Learning from Experience – Lesson 7
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
🔗 Link to the original lecture
🔗 Link to the transcript on Sofer.AI
Table of Contents
- Induction and abduction in learning from experience
- A fortiori reasoning, two formulations, and refutation
- The theory behind a fortiori reasoning and the unity of rows and columns
- The common denominator as scientific generalization and Ockham’s razor
- Induction, abduction, and a practical difference at the end of the lecture
Summary
General overview
At the end of the previous lecture, two types of inference from particular cases are presented: induction and abduction. Induction is a direct generalization from observed cases to a general law, while abduction moves from observed cases to constructing a theory that explains them, from which general conclusions are then derived. It is argued that in science most of the work is abductive, because a substantive theory includes unobserved theoretical entities and explains the facts, whereas a phenomenological description only describes the observed behavior. Through an analysis of a fortiori reasoning and its vulnerability to refutation, it is explained that a “simple” inference that skips over the theory looks as though it contains two independent arguments, but in fact both depend on a shared theoretical parameter, and therefore one refutation brings down the entire a fortiori argument. Ockham’s razor is presented as a tool for deciding between possible theories by preferring the simplest one, and this is demonstrated both in a fortiori reasoning and in inference by common denominator, as a model for the way science builds generalizations.
Induction and abduction in learning from experience
Scientific induction is defined as an inference from observed cases to a conclusion about a general class, without an explicit move through theory, even if in practice some unconscious theoretical assumption lies behind it. Abduction is defined as moving from observed cases to creating a theory that explains the cases, from which the general law and applications to additional cases are derived; therefore abduction includes induction, but not vice versa. The example of sunrise presents induction as the habit of generalizing from past to future, in contrast to abduction, which explains sunrise through the earth’s rotation and the law of inertia and thereby infers what will happen next. The example of black-body radiation presents a move from a phenomenological description in a mathematical formula to a substantive theory that includes quantum components, and emphasizes that a substantive theory also relies on theoretical entities such as fields and particles that were not directly observed.
A fortiori reasoning, two formulations, and refutation
The table of tooth-and-foot versus horn in the public domain and in the damaged party’s courtyard is formulated as an a fortiori argument in which it is known that tooth-and-foot is exempt in the public domain and liable in the damaged party’s courtyard, while horn is liable in the public domain, and the question is what the law of horn is in the damaged party’s courtyard. Two forms of a fortiori reasoning are presented that appear to be two different inferences: a “row-based a fortiori” that builds a hierarchy between the damaged party’s courtyard and the public domain from the top row and applies it to the bottom row, and a “column-based a fortiori” that builds a hierarchy between horn and tooth-and-foot from the public-domain column and applies it to the damaged party’s courtyard column. It is argued that a column refutation undermines the assumption that horn is necessarily more severe than tooth-and-foot and therefore seemingly topples only the column inference, while a row refutation undermines the assumption that the damaged party’s courtyard is necessarily more severe than the public domain and therefore seemingly topples only the row inference. It is shown that this intuition leads to the expectation that both a row refutation and a column refutation should be needed to topple an a fortiori argument, but in fact in the Talmud one refutation topples the whole argument, with mention of exceptions in tractate Bava Kamma and in tractate Chullin when the values are not “one” but “half,” and issues of “it is enough” come in.
The theory behind a fortiori reasoning and the unity of rows and columns
A parametric theory is presented in which tooth-and-foot has a property with intensity alpha and horn has it with intensity two alpha, while in the public domain two alpha is required for liability and in the damaged party’s courtyard alpha is enough for liability. It is argued that the need to explain the three given data points in the table forces one to formulate the severity of the domains in the same “language” of the same parameter that formulates the severity of the damaging agents, and therefore the row-based a fortiori and the column-based a fortiori are not independent but rely on the same theoretical parameter. A possibility is presented for an alternative theory with two parameters, alpha and beta, in which horn and tooth-and-foot differ by different parameters and the domains depend on different parameters, and this may lead to a different conclusion in the fourth cell. Ockham’s razor is declared decisive in favor of a theory with one parameter when it explains the data just as well as a theory with many parameters, and therefore the conclusion in the fourth cell is accepted as “one.” Refutation is explained as introducing a datum that forces a move to a multi-parameter model and breaks the ability to decide, so that refutation does not prove the opposite result but leaves a question mark, because there are equivalent models that lead to different results.
The common denominator as scientific generalization and Ockham’s razor
The inference of common denominator is demonstrated through keys and a phone falling to the earth, and inferring that a book will also fall, while presenting competing properties such as “green,” “electronic,” “rectangular,” and “paper” that create confusion without a theory. It is argued that common denominator focuses on the shared property of the two source cases, such as mass, and from that infers about the target case, but one can also build an alternative theory that explains the data using a disjunction of two parameters such as “green or electronic.” Ockham’s razor is presented as a preference for a theory with a single parameter that explains the facts over a more complex theory, and therefore the theory is chosen that “everything with mass falls toward the earth,” from which the conclusion about the book is derived. The example of black ravens shows how the shared property “raven” is preferable as an explanation over a more complex theory such as “in Australia or in South America,” and how a refutation of common denominator arises when an alternative shared property is proposed that does not exist in the target case and generates an equally simple theory but with the opposite conclusion.
Induction, abduction, and a practical difference at the end of the lecture
It is argued that inferences such as a fortiori reasoning and common denominator demonstrate the structure of science: collecting facts, building an explanatory theory through abduction, choosing the simplest theory by means of Ockham’s razor, and then deriving phenomena or laws as induction derived from the theory. At the end, a distinction is discussed between practical differences that emerge from a distinction and practical differences that demonstrate the distinction, using the example in which “you can betroth a woman with this” does not explain why lesser sancta are the owner’s property but is only a derivative. The distinction between a positive commandment and a prohibition is presented such that consequences like punishment are “only” results, while the example of rape and the state of a house without a parapet illustrates the theory that distinguishes between a negative state defined by a prohibition and a positive state defined by a positive commandment. The connection to the lecture is formulated as a demand for a theory that explains laws and not merely a list of consequences, so that abduction provides the explanation and induction provides the derivatives.
Full Transcript
[Rabbi Michael Abraham] Okay, let’s begin. At the end of the last few lectures, in the last lecture I spoke about two kinds of generalization, or inference, from particular cases toward something more general. One kind of inference is called induction, and the second is called abduction. Induction—scientific induction—is not mathematical induction, the thing they teach in high school. Mathematical induction is deduction. But scientific induction is basically taking a few cases and inferring from them a conclusion about an entire class of cases, where the cases I saw are particular examples of the general class. Abduction is basically moving from the cases I saw to creating a theory. The theory, of course, will then give me the general law and the applications to the other cases, but I’m going through a theory. So really the second type includes the first, while the first does not include the second. Meaning, if I do abduction, then I basically find the theory that explains the cases I observed. Once I have a theory, of course, from that theory I can derive general conclusions about all the cases, the whole relevant class. And therefore induction too, in the end, comes out of abduction—the general law as well. But if I do induction, then induction is simply moving from particular cases to a general law. I’m not making that move through a theory, and so induction does not include abduction within it, at least not explicitly. Very often, unconsciously or not explicitly, when I do induction there is some assumption sitting behind it about the theory underlying the facts. But basically it isn’t conscious. Yes? I mean, if I say: I see one donkey that is mortal, so I say all donkeys are mortal. That’s a generalization from one case to all cases. That’s induction. There’s no theory behind it, at least not explicitly. Of course, in some unformulated or implicit way I’m assuming that the fact that the donkey I saw was mortal is connected to its being a donkey. In other words, it follows in some sense from its characteristics, characteristics it shares with all the other donkeys. And so there is always some theory underlying induction. But when I do induction, usually I’m simply making a direct generalization from the particular cases to the general case. I’m not passing through the theory. When I do abduction, I posit a theory. Say, in the scientific world, usually we deal with abductions, not inductions. Induction can be a first stage. Meaning, I do, say—the simplest thing, maybe the clearest example of this—again, I won’t go into details—is black-body radiation. Black-body radiation: Einstein found, based on the findings before him, a phenomenological theory. He simply described in a mathematical formula what the radiation of a black body looks like. And then, from the phenomenological description, he found the theory that governs it. He claimed there were quantum matters involved there. So the generalization is the phenomenological theory. The phenomenological theory basically tells me what will happen in all cases based on the few cases I observed. It’s called phenomenological because it describes the behavior, the facts I will see in the world. A substantive theory is a theory that tells me what the theory is that stands behind those behaviors. And the theory includes within it, among other things, entities that can’t be observed, that I haven’t seen. I only assume their existence. Various objects, particles, photons, electrons, all kinds of things like that—particles no one has ever seen. But they are theoretical entities, whose existence I assume based on the facts I saw, and the theory I built, which incorporates these entities and various principles governing the relations between them, basically gives me all the facts, the generalization. Okay. Now this is very important when we come to talk about learning from experience. Because in the end, learning from experience can’t ignore, can’t focus only on induction. I saw—for example, when I see the sun rise every morning, I say it’ll probably rise on future mornings too, because every morning until now that’s what happened. That’s induction. But I can build the theory behind it. What does that mean? I say: the earth rotates at a constant rate of twenty-four hours, okay? Therefore every twenty-four hours there is a sunrise. Now there’s the law of inertia. The law of inertia says that if the earth is rotating and no force acts on it, it will continue to rotate. And since that’s so, then obviously tomorrow too the sun will rise. You understand that here it’s not just inferring from what happened until now to what will happen tomorrow morning and onward. It’s understanding the theory behind the facts I’ve seen until today. And once I have the theory, I can also infer the conclusions about what will happen tomorrow, the day after, and in the days after that. So in the end I reach induction, meaning I reach the general law that deals with the whole set of facts on the basis of the particular cases I encountered. But I get there as a derivative of the theory. Therefore this is actually abduction. A person—think of a child. A child knows that every morning the sun rises. He has no idea why that happens. He doesn’t understand that the earth rotates and all those things. He simply sees that the sun rises every morning. Now since he’s gotten used to the sun rising every morning, it’s obvious to him that it’ll happen tomorrow morning too, without thinking about why it happens, what theory explains it. It’s simply a direct move from the particular cases to the general law. That’s called induction. When a scientist looks at it, he also begins like the child with induction—he says it’s obvious to me this will happen in the coming days too. But in truth that’s a good question: what exactly is the theory that causes this whole business to happen? Then you formulate for yourself that the earth rotates, and the law of inertia means it won’t stop rotating if no force acts on it, and therefore it’s clear to you that the sun will rise on the coming days too. In the end you get to the same facts the child got to, but you get there from a theory that explains those facts. Therefore you’ve done abduction, not induction. Now, at the end of last time I started discussing a fortiori reasoning. My goal was really to use that to show the difference between abduction and induction, the significance of abduction. So let’s go back for a moment to what we did there. This time I’ll do it with a blank board; I think that’ll be better.
[Speaker B] Okay, so like this: we have tooth-and-foot—sorry for the primitiveness—and this is horn. This is the public domain, and this is the damaged party’s courtyard. Okay, now and this—
[Rabbi Michael Abraham] This is the structure of every a fortiori argument in the language of the Sages, and not only in the language of the Sages but also in life. But let’s take this example so it’ll be more tangible. In every a fortiori argument I have the following data. This is zero, this is one, and this too is one. And here I have a question mark. Okay. Meaning, I know that tooth-and-foot is exempt in the public domain, exempt in the public domain and liable in the damaged party’s courtyard. Horn is liable in the public domain—in fact it’s liable for half, but I’m ignoring that for the moment. So it’s liable in the public domain, and the question is what the law of horn is in the damaged party’s courtyard. That’s the question. So we derive an a fortiori argument. What a fortiori? “If tooth-and-foot, which is exempt in the public domain…” That’s one formulation. A second formulation: “If in the public domain, where tooth-and-foot is exempt, horn is liable, then in the damaged party’s courtyard, where tooth-and-foot is liable, is it not logical that horn should be liable there?” And again, the answer is one. Now last time we examined whether these two formulations are two different formulations of the same argument, just two different ways of expressing the same inference, or whether they are two different inferences. Let’s formulate this a bit more neatly. I said there’s a row-based a fortiori. The row-based a fortiori starts from this row and infers to this row. Okay? Basically it goes like this. In this row I see that the damaged party’s courtyard is more stringent than the public domain. More stringent means it’s easier to impose liability in the damaged party’s courtyard than in the public domain, right? You see that in this row. So that’s the inference from the row. I go down and apply that same hierarchy to the bottom row. I say: if in the damaged party’s courtyard it’s easier to impose liability than in the public domain, and horn is liable in the public domain, then certainly it will be liable in the damaged party’s courtyard. Because the damaged party’s courtyard is always more stringent than the public domain. Just as here, so too here. That’s what I called the row inference. Similarly, there’s also a column inference. Now I make this inference. I start from this column, end with this column, and the move is like this. What does that mean? I start from this column. In this column I see that horn is more severe than tooth-and-foot, right? Because in the public domain tooth-and-foot is exempt and horn is liable. So horn is more severe than tooth-and-foot. Now I move to the damaged party’s courtyard, here, and I apply that same hierarchy to this column. So in this column too I say that horn should be more severe than tooth-and-foot. And if tooth-and-foot is liable, then horn, which is more severe than it, certainly should be liable. So I build the hierarchical relation here, transfer it here, and then infer the conclusion here. Exactly as I did here, only here it works across columns and here it works across rows. Now, in every a fortiori argument you can always do this. There are always these two formulations. Because ignore tooth-and-foot and horn and all the specific content. You can call them X, Y, A, B. And it will always be zero here, one here, one here, and a question mark here. And that question mark is replaced by one. The correct answer is one. Okay? Every a fortiori argument is like that. Now let’s see what exactly this means. The row-based a fortiori—let’s look at that a fortiori argument—this is the row-based one. What is it really saying? It’s basically saying: I build a hierarchy relation, I take these two data points and from them build a hierarchy relation. What do I say? The damaged party’s courtyard is more stringent than the public domain. I see that from the top row. Then I apply it below. So what I really have here is a hierarchy relation built from two of the data points, and then an application of it to the third data point. That’s how I reach the conclusion that here too there is a one. Okay? The hierarchy relation basically says the damaged party’s courtyard is always more stringent than the public domain. What happens here? Here the hierarchy relation is built on the right-hand column: that horn is more severe than tooth-and-foot. And now I apply it here. If horn is more severe than tooth-and-foot, then in the damaged party’s courtyard, where tooth-and-foot is liable, horn certainly will be liable. Now notice that the assumption of the row argument, the first argument, is that the damaged party’s courtyard is more stringent than the public domain. That argument doesn’t need, or doesn’t require, the question of what the relation is between tooth-and-foot and horn. Right? I never assumed anywhere that horn is more severe than tooth-and-foot if I’m dealing with the row argument. It doesn’t enter anywhere. The relation between tooth-and-foot and horn doesn’t affect the inference. Whether horn is more severe than tooth-and-foot or less severe than tooth-and-foot, I can still say that in this row we see that the damaged party’s courtyard is more stringent than the public domain, so as for horn—I don’t care whether it’s lighter or more severe—the damaged party’s courtyard is still always more stringent than the public domain. You see that the relation between these two, the relation I built here, plays no role in this inference, in the row inference. And similarly the other way around: in the column inference, I assume that horn is more severe than tooth-and-foot from this column. Nowhere does the question enter of which is more stringent, the public domain or the damaged party’s courtyard. It plays no role in the row inference. Therefore, seemingly, these are two different inferences. What’s the indication of that? Let’s see. Suppose I now found another domain in which the opposite relation holds—what’s called a refutation, yes? We spoke about the moon. Some other domain we don’t know, it doesn’t matter right now, some other place, not the damaged party’s courtyard and not the public domain, where tooth-and-foot is liable and horn is exempt. Those are the data. Now I ask what happens now. This thing is called a refutation. Why is it a refutation? Because if you look at the a fortiori argument of the columns, you’ll see that it’s built on the assumption that horn is more severe than tooth-and-foot, right? From this column. But if you look at that column, you’ll see that this isn’t true. Here tooth-and-foot is more severe than horn. That means the relation between tooth-and-foot and horn is no longer unambiguous. And therefore you can’t apply it here and say that if tooth-and-foot is liable in the damaged party’s courtyard, then horn certainly is liable, because it’s more severe. Not true; it’s not certain that it’s more severe. The relation between them is now a question mark.
[Speaker C] Fine, but you need to see which one the damaged party’s courtyard is more similar to.
[Rabbi Michael Abraham] Right, but I don’t know which one it’s more similar to. So therefore I say: I’m in doubt, I can’t infer conclusions. A refutation does not prove that horn is exempt in the damaged party’s courtyard. What it shows is that you can’t prove that it is liable. I remain in doubt. That’s what’s called a refutation. There’s an asymmetry between refutation and a fortiori reasoning. A fortiori proves a result. Refutation doesn’t claim the opposite result, but only says: this result, you can’t be certain it’s correct. Okay, so this refutation that I added here regarding the moon refutes the a fortiori argument of the columns, because one of its assumptions was that horn is necessarily more severe than tooth-and-foot. Here you see that this isn’t true, or at least not necessarily true. What does that do to this row-based a fortiori argument? Nothing. The row-based a fortiori argument says the damaged party’s courtyard is more stringent than the public domain, right? Does this column undermine in any way that assumption, that the damaged party’s courtyard is more stringent than the public domain? No, it says nothing about that. Therefore I can continue to infer that if horn is liable in the public domain, then in the damaged party’s courtyard it certainly will be liable. Because in the damaged party’s courtyard they impose liability more readily than in the public domain. Therefore this refutation knocks down the column-based a fortiori, but not the row-based one. In order to knock down the row-based a fortiori, I really need to add a refutation like this. Say, find some damaging agent—I don’t know what—let’s call this damaging agent “ear,” all right? Just for the—where this would be one and this would be zero. Now how does that refute? It refutes the row-based a fortiori. Why? Because in the row-based a fortiori I assume that the damaged party’s courtyard is more stringent than the public domain on the basis of this row. But here suddenly I see that that’s not necessarily true. Here, on the contrary, the public domain is more stringent than the damaged party’s courtyard. Therefore the relation between them, the relation I assumed here, has fallen; it’s not necessary. Consequently you can’t apply it here. Therefore you can’t decide that horn too will be more severe, that with regard to horn the damaged party’s courtyard will be more stringent than the public domain. The question is whether horn is like ear or whether horn is like tooth-and-foot. Since I don’t know, the a fortiori falls. Meaning, in order to topple the row-based a fortiori I need to add what we’ll call a row refutation. In order to topple the column-based a fortiori I need to add a column refutation. Now what would we expect to happen when the Talmud raises a refutation against an a fortiori argument—say a row refutation, a column refutation? I would expect that the column-based a fortiori would fall, but the row-based a fortiori would remain in force. Therefore if they raise such a refutation, that won’t be—yes? That’s not here. I’m looking only at this; there’s a column refutation. The moment someone raises a column refutation, the conclusion of the a fortiori argument still remains, the answer here is still one. Not because of the a fortiori of the columns, but because of the a fortiori of the rows, this a fortiori here, because it isn’t affected by this kind of refutation. Therefore, really, in order to refute an a fortiori argument you should always have to bring both a column refutation and a row refutation. Without that you haven’t brought down the a fortiori, because it’s enough for one of the formulations to remain in order to prove the result, right? That here it says one—you don’t need both. It’s enough that one of the arguments is correct. The second says I don’t know, but this argument says I do know; the answer is one. So it’s enough for me that one of the arguments is correct in order for the answer here to be one. So really, in order to topple an a fortiori argument, I should always have had to bring a row refutation and a column refutation. It turns out it doesn’t work that way. In the Talmud they bring either a row refutation or a column refutation, and the a fortiori argument falls. Nobody says, wait a second, you knocked down the row formulation, let’s move to the column formulation—or the reverse. No, the a fortiori falls. I said there are two places where that’s not so, in tractate Bava Kamma and in tractate Chullin, but in both those places the table is not—what’s written here isn’t one but half. And then there’s an issue of “it is enough,” and so on; I’m not getting into that now. Maybe I will—just one thing I will say. Suppose it says half here; that also sharpens the difference between the arguments. Look, if half is written here, what will the row-based a fortiori give me? I did this last time too. What will the row-based a fortiori give me? The row-based a fortiori says that the damaged party’s courtyard is more stringent than the public domain. That’s from the top row. Now I come down here; here too the damaged party’s courtyard is more stringent than the public domain. So if horn in the public domain is liable for half, then in the damaged party’s courtyard—well, also there it will be liable for half. So the result is half, not one. But the column-based a fortiori—look—here it says that horn, from here we see that horn is more severe than tooth-and-foot, from the right-hand column. Now I move here; here too horn is more severe than tooth-and-foot. So if tooth-and-foot is one, then horn is one. You see? The result is one, not half. What does that mean? That these really are two different arguments. The fact is that their results are different. These aren’t two different formulations of the same argument, because otherwise I would expect the result too to be the same. If argument A gives me a result of half and argument B gives me a result of one, that means these are probably two different arguments, not two different formulations of the same argument. So I’m using this half here only to show that we’re dealing with two different arguments. Okay, so now the riddle we’re left with is: why in the Talmud, when they bring a column refutation or a row refutation, one of the two, does the a fortiori argument fall? And nobody says, okay, let’s use the perpendicular formulation. If you refuted the rows, I’ll use the columns; if you refuted the columns, I’ll use the rows. No, it doesn’t happen. Why not? So here—
[Speaker C] Rabbi, I think you switched something somewhere. What?
[Rabbi Michael Abraham] I think you switched something somewhere.
[Speaker C] You said that adding a row is a refutation of the row inference. Yes.
[Rabbi Michael Abraham] It’s a refutation of the column inference, no?
[Speaker C] No, no.
[Rabbi Michael Abraham] Adding a row here, where here it says one and here it says zero, refutes this row, where here it says zero and here it says one. It refutes the row inference.
[Speaker B] The row inference is this. Wait, now how do I—how do I—okay, wait. Does any one of you know how I restore—okay,
[Rabbi Michael Abraham] Now look. I’ll try to explain why these two arguments really are one argument, or why they are both refuted together. Let’s add this here, okay. Look, now I’m
[Speaker B] looking at this table
[Rabbi Michael Abraham] and I’m going to try to explain the inference in a more detailed way. So I’ll say it like this. Look. Let’s look at the row inference. In the row inference I begin with the fact that the damaged party’s courtyard is more stringent than the public domain. From the laws of tooth-and-foot I infer that the damaged party’s courtyard is more stringent than the public domain. What does that mean? It means that in order to impose liability in the damaged party’s courtyard, something less is required than what is needed to impose liability in the public domain. Suppose I denote tooth-and-foot by alpha—
[Speaker B] I think you took a color instead of a pencil.
[Rabbi Michael Abraham] What? You took a color. Yes, doesn’t matter. Look. Suppose tooth-and-foot has some property alpha. Okay? And now I say that horn is more severe than it—let’s look at the columns—horn is more severe than it. That means horn has the property two alpha. Okay? Now let’s try to explain the data according to this property without identifying what the property is. So I say as follows: in the public domain, tooth-and-foot is not liable. Why? Because alpha, the intensity alpha, is not enough to impose liability. Only something very severe will manage to impose liability in the public domain, right? That means that in order to impose liability in the public domain you need an intensity of two alpha. Agreed? Tooth-and-foot, which has intensity alpha, won’t impose liability in the public domain because you need at least two alpha. Horn, which has two alpha, will indeed impose liability in the public domain. What happens in the damaged party’s courtyard? In the damaged party’s courtyard, tooth-and-foot does impose liability. That means alpha alone is enough there, right? Since tooth-and-foot, which has alpha, imposes liability. Now I ask: what will horn do in the damaged party’s courtyard? It certainly will impose liability, because if alpha is enough, if you have two alpha then certainly you will impose liability. Therefore the answer is one. But notice what I got. What I got is that the relation between tooth-and-foot and horn is formulated in the same parameter as the relation between the public domain and the damaged party’s courtyard. Because if there were no connection between the public domain and the damaged party’s courtyard—suppose this were alpha and this were beta, here this would be beta—
[Speaker B] here this would be beta, okay? So what am I saying?
[Rabbi Michael Abraham] So if this were beta, then there would be no explanation for this, right? Alpha can’t impose liability because you need beta, and tooth-and-foot doesn’t have beta, it has only alpha. So this is zero. Here it has alpha, so it imposes liability because here there is alpha and here alpha is enough to impose liability. I can explain the top row. But I won’t be able to explain the bottom row. Because in the bottom row, what are you telling me? That horn is more severe than tooth-and-foot, and therefore horn has an intensity of two alpha. But explain to me why in the public domain it is liable. There you need beta. It doesn’t have beta. Therefore it’s clear that the severity of—the properties of the public domain must also be formulated in the language of alpha, and not in the language of beta. The same language in which I formulate these properties, I have to formulate these properties as well. Suddenly you see that the vertical a fortiori and the horizontal a fortiori—the rows and the columns—it’s really the same a fortiori. When I say that in the damaged party’s courtyard it’s easier to impose liability than in the public domain, I’m really saying that in order to impose liability in the damaged party’s courtyard, the alpha level is enough. In order to impose liability in the public domain, you need the level of two alpha; it’s harder to impose liability there. And this is formulated in the same units in which I formulate the severity of horn as against tooth-and-foot. If it were not the same units, there would be no explanation for this table. And in fact, all these alphas that I marked here are the explanation of the a fortiori that arrives here at the answer one. Basically, behind the inference that places the number one here sits a theory, and the theory says that there is some characteristic called alpha, which tooth-and-foot has at intensity one and horn has at intensity two. And with regard to domains, this is the important parameter for imposing liability, except that in the public domain you need an intensity of two alpha in order to impose liability—less than that won’t impose liability—and in the damaged party’s courtyard an intensity of one alpha is enough to impose liability. You suddenly see that the same axis that determines the hierarchy between horn and tooth-and-foot is the same axis that determines the hierarchy between the public domain and the damaged party’s courtyard. Which means that the row-based a fortiori—the a fortiori from here to here—is not independent of the column-based a fortiori. Both assume the existence of the parameter alpha. And if you want to formulate the column-based a fortiori, it’s not enough for you to say that horn has two alpha and tooth-and-foot has alpha. You also have to say that the damaged party’s courtyard is alpha and the public domain is two alpha. You need to assume something about the row too in order to make the column-based a fortiori, and vice versa.
[Speaker C] So Rabbi, when we do abduction in the end, the row inference teaches a hierarchy relation for both the columns and the rows.
[Rabbi Michael Abraham] Correct, exactly. In other words, when I do the a fortiori argument, when I do the a fortiori in the way it’s usually done, that’s parallel to induction. It’s really an analogy, not induction, because it’s not a conclusion to a general law but a conclusion to one more case. From these cases I infer to one more case, so in principle it’s an analogy. Okay? A certain kind of analogy. So when I make this analogy, I’m basically saying: if horn is liable in the public domain, then it’s liable in the damaged party’s courtyard, without entering the question of what the theory is behind this analogy. But if I don’t look at the theory and only make the inferences from case to case—yes, just as I do induction and ignore abduction—you see that I can miss something. Because on that simple level, that phenomenological level, it looks as though the two arguments of the rows and the columns are two different, independent arguments. Then the refutations would have to be doubled in order to topple the a fortiori, but that doesn’t fit the Talmud.
[Speaker D] But you’re concluding—sorry—you’re concluding that there’s a shared parameter for both the column and the row, for both the columns and the rows, because the Talmud doesn’t preserve the a fortiori when it’s refuted by only a column being refuted or a row being refuted, so you’re sort of concluding that the alpha parameter is shared by both the columns and the rows because the Talmud didn’t want to preserve the a fortiori even though—whoa, no, no, not because of that.
[Rabbi Michael Abraham] No, otherwise that would just be begging the question. No, that’s why I showed that it has to be—suppose there were only the row-based a fortiori, I could still show you that you have to attach the alpha parameter to both the columns and the rows. And the result is that we now understand why, when the Talmud refutes the rows, it is basically throwing out the columns as well. That’s a result.
[Speaker D] But it seems to me like a matter of wording. You changed the wording at the beginning when you wanted to prove that the a fortiori could remain if I relate to a row or a column—you formulated it in a certain way—and afterward you… To say that tooth-and-foot is more when horn is more severe than tooth-and-foot in the case of an innocuous ox, that’s not correct, because it really depends on the domain, it depends on where it takes place. What I want to say is that I’m not inferring from the fact that the Talmud doesn’t refute, doesn’t preserve the a fortiori even though seemingly it could preserve the a fortiori because the refutation is only against a row or a column—I don’t need to get from there to the claim that therefore it’s a shared parameter in both directions, but rather that in the matter itself you can’t ignore the situation, the meaning of the things, that I can’t just throw out horn as such. If I go to the row, the damaged party’s courtyard is more stringent than the public domain. It depends. It depends, depends on what we’re talking about.
[Speaker B] And if it depended, then there wouldn’t be an a fortiori. Because if it depended, then there is no a fortiori. Let me bring this back for a second. Look, if it depended, then I would say like this: suppose this has beta—
[Rabbi Michael Abraham] and this too has beta. Fine? Or actually, you know what, let’s make it two beta. Fine? Meaning tooth-and-foot is more severe than horn in terms of betas, and horn is more severe than tooth-and-foot in terms of alphas. Okay, let’s assume that’s the case. Then I say: in order to explain these laws, these, I can explain them by saying that here two alpha is required and tooth-and-foot has only one alpha, and these laws I’ll explain because here it’s two beta. But then there’s no a fortiori. Or you know what, you can even erase this beta—there’s nothing. Then there’s no a fortiori. How do you know that the answer here is one? My claim is that the moment you make an a fortiori argument, it means you assume—suppose you made the row-based a fortiori, the column-based one—then you assume that horn is more severe than tooth-and-foot because this is two alpha and this is alpha, in some parameter that I don’t know what it is, but horn has more of it than tooth-and-foot does. I mark that as two alpha and alpha. My claim is that this doesn’t need to pass over to the row-based a fortiori. Stay with the column-based a fortiori. Even then you need to write two alpha here and alpha here, because otherwise you won’t be able to formulate even the column-based a fortiori. Because if you look, say, at what exactly I assumed in the question: in the question I assumed that in the column-based a fortiori all I’m saying is that horn is two alpha and tooth-and-foot is alpha. I establish a hierarchy between these two, but don’t need anything about the hierarchy between these two. Agreed? That’s my status quo, right? And that’s why the a fortiori arguments looked independent to me. I imposed a hierarchy between these two and I don’t need the hierarchy between these two—that’s a different argument from establishing a hierarchy between these two without needing the hierarchy between these two. That was the question. What do I answer? A very simple answer, without assuming that the Talmud refutes it in only one direction and therefore it’s refuted. That will be the result—I’ll explain that in light of my conclusion from here—but my conclusion from here I derive from logic alone. What does logic say? Suppose horn has the characteristic two alpha and tooth-and-foot has the characteristic alpha, therefore horn is more severe than tooth-and-foot. Still, in order to explain the facts—these are the facts, right? Now I want to explain the facts, so how do I explain them? I say: tooth-and-foot in the public domain is not liable. Why not? What is needed in order to impose liability in the public domain? Give me the theory. Two alpha. It has to be two alpha, right? Because if alpha were enough, then it would be liable here, so it has to be two alpha. You see that I reach this conclusion from considerations of logic alone. And the same thing here. In order to explain that in the damaged party’s courtyard tooth-and-foot is liable, I have to assume that in the damaged party’s courtyard alpha is enough to impose liability, right? Because it has alpha and it is liable. So you see that I used—I used only the a fortiori of the columns, not of the rows, and still a hierarchy was created on the row as well. Do you understand what I’m saying? The column-based a fortiori alone assumes a hierarchy in both the columns and the rows. It’s not true that the column-based a fortiori assumes a hierarchy only in the columns, and the row-based a fortiori assumes a hierarchy only in the rows. No. Each of the two formulations assumes a hierarchy both on this axis and on this axis. Because otherwise—yes?
[Speaker E] Why are you assuming that when you want to impose liability on horn you need two alpha, a greater intensity, and not that maybe there’s actually another parameter here, so that horn is liable because of that, and that’s beta?
[Rabbi Michael Abraham] That’s an excellent question, an excellent question, and let me answer it now. Basically what he asked… is that Ockham? Wait. What?
[Speaker C] But that’s what
[Rabbi Michael Abraham] you said before,
[Speaker C] why not assume there are two parameters?
[Rabbi Michael Abraham] Right. I’ll get to that in a moment. Alpha. Look. That’s also a possible theory. Right? Tooth and foot have an alpha property, horn has a beta property. For the public domain you need beta in order to obligate; for the injured party’s courtyard you need alpha in order to obligate. You see that all the laws are explained, right? That’s also a possible theory. Agreed? Tooth and foot are not liable in the public domain because they have alpha, but you need beta; they don’t have beta, so they’re exempt in the public domain. And in the injured party’s courtyard they are liable, because to obligate in the injured party’s courtyard you need alpha, and tooth and foot really do have alpha. Horn has beta, and therefore it is liable in the public domain, because in the public domain whoever has beta is liable. In the injured party’s courtyard you need alpha, and horn doesn’t have alpha, it has beta. Therefore the answer here should have been zero. If that theory… but then it’s not an a fortiori inference? Wait a second. If that theory is correct, then the correct answer here is zero. If the previous theory is correct, the previous theory basically says that beta is two alphas. Right? The second parameter is basically two alphas. If the theory that beta equals two alphas is correct, then the answer here is one. Do you understand what I’m saying? Now the question is, okay, so which theory is right? We have two theories, each of which leads to a different conclusion. Which of the two theories is right?
[Speaker C] Occam’s razor. What? That one parameter is enough. Right. Why assume there are two parameters?
[Rabbi Michael Abraham] If I have a theory with a single parameter that explains all the empirical facts—which are these three facts here, right—and I also have a theory that requires two parameters and it too explains these facts, I’ll choose the simpler theory, Occam’s razor. Meaning the theory with the single parameter, and therefore in the end the conclusion is that the theory is this one, that this is the correct answer. That’s it, that’s the right answer, and therefore the right answer is one and not zero. Occam’s razor decides it. Now you see, what we’re seeing here is the whole Torah in a nutshell. Basically what I showed you here is how science works. Science works this way too. I have three data points, the three facts that I see in the Torah, right, so from the standpoint of Jewish law these are empirical data. It’s written in the Torah. Now I ask myself a question: what will the law be in that case? In parallel, in science I ask what will happen in such-and-such a situation that I haven’t observed. Right? That’s basically the parallel scientific question. What do I do in order to answer it? I build a theory on the basis of the facts known to me, namely the three facts in the a fortiori table. That’s abduction. And from that theory I infer the conclusion—what will happen in the fourth case. That’s exactly what I did here. I take the facts, build a theory that explains the facts, and from that theory derive what the law will be in the fourth case. Now of course for every set of facts, as we saw in the graph if you remember, for every set of facts there are infinitely many possible theories that explain them. Here too. I could have produced theories with beta, gamma, delta, as many as you want. There are theories here, infinitely many theories. How do I sort out which theory is the correct one in my eyes, or which theory I’m going to use? The simplest one, Occam’s razor, the straight line. Okay, that’s basically the simplest theory. The simplest theory in the context of the a fortiori argument is the alpha-two alpha on the axis, on the two axes, on the columns and on the rows. Once that’s the simplest theory, I assume it’s the correct one. Once it’s the correct one, the answer in the fourth case is one. Now, this really illustrates almost everything we’ve talked about until now. Because on the one hand there’s the process of induction. In our case, what corresponds to that is: ask a person in the study hall, present him with this table of tooth and foot, horn, injured party’s courtyard, and public domain, with the three data points, and he’ll tell you the fourth answer is one. But he has no theory behind it; he just does the induction directly. Then you ask him, wait, why do you think the answer is one? Induction—because there were certain cases where that was the law, so I generalize to the general case. In this case it’s not generalization, as I said earlier, but it parallels inductive generalization, okay? Without explaining to you what the theory is that stands behind it. Then a scientist comes along and says, wait, let’s think for a moment about what’s behind this whole business; let’s build a theory that explains the known facts. From the known facts I build a theory—for example the straight line on the graph we talked about in previous classes. From the theory I’ll infer a conclusion about a case I still haven’t observed, or in our case, the fourth square in the table. From the theory. How do I choose the theory? After all there are infinitely many possible theories. The simplest theory—that’s Occam’s razor. So here I did induction, I did abduction, and I showed the application of Occam’s razor. The application of Occam’s razor always operates on the plane of abduction, not induction. Because induction basically says there is one induction. That is, if I saw two donkeys, then all donkeys are mortal, okay? That’s induction. The theory behind the induction could be many possible theories. And arriving at the theory—that’s abduction, not induction. Among the possible theories I choose the simplest one, and Occam’s razor governs abduction, not induction. Of course, because once it determines the correct abduction, the correct induction follows automatically. Because once you’ve established the theory, it also tells you what will happen in the additional cases, as with us. Once I chose the simplest theory, alpha-two alpha, alpha-two alpha, I know that the answer is one. So I can do the induction after I have the theory in hand. Clear? Now, I’ll do one more thing, one that’s closer to what happens in science. And what
[Speaker C] I’m—Rabbi, can we see a refutation with these parameters? What? Can we see how a refutation works with these parameters?
[Rabbi Michael Abraham] Ah, yes, yes, let me show you how a refutation works. Let’s go to the moon. The moon of eta. Okay, that was a refutation of the column. Now let me show you how it refutes both a fortiori arguments. Once I put this refutation here, what did I actually say? I said that here you can no longer say that this is alpha and this is two alpha, right? The severity relation is no longer clear, because if that were the model you couldn’t explain this. Agreed? Rather, something else must be going on here. A second parameter is forced on me. I have no option to build a one-parameter model. With one parameter you won’t succeed in explaining a table of this type when in the matrices, yes, you have two rows that are independent of one another. When there are two rows independent of one another, that means that the basis spanning them is at least two—the rank of the matrix. So that means you need two parameters, basically, in order—at least two parameters—in order to explain this matrix. Consequently, it’s also not true to say that this is two alpha and this is alpha. Because think: if I now stay with that theory—that was the theory before, because that’s the theory of the row—it can’t be correct, because if this is found this way, this it explains, but what will this be? Also alpha, right? Because tooth and foot are liable. Okay? And beta—horn has beta—so here it’s really zero because you need alpha. But notice what happens here. I have no explanation. How is this one? Rather, I have to add two alpha here. This has both beta and two alpha, and therefore this is one. But if it has two alpha then this too will be one. So that can’t be; it won’t explain this law. So either it can’t explain this law or it can’t explain this law; you have no model that can explain it. I can build a model with two parameters that explains it. But the two parameters have to be here too, not only here. Meaning, if here you keep one parameter—one parameter for each column—you won’t succeed in explaining the table. It shifts to two parameters. Once it shifts to two parameters, then you can see that both the a fortiori argument of the rows and the a fortiori argument of the columns have collapsed. Therefore you don’t need an additional row refutation to topple the a fortiori argument of the rows. I’ve already shown that it can’t be alpha and two alpha, with no beta parameter in play here at all. It can’t be. Here too there has to be a beta parameter in play. And basically what is called for here is: this is beta, this is alpha. Right? If I put alpha here, beta here, alpha here, beta here, then it explains this law, it explains this law. This is zero because here there is alpha and you need beta. It explains this law, here there is one, and it explains this law because here there is beta but you need alpha in order to obligate—and the law as well, it explains all the laws. You see? This model explains all the laws. A two-parameter model that explains all the laws. What should be here now? Zero. Zero. There’s your refutation. Now it refutes—you see that it also refutes this relation. Here too it’s no longer two alpha but rather beta. So the column refutation refuted not only the a fortiori argument of the columns, but also the a fortiori argument of the rows. Once you enter abduction, you suddenly understand what stands behind these things, and then all the puzzles are solved immediately.
[Speaker C] So why, by means of the refutation, don’t we say that in fact the law really is that it’s zero? What? Why, by means of the refutation, don’t we say that in fact the law there is zero and not one?
[Rabbi Michael Abraham] Because I can find you a two-parameter model in which the answer will be one. What I wrote here is one model: alpha, alpha, beta. But I could also build a model in which the answer would be one. And that too would be a two-parameter model, meaning this model would not be preferable to it. And once you have two models that are equivalent, one gives one and one gives zero—that’s the meaning of a refutation. You’re right that if the model that gives zero is always more complicated than the model that gives one, or the model that gives one is more complicated than the model that gives zero, then this refutation is not a refutation but a counterproof. It’s a proof that here it is zero and not one; that’s not a refutation. A refutation means it remains a question mark. If you proved that here it is zero, you could actually call it a light-and-a fortiori, not an a fortiori. You proved a leniency here, not a stringency, but still you proved it, and that’s an argument, not a refutation. But that’s not what happens. Now I’m not going into all the—there’s a mathematical way to do this in a completely orderly fashion. I’m not going into all those algorithms here. We once wrote a book about it, but for our purposes here, what I’ve shown you is enough. I’m showing you that once you do the abduction, once you enter into alpha and beta, these are basically the theoretical entities—like the electron, the field, all those things. They’re parameters that I don’t see, I don’t know what they are, they’re there, but I assume they’re there in the background because that’s what explains the facts for me, the data. And after I posit the theory I can infer from it the conclusion: what is the law in this case? So really the alphas and betas are the theory. Abduction is moving from the laws.
[Speaker B] Okay, maybe just to reinforce this a bit more, I’ll
[Rabbi Michael Abraham] do another inference that is simply really similar to scientific generalization. This is really how science works. So look, this is the inference of the common denominator. I’m not going to do it with a table because with a table it’s more complicated, though you absolutely can do it by exactly the same method. But I’ll do it in a more intuitive way. The common denominator basically tells me—let’s do the common denominator scientifically, as a scientific inference, not a Talmudic inference, but it’s the same thing, it looks exactly the same. Let’s say I want to infer the law of gravity, okay? I want to study the law of gravity. So I say: I look at a certain object, let’s say my phone, I let go of it and it falls to the earth. Then I take this bunch of keys, and it too falls to the earth. So I say okay, then
[Speaker B] I’m now constructing an inference of this sort, look. I have here some—these are keys. Okay? I’m writing here with my disgusting mouse, so sorry for the scribble. Keys, and this is a phone. Okay? This is a phone. Now I say: the phone and the keys, each of them fell to the earth.
[Rabbi Michael Abraham] Conclusion: this book too, which I’m holding here and will let go of, falls to the earth. Let’s say this—the conclusion is actually about a whole group of objects, but here I’m doing it as an analogy, not scientific induction, scientific analogy. The book too will fall to the earth. That’s the inference. Now I say this: let’s see. If the keys fell to the earth, apparently the book too will fall to the earth. Someone says no: what about the keys, which are green? Fine? This is green. The phone is not green, right? So that property doesn’t hold for it. And the book is not green. So I say maybe the keys fall to the earth because they are green. The book is not green, so it won’t fall to the earth. Remember the example I gave from Koch and Semmelweis? Yes? If you don’t know a theory, you have no idea which facts may be relevant. Is the fact that the key is green, the color of the key, relevant to its falling to the earth? I have no idea, I don’t know. So I check. I say okay, the key is green, maybe that’s why it falls to the earth, so the book doesn’t fall. So I say the phone will prove otherwise, because the phone is black. Someone says no, the phone—sorry—the phone is square, okay? The phone is square, rectangular more precisely.
[Speaker B] It is rectangular, and the book too is rectangular. So excellent, then I have a rectangle here, and therefore it falls to the earth. Someone says no, the problem with the phone is that it is an electronic device and the book isn’t, not an electronic device. And the book is not an electronic device. I put a line over it, meaning it is not an electronic device. Fine? The book is also not green. In the keys, what is similar to the book? That they contain paper, okay? The keychain has, here,
[Rabbi Michael Abraham] a piece of paper, and the book too has papers, so the book also has paper. Sorry for the mess. Okay, now the thing is built like this, look. The keys have paper and the book also has paper, so let’s learn from the keys to the book. Someone says no, the keys have the property that they are green, and the book is not green. Fine, so then let’s learn from the phone. The phone is rectangular and the book is rectangular. He says yes, but the phone is not electronic—the phone is electronic and the book is not electronic. So now you can’t learn from either of them. So I say fine, but since both the keys have mass and the phone has mass and the book has mass, then I can learn from the two of them together that the book will fall to the earth, and in fact I can learn that everything with mass falls to the earth. Now of course this is called the common denominator, right? The separate sides can’t teach me; the common denominator means the property that both sources have, which the target—the book—also has, and therefore that is what determines it. Now here too, exactly as we saw with the a fortiori argument, I can ask myself: there are two possible theories here—or more than two, but at least two possible theories—that explain the data I know. The data I know are these two. Right? This is the square I’m trying to fill in: what will happen to the book. These are my two data points. So now I want a theory that explains those two data points. I have two possibilities. One theory says this: everything that has mass falls to the earth. The keys have mass, the phone has mass, so Theory A is that what determines falling to the earth is having mass. Fine? That’s Theory A. Theory B is that what falls to the earth is either an electronic
[Speaker B] device or something green.
[Rabbi Michael Abraham] I have two theories. Either this theory is correct or this theory is correct. What’s the difference between them? If this theory is correct, then it explains why the keys and the phone fall, because both of them have mass. So the book also has mass, and therefore it too will fall. So the result of that theory is that the book will fall. One—I’m basically marking a one here, it falls. But if this theory is correct, meaning the keys and the phone fall—the keys because they are green and the phone because it is electronic—if that is the theory that actually explains why those two fall, then the conclusion is that the book will not fall, because it is neither electronic nor green. Now it’s very important to understand which of the two theories is correct, because that will determine whether the book falls or not. And again, how do I choose the theory? Occam’s razor, of course. I’ll choose the theory that has a single parameter explaining the facts. Why choose a theory with two parameters explaining the facts? Either this or that. I prefer the theory that says: everything that has mass. Even though this explains the two facts and that also explains the two facts. But since this is the simpler theory, I choose it. Once I’ve chosen it, you see that the conclusion is that the book too will fall to the earth. This is really parallel to what we do in a fortiori reasoning in every respect. I’m basically looking—I’m doing abduction. The green and the electronic and all those properties, that’s basically the abduction. I’m building the theory of why objects fall to the earth. Now I have many possible theories like that; I choose the simplest one, that’s Occam’s razor. And from that simplest theory I infer the conclusions: what will happen with other objects, whether they will fall to the earth or not fall to the earth. Think, for example, yes, about how one actually arrives at the law of gravity this way, right? I look and see that all sorts of objects fall to the earth: a stone falls to the earth, and a pencil falls to the earth, and a phone falls to the earth, and a chair falls to the earth. And what is common to all of these? What you’ll try to produce can be either—say I observed ten objects that fell to the earth. Now I have several possibilities. Either to assume that in each one there is some property, a property that caused it to fall to the earth, or to say that perhaps they all have some shared property that is what is responsible for the falling to the earth. The shared property I find in all of them is that all of them have mass.
[Speaker D] What if there are several shared properties?
[Rabbi Michael Abraham] Then I really can’t know. If I find another property shared by all of them, I have no empirical way to know. I can try to think logically, try to do further measurements, but I have no empirical way to decide. Because I have several theories and all of them are equally simple. If you remember the—this is really the refutation of the a fortiori argument, right? What happened in the refutation of the a fortiori argument? Once we added the refutation, it turned out that there were two theories: one leads to the answer one, one leads to the answer zero, but both are equally simple. What you just said is a refutation of the common denominator. Because if there is a property shared by the two sources that is not found in the target, then you can attribute the fact that they fall to the earth to that property, not to the fact that they have mass. And a property shared by the two sources—that is exactly how a refutation of an inference from the common denominator is structured. And why is it a refutation? Because it offers me a theory that is equally simple. It too is a one-parameter theory, it too explains all the facts, but the result for our purposes, for the book, is the opposite. And since it is equally simple, I have no way of deciding which of the two theories is correct. Therefore the scientific conclusion can be inferred only if I have a uniquely simplest theory. The simplest theory is one theory. It has no competitors. Or at least no competitors that give different results. And let’s say mass is the shared property of all the objects I know that are drawn to the earth. I don’t think there is another relevant shared property among them, and therefore I choose this theory—that mass is what is responsible for falling to the earth. Now, once I’ve chosen that property, I already have a theory, that apparently mass is attracted to another mass, to the earth. So that is the theory, basically. So here I did abduction. From the facts that I saw—things being drawn to the earth—I did abduction, found a theory, and from that I can infer the conclusions. The common denominator is simply scientific generalization. That’s what the common denominator does. Scientific generalization always works with the logic of the common denominator. Let’s say I want to know whether all ravens are black. That’s Hempel’s example that we talked about. So I see one raven—say I go to Australia and I see a raven there and it’s black. Interesting. I say fine, but maybe that’s because it’s in Australia; maybe only Australian ravens are black? So I go to South America and I see that there too the ravens are black. I say ah, so apparently Australia can’t be the relevant parameter—so what is? That they are ravens. So that means that apparently the fact that you are a raven is what causes you to be black. Not that you are in Australia or in South America. That is exactly the common denominator. You take what is common to all the examples you saw, and not the special properties that this one from Australia has and that one from South America has. After all, you could have built a theory saying maybe ravens are black only if they are in Australia or South America, but nowhere else. That too explains all the facts. It is an alternative possible theory. But it is more complicated. Instead of attributing everything to the fact that they are ravens, I say no, no—it’s only ravens either in Australia or in South America. I’m introducing two parameters. That is a more complicated theory. It’s better to attribute everything to a single parameter; that’s simpler. The common denominator and scientific generalization are exactly the same logic. And the way to understand how scientific generalization is done is the analysis I did here. You can do the same analysis I did for a fortiori reasoning for the common denominator too, with a table and everything. It’s a completely general method. And this is basically an explanation of how a scientific theory is built. How scientific generalization is built. How one makes generalizations. It’s interesting, because usually people are used to thinking that generalization is a creative act. There’s no systematic method for making generalizations. You see examples—after all, you can generalize them in lots of ways. There’s something that seems to you… yes, if you’re creative then you come up with some generalization that sounds reasonable to all of us. What I’m claiming is that there is a systematic way to make generalizations. You don’t need to do especially creative gymnastics. There is a systematic way to arrive at the correct generalization. Build all the generalizations, choose the simplest one. If you know how to build all the generalizations systematically, choose the simplest one—that’s the correct generalization. Now, in tables of the sort of a fortiori argument, common denominator, and as much complexity as you like, it can be shown how systematically—systematically to build all the theories and how to find the simplest theory. In this case it’s simply with as few parameters as possible, which is Occam’s original razor, right? Occam’s original razor is that the theory should contain as few entities as possible. Or in other words, as few parameters as possible; those are the theoretical entities. Okay, so again, I’m not going to get into all the mathematics of the matter and all the more complicated examples. I hope these two examples, of common denominator and a fortiori argument, have clarified what the meaning of abduction is, how Occam’s razor guides me in choosing a theory, how through Occam’s razor I do the abduction, and how, after I’ve done the abduction, I can derive the induction from that—that is, the law or the phenomenon. From the theory you can derive the phenomena, or the laws in the halakhic context. Okay? Good, we’ll stop here. Does anyone want to comment or ask? Fine.
[Speaker C] Rabbi, can I ask an unrelated question? What? Can I ask a question unrelated to this?
[Rabbi Michael Abraham] Okay.
[Speaker C] I asked you on the website that you distinguish between practical differences that come out of a certain distinction and practical differences that show a certain distinction. And you answered me something, but I didn’t quite grasp it. I didn’t understand the difference between them. Basically every practical difference comes out of the distinction, so how can a practical difference show the distinction?
[Rabbi Michael Abraham] No, there’s a practical difference—for example, I talked about this: when they tell me that offerings of lesser sanctity are the owner’s property—Rabbi Yosei HaGlili says that offerings of lesser sanctity are the owner’s property. Now those offerings of lesser sanctity have to be sacrificed, and there is a very clear procedure for what has to be done with them. So in what sense is this the owner’s property? So people—I once discussed this with my study partner—he said to me, what do you mean? Because you can betroth a woman with it. That’s not an answer. I’m asking: after all, because it is the owner’s property, you can betroth a woman with it. But I’m asking why it is the owner’s property. Why can you betroth a woman with it? That doesn’t explain anything to me if you bring me a practical difference. Okay? But in our context, you were talking there about the difference between a positive commandment and a prohibition, right? And you said: the difference between a positive commandment and a prohibition—I can tell you what the difference is: for a positive commandment you are not punished, and for a prohibition you are punished; or for a prohibition you need to spend all your money and for a positive commandment you don’t need all that. That says nothing. I’m asking: why is there a difference between a prohibition and a positive commandment that causes all those consequences? But if I tell you that the prohibition points to a negative state and the positive commandment points to a positive state, and I demonstrate that through the question of what happens if I was coerced and did not act—then you understand that this is not just a consequence, it is a consequence that demonstrates the distinction. By the way, it’s very connected to the class.
[Speaker C] Yes, exactly, it’s about today. Yes, that’s why I thought of it.
[Rabbi Michael Abraham] Because basically the consequences are exactly the induction. Meaning, tell me what laws come out. I say yes, but what is the theory because of which those laws come out? I want the abduction, not the induction. Basically I should have answered you on the website that on page four hundred seventy-seven it’s abduction and on page four hundred seventy-six it’s induction.
[Speaker C] Okay, but you bring a consequence that shows that this consequence follows from the distinction, but does it also show it?
[Rabbi Michael Abraham] Yes. With the other consequences, if I told you that for a prohibition you are punished and for a positive commandment you are not punished, yes,
[Speaker C] that’s just a consequence.
[Rabbi Michael Abraham] Yes, what does that mean? It doesn’t explain to me at all the difference between a prohibition and a positive commandment. But if I tell you, look, if you were coerced and did not perform the positive commandment, then you still did not perform it—coercion is not as if one acted—you did not fulfill the commandment. But if you were coerced and did not violate the prohibition, and you violated the prohibition under coercion, then you violated the prohibition, but under coercion. Okay? Or with a parapet: if you received a house without a parapet, then you did not violate the prohibition that you have a house without a parapet, and you also did not fulfill the positive commandment to erect a parapet. Now you understand that this nicely demonstrates the theory, the abduction, the difference between a positive commandment and a prohibition. It’s not just a consequence like punishment. Okay, thank you. Good. Okay. Thank you very much. Sabbath peace. Sabbath peace.