Learning from Experience – Lesson 6
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Table of Contents
- [0:00] The problem of induction and the uncertainty in experience
- [1:11] Three modes of inference in logic
- [3:05] The presentation of a representative sample in induction
- [10:55] Introducing abduction as a scientific tool
- [13:33] The difference between a law of gravitation and a theory
- [17:12] Abduction as the basis of induction and theory
- [25:59] Wittgenstein’s example: a number series and theory
Summary
General Overview
The text presents the problem of induction and the uncertainty built into inference from experience, and argues that the move from observations to laws of nature requires intellectual principles that are not learned from experience itself. It distinguishes between deduction as a necessary inference that adds no information, and analogy and induction as uncertain inferences that do add information and therefore serve science. It adds the concept of abduction as an inference from facts to a theory that explains them, and emphasizes that science is mainly interested in testing theories and not only phenomenological generalizations, especially when correlation does not settle causation. It concludes with an illustration through an analysis of the a fortiori inference, presented as a tabular structure with two separate formulations (rows and columns), and raises a difficulty in the literature of the Sages: a refutation knocks down the entire a fortiori argument without moving to the alternative formulation.
The Problem of Induction and the Need for Non-Observational Principles
The text states that inference from experience does not rest only on observation but requires logical or philosophical principles “that we bring from home” and do not derive from experience. It presents this as an explanation for why accumulating information about the world cannot suffice with describing particular cases alone, because the move from them to a general rule is an act of thought that introduces uncertainty.
Three Modes of Inference: Deduction, Analogy, and Induction
The text divides inference into three types: deduction as inference from the general to the particular, analogy as inference from one particular to another, and induction as inference from the particular to the general. It presents deduction as necessary because the conclusion is contained in the premises and adds no new information, and therefore standard logic and mathematics deal with it. It presents analogy and induction as inferences that are not necessary, because the conclusion adds information beyond the premises and therefore involves speculation, illustrating this with donkeys and with examples about human mortality.
Science versus Logic and Mathematics: Certainty versus Adding Information
The text argues that deduction “never adds information for me” and therefore is not the central tool of science, whereas science is meant to add information about the world and therefore relies on analogy and induction. It connects certainty with the absence of informational novelty, and brings “the hot air balloon joke” as an illustration that the precision and certainty of the mathematician come from the fact that he does not innovate beyond what was already assumed, and therefore “doesn’t help us at all” in expanding empirical knowledge. It adds that science “can never be certain” and connects this to Popper and the lack of absoluteness in scientific generalizations.
Abduction: From Facts to Theory and Theoretical Entities
The text presents abduction as the inference involved in building a theory out of a collection of observations, where the theory yields the general law that induction gives in phenomenological form. It illustrates this with a graph of five measurements that makes it possible to generalize to a straight line, but argues that science does not stop there and instead asks “why” and looks for a theoretical explanation, for example in the distinction between the law of gravitation and a theory about a gravitational field. It states that theories create “theoretical entities” such as gravitational force, gravitational field, and gravitons, which are not observed directly but are proposed in order to explain phenomena. It distinguishes between a “phenomenological theory” that says what will be seen and an “essential” theory that explains the facts by means of concepts and principles that are not seen, and argues that abduction contains induction within it, but induction does not contain abduction.
Uncertainty, Multiple Theories, and Testing Theory through Creative Experiments
The text argues that any set of data can be interpreted through “many theories,” and therefore moving from cases to theory is “a rather delicate business,” and the root of the uncertainty in induction lies in the uncertainty in abduction. It uses Wittgenstein’s example “three, five, seven” to show that a theory of “the odd numbers” yields nine, whereas a theory of “the prime numbers” yields eleven, and no additional number of measurements eliminates the possibility of infinitely many generalizations. It states that science tests theories through tests that distinguish between the predictions of different theories, and not by more measurements of the same kind, illustrating this with Leibniz’s clocks by moving the hands as an experiment that decides between directions of causality or external synchronization.
Correlation versus Causation and the Direction of the Connection
The text presents the statistical distinction between correlation and causation and emphasizes that correlation is a phenomenological law that does not determine what causes what. It brings an example of a professor from the Technion who wrote a letter to the editor of Haaretz and argued that investment in higher education causes GDP to rise on the basis of a correlation between investment and GDP, and presents the opposite possibility, according to which a high GDP makes investment in higher education possible. It illustrates the same mistake with the example “it’s not worth going on a diet… diets make you fat,” and explains that the direction could be that being overweight leads to dieting, and expands this to the example of smoking and cancer, where a correlation between smoking and cancer does not decide whether smoking causes cancer or cancer causes a tendency to smoke. It states that the practical implications depend on the correct causal theory, and therefore there is a need for experiments that generate a distinction between the possible directions.
A Fortiori Inference as an Illustration of the Relation between Facts and Theory
The text proposes to “illustrate” the relation between induction and abduction through an analysis of an a fortiori inference, and brings the a fortiori argument in Bava Kamma 25 concerning categories of damage: tooth and foot are exempt in the public domain and liable in the injured party’s courtyard, horn is liable in the public domain, and the question is what the law is for horn in the injured party’s courtyard. It presents a table with two damagers and two domains, with three known data points and a fourth missing place, and shows two separate formulations of the a fortiori inference: a “columns” formulation that derives a hierarchy between the damagers from the public domain and applies it in the injured party’s courtyard, and a “rows” formulation that derives a hierarchy between the domains from tooth and foot and applies it to horn. It argues that these are two arguments that are “completely different, independent,” and shows that a refutation that adds another domain can collapse the columns formulation without touching the rows formulation, and that the law of “dayyo” when horn in the public domain is half-damages may yield a different result in the two formulations.
A Difficulty from the Talmud: A Refutation Topples the Entire A Fortiori Argument
The text states that in the literature of the Sages, “the moment a refutation arises, the a fortiori argument falls,” and there is no move from the rows formulation to the columns formulation or vice versa, even though according to the analysis these are two independent arguments. It notes that there are “two places” that are exceptions, where they make such a move when the table is not symmetric, but generally, in refutations of a symmetric a fortiori argument, every refutation topples the whole inference. It presents this as a central question of “why,” one that seems to undermine the claim that these are two separate arguments, and concludes by saying that next time he will continue explaining and connect this to abduction versus induction.
Full Transcript
[Rabbi Michael Abraham] Up to now I’ve been dealing with the problem of induction and the difficulty, or uncertainty, that accompanies inference from experience. I tried to show that inference from experience involves the use of rationalist principles, not only observations. And that basically means that when we accumulate information about the world, when we collect information about the world, we can’t make do with observation alone. We need to use principles—call them logical or philosophical—that we bring with us from home. We do not derive them from experience. I spoke about the intuition that underlies the issue, and I just want, in a few more sentences, to place this within a context. Basically, in logic we usually divide things into three modes of inference: analogy, induction, and deduction. Deduction is the logical argument, the classic kind of logical argument. Right? All human beings are mortal, Socrates is a human being, conclusion: Socrates is mortal. That is an inference from the general to the particular. Right? I assume that all human beings are mortal, and the individual, Socrates, is one of those human beings, so he too must be mortal. An inference from the general to the particular is a necessary inference. That’s what standard logic deals with. That’s what it is—an inference from the general to the particular. Analogy is an inference from one particular to another particular. Say, if this donkey is mortal, then that donkey is probably mortal too. That is obviously an inference that is not necessary or certain, because the fact that I saw something about one object doesn’t mean the same thing will also apply to another object, or that another object will behave exactly the same way. I assume that if they’re similar then probably their behavior will be similar too, but maybe that’s true and maybe it isn’t. It’s always something uncertain. Unlike deduction, which is certain, in analogy there are always some doubts that accompany it. And induction—right, so we said deduction is from the general to the particular, analogy is from particular to particular, induction is from particular to general. Okay? So that basically means these are all the possibilities there are. Apparently there’s no other possibility. From particulars to a general, or from a particular to a general—that’s basically what we call generalizations. We encounter one particular or a few particulars, and we assume that what we saw regarding those particulars is true regarding a broader whole of particulars, an entire group of which these are only examples. And here, of course, we have to talk about a representative sample. That is, the few examples I saw are supposed to be a representative sample of that general category, from which I can infer the properties of the whole. For example, if I saw that Socrates is mortal, and I saw that Jacob is mortal, and that Mahmoud is mortal, then I assume that all human beings are mortal, my assumption being that the examples I relied on were not special cases. They were a representative sample in the sense that what happens with them probably happens in the group as a whole. That’s what is called induction, or generalization, and this too is of course an argument that is not a necessary argument, because it could be that these examples are in fact exceptional examples, and what happened in them does not happen in the whole set. Therefore analogy and induction are inferences that are not certain, not necessary. The conclusion does not necessarily follow from the premises. In deduction the conclusion necessarily follows from the premises. I think I once talked about this, that the reason why in deduction the conclusion necessarily follows from the premises is because… right, that’s the hot air balloon joke. It’s basically because the conclusion doesn’t tell me anything beyond what is already in the premises. Because if I assume that all human beings are mortal, if I cash that out in detail, basically I’m assuming that Jacob is mortal, Socrates is mortal, Mahmoud is mortal. In short, I say all human beings are mortal, but that’s really just a collection of a great many assumptions. So of course the conclusion about Socrates, that Socrates is mortal, is simply one of the assumptions hidden inside the statement that all human beings are mortal. So in effect I already assumed that conclusion. Right? So that’s the emptiness of the analytic. So therefore the necessity in deduction—the conclusion necessarily follows from the premises because it is actually already inside the premise, embedded within it. And therefore, therefore, it is necessary. By contrast, in analogy and induction, the conclusions are not embedded in the premises. They contain additional information beyond what was in the premises, and therefore this has to be something a bit speculative. That is, it can’t be certain; it’s not definitely true. If I say that this donkey is mortal, therefore that donkey is also mortal, then the premise is that this donkey is mortal. The conclusion is that that donkey is mortal. That conclusion is a fact that was not embedded in my premises. My premises dealt with this donkey; the conclusion deals with that donkey. So maybe it’s not true. Since the conclusion contains information beyond what was in the premises, this inference is not certain. And in induction of course it’s the same. If I say that this donkey is mortal and that donkey is mortal, then probably all donkeys are mortal. So again, the statement that is the conclusion contains more information than I had in the premises about this donkey and that donkey. And therefore there is a measure of speculation here. Therefore, I am adding information beyond what I had in the premises, and so there is something uncertain here. This is another way of illustrating our need to resort to things beyond experience. That is, if the laws of science merely described what our experience tells us, we wouldn’t need generalizations. But the laws of science contain more general information, and what I saw in experiments were specific cases, particular cases. And since that is so, the move from the particular cases to the general law is basically a kind of generalization. And that generalization is of course not an act of observation but an act of thought. And therefore the claim is that when I arrive at the laws of nature, when I make my scientific generalizations, the move from the particular to the general law includes, in addition to the observations that gave me the particulars, certain intellectual moves—logical, rationalist, if you like. And that is what introduces the uncertainty. That is what introduces the fact that you are not sure, cannot be certain, that it’s true. Here I basically said that I divided inferences into three types. The necessary inferences are deduction, and logic and mathematics deal with them. They deal with deductive inference. Analogy and induction are the tools of science. Science deals with accumulating information about the world. Deduction never adds information for me. Right? Deduction simply extracts information from the premises I had at the start. I knew at the start that all human beings are mortal; from that I extract the conclusion that Socrates is mortal. In other words, the conclusion is a fact that was really embedded in the premises; there is no new information in it beyond what was in the premises. And that’s always the case in logic and mathematics. The conclusion does not contain more information than the premises, although it’s not always easy to extract the conclusion from the premises. But if you dig hard enough, in the end you see that it’s there in the content. That’s the meaning of the certainty there is in logic and mathematics. Science deals with accumulating information about the world—yes, science comes from the word knowledge. It deals with accumulating information about the world. Accumulating information about the world, by definition, is something that is supposed to add information beyond what I had at the start. And since that is so, it cannot use the tools of logic and mathematics—or only the tools of logic and mathematics. There has to be something more here that enables me to add information and not merely to extract existing information that is already in my hands. And therefore analogy and induction, which are the two additional kinds of inference, are the tools of science. They are non-necessary tools, and they add information. Right, I said that this is the hot air balloon joke, which says that the mathematician is perfectly precise or perfectly certain, and therefore is no help to us at all. Those are exactly two sides of the same coin. That is, precisely because he is completely certain, therefore he is no help to us at all, because his certainty stems from the fact that he adds no information beyond what I already knew. That is why it is certain. So therefore anything certain, by definition, adds nothing to me, tells me nothing new. That is why it is certain. It is certain because it contains no additional information beyond the information already in my possession, so from my point of view it is safe. The moment I add information beyond the information I already have, then it is no longer safe. At that point I may be mistaken in that addition of information, in the generalization, in the move I am making beyond. And therefore science can never be certain, can never be secure, can never be proven—we talked about Popper. And therefore the tools that science uses, the basic tools science uses—sorry—are analogy and induction, not deduction. Logic and mathematics deal with deduction; science deals with analogy and induction. So this is a kind of summary, from another angle, of what we’ve seen so far. But now I want to add another point, a very important point, which in fact we already encountered without putting it explicitly on the table. And here I want to talk about what is usually called abduction. What is abduction? Apparently there is analogy, induction, and deduction. Supposedly only three ways. What else could there be? From the general to the particular, from the particular to the general, or from one particular to another. General to general and particular to particular are the same thing—that is, at the same level; that’s analogy. Supposedly there can’t be anything beyond that. So what is abduction? Abduction is this: when we make a scientific inference, when we build a scientific theory, we basically take our observations and try not only to find—I talked in previous lectures about the graph. Right, remember the graph I showed here. Here it is, the graph that has accompanied us in the last few lectures. Basically this graph tells me—I’m measuring the relation between acceleration and acceleration—right, this is the x and force is the F, and I measure the relation between them, and these are the circles I measured: one, two, three, four, and five. And now I ask myself what the law is, I ask what the general law is. So I explained that the general law is the straight line that stitches together these five points, and I explained why other lines that stitch them together are less likely—that is, why the straight line, which is simpler, is also the more correct one: Occam’s razor, actualism, informativeness, all the things we’ve talked about until now. But this straight line I’m talking about here is really only a simple generalization of the particular cases. It does not constitute a theory in the full sense, in the scientific sense of the word. It merely says: the same thing I saw in these five observations will be true in every other observation, and therefore I say that this straight line is continuous. That is, anywhere you make a measurement, you will get a force corresponding exactly to the acceleration that lies on this line. In other words, what you encountered in these five points is not an exceptional case, but a representative sample; it is true of all the points. At every acceleration and force you measure, the relation between them will always be a direct relation. The force equals mass times acceleration. That is a simple generalization. Science may usually begin with simple generalizations, but it does not end there. Science basically deals with building a theory. So I ask myself what lies behind—say I’ve drawn this straight line, and now I ask myself: let’s try to find an explanation for why the line is straight. Why is there really a direct proportion between force and acceleration? Then I need to begin looking for theories that explain this. If we talk about the law of gravitation it’s easier than the second law of Newton, because Newton’s second law really is not a theory; it’s a simple inductive generalization. But the theory of gravitation is already a theory. What do I mean? I can say that there is some relation between the motion of masses—that is, when there are two masses located in certain places, I can say what acceleration each of those masses will develop, or what force each such mass exerts on the other mass. But I can do this on two levels. One level is to say that every two masses attract one another and cause such-and-such an acceleration, the product of the masses divided by the square of the distance. Okay, that is basically the force the masses exert, and from here one can understand what the acceleration of each of these masses will be. But now I can go one step further. So far that is only a description. I saw certain cases in which two masses attracted one another with the force I described before, so I say that probably every two masses behave in this way; so I generalized it—that’s the straight line, right?—I generalized it to all situations in which there are two masses in space. But now I ask myself, okay, and why does this happen? Okay, they attract each other—why does this happen? And then comes a theory, the theory of gravitation. What does that mean? The theory says that every mass actually exerts a force, and that force attracts the other mass, and therefore produces the acceleration we see. So here there is already an explanation of the acceleration. It is not just a statement that there will be an acceleration of such-and-such magnitude. That would be only a simple generalization. The theory comes to explain why the law is actually this straight line, or the law of gravitation, or whatever it may be. That is, I think I spoke about this—maybe even in this series, I no longer remember—that there is a difference between speaking about the law of gravitation and speaking about the force of gravitation. The law of gravitation means that when there are two masses standing in space at certain positions, they will develop such-and-such accelerations. That is the law of gravitation—or a certain force acts on them. But I can claim that what causes the movement of these masses is actually a gravitational field created by one mass, which exerts a force on the other mass, and therefore it moves. You understand—here already there is a theory. That is, I am proposing some theory that explains the facts. A generalization deals only with the facts. A theory explains the facts. Now this is a different kind of generalization. That is, I can take the facts and see, say, that all bodies with mass that I release into the air fall toward the earth. I can also understand that they fall with a certain acceleration, measure that acceleration, and then say okay, I’m making a generalization: all bodies with mass will fall toward the earth with that acceleration, even though I saw only ten of them, say for the sake of discussion. My claim is: no, all bodies with mass will fall toward the earth. That is a generalization, but it is a generalization at the simplest level; that is induction. Induction means: from the particular examples I saw, I claim that this is true for the whole set; the sample I saw is representative, what happens in it happens in the whole set. Abduction says: take these specific cases and build a theory from them. The theory will yield the general law, the straight line, or the law, the facts I describe through induction. Abduction stands at the base of induction. Abduction is the theory that explains why the generalization I arrived at should look this way and not another way.
[Speaker B] So Rabbi, can you hear? Rabbi, how does actualism fit with abduction? Does it reject it?
[Rabbi Michael Abraham] Yes, it doesn’t reject it. It just claims that this is a statement about us, not about the world. In the world itself there really is no gravitational force; it is simply convenient for me to speak in language as if there were a gravitational force here.
[Speaker B] No, but if there were a gravitational force, you could predict something, you could make predictions.
[Rabbi Michael Abraham] It does not reject predictions. At the moment, one of my claims against actualism is precisely that we are investing billions of dollars in order to try to measure the existence of gravitons. Gravitons are the particles of the gravitational field, the ones that carry the gravitational force. Okay? Now if two masses attract each other without the theory, only the generalization, then there are no gravitons, no gravitational field, no anything—there are simply two masses standing opposite one another that will move in such-and-such a way. There is no theory. If I say there is a theory, that means that between those two masses there is a force or a gravitational field, and this field, just as the electromagnetic field is carried by photons, this field is carried by gravitons. Okay? Then I say that these two bodies are actually exchanging gravitons, and these gravitons that run rapidly from one body to the other are what cause the second body to move. But that is a result of the theory. The theory predicts the existence of gravitons. Induction does not need this. Induction says that that body will start moving; I have no explanations, I just say: if this happens in the bodies I saw, then it happens in all bodies, without explaining why. When the theory explains why this happens, it often produces what are called theoretical entities. Theoretical entities mean things like the force of gravitation, or gravitons, or an entire world of entities that none of us has seen, and maybe never will see, but we assume they exist there because that is what explains the phenomena I observe. So in science there is what is called a phenomenological theory. A phenomenological theory tells me what I will see. I will see that this body begins to move with such-and-such acceleration. The substantive theory—not the phenomenological one—is a theory that explains the facts I see. But it itself uses entities and principles that I do not see. I create an entire theory, which is a collection of concepts with relations among them, and principles, and how they appear in the world, and all these things, and I claim that this is basically it. Basically this set of entities exists there in the background, and it is what produces the phenomena that I arrived at through induction. So you understand that this is a deeper step than induction. Induction is simply to describe what I will see in all situations; these are just facts that I observe directly. I say this body will move like this, that body in such a situation will move like that. That’s all I’m saying. That is induction. From particular cases I make a generalization and say this is a general law, it is true for all cases. Abduction says: I take the particular cases I observed, and I build from them a theory that explains these cases. From that theory I derive the general law—what will happen in other cases. The result of induction, the phenomenological law, is a derivative of the theory. I can arrive at it directly without paying attention to the theory, simply by assuming that what happens in particular cases happens in all cases. And I can explain why it happens in the particular cases, and then that will also show me that it happens in all cases. Okay, that is what is called abduction. Induction goes from facts, from a small set of facts, to a large set of facts, right? Induction is from the particulars to the general. From a few particular cases I observed to all cases. That is called induction. Induction moves from specific cases to many cases, to a group of cases of which these are only examples. Abduction takes me from a set of cases to a theory—not to a larger set of cases, but from a set of cases to something altogether different, to a theory. Of course, once I have a theory, I can also know what will happen in the other cases and derive from that the general law as well, the result of induction. In other words, abduction includes induction within it, but induction does not include abduction within it. That is, I can say what will happen in all cases without committing myself to what theory explains it. I simply say: I make a generalization—from these cases it’s probably true for all cases. But if I state the theory, then from the theory there will also come the law of what happens in all cases. In other words, from abduction one can also derive induction, but not the other way around. The theory dictates the outcome; the outcome does not dictate a theory. We spoke about the fact that there can be several theories that explain the same outcome. Okay? And therefore philosophers speak of another type of inference, which is not analogy, not induction, and not deduction, and that is what is called abduction. Abduction is from cases to theory. Now, we often use this, but one has to be very careful in using it. And therefore it is easier for me to explain this in terms of abduction than in terms of induction. In induction too we saw that the generalization is not secure; one has to be careful with generalizations because they can be made in many forms. Remember? There is the straight line and there is the dotted line; there are many ways to generalize from the five examples I observed. So the fact that I decide on the straight line is nice, but it’s not certain that I’m right. I have to check myself, do another experiment, see exactly what happens. So there can be a mistake there. Okay? Why can there be a mistake? Because the theory on the basis of which I made the generalization may not be the correct theory. In the final analysis, behind all these generalizations sit theories. And those theories are either correct or not correct; one has to be very careful. You have certain examples, you build a theory on the basis of those examples, and from that theory you derive what will happen in other cases. But who says your theory is correct? In other words, this set of examples can be interpreted by many theories. You chose one, perhaps the simplest one, and therefore you derive from it what will happen next—but it is not certain that this theory is the correct one. In other words, moving from cases to theory is a rather delicate business and needs to be done very carefully. Therefore, the examples I’m talking about—right, for example what I said in connection with the graph—let’s go back for a moment again to the graph, the Potoczky graph. So I say: here in this graph I have five results, one two three four five. I can generalize them through the straight line. That is one kind of induction. But this too is a kind of induction—the dotted line also passes through these five examples, right? And of course there are infinitely many more such lines. So this expresses the fact that induction is not something secure. It’s something you need to do carefully, and the question whether the simplest is the most correct or not—we spoke about that before. But in principle, even if it is the most correct, clearly it is not certain that it is the correct one. That is, you may be mistaken. Okay? What is the difference between all these generalizations? Behind each such generalization sits a different theory. In effect, implicitly, when I assume the straight line I am assuming a certain theory; when I assume this line, I am simply assuming a different theory. And that theory will yield the dotted line, while the other theory will yield the straight line. Every theory will give me a different type of generalization, and therefore the uncertainty I have in the induction I perform actually stems from the uncertainty in abduction, in the question of which theory governs this general law. Okay, is that coming out clearly? I hope I’m being clear. Yes or no?
[Speaker B] Clear, clear.
[Rabbi Michael Abraham] Yes, fine, okay. Understood. So the important point is that when we make a generalization from particular cases to other cases, basically many times in some implicit way, and sometimes explicitly—in science they usually try to arrive at an explicit formulation of the theory—we are assuming a theory. There is a theory standing behind our conclusion about what the general law is. If you remember the examples I gave about following a rule in Wittgenstein. Right? I said: three, five, seven—what is the next number? Right, that was one of the examples I gave. So there were suggestions that it would be nine, and there were suggestions that it would be eleven. What is the difference? Three, five, seven—someone who said the next one is nine is assuming a theory that this series is basically the odd numbers. He is actually proposing an explanation for the series three, five, seven. What is the explanation? Simply that it is the series of odd numbers. According to that theory, he can guess that the next number will be nine, the next odd number: three, five, seven, nine. So you see that once you determine the theory, from the theory you extract your generalization, your general law. What will the next number be? Nine, and after it eleven, and after it thirteen—you can derive all the numbers. But basically, maybe without even noticing it, or without formulating it to yourself, you assumed a certain kind of theory, a theoretical explanation standing at the basis of the three examples you received: three, five, seven.
[Speaker B] But here you’ve already generalized several things, you generalized three, five, seven. Meaning, in a situation where you are going to generalize several things and you don’t know which of them to generalize, then aren’t you already coming with a theory?
[Rabbi Michael Abraham] No, no, I’m not coming with a theory. I’m saying: I have three, five, seven—that’s a given. I ask you what the next number is. Okay, so here—nine. In fact, even if he didn’t formulate it to himself, he actually built a theory. The theory is that three, five, seven is simply the series of odd numbers, and from that theory he derives that the next number will be nine. And now someone else tells you no, the next number is eleven, not nine. What does that mean? Why did he arrive at eleven? Because he built a different theory. He says three, five, seven is not the odd numbers but the prime numbers, and the prime numbers are three, five, seven; nine is not prime, the next prime is eleven.
[Speaker B] Okay, and that’s true when you already have several data points. Right, you always build a generalization out of data.
[Rabbi Michael Abraham] Okay, but when you come to collect several data points…
[Speaker B] You always build a generalization out of data. Okay, and when you come to collect the data…
[Rabbi Michael Abraham] Like with a graph: in a graph I have five data points that I measured, and now I want to know the general law. So there are several ways to derive a general law from those five data points. Okay? But that will depend on the question of what my theory is. Every theory will generate a different generalization. So in the example of 3, 5, 7, that’s like the equivalent of the data points I measured. I measured 3, 5, 7. Now they ask me what the next number is. It depends on the theory. If my theory is that these are the odd numbers, then the next number will be 9. If my theory is that these are prime numbers, then the next number will be 11. Okay? That’s an example of the issue behind Wittgenstein’s rule-following problem. What lies behind the problem is that you can’t know, from the three numbers 3, 5, and 7, what the correct theory is. Maybe it’s the odd numbers, maybe it’s the primes. You have no way of knowing, because both theories fit the three examples you measured, the three cases you encountered. So you have no way to know; you can build this theory, you can build another theory.
Now in science too, that’s how it works. For every collection of facts we encounter, you can build lots of theories that explain those facts, and every theory will give me a different general law or a different induction. Like in the graph: the straight line is the result of one theory, that dashed line is the result of another theory. It’s exactly the same thing as Wittgenstein on 3, 5, 7. You take certain data and try to build from them a general law. When you build a general law, you’re always assuming some kind of theory. Sometimes it’s unconscious, sometimes you don’t formulate it explicitly to yourself, but basically you’re always assuming some kind of theory. And the question whether that theory is correct, or unique, or whether there could be other theories—that’s always a question you need to ask yourself. Why are you choosing this theory rather than another, and why?
That’s why, in science, in order to test whether we chose the right theory, we put it to an additional test. Let’s take the next measurement and see whether it gives me 9 or 11. If it gives me 9, that means the series is probably the odd numbers. If it gives me 11, that means the series is the primes. If it gives me minus 7 and a half, then that means this series is really something else altogether. That’s why I do experiments. But as I said, those experiments can’t really help in any essential way, because after I’ve measured one more result, now I have four results: 3, 5, 7, and minus 7 and a half. Those four results too I can generalize in infinitely many ways. So no matter how many measurements I make, it won’t help. I’ll always remain with infinitely many possibilities. And then of course I always choose the simplest one, and I said: the simple one is the correct one—actualism, informativeness, and everything we discussed earlier.
But here I want to add something else that we haven’t spoken about up to now, and that is that behind every such option there stands some theoretical assumption. What is the theory that stands behind this whole story? And this is a very important point, because science very often makes generalizations, but if there is no theory in the background of the generalizations, then those generalizations are problematic generalizations. Right—the classic example is what statisticians always talk about: the difference between correlation and causation. You can find a correlation between two variables, but that doesn’t necessarily mean that one is the cause of the other. Correlation can be explained in many ways, not necessarily causally. Therefore when you find a correlation, all you’ve really found is some phenomenological theory. You’re describing to me what happens. But the theory has to explain what happens.
And in the theory it could be that A is the cause of B, or it could be that B is the cause—for example, Leibniz’s parable of the two clocks. Leibniz says: look at two clocks. These two clocks always show exactly the same time. Now, that’s a fact. And so if you ask me what these clocks will show in two days, then if this clock shows five o’clock, the second clock will also show five o’clock, because I make a generalization—I say they always show the same time. But you understand that here there’s no theory. It’s a phenomenological law; it only says what will happen. Now I ask: wait a second, but why do they really show the same time? Give me an explanation for why they show the same time.
Now there could be many explanations here. For example, clock A is the causal factor behind what clock B shows. Clock B simply reads clock A and copies it. That’s one possibility. A second possibility is the opposite: clock B is the causal factor behind what clock A does. Right? Also a theoretical option. A third possibility: neither of them is the cause of the other; there is a clockmaker who adjusted both of them and makes sure they stay synchronized. That’s a third explanation. A fourth explanation is that the fit between them is accidental, and this really isn’t a representative sample, and it will break tomorrow morning. How do I test?
So these are different theories, all of which can explain the same facts, but they may have different implications. For example, if I say that clock A is the cause of clock B, then I need to see what happens when I change clock A. I move the hands. Presumably that should also move the hands of clock B, right? The same change should happen in clock B, because clock A determines what clock B will show. But if clock B is the cause of clock A, then when I move the hands of clock A, nothing will happen to the hands of clock B. Because they are the cause of A; they are not A’s effect. So that’s one way, for example, to test which of those two theories is correct: simply move the hands of one of the clocks and see what happens in the other clock. Okay? That’s a way science tries to test what theory lies behind the correlation.
Correlation is a fact—let’s call it an empirical fact. The two clocks always show the same time. But behind this empirical generalization there can sit lots of theories: clock A causes clock B, clock B causes clock A, there is some third thing synchronizing them both, it’s just an accidental fit, maybe it’s a fit that happens today but tomorrow there will be a gap of an hour and the day after a gap of two hours, and so on. There can be many theories that explain this fit, this correlation. Science is interested not in the correlation but in the theory. What is the theory? What is really happening between these two clocks?
How does science proceed with respect to theory? It simply tries to do experiments that test the theory. The experimenter will move the hands of clock A and see what happens to the hands of clock B, and from that he’ll be able to rule out certain theories and leave others standing. So you understand that this works not on the level of the generalization of what will happen. As for what will happen—that they always go together—everyone agrees on that. And still, behind that there can be lots of theories; each theory will yield something different, and there may be experiments that look different according to theory A, B, or C. In fact, science deals with theories, not facts. For science, facts are only a means to clarify what the correct theory is.
And I’m dealing with theory, so I need to design an intelligent experiment that will show me which theory is correct—not what the facts are. Every experiment I measure in the future will show me what the facts are. But what the facts are is not what interests us; if the two clocks show the same thing, then they’ll keep showing the same thing. What interests the scientist is why they show the same thing. What is the theory behind it. And for that, it’s not enough to measure them tomorrow and the day after and the day after that. What good will that do me? It won’t help me at all; I’ll just keep seeing that they show the same time. If I want to test the theory, I need to do a creative experiment. An experiment that will be a touchstone: according to theory A there should be one result, according to theory B there should be another result. Then the experiment can help me decide which theory is the correct one.
And that is basically what a scientist does. A scientist deals in abductions, not inductions. The scientist wants to understand, from the facts he observed, what the theory is. What is the law of nature, what is the correct theory—not what the general law is. The general law is the result of the theory. Okay? So here the question really arises: how exactly do we derive a theory from data? And I’ll already jump ahead a little, because we’ll talk about various fallacies. Many times there are fallacies in the way we infer conclusions from data. We have certain data, we make some generalization or draw a conclusion—mistake. Usually the mistake stems from the fact that we assumed a certain theory that explains the data, but there could be another theory that explains the data and would yield different results in different contexts. And somehow it seems obvious to us that if the data are these, then this is probably the theory.
Very often we don’t even give ourselves any real accounting. And then obviously that’s the theory, and therefore we also infer the general conclusion. But wait, wait, wait. You assumed, without noticing, some kind of theory. But maybe the theory is a mistaken one? Maybe the real theory is a different theory? And then your generalization is an incorrect generalization. Okay? There are examples of this. All the fallacies we deal with in statistical inference—like the law of small numbers, what I wrote about in certain columns, and we’ll get to that later—they are all really this issue: reliance on an incorrect theory that seems to explain the facts, but there may be other theories that also explain the facts, and one needs to be careful before deciding that this particular theory is the correct one. These are errors in choosing the correct theory out of the data set I observed or was given. Okay? The move from data to theory is a problematic move. And very often we make it rashly.
And if I want to talk about the focal point of error in learning from experience—and that is the topic of this series—when we make mistakes in learning from experience, it is simply because we adopt a theory too hastily. We assume: look, these data fit the theory perfectly, so obviously this is the right theory, and therefore it’s clear to me what the general law is, and everything is wonderful. Whoa—hold on. There’s also another theory that could explain it. Right? 3, 5, 7. Ask people what the next number is, and they’ll all tell you with a little smile: obviously 9. Why is it obvious? Because you’re assuming that the theory is that this is the series of odd numbers. Then someone says: wait, but maybe this is the series of prime numbers? Boom—and the person stops. It never even occurred to him that after 3, 5, 7 the next could be 11 rather than 9.
Very often our minds make generalizations on the basis of a theory that seems simple or natural to us. We don’t even notice that we relied on some kind of theory, but it seems obvious to us that this is what’s right, so we determine the general law, immediately draw conclusions very hastily, and everything is wonderful. Right, all the spurious correlations of… yes, the example I once gave in one of my books—I think it was in Shtei Agalot, I no longer remember, or in Ve’et Asher Yeshno. There was a professor from the Technion who wrote a letter to the editor of Haaretz, a letter to the editor. He said that the State of Israel must invest more in higher education. Why? Because investment in higher education causes GDP to rise. And his proof was that in countries around the world that invest more in their higher education, their GDP is higher. QED. Where’s the mistake? What do you say?
[Speaker B] And what is GDP anyway?
[Rabbi Michael Abraham] Gross domestic product. How much that country produces. The output, employment, economic productivity is greater. The mistake is in his theory, because he assumes that investment in higher education boosts GDP or raises GDP. But it could also be the other way around. It could be that GDP drives investment in higher education. Countries that are rich enough can invest a lot of money in higher education. They have enough money. Countries that are not rich won’t invest in higher education because they don’t have enough money. That too explains all the correlations he presented there. It explains in exactly the same way the link or the correlation between the wealth of countries, in terms of the size of their GDP, and the extent of their investment in higher education. It could be that because they are rich they invest in higher education, or because they invest in higher education they are rich. Both theories are possible. From the data he brought, you cannot know which of the theories is correct.
Now why does that matter? Because if the correct theory is mine and not his, then investing in higher education won’t help. According to the theory I’m proposing, higher education doesn’t increase GDP. On the contrary: increasing GDP will lead to increased investment in higher education, but not the reverse. Or in other words, yes, as people say, as our sages said: it’s not worth going on a diet, because people who go on diets are fat. Dieting makes you fat. A fact: everyone who goes on a diet is fat. Thin people don’t go on diets. Right? So you see that dieting makes you fat. What’s the mistake? The mistake is that dieting doesn’t cause obesity; obesity causes dieting. The direction of the influence, or the direction of the correlation, is reversed.
Now when I look at the facts I see: everyone who diets is fat; everyone who doesn’t diet is not fat, let’s say for the sake of discussion, okay? The correlation exists. And we immediately jump to the conclusion: ah, that means dieting makes you fat. No, not true. It could be that fat people go on diets, or that rich countries invest in higher education. The direction of the correlation is very important. You see that smoking causes cancer. How do you know smoking causes cancer? You take a group of smokers and a group of non-smokers, and you ask how many of them have cancer and how many of those have cancer. Suppose that in the smokers’ group there are significantly more cancer patients. But that could also stem from the fact that cancer patients have a tendency to smoke—not that smoking causes cancer. That too could explain the matter.
To determine which of the two theories is correct, it is not enough to take two groups and see whether in the cancer group there are more smokers, or in the smokers’ group there are more cancer patients than in the non-smokers’ group. Because it could be that the correlation is reversed: those who have cancer have a greater tendency to smoke. For some reason the illness causes them to feel a need to smoke. Okay? Well, if that’s true, it’s very important to know it, because it means that if I smoke, it doesn’t cause me to get cancer. There’s no connection. According to the suggestion I’m making, smoking doesn’t increase the chance of getting cancer. Rather, if I have cancer, I have a tendency to smoke. It’s just like it isn’t correct to stop dieting in order to lose weight. Right? Exactly the same way. Why? Because it’s not true that dieting causes obesity; rather obesity causes dieting, not dieting causes obesity. Okay?
So the direction of the correlation is very important, because it also has practical implications. It’s not only a theoretical question. But notice that the same induction, the same general law that says everyone who diets is fat and everyone who doesn’t diet is not fat, is correct by everyone’s lights. That’s a correlation we measured. Those are facts. But from those facts one can infer many different kinds of conclusions. Each kind of conclusion stems from a different theory. And what matters is what the correct theory is, not what the correlation is. The correlation is the facts we observe. The correlation may be correct. But still, the conclusions we draw depend on the theory. And what matters for science is to check what the correct theory is.
The correlations we discover in phenomenological research are only an initial indication. Once we discover that there is a link between cigarettes and cancer, that is an interesting link. But we still haven’t finished the job, because all it says is that now it could be either that cigarettes cause cancer or that cancer causes smoking. And it’s very important to know which of those two is true, in order to tell people whether it’s worth quitting smoking or not. Because if cancer causes smoking, then there is no reason to stop smoking. Whoever wants to smoke can smoke. It doesn’t increase… I can take, for example, a group of people with similar characteristics and cause people to smoke and see whether the percentage of cancer patients rises. For the sake of discussion, of course, that’s an experiment you’re not allowed to do, but on the level of principle, yes? In other words, I cause people to start smoking and then I check whether it affects anything, whether the cancer rate rises. If it does, then I deliberately caused them to smoke; it’s not that because of the cancer they wanted to smoke. And that increased the rate of cancer patients. That’s the kind of experiment that can reveal to me that smoking causes cancer and not cancer causes smoking. Exactly like with Leibniz’s clocks. If I want to know whether clock A causes B or B causes A, how do I test it? I’ll change the time on clock A and see whether clock B changes. If it doesn’t change, that’s a sign that B causes A and not A causes B. It’s exactly the same thing.
But all of these—notice—are experiments we do in order to test abductions, not inductions. The inductions we already know. The induction is that these clocks will always show the same time. I made a generalization. Up to now I saw that they show the same time, so I generalize: these clocks always show the same time. That’s the induction. Now let’s assume that induction is correct. Even then, on the level of the abduction, the theory, there are several possible theories here. In fact infinitely many. But there are at least several natural possible theories. And in order to decide between them, it is not enough for me to check more and more moments in which those clocks are synchronized or not synchronized. I need to do more creative maneuvers in order to try to test which theory is correct: deliberately move this clock, stop smoking cigarettes or start smoking cigarettes and see what that does to cancer, all kinds of things of that sort—in effect to run regressions in order to see whether there is a relationship between variables, whatever variables we are measuring, and not just check correlations. I need to do more sophisticated research maneuvers in order to sift and test my theories, to see what the correct theory is.
Now I want to give a demonstration. I want to demonstrate this transition, the relation between induction and abduction, or between theory and facts. And I’ll do it through something I already discussed here at length in the past, quite a few years ago. I want to analyze the inference of kal va-chomer, an a fortiori inference. Okay, let’s try to think about a kal va-chomer. So I’ll say: how is a kal va-chomer actually structured? Let’s take the kal va-chomer in the Talmud in tractate Bava Kamma 25 regarding primary categories of damages. We have three data points that we know from the Torah: tooth and foot. Right? There are four primary categories of damages—well, not four, actually there are more. There are several primary categories of damages, among them the primary categories related to an ox that causes damage: tooth, foot, and horn. Okay? Tooth is when it eats or derives benefit from the damage; foot is when it walks and damages in the course of walking; and horn is when it damages intentionally, with intent to damage. It gores with its horns, or pushes—in other words, its purpose is to damage. It’s not doing it just while walking, or in order to enjoy itself, or something like that, but in order to damage. That is called horn.
Now these three categories have several data points that I can learn from the Torah. I know that in the injured party’s domain, tooth and foot are liable, and in the public domain, tooth and foot are exempt. Horn in the public domain is liable. Now the question is: what is the law of horn in the injured party’s domain? That is not written. So I don’t know what its law is. So what do we do? We make a kal va-chomer. I’m ignoring right now the fact that horn pays only half damages; that’s not important at the moment because it adds another complication here. Let’s say for the sake of discussion that it is liable; I don’t care right now how much liable. So I ask myself: what is the law of horn in the injured party’s domain?
So let’s look for a moment—I’m drawing it as a table and putting these data into the table. Let’s say, this doesn’t really describe tooth and foot and horn, but this is just the table I happen to have at hand. So let’s say I have here tooth and foot, okay? And horn. Okay? This is the injured party’s domain, sorry, this is the injured party’s domain. And this is the public domain. So tooth and foot are exempt in the public domain—not liable—and liable in the injured party’s domain. Horn is liable in the public domain, and I ask myself whether it is liable in the injured party’s domain. This is the classic table of a kal va-chomer. Every kal va-chomer in the Talmud is built like this. Okay? So horn, tooth and foot—those are words. In practice, I have here two damagers, M and H. I have here two domains, A and N. And now the data are these: zero here, one here, and one here. I ask myself what the law will be here. So they tell me: obvious, kal va-chomer. Obviously the law here will be liable. Right? Horn will also be liable in the injured party’s domain.
Why? How is it structured? You can formulate it in two ways, and both are built on the same logic. The first form—let’s call it the columns form. What does that mean? Let’s look for a moment at the right-hand column. In the right-hand column we are talking about the public domain. Tooth and foot are exempt in the public domain; horn is liable in the public domain. From this column I can understand that horn is something more severe than tooth and foot, right? Because since tooth and foot are exempt and horn is liable, horn is more severe than tooth and foot. You see that from this column. Now let’s apply that to this column. If horn is more severe than tooth and foot, and tooth and foot are liable, then horn, which is more severe, certainly must be liable. Therefore here there will be a one. Okay?
Now let’s do a different kal va-chomer. Let’s look at this row, the top row. The row of tooth and foot, where in the public domain it is exempt and in the injured party’s domain it is liable. Looking at this row, I claim that the injured party’s domain is more severe than the public domain, right? That’s what I see in this row. Or, it is easier to impose liability in the injured party’s domain than in the public domain. Since tooth and foot—which are not enough to incur liability in the public domain—are liable in the injured party’s domain, it is easier to impose liability in the injured party’s domain. So if horn is liable in the public domain, then all the more so it is liable in the injured party’s domain. Because in the injured party’s domain it is easier to impose liability than in the public domain. Okay?
So what this means is that the kal va-chomer is based on three data points, and I want to infer the fourth data point from them. What do I do? I use two of the data points to generate a certain hierarchy, and then I apply that hierarchy to the third data point. For example, in the columns, I say the hierarchy is that H is more severe than M, right? We see that H is more severe than M. Now that is a hierarchy rule. Now I apply this hierarchy rule to this data point. I have here a data point that M is liable in this domain, in domain A. M is liable. And the hierarchy rule is that H is always more liable than M. So if M is liable, then H certainly will be liable. In other words, I take two out of the three data points, build from them a hierarchy rule, and then apply that hierarchy rule to the third data point.
I do the same thing in the rows-based kal va-chomer. I go to the top row and build a hierarchy rule: A is more severe than N, right? And now I take that hierarchy rule and apply it here. H is liable here in N, right? That’s one. So if A imposes more liability than N, then if H is liable in N, then in A it is certainly liable. Again, I am applying the hierarchy rule I learned from this row to this data point. And the conclusion is that here there will be a one. That is how one makes a kal va-chomer.
Now, as I have described the kal va-chomer here, we are really dealing with two completely different and independent arguments: the columns argument and the rows argument. Right? The rows argument assumes that A is more severe than N. It assumes nothing about their relation to H. Right? It is enough for me to assume that A is more severe than N. That’s it. I do not need to assume anything at all about their relation to H. Agreed? If I know that A is more severe than N, and H incurs liability in N, then A certainly incurs liability in H—because it is more severe than it. I have nowhere here assumed that H is more severe than M. I assumed that A is more severe than N. In the columns argument it works the other way around. I assume that H is more severe than M, from this column. I assume that H is more severe than M and apply that here. I do not need to assume anything at all about the relation between A and N. The assumption I made in the previous formulation is not needed in this formulation. In other words, these are two arguments based on… that here there should be a one. Two ways of proving that here there should be a one: either by way of the rows or by way of the columns, and they are independent.
Now let me show you what it means that they are independent.
[Speaker C] Suppose, for example, I now come up with a refutation of the kal va-chomer. What does that mean? I now find another column, okay? Another column, in which the relation is zero here and one here. Let’s see that below for a moment. I have it here somewhere. Here.
[Rabbi Michael Abraham] You see? This is the kal va-chomer I drew before, these four data points, and now an additional column has been added here. This column says—let’s say in that case this is tooth and foot and this is horn, this is the public domain and this is the injured party’s domain—so this is the moon, okay? Another domain, doesn’t matter. I found some domain in which horn is exempt but tooth and foot are liable. Suppose I found such a thing. This is a refutation of the kal va-chomer. Right? Why does it refute the kal va-chomer? Let’s try to think. Why does it refute the kal va-chomer? Because the kal va-chomer was based on the assumption that from this column we see that H is more severe than M, right? That was an assumption of the kal va-chomer. But here we see that that is not true. Here M is more severe than H. Or in other words, there is no simple relation between M and H. And therefore here, when you get to this column, you cannot assume that H is necessarily more severe than M; it depends whether it’s like this or like that. You can’t know. And that is the refutation of the kal va-chomer.
Now I’ll ask you: why shouldn’t I use the rows-based kal va-chomer? The rows-based kal va-chomer does not assume that H is more severe than M—that’s what this refutes. It assumes that A is more severe than N, right? It assumes a relation between these two: A is more severe than N, and therefore here too, if M is liable, then A certainly will be liable. This refutation has no effect on that in any way whatsoever. What do I care that this thing exists here? What is this supposed to prove—that A is not more severe than N? No. A is more severe than N; what does P have to do with it? So what comes out is that if I add such a refutation, it indeed refutes the columns formulation, but the rows formulation can remain intact.
[Speaker B] No, no. You see that P is more severe than N, and despite that, in H’s row it is more lenient.
[Rabbi Michael Abraham] Correct. So that means with respect to P you can’t learn a kal va-chomer; so what does that mean? Fine. So P has no simple relation to N. But does that mean that A is not more severe than N? If I came to check which is more severe, P or N, you’re right, I couldn’t know. Because from M’s standpoint, P is more severe, and from H’s standpoint, N is more severe. I agree. So there is no simple relation between N and P. But the relation between N and P doesn’t interest me. What interests me is the relation between A and N.
[Speaker B] No, but the relation between P and N teaches you that even though in one property something can be more severe, in another property it can be more lenient.
[Rabbi Michael Abraham] Thank you very much! And so on—then don’t use kal va-chomer anywhere in the Torah. Because if you find that one thing is more severe than another, fine, but maybe there are other properties in which it’s not more severe. So why do we need refutations? It would be enough if I made a refutation on one single kal va-chomer anywhere in the Torah to say that there are no more lighter-and-heavier arguments anywhere in the Torah. Because this doesn’t really refute this kal va-chomer. It only says that in general, kal va-chomer is a dubious move. Okay.
[Speaker B] So it’s not…
[Rabbi Michael Abraham] But it doesn’t work like that. We do use kal va-chomer. A refutation like this refutes this particular kal va-chomer; it doesn’t throw the entire concept of kal va-chomer out of Jewish law. So what is happening here? How does it work?
Let me perhaps show you another implication of the fact that we are dealing here with two different arguments. I mentioned earlier that actually the correct datum here is not one but a half. Horn in the public domain is liable for half, not one. Tooth and foot in the injured party’s domain are liable for full damages. And this is liable for half damages. A harmless horn, like in the first acts of goring, is liable for half damages. Imagine for a moment that here it says half.
Now in the columns-based kal va-chomer, it goes like this: if here it says half, half is still more than zero, right? So H is more severe than M. Agreed? But if H is more severe than M, and here M is one, then H is certainly one. Right? Because it has to be more than the one that is here, so the result is one. But now let’s look at the rows formulation. The rows formulation says that A is more severe than N, right? Let’s go down here. Here it says half. So here too A is more severe than N. Therefore what should A be? Half, not one—or at least half. But dayo, enough, right? You can’t make it more than half because you can’t know. Half is what you have here.
Notice that the result of the kal va-chomer can be different depending on whether I use the columns or the rows. That is an indication that we are indeed dealing with two different and independent arguments. It’s not the same argument in two languages; it’s two different arguments. They can have different results. So in short we have two indications that these are different arguments. First of all, we explained substantively why: because the rows argument in no way assumes any relation between these and H, and the columns argument in no way assumes any relation between A and N. So on the merits these are two different arguments. But we also have two indications that this is really so. One indication is that this refutation refutes the columns formulation, but it doesn’t touch the rows formulation. So if I refuted this formulation and the other one remains valid, then they are not equivalent formulations. If they were equivalent, then once one falls, the other would fall as well. And the dayo principle, as I said: if here it says half, then in the rows formulation the result here is half, whereas in the columns formulation the result here is one. And once the results are different in the rows formulation and the columns formulation, that means it is not the same argument—the rows argument and the columns argument. The fact is that its result is different. Okay?
So in the end the conclusion is that the kal va-chomer actually hides behind it two arguments, not one. And each one by itself is enough to prove the result. It is enough that one of them is correct. And these are two different and independent arguments.
Now comes the big question. In the entire Talmud, in all of rabbinic literature—not only the Talmud, in all of rabbinic literature—you will never find that when they raise a refutation against a kal va-chomer like this one, they say: okay, the columns refutation fell, let’s use the rows refutation because the rows refutation still remains valid. A refutation like this does not attack the rows formulation, only the columns formulation. So the columns-based kal va-chomer fell, but the result is still correct because of the rows-based kal va-chomer. No. In the entire Talmud, in all rabbinic literature, the moment a refutation arises, the kal va-chomer falls. We do not say: okay, you knocked out the rows, let’s use the columns; or the other way around. According to what I described earlier, in order to refute a kal va-chomer you would really need a table like this. The refutation P refutes the columns-based kal va-chomer, right? Because here we see that the hierarchy between M and H is not correct; it is not true that H is always more severe than M. And from here we see that this refutation refutes the rows, because from here we see that A is not more severe than N, because here for example it is the reverse. If I have both of these refutations, then the kal va-chomer falls. But if I have only one of the refutations, then it knocks out the formulation parallel to it, but the perpendicular formulation remains valid. Because they are two different arguments.
But it turns out that in rabbinic literature that is not how it works. Any refutation you bring—either a refutation like this or one like that, whether P or X, no matter which—topples the kal va-chomer. We do not say: wait, you knocked out the rows, let’s use the columns, or vice versa. You knocked out the rows—the kal va-chomer falls. You knocked out the columns—the kal va-chomer falls. There are two places where they do make such a switch, saying: you knocked out the rows, let’s use the columns, and vice versa. In those two places the table is not symmetrical. And in this table, what is written here is not one but half, or whatever, or something that is not symmetrical with this. Only in the two unique places in rabbinic literature where they switch a kal va-chomer is when the table is not symmetrical.
Now I have an explanation for that, but it does not interest us here. What matters for us here is that when the table is symmetrical—and that is true in almost all the a fortiori arguments in rabbinic literature—a refutation of one of the directions, whether a row-refutation or a column-refutation, knocks out the kal va-chomer and that’s that. They never say: you knocked out the rows, let’s use the columns, or vice versa. The question is why. On the face of it, this really means that what I said earlier is not correct: that the rows-based kal va-chomer and the columns-based kal va-chomer are two different and independent arguments. Apparently, if one falls then the other falls too. But why? Because when I explained the logic earlier, it seemed perfectly clear that these are two independent arguments, with no connection between them. Why, if one falls, does it automatically bring down the other with it, and vice versa?
[Speaker D] So here, it takes a little time to explain the continuation.
[Speaker E] I just want to ask a question. In this whole thing you’re doing, you’re basically playing a mathematical game. And I innocently thought that in a kal va-chomer it’s very important to know the content, not to do a mathematical exercise here. Because for example when you added another datum about the moon, I immediately wanted to ask: wait, but is the moon in a private area on the moon or in a public area on the moon? Because that has significance.
[Rabbi Michael Abraham] No, let’s say it’s neither private nor public, because there are no people there. I deliberately took something that, if anything, is another domain.
[Speaker E] But I think—is it really possible to analyze a kal va-chomer purely in the form of this table without relating at all to the content?
[Rabbi Michael Abraham] That’s a fascinating question, and in fact we devoted an appendix to this book that deals with these matters. I’ll comment on it later, after I explain what I want to explain here. Very good question. I’ll come back to it; I just need a few more things in order to address it. Okay. So look—well, I won’t continue now because I need time in order to do the next part. We’ll continue next time. I’ll summarize a bit of what I said about kal va-chomer next time, and then I’ll try to propose the explanation, and you’ll see why it is connected to abduction as opposed to induction, which I discussed in the previous part of the lesson. Okay, so let’s stop here. If there are any questions or comments.
[Speaker D] Okay, so goodbye, Sabbath peace.
[Speaker C] Have a good month. Sabbath peace.