Topics in Talmudic Logic, Lecture 1
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Table of Contents
- The structure of the course in Talmudic logic
- Aristotle and the conceptualization of logic
- Formal logic and content-based logic
- The value of conceptualization as a toolbox and mechanization
- The Brisker conceptual framework as a parallel example
- Ponevezh and Slabodka as models of a closed versus open method
- A division of forms of inference: analogy, induction, deduction
- Deduction as certainty because it contains no novelty
- Mathematics as precise because it adds no information
- Ranking inferences, information versus certainty, and science
- The tricky relation between analogy and induction, and a proposal to reduce them to one path
- Abduction as movement from cases to theory
- Applying mathematics to the world as positing a model that can be refuted
- A fortiori reasoning, “included in two hundred is one hundred,” and refutation in application
- Claim versus argument, validity versus truth, and the role of logic
- Mathematics as dealing with “if-then,” and the move to Talmudic thinking
- Talmudic logic as the mechanization of analogy and induction
Summary
General overview
The speaker presents a course on topics in Talmudic logic. In the first semester, the focus is on forms of inference such as a fortiori reasoning, binyan av, and refutations; in the second semester, the course moves to topics such as the relation between prohibition and positive commandment, and between general and particular, while trying to build a systematic toolbox similar to the conceptualization that took place in formal logic. He defines logic as the study of forms of inference, explains Aristotle’s contribution in conceptualizing valid forms of inference that do not depend on content, and develops a distinction between formal logic and content-based logic. He argues that deductive inference is necessary דווקא because it adds no new information, whereas analogy, induction, and abduction are forms of inference that add information but give up certainty. From this comes the motivation to mechanize and formally formulate even “soft” inferences, as, in his view, the logic of the hermeneutic methods does.
The structure of the course in Talmudic logic
The speaker says he was involved in writing a series of books on Talmudic logic, and he is choosing only some of the topics. He states that the first semester mainly deals with Talmudic and midrashic forms of inference, including a fortiori reasoning, binyan av from one verse and from two verses, refutations, and an attempt to build a general theory, and he notes that this part will be “a bit mathematical, but it’s not too bad.” For the second semester he sets topics such as the logical relation between prohibition and positive commandment, deontic logic, and general and particular in contexts such as defining sets.
Aristotle and the conceptualization of logic
The speaker attributes to Aristotle, in his Organon, the central conceptualization of logic as a general system of rules of inference, even though before Aristotle people were already intuitively drawing valid conclusions. He argues that Aristotle identified a uniform formula across a range of arguments, so that validity comes from the structure and not from the contents. He presents the argument pattern “Every X is Y; A is X; therefore A is Y” and defines it as a template that yields a valid argument no matter how the variables are filled in.
Formal logic and content-based logic
The speaker explains that formal logic is structural logic, in which validity depends only on form, and therefore can be applied in any context regardless of subject matter. He distinguishes this from content-based logic, in which necessary truths arise from the meanings of the concepts, such as “Every bachelor is unmarried,” so replacing the terms breaks the truth. He states that in standard usage “logic” refers mainly to formal logic, while content-based logic usually appears as a side note in introductory books.
The value of conceptualization as a toolbox and mechanization
The speaker says that conceptualization and formalization make possible a shortcut, a systematic classification of kinds of inference, and an understanding of dependencies between rules, and they also make mechanization possible by setting rules up for mechanical application. He argues that computers could not exist without an understanding of rules that are not content-dependent. He objects to formalization for its own sake, and argues that formalization has value only if it makes possible operations that cannot be carried out without it, similar to the use of mathematics in science. As a test for Talmudic logic, he sets the ability to extract tools that can be added to the toolbox so they can be applied in other contexts as well.
The Brisker conceptual framework as a parallel example
The speaker presents Rabbi Chaim as someone who did not “invent” the style of yeshiva analysis, but rather conceptualized a set of distinctions and universal analytical tools, similar to Aristotle in logic. As an example, he mentions the Pnei Yehoshua, who at times produced clearly “Brisker” analysis, and he emphasizes the historical difficulty of attributing ideas to their owners when those ideas also appear in earlier writers. He portrays conceptualization as creating code names and fixed tools like object and person, sign and cause, and two laws, which allow one to solve difficulties by “paging through” a toolbox.
Ponevezh and Slabodka as models of a closed versus open method
The speaker describes a sociological and scholarly difference in which Ponevezh is perceived as structured and mechanistic, with a “locked” set of tools, while Slabodka is open and less structured. He argues that the closed structure allows a person who is not a genius to apply the same patterns to any topic and give lectures whose shape is predictable in advance, and therefore “most roshei yeshiva came out of Ponevezh.” He tells a story about a lecturer who was a student of Rav Shmuel and taught tractate Sukkah in a way that almost overlapped with the published “Shiurei Rav Shmuel,” and he interprets this as the success of a method that reproduces moves from a toolbox.
A division of forms of inference: analogy, induction, deduction
The speaker divides inferences into analogy, induction, and deduction according to “level of integration” or level of generality. He defines induction as movement from particular to general, deduction as movement from a general rule to a particular included within it, and analogy as movement from particular to particular or from general to general on the same level. He states that deduction is a necessary inference, in the sense that if the premises are true then the conclusion must be true, whereas analogy and induction are non-necessary inferences that add information and therefore carry risk.
Deduction as certainty because it contains no novelty
The speaker explains that the necessity of deduction comes from the fact that the conclusion contains no information beyond what is already included in the premises, and therefore anyone who accepts the premises is “forced” to accept the conclusion. He uses illustrations such as “All human beings are mortal; Socrates is a human being; therefore Socrates is mortal,” and his claim that the conclusion is already embedded in the general premise, which is just a shorthand for a list of particulars. He argues that every valid argument in this sense “begs the question,” and mentions “And Abraham went” as a joke illustrating that the issue is not validity but triviality, and the fact that the conclusion is already contained in the premises.
Mathematics as precise because it adds no information
The speaker links deduction to mathematics and argues that mathematics is certain and precise because it does not add new information, but rather “reveals” information that was already latent in the premises, even if genius is required to extract it. He uses the hot-air-balloon joke to say that mathematics is precise but “doesn’t help us at all” in the sense that it adds no new information, while qualifying that it is still practically useful because human beings cannot extract all the consequences on their own. He presents “the emptiness of the analytic” as the claim that logic and mathematics, both in form and in content-truths like “every bachelor is unmarried,” do not produce new information.
Ranking inferences, information versus certainty, and science
The speaker suggests that degree of validity stands in inverse relation to the amount of information an inference adds, and formulates a kind of “logical uncertainty principle” in which “the product of the amount of information and the degree of certainty” is constant. He states that science is based on non-deductive inferences and therefore its conclusions are not certain, whereas mathematics does not get replaced and is not refutable if the proof is valid. He presents the idea that someone who demands absolute certainty will not be able to accumulate information, and from this he explains why philosophers arrive at skepticism if they reject analogy and induction.
The tricky relation between analogy and induction, and a proposal to reduce them to one path
The speaker argues that the distinction between analogy and induction is not sharp, because analogy can be seen as hidden induction followed by deduction, and induction can be seen as a collection of analogies. He proposes a model in which “there are not three forms of inference” but really one analogy that breaks down into two stages: rising to a generalization by induction and descending to a particular by deduction. He notes that this connects to Talmudic questions such as the relation between binyan av from one verse and from two verses, and he hints at the possibility of a dispute among medieval authorities (Rishonim) in understanding the relation between analogy and generalization.
Abduction as movement from cases to theory
The speaker adds, “as a side note,” abduction, and attributes the concept to Peirce. He distinguishes between induction, which generates a descriptive general law, and abduction, which generates an explanatory theory with theoretical entities like gravitational force, gravitons, and electrons, which are not directly observed but only through their consequences. He states that science does not make do with phenomenological description but aims for explanation, and the move from cases to theory is more speculative than simple generalization.
Applying mathematics to the world as positing a model that can be refuted
The speaker argues that mathematics does not make claims about the world and therefore cannot be refuted by experiment; what can be refuted is the claim that the world fits a certain mathematical model. He presents the example of vector addition of forces to show that a failed experiment does not refute “ten plus ten equals twenty,” but rather refutes the assumption that the physical situation is described by arithmetic addition. He also uses Euclidean geometry to say that the sum of the angles in a triangle is not necessarily 180 degrees in a curved world, and explains that the question of which geometry describes the world is a question in physics, not in mathematics.
A fortiori reasoning, “included in two hundred is one hundred,” and refutation in application
The speaker mentions Chaïm Perelman and a story about “the Vandervelde law” in Belgium, which forbade selling workers “more than two liters of wine,” and the buyer’s claim that ten liters are specifically permitted because “the law forbids two, not ten.” He describes how the judge accepted the buyer’s argument and explained that ten liters can be seen as an investment rather than immediate consumption, and therefore application depends on purpose and context, not only on formalism. He states that the a fortiori reasoning of “included in two hundred is one hundred” can be refuted in practice, and connects this to the discussion of “we do not administer punishments based on logical derivation,” to claims by later authorities (Acharonim) about an a fortiori inference that admits no refutation, and to examples such as “he passes some of his children to Molech” versus “all his children to Molech,” along with other examples in which it appears that “there is no refutation,” but in practice the application assumptions can be challenged.
Claim versus argument, validity versus truth, and the role of logic
The speaker defines a claim as a sentence that can receive a truth value of true or false, and an argument as a structure that derives a conclusion from premises, so that an argument is judged as valid or invalid, not as true or false. He demonstrates that an argument can be valid even if all the claims in it are false, and an argument can be invalid even if all the claims in it are true, but it cannot be valid with true premises and a false conclusion. He states that this is the connection that justifies logic: it provides a toolbox for deriving true conclusions from true premises by means of valid patterns of inference.
Mathematics as dealing with “if-then,” and the move to Talmudic thinking
The speaker states that a mathematician deals only with the link between premises and conclusion, not with the truth of the premises, and therefore the answer depends on the axioms given. He explains that someone working in physics, law, or other fields posits premises as given for external reasons and then uses logical and mathematical tools to derive conclusions. He concludes by saying that information is added specifically through analogies and inductions, but since they are not certain, a tension is created that leads either to skepticism or to the development of non-deductive logics that try to refine soft inferences.
Talmudic logic as the mechanization of analogy and induction
The speaker argues that the aspiration of what will be taught during the semester is “to mechanize analogies and inductions” and to try to give them a formal formulation, presenting this as “squaring the circle.” He connects this to the claim that this is exactly what lies at the foundation of “the logic of the hermeneutic methods,” and concludes that the challenge is to propose a formal path that will allow systematic use of a Talmudic toolbox similar to what Aristotle did for formal logic.
Full Transcript
[Rabbi Michael Abraham] Okay, so the topic is defined as topics in Talmudic logic, and there are all kinds of topics. I was actually involved in writing a series of books on this subject, and I’ll simply choose a few of the topics; we won’t be able to cover all of them. I’ll start in the first semester, after a bit of an introduction about logic and what logic is and what it means for us, we’ll get into the Talmudic forms of inference, which are really not only Talmudic but maybe even midrashic: a fortiori reasoning, binyan av from one verse, from two verses, refutations, and some attempt to build a more general theory. I did this here a few years ago. It’ll be a bit mathematical, but it’s not too bad, no need to be frightened by that. And with that I hope to finish the first semester. In the second semester we’ll get into other topics, like the relation between prohibition and positive commandment, meaning the logical aspects of that issue, maybe deontic logic and things of that kind, general and particular, maybe the definition of sets. There are various somewhat smaller topics. The topic of the first semester will mainly be forms of inference. Okay, but I want to start with a bit of an introduction about what logic is. If there’s anyone here who studied it and knows, maybe knows more than I do, then I already ask your forgiveness, but just to synchronize things I’ll still give this introduction. Okay, the founder, or the first and main conceptualizer of this field of logic, is considered to be Aristotle in his book Organon. But logic deals with forms of inference: if every X is Y and A is X, then A is Y, and so on, all kinds of forms of inference of that sort. But you have to understand that Aristotle didn’t invent logic, not only in the sense that I’m not talking about the question of whether logic is a law in the object or a law in the person, meaning whether there really is logic in the world or whether it’s only our way of looking at things. Rather, even if we look at our way of looking at things, it’s clear that even before Aristotle everyone knew how to infer the conclusion that if all tables are brown and this is a table, then it’s brown. Aristotle didn’t innovate that; it’s clear to me anyway—I didn’t ask anyone who was alive then, but I assume it was clear before Aristotle too. What Aristotle did was basically conceptualize the logical rules. In other words, they were being used intuitively before him as well, but he was the one who—again, this is all simplistic, there was stuff before him too—but let’s say as an illustration, or as a major stage in the process, he was the one who understood that there is some general system of rules here. I’ll give you an example. If I said earlier: if all tables are brown and this thing is a table, then this thing too is brown. Everyone knows that. All human beings are mortal, Socrates is a human being, conclusion: Socrates is mortal. Right, that’s the worn-out example everyone always brings in this context. Or whatever you want: all frogs have wings, this lectern is a frog, therefore this lectern has wings. That too is a valid inference. And people did that before him too. But Aristotle noticed that in all these contexts there is some uniform formula, a uniform form. And that is basically the conceptualization he made of logical inference. In other words, he distilled from the forms of thinking that were used even before him some set of abstract rules that appear in all contexts. Why do they appear in all contexts? Because he noticed that these inferences aren’t about lecterns or frogs or wings. It’s simply a formal structure—form in the foreign term means shape, right?—a formal structure that says that what you plug into these variables doesn’t matter; in the end the argument will always be valid. When I say a structure like this: every X is Y, A is X, conclusion: A is Y. Right? Those are the two premises and that is the conclusion. That’s not an argument, that’s what’s called an argument form. An argument form is like a number pattern that you learn in school; a number pattern, you know, if you put something like this, this plus this equals—equals eight, that’s a sentence pattern. Sorry, that’s a sentence pattern. A number pattern is this, okay? Let’s say this plus three. Whatever you fill in here will give you some number, okay. That’s a number pattern. If you put something like this in it, then it becomes a sentence pattern. Put one here and four here and it comes out, gives you a true sentence, a true proposition. Okay. So a pattern is something that, when you fill it with certain content, turns into a proposition. That’s a sentence pattern. A number pattern is something that if you fill it with certain content turns into a number. This thing is an argument pattern. When you fill in X, Y, and A with certain things—for example, A is Socrates, X is human beings or a human being, Y is mortal. Every X—everyone who is a human being is mortal. Socrates is a human being. Conclusion: Socrates is mortal. So after we filled the variables A, X, and Y with certain content, we got a valid argument. Valid means the conclusion follows from the premises. Okay. But we could just as well fill it in with tables, brownness, and this object. Let’s say this is A. Okay. Everything that is a table is brown, this object is a table, conclusion: this object is brown. And any filling you put into those variables will give you a valid argument. So what does that mean? It means that this structure is a valid structure regardless of the contents I put into it. Its validity comes from the structure, not from the contents involved. Okay. That’s why it’s called formal logic, structural logic. Form means shape, right? Formal logic is structural logic. That means the validity of the argument is validity that comes from its form, not from its contents. There are parts of logic—this is how it’s usually taught, yes, when people teach logic, mainly in philosophy, because logic is taught both in mathematics and in philosophy. When they talk in philosophy they also talk about content-based logic. What is content-based logic? For example, when you say: every bachelor is unmarried. Now that too is a necessary statement, right? Every bachelor is unmarried. But here the validity—or the truth—of this statement, that tautology, the necessary truth of this statement, comes from the concepts inside it, not from its form. If you replace bachelor and unmarried with something else—every lectern is not made of wood—that’s not true. Okay. Meaning, it’s not true that a structure like this of a sentence or an argument, whatever you place into the variables, will give you a true proposition or a valid argument. So this is an argument—or rather a claim—that is indeed necessarily true, it is tautological, but its tautological character or its truth comes from the contents involved in it, and therefore it is called content-based logic, not formal logic. Formal logic does not depend on content; it depends on form, on structure, on the form. Okay. Formal logic. But content-based logic is logic that depends on content. What is generally accepted, when people speak about logic, is formal logic. Content-based logic—in introductory books they always say, yes, there is also content-based logic, and there are all kinds of philosophers who talk about content-based logic—but in formal work, mathematicians, when they really do logic, almost never deal with that, I think, at least as far as I know. They deal with formal logic. Let’s get a little deeper into this. What does this thing actually mean? This thing actually means—maybe before we get into it—what is the significance of Aristotle’s conceptualization? In other words, what exactly did Aristotle do, and what is the significance of what he did? Why is what he did important? After all, even before him people knew how to do that kind of analysis or present such inferences. But Aristotle noticed that there is, as I said earlier, some kind of general structural tool here that doesn’t depend on the contents, and therefore it can be applied in many contexts with the same form of analysis. You don’t have to check it again every time. We already know—it’s always valid. No matter what you put in, no matter how you fill the variables. Okay. So first of all, that can sometimes be a shortcut, because it means I can draw a conclusion that there is some pattern which is a valid argument pattern, and now whatever we put inside it will be fine. You don’t have to check every time whether it’s valid or not. So it saves time. That is, it’s useful for brevity. But it’s more than that. Once you conceptualize and formalize structures like these, then first of all you can classify them. You can understand what kinds there are and what kinds there are not. Which depends on which, and which does not depend on which. For which one can suggest alternatives and for which one cannot. You can mechanize it, for example. The whole subject of computers could not have existed without Aristotle’s conceptualization. We could have been world champions in logic; you still couldn’t have built a computer if we didn’t understand that there is some formal set of rules here, rules that don’t depend on content and that can somehow be symbolized. They can be written in a mechanical way for mechanical implementation by a golem, yes, by a computer. If we just knew how to do everything, that wouldn’t be enough. In other words, the conceptualization is very important. I’ll maybe give an example of this. There was once, a few years ago, a conference here on the Pnei Yehoshua. So I spoke there about buds of Brisker thinking in the Pnei Yehoshua. You can see in several topics that the Pnei Yehoshua did really fully Brisker analysis. Not just buds—it’s completely there. It’s buds only in the sense that it appears in only a few places in his writings. But in those places it’s absolutely unmistakable. So what does that mean? That the Pnei Yehoshua invented what we today call yeshiva-style learning, the yeshiva analytical style? Not Rabbi Chaim? No, it doesn’t mean that. I gave an introduction there before I brought the examples, and I said there’s a built-in problem when we deal with the history of ideas. The development and history of ideas—it’s always the question: where did an idea come from, who influenced it, who shaped it, and so on. It’s very hard to put your finger on a yes-or-no answer. Who shaped it and who didn’t, who influenced it and who didn’t, or to whom the idea belongs. Who owns it? To whom should we attribute the idea? Is this Brisker thinking or Pnei Yehoshua thinking? How are we supposed to relate to it? It’s very difficult. There’s no doubt that Rabbi Chaim invented nothing. Every type of consideration Rabbi Chaim made, I’m sure—I didn’t check, but it’s clear to me—exists in one form or another among his predecessors too. What Rabbi Chaim did was the conceptualization. Rabbi Chaim basically said: look, there is some set of distinctions or forms of analysis that are universal. They appear in a great many topics; almost every topic can potentially be analyzed using this set of tools. And then he says: look, there are object and person, sign and cause, two laws, all the standard yeshiva analytical forms. But—you see, I already said it through names, code names. For every kind of analysis we already have code names, we know. That is exactly the conceptualization Aristotle did for logic. What Rabbi Chaim did was simply conceptualize a set of rules, formulate them as rules that stand on their own, even though before him too people did these kinds of analyses, like the Pnei Yehoshua I mentioned earlier. And what does that mean? As I said, what Aristotle helped us with when he conceptualized logic. In other words, maybe now we have a toolbox that contains a defined set of tools, so that if I come to some topic and say I get stuck on a difficulty, I can now page through the toolbox, look for the right tool, and use it to solve the problem. Many times I would say this at the opening of the yearly lectures when I was teaching classes that learned Talmud with me, or in yeshiva when I taught in Yeruham. I would give them an introduction, kind of half-sociological, about my cheerful period in Bnei Brak, where there was a very interesting phenomenon. There were two major important yeshivot, Slabodka and Ponevezh. Today the map is a bit different, I think; first of all there are two Ponevezhes. But Slabodka and Ponevezh. And there were two differences between them. One difference was that most roshei yeshiva came out of Ponevezh; very few came out of Slabodka. The second difference was that Ponevezh was much more square, much more rigid. In Slabodka they learned in a more balabatish way; in Ponevezh they were kind of formalist, Brisker types. Not completely—it was more Rav Shmuel, but never mind, those are nuances already. In short, the Ponevezh style of analysis is more mechanistic. Meaning there is a set of rules—by the way, a very locked one. There is a certain kind of thing a Ponevezher simply is not willing to hear. He won’t even consider it; his ears just close when he hears something that doesn’t fit the pattern he knows. Among the Slabodka people—what bothered me, by the way—I spent some time in the Chazon Ish kollel, which is usually from the Ponevezh school. I sat there a few months, so generally I learned in the Ponevezh school. And there what bothered me was that anarchy. There were no analytical tools at all, everyone says, seems to me like this, seems to me like that, here, there—not structured, like you see in the Chazon Ish. In the Chazon Ish you see he is very unstructured, okay? And after that I would say in those introductions that there is a connection between these two characteristics. Yes, one characteristic is that in Ponevezh it’s structured, in Slabodka it’s open; the second characteristic is that in Ponevezh almost all the roshei yeshiva came out, and in Slabodka not. And the point is that the moment you receive a closed set of rules, you have a toolbox. Once you acquire it after a number of years of skill, you can apply it to any topic relatively easily. You don’t have to be a genius for that. Again, of course talent matters, but you will be able to give a general lecture on any topic. You have a set of tools and you… I would go hear lectures during intersession breaks; roshei yeshiva would always come and give lectures in various places, and I’d go for my own enjoyment to all kinds of such lectures. After a few years, when a lecture would start I could tell you how it would end. What he would ask, what the practical difference would be, and where he’d arrive at the end. Truly. Truly. And it’s not because I’m some great genius, but because once you catch the shtick, the form of thinking, then you know what he’s going to say. There won’t be many surprises there. And therefore—I’ll maybe give you another example, since I’m already talking about this—my Talmud lecturer, I studied in Netivot Olam there, which is a yeshiva for ba’alei teshuva, and my lecturer was a student of Rav Shmuel. We learned tractate Sukkah with him one year, and in the middle of that year the first volume of Shiurei Rav Shmuel on tractate Sukkah came out. So I bought the book, opened it, and it was embarrassing. Because it was simply Rav Shmuel’s lectures. We had heard Rav Shmuel’s lectures from him, and he didn’t say—he didn’t say that he was taking them from Rav Shmuel; he presented them as if these were his own moves. It was awkward, we were debating, yes, no, but I said I have to ask. So I went up to him and said: Rabbi, did you know that a book of Shiurei Rav Shmuel on tractate Sukkah came out? So he said to me: Yes, interesting, I didn’t know. I said, look, the truth is that when I look in the book carefully—it was a bit embarrassing—the lectures are really one to one. What do you say? Unbelievable. He had to say, bring me the book, he wanted to see the book. Amazing. He says, I’ll tell you: I never learned tractate Sukkah with Rav Shmuel. I never heard from him lectures on tractate Sukkah at all. I believe him; he wasn’t fooling me. He was so polished in the method that when you apply it to any topic, you yourself will say what Rav Shmuel said. Meaning if you are sufficiently polished, you will yourself say the lecture that Rav Shmuel said, more or less. There can of course be differences, but broadly it will be the same lecture. Because it’s a pattern of thought. Yes, it’s exaggerated of course. I’m presenting things in a slightly exaggerated way, but it really was very similar, very similar. And by the way, he was overjoyed when he discovered this, because it means he succeeded. Meaning, the model is to be a duplicate of Rav Shmuel. That’s the Ponevezh model. Meaning if you manage to be another Rav Shmuel, that is the peak of success. And that is really the demonstration of what I described earlier. Now in such a situation, obviously most roshei yeshiva will come from Ponevezh, because even if you are a mediocre person you can be a rosh yeshiva; you’ll give an excellent general lecture on any topic, even if you are mediocre. To be the Chazon Ish you have to be a genius. You have to be a genius because you invent for yourself what you say. You are not going by patterns with a well-defined toolbox. Now of course in Ponevezh too there are talents, maybe even more, I don’t know, I’m not getting into judgments now, and talent matters. I’m presenting things in an extreme way, but there’s a great deal of truth in it. Meaning there is something here such that if you acquire the tools, you can already work with them. And it’s not because you are smarter—on the contrary, even if you are less smart, you have a toolbox at your service and you can use it. That is exactly what Aristotle did for logic. Meaning this is the… What?
[Speaker B] Mental open-source code.
[Rabbi Michael Abraham] Why is that specifically connected to open source?
[Speaker B] You break into every code, you’re not locked inside the language that…
[Rabbi Michael Abraham] No, you are locked inside a language; you’re just making the code open. The code is written in a defined language. You don’t write a language; you write code in a given language. Okay, never mind. In any case, Aristotle’s conceptualization basically gave us some toolbox that previously had to be used intuitively. Now that we have the toolbox, now you don’t have to be Aristotle anymore for this. I have a toolbox, I know it, and in any topic—not necessarily Talmudic, also philosophical, conceptual, whatever—I can already use this system of tools. I already know. I don’t need to do the calculation again: what’s a necessary condition and a sufficient condition, and if A then B, then if not B then not A, and I already know how to draw the conclusions. I’ll turn off this destructive device. Wait, actually I’m not allowed to turn it off, it’s recording. Someone has to bring me back. Okay, so in any case, that is the meaning of conceptualization. Now why am I saying this? Because in the end, when I want to conceptualize logical tools in Talmudic thinking, I want to do exactly the same thing. There are a lot of people who really, really enjoyed—it was in the first issues of Higgayon, which later became Bedad. There were many articles there that drove me crazy. Why? Because there are people who have… I call them formalophiles. They just die for formalization. Meaning, you take some topic, you write it in the form of X and Y, and wonderful, you’ve reached the fulfillment of your life’s ambitions. Nobody cares whether you write it in English or in mathematics or in Turkish. If the formalization helps you, then yes. Sometimes you do a formalization that helps you do things that without the formalization you couldn’t do. Fine? Like formalization in science. If you describe things mathematically, scientific laws in mathematics, you haven’t done anything—only if the mathematics gives you the ability to use it in places where without the mathematics you couldn’t do that, then formalization has value. Fine? Here too it’s the same, because the goal is to set up some toolbox for use, so that you can also use it in other contexts. So when we get to the tools of Talmudic logic as well, in the end that is how it will be judged. Meaning I want to see that I really extracted tools here that were added to my toolbox, and now I can use them. I don’t need to think creatively each time, intuitively, how to solve the problem or how to analyze the problem. I already have a toolbox. Of course it’s not complete, but each person has to add more tools to the toolbox and make it available for public use. Okay. All right, so that’s in principle. When we talk about inferences—I’m going back to formal logic or the logic of inferences—when we talk about inferences, it is customary to divide them into three types: analogy, induction, and deduction. The differences between them have to do with the level of integration or the level of generalization. Meaning, induction is movement from particulars to a general rule, from particular to general. Deduction is movement from a general rule to a particular or particulars. Analogy is movement from particular to particular or from general to general. Fine? That stays on the same level of integration; it stays on the same level, the level doesn’t change. So for example, if I say all donkeys are mortal, therefore all horses are mortal as well—what is that? Analogy, right? Even though I’m making an analogy between groups, between rules, but it’s from general to general, not from particular to general and not from general to particular, but it stays on the same level. So that’s an analogy. Just as if this donkey is mortal then that donkey is mortal too—that’s also analogy, right? Because it’s from particular to particular, it stays on the same level. What happens if this donkey is mortal, therefore all horses are mortal—what is that? You tried everything except the right answer.
[Speaker C] Is that because you’re saying the donkey is a species within the family?
[Rabbi Michael Abraham] No, I’m not saying that. It’s analogy. You move from a particular to a general, but it’s not a general that contains the particular. You’re simply making an analogy from this particular to the collection of particulars that make up the general category of horses, a collection of analogies, that’s all. But in principle it’s an analogy. Generalization is if this donkey is mortal—induction—if this donkey is mortal, then all donkeys are mortal. That’s induction. Why? Because I take a particular and infer from it a conclusion about an entire class of which this particular is a part, it belongs to it. Fine? By contrast, if I go from this particular to another class that it is not itself part of, that’s not induction, it’s analogy in principle. What? Yes, an analogy between a donkey and a horse, only applied to all horses, each one of the horses. By the way, analogy does not always have to be correct. Fine? Horse isn’t written with a chet and donkey is—okay. Fine. There are differences between donkeys and horses too. Just yesterday someone told me that donkeys are considered more intelligent animals than horses. By the way, when you speak about a person and call him a donkey—relative to human beings, donkey is an insult; relative to horses, donkey is praise. Okay, in any case, deduction is movement from a general rule to a particular, but again, to a particular that is a particular within the general rule being discussed. If all human beings are mortal and Socrates is a human being, he is one of that class, then Socrates is mortal. That is movement from a general rule to a particular that is inside it, that belongs to it, it is an element within it. Therefore this is deduction. Okay. What is the relation between these three things? Usually—not usually, it is accepted to say—that deduction is a necessary inference, right? If all human beings are mortal, Socrates is a human being, conclusion: Socrates is mortal. A necessary inference. What does necessary mean? If the premises are true, then necessarily the conclusion is true too. Okay? That does not mean the conclusion is true—only that if the premises are true, then the conclusion is true. Analogy and induction are non-necessary inferences. Right? I make a comparison, but I can also be mistaken; not every comparison is correct. It may be right, but there is nothing necessary about it. Fine? On the other hand, it’s probably not just a shot in the dark. Right? It’s not random. There are analogies that make more sense and analogies that make less sense. Okay? Without getting into how we know that. I’ll talk about that later, because it relates directly to our topic. How do we know that? Right now I’m not getting into it yet. So I’m saying: analogy too is a kind of inference, but it’s a softer inference. It’s soft in the sense that it isn’t necessary. Fine? Induction too, right? A soft inference. It may be correct, it may be incorrect, but it makes sense, meaning it is a mental tool that we use. If I had to rank the validity of these inferences—analogy, induction, and deduction—how would you rank them?
[Speaker D] Deduction first.
[Rabbi Michael Abraham] Deduction is the strongest, it’s certain, right? What about induction and analogy? Analogy is second and induction is third—does everyone agree? Why?
[Speaker D] Induction is second. Why? Because when you have—it’s more reasonable—if you learn from several particulars that all behave the same way, then it’s reasonable to infer from that to everything, to the particular…
[Rabbi Michael Abraham] It doesn’t matter—even from one particular I learn to a general rule, not from several particulars. That’s not essential. You can do induction. Again, induction here is not what they teach in mathematics under the name induction, of course, right? The thing called induction in mathematics is deduction. Meaning, the induction I’m calling scientific induction—that’s generalization. I saw one example, I saw two examples, so I assume it’s true for the whole group. Okay? What’s called generalization.
[Speaker E] Why would you put induction before analogy? I don’t…
[Rabbi Michael Abraham] You could say that analogy is a particular case of induction.
[Speaker E] Why? Because you compare the particular to part of the class and then you say…
[Rabbi Michael Abraham] Okay, I’ll get to that in just a moment. Look, in another second I’ll get there. Look, in principle the point is this. In deduction, let me ask a different question: why is deduction necessary? Imagine an alien arrives from the Little Prince’s planet, suddenly drops down here, together with the baobab from over there. He comes down here, and I say to him, listen, all human beings… I’m introducing him to the creatures walking around here on the globe. I say to him: look, human beings are mortal. All of them. And Socrates—you see him, that fellow over there—he’s also a human being. So you should know that he is mortal. He looks at us: why? Who says? What do you mean? You agree that he’s human, right? And that all human beings are mortal—you agree with that too? Yes, yes, of course. Well then, Socrates is mortal. Why? Who told you? He accepts the premises but doesn’t accept the conclusion. What do you do with someone like that besides hospitalizing him? How can you explain his mistake to him, if at all? Why does he have to accept the conclusion if the premises are true?
[Speaker G] Socrates is within the set of human beings, part of…
[Rabbi Michael Abraham] He does see that—he said Socrates is one of the human beings, what do you mean? He admitted the premise. “There’s no other possibility” is the slogan. He says: here, there is another possibility, I just don’t… everything’s fine, I just don’t see that he’s mortal. Formal logic is, once again, a declaration. What do you say to this guy? That he isn’t rational.
[Speaker H] He doesn’t have your rationality.
[Rabbi Michael Abraham] That’s below the belt. That’s attacking him, not making an argument. The question is: what argument do you make? Look, let’s ask it differently: by virtue of what am I convinced this is true? Forget this fellow. I’m convinced it’s true because the conclusion contains no information beyond what was already in the premises. Right? If you look at the information in the premises, what information is there? When I say all human beings are mortal, let’s translate—dictionary: Socrates is mortal, Yankele is mortal, Dudu is mortal, Ahmad is mortal—everyone, right? Socrates in particular is also one of them. In one general sentence I say all human beings are mortal, and that’s just a way of expressing it. In principle, what I really said here was a whole collection of many claims, one of which was that Socrates is mortal. So it’s no wonder that the conclusion, “Socrates is mortal,” follows necessarily from the premises. Anyone who accepts the premise has to accept the conclusion, because the conclusion is inside the premise. The information in the conclusion is simply part of the information in the premises. In other words, the validity of deduction stems from the fact that there is no novelty whatsoever in deduction. There is no information in it beyond the information we had when we set out, if I think of this as a path of inference, as walking along some road. You start from premises and walk toward the conclusion. When I set out, I knew the premises. Okay? Now I walk to the conclusion. But when I walked to the conclusion, like Winnie-the-Pooh, right, who discovered that the tracks he found were his own—what happens is that when you reach the conclusion, you’re actually discovering information that was already in your hands when you set out. Nothing new has occurred to you. It’s like that yeshiva joke about Abraham and Jacob, right? Who said every Jew has to walk around with a hat? It says, “And Abraham went.” Well, a Jew like him obviously didn’t go around without a hat, right? So if Abraham went with a hat, then each of us, his faithful sons, also has to walk in his ways—to go with a hat. Which is what we wanted to prove. What do you say about that proof?
[Speaker I] That it’s not necessary that it…
[Rabbi Michael Abraham] Why not? A Jew like him didn’t go without a hat—come on. First of all, that argument is a valid argument. That argument is a valid argument for someone who
[Speaker I] accepts the premise.
[Rabbi Michael Abraham] Yes, every argument is like that. An argument is always for someone who accepts its premises, right? He can argue with the premise, fine. That’s always true; with every argument that’s true. There’s no problem with this argument, just so you know. No problem at all—a very good argument. The problem you feel here is what’s called begging the question. Begging the question means you want to prove a conclusion, and you take that very conclusion as one of the premises on which you build your proof. So in fact you gained nothing, because you already assumed the conclusion inside the premises. So you didn’t need to walk the road to get to the conclusion, because it was already in your hands when you set out, right? That’s the problem people feel in this argument. But notice: every logical argument is like this. Every logical argument is like this. When I say all human beings are mortal, Socrates is a human being, therefore Socrates is mortal—they say this in every philosophy or logic class around the globe, and nobody laughs. But when it’s about our father Abraham, everyone laughs. Such antisemites. Socrates—everything’s fine. But Abraham our father, everyone laughs. Why? Deduction. No, both are deduction.
[Speaker J] But we accept the premise…
[Rabbi Michael Abraham] Fine, you don’t accept the premise there either—you could accept the premise there too, so what? But when I’m demonstrating an inference to you, the inference doesn’t depend on the premises. An inference is always based on premises. If you accept the premises, this is the conclusion. That’s true in every context. Or in other words, a valid logical argument is a logical argument whose conclusion adds nothing new. It’s an argument that begs the question. Every valid logical argument begs the question. Every valid logical argument, without exception. If it didn’t beg the question, it wouldn’t be valid. Because its validity comes from the fact that the conclusion is actually already located or embedded within the premises; that’s why it’s valid. That’s why you have to accept the conclusion if you accepted the premises. But that means the argument actually doesn’t teach you anything new. Right? That’s the joke about the hot-air balloon that I wrote at the beginning of Two Wagons, my book. I said that people—right, they usually tell this joke in the mathematics faculty, and later I heard it in other places too—that two people got lost in a hot-air balloon. For several days they didn’t know where they were. They see someone plowing a field below. They say to him: tell us, can you tell us where we are? “Above my field.” So the fellow up in the hot-air balloon says to his friend: that guy down there is definitely a mathematician. Why? Two reasons. A: what he said is completely precise. And B: it doesn’t help us at all. In other words, that’s exactly the point—it’s actually the very definition of mathematics. Why is mathematics certain, necessary, and completely precise? Because it doesn’t help us at all. Or, “doesn’t help us at all” not in the practical sense—it’s not that it isn’t practically useful—but it doesn’t help us at all in the sense that, in a logical inference or a mathematical proof, what is a mathematical proof? A mathematical proof is that I give you a set of premises and from them I can derive the conclusion. But if I derive the conclusion by logical means from the premises, that means it was already in them. Otherwise it wouldn’t be mathematics; it would be analogy or something else. It wouldn’t be necessary inference. The necessity of the inference comes from the fact that the conclusion is actually already inside the premises, right? And that means—why is mathematics certain and completely precise? Because it doesn’t help us at all. Because it doesn’t add information to us. In other words, “doesn’t help us at all” is an imprecise expression. Rather, it doesn’t add new information. It just clarifies for us, better, the information that is already in our possession. That’s the point. Think about geometry. We start from the axioms of geometry—Euclidean geometry, say. There is a set of premises, and from them we derive all sorts of propositions, conclusions, theorems. Okay? Now, if we can prove them on the basis of the premises, then that means these propositions are somehow embedded inside the premises. Right? Because otherwise it wouldn’t be a proof; it wouldn’t be mathematics. Mathematics means that anyone who accepts the premises has to accept the conclusion. I have a proof. That’s what a proof in mathematics means, okay? So that actually means you didn’t tell me anything new. All of geometry contains nothing new—take the four axioms and that’s it.
[Speaker F] But there are conjectures that have no solution.
[Rabbi Michael Abraham] We’re not talking about conjectures. Conjectures aren’t mathematics. Conjectures are a trigger for mathematical work. Mathematics is proved things. But maybe there’s novelty there for everyone? When you solve them. Once they’re solved, there’s already a proof; then it stops being a conjecture. The claim is that what mathematical proof does is expose some piece of information that was already inside the bulk of the information I had in my hands; I just wasn’t aware of it. And therefore, when I say it doesn’t help us at all, that’s nonsense, because none of… I assume, at least I, would not have arrived at all of geometry without learning it from someone. I assume most people sitting here are the same—allow me, with all due respect, to assume that too. Right? So that means it does help us learn geometry. But it doesn’t innovate in the essential sense. There is no information in the theorems beyond what was already embedded in the premises. It’s just that it takes nontrivial work to extract, to draw out, that information from the premises. And not everyone can do that; sometimes it really takes genius. Okay? So it helps us in the practical sense. But in the essential sense, if I ask whether it added information, the answer is no. It didn’t add new information. That’s the definition of logic and of mathematics. It’s something that doesn’t help us at all, and that’s why it’s completely precise. This hot-air balloon joke isn’t a joke; it’s the definition of mathematics. Okay? So that basically means that a logical argument—and for me mathematics is also a logical argument; there are a few philosophical landmines here but let’s leave that aside—the logical argument is an argument that doesn’t add new information, and then it can be a valid argument. And therefore Abraham our father and the hat is a fully valid, top-grade kosher logical argument. Top-grade kosher.
[Speaker F] And it
[Rabbi Michael Abraham] begs the question, because I assume that Abraham our father wore a hat when I say that a Jew like him obviously didn’t go without a hat, and I assume that everyone has to walk in the ways of Abraham our father, so obviously the conclusion that I need to wear a hat is also in there, right? To put on a hat. So it was already inside the premises. Very good. But with Socrates and mortality too, the conclusion was inside the premises. From the mere fact that this is a valid argument, it follows that it begs the question. Contrary to what they teach in every first logic class, begging the question is not a fallacy. Begging the question is the definition of a valid logical argument. If it doesn’t beg the question, it isn’t valid. It’s just that there are arguments that are trivial. Now, if I know that if Abraham went with a hat, then Abraham went with a hat—that’s a valid argument. P implies P. That’s a valid argument, but it’s a trivial valid argument. Okay? That’s obvious. Right? So I don’t need the geometry teacher to tell me that. Fine, but it’s a valid argument. It’s just a simple valid argument. It’s unnecessary. Where do I need logic? In places where the inference is a complex inference. It’s still an inference, and after they present it to me I’ll see: yes, the conclusion is inside the premises. It just has to be presented to me because I wouldn’t have gotten there on my own. Therefore it’s not trivial, and therefore it’s worth learning. But that’s not a fundamental difference; it’s a difference of difficulty, of level of complexity. That’s all. On the fundamental level it’s exactly the same thing. So if we return to our topic: we spoke about deduction, and deduction is necessary, certain inference. Its certainty stems from the fact that it teaches us nothing new. If that’s the advantage deduction has, and I now need to place analogy and induction below deduction, then what comes after deduction is analogy, and induction is at the bottom of the ladder. Why? Because analogy adds new information; therefore it’s not deduction. I say: this lectern is brown. That one is a lectern, so it too is brown. Okay? What are my premises? This is a lectern, first premise. This lectern is brown, second premise. And that is a lectern, third premise. From this I infer that that one is also brown. Right? The inference that it’s brown is not in the premises in any way. Therefore you will not succeed in proving logically that this thing is brown. That’s analogy, not deduction. Okay? Sometimes you’ll be right, sometimes you won’t. You’re taking a certain risk. But what you’re doing is adding information beyond the information embedded in the premises. Therefore it’s not deduction. But how much information did you add? In analogy you added information about one item. In induction you take a premise that is one item and make a crazy speculation about infinitely many or a huge number of items. So it’s much more speculative. Here you commit yourself only about the nature of one object. There you’re saying something about a whole group of objects. Much more speculative, much weaker. Or in other words—one second—in other words, I’ll qualify this in a moment; this is only a didactic move. So what I want to say is the following: the degree of validity of an argument stands in inverse proportion to the amount of information it adds. The more speculative it is, the more information it adds beyond what was embedded in the premises when I set out, the more questionable it is. Right? If it adds almost no information, then it’s stronger. If it adds no information at all, then it’s certain. Okay, so there is some kind of game here—it’s the logical uncertainty principle. The uncertainty principle—you know that for uncertainty in position and velocity, their product is constant. Meaning, if there’s great uncertainty in position, there’s little uncertainty in velocity, and vice versa. That’s the uncertainty principle in quantum theory. The same thing applies to information and certainty. The product of the amount of information and the degree of certainty is constant. If you add a lot of information, certainty becomes very small. If you want a lot of certainty, add little information. Be less speculative, okay? There’s some interplay between these two things. Someone who wants to play it safe—that is, to be absolutely certain only—won’t manage to add any information. He’ll die with the information he was born with. Fine? You can’t add information without paying in the currency of certainty. You can’t, by definition. Therefore, for example, science, which is based on analogies and inductions and not on deductions like mathematics—therefore science, its conclusions are never certain. They may be well-grounded, they may be strong, they may be convincing, but certain? No. A scientific theory can always be refuted at some stage and replaced. In mathematics that doesn’t happen. There’s no old mathematics and new mathematics. There may be more efficient forms of formalization, but there isn’t… If it was a proof, it’s a proof; nobody can refute it. There could be a mistake in the proof, perhaps, but that’s just a mistake; it turns out retroactively that it wasn’t correct—you just missed something. But if there’s something proved, then it’s proved. There’s no way around it; it’s certain. Okay, science is not certain. Why? Because it uses analogies and inductions, not deductions.
[Speaker K] Is extrapolation basically induction?
[Rabbi Michael Abraham] Yes. Now interpolation is exactly the visual picture of induction, because you say—you’re not—extrapolation, sorry, not interpolation. Extrapolation. Meaning, you go outside the region you know and make some claim about the surroundings, so you’re extending. On the contrary, interpolation is parallel to deduction, basically. You say—not exactly, because interpolation is between two points to say something about what’s in the middle—but if I know about the whole segment, not just two points on the two sides, about the whole segment, and I say something about a point in the middle, that’s deduction. Okay? So apparently the ranking is deduction, analogy, and induction. But it’s not so simple—I said I’d qualify that. Why? Yes.
[Speaker L] If I compare between two similar things, is there some reason I wouldn’t get the standards of both of them?
[Rabbi Michael Abraham] Okay, then make more lecterns, fine. But it could be that you’re mistaken—you’re right only about some of the standards, not all of them. You could be wrong, after all; you can always be wrong. So maybe in some analogies you’ll be right, in others you won’t, whereas with one lectern you’re taking fewer risks. Less risk, but that doesn’t
[Speaker H] mean it’s stronger.
[Rabbi Michael Abraham] Why not? If it’s less risk, that means it’s better.
[Speaker H] Because I had no reason to say that this lectern is less.
[Rabbi Michael Abraham] It’s like economics—that’s what’s confusing you. In economics, after all, we know that return is proportional to risk. If the risk is high, the return can be high, right? But not the expected return. Expected return is something else—you have to be careful with that. Here too, by the way, if you pay in the currency of certainty, you can gain a lot of information. That’s the return. You have to pay for it in the currency of certainty. If you don’t pay in the currency of certainty, you won’t get any return. You can stay only with what you know for certain, and then you won’t know anything. You’ll know what you knew when you came into the world—which means nothing. Okay, so why do I say that the ranking between analogy and induction isn’t so simple? It’s connected to what you said earlier. Because when I make an analogy between this lectern and that lectern, I’m not really assuming anything beyond the fact that this is a lectern and that is a lectern, right? They have specific properties, but I’m not assuming that; I’m assuming that this is a property characteristic of a lectern as such, right? But then in fact what I’ve done here is generalize to all lecterns, and in particular to that lectern, right? That’s really what I did. So therefore, in practice, analogy is nothing but an application—you said it, or someone said earlier—a specific application of a hidden induction. Basically I took this lectern and said: this is true of all lecterns, and in particular of that lectern. Right? But on the other hand, one could also say the opposite. How do I know this is true of all lecterns? Because I say: this is similar to this lectern—ah, that’s also similar to that lectern, and also to that one—ah, it’s similar—so in fact induction is nothing but a collection of analogies. The relationship between analogy and induction is very tricky. By the way, I think there’s a dispute among the medieval authorities (Rishonim) about this, about the relationship between analogy and induction, regarding an a fortiori paradigm built from two verses and an a fortiori paradigm from one verse. Maybe I’ll talk about that later. Basically I think the more correct way to view these three modes of inference—and all this will connect for us later; it’s an important introduction, I want you with me—all this connects; all of this in the end, it seems to me, should be presented as follows: there aren’t three modes of inference. There’s only one. Ultimately, there are only analogies. But we do the analogy in two steps. We do analogy by means of a first step, which is induction, and a second step, which is deduction. When I want to make an analogy, I see that this lectern is brown. I want to draw conclusions about that lectern. How do I do it? I say: this lectern is brown. It seems to me that being brown is a lectern-like property. So all lecterns are brown. What did I do here? Induction, right? Now I’m at the stage where I know all lecterns are brown. Now I say: that one is also a lectern. Now do you see what I’m doing now? Deduction. I’m going from the general to the particular. That means that in fact it isn’t three modes of inference. I’m always making an analogy. It’s just that you can break the analogy into two steps, two stages. In the first stage I do induction from the particular to the general, and in the second stage I come down from the general to another particular. Right? Because what’s happening? When I make inferences, in the end what I want is to accumulate information. I have some information about this lectern; I want to know what’s going on with that lectern, right? I want to accumulate information. How do I accumulate information? How do I make analogy? I do induction—I say, this seems to me to be a lectern-like property—and afterward I do deduction. So in fact the relationship is not three modes of inference; rather, you can write it this way: analogy = induction + deduction, or followed by—“plus” here is only symbolic. Fine? Meaning, induction followed by deduction. Okay? And so there aren’t three modes of inference; there is only one. We are making analogies all the time—that’s what we’re really doing. It’s just that analogies break down into two stages, that’s all. You can take one section of the—if you draw it like this: I have one example and I want to infer from it a conclusion about the second example, so I go up to the general, and from the general I go down to the particular, and that’s how I make the analogy. Right? So analogy is built by what? By rising to the general—which is induction—and this is deduction, and that whole move is in fact analogy. Right? We’re only making analogies all the time.
[Speaker M] The three sides are analogy. What?
[Rabbi Michael Abraham] The three sides are basically analogy. The route from here to here is analogy. When I go directly, I’m making analogy; when I go through this, I do it in one step as induction and a second step as deduction. But still, in the end, when I say all human beings are mortal, what I’m really saying about each one of them is that he is mortal. That’s a shorthand definition, so I say all human beings are mortal. Okay? So therefore now the question of who is first and who is second is a little hard to define here, but in principle it’s a triangle. It’s not three modes of inference. It’s some kind of circuit where each time you can walk either from here to here, or from here to here, or from here to here and then to here. Choose which route you take, but we are always on some such triangle along which we walk. That’s the whole story. Now science basically deals with this. Science deals with this. Mathematics deals with this. Right? These are the necessary inferences. Science makes generalizations, arrives at general laws in science. And now if we want to infer a conclusion from a particular fact we know to another particular fact, or from a law to a law, then in these two steps I use science plus mathematics in order to reach a conclusion about the world or about the information that interests me. Okay? That’s basically how things work. All right, there’s also abduction, but I’m not going into that now. Maybe we’ll get to it later. Okay, I’ll say one sentence about it anyway, because we’ll use that too. I see abduction this way—I’d say, you can describe it in a few ways. Peirce, I think, first introduced the concept. Abduction looks a bit like induction, but it’s not exactly that. In scientific theory, for example, think of the law of gravitation. So I see that this marker—if I let go of it—it falls to the ground. And I see that this also falls to the ground. Fine? And what will happen to this if I let go of it? It too will fall to the ground. How do I know? Because if both of them fell, this one will also fall. Why? Because there is a general law that says that all objects, say, with mass fall toward the earth. Fine? But that law, when you look more deeply and enter the theory of gravitation, it’s not just that all objects fall toward the earth. Really, every two masses attract one another, not just the earth. In addition, there is gravitational force and we have a mathematical description of it, and there are particles that carry that force—gravitons, okay? At least that’s what people believe; nobody has observed them yet, but that’s the idea: that there are particles that carry it. There is a theory with theoretical entities that actually explains this principle. Science doesn’t suffice with saying all bodies with mass fall toward the earth. That’s a phenomenological description; that’s not science. Science is the explanation of why this happens, or the theory standing behind this phenomenon. Within this theory there are what philosophers call theoretical entities. Theoretical entities are things like gravitational force, like gravitons, like all sorts of such things that nobody has ever seen. By the way, even the electron is in many senses a theoretical entity. Nobody has really seen an electron directly. Today there are claims that maybe people do, but it’s a philosophical question whether that really counts as seeing the electron or not.
[Speaker K] Yes, exactly that.
[Rabbi Michael Abraham] Exactly—that’s what I’m talking about. It’s not a simple philosophical question. You don’t see electrons with your eyes; you see the consequences of electrons. In any case, in science we move up from the examples we observed, the particular cases we observed, to a theory—not to a generalization. A generalization would be, say: we saw these objects falling to the earth, so I say all objects with mass fall toward the earth. That’s induction. Science doesn’t do only that. Science looks at why all bodies fall toward the earth. Ah, because there is something in mass that causes it to be attracted to another mass. And there are gravitons exchanged between them. And there is an entire theory that emerges from all these things. Therefore the move from particular cases to a theory is what’s called abduction. The theory explains the general law that I obtain by induction. The general law that all bodies with mass fall toward the earth on the basis of examples—that’s induction. But science doesn’t do only inductions. Those are what are called phenomenological theories. Phenomenological theories are theories that only describe. That’s phenomenology; that’s induction. But science not only describes; science also explains how it happens. Okay? And here we are dealing with abduction, not induction. Abduction is the move from examples to theory, not to a general law. In a certain sense that move is even more speculative than moving from examples to a particular case. But I’ll talk about that another time. Now let’s look a bit more at deduction. Because that is basically what people usually identify as logic. I mentioned earlier that analogy and induction are tools used in scientific thinking. Abduction too. Deduction—that’s mathematics. Okay, that’s the disciplinary division among these things. Does mathematics say anything about the world? Depends what you call
[Speaker K] the world. Sometimes you can flatten it, take things…
[Rabbi Michael Abraham] So in principle, no. It says nothing about the world. A scientific theory is defined—at least after Popper—as a theory that is falsifiable. Okay? At least that’s a necessary condition for a theory, even if not a sufficient one, for a scientific theory: that it be falsifiable. Meaning, I can propose an experiment that will put the theory to the test. Either it will pass or it won’t. Then the theory is scientific. If it passed the test, then it was corroborated—that’s already not Popper. If it didn’t pass the test, then it was refuted—that is Popper. Okay? But the theory “all fairies have three wings” is not a scientific theory. Why? There’s no way to propose an experiment that will confirm or refute that theory—confirm, refute that theory. Okay? Now the question is: let’s look at a mathematical theory. Two plus three equals five. Nice theory. A nice mathematical theory. Is this a scientific theory? Yes, you can corroborate it. How? You
[Speaker J] take something countable and…
[Rabbi Michael Abraham] Say apples. You take two apples, put them in a basket. Take another three apples, put them in the basket, count how many came out. If it comes out seven, then I’ve refuted the proposition two plus three equals five, the theory. If it comes out five, then I’ve corroborated it, right? Apparently. But actually no. Again, there’s another bit of philosophical dispute here; in my opinion that’s not correct. Why? When I taught these things in physics, in a mechanics course, in the first tutorial I asked them exactly this question. Can you propose an experiment that would refute or corroborate the theory that two plus three equals five? Usually they gave me this answer. And then I said to them: look, I’ll propose an experiment that refutes the theory that five plus five equals ten. A simple experiment. I take a body and apply to it a force of ten newtons northward. Newton is a unit of force, doesn’t matter. Fine? Ten newtons. Now I take another force and apply to it another force—ten newtons eastward. Fine? What is the total force acting on the body? What’s the velocity, as it were?
[Speaker N] What’s the vector? Ten times the mass.
[Rabbi Michael Abraham] Ten times square root of two, right? Right. Newton already takes the mass into account. So… this squared plus this squared, then square root. Fine? So there you go, look: ten plus ten equals fourteen-something. I’ve refuted the theory that ten plus ten equals twenty. Right? So why are you still using this refuted theory that ten plus ten equals twenty?
[Speaker O] You said you used addition here as adding. Okay. But there is a certain sense of adding that you can use.
[Rabbi Michael Abraham] What did I actually do in this experiment, supposedly? I refuted not the law that ten plus ten equals twenty. I changed the concept. Rather, I claimed that this addition is not arithmetic addition. This is not the “plus” that arithmetic talks about. Right? Rather, you need vector addition, and then you do vector calculation. Right? Meaning, I have a mathematical statement, I did an experiment, it was refuted. Did I give it up? No. Rather, I explained: no, no, that’s not what’s called “plus,” that’s not the relevant addition. But understand that this is exactly why you will never refute mathematics. Let’s take your experiment. I put two apples in a basket, then add another three apples, and it comes out seven. It comes out seven—Houdini did his magic and got you seven. Fine? From that point on, do you agree that two plus three doesn’t equal five? You’ll never accept that in your life, right? You’ll say there was an error in the experiment, I don’t know what; he didn’t actually add apples, there were more apples there, there was a hidden hole, who knows. Right? He fooled you when he took out the apples; they came from his sleeve, not from the container. You will never give up the proposition that two plus three equals five. Why? Because it is not subject to a test of refutation. Two plus three equals five is not subject to a test of refutation. No experiment you try will refute it. It cannot be refuted. Any failed experiment, you’ll explain in a thousand ways: no, no, that’s not what “adding” means. No, there was an error in the experiment. No—it is not subject to a test of refutation. Why? Because the theory says nothing about the world.
[Speaker M] A mathematical theory.
[Rabbi Michael Abraham] Mathematics says nothing about the world. What you refuted is the physical assumption that the addition of forces is described by the mathematical theory called arithmetic. That is a claim in physics. The claim in physics says that arithmetic is—that physical forces are a model of the theory called arithmetic. Yes, it is an application of arithmetic theory. Okay? That is what we refuted. But that is a claim in physics, not in mathematics. Whether the world fits this theory or that theory has nothing to do with mathematicians. Physicists have to see whether it fits or doesn’t fit. Meaning, what I refuted here is a claim in physics. The same thing with apples in the basket. If I put in two and then another three and I get seven, I will never conclude that two plus three equals seven. At most what I’ll say—assuming I found no flaw in the experiment—is that adding apples into a basket cannot be described by arithmetic addition. At most. Right? But that is a claim in physics: whether adding apples into a basket is described by arithmetic addition or by some other mathematical doctrine. Right? That is a claim in physics, not in mathematics. Because only physics stands open to refutation; mathematics does not. I’ll give you an interesting example that jumps ahead a bit, but maybe you’ll see it more sharply this way. There is a legal philosopher named Chaim Perelman. Have you heard of him? He was a Jew from Belgium, University of Antwerp I think, I don’t remember, something like that. And he deals with rhetoric and philosophy of law. Some of his books were also translated into Hebrew; I think Ora Gringard translated them. And in one of the books he cites a law that was accepted in Belgium, called the Vandervelde law—some local name, I don’t know exactly. And that law said the following: it is forbidden to sell workers in a pub, in a bar, more than two liters of wine. Forbidden. Not forbidden to sell two liters—more than that. It is forbidden to sell workers two liters of wine. Fine? The rationale of the law was that workers bring their weekly salary; the idea was that they shouldn’t waste it in the bar but bring it home. If you want to drink a glass, drink a glass, but don’t waste your whole weekly salary. You buy about two liters of wine or something like that, I don’t know. Anyway, that was the law. A nice fellow came to the pub and said: I want ten liters. Ten liters of wine. The seller told him: the law forbids it, I can’t sell it to you. He said: no, the law forbids two; I want ten. So he told him: in a greater amount, the lesser amount is included. Meaning, ten is two plus eight. I sold you two and I did that five times. Fine? That’s what’s called an a fortiori argument of “the greater amount includes the lesser,” and some later authorities (Acharonim) say that this is an a fortiori argument with no possible refutation. Therefore, for example, with respect to the rule that one does not administer punishments by logical derivation, some later authorities say that one may administer punishment on the basis of an a fortiori argument of “the greater amount includes the lesser,” because it cannot be refuted. It went to court, as he recounts there—well, that’s how he tells it—and the one who won was the buyer. The judge said that the seller had to sell him the wine, the ten liters of wine. The law forbids two, not ten. Why? He explained. A person has freedom of occupation—I’m speaking in our language now, right?—and if a person wants to invest in wine, ten liters, not for immediate consumption, what, is he forbidden? He wants to get into the wine business; this is an investment. With that he’ll bring a livelihood home, I don’t know. If he wants to get drunk on two liters of wine, three liters of wine, you may not sell it to him. But a person who takes his savings and buys a stock of wine—the law cannot forbid him from doing that. So sell him ten liters. Two is forbidden, ten is permitted. What does that mean? When you look at this formally, there is no refutation. If you sell him ten, then in particular you sold him two, right? But when you look at the application in the world, it is never pure mathematics. You always have to look at the context, the meaning of the concepts, what it is trying to achieve—teleological interpretation and all those debates—what it is trying to achieve. And then it may be that even something that, on its face, seems completely impossible to refute, can in fact be refuted. Let me perhaps give you another example. There is the rule I mentioned earlier that one does not administer punishments by logical derivation. According to most views except Maimonides, you don’t punish on the basis of an a fortiori argument. Maimonides says from all the interpretive principles by which Torah is expounded—but in the simple reading, even in the Talmudic texts it seems that you do not punish on the basis of an a fortiori argument. Now I mentioned that some later authorities want to argue that an a fortiori argument of “the greater amount includes the lesser,” like the ten and the two, yes, one does punish by it. Why? Because in their view, the reason one does not punish by logical derivation is the concern that there might be a refutation. Therefore you don’t punish by logical derivation. But here there is no concern that there might be a refutation. With “the greater amount includes the lesser,” you’ll never in your life manage to refute such a thing. Okay? But there are other explanations for why one does not punish by logical derivation. For example, that the punishment for the lesser thing is insufficient to punish for the greater thing. There is a commentary of Kesef Mishneh on the rule about one who passes some of his sons through the fire to Molekh, but not all of his sons to Molekh. One who passes all of his sons to Molekh is not punished, and one who passes some of his sons to Molekh—only some of them—is punished. This is an a fortiori argument of “the greater amount includes the lesser.” It’s like the two and the ten. Right? There it may be—the Kesef Mishneh discusses it there and brings two possibilities, these two possibilities. In any case, what happens here? Notice that the second explanation is a refutation of the first explanation. What does the second explanation say? Against the mathematics I have no refutation. “The greater amount includes the lesser”: when you gave two hundred, then of course in particular you gave one hundred. There’s no refutation of the mathematics of that. But in life there may still be a refutation. If the meaning here is to assign punishment, it may be that the lighter punishment is insufficient for the more severe offense, and then you can’t give it, even though this is an a fortiori argument that supposedly cannot be refuted. What does that mean? There—you’ve just refuted it. Because after all, the result of that a fortiori argument is that one should punish the more severe thing with the punishment given for the lesser thing, right? I just showed you that one should not. But there is no refutation of that a fortiori argument. And yet there is. This is another example of the fact that whenever you apply mathematics to life, there is always some assumption beyond the mathematical assumption. And that assumption says that this mathematical model or this mathematical doctrine can be applied to life. Life is a model of that theory—that’s what mathematicians call it. Okay? That is an assumption in physics, not in mathematics, and that can always be refuted. Therefore an a fortiori argument of “the greater amount includes the lesser” can certainly be refuted. Those later authorities who say it cannot are mistaken. It certainly can be refuted.
[Speaker P] That’s not physics, it’s sociology.
[Rabbi Michael Abraham] Doesn’t matter. By “physics” I meant scientific claims about the world as opposed to mathematics. That’s what I meant by physics. Exactly like here. When you move from here to here, you first need to do induction and then there will be deduction. But it may be that here you did something that can be refuted. We’ll see that later in a fortiori arguments and all these things—we’ll really see all these principles.
[Speaker P] Do you understand the tribe that does not acquit
[Rabbi Michael Abraham] So it actually comes out—no, that’s a different passage. There too you could maybe see it as a refutation of an a fortiori argument of the “included in two hundred is one hundred” type. If twenty are enough to convict, then how could twenty-three not be? So there you go, exactly, right, that too is a refutation of an a fortiori argument of the “included in two hundred is one hundred” type. There’s also, by the way: if one is liable for tearing, then for opening certainly all the more so? If one is liable for opening, then for tearing certainly all the more so? Because if you’re liable for opening, that is included within tearing. After all, opening is removing the top layer; tearing is removing everything, and in particular the top layer. That’s an a fortiori argument of the “included in two hundred is one hundred” type. And about that they say: punishments are not derived by logical inference. There’s a dispute between the Babylonian Talmud and the Mekhilta at the beginning of Bava Kamma, and according to some views. In short, here it’s simply a mistake. An a fortiori argument of the “included in two hundred is one hundred” type can be refuted. Period. Anyone who says otherwise is simply mistaken. It can be refuted, because its mathematics cannot be refuted—that’s true, mathematics can’t be refuted—but when you apply it, those assumptions may turn out not to be correct; they can be refuted. This is not mathematics. The assumption that this mathematics is the right tool to describe this piece of life, this piece of law, this piece of physics or chemistry or whatever it may be—that is an assumption, but it’s a scientific assumption, not a mathematical one, and that can always be refuted. Okay? Therefore there is nothing in the world that touches our world that cannot be refuted. There isn’t—simply isn’t. Okay? And that basically means, if I go back to the beginning, that mathematics does not make claims about the world. Mathematics makes no claims about the world. Mathematics says: if these assumptions are true, then these are the conclusions that follow from them, as in geometry. Now you can decide whether that mathematical theory describes the world or not—that’s a physicist’s question, not a mathematician’s question, and of course it’s a question where you can be mistaken and where the claim can be refuted. It’s a physical question, not a mathematical one. Think about geometry. Geometry says that the sum of the angles in a triangle—in Euclidean geometry—the sum of the angles in a triangle is one hundred and eighty degrees. Fine. Does that mean that if I draw a triangle in the world, it will necessarily have one hundred and eighty degrees? The answer is no. Not only is it not necessarily so—not only is it not necessarily so—it won’t be so. It’s simply not true in our world. A closed triangle with a lock. Not true. Because our world is a curved world; in curved space, Euclidean laws do not apply. Right, it won’t be a triangle in straight space, it’ll be a triangle in curved space, and in such a triangle the sum of the angles is not one hundred and eighty degrees. But that is a claim in physics, because the physical claim that says Euclidean geometry describes our world—that is not a mathematician’s claim; it’s a physicist’s claim. He is making claims about the properties of the world. Now it may be that he is mistaken; that can be refuted. The mathematics cannot be refuted. Given the assumptions of Euclidean geometry, the sum of the angles is one hundred and eighty degrees—that’s mathematics. Those are the assumptions of Euclidean geometry. No, that’s the assumptions of Euclidean geometry. Now the question whether our world is in fact flat—that is, whether Euclidean geometry is the right tool to deal with it—that is a question in physics. You have to test it experimentally, measure, and see, and there is the possibility that you’ll be mistaken. You’ll see examples but still be wrong, because you made a bad generalization or something like that. That’s a question in physics. Therefore—and this is a very important point—every application of a mathematical theory to something scientific, theoretical, legal, whatever it may be, social science, humanities, natural science, whatever—when you apply it to factual claims, you are always making application assumptions beyond the mathematical assumptions, and those application assumptions are always vulnerable to attacks, to refutations. We’ll see this with a fortiori arguments as well. Okay? So that’s regarding the relation between logic and life. Logic and mathematics do not deal with life. In order to apply them to life, you need to assume some assumption that is itself not a mathematical assumption or a logical assumption, okay? It’s itself an assumption connected to the field you’re dealing with, and it stands the test of refutation. It’s a scientific assumption; it has to be checked. Good. Now I’ll continue. How do we move forward with logic? So look, in deductive logic—and now we are not in life, we are in the Platonic world; I’m a Platonist—so in the Platonic world where the truths of mathematics exist in some sense, okay, we are not dealing with our world and applications and so on; we are dealing with the theory, not the model. So in that Platonic world, the argument is of course a necessary argument: if these are the assumptions, then this is the conclusion, everything is fine. Is the conclusion really true? I don’t know; it depends on whether the assumptions are true. Okay? All I can know is that if the assumptions are true, then the conclusion is true. That is basically the difference between the validity or invalidity of an argument and the truth or falsity of a proposition. Do you know what the difference is between a proposition and an argument? A proposition is a proposition—it’s a sentence that claims something. A sentence that can receive a truth value or a falsity value is called a proposition; that’s the Aristotelian definition of a proposition. A sentence that says, “What time is it now?”—that too is a sentence, an interrogative sentence, but it is not a proposition; it claims nothing. You can’t say that this sentence is true, and you can’t say that this sentence is false, but it is a sentence. The sentences called propositions are a subset of the set of sentences—those sentences about which I can say that they are true or false, in principle. I won’t always know whether it is true or false, but in principle it admits the evaluation or logical judgment of truth or falsity; that is called a proposition. An argument is something else. An argument is a logical structure that derives one proposition from a collection of other propositions—what in another context you might call a proof, if you like. The other propositions are the premises and the conclusion. Both the premises and the conclusion are propositions. The structure. Good. Now take the argument: all human beings are mortal; Socrates is a human being; conclusion: Socrates is mortal. Is this argument true? The answer is no. True and false are not categories that apply to an argument. About an argument we should speak in terms of invalid or valid. Invalid or valid means: does the conclusion necessarily follow from the premises? That is called a valid argument. If it does not necessarily follow from the premises, then it is an invalid argument. Not that it is necessarily untrue, but that it does not necessarily follow from the premises. An argument that is not valid is an invalid argument. That is the accepted terminology. About propositions we say that they are true or false. What is the connection between the truth or falsity of a proposition and the invalidity or validity of an argument? The truth is, there is no connection. I gave you an example before. All frogs have wings—premise. Second premise: this lectern is a frog. Conclusion: this lectern has wings. Rabbi Elisha’s winged lectern. At the logical level, this argument is valid. Anyone who accepts the premises must accept the conclusion. If I ask whether this argument is valid, the answer is yes. If I ask about the propositions that compose it, whether they are true or false—the answer is: all of them are false. Both premises and the conclusion are false. That means there is independence between the truth and falsity of propositions and the validity and invalidity of arguments. There can also be the opposite kind of argument, where both the premises and the conclusion are true and the argument is invalid. These windows are transparent. This fluorescent light shines white. Therefore the Queen of England is—what was her name? Elizabeth. All the propositions are true, but the conclusion does not follow from the premises; it does not follow necessarily, and in fact does not follow at all from the premises. So all the propositions are true propositions, but the argument I built here is an invalid argument. The conclusion does not necessarily follow from the premises. So apparently there is complete independence between the truth and falsity of the propositions that make up the argument and the validity or invalidity of the argument itself. So why do we need to define arguments at all? Who cares? What interests us is what is true and what is not true, right? What we want to know about are propositions, whether they are true or not true. Arguments are only a tool. That is Aristotle’s conceptualization. Why is it important? Why is it useful? Because there is a certain connection between the judgment of propositions and the judgment of arguments. One connection—but it is important enough to generate logic. And that connection says that if the premises are true and the argument is valid, then the conclusion is also necessarily true. There is some link between the validity of the argument and the truth or falsity of the proposition. It is not true that they are completely independent. There can be a valid argument all of whose premises and conclusion are false. There can be an invalid argument whose premises and conclusions are true. That can happen. But there cannot be a valid argument whose premises are true and whose conclusion is false. That cannot happen. So it is not the case that there is complete independence between the validity or invalidity of an argument and the truth or falsity of propositions. This is precisely where logic comes in. The role of logic is to give me a toolbox—there was once some film about Officer Training School 1. My children showed me that the commander of Officer Training School 1 was speaking with cadets on television; there was some series about Officer Training School 1, I don’t remember what it was called. There the commander of Officer Training School 1 said to the cadets in a summary talk: I want you to leave here with one thing—a toolbox. Literally, that’s what he said. So one thing: a toolbox. It’s the same thing here. In other words, what logic gives us is some collection of tools that can serve us for what? For taking true premises and deriving from them other conclusions about which I don’t know whether they are true or not. But if I have a valid logical argument that leads from the premises to the conclusion, and I know that the premises are true, then no problem—I can infer the conclusion, that the conclusion too is true. That is the use of arguments. For us, that is the use of arguments. Or in other words, arguments are a toolbox by means of which one can build proofs. Logical arguments or logical inferences, patterns of logical inference, are a toolbox by means of which one can build proofs. Proofs are taking premises and deriving from them a conclusion. To ask whether the premise is true or not true is an illegitimate question in mathematics. Check for yourself whether the premise is true or not; you cannot ask the mathematician that question. It is not within his mandate as a mathematician. Ask a mathematician directly whether the sum of the angles in a triangle is one hundred and eighty degrees. A mathematician should immediately answer you: I have no idea. I have no idea; it depends on your assumptions. If your assumptions are Euclidean, then yes. If your assumptions are non-Euclidean, then no. I can build a geometry for any angle sum you order. No problem at all. An angle sum of minus two hundred and thirteen degrees—you can also build a geometry for that, in principle. Fine? Or whatever you want. So all the mathematician can tell you is: give me your assumptions, and I know how to derive the conclusion from them. Mathematics deals only with “if-then.” It deals neither with the “if” nor with the “then,” but only with the relation between the “if” and the “then,” between the premises and the conclusion. Now a mathematician—or rather a physicist, sorry—or anyone working in another field, a jurist, whoever it may be, can come and say: okay, these assumptions are true. I know that from one consideration or another. The legislator determined it, common sense says it, observation says it, whatever. I know these assumptions are true. Now I take the logical tool, or whatever mathematical instruments are in the toolbox, and I can derive conclusions from them and infer that those too are true. I am making an inference. So if you put at my disposal a toolbox—that is what the mathematician and the logician deal with—then now I can learn more and more propositions that are true or not true. Or in other words, discover more and more items of information—though if the tool is mathematics, then of course only to uncover information that was already there for me, not to add new information. To sharpen or uncover information that was already latent in what I had from the start. That is basically the general meaning of logic. And then the claim I actually want to end with is what in philosophy is called the emptiness of the analytic. The emptiness of the analytic. That is to say, analytic arguments are empty. Empty of information. Meaning: an analytic argument, a logical argument that carries me by means of a valid argument from premises to conclusion, never adds information for me.
[Speaker B] Do you mean formal?
[Rabbi Michael Abraham] Suppose so. But also “every bachelor is unmarried”—that’s not formal; that’s content logic, and it also doesn’t add information. So it’s not only formal. The formal satisfies this, but it’s not only the formal that satisfies this. So the emptiness of the analytic means that logic in its mathematical sense, necessary mathematical arguments, cannot add information for me. But that raises a somewhat embarrassing question. So how do we add information? The answer is probably with analogies and inductions. Right? There is no other way. Only those are forms of inference that add information for us. But that is why so many philosophers and thinkers find it so troubling, and say that one in fact arrives at a kind of absolute skepticism. One arrives at absolute skepticism because they do not trust those methods. Whatever is not proven, whatever is not mathematical or logical in the strict sense of the word, in the rigorous sense of the word, is arbitrary. I am built this way, so I think this analogy is correct; someone else is built differently and thinks this analogy is not correct. You cannot prove anything of that sort, so you can assume it, or you can choose not to assume it. Soft modes of inference, on the one hand, are the only modes that can add information for me, but on the other hand they are soft—they are not certain. So somehow I am torn. I want certain information. There is no such creature. Certain information is an oxymoron. Because if it is certain, then there is no information here, and if there is information here, then it is not certain. That is the principle of certainty I mentioned earlier. There is no certain information. So there are those who say: okay, if there is no certain information, then I don’t believe in anything, so there is no information, and I become a skeptic. Narratives, and today they call it by all kinds of complicated names—skepticism is what matters, nothing beyond that. The other approach basically says: no. Who said information has to be certain? On the contrary, information cannot be certain. But still, there is information that is more reasonable and information that is less reasonable. Soft arguments have significance too. True, they do not give me certainty, but they do give me information. And perhaps I even have a way to improve soft arguments, to make them more correct or less correct. Certain eliminations. What is called inductive logic or non-deductive logics. Okay? And then these are basically ways of trying to create supposedly formal logical tools in order to mechanize analogical and inductive reasoning, which on its face seems impossible. Mathematics is only deductions. To make mathematics out of the other tools is something that, on the face of it, seems impossible—like squaring the circle. I think that is exactly what Talmudic logic does, by the way. And now I’ve reached the opening point after this whole introduction. Because I think that what we are going to study this semester really is an attempt to mechanize analogies and inductions. Which is almost basically doing the impossible, squaring the circle. Okay, that is basically the point where I’ll stop. Let’s stop here. Okay, so you can formalize it, you can turn it into mathematics and let a computer do it. How will you get a computer to do analogies or inductions? That is not—it’s something seemingly human. Today computers do it, but that’s another discussion. It’s something that is seemingly human. There is no way to mechanize it. It requires something of another kind, and that’s why it really is a new phenomenon in the world of computing, relatively new. Computers that do analogies and inductions. Because until recently computers knew how to do mathematical computations, computations that are straightforward. Give me the assumptions, tell me what the algorithm is, and I’ll tell you—I’ll do the calculation quickly. Okay? Is it possible to mechanize analogies and inductions? In computing, when a computer does analogy and induction, it does not do it by means of a mechanism. It does it in certain indirect ways, as far as I know at least. I’m going to try to suggest a way that mechanizes analogies and inductions, and I claim that this is what underlies the logic of the methods of exposition. Okay? So here I’ll stop, because I’ll still explain all this, but let’s stop.