money and intercourse will prove it, and the argument comes back around. This is not like that and that is not like this. Here “this” is of course two things together—that is, a document as against money and intercourse together, yes? The common denominator between them is that they acquire against her will; so I too will bring the canopy, which acquires against her will. And now they refute the larger common denominator. What about the common denominator between them, for they exist against her will? All three teaching cases—intercourse, money, and document—exist against the woman’s will. Money? Yes, in the case of a Hebrew maidservant. The Talmud talks there—there is a case of against her will. Not always, but in principle it works. With a canopy it does not work against her will, so that’s a refutation of this kind. You see? One one one. That refuted the small common denominator. Now here we make a refutation of the larger common denominator. Here, here, and here there will be one, and here there will be zero. Right? Because that really says these two have this property, this one has this property, but the case we are interested in does not have this property. Okay? Then the Talmud says: “And Rav Huna says, we do not find money in marriage against her will.” Rav Huna basically argues that money cannot work against her will, because that is in the case of a Hebrew maidservant, not in marriage. The doorpost and the mezuzah? Why are you mixing a Hebrew maidservant with marriage? Those are two different things; it’s a doorpost and a mezuzah. And so in fact he changes nothing; he says: take the table you wrote here—money, which is here—you wrote a one here, put a zero there. Because in marriage there is no “against her will” with money. There is “against her will” with a Hebrew maidservant, but not in marriage. And that’s where it ends. And from this the Talmud derives the matter. Now, how do I know that indeed the result after this whole story—
[Speaker E] You studied all those years in yeshiva without this table?
[Rabbi Michael Abraham] What? You have to follow the logic. But you know, that logic collapses. No—when you understand the… That’s why when the Sages did it, they didn’t use a table; rather, they had building blocks. You understand the logic of the common denominator, you understand the logic of binyan av, you understand the logic of an a fortiori argument. What you’re doing now is generalizations—you’re now making a common denominator between a common denominator and a binyan av—so you can follow it, but it can become terribly confusing, and it… sometimes there are things that are mistaken, because if you don’t do the calculation, intuition says yes. I had to give a lecture like this at Tel Aviv University in computer science; they had me lecture on this issue. And the Sabbath before that my son came to me from Mercaz—let me finish with this. He came to me from Mercaz and said to me, “Listen, we have some a fortiori argument in Tosafot; nobody understands what’s happening here.” It’s not logical—a three-by-three table, not even a complicated table, but nobody understands it, it doesn’t make sense. So what if this one has this—how did Tosafot, what? So I told him, “I don’t know, let’s draw the table.” I did the calculation and said, “There—what, obviously, the result is one.” But with a three-by-three table they already couldn’t keep track. And he says, “The whole yeshiva, with all the prodigies, everyone.” So it’s not that I’m smarter than they are; I probably also wouldn’t have understood it, just like them. But that’s exactly the advantage of a computational method. When you have a method, then even when you lose the intuition, the method tells you what’s right. And it works everywhere I’ve tested it. Now exactly how we’re going to see this within—