Topics in Talmudic Logic, Lecture 4
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Table of Contents
- Logical hermeneutic rules and textual hermeneutic rules
- A fortiori inference: structure, refutation, and tables
- The decision algorithm: two tables and a simple theory
- “Microscopic” parameters behind halakhic facts
- Binyan av: analogy and the claim of simplicity
- Scientific analogy and non-deductive inference
- Binyan av from two texts and the common denominator: elimination of parameters
- The refutation of “a stricter side” in the Talmud and Tosafot
- The decisive distinction: real-world properties versus stringencies that are themselves laws
- Conclusion and continuation
Summary
General Overview
The lecturer divides the hermeneutic rules into two types, logical rules and textual rules, and argues that the logical rules are general forms of inference that also appear in science, law, and non-deductive reasoning in general. He develops an algorithm in which one fills in a missing cell in a table of laws by comparing two competing theories and deciding based on theoretical simplicity in the spirit of Ockham’s razor, assuming that behind halakhic facts there sit “microscopic” parameters that are not visible in the text. He uses a fortiori inference, binyan av, and the common denominator to show how refutations undermine an inference when they force a shift from a one-parameter theory to a multi-parameter theory, and he explains the difficulty with the refutation of “a stricter side” in the Talmud as a difference between a refutation based on real-world properties and one based on stringencies that are themselves legal rules.
Logical hermeneutic rules and textual hermeneutic rules
The lecturer says there are logical hermeneutic rules such as a fortiori inference and binyan av, as opposed to textual hermeneutic rules such as verbal analogy, general and specific formulations, and structures that rely on an extra word, a keyword, a shift from singular to plural language, or identical use of a word in two places. He defines textual rules as requiring a trigger from the text itself, whereas logical rules need no textual hint, only an understanding of the meaning of the law and the creation of analogies and hierarchies based on reasoning. He argues that refutation belongs to logical rules because it undermines the claim of similarity or hierarchy on which the inference rests, whereas in textual rules there is no refutation of the structure itself, such as verbal analogy or general-and-specific, because the text itself forces the interpretation.
A fortiori inference: structure, refutation, and tables
The lecturer describes a fortiori inference as a process of constructing a hierarchy-rule from data, and distinguishes between an a fortiori inference based on one datum and a Talmudic a fortiori inference that seems to be based on three data points but in fact uses two of them to derive a hierarchy-rule, leaving a rule plus one datum. He presents a fortiori inference through a two-dimensional table and explains that there is a “columns” formulation and a “rows” formulation, then shows that a refutation of type C can seemingly knock down one formulation without knocking down the other. But in the Talmud they do not “rotate” an a fortiori inference, and so the conclusion is that the two formulations are, at depth, the same argument. He explains this through the language of parameters and strengths on a single axis, in which one particular filling of the table can be explained by a consistent one-parameter theory, whereas an alternative filling requires two independent parameters and is therefore more complicated.
The decision algorithm: two tables and a simple theory
The lecturer proposes a general algorithm in which there is a table of zeros and ones with one unknown cell. One builds two alternative tables by filling it with 0 or 1, and formulates for each table a theory that explains all the data. He says the preferred theory is the simpler one, and “simplicity” is identified as the minimum number of parameters or relevant quantities needed to explain the data. He connects this to Ockham’s razor, explains the historical wording in terms of a minimum number of “entities,” and emphasizes that in his application we are dealing with theoretical parameters that explain the facts, not observable objects.
“Microscopic” parameters behind halakhic facts
The lecturer argues that the laws are “macroscopic” phenomena, while behind them sit theoretical parameters that are not visible to the eye, similar to the electron in physics, which is inferred from the fingerprints it leaves in phenomena. He says there is no need at this stage to identify exactly what alpha and beta are; it is enough to assume that such parameters exist in order to explain why a certain array of laws is stable and why refutations alter the structure of the inference. He illustrates possible identification of such parameters through language like “its intent is to damage,” “the beginning of its making is for damage,” and distinctions between the public domain and the injured party’s courtyard as features that might explain laws.
Binyan av: analogy and the claim of simplicity
The lecturer presents binyan av as a table similar to an a fortiori inference but without hierarchy, rather an analogy “on the same level,” in which similarity between two cells leads to filling in the missing result. He explains that here too one builds two tables and prefers the simpler theory, and remarks that in the case of binyan av both theories can be one-parameter theories, so he proposes a candidate according to which the number of values on the parameter scale affects simplicity. He adds that this is only a “hint: actually no longer,” and says that later it will turn out that the number of values is not the main criterion in the early stages. He gives a concrete example from Kiddushin 5 on the question of whether a wedding canopy effects acquisition, with an a fortiori argument from money and a binyan av from intercourse, and notes that later a common denominator will also be built from those derivations.
Scientific analogy and non-deductive inference
The lecturer compares halakhic inference to scientific inference by way of induction and abduction, and defines analogy as “induction plus deduction” through rising to a theory and returning to predict a new case. He demonstrates this with the example of falling to the earth, where color is irrelevant but mass is relevant, and parallels that to the question of which features in a halakhic situation are relevant to liability or exemption. He adds an example of the psychometric exam and estimating academic success as a structure of a fortiori inference and analogy, and explains that refutation parallels a new datum that forces one to add another parameter and break an earlier generalization.
Binyan av from two texts and the common denominator: elimination of parameters
The lecturer explains that binyan av from two texts is needed when one source is not enough because of a refutation, so two source cases are used in order to carry out the move of “the law returns; this case is not like that case,” and rely on the common denominator. He interprets the common denominator as choosing a shared parameter Z after eliminating other possibilities, and explains that this elimination is similar to isolating a chemical component in a medical experiment where one tries to identify what cures by comparing cases. He asks why we should not prefer a theory in which the obligating factor is “Z plus something else,” and answers that the decision is once again made by Ockham’s razor, because a one-parameter theory is simpler than a theory that requires a combination of X or Y.
The refutation of “a stricter side” in the Talmud and Tosafot
The lecturer brings Ketubot 32, where Ulla learns that “wherever there is monetary payment and lashes, he pays money and does not receive lashes” from injuring another person and conspiring witnesses by means of the common denominator, and the Talmud refutes it: “what is unique to their common denominator is that they have a stricter side,” and answers with a different derivation. He quotes Tosafot: “This is difficult, because if so, we will never derive anything from the common denominator anywhere, since in all cases one can refute either by saying there is a stricter side or a lighter side,” and presents the problem that if this refutation is principled, then there is no common denominator in the Torah. He notes that a similar structure appears also in Makkot 4, and that there is a long Ritva there with several answers that he does not find convincing.
The decisive distinction: real-world properties versus stringencies that are themselves laws
The lecturer argues that the solution is the difference between a common denominator like the one in Bava Kamma 6, learned from fire and pit through properties such as “the beginning of its making is for damage” and “another force is involved in it,” and the common denominator in Ketubot and Makkot, where the “stringencies” are halakhic laws such as “liable for five payments” or “they do not require prior warning.” He explains that when the refutation points to laws, it may be that the two different laws stem from one shared microscopic parameter, and therefore there is no clear advantage to the theory of Z over “X or Y” in terms of microscopic parameters. From this comes the justification for the refutation of a stricter side. He concludes that understanding the theoretical layer behind the laws explains both why we do not rotate an a fortiori inference and how the refutation of a stricter side works without canceling the very existence of the common denominator as a hermeneutic rule.
Conclusion and continuation
The lecturer sums up that the goal of the lecture is to show indications that theories sit behind halakhic inferences, and not to see the derivations as “hocus-pocus,” and says that later he will return to a fortiori inference, binyan av, and refutations in order to build the algorithm more systematically. He ends with a question about a topic in which Rabbi Tarfon and the Sages “reverse” an a fortiori inference with tooth-and-foot and horn, and answers that this is a special case connected to half-damages and that he will explain it if he gets to it.
Full Transcript
[Rabbi Michael Abraham] Okay, we were at Rafael. Good. We were discussing the topic of modes of inference that we can begin to examine through the hermeneutic rules, as rules by which the Torah is interpreted. I said there are two—right, actually, David, now that you remind me, if I make some kind of board, would someone be willing to photograph the boards? Okay, can you? Are you recording the lecture? I’m recording it, but I don’t have a picture of the board. If there’s a board, then I—
[Speaker B] I’d be very happy if you’d send me that. Okay. Fine. No, it’s not connected to us.
[Rabbi Michael Abraham] In any case, so I said that the hermeneutic rules by which the Torah is interpreted can be divided into two types. There are the logical rules and there are the textual rules. The logical rules are a fortiori inference, binyan av from one text and from two texts, refutations, and various combinations of those. But basically those are three rules in the fundamental sense; the rest are combinations. The other rules, aside from two rules—two texts that contradict one another, maybe—are textual rules. What’s the difference between logical and textual rules? With textual rules, there has to be something in the text that tells me to make the derivation. Some sort of hint: a keyword, some redundancy, a shift from singular language to plural language, use of a similar word in two places—right, verbal analogy. There has to be some kind of hint in the text that tells me to interpret. Just from reading the text, I can’t do anything. It doesn’t say anything. I’m simply reading the text, okay, and I have no way to make a derivation by verbal analogy unless I see that here and here the same word appears. If the Torah uses the same word here and here, then I make a comparison; that’s what’s called verbal analogy. In contrast, with the logical rules, no hint in the text is needed, and there also is no hint in the text. You don’t need an extra word and you don’t need anything like that. What you need is to understand what the text is saying and make analogies on that basis. Say the text states a certain law; from it I learn another law by a fortiori reasoning or by analogy. I’m not learning it because the verse is phrased a certain way, but because there is a similarity between this law and that law, and therefore I think that if this one is true then that one is also true. I’m basically making an analogy, okay? Or an a fortiori inference, which is also a kind of analogy. So the logical rules are rules that do not have a trigger in the text itself that causes me to make the derivation, unlike the textual rules. And the flip side of that same coin is that you won’t find a refutation against a textual rule. There is no refutation against a verbal analogy or against a general-and-specific formulation. There is refutation against binyan av, against a fortiori inference, against binyan av from two texts—against those there are refutations. And why? Because the refutation says that what you thought was similar may actually not be similar, and the assumption of the inference is precisely that assumption that it is similar. So if you show me that it isn’t similar, the inference collapses. In contrast, if the basis of the inference is a textual phenomenon—something extra, use of the same word in two places, a shift from singular to plural, right, specific and general, things like that—what relevance do refutations have? I didn’t assume anything about a connection between one thing and another, so there is nothing to refute in that context. The text forces me to make the derivation, regardless of what refutations I do or do not find. So those are two sides—remind me of your name?
[Speaker B] Yosef bar Rabbi.
[Rabbi Michael Abraham] Okay. So really these differences are two sides of the same coin. Now I want to focus on the logical rules, because the logical rules—later on, maybe, later this year we’ll also have some section on textual rules—but right now I’m talking about logical rules. And in my view, these forms of inference that underlie the logical rules—a fortiori inference, binyan av from one text and from two texts—are really the same modes of inference that we use in many other fields too, maybe all the other fields. And in a certain sense there is here a certain basis from which one can build, I don’t know if all, but almost all non-deductive inference—in science, in law, anywhere. That’s why I said we’re going to study these three rules in a bit of detail, and afterward I’ll try to show you that on the basis of these three rules one can systematically build a logic of non-deductive modes of inference. So last time I focused on a fortiori inference and said that I distinguished between several types. Types of a fortiori inference: there is a fortiori inference on the basis of one datum, an a fortiori inference of “if the larger amount includes it, then certainly the smaller amount,” and an a fortiori inference where not “if the larger amount includes it, then the smaller amount,” but in any case it’s based on one datum from which I infer a second datum, while in the background there is a hierarchy rule, an assumption of hierarchy. And in the Torah itself, usually it’s an a fortiori inference based on one datum. In the Gemara we saw that it’s an a fortiori inference based on three data points, but I said that when you look at how that inference is actually built, it is really the same thing as ordinary a fortiori reasoning, with one difference: two of the three data points are used to generate a hierarchy rule. After I’ve used them, I throw them away; now I have a hierarchy rule plus one datum. A hierarchy rule and a datum—that’s ordinary a fortiori reasoning. Okay? So the difference between Talmudic a fortiori reasoning and ordinary a fortiori reasoning is basically in the way we arrive at the rule, at the hierarchy rule. In the Torah, in ordinary a fortiori reasoning, the hierarchy rule comes from reasoning. In Talmudic a fortiori reasoning, the hierarchy rule comes from applying it to two data points, two of the three data points. What did you say your name was again? Neriya. How? Neriya. So once there is a hierarchy rule and a datum, we are basically doing a fortiori reasoning. So I moved on to show how the Talmudic a fortiori inference actually works in more detail. I’ll recap it because I’ll use it later too. There’s no black marker here. So I said this: every Talmudic a fortiori inference is basically built in the following way. Here we have, say, act A and act B, and here we have, say, A and B, these big ones, okay? These are domains and these are damagers. For the sake of discussion, if we speak about tooth-and-foot and horn. Tooth-and-foot and horn. And here it’s the public domain and the courtyard—
[Speaker B] of the injured party.
[Rabbi Michael Abraham] Okay? But I’m using A’s and B’s to show that this is completely general, meaning I’m not assuming anything special here specifically about torts. Now in every a fortiori inference, usually—not usually, in every a fortiori inference—the table looks like this. I said there could be a half here, but right now I’m ignoring that. And here I have a question mark. I don’t know what the law is. And when I make an a fortiori inference as I described earlier, I’m basically saying: from these two data points I infer a hierarchy rule that B is stronger than A, right? In other words, in this case horn is more severe than tooth-and-foot. So I used those two data points. Now I take this datum and I say: if tooth-and-foot here in B is liable, and there is a hierarchy rule that B is greater than A, stronger than A, then this too will be liable, right? So this too is liable. Alternatively, I can take these two data points—this is an a fortiori inference by columns. An a fortiori inference by rows: I take these two data points, from them I infer that B is greater, stronger, than big A, right? Then I ignore that, because I used it to generate the hierarchy rule. Now I have one datum here, and from it I infer that here too the result is liable. Then I asked: are these two different formulations of the same inference? And the answer was no. Seemingly no. Why? Because if I introduce a refutation here with some C—some other domain, the moon, right? Some other domain where the rule is like this, where the data are like this—then that refutes the hierarchical relation I built on the basis of this column. Because you can no longer assume that this damager is more severe than that damager, right? Because here the relation is reversed, and therefore you can’t infer a conclusion here; it remains open. On the other hand, that does not refute the relation between these two, right? And therefore I can continue making the inference by rows even though there is a refutation. And the same thing if I make a refutation of a row: it would refute this inference but not this inference of the columns.
[Speaker B] You don’t find that either.
[Rabbi Michael Abraham] Why? Right, and then I said that apparently these are nevertheless two formulations of the same argument, and I tried to explain why. And then I said—and this is the point I want you to pay very close attention to—basically I said the following. Now I’m already summarizing directly what I said.
[Speaker B] Are you going to photograph this or what?
[Rabbi Michael Abraham] No, that was from last time, so it’s not essential.
[Speaker B] Tell me exactly where you are.
[Rabbi Michael Abraham] Okay. So look: basically what I’m saying is that we need to make—we need to make two tables. I’m just returning to the algorithm that… okay? I make two tables. Here I put one and here I put zero. And I want to know which filling is more correct, which we said means simpler. So I compare the two, and whichever is preferable is the correct answer. Instead of saying let’s explain these three and derive this result from them, I showed last time that this is really the same thing. And then I say this: the way these things are filled—if, say, alpha has a value or strength or severity at the level of one alpha, alpha is a kind of axis—then there is one alpha, two alpha, three alpha, and so on. These are intensities along that axis, the alpha axis. Then B has an intensity of two alpha, right? Because it is stronger. Now here—pay attention—the intensities here mean: what intensity is needed in order to obligate. So if here there are two alpha, that means this is easier than this. Do you understand why? This is more severe than this. But when here there are two alpha and here alpha, that means this is easier than this. Why? Because only with an intensity of two alpha would you be liable there; one alpha is not enough to obligate there. And to obligate there it has to be very severe, because it’s an easier place. In the public domain it is hard to impose liability; it’s an easier place. It’s very hard to obligate there, because I have a right to walk there—what do you want, to obligate me for what I do there? That’s how all the medieval authorities explain the exemption of foot in the public domain. So therefore the relation is alpha and two alpha, and here it reverses—but the meaning also reverses. This damager is more severe than that one because it is two alpha as against one alpha. This domain is more lenient than that domain, because in order to impose liability there you need a damager with intensity two alpha. An ordinary damager won’t be liable there. So therefore, how do I know that this solves the problem? If this has intensity alpha, here that is not enough to obligate, because here you need two alpha in order to obligate—so zero. Here alpha is enough, so one. This has an intensity of two alpha, so it creates liability even here, not only here. In both places, right? So I succeeded in explaining this table with a single parameter and two values, right? Alpha and two alpha. What happens here? What happens here is that there is no way, as you can easily see, no way to explain all these data with a single parameter. There is simply independence between the two columns or between the two rows; there is no way to do it. It’s a two-dimensional problem. In other words, there are two basis vectors here. So therefore, if I define this as alpha and this as beta—this is beta and this is alpha—notice that alpha and beta means there is no severity relation between them. These are two parameters from totally different conceptual worlds; they do not speak to one another. Between alpha and two alpha there is a connection, right? Between alpha and beta I use different notation to say there is no connection here. These are two completely different parameters, playing in completely different fields. And then I say: B has the characteristic beta and does not have the characteristic alpha. So why is it liable here, in the courtyard—in the public domain? Because in order to impose liability in the public domain, what you need is beta, and it has that. Therefore it is liable. But this one does not have beta, and so here it is exempt. It has alpha—who cares? Alpha is not relevant for imposing liability in the public domain. Okay? In contrast, in B, what is needed to impose liability is specifically alpha, and that is what is written here. Therefore whoever has alpha is liable, but whoever has beta and does not have alpha is exempt. That is the natural explanation of this table. You need two parameters here. Fewer than two parameters are impossible. There is no correlation. Okay?
[Speaker B] Exactly.
[Rabbi Michael Abraham] So what I’m saying now is: now I need to decide which table is more correct. What does “more correct” mean? More correct—I don’t know—what is more plausible? After all, these three are fixed data. It’s written in the Torah. Now I ask: what is more plausible to fill in here—one or zero? The answer is: it is more plausible to fill in one here. Why? Because the theory behind that is a one-parameter theory. It has only one parameter. The theory that explains this assumes that two parameters are floating around here. And that is less simple. It could be, but it is less simple. Whoever wants that—the burden of proof is on him. So if someone wants to show that there are two parameters here, let him bring proof. If I have an explanation with one parameter, there is no reason to assume that two parameters are involved. This is basically Ockham’s razor, which originally was even formulated as—yes, this is an interesting note—originally Ockham, William of Ockham, was a Christian scholar, and he formulated the principle of his razor as saying that a theory containing fewer entities is simpler, preferable. Fewer entities. Entities—if you have one theory in which you posit an electron, a positron, and who knows what else, and the other theory explains everything with only positrons, then it is simpler. There are fewer theoretical entities you need in order to explain things. He meant the Holy One, blessed be He. There’s a booklet by—by Kellner. Do you know it? Something like generality and— I don’t remember what it’s called—on fanaticism and something, pluralism and generality, or something like that. There his basic claim is that the theory of the Holy One, blessed be He, is the simplest theory, and therefore it is true, because one being explains everything. To my mind that’s nonsense, but that is basically the claim, because it explains nothing. In any event, Ockham’s principle originally was minimum number of entities. What we do with it today is something else. Today, in philosophy generally and in science, when people speak about Ockham’s razor they mean the simplest theory, not necessarily the theory with the smallest number of entities. And what I’m showing here is that theoretical simplicity is—
[Speaker B] very—
[Rabbi Michael Abraham] very similar to the requirement of a small number of entities. Here these are not entities but parameters, or types of quantities with which I build the theory. But again, we have basically returned to the smallest number of elements. These elements are not entities; they are parameters, types of relevant quantities. But it is still very much the case that the two formulations of the razor principle come very close together. Minimum entities, or minimum principles, or minimum relevant quantities, kinds of relevant quantities—they are all criteria for the simplicity of a theory. Okay? So therefore my claim is that this is the basis of the whole issue. There is here simplicity of theory. We are building a theory with, let’s call them, microscopic parameters. I call them microscopic in the sense that the data written here are halakhic facts. Tooth-and-foot in the public domain—so you have an ox, you have a damaging agent of tooth-and-foot, there is an injured party, there is a courtyard—these are all macroscopic things, things from our world. What I’m claiming is that in the background of these laws there sit certain theoretical parameters. They are not parameters you see in the physical world. They are some properties of the damagers or the domains because of which these laws exist. So I call them microscopic parameters, in the sense that they are not the sort of things I see with my eyes. These are theoretical parameters, like the electron. With your eyes you see various phenomena. The assumption that there is an electron is an assumption that explains the phenomena. Nobody ever saw an electron. Okay? An electron is, in principle, a theoretical entity. So I call it a microscopic parameter in that sense—it’s not the sort of thing you see with your eyes. But I do assume that they exist because I see their fingerprints, I see what they produce. Okay? So in that sense, here too it’s the same. I build a theory with all kinds of parameters, alphas, betas, and the like, and I offer an explanation of the scriptural facts, the facts I find in the Torah, through that theory. And by the way, this theory contains parameters that I am not identifying. I haven’t defined who alpha is and who beta is and what exactly they express. We’ll get later to the question of how one identifies them at all. But for now I don’t need to identify them at all. I assume there is such a thing, and that’s enough for me. I don’t need to identify who alpha and beta are. Okay? So basically now the conclusion is that I build a theory for filling the cell with zero and for filling it with one in the box with the question mark. I check which theory is simpler, and for me that is the correct theory. Now let’s try to apply this immediately to binyan av. Okay? In binyan av what happens is again—this is universal—the table is the same table. That’s what is nice about these labels: in every sugya you go through, the table will be the same table. They speak about different concepts, but the table is the same table, and in that sense this is exactly like Aristotle. Remember, I talked about the conceptualization Aristotle did for logic. He says: all sorts of kinds of statements with all sorts of words and concepts—at the end of the day, turn them into a schema of X and Y, and it’s all the same schema. That’s the whole idea of conceptualization. I’m saying the same thing here. I’m basically making a logical conceptualization of non-deductive inferences. So I say that the table of binyan av is like this. This is the difference between binyan av and a fortiori inference. This is what it looks like. Really the only difference is right here. Here it is one and here it is zero. And then I say: binyan av says this is similar to that, therefore here too I’ll make it similar. So just as this is one, this too is one. Right? In a fortiori inference there is hierarchy. The hierarchy is that this is more severe than that. And then if here there is one, then here there will certainly be one. Here it is not hierarchy; it is analogy, on the same level. So I say: if this is similar to that here, then why shouldn’t I assume that this is similar to that here as well? Therefore—so if the law of A exists—say if I knew that tooth-and-foot is liable in the public domain and liable in the courtyard of the injured party, and horn is liable in the public domain, but I don’t know what it would be in the courtyard of the injured party, then I make an analogy. I say: if with horn there is no difference between the courtyard of the injured party and the public domain, why assume that with tooth-and-foot there is a difference between them? So I make an analogy. You understand that this is basically the same thing as a fortiori inference, and what distinguishes binyan av from a fortiori inference is the filling of this square.
[Speaker B] Yes, but here you necessarily have one parameter.
[Rabbi Michael Abraham] Wait, in a second, in a second we’ll see. You’re right. We’ll see in a moment. So now I’m trying to analyze it in the same language in which I analyzed a fortiori inference. Okay? So I do two fillings, because I want to build some general algorithm here. So I do two kinds of filling. This is when I fill it with one. Okay? Two kinds of filling. Now I want to know which one is simpler. I already have a clue, right? I know that binyan av is a rule by which the Torah is interpreted, meaning the correct filling is one. I already know that. The Sages have already revealed to me that the first filling is one. But I’m trying to understand how they got there. Okay? Why is this preferable to this? That is basically the theory I’m looking for. So let’s see. In this case, as you correctly noted, both theories will be one-parameter theories. Now both of the—you see, neither in this table nor in this table are there two independent columns. Right? Therefore here it will always be one parameter. So, for example, here I would explain it like this. This is really a classic analogy. Right? Basically this is identical to that. Theoretically it is identical to that. If that is alpha, this too is alpha. And this too is identical to that. I have no reason to assume there is any difference between them, so it is all one. There’s nothing—the data, the facts—give me no hint that I need to introduce some additional distinction here. Everything is the same. This is a world in which all the damagers are the same, all the domains are the same, everything is the same. Okay? So that’s this table. What happens here is a little more complex, and if you look at it and see it here, you can see that it is exactly the same thing. Just switch the rows and the columns, call A B and B A, and you get this table. Right? Exactly the same thing. So it’s clear that what there is here is two alpha and one alpha, and here too there are two alpha and one alpha. Right?
[Speaker B] It’s no more complicated than that.
[Rabbi Michael Abraham] Just switch the columns and rows, but it’s clear that that is the explanation. Okay? Except for what? Now the question is why the Sages tell us that this is simpler than that. In both cases the theories are one-parameter theories. Right. So the first candidate—we’ll see later whether it really holds water, hint: not anymore, but for now I’m saying the natural candidate to explain this is now to talk about the number of values there are on the alpha scale. Okay? Here the number of values has to be two. You can’t explain this with a single value of alpha. You won’t manage to explain it. This can be explained with a single value of alpha. Okay? So basically this is still simpler than that. And therefore in binyan av too, the preferred filling is one. Okay? That is basically the analogy… that is basically the parallel analysis I make for binyan av as against a fortiori inference. We’ll see later that this doesn’t really work. The number of values doesn’t really matter. At least not in the early stages of the discussion.
[Speaker E] So like, what would be an example like that, of binyan av that you can actually kind of grasp? A concrete example of binyan av? I didn’t understand. I want an example of binyan av, a concrete example.
[Rabbi Michael Abraham] With concepts? You want me to tell you specific A and B? Yeah, just some example. In the Gemara in Kiddushin 5, the Gemara makes a binyan av. The Gemara asks whether a wedding canopy effects acquisition. Acquisition means whether betrothal can be done by means of a wedding canopy.
[Speaker B] Okay? And Abaye says that a wedding canopy effects acquisition by a fortiori inference.
[Rabbi Michael Abraham] Yes. So that’s the sugya there. So we ask whether a wedding canopy effects acquisition. According to the halakhic ruling, it does not. Rav Huna says it does. So I learn it by—it starts with a fortiori reasoning, doesn’t matter, but an a fortiori inference from money. But afterward they make a binyan av from yevama—from intercourse. Okay? Why? Because intercourse effects both betrothal and marriage. Say if you call this intercourse: intercourse effects both betrothal and marriage. Fine? Now a wedding canopy effects marriage. The question is whether it effects betrothal. Why not? If intercourse effects both, then a wedding canopy should also effect both. Okay? The a fortiori inference from money, by contrast, works like this: because money does not effect marriage. You can’t do marriage with money, but betrothal yes. Okay? Now a wedding canopy specifically does effect marriage—that I know. But betrothal I don’t know. So we make an a fortiori inference. Money shows me that effecting betrothal is easier than effecting marriage. So if the wedding canopy effects marriage, which is harder, then certainly it effects betrothal. That’s a fortiori reasoning. Do you understand? And afterward, by the way, they make a common denominator, meaning they combine the a fortiori inference and the binyan av and generate from them some kind of common denominator, and we’ll still see how these things work. That is, it keeps building and becoming more and more branched out, or more complex. Okay, so for now, for now I’m taking a picture—
[Speaker B] You’re taking a picture.
[Rabbi Michael Abraham] So for now, what I want to claim is that this algorithm works. Meaning, when we need to fill in our puzzle, it is really always a table filled with data of zero or one. One of the cells has a question mark and I need to know how to fill it. I make two tables, in one of which I fill in zero and in the other I fill in one. I build a theory that explains each of the two tables, and I check which of the two theories is simpler—that is the answer. Okay? That’s the algorithm for now. That is also the algorithm at the end; it’s just that what “simpler” means is going to become a little more complicated for us. Okay? But that is basically the method of work.
[Speaker B] Now, do I need to write here a fortiori inference and here verbal analogy—
[Rabbi Michael Abraham] That’s—
[Speaker B] not verbal analogy, it’s binyan av. Binyan av, sorry.
[Rabbi Michael Abraham] Okay, for the benefit of future generations, a fortiori inference—
[Speaker B] and this is binyan av.
[Rabbi Michael Abraham] Good. Now I’m leaving aside kal va-chomer and binyan av for a moment. Before we continue with the theory, I want to persuade you from another angle that behind the facts there is always a theory. Physicists don’t need to be convinced of this; for them it’s some assumption that nobody even thinks to put to any test. Meaning, it’s obvious to them that once you encounter some facts, there has to be some theoretical explanation that accounts for them. It didn’t have to be that way. You assume that there are general laws, laws of nature, and that everything happening around us is simply a result of the laws of nature. Right? That’s what physicists usually assume—really all science, and especially physicists. So I’m claiming the same thing in the halakhic context. When the Torah states some Jewish law, what it is really determining is that in a certain situation, a certain halakhic norm applies. Okay? You have to pay, you’re exempt from paying, impure, pure, whatever—it doesn’t matter—some normative determination that applies to a factual situation. My claim is that behind this sits a theory. The theory says: take the factual situation, see who is involved there—the damagers, the domains, right? The public domain and the injured party’s courtyard. Understand that within the damagers there are components. For example, intending to cause damage is a component. Beginning as something made for damage is a component. That can be the alphas and betas. Okay? Someone whose intent is to damage—that’s very severe, so therefore he’ll be liable in the public domain, for example, I’m just saying. Okay? So that means alpha is what’s responsible for liability in the public domain. Fine? But someone who damages in the course of normal walking, like foot damage that happens while walking in the public domain, is exempt. That’s beta. Beta is not relevant to liability in the public domain. So here’s an example of identifying the microscopic parameters. Meaning, I’m basically claiming that there are features of the situation because of which the laws were determined as they were determined. In other words, it’s because of those features of the situation. That’s what I’m saying; that’s really the claim. And understand that what I’ve done here is prove it to you. Right? That suits mathematicians—they usually prove something everybody understands is obvious, as if—it can take them a whole semester to prove something nobody understands why it even needs proving, it’s obvious, no? So here too, same thing. What I’m trying to do is prove to you that behind the laws there sits a theory. How did I prove it to you? Simply empirically. Meaning, I showed you that you won’t be able to explain this unless you assume that behind it there is a theory. Why? Because you won’t be able to show me why a pircha, a refutation, doesn’t merely rotate the kal va-chomer but knocks it down completely. Right? That’s really the claim. If you don’t adopt the fact that behind this there is some theoretical explanation with alphas and so on, then these are two different arguments. After all, I showed you that it’s the same argument, right? I showed you that the hierarchy between little A and B is the same hierarchy as between big A and B. Therefore the argument of the rows and the argument of the columns—the rows argument is the same argument in two different dressings, that’s really the claim. So then what I asked is no longer difficult: why is it that the Sages never just rotate a kal va-chomer, but when there’s a pircha it falls apart? Okay? Why? Because it really is the same kal va-chomer. If you put a pircha here, what have you shown? That the hierarchy between these two cannot be explained by alpha and two alphas; there has to be some beta here as well. Right? Ah, but if so, then between these two as well it’s no longer alpha and two alphas. So in fact you’ve also knocked down the row-based kal va-chomer, not only the column-based one. Maybe I didn’t sharpen that enough last time, I don’t remember, but that’s the point. Do you understand? Meaning, the previous move was meant not only to understand kal va-chomer, but within the broader framework, what I wanted to get from it was to convince you that behind the laws there sit theories. In fact, the inference from fact to fact is made on the basis of a theory, exactly as in the scientific context. I talked about abductions, if you remember, in the introductions. All the introductions will be relevant, if you remember what we talked about there. So I said there that when I make a comparison between two things, I assume something about what is relevant to what I’m talking about. Say, if this falls to the earth when I let go of it, and this falls to the earth when I let go of it, I need to assume that this too will fall to the earth if I let go of it. I need to assume that what’s relevant in this thing for explaining the fall—for example, its color is apparently not relevant, right? And in fact these two are not the same color. So why, how do I know to infer from here to here? Because I assume that color is not the determining thing. Right? So what is? Its having mass. Right? Mass is the alpha. Mass is the feature of the situation, or of the object, because of which the physical phenomena occur. Exactly like in the halakhic context, where alpha and beta are the theory or the theoretical characteristics because of which the halakhic determinations occur. That’s why one is liable in the public domain; that’s why one is exempt in the injured party’s courtyard. That’s why horn has this, while tooth and foot do not. Same thing. This has mass; a photon has no mass; this has color; something else has no color. So those are various characteristics. Those characteristics generate physical phenomena. Okay? The move from the phenomena to the characteristics is theoretical generalization. The move from one object that falls to earth to all objects that fall to earth—that’s induction. But the move from the fact that objects fall to earth to the theory that masses attract one another—that’s abduction. Right? And what I want to say is that what I’m doing here is the same thing. I’m making an analogy from this to that, or from this to that, but behind the analogy sits the abduction. I’m basically assuming that there is some theory here, and it justifies the analogy I’m making. And what is the theory? The theory that there is some component alpha—I don’t know what it is—but there is some such component, which exists at strength one here and strength two here, at strength two here and strength one here, and given that there is such a component, I can explain all the halakhic facts to you. And that’s confirmation, just like in science. If it explains the facts, apparently it’s a correct theory. So if it’s a correct theory, then I’m also prepared to infer a conclusion from it. Yes, from a correct theory what do I say? If mass is the cause of falling, now I say: this too has mass. What do you say—will it fall or not? It’ll fall. Because if the theory is correct, if mass really is the true factor causing the fall, that’s the alpha, okay? Then anything that has alpha will also fall. Right? So I’m saying the same thing here. If the explanation really is this explanation, then future predictions too—for example, what will the law be here? Suppose we found a manuscript in which the law of horn in the injured party’s courtyard is also written. It doesn’t say so in the Torah, but we found some ancient manuscript where it is written. Let’s bet on what will be written there. My claim is that it will say one. Why? Because if these three are correct, I built a theory, I made an abduction. After I made the abduction, I go back from the theory and infer this conclusion from it. What we saw, remember? Here it even says it: analogy is induction plus deduction. Okay? That’s really a very similar move. I go to the theory—that’s sort of the induction—and then I take from the theory an implication, so I do the deduction. I come back down to this particular case and say: then here it has to be one. Or here too it has to be one.
[Speaker B] And if they discover there that what’s written is actually the opposite?
[Rabbi Michael Abraham] Then that’s a pircha, just like in science. If you discover, for example, a body…
[Speaker B] Right,
[Rabbi Michael Abraham] If you discover, say in science, that there is a body with mass that does not fall to the earth, that would be a pircha exactly like here—it’s exactly the same thing. What are you really saying? Body A falls to the earth and doesn’t fly. Body B flies, so it surely will fall… okay, I didn’t do that well. So it will surely fall to the earth. You understand? That’s the form of thought. And if you find some body that falls to the earth but also flies, then that would refute it. Or the opposite. Understand? It’s exactly the same thing. It’s all the same thing—it really doesn’t matter what field you’re in. It’s all the same form of thought. Another example—I don’t remember if I mentioned this, because in the book I do this through these examples in True and Unstable. I talk there about evaluating students’ abilities. A psychometric exam, okay? I want to make a test that will evaluate how much a candidate will succeed in history studies, okay? So I want to ask what—to build some sort of screening test, whom to accept and whom not to accept. And let’s say I test success—he already studied geography. I want to infer. To test his success, or theirs—a group comes to me. I say this: look, I know, I have a few people I already have experience with. All those who succeeded in geography succeeded in history. But there are some who did not succeed in geography and did succeed in history. What does that mean? That history is easier to succeed in, right? Now, if I test the new candidates on geography—or I look at the grades they got in geography—that will be a good predictor for history, right? Because whoever succeeds in geography certainly succeeds in history. Right? That’s a kal va-chomer. Okay? Now, if I want to do this for all subjects in the university, then I build a test that is not any one of the subjects, but rather includes within it all the abilities—or as much as possible, the abilities relevant to study in all departments, right? And that will be my predictor. And my claim now is that what I’m doing is probably not a kal va-chomer but an analogy. And I say: whoever got 600 on the psychometric exam will succeed in gender studies—unless he’s overqualified for it. But someone who gets 700, then all the more so he’ll succeed there. Right? Why? Because I assume there is some relation between success on the psychometric exam and success in gender studies. And that relation in this case is probably analogy, not kal va-chomer. Okay? It’s even lighter than light and heavy. So the psychometric exam is simply an application of some theory. I checked the relation between geography and history, the relation between chemistry and physics, between physics and history, and suddenly I saw that these and those abilities are responsible for success in all subjects. I built from those abilities some kind of test. But for that, of course, I first had to identify my alphas and betas in order to understand what skills are required.
[Speaker B] There’s a majority, there isn’t a majority.
[Rabbi Michael Abraham] Fine, you never succeed one hundred percent. You never succeed one hundred percent. We talked about this—that this is the logic of uncertain businesses. That’s exactly the point, but we’ll come back to it. It’ll come back to us a lot more.
[Speaker B] Maybe probability?
[Rabbi Michael Abraham] So what I’m actually doing is building some sort of table, and I say this: suppose this is success on the psychometric exam and this is success in history. I check candidates in history. I say, I have Reuven. I know that Reuven didn’t succeed on the psychometric exam, but in history he did succeed. So that means that history is easier to succeed in than the psychometric exam. The psychometric exam is a good measure—maybe even too good. Here, someone succeeded even though he didn’t pass the psychometric exam. Fine. And whoever does pass the psychometric exam—I have no problem with him at all, right? So it’s a good measure. So I determine: the psychometric exam will be the criterion for admission to history studies. Okay? And then I do a kal va-chomer. I say: if Reuven, who did not succeed on the psychometric exam, succeeded in history, then Shimon, who does succeed on the psychometric exam, I assume will also succeed in history. Okay? I do a kal va-chomer. If I make an analogy, say, between history and chemistry, then I can do it through a binyan av. Doesn’t matter. Say I don’t use a psychometric exam. I look at his matriculation grade, okay? If in his matriculation studies he succeeded to such-and-such a degree, I assume that in university studies he’ll succeed similarly, for the sake of discussion. So that’s a binyan av. Okay? It’s the same things. The exact same things everywhere. By the way, there are also pirchot, refutations. Now someone else comes who succeeded in geography and did not succeed in history—he knocks down my whole theory. Now it’s no longer true that between history and geography there is one parameter which is alpha in geography and two alphas in history. No. There is probably another parameter there that can behave the opposite way—such that in history it helps and in geography it hurts, and vice versa. Understand? That’s what makes a pircha. The pircha is basically saying that it’s apparently not the same talent, geography and history. It’s a different kind of talent. And therefore you can’t infer from success in geography to success in history, because perhaps there is an additional kind of talent you need for history that you don’t need for geography. And vice versa. Then you can’t infer. And that would be a table of this sort. Okay? You can’t infer.
[Speaker D] Can a pircha on a binyan av turn it into a kind of kal va-chomer? What? Can a pircha on a binyan av turn it into a kind of kal va-chomer? Right.
[Rabbi Michael Abraham] In principle it can, but that doesn’t happen. Meaning, we’ll analyze all these things. Right now I’m just building this step by step. Okay, so all this was an introduction, because I actually want to prove to you in yet another way—and this time from the tzad ha-shaveh, or from a binyan av from two verses—the necessity of, or the conclusion, that in fact at the foundation of halakhic inference, or at the foundation of the relation between halakhic determinations, halakhic facts, there sits a theory. Okay? Let’s do it this way. I now want to talk about binyan av. Actually we learned kal va-chomer, we learned binyan av. We’ll come back to them, but for now I’m just putting the tools on the board. After that we’ll start playing chess. Okay? So now I want to talk about a binyan av from two verses. What is a binyan av from two verses? I have two sources that teach, okay? And one thing learned. And I want to learn something about the learned one from the two teaching sources. Why do I need two teaching sources? Because each one by itself doesn’t manage to teach me. There is a pircha, right? It doesn’t manage to teach me. Okay. So therefore I somehow need both of them. That’s the general structure of how this thing is built. So let’s formulate it now a little more precisely. You see there are no rules. There are markers that this eraser erases, and markers that it doesn’t.
[Speaker B] The red erases better than the black.
[Rabbi Michael Abraham] Yes, so you’re saying color is a relevant parameter here. It’s not enough that it’s a marker; the question is what color. There really are two relevant parameters here. Okay. The first lesson in logic is about to turn into the most
[Speaker B] basic lesson in chemistry. A bit artistic.
[Rabbi Michael Abraham] In medicine too it works this way, you have to understand. Let’s say I want to know what cures a cold. I give a person hot soup to drink and I give him an apple to eat. Now I ask myself which of them will help. Okay? So at the moment I don’t know.
[Speaker B] This will help his heartburn and that will help his grandmother.
[Rabbi Michael Abraham] It’ll help his grandmother, never mind, but I want to know which of them will help. Say I give him soup to drink and it improves his condition. The apple doesn’t help at all. A cream cake and he becomes healthy on the spot. All kinds of things—I try all kinds of things. How do I build a theory from this in the end? How do I know now whether some other food, about which I need to guess—say, a banana—will help or not help? I say: I do a chemical analysis. I basically ask what’s in the soup, what’s in the apple, and I try to isolate what characterizes those things that help as opposed to those that don’t help. Scientific elimination, right? In other words, I build a theory and decide which parameters—in this case the chemical elements—which parameters need to be present in order to perform the action. Then I say, ah, for the banana now I no longer need to test. I know: if the banana contains this and this and this component, it will probably also help, and if not, then not. Okay? Everywhere it’s all built the same way. It’s all one workshop. So what I now want to do is learn binyan av from two verses. Now in a binyan av from two verses, what basically happens is the following. We have—wait. We’ve had bad experience with blue, right? Not to learn from experience when that’s our topic would be doubly outrageous. This is one. On the other hand, you can’t see this, though it erases nicely—you just can’t see it. The law of conservation of productivity. Okay.
[Speaker B] So here there’s A and B
[Rabbi Michael Abraham] and here I have C. Okay? You won’t see this on the camera, right?
[Speaker B] I hope the camera will see it.
[Rabbi Michael Abraham] The most important thing is the camera. You’re not a factor here.
[Speaker B] I’ll just erase it afterward.
[Rabbi Michael Abraham] We are slaves of technology; technology has long since surpassed us. Okay? So these are my two teaching sources. Now I want to learn a certain law—say law A exists here and exists here too. Fine? Or you know what, let’s make it law A, it exists here and it exists here too. Say for example these are two damagers. Okay? So I say this is a pit and this is fire, or whatever, fine? So I say both pit and fire are liable, right? They have to pay if a pit and fire caused damage. Now I want to know what the law is regarding his stone, knife, and load that he left on top of his roof, and they fell in a normal wind and caused damage. The Talmud in tractate Bava Kamma 6a learns this from the tzad ha-shaveh of pit and fire. Okay? That they caused damage after coming to rest. So this is pit and fire. So how do I learn it? I start like this. Let’s learn from A, let’s learn C from A. Why? Because there is probably something common to C and A, or at least that’s what I assume. Okay? So I say: no, but there is another feature here, Y, which C doesn’t have. Y—roof, C doesn’t have it. Okay? So therefore there is a pircha on this derivation, right? By the way, this derivation can be either a binyan av or a kal va-chomer, it doesn’t matter. There are several kinds of tzad ha-shaveh; actually there are three. There is a tzad ha-shaveh built on two kal va-chomers, a tzad ha-shaveh built on two binyan avs, and a tzad ha-shaveh built on a binyan av and a kal va-chomer. Okay? We’ll talk more about that too, all these things—for now we’re just in the introduction. So this doesn’t work for us. Fine, then let’s learn from here. This one also has Z and we learn to here, same thing. And here the assumption is that really Y is not here—otherwise it doesn’t help us at all, right? We’re trying to learn from here, and this Y pircha cannot be raised here. Fine? So one says: true, but here there is X, and here there is no X. Okay? So that refutes it. Now I say: here too there is no X, right? And then I say: from here too you can’t learn, we refuted it, right? What one has, the other has not. Then they say: the law returns. This is not like that, and that is not like this; the common side in them. What is the common side in them? Z, right? And the common side in them exists also
[Speaker B] in the learned case.
[Rabbi Michael Abraham] And since that’s so, if the common side is what determines, then in fact law A also applies in C—that one must pay. That is law A, that one must pay. What?
[Speaker B] Because of the Z.
[Rabbi Michael Abraham] What’s called the common side. What is the common side?
[Speaker B] The common side is the parameter.
[Rabbi Michael Abraham] The common side is the parameter. Meaning, if you now call them alpha, beta, and gamma, then in these two there is alpha, and in this too there is alpha. This one doesn’t have beta; that one has beta. This one has gamma; that one doesn’t have gamma. Okay? So once again what stands behind this is a theory of parameters. And what we are saying is this. In practice we are doing elimination of parameters, exactly like I did in the table. Later I’ll also do this literally in a table so you’ll see it’s the same thing. But for now I’m just explaining it intuitively. So what I’m saying, really, is that I’m doing elimination here. I say: look, I’m learning from here. When I learn from here, what am I assuming? That Z is the important, determining factor. And Z exists here too. Because of Z it is liable, right? That’s the tuning factor there, yes, the chemical component of this alpha that causes liability in torts. So I say: if here it has it and here it has it, then probably this too will be liable in torts. Wait, one second—but from here you can’t infer that. Maybe Y is what obligates? You can’t. So let’s go here. Here Y is absent, and yet it is still liable. So apparently Z is still the obligating factor. No, no—wait. But here there is helium. Okay? Here there is helium. Maybe that’s why? And here there is no helium. So maybe that’s why—sorry, here there is helium, therefore you can’t learn from here to here. Maybe helium is a relevant parameter? No, no, what are you talking about? Look here. Here there’s no helium, and nevertheless they’re liable, right? Law A applies here. So that means I’ve done elimination. It cannot be the parameter Y, because here it isn’t present and liability still exists. It cannot be the parameter X, because here there is no X and nevertheless there is liability. So if I ruled out the possibility that Y is
[Speaker B] what obligates
[Rabbi Michael Abraham] and the possibility that X is what obligates, I’m left with Z as what obligates. That’s elimination. And therefore I can infer that here, despite the fact that there is no X and no Y, since there is Z, one is liable. Right? That’s how the tzad ha-shaveh is built. Okay? Now I have a question for you. After all, you can always ask in a situation, right?
[Speaker F] It’s more likely that it’s Z plus something else. What? More likely that what obligates is Z plus something else. For example what?
[Rabbi Michael Abraham] Which, for example what?
[Speaker F] In both A and B there is Z plus something else. Right, and that’s why it’s liable. Ah.
[Rabbi Michael Abraham] That’s my next sentence. Here it would be natural to raise a question. The logic here is problematic. Why? Because I proposed a theory. One theory says that the obligating factor—let’s say A is the liability to pay for damages, okay? The obligating factor is Z. Then I said, wait, maybe—I started at the beginning, yes, when I’m here, I say there are three parameters here: Z, Y, and not-X—it doesn’t matter, X isn’t present. Okay? I say: which one obligates? I don’t know. Let’s try the hypothesis of Z. Okay? Z obligates, and therefore I’ll infer that here too it is liable. Wait—but from there you can’t infer that. Maybe Y is what obligates? You can’t. So let’s go here. Here there is no Y and yet it’s still liable. So apparently Z is what obligates after all. No, no—but here there is X. It could be that X is what obligates—not Y but also not Z, maybe X? One says: no, but here you see, here there is no X and yet they are still liable. So X also is not what obligates. So if I ruled out the possibility that Y obligates and the possibility that X obligates, I’m left with Z as what obligates. That’s elimination. And therefore I can infer that here, despite the absence of X and Y, since there is Z, one is liable. Right? But here the question always arises: who told you? Maybe what obligates is W,
[Speaker B] X or Y.
[Rabbi Michael Abraham] which is
[Speaker B] whatever it may be.
[Rabbi Michael Abraham] If you have either X or Y in you, that also obligates.
[Speaker B] Relative to A, Z is the simplest. Or. So that creates the fewest difficulties. Z is the simplest.
[Rabbi Michael Abraham] Wait, wait, let me note this for a second. What I actually want to claim here is that here too you see Occam’s razor. Because true, there are now two competing theories. Either Z is what obligates, or one of the two—X or Y—obligates. Okay? How do I decide between them? Exactly as we learned with kal va-chomer and binyan av. I claim that this contains fewer parameters. And therefore it is simpler, and therefore this is really the preferable theory, so I build it. Meaning, you see that here too I did, in a different way, exactly the same logic. Later I’ll do it in exactly the same way as well. But here I’m just trying to show you how this whole business works. Okay? Now of course, if I were looking at these labels, in principle I could call this parameter X-or-Y; I can define it as W. And what is a person? A person is a collection of cells and many things, right? And still, from my point of view all that whole complex is one thing. I can define the complex X-or-Y as a certain parameter. It could be that this parameter too is composed of things. Okay? Then you can’t know who is more—indeed you can’t even know which is simpler. So true, and this is a remark that has to be understood; I’ll answer it when we get to doing this in the form of a table. But at this level I say: in principle you’re right, that’s true. But here I already know that it is made of two parameters. Here maybe it is composed, maybe not—I don’t know. So if I have to decide which is simpler, it still seems reasonable to say that this is simpler. If you show me that this too is made of two parameters, that really will be a non-simple question, and in a moment I’ll comment on that.
[Speaker B] If you assume it’s W, then from W to C you don’t have any W at all, and you can’t infer anything. C—you can’t infer.
[Rabbi Michael Abraham] Exactly. Therefore I say: if you adopt this theory, then
[Speaker B] it doesn’t work at all.
[Rabbi Michael Abraham] Wait, that’s exactly the point. If you adopt this theory, then in C law A will not apply, because in C there is neither X nor Y. If you adopt this theory, then in C the law will be A. Therefore exactly, I’m saying that I have to decide which of the two is the correct theory. And I choose the simpler one, and then the conclusion is that in C indeed the law applies, and one has to pay for the damage and the pain and so on and so on. Okay? Now I want to ask you an interesting question.
[Speaker G] You shot an arrow at the target—arrow and bird. What? I didn’t notice what happened. You marked a target, and afterward you drew the circle around it.
[Rabbi Michael Abraham] What target? I didn’t understand.
[Speaker G] You said you want to prove that this thing is equal
[Rabbi Michael Abraham] to this. What is equal to what?
[Speaker G] That there are equal parameters in A and B for C.
[Rabbi Michael Abraham] But I didn’t prove that—I showed it.
[Speaker G] But what do you mean you showed it? Who says that parameter Z is the relevant parameter regarding C?
[Rabbi Michael Abraham] Here, I showed it to you, because otherwise the second theory is more complicated. This is the simplest theory because it depends on one parameter. I didn’t assume it. The only thing I assume is Occam’s razor—that the simpler theory is the correct one. But from there on, it’s calculations. I simply look at which theory is simpler, which contains fewer parameters, and that’s all.
[Speaker C] So then what again solves the problem of, say, really defining W?
[Rabbi Michael Abraham] I said that maybe, if you show me that inside Z there are also two parameters, that it too is some composition of two parameters, then maybe you’ve really refuted the binyan av, the tzad ha-shaveh. But here I see there are already two parameters; I checked, I know. With the second one you say maybe. Maybe they go to the grocery store, maybe—you can always say maybe you’re wrong. Fine, in the meantime, if I have to assume which of them is preferable, then this one is preferable. Now look, there is something very surprising that comes up in light of what I’ve said here. Because what I really want to claim is that the simpler theory is the correct one. And if I now have to decide whether parameter Z is what generates the law of liability to pay, or parameter X-or-Y, then I prefer the theory of parameter Z alone, right? Except that in the Talmud, in several places, the Talmud itself makes a move that seems to contradict this. In tractate Ketubot 32 the Talmud says this: “Evidently Ulla holds that wherever there is monetary payment and lashes, one pays the money and is not flogged.” Yes? Ulla holds that if there is monetary liability and lashes, you pay and are not flogged.
[Speaker B] Kim lei be-rabba minei, kim
[Rabbi Michael Abraham] Kim lei be-rabba minei—meaning, you pay and are not flogged. Okay?
[Speaker B] Why does it come out the opposite?
[Rabbi Michael Abraham] Whether it should or shouldn’t, never mind, that’s what Ulla holds. “From where does Ulla derive this?” Where does Ulla know it from? The Talmud says: he learns it from one who wounds his fellow. Just as with one who wounds his fellow, where there is money and lashes, he pays the money and is not flogged, so too anywhere there is money and lashes, he pays the money and is not flogged. In the case of one who wounds his fellow, there is both money and lashes, because he committed an offense—one who wounds also deserves lashes—but he also pays as one who caused bodily injury, and there the law is that he pays; that is clear, the Torah itself says so. So therefore I make from it a binyan av and say: as here, so in all the other places. The Talmud says: no—what about one who wounds his fellow, since he is liable for five things? There is a special stringency in one who wounds his fellow, unlike ordinary property damage, where for monetary damage I pay only the damage itself; one who wounds his fellow pays five things: lost time, medical costs, pain, shame, and damage. So he has a special stringency. What does it mean that he has a special stringency? Notice, it means that if A is, say, one who wounds his fellow—so A is one who wounds his fellow, Z is that there is money and lashes, and therefore the assumption is that every offense that has both money and lashes—well, you pay and are not flogged—that’s the Z. This is an offense that has money and lashes. But no—who says so? Maybe this applies only to such an offense that also has liability for five things—that’s the Y. And here there isn’t that liability, there isn’t this stringency of five liabilities, so therefore you
[Speaker B] can’t learn from there.
[Rabbi Michael Abraham] So the Talmud says: rather, he learns it from conspiring witnesses. That’s the B. We learn from conspiring witnesses, that what? Just as conspiring witnesses involve money and lashes, and one pays money and is not flogged—there too they have Z. It’s money and lashes, and in fact A exists there too, that he pays money and is not flogged. So too anywhere there is money and lashes, he pays money and is not flogged. So that is the derivation. We gave up on A and learn from B. The Talmud says: what about conspiring witnesses, since they do not require prior warning? In B there is a special stringency, X. They are punished without prior warning. For no other offense in the Torah are people punished without prior warning; conspiring witnesses are. So we see there is a special stringency here. So maybe specifically there one pays and is not flogged, but in other places one both pays and is flogged. Of course there are many problems here, because at first glance this should be a leniency, not a stringency—paying and not being flogged should be a leniency, not a stringency—but
[Speaker G] the case is very special.
[Rabbi Michael Abraham] Never mind—the case is special. Precisely because of that you can’t derive from it, because it’s a special case. The Talmud says: rather, he learns it from both of them. What is the common side in them? That there is money and lashes, and one pays money and is not flogged. So too anywhere there is money and lashes, one pays money and is not flogged. And then we came back and said: apparently the theory is Z, not X or Y. I have two possibilities. So I say: apparently it’s not X or Y, but rather the common side, the Z that is found in both. And if so, then in every place where there is Z—that is, where there is monetary liability and liability to lashes—then indeed the law will be A: one pays and is not flogged. Up to here this is a regular structure of the tzad ha-shaveh. Now the Talmud continues: what about the common side in them, since they each contain an element of stringency?
[Speaker B] You can’t learn the lenient from the stringent, or the stringent from the lenient. Meaning, they each have some stringency.
[Rabbi Michael Abraham] He’s simply asking exactly that. It doesn’t work. Who says the theory is Z? Maybe the theory is X or Y. In each of them there is some stringent aspect relative to the learned case. This one has X and this one has Y. So who says that specifically the common side is the relevant parameter? Maybe דווקא the differing sides are the relevant parameter—X or Y. Or in other words, maybe this is the theory and not this theory. And then what’s the difference? If the theory is X or Y, what will the law be in C? It has neither X nor Y, so in C he won’t pay; law A won’t apply. Now this is a very puzzling thing that the Talmud is doing here. It makes sense, but it is very puzzling. Tosafot already asks this. Tosafot says: this is difficult, because if so, then we will never again learn from a tzad ha-shaveh anywhere, since in all of them one can raise a refutation—either an aspect of stringency or an aspect of leniency. After all, if this pircha is relevant, then there is no tzad ha-shaveh in the Torah. We have a law given to Moses at Sinai with the thirteen hermeneutic principles by which the Torah is interpreted, and one of them is a binyan av from two verses. By the way, some say that even a binyan av from one verse is a tzad ha-shaveh. But a binyan av from two verses is certainly a tzad ha-shaveh. According to the Talmud here, there is no place at all for this hermeneutic principle. Every time you make this kind of derivation—and this is the pattern that always appears, this isn’t something unique to this case—every time you do it, if you accept this pircha of who says Z is the determinant, maybe X or Y is the determinant, then you have erased the tzad ha-shaveh from Jewish law. There is no such thing. Do you understand? That is basically what he is asking. He says: wait, who told you? You did not prove that it is Z; maybe it is X or Y. What do I say lies behind this? Basically the point is what I said earlier. Earlier I asked this question and said: yes, but Z is a simpler theory. So if I now translate that to here, the fact that there is a tzad ha-shaveh in the Torah, a law given to Moses at Sinai—what does that mean? Apparently, true, there is indeed the possibility that the theory is X or Y and not that the theory is Z. But Z is simpler, and therefore I am not troubled by the option of interpreting this through the theory of X or Y. That’s how I explain the tzad ha-shaveh in the Torah. If that’s the case, then what are you refuting here? Here you say to me: no, maybe it’s X or Y. But the whole tzad ha-shaveh tells you: I’m not troubled by more complicated theories when I have a simple theory. That’s basically what Tosafot is asking. Right? Tosafot is really asking: why are you refuting me with some complicated theory? The whole idea of being told there is a tzad ha-shaveh in the Torah means: I’m not troubled by more complicated possibilities; the simplest option is for me the correct one until proven otherwise. So what do they want here? The same thing appears in tractate Makkot; I didn’t bring it—what time is it now? Five twenty
[Speaker H] eight. So the Talmud isn’t trying to refute the what? So the Talmud isn’t trying to refute the previous move, with the aspect of stringency? No, no—the Talmud accepts it.
[Rabbi Michael Abraham] The Talmud says there—there the Talmud says: rather, Ulla learns “under” from “under,” and it moves on somewhere else.
[Speaker B] Meaning it accepts this as a rejection?
[Rabbi Michael Abraham] Yes, yes, as if there is a greater stringency here.
[Speaker B] That’s what Tosafot asks. That’s what Tosafot asks. Why? A question.
[Rabbi Michael Abraham] I haven’t yet said what Tosafot’s answer is.
[Speaker B] Now
[Speaker H] have we answered
[Rabbi Michael Abraham] Tosafot is asking about the initial assumption. How did the initial assumption understand at all the tzad ha-shaveh in the Torah in general? Right, but
[Speaker H] does the Talmud also accept the rejection in the conclusion? Yes. Yes. So then—we don’t know how to explain the initial assumption at all? No
[Rabbi Michael Abraham] I said what Tosafot answers; I only brought his question. So even if I tell you what he answers, it won’t help you, but what I answer will help you. The point is that what lies behind Tosafot’s question is that from the very fact that there is a common denominator in the Torah, we’re basically saying that, as far as we’re concerned, the simplest theory is the correct one. Okay? Now notice, the common denominator—I’m going back to the scientific analogy. In a scientific generalization we do exactly the same thing. What I said before: this falls to the ground, right? So what about this—will it also fall? Yes? No. What does it have that it’s made of paper? Maybe that’s why it falls? I don’t know. Right, I’m starting with a blank slate, tabula rasa, I’m not assuming anything. I want to derive conclusions from the facts, to work scientifically. Okay? So what does this have, that it’s made of paper? Then I say: the eraser will prove it—it isn’t made of paper, and it also fell. You see? So this will probably also fall. Suddenly: what does the eraser have, that it’s green? Right? And this one is blue. So maybe it fell because of that? So I say, and this one isn’t green, and then the argument comes back around. This is not like that, and that is not like this: this one is green and that one isn’t green, this one is made of paper and that one isn’t made of paper. Their common denominator is that both have mass, so apparently what causes the fall is the common denominator—it’s the mass, the z—not the x or the y, not the green color now, or green color, or being made of paper. No. Rather, their common denominator is that they have mass. Now what is the Talmud actually asking? Who says so? It could be that really it isn’t mass that determines it, but rather anything that is either green or made of paper falls to the ground. That’s basically what it’s saying. Now this question is difficult for every scientific generalization—you have to understand, it’s exactly the same logic. There’s nothing special here about the Torah; it’s all the same. In law, in science, in anything, everywhere—these are the non-deductive forms of inference that we use everywhere. And therefore I’m trying to show you that what we’re clarifying here is simply the form of human thinking in general. It has nothing specifically to do with one interpretive rule or another, or specifically with interpretation, or specifically with Torah. How do we think? What is it based on? Okay. So basically the question is how generalizations are made. A prototype built from two verses is a generalization. Why is it a generalization? Because how is a generalization structured? A generalization is built exactly the way I said here. I take two examples, right? Fine, this is a unique example—who says you can generalize to all things with mass? Here, I’ll show you. No, no, that too is a unique example—it’s green. How can you generalize to all masses? Maybe only all green objects, or those made of paper, fall to the ground. And then I say: no, no—the common denominator. What does the common denominator mean? It means making a generalization. What does making a generalization mean? It means explaining that some of the features—the alphas and betas—that both of these have are irrelevant to the phenomenon, so I erase them. What remains is elimination, and I’m left only with the relevant parameters. Okay? Then I’ve made a generalization. Because you understand that the fewer parameters I have, the more objects the rule applies to, right? Because if you require fewer parameters, then more objects will meet that requirement. When I say—yes, this is information theory—when you say “the first prime minister of the State of Israel,” that’s Ben-Gurion; there’s only one such person, right? Now if you say “the first prime minister of a state,” there are many more, right? Now “the first prime minister”—not only of a state, maybe a principality too, all kinds of things—there are even more, right? Now I say “prime minister” without “first”—there are even more. Now I say “head” without “minister”—he’s the head of something, okay, not that kind of head, the head of something, just not necessarily of a government—there are many more, right? Meaning, the more characteristics I remove, the more I enlarge the group to which those characteristics belong, the group characterized by them. Okay? Therefore a generalization always, by definition, when we make one, means erasing irrelevant characteristics. That’s what it means to generalize.
[Speaker B] To broaden, to enlarge, not to narrow.
[Rabbi Michael Abraham] To generalize means to erase irrelevant characteristics, reduce the number of characteristics, and expand the number of objects. It’s the same thing.
[Speaker F] Now when we talk about science, what the Rabbi is saying, then we really have tons and tons of objects, and to find what is unique to each one, that really is enormous. But when you’re talking about a prototype built from two verses, you only have two verses—it’s not that many.
[Rabbi Michael Abraham] It doesn’t matter, it’s only a quantitative question.
[Speaker F] Right, but intuitively it seems—since when?
[Rabbi Michael Abraham] From five objects? From how many? So I’m saying, the Talmud says, Jewish law says: from two. Therefore, by the way, that really is the dispute there between Rabbi and Rabban Shimon ben Gamliel.
[Speaker F] We think—thinking is easier for us to grasp with, about all the…
[Rabbi Michael Abraham] No, we think that way here too. If you have two examples and two possible ways to explain them, you’ll prefer the simpler option, right? Here the preference won’t be decisive. But a majority of 51% is still a majority. If you have to decide between these two, I would still decide in favor of this. Right? And that’s exactly Rav Chaim—you know—the signs of an imbecile, or Rabbi and Rabban Shimon ben Gamliel about a presumption after three times. According to Rabbi it’s after twice; according to Rabban Shimon ben Gamliel it’s after three times. Say an ox gores twice or an ox gores three times, or a woman sees blood on a certain date twice or three times, and so on. It’s exactly the same thing. Meaning, when you see three things, each one has separate characteristics. But you say: if it happened three times, apparently the special characteristics unique to the three occurrences are not the important ones. What is important? The common denominator, what appears in all three cases. And apparently that’s what determines it. That’s how you build a theory everywhere. It’s true everywhere. So this question of the Talmud is a very difficult question. Right? What is the Talmud… the refutation… it’s not even a question, it’s the Talmud’s whole conception. “What is special about these two, that they have a stricter aspect?” So there is no common denominator in the Torah. Or let’s formulate it this way now: there are no generalizations in the world. And according to this Talmudic passage, you can’t generalize anything.
[Speaker B] But when there are several shared parameters, how do we choose which one is the relevant parameter?
[Rabbi Michael Abraham] One of them.
[Speaker B] Let’s say if ox and pit had both begun with the letter s, nobody would say that the common denominator of ox and pit is that they both begin with the letter s.
[Rabbi Michael Abraham] Because that seems irrelevant to us. We do elimination on the basis of reasoning too. But in the selection of parameters—I spoke one of the previous times—exactly. I’ll come back to that, it’s also an important point. I spoke about it one of the previous times, about Carr the historian and Semmelweis. Hempel—as it were, with Hempel’s example—where I tried to show exactly that you have to come with some intuition in order to do elimination, otherwise you won’t get anywhere. We’ll return to that—we’ll come back to it here too. Everything I said in the introduction will come back here as well. Okay. So now, what do we do—what do you say? There is a long Ritva in Makkot; it has the same structure. A Talmudic passage in Makkot 4a, the same structure, also like this, and in the end they refute it: what is special about these two, that they have a stricter aspect? The Talmud there claims that this is a dispute among tannaim, whether one can raise the refutation of a stricter aspect or not. So that eases the problem a bit, but still, according to at least one of the tannaim who does raise a refutation of a stricter aspect, there is no common denominator in the Torah. What… so, a law given to Moses at Sinai—you don’t think he disagrees with that. So what, how do you explain it? So there is a very long Ritva there in Makkot 4a, and he brings four or five explanations from different medieval authorities (Rishonim), four or five explanations for why this is so—why despite everything one does raise here a refutation of a stricter aspect; that this is a special case, an exception, all kinds of reasons. None of them is convincing. What you’re saying here is also one of the explanations brought there; none of them is convincing. I’ll tell you what the explanation is. I think the explanation is obvious, based on what I said earlier. There is a very great difference between the common denominator in Bava Kamma 6a and the common denominator that I read here. Do you know what the difference is? In the Talmud in Bava Kamma 6a: his stone, his knife, and his burden that fell from the top of the roof and caused damage. So they derive that from fire and pit. Fire—because the wind carried them just as the wind carries fire; and pit—because when it lands below it causes damage like a pit. People step on it, or when they fall, or they are harmed. Okay? So there’s a common denominator from fire and pit, and you can’t derive it from fire alone and you can’t derive it from pit alone, so you make a common denominator and derive it. What is different there from here? One thing. Exactly. In Bava Kamma the parameters being used are properties. What is special about a pit? That from the start it was made to cause damage. What is special about fire? That another force is involved in it. These are real-world properties of the fire and the pit, what I called microscopic parameters. Those are the alphas and betas. Right? Here, both in Makkot and in Ketubot, that is not the case. What are the characteristics x, y, and z that we called here? That it involves liability for five categories of damage, that no prior warning is required—these are laws. They are not factual characteristics. If you were to tell me, what is special about conspiring witnesses? That they are very wicked and want to kill him—fine, that would be a microscopic parameter, because that would be a fact, part of the characteristics of the act of a conspiring witness. But here you’re not telling me a characteristic of a conspiring witness; you’re telling me a law that applies to a conspiring witness. That’s the whole difference. Do you understand why that makes a difference? Why is it a difference? I’ll explain. If these things are microscopic parameters, then I really do need to decide which theory is simpler, and it’s obvious that this is the simpler theory. But if x, y, and z are laws, then I’m basically saying the following: what causes me to pay and not receive lashes? Is it the fact that there is on me a monetary liability and lashes—that that is the z? Or that no warning is required, or that I am liable for five categories of damage?
[Speaker B] There is a property shared by both of them.
[Rabbi Michael Abraham] Ah! It could be that there is a shared property of both conspiring witnesses and this case, such that in the case of conspiring witnesses it causes the fact that no warning is needed, and in the case of one who injures his fellow it causes the fact that he is liable for five categories of damage. But the property, the characteristic, the alpha here—that’s the same alpha, it’s just that in different contexts it creates different laws, different halakhic stringencies. So therefore you really can’t know; it may be that behind both of these there sits one single characteristic, while behind that there sits beta. If so, then this is no longer necessarily more complicated than that. If you’re talking about characteristics, then you compare this theory to that theory, and then it is obvious that this is simpler, right? Because here there is one parameter and there there are two. The only question is how many parameters there are here and how many there are there—how many parameters, not how many laws. You have no proof that there isn’t one parameter here, that there are only two. These are two laws, two stringencies, but behind both of them there could be one parameter of stringency. In the case of conspiring witnesses, say, and in the case of one who injures his fellow, both involve someone who wants to harm another person, and interpersonal wrongdoing is as severe as it gets. Fine, let’s say, for the sake of discussion. Fine? So I say, look—then in the case of conspiring witnesses, because it’s so severe, no warning is needed. Why is it severe? Because he is harming another person. And in the case of one who injures his fellow, because it’s so severe—why? For the same reason, because he is harming his fellow—therefore one who injures his fellow is liable for five categories of damage. Okay, but the reason why the two different stringencies exist in the two contexts is one microscopic reason, because these stringencies are halakhic stringencies; they are not properties of the things themselves. And since that is so, you have no proof that this is more complex than that; it could be that behind this there sits one microscopic parameter. What is the dispute between Rabbi Yehudah and the Sages? Rabbi Yehudah and the Sages—one here raises a refutation of a stricter aspect and one does not. The one who raises a refutation of a stricter aspect is basically saying: if these two are laws, then behind them there may sit one microscopic parameter—you have no proof. The other says: look, maybe so, but if I have to assess things in light of the data I have, so long as you don’t show me that one parameter is responsible for both of these, my assumption is that this is simpler. You can understand that. Those are the two sides here. What is important to me, of course, is to explain the side that does raise the refutation of a stricter aspect. The side that does not raise the refutation of a stricter aspect is the simple one. The side that does raise the refutation of a stricter aspect is the… Now check throughout the Talmud: there are a few examples—not very many, but there are several, I’ve definitely seen five or six—in all of them the refutation of a stricter aspect is raised where we are dealing with laws and not with characteristics. In all of them. Okay? And then it’s simple, it’s just the logic. Now what do we see from this? Again, for me, beyond understanding what a common denominator is, I’m trying to show you through this that behind inferences about laws, or deriving one law from other laws, there sits a theory; there sit microscopic parameters. And someone who doesn’t understand that behind facts there sit theories, and instead treats it like some kind of hocus-pocus—if this is true then that is also true—he’ll get stuck. He’ll get stuck exactly here; he won’t understand what the refutation of a stricter aspect is, he won’t… you won’t understand how this whole business works. But if you understand that behind laws, behind halakhic determinations, there sits a theory, then you understand why an a fortiori argument is not reversed, you understand why one raises a refutation of a stricter aspect, and why that does not contradict the common denominator. You see, these are different indications of the fact that behind inferences there sit theories. Okay? That’s what was important for me to show today. And now we’ll go back and we’ll begin a fortiori inference, prototype building, and refutations, with the theory and the… we’ll build the algorithm in a more systematic way. Okay, I’ll stop here. Did you photograph this? The board?
[Speaker G] Without the z and the beta, but this, what he hinted at…
[Speaker F] Fine. Explain last week why in Bava Kamma, with tooth and foot, Rav Tarfon and the Sages do reverse the a fortiori argument? Is it because of the half-damage there? Again? With tooth and foot and with horn, Rav Tarfon and the Sages do reverse the a fortiori argument?
[Rabbi Michael Abraham] There is a Talmudic topic… because of the half… because of the half-damage, Rabbi. Fine, I’ll explain why—if I get to it, I’ll explain.
[Speaker F] Is that a special case there…
[Rabbi Michael Abraham] If I get to it, then I’ll explain.