Conceptual Analysis – Lecture 13

[Rabbi Michael Abraham] Today I want to start dealing with analysis and synthesis of concepts, and really these are two kinds of tools that are in the toolbox of conceptual analysis. Analysis, in a free translation, means breaking something down, so literally conceptual analysis is analysis. But when we talk about conceptual analysis, we’re talking about analysis and synthesis, and these are really two moves that go in opposite directions. Kant already spoke about analytic propositions and synthetic propositions. An analytic proposition is a proposition that analyzes the concepts involved in it. Say, when I say “this ball is round,” in order to know that proposition I don’t really need to make an observation, I don’t really need to look at the ball, because a ball, by definition, is round. So I can determine that without observation, just from analyzing the concept “ball.” I’m not going into Kant’s nuances right now—whether synthetic a priori propositions and analytic ones are not the same thing. As far as I’m concerned, I’m just using the concepts without getting into finer resolutions. So that’s an analytic proposition. What about “this ball is heavy”? That proposition is a synthetic proposition. I can’t know that this ball is heavy from the concept “ball.” There are heavy balls, there are non-heavy balls. From the definition of the concept “ball” itself, I can’t derive that it’s heavy. So how can I know it? By some other consideration—maybe observation, whatever—but some other consideration, not just analysis of the concept. Therefore that proposition is called synthetic. The analytic proposition is an analyzing proposition, a proposition that comes from analysis of the concepts involved in it. Synthetic propositions are propositions that come out of a synthesis between the definition of the concept and additional sources of information, for example, or additional concepts. Now, that’s with respect to claims. I want to say that the same thing can also be done with respect to concepts. I can do synthesis and analysis of concepts, not of claims. What does that mean? I can produce one concept from another concept by analysis, and I can produce a concept out of other concepts by synthesis. To illustrate these two things, I want to begin with the topic of the common side. When we read the baraita of the thirteen principles, the baraita of Rabbi Ishmael, it begins: a fortiori reasoning, verbal analogy, building a parent category from one text, building a parent category from two texts. What is building a parent category from two texts? The accepted explanation is that this is what’s called learning from the common side, or “what is common to them”—those are the two names. What does that mean? It means a method of learning where I have two teaching sources and not one. That’s in contrast to building a parent category from one text, or building a parent category from one text—there are variant readings in the baraita there—but building a parent category from one text is when I learn from one source. For example, the relationship between primary categories and derivative categories. The relationship between primary categories and derivatives is basically a relationship of building a parent category. Right—if I have, say, the primary category of carrying out, then I can say I have a derivative category, which is bringing in. Carrying out is from a private domain to a public domain; bringing in is from a public domain to a private domain. It’s not clear that this relationship is primary category and derivative—there’s a dispute among the medieval authorities (Rishonim) about that—but I’m only bringing it as an example. Why? What’s the idea here? I look at this concept called carrying out and I say: you transfer an object from a private domain to a public domain. Then I say: really there’s no principled difference between that and bringing it in from a public domain to a private domain. You’re changing the location of the object from one domain to a different domain. It doesn’t matter whether the direction is this way or that way. “What difference is there between bringing in and taking out,” as the Talmudic text says,

[Speaker B] and therefore we learn from this that if carrying out is a primary category, then bringing in is also forbidden. Rabbi, maybe there is a difference? If you take something out from a private domain to a public domain, there I can’t move it around freely inside the public domain anymore, but from the public domain to the private domain I can move wherever I want within the private domain. And that’s not the same thing as there, where I’m kind of limiting it the moment—I’m sort of making it set aside there, I can’t move it.

[Rabbi Michael Abraham] In the private domain too, you’re basically making it set aside. It has to stay stuck in the private domain and not leave it.

[Speaker B] Yes, but I can move around

[Rabbi Michael Abraham] as much as I want in the private domain. In the private domain—but its size, its size can be four handbreadths. A private domain is already four handbreadths. In the public domain you can move it four cubits. Okay. So I don’t think that on this path lies our glory. Meaning, by reasoning alone it doesn’t seem to me there should be a difference, and the Talmudic text too—I’m in good company—the Talmudic text too says there’s no difference. I’m only bringing this as an example of the fact that in the Torah a primary category is written, and we learn from it by what’s called building a parent category. That is, I learn the derivative from the primary category. Why? Because the derivative resembles the primary category, so if the primary category is forbidden, then the derivative is also forbidden. But building a parent category from two texts is when I have a derivative that, in order to learn it, I need two primary categories, not one. Now, in the laws of the Sabbath there’s no such thing—we don’t find such a thing. I even asked people more expert than I am. There is no derivative of two primary categories in the laws of the Sabbath. But in damages we know there is. And in damages there is a derivative of two primary categories, and therefore I want to get a bit into this in order to show the method—what I want to show is the relationship between analysis and synthesis. So let’s begin with the first Mishnah in Bava Kamma.

[Speaker C] So Rabbi, just a second—in your terminology, analysis and synthesis are like in Kant’s language, right?

[Rabbi Michael Abraham] It’s in human language, not specifically Kant’s. Kant used the concepts of analysis and synthesis for his own particular needs—he called them analytic propositions and synthetic propositions—but the concepts analysis and synthesis don’t need Kant in order for us to use them. So right now I’m not using them specifically in the Kantian sense. I only used Kant to illustrate what we’re talking about, not because Kant created these concepts. He created a certain use of them. So I want for a moment to go into the first Mishnah of Bava Kamma. I’ll share the text here. Okay. “There are four primary categories of damages: the ox, the pit, the grazer, and the fire. The ox is not like the grazer, and the grazer is not like the ox. And neither this nor that, which have life in them, are like fire, which has no life in it. And neither this nor that, whose way is to move and cause damage, are like the pit, whose way is not to move and cause damage. The common side among them is that their way is to cause damage and their guarding is upon you, and when damage occurs, the one who caused the damage is liable to pay compensation from the best of his land.” So basically the Mishnah tells me there are four primary categories of damages. There’s an Amoraic dispute about what exactly these four categories are—Rav and Shmuel disagree here—not important for our purposes. But after that the Mishnah goes on to make distinctions among the different primary categories. What does it mean to make distinctions among the different primary categories? When I say “the ox is not like the grazer,” what do I mean? I mean that if only the grazer had been written, I couldn’t have learned the ox from it, because the ox is not like the grazer; there isn’t enough resemblance between them. Therefore the ox cannot be learned from the grazer, and that is why the Torah had to write this primary category of the ox, because the primary categories are what the Torah writes. So when the Talmudic text—when the Mishnah says “the ox is not like the grazer,” it is really coming to neutralize a possible building of a parent category. If the ox and the grazer really were similar, then writing the ox would have been unnecessary. Let it write the grazer, and I would learn the ox from it. Okay, so that is building a parent category—the negation of building a parent category. So we have four primary categories, and each one is needed; you can’t learn it from the others. The Talmudic text develops the “each is needed” analysis in more detail. Once I have the four primary categories, what now? Now the Mishnah comes and says: “The common side among them is that their way is to cause damage, and their guarding is upon you, and when damage occurs, the one who caused the damage is liable to pay compensation from the best of his land.” By the way, there’s a version of the Rif here—one second—the Rif reads the Mishnah a bit differently. Yes, the Rif reads, look here—do you see the Rif? You do, right? Okay. “The common side among them is that their way is to cause damage, and they are your property, and their guarding is upon you.” The Rif adds that it is also your property. “And when damage occurs, the one who caused the damage is liable to pay compensation from the best of his land.” So basically the Mishnah concludes with “the common side among them,” among them meaning all four primary categories. What does “the common side among them” mean? It means: what is the shared factor that characterizes all of them? I have ox, pit, grazer, and fire—yes, I have fire, pit, horn, tooth, foot, all the various primary categories of damages. What do they have in common? There are all kinds of differences among them. For example, a pit is created initially for damage; it has various laws of this kind and that, and it’s in the public domain. Fire is exempt for concealed objects; fire does move with the wind. Each one has special characteristics. But there are certain characteristics that are shared by all of them. That’s what in the language of the Sages is called the common side. Meaning, there are differing aspects among the primary categories, and there are aspects that are equal, shared by all of them. If you want, you can call it intersection and union. The intersection between two primary categories will give me the common side. Say primary category A has characteristics A, B, C, and primary category B has characteristics A, C, D. Then the intersection is A and C, right? Those are the two shared characteristics of primary categories A and B. That is called the common side. The common side of the two primary categories is the shared characteristics. What is the differing side? The differing side is the different characteristics: B for A and D for B. Okay? That’s the differing side. Of course this is my term; there’s no such term in the Talmudic text, but we’ll see later that it will serve us. So the common side means the shared characteristics. I go back to the Mishnah in Bava Kamma: the shared characteristics of all four primary categories are that their way is to cause damage, and they are your property—belong to you—and their guarding is upon you. The Torah requires you to guard them so they won’t cause damage, and therefore when damage occurs, the one who caused the damage is liable to pay compensation from the best of his land. On the face of it, the Mishnah is really coming to tell me: look, four primary categories are written in the Torah. Now you may encounter other kinds of damage that aren’t written in the Torah. What do you do with them? So the Mishnah says: look, I’m summarizing for you the rule that emerges from the four primary categories. What do they all have in common? They are your property, their way is to cause damage, and their guarding is upon you. So now you have a rule you can use. Any damager that comes before you—if it satisfies these characteristics, if it has these characteristics—you’ll be liable to pay if it caused damage. Okay? Basically the common side is the rule that emerges from the four examples. The examples are examples, and there is a shared rule. By the way, the Talmudic text on page 5 says that each of the primary categories has special laws. For horn, the first three times you pay only half-damages, not full damages. A pit is exempt for vessels, an ox—

[Speaker B] “An ox” and not a person, “a donkey” and not vessels.

[Rabbi Michael Abraham] “An ox” and not a person, “a donkey” and not vessels. So that comes to exclude vessels and a person. Fire is exempt for concealed items; tooth and foot are exempt in the public domain. Meaning, each of the primary categories has special exemptions. Okay? But there is something shared by all of them on the halakhic / of Jewish law plane, and that is that one must pay if they occur, except for each one’s special cases. This intersection among all of them can be learned for all other kinds of damages. Now a case comes before me and I need to decide whether payment is required. I look: if it is someone’s property, and its way is to cause damage, and that person really was supposed to guard it—if that’s the case, then that person will have to pay. Okay? I now have a rule that saves me the work. I don’t need to check whether it resembles tooth, whether it resembles horn, whether it resembles fire, pit—who it resembles. No, no need. Once it satisfies these characteristics, then I know he must pay. I’ve received the rule. By the way, the Talmudic text says: then why were they needed? For their special laws. The Talmudic text on page 5 says that you still need to write the four primary categories. Why? Because in each of the primary categories, beyond the fact that one must pay, there are also special exemptions—the ones I mentioned before. So for example, if there were—if there were something that is a derivative of fire, yes? If there were something that is a derivative of fire, in that case it would be exempt for concealed items, because just as fire is exempt for concealed items, so too it would be exempt for concealed items. Therefore, when a case comes before me, it isn’t enough to see whether it satisfies the common side, the general principle. I also have to see whether it resembles one of the specific primary categories. Why? Because the basic obligation to pay I really can learn from the general principle. If it’s someone’s property, its way is to cause damage, and he was obligated to guard it, he has to pay. But it could be that in certain cases it will still be exempt. If it resembles fire and the damage was to something—an object concealed inside something else, then it will be exempt. If it doesn’t resemble fire, then it won’t be exempt. If it resembles horn, then he’ll have to pay half-damages the first three times. If not, then not, right? And so on. So there is also significance to the resemblance to each primary category separately, but there is also importance to the rule. The rule first of all tells me when and who must pay. After that I need to check whether there is some special exemption, and here I examine it in light of its resemblance to one of the primary categories.

[Speaker D] Does that mean the rule doesn’t apply to the four?

[Rabbi Michael Abraham] It does apply to the four—what do you mean, it doesn’t apply? It’s just that additional rules also apply to them, rules that qualify this rule. In special cases one doesn’t have to pay, but the basic obligation to pay exists in all four.

[Speaker B] Their common denominator.

[Rabbi Michael Abraham] Yes. And therefore, any damager that comes before me, if it satisfies that common denominator, then first of all he has to pay. Now we need to check whether it has one of the exemptions, and here I need to check not only the general characteristics but whether it resembles one of the specific primary categories, and then I’ll see whether it has a particular exemption or not. But the basic obligation to pay emerges from the rule. Okay. Now I want to move to the Talmudic text; I’ll close the Rif for a moment. That’s the Mishnah on page 2. I’m moving to the Talmudic text on page 6a. “The common side among them comes to include what?” What does it mean, “The common side among them comes to include what?” Why did the Mishnah conclude with this formulation of the rule of the common side? It listed the four primary categories, and it ends with this sort of summary: “The common side among them is that they are your property and their guarding is upon you and their way is to cause damage, and when damage occurs, the one who caused the damage is liable to pay compensation from the best of his land.” Why is that needed? Now, I’ve already spoken about this more than once in the past, and I’ll say it here just as a brief note. This is a very strange question. A very strange question, because I would ask the opposite question: why do we need the examples? Just tell me the rule. Give me the rule, and then I know what to do. What do I need the examples for? The examples only served me in order to generate the rule. Once I already have the rule, what do I need the examples for? Just tell me directly: anything that is your property, whose way is to cause damage, and whose guarding is upon you—if it causes damage, you are liable to pay compensation from the best of your land. Why do I care about the four examples from which I derived this rule?

[Speaker B] But the Rabbi says that the four examples come…

[Rabbi Michael Abraham] Wait, what again?

[Speaker B] The Rabbi said that the four examples come…

[Rabbi Michael Abraham] Right, so I already answered that earlier. I already answered that earlier—that in fact there are special laws that characterize each of those primary categories. But the truth is that there are places where the Talmudic text asks this and that solution doesn’t exist. In this case it really does, but in other cases it doesn’t, and still the Talmudic text asks, “The common side comes to include what?” What does that mean? I may talk about that later in another context, but here I’ll note it just as a remark. In general, in the legal world there are two basic approaches to how this system is supposed to function. They call it positivism and, say, anti-positivism—I don’t want to get into natural law and specific things right now. What does that mean? Positivism means a way of looking at the system as a system built like a logical system, an axiomatic system. Anyone who knows—who studied geometry in high school—there are axioms, and from the axioms we derive theorems, and from those theorems we derive more theorems, and so on. There is some hierarchical logical structure that begins with axioms and from them derives various conclusions and so on. The legal system is basically supposed to operate in the same way. The legislator has to establish the rules, and the judge, in a case that comes before him, has to see how one can derive… And according to this approach, the legal system has to be formulated as a system of rules. By contrast, there is an approach—say, in German law, which is the more prominent example of a positivist approach, although in the last generation this too has weakened there, because they understand it can’t work. The opposing approach—we’ll call it the casuistic approach, from cases. That’s the British approach. And the British legal system is basically not built on a system of rules, but on cases and examples, and the judge’s job is not to derive by deduction a conclusion about a private case from a general law, but to make an analogy. There were previous cases in which such-and-such a ruling was given; this case resembles that case or that case, and according to that resemblance I determine what the ruling will be in the case before me. The basic tool of casuistic law is analogy. The basic tool of positivist law is deduction. Deduction is deriving a specific conclusion from a general law. Analogy is deriving a specific conclusion from another specific conclusion. From one example I make an analogy to another example, as opposed to deriving a specific conclusion from a law. For example, I say all tables have four legs; the object before me is a table; conclusion: the object before me has four legs. That is deduction. Why is that deduction? Because here we are going from the general to the particular. We have a general principle: all tables have four legs. This particular item, this table, is included in the group that the general law is talking about, so this table too has four legs. This is called a necessary inference, deduction, where one learns the particular from the general, okay? Analogy is learning one particular from another particular. For example: this table has four legs; that object I see over there is also a table; so apparently that one too has four legs. That’s not deduction, that’s analogy. Why? Because I’m learning the rule in that specific case from the rule in this specific case. I’m learning one private case from another private case that resembles it. That’s analogy, not deduction. Deduction is learning a case from a general law. Analogy is learning a case from a case. So that basically is the distinction between positivism and casuistry, yes, the case-based approach that relies on cases, on specific cases, okay?

[Speaker B] So when we—just a second, Rabbi—when we have many examples, doesn’t that eventually become a kind of rule?

[Rabbi Michael Abraham] Usually, in order to learn from the examples, we basically create a rule out of them. You know what? Since you asked, I’ll continue a bit with this issue.

[Speaker C] But Rabbi, when halakhic decisors make one case resemble another case, right,

[Rabbi Michael Abraham] that’s analogy.

[Speaker C] that’s the approach of analogy?

[Rabbi Michael Abraham] Right, I’ll get to that. But I want to expand a bit more on the logical concepts. The issues have already come up, and I really think it’s worth expanding. In logic, people usually divide arguments, inferences, into three kinds: deduction, induction, and analogy. Deduction is deriving a particular from a general rule; analogy is a particular from a particular; induction is deriving a general rule from particulars, okay? From this table having four legs, and that table having four legs, and a few examples I’ve seen, I derive from this a general law that all tables have four legs. Okay? That’s induction. Okay? Deduction is a necessary inference. Anyone who accepts the premises is forced to accept the conclusion; he cannot argue with the conclusion. Analogy and induction are not necessary inferences. Right? I can accept that this resembles that, and I can also refuse to accept that this resembles that. I can decide that from these examples there emerges a rule that all tables have four legs, and I can say: who told you that? Maybe in those particular three tables you saw, they had four legs, but there are other tables that have three legs, five legs, or no legs at all. You can never really be sure. We also use analogy and induction, but these are not certain tools. Maybe it’s true, maybe not. Deduction is a certain tool. Now, there is a famous critique of deduction by John Stuart Mill. Mill claimed that this view, which says that deduction is a necessary inference, is an illusion. Why is it an illusion? Let’s return to tables. What does the deductive inference say? All tables have four legs, and therefore this object before me, which is a table, also has four legs. Mill asks: how do you know that the table before you has four legs? Because of the premise that all tables have four legs. And where do you know the premise itself from? But without knowing the premise, you also can’t know the conclusion. So where do you know your premise from? How do you know that all tables have four legs? Apparently from some induction. I encountered a few tables, saw that each had four legs, and inferred some conclusion that apparently all tables have four legs. Which means that deduction secretly contains, without telling us, an induction at its base. Since the rule on which deduction is built, the general law on which deduction is built, is itself produced by way of induction. Right? Because otherwise I have no way of knowing. How do I know? I haven’t seen all the tables in the world. How do I know that all tables have four legs? So Mill basically says: if you can’t be sure of induction, then you also can’t be sure of deduction. Since your confidence that this table has four legs depends on whether you’re sure that all tables have four legs. But the statement that all tables have four legs is a result of induction. You can’t be sure of that. And if you can’t be sure of that, then you also can’t be sure of the conclusion that follows from it.

[Speaker C] Of course, when people talk—wait—not every deduction hides an induction within it?

[Rabbi Michael Abraham] What? I can’t hear.

[Speaker C] Doesn’t every deduction hide an induction within it? Isn’t there an induction standing behind every deduction?

[Rabbi Michael Abraham] Yes, that’s exactly what Mill says.

[Speaker C] For example, Socrates is a human being and he is mortal. So that human beings are mortal—they die—that doesn’t come from induction? It comes from knowledge.

[Rabbi Michael Abraham] And what—where does that knowledge come from? From induction. Because you saw human beings, you saw that they all die, so you infer from those examples that all human beings die. That’s induction. You have no other way of arriving at a general law.

[Speaker C] So that’s certain induction.

[Rabbi Michael Abraham] No, of course not. Nothing here is certain. Who told you that I’ll die? Have you ever seen me dead? You saw other people in the past who lived and died. Maybe I’ll live forever. How do you know I won’t? It’s all induction. Nothing is certain; everything is induction. Every scientific claim is based on induction.

[Speaker C] For example, the Pythagorean theorem.

[Rabbi Michael Abraham] The Pythagorean theorem is not—that’s mathematics, not science. So the claim—someone asks here—even intellectual observation, observation with the mind’s eye, is also not something certain, just as no observation is certain. Therefore, no—we don’t have tools that can give us full certainty about anything. In any event, what I’m saying for our purposes is this: when people speak about the necessity of deduction, of course they do not mean to say that the claim that this table has four legs is certain. Because it isn’t true that it’s certain. What they mean is that it follows with certainty from the premise. That’s where the certainty lies. Not that the claim itself is certain, but that its derivation from the premise is certain. Okay? Is that clear? So basically it comes out like this. Or maybe I’ll add one more sentence. Where does this certainty of deductive derivation really come from? If all tables have four legs, and this thing is a table, conclusion: this thing has four legs. Why—where does that conclusion come from? Why is it so obvious? Why can’t one argue with it?

[Speaker B] Then they won’t call it a table, they’ll call it something else.

[Rabbi Michael Abraham] What they call it is a matter of convention.

[Speaker B] No—how did we say it? If it’s a table then it has four legs. What does “if it’s a table” mean? That’s what the Rabbi said, right? What does “if it’s a table” mean?

[Rabbi Michael Abraham] I’m saying: Premise A, all tables have four legs. Premise B, this thing before me is a table. Conclusion: this thing before me has four legs. Okay? That’s the argument I’m talking about. Now I ask: someone who accepts the two premises—some alien comes to you, lands on you from the moon, okay? You tell him the two premises. Know that all tables have four legs. He says okay, understood. You say, know that this object before me is a table. He says, I understood that too. Now you say to him, so you understand that this object before me has four legs? He says no, I don’t understand. Why do you think—what do you mean?

[Speaker C] If A=B and B=C then A=C, but that’s not equality.

[Rabbi Michael Abraham] There’s no equality here. It’s inclusion, not equality.

[Speaker B] It’s just a certain premise.

[Rabbi Michael Abraham] The point, I think, is this: what if someone like that were standing here in front of you—how exactly could you explain his mistake to him? It’s obvious that he’s wrong, but how can you explain the mistake? It seems to me that what you have to say is this. If you really say that all tables have four legs, let’s break that down into small change. What does that mean? You accepted that statement, right? That’s one of the premises. Let’s break that premise down into small change. It means that table A has four legs, table B has four legs, table C has four legs, and so on for all the tables in the world. In other words, the claim “all tables have four legs” is nothing more than a collection of many, many claims about individual tables. Right? We’re just saying it briefly. One of those tables is the table in front of me. So when you said that all tables have four legs, if you unpack what you’re saying, then you already said that this table too has four legs, because it’s one of the tables. So you’ve already accepted that this table has four legs. You can’t both accept that and not accept it—that’s a contradiction. That’s what you can say to that alien. And what this really means—the joke I always remember in this context—is the yeshiva joke that says: let me prove to you that every Jew has to wear a hat. Why? Because it says, “And Abraham went,” right? Now a Jew like him obviously didn’t go around without a hat. We all understand that, right? Therefore, if Abraham went with a hat, then we all have to walk in Abraham’s ways, so we too have to go with a hat. Which is what we wanted to prove. That’s it, that’s the proof. Now they tell this as a joke in yeshivas, but really let’s think for a moment about what’s problematic in this proof.

[Speaker B] Who says Abraham went with a hat?

[Rabbi Michael Abraham] A Jew doesn’t go without a hat, think about it. It can’t be.

[Speaker B] Why not? Because he’s a Jew.

[Rabbi Michael Abraham] Can you imagine a Jew going without a hat?

[Speaker B] Yes—like Abraham.

[Rabbi Michael Abraham] He went with a hat.

[Speaker B] Who says Abraham went with a hat?

[Rabbi Michael Abraham] Because a Jew doesn’t go without a hat.

[Speaker C] But that’s what he’s trying to prove.

[Rabbi Michael Abraham] Ah—so the claim is that there’s a fallacy here called begging the question. Begging the question means this: you want to prove a certain claim—that’s what you’re trying to prove. You can’t take the very claim you want to prove and put it in as a premise in the argument. Right? If you assume what you want to prove, then what good is your proof to me? Okay? So there’s really a fallacy of begging the question here. But what this actually means—let’s go back to the tables. Do you understand that the proof about the tables is also begging the question? What am I saying? All tables have four legs, this object in front of me is a table, conclusion: this object has four legs. Same thing. Do you understand that this too assumes the conclusion? Because I assumed that all tables have four legs, and this is one of the tables, so I assumed the conclusion.

[Speaker C] No, here I’m not trying to prove that all tables have four legs, I’m—

[Rabbi Michael Abraham] Trying to prove that this table has four legs. Fine—but I assumed that this table has four legs.

[Speaker C] No, that all tables have four legs.

[Rabbi Michael Abraham] When I said that all tables have four legs, I unpacked that claim earlier. I said that it’s really a collection of many, many claims about individual tables, so in effect I assumed that this table has four legs. It’s an individual one. I didn’t say it explicitly, but you understand that I really assumed it. Or in other words, if someone thought that this table does not have four legs, then he obviously wouldn’t accept the premise that all tables have four legs, so I couldn’t prove to him what I want to prove, right? Why can I prove it to him? Because he already assumed it from the start. So what does that help? That’s begging the question. Now understand—what?

[Speaker B] Is it always like that? Is it always like that?

[Rabbi Michael Abraham] Always. Every valid logical argument is an argument that assumes the conclusion, by definition. I’ll say more than that: a logical argument is valid because it assumes the conclusion. After all, I asked you earlier: why is someone who accepts the two premises forced to accept the conclusion? The answer is because the conclusion is contained within the premises. So once he accepted the premises, he automatically accepted the conclusion as well. So of course he has to accept the conclusion—it’s included in the premises. That’s the reason a deductive argument is a valid argument. And you have to accept its conclusion, because someone who accepts the premises must accept the conclusion. And the reason for that is that there is nothing in the conclusion—

[Speaker B] Beyond what was in the premises.

[Rabbi Michael Abraham] The conclusion doesn’t add information beyond the information latent in the premises. And that’s why I have to accept the conclusion, because I already accepted that information when I accepted the premises. What characterizes deduction—the move from the general to the particular, necessary inferences—is that they really don’t add information. That connects to another joke, the famous joke about the hot-air balloon. I’ve made quite a living off that joke already. Two people lose their way in a hot-air balloon; they don’t know where they are. At some point they pass over some field, where a man below is working in his field. So one of them shouts down to the guy below, “Tell us, where are we?” So the man looks up and says, “Above my field.” So the fellow in the balloon says to his friend, “That guy down there is definitely a mathematician.” Why? Two reasons: first, what he says is completely certain, necessarily true. Second, it doesn’t help me at all. The characteristic of mathematics is that what it says is necessarily true—and it’s also no help whatsoever. And I think behind that there’s a very serious point. The reason a logical argument or a mathematical argument is necessarily true is because it doesn’t add information—it doesn’t help me at all. That’s the reason logical arguments are valid, necessarily true, because they don’t help me at all—that is to say, they don’t add information beyond the information I already accepted when I accepted the premises. So people may ask: then why study mathematics, why study geometry? We see that it helps us. Of course—it helps us practically. Meaning, it doesn’t really add information, it doesn’t add any information at all, but there’s a great deal of information that is within me and I don’t know how to get to it, I don’t know how to use it. So math lessons help me extract more and more information that is already inside me. That’s the role of a math lesson, or a logic lesson, or a logical argument, or a mathematical proof. All of those are tools that help me bring out more and more things that I already know.

[Speaker F] But why do I need the deductive argument at all in this case? Don’t I need to use an argument like that only when I have a question? Here I don’t have a question—I see the table, I see four legs, so why do I need—

[Rabbi Michael Abraham] No—say I see it from far away. I see that it’s a table but I don’t see how many legs it has. Fine, just an example, it doesn’t matter.

[Speaker C] So basically you’re saying that every logical argument contains begging the question. Right.

[Rabbi Michael Abraham] So it follows that a logical argument, by definition—something that has a logical proof, by definition—doesn’t add information. Of course it can help me, because in mathematics too, even someone who knows the axioms will find it very difficult to prove the theorems derived from them. But after it’s proven to him, he’ll understand that the theorem was already there inside the axioms; it was just hard for him to extract it from there. Think of a safe locked with a very sophisticated lock. Then someone comes along and manages to open the safe, and now he gives me everything inside it. The safe is mine, so everything in it was already mine beforehand; I just couldn’t use it because I didn’t know how to open the safe. Someone came and helped me open it. So what did he do? He didn’t add property to me; that property was mine already. He just made the property accessible to me, he enabled me to use property that was already mine from the start. That’s the parable I use to explain what logic does or what a proof in mathematics does. It basically makes information accessible to me that I already had from the outset.

[Speaker B] Is the Rabbi saying that everyone is capable of being a mathematician?

[Rabbi Michael Abraham] No, absolutely not. I said the opposite.

[Speaker B] But if everyone—

[Rabbi Michael Abraham] If everyone were capable of being a mathematician, there would be no mathematics in the world. The whole idea of mathematics is that even though all of us have all the information, very few of us are smart enough to extract that information from within ourselves, to open the safe, and therefore not everyone can be a mathematician.

[Speaker B] Not everyone has the key.

[Rabbi Michael Abraham] Right, you have to be a master locksmith who knows how to open safes. Everyone has the safe, and everything is inside all of us—inside each of us there is all the mathematical information. Mathematical information. But very few of us know how to get that information out of there.

[Speaker C] We have that information only if we have the premises. But if we don’t know the premises at all—

[Rabbi Michael Abraham] If we don’t have the premises, then no mathematician will help us. A mathematician only helps us take the premises and derive the conclusions from them. So in any case, the mathematician won’t do that part for you. Everything he— You ask, you know, here’s the example: I ask a mathematician what the sum of the angles in a triangle is. So he’s supposed to answer me: that depends on your axioms. He can’t just say one hundred eighty. In Euclidean geometry, yes, on a flat surface, flat space, the answer is that the sum of the angles is one hundred eighty. In a different geometry, the sum of the angles is different. Now, a mathematician is not authorized to tell me what my axioms are. I have to decide that myself. The mathematician knows how to tell me: if those are your premises, I’ll show you what the sum of the angles in your triangles is. Okay? So the mathematician can’t give me the premises. The assumption is that the premises already exist within me. Now the question is what the mathematician does. A mathematician helps me extract more and more information out of those premises. But who gives me the premises? I do. Okay? So the claim is that what distinguishes deduction from analogy and induction is basically the question whether the inference adds information. A deductive inference does not add information, and therefore it is certain. That’s why the conclusion follows necessarily from the premises, because there’s nothing new in it. There is no information in it that wasn’t in the premises. Analogy and induction are two non-certain modes of inference, right? We saw that earlier. You can argue about them; it’s not certain that they’re correct. Why? Because each of the inferences in analogy and induction adds information. Again, I have one table that has four legs. I see another table from far away. I say: how many legs does it have? I make an analogy. I say: if this is a table and that is a table, if this one has four legs then that one also has four legs. Do you understand that in my premises I did not have the information that that table has four legs? I only had information about the table in front of me. I just think that the other table resembles this one, and I infer that it too has four legs. That conclusion contains new information that I did not have when I started out. So an analogical inference… an analogical inference adds information. Since it adds information, it may be wrong. It tells me something new. Who knows—maybe it’s right, maybe it isn’t. I can’t be certain. Same thing with induction. I saw one table that had four legs, I saw another table that had four legs, and from that I conclude that all tables have four legs. Do you understand that the conclusion contains more information than the premises? My premises are about several individual tables that I saw. In the conclusion I’m talking about all existing tables. So the conclusion contains more information than the premises, which means information was added to me in an inductive inference. If so, it cannot be necessary. Or in other words, I call this the principle of logical uncertainty. The principle of uncertainty—you know that the uncertainty principle in quantum theory says that the uncertainty of position and the uncertainty of velocity are inversely related. Their product is constant. Meaning, if I know exactly the position of a certain particle, I can’t know anything about its velocity. If I know exactly its velocity, I know nothing about its position. If I know its position within about a meter this way or that, then there’s some range of velocities I can know about it—that it lies between such-and-such a velocity and such-and-such a velocity. The more precise my information about position is—the smaller the range of uncertainty about the position—the greater the uncertainty about the velocity, and so on. The mathematical description of this says that the uncertainty of position multiplied by the uncertainty of velocity is constant. So if one is small, the other is large; if the second is small, the first is large. Because their product has to remain constant. Right? I claim that there’s a similar relation between the degree of certainty and the amount of information an inference adds. The product is constant. Meaning, if the inference adds me a lot of information, my certainty is very low. If the inference adds me a little information, my certainty can be very high. If the inference adds me no information at all, the probability… the uncertainty in the added information is zero, so the certainty can be full.

[Speaker C] And which is better—to add knowledge or to add certainty?

[Rabbi Michael Abraham] Exactly. And whoever wants— in other words, you know, postmodernism, I’m going a bit broad here but I think it’s relevant. Suddenly I realized it is relevant to our topic. Postmodernism. We’ve almost forgotten Bava Kamma; we’ll get back to it at the end, but I’m saying this because we’ll use it later. So the postmodern view says that what is certain, what comes out of a logical argument for which you have a proof—that’s perfectly fine. Everything else is subjective, it’s a narrative. Everyone has his own narrative. Okay? What they’re basically saying is that they’re not willing to accept anything unless it is certain. Whatever isn’t certain, I’m not willing to accept—it’s subjective. Do whatever you want with it. Now you understand that if that’s really so, then you’re left with no information at all. Because if you’re willing to accept only things that are certain, then certain inferences are inferences that add no information to you, by definition. So how will you ever discover your first piece of information? Where will you get it from? Every logical inference that serves as a proof for some claim is based on premises. And where do you know the premises from? Do you have a proof for them? And what about the premises of that proof—where do you know them from? In the end, in the end, you’ll be left with something for which you have no proof. And if you’re not willing to accept something for which you have no proof, you won’t be able to accept anything. And so postmodernism really does arrive at intellectual nihilism, yes, complete skepticism. You can’t know anything. No one is more right than anyone else. Okay? Why? Because they want absolute certainty. Someone who wants absolute certainty is left with zero information. The only way to deal with postmodernism, or with that kind of skepticism, is to understand that in order to gain information you have to pay in the currency of certainty. There’s a price. You want information? Give up the level of certainty you demand. Someone who demands absolute certainty will remain without information. Someone who is willing to accept things even if they are not certain—maybe they’re probable, eighty percent, ninety percent, sixty percent, I don’t know. I’m willing to compromise. If you’re willing to compromise on the level of certainty, you have a chance of accumulating information. If you’re not willing to compromise on the level of certainty, you have no information. You want all the information in the world? You have no chance, because your certainty will be zero. So you’ll know all the information in the world with zero certainty—which means you won’t know anything. It reminds me of some speech I once heard by Bibi. Bibi explained why raising taxes is a bad thing. And it’s nice. Today I’m a bit associative, sorry, but wow. He explained why raising taxes is a bad thing. Let’s try this in an Indian style, okay? I have a coordinate system. Here is money—the money in the state treasury, okay? And here is the tax rate, taxes, okay? The tax rate. Now the tax rate is between zero and one. Right? Here is zero, let’s say here is one. Okay? And I want to know how much money I have at each tax rate. If the tax rate is zero—yes? They impose zero percent tax on all workers—how much money is in the state treasury? Zero, right? We’re here. If the tax rate is one, now—

[Speaker C] One hundred—

[Rabbi Michael Abraham] Percent—how much money is in the state treasury? Zero, of course. Why? Because if I earn nothing, I’m not going to work. If I pay everything in taxes, then my salary is zero. If my salary is zero, I’m not going to work. So with a one hundred percent tax, the state treasury also has zero, right? So that means the graph looks something like this. Which means there’s some tax rate in the middle that gives the maximum money in the state treasury. Now if I’m located—say research shows that it’s something like thirty percent. Okay? So here is thirty percent. Okay, zero point three. Okay, this is zero point three here. Now suppose our tax rate is here and we want to increase the state treasury. Then we need to raise the tax rate, right? If we raise it to here, the money in the state treasury will increase. But if we’re here, in a country whose tax rate is high—say zero point four, forty percent tax—then in order to increase the state treasury we need to lower the tax rate. In short, the socialists who think that the solution to every problem is raising the tax rate are wrong. It depends where you are relative to the optimum. Okay? Sometimes lowering the tax rate increases the state treasury, the amount of money the state has. It’s actually quite simple, but I think it’s a very nice argument that shows it in a very simple way. Right? Basically something similar exists here too. There’s a trade-off between the amount of money and the tax rate; here too you can say the product is constant. Okay? Therefore the claim is that something similar happens between certainty and the amount of information. If you want infinite information, you have zero certainty—which means you have zero information. You want all the information in the world. All the information in the world means that you have no standard of certainty at all. You’re willing to accept everything. The moment you’re willing to accept everything, you know nothing. You have zero information. If you want to be with zero information—if you want to be with zero information—then zero information means that you can actually have complete certainty, but about nothing. So you still remain with zero information. Meaning that at both ends of the graph you have zero information—whether you want absolute certainty or whether you don’t care about certainty at all, you remain with zero information. And there’s some optimum point you need to pass through, where you’re willing to compromise on the amount of certainty in order to gain information. Something like the argument I presented earlier can be presented here too. And therefore rational people, not postmodern people, understand—like scientists, for example, because scientists can’t be postmodern. So anywhere at the university where they tell you that there is a postmodern approach, know that this is not science. It’s nonsense. It’s simply because that means they either gave up certainty or demand one hundred percent certainty. Therefore, in the end, a scientist understands that in order to accumulate information one must give up certainty. You can’t demand one hundred percent certainty. So you’re willing to accept analogy and induction too, not only deduction. Also things that don’t give you full certainty. Of course, now you have to develop tools for how to control, how to supervise the information you’re accumulating, because it isn’t certain. You need cross-checking, you need to test, to do an experiment that tests my theory, to see that I’m not just walking blind down the road, because here there is something uncertain. And if something is uncertain, then you develop various methods—and that is the whole scientific method. How do we develop methods that succeed in giving us control over our information gathering even though it isn’t certain, so that it won’t be nonsense? Okay? That is basically the scientific method. So now what I want to claim is that if someone wants to do only deductions, to remain only in full certainty, he’ll be left with zero information. Right? It’s actually impossible to do only deduction. You can’t rely only on deductions. Say, translated into the legal world—not a one-to-one translation, but the idea is similar—someone who wants to produce a pure positivist system, where the legislator sets the rules and the judge is basically just a device: he takes the rules and applies them to the specific case before him. There is no chance of building such a legal system. By the way, that is what ultimately led, in a certain sense, to the Holocaust. And this came up in the Nuremberg trials, when the Nazis said, “I obeyed orders.” What does “I obeyed orders” mean? There’s a law in Germany that one must obey orders. The commander gave an order, conclusion: one must obey the commander. Which is what we wanted to prove. That’s a positivist approach. And what did the judges argue against them? This was the great innovation of the Nuremberg trials. Use your head. There is also common sense. It’s not enough to have logical rules and legislation. And therefore in German law—it’s no surprise that German law is positivist. That’s the Yekke character. That’s why Yekkes talk about the science of law. For them law is a science. Science meaning mathematics. Pure logic. In contrast, in the West, the British-American world—mainly British in this case—the law is the common law. Meaning, we have cases and we make analogies. We don’t believe in rules. Okay? Rules can’t do the job. The judge always has to use his head. He can’t be a computer that merely extracts conclusions logically from the rules set by the legislator. It won’t work. No case that comes before you is a simple case where you can derive the answer from the rules. Anyone who thinks there are such cases is simply mistaken. There isn’t a single such case, in my opinion. There is no case that comes to court in which there isn’t still something you have to think about—maybe it’s not exactly similar to the law, or maybe it is. There are no clear cases. Any lawyer will tell you that. And therefore this positivist hope, which basically parallels the deductive approach—the approach that sees absolute certainty as the criterion for accepting things—is an approach that cannot work. You have to use analogies and inductions. You have to use a system of rules that is softer, less necessary, not certain. There’s no choice. You can’t function without it. That’s what leads us to judicial legislation and all these controversies in the world of law. In any case, if I return to the Talmud, someone mentioned this earlier—Moshe Eliyahu, I think you mentioned it earlier—the basic conception in the Talmud is a casuistic conception. In the Mishnah and the Talmud they bring cases. Not rules, almost never. Here and there there are rules, very few. Usually it’s a case: someone did such and such, his law is such and such. It’s not a rule. Usually a normal Mishnah says: “There are four primary categories of damages: the ox, the pit, the maveh, and the fire.” The common denominator does not appear in most mishnayot. Here in Bava Kamma they did us a favor. They gave us the rule—the common denominator—and you understand that this is the rule that comes out of the examples. They did the induction for us. Wait—they did the induction for us and basically said: we have four primary categories that the Torah wrote, and from them we derive the rule that all of them are your property, their guarding is upon you, and they are prone to cause damage. And that’s the rule. Now we can work with the rule; we don’t need the details. But that’s not true. Because the Talmud—even in a place where the Mishnah departed from its usual pattern and gave us the rule—the Talmud is not impressed by that at all. The Talmud asks: who needs this rule? I have examples. What does that mean? The Talmud has no faith in rules. The Talmud doesn’t believe in positivism. There are no rules. With rules you won’t get anywhere. Rules can be interpreted in all sorts of ways. When the case comes before you, if you work only with a rule, you won’t know what to do with it. You have to use the details and make analogies. Therefore even when the Mishnah brings rules, the Talmud sees the rules as something superfluous, and it prefers the examples, not the rules. Because from the examples I’ll make an analogy and I’ll know what happens. Maybe I remembered one more thing; I see today’s class has gone in the direction of these introductions, so I’ll just clarify and sharpen the point a bit more. Look, Wittgenstein—many considered him the greatest philosopher of the twentieth century, Ludwig Wittgenstein. He was basically—yes—a colleague-student of Bertrand Russell, one of the positivists. By the way, there’s an early and a late Wittgenstein, he underwent certain changes, but he’s considered to belong to the positivists. The later Wittgenstein, who is somewhat less positivist, if at all, argues—this is a famous argument of Wittgenstein called “following a rule”—and Wittgenstein claims that rules are a fiction. What does that mean? Think of a case where I give you a question on the psychometric exam. I tell you: one, two, three, four, five, six, blank line—what’s the next number? I assume all of you will say seven, right? Otherwise you’re not getting into the university, you’re an idiot. Fine? The correct answer is seven. The alien says: what do you mean? Why is the correct answer seven? Because there’s a rule: one, two, three, four, five, six, obviously the next one is seven. How do you know that’s the rule? Because you take the cases one, two, three, four, five, six, generalize them, create a general law from them by induction, and then apply it to the next case, right? In mathematical language, this rule is called f of n equals n. Right? In the first place sits the number one, in the second place sits the number two, in the third place sits the number three, and so on. There’s another law, for example: one, four, nine, sixteen, twenty-five, and so on. Here the law is f of n equals n squared. In the nth place sits n squared; in place one sits one squared; in place two sits two squared, which is four; in place three sits three squared, which is nine, and so on. Okay, this is basically the kind of law where we see the first examples, generalize it, create a general law from it, and now we also know what will be in the next place. Okay? Now I ask you the following question. Three, five, seven—what’s the next number? Nine. Everyone agrees? Nine? I say eleven. Primes.

[Speaker C] Right?

[Rabbi Michael Abraham] If the rule is odd numbers, then it’s three, five, seven, and the next is nine. But if the rule is prime numbers, then it’s three, five, seven—and nine isn’t prime, so eleven is next. Which rule is correct? Both are correct; I don’t know. From the examples, the first three examples, you could have derived this rule, or you could have thought the rule was different. There is no correct answer here. Now I’ll tell you: if it says three, five, seven, I can also write after that minus two point four, and that will also be correct. Why? Because I can show you that there’s a rule that gives you that. In the first place there’ll be three, in the second five, in the third seven, in the fourth minus two point four. It’s not even that hard to build such a rule—it’s four equations with four unknowns. I can build that rule quite simply. So basically this alien says to us: what do you want from me? I built a rule and the next number is minus two point four, and here is my rule. What advantage does your rule have over my rule? This is a rule and that is a rule. After all, what we’re given is only the first three examples. We derive the rule from the examples—that’s induction. Induction is not necessarily correct. You make one induction, he makes another induction. Who’s right? There is no right—what does right even mean? So in fact this alien who says minus two point four ought to be admitted to the university exactly like me. He’s not an idiot at all—he just thinks differently from me, that’s all. Look what a dream of postmodernism fulfilled itself here. Everyone is smart, everyone is right, no one is wrong. Right? Everything is right, everything is fine. We have this feeling that if we work with rules, it’s impossible to go wrong. Wittgenstein says that’s nonsense. Exactly like Mill’s argument against deduction. Wittgenstein is basically saying: how do you construct the rule itself? You construct the rule itself by induction. Say I teach children to count, in first grade, second, third—I teach one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, twenty, thirty, forty, one hundred, one hundred and one, and so on up to one thousand. Okay, I got to one thousand. Now I say to the child, okay, what’s the next number? And he says minus two point four. I don’t understand. One, two, three, four, five up to one thousand—what’s the next number? Minus two point four. What’s unclear here? I tell him, don’t you understand? It goes up by one each time, so the next number is one thousand and one. No, he says, what do you mean? The next number is minus two point four. I claim the rule is one, two, three up to one thousand, and from one thousand you drop to minus two point four and start counting again. So the next number is minus two point four, and now minus one point four, minus zero point four, keep going up by one all the time. That rule also fits the data you gave me. So what’s wrong with what I’m saying? It’s a perfectly fine rule. So who is right? No one is right. His mind is simply built differently from mine. In other words, what the psychometric exam tests is not whether you’re smart or stupid. What it tests is whether you think like me. When I admit students to the university, I want them to be able to understand what I’m saying to them, and for that their minds have to work like mine. If their minds work differently from mine, then maybe they’re just as smart as I am, but I won’t be able to teach them anything. Because when I teach them numbers up to one thousand, their next number will be minus two point four. They don’t understand as I do that after one thousand comes one thousand and one. Someone whose mind is built like mine—if I teach him from one to one thousand, he’ll also know how to keep counting. Fortunately, in basic things all human beings think similarly. And therefore we can study mathematics and physics and science and all those things, because our minds are similar. But only because of that. Anyone who thinks mathematics has an advantage over other fields is simply mistaken. You could not teach mathematics to someone whose mind was built differently from yours—nothing at all. You couldn’t teach him anything. When you teach him, you always teach him through generalization from examples. There’s this illusion as though mathematics is going from the general to the particulars. But in order to explain the rules to a person, you always use examples from which he makes an induction and creates the rule. You can’t teach him the rule directly. How will you explain the rule to him? You give him examples to show him what the rule is. But if from those examples he understands a different rule, then you failed to communicate the rule you wanted to communicate. In other words, the use of rules seems very tempting. But in fact it’s an illusion. We can’t really use rules. Even when we use rules, in deduction, what we’re really doing is induction. Only what? Only that we all make similar inductions, so it looks to us like deduction. And this argument of Wittgenstein has far-reaching implications not only for understanding mathematics, but for everything we think about in the universe, in all fields. Because it basically tells us: friends, there is no such thing in the world as pure logic. There is no such thing. Always, always, always, these rules involve some generalization from examples. The question is whether you understand the rule the way I understand the rule. In the end, in the end, there is always some uncertainty here. In the end, in the end, the certainty is never full. Even in mathematics, certainty is not full. Just as Mill argued against deduction. And I think this explains very well why, just as in English law, so too in the Talmud there is no trust in working with rules. And we all know this. There are rules in the Talmud, then the halakhic decisors come and want to apply the rules, and suddenly the arguments begin. Why? Which rule to apply, and how to apply it, and what the rule means, and whether it fits here or fits there—and therefore the use of rules doesn’t really help much. Until you’ve seen it through examples and understood what the rule really says, the rule by itself is worth nothing. And that’s what the Mishnah in Bava Kamma does. The Mishnah in Bava Kamma begins with four examples, and afterwards concludes with the rule. The Talmud says the rule is superfluous. I have the examples—who needs the rule? What do you mean, the rule is wonderful—they already gave you the induction, the best thing in the world, now all you have to do is work like a computer. You have a rule, extract the point from it. Here, I’ll give you one or two examples. The rule now is: anything that is your property, whose guarding is upon you, yes, and that is prone to cause damage—you have to pay. What about a guardian? A guardian who was negligent with an animal, and the animal caused damage—is he obligated to pay? No, because it isn’t his property. It isn’t his property. But the law is that he is obligated to pay. So what happened to the rule? Throw the rule in the trash; a rule is worth nothing.

[Speaker C] But every rule has an exception. Exactly.

[Rabbi Michael Abraham] That means this rule isn’t really a complete rule. It isn’t sweeping.

[Speaker C] It’s a pseudo-rule.

[Rabbi Michael Abraham] It’s not really a rule.

[Speaker C] There isn’t any rule in the world—every rule has an exception.

[Rabbi Michael Abraham] Right. And that basically means that the rule is only an approximation. It’s not really—the rule is not the truth. The rule is an approximation that usually helps us work, but you constantly have to be careful not to work only with the rule. Always, always accompany it with examples in order to see how to apply the rule correctly and how not to. That is exactly Wittgenstein’s lesson, and that is exactly what the Talmud does in Bava Kamma. It gives me the examples and gives me the rule. But the Talmud says: so why do I need the rule? Since you already gave me the examples, the examples are enough. The rule alone can’t work. But the examples alone can. Give me the examples, I’ll do the induction, derive the rule, and everything is fine. And then the Talmud begins to bring what this rule nevertheless comes to teach. Okay? But that’s why the Mishnah deals in cases; the Mishnah is casuistic. It doesn’t believe in rules. And even when it already brings rules, the Talmud doesn’t take that seriously, doesn’t understand why it’s needed. Okay? I’ll perhaps finish with a Talmud passage in Kiddushin. The Talmud says there: every positive commandment dependent on time—women are exempt, except for three or four. Then the Talmud asks: wait, but there’s also another exception—hakhel. So the Talmud says, yes, one does not derive from general rules, even where it says “except.” Now look: if it had said every positive commandment dependent on time, women are exempt, and I found an exception, I’d say fine, they stated the rule, but obviously here and there there are cases that don’t fit. I understand. But here it says “a place where it says except.” And you say every positive commandment dependent on time, women are exempt, except for A, B, C, and D. So they already counted the exceptions. Now I say there’s also E, and they say don’t make a whole issue out of it—so there’s also E. Why make a big deal out of everything? Meaning, even in a place where the formulation of the rule is the most precise formulation possible—it says “except.” All this and this, except A, B, C, and D—even there don’t get excited about it; there may also be E. I think that’s a wonderful expression of contempt for rules. Even where the rule looks like a precise rule. There’s a rule in the Talmud that Jewish law follows Abaye in Ya’al Kegam, right? In disputes between Abaye and Rava, Jewish law follows Rava except in six cases, Ya’al Kegam. You know that there are quite a few additional cases where the ruling follows Abaye. Maimonides here and there in the Mishneh Torah does not rule like Abaye according to some commentators.

[Speaker B] The Rabbi spoke about “its end was seized in punishment.”

[Rabbi Michael Abraham] In “its end was seized in punishment” there are those who rule like Abaye, right? In “do not form factions,” Maimonides rules like Abaye against Rava—two courts in one city. So what happened to the rule? Again, it’s a rule where it says “except.” In all places the Jewish law follows Rava except for Ya’al Kegam—yes, and except for a few other things. In short, don’t make too much out of rules—that’s the conclusion. From here we’ll continue next time.

[Speaker B] More power to you, thank you very much.

[Speaker C] Rabbi, can I ask just one quick question? I can’t hear. Can I ask just one quick question? Yes, yes. Basically you’re saying that in the lesson you said that all premises come from induction, right?

[Rabbi Michael Abraham] I didn’t say that all premises come from induction, but rather that general premises come from induction. Meaning, when I make a general law— all tables have four legs, or all people are mortal—that comes from induction. But there are premises that do not come from induction.

[Speaker C] And what do we do with them?

[Rabbi Michael Abraham] What do you mean, what do we do with them? For example, my intuition can also give me premises. Not induction from cases. I have an intuition that there is a God, okay? So my premise is that there is a God; naturally, if He commanded me, I have to fulfill what He commanded. How do I know there is a God? That’s not induction. It’s not a general law that comes from examples I saw.

[Speaker C] Like I said, in mathematics there is, for example, the Pythagorean theorem, and that’s a premise that doesn’t come from induction—that’s a premise that comes from a proof.

[Rabbi Michael Abraham] No, there’s no such thing as a premise that comes from a proof. The Pythagorean theorem is not a premise; the Pythagorean theorem is a theorem. The axioms are the premises. The axioms of geometry are the premises.

[Speaker C] The axioms certainly have proofs. What do you mean?

[Rabbi Michael Abraham] Can you prove to me that exactly one straight line passes between two points? If you can, you’re about to win a prize, a very valuable prize. Or that two parallel lines never meet—can you prove that? Those are Euclid’s axioms. There’s no way to prove them. We assume them, and from that we prove the Pythagorean theorem, for example.

[Speaker C] So why do we assume that if there’s no proof for it?

[Rabbi Michael Abraham] Exactly. That’s what the postmodernists argue. They say that what has no proof is unacceptable. And I say that’s not true. If it’s reasonable, then that’s perfectly fine by me. My intuition tells me so. But that’s not necessarily the result of induction; it’s just what my intuition tells me. Fine, and I’ve written books about intuition and about these problems of certainty. Anyone else?

[Speaker E] According to the Talmud, if not rules, then what do we have?

[Rabbi Michael Abraham] We have examples, and we make analogies—we compare one thing to another.

[Speaker E] So analogy is okay?

[Rabbi Michael Abraham] Of course.

[Speaker C] So is it true that Maimonides is a positivist, because Maimonides did…

[Rabbi Michael Abraham] He may be a bit more inclined toward systematic thinking, but he’s far from being a positivist. In the Mishneh Torah, you’ll find almost no rules. It’s all cases.

[Speaker C] In the Mishneh Torah I won’t find any rules?

[Rabbi Michael Abraham] You’ll find some, but only a few. Most of the laws are cases. Open it up, flip through it, and you’ll see.

[Speaker C] At the beginning of each set of laws, in every chapter, there’s a generalization there.

[Rabbi Michael Abraham] There are occasional generalizations here and there, but they won’t help you at all without the cases.

[Speaker G] Maimonides also doesn’t make analogies, right, Rabbi?

[Rabbi Michael Abraham] Of course he doesn’t make analogies. Maimonides gives halakhic rulings; he’s not in the business of argumentation. We do the argumentation. Maimonides gives the halakhic ruling. Where did he get it from? About that, you can debate. He doesn’t write where he got it from. Anyone else? Thank you very much. Okay then, Sabbath shalom.