Session on Sparks of Derash Logic: The Logical Principles of Derash as a Basis for Non-Deductive Logic – Rabbi Dr. Michael Abraham – Bar-Ilan University
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
🔗 Link to the original lecture
🔗 Link to the transcript on Sofer.AI
Table of Contents
- [0:10] Presenting the goal of the first paper
- [1:16] Dealing with formalophobia and formalophilia
- [2:59] Introduction to the first book
- [4:18] A legal example of an a fortiori inference
- [7:13] The question of the symmetry of an a fortiori inference
- [10:13] An example of asymmetry – grades
- [12:28] The original a fortiori inference in the Talmud
- [16:35] The need for a language for hermeneutic rules
- [22:33] The simple model and its parameters
- [26:46] Unambiguous inference and clear advantage
- [27:52] Correcting an error in the table and its implications for the model
- [29:07] The connection between an a fortiori inference and a refutation in the Talmud
- [30:19] The functions of the model: filling in and presenting a theory
- [31:49] A mechanical algorithm for derivations and non-deductive inferences
- [33:20] Applying Occam’s razor to models
- [34:24] The artificial intelligence dilemma at Tel Aviv University
Summary
General Overview
The joint work of Roshi, Edgar, Babi, and the speaker defines a double goal: importing new techniques to decode the Talmud, and exporting insights from it to other fields, while distancing itself from formalophilia and emphasizing that formalization is a tool for understanding, not a value in itself. The speaker presents the first book on logical-deductive inferences through the a fortiori inference, shows problems of symmetry, *dayo*, and refutations, and argues that the Talmud treats an a fortiori inference as symmetric even when it seems to rest on different assumptions. He proposes a matrix language and a “chemical analysis” of concepts based on Occam’s razor and graph-simplicity measures in order to determine how to fill in a missing cell, defines a refutation as equivalence between possible fillings rather than as a decision for zero, and shows how this also solves examples from Tosafot. In the end he argues that the model provides a mechanical algorithm for non-deductive inferences that imitates rational human thought, connects this to scientific generalization and the problem of induction, and sketches future directions such as continuous data, data mining, and the mathematical space of the matrices.
The Goals of the Project: Import, Export, and Caution About Formalization
The speaker defines the goal of the project as twofold: import means using newer techniques to decode things that appear in the Talmud, and export means finding various insights that emerge from the Talmud and seeing whether they can help other communities or other fields of research. He presents “formalophobia” as a fear of formalism, and sets against it “formalophilia,” the view that translating things into the world of x’s, y’s, and mathematics is itself a kind of redemption. He argues that in his opinion formalization is not an end in itself, and that it should be done only if it helps us understand the material; merely translating something into formulas and mathematical mechanisms has no value in itself.
The Four Books and the Logical Fields
The speaker describes four books—or three, with the fourth in preparation. The first deals with logical-deductive inferences. The second deals with some of the textual hermeneutic rules, such as general and particular, particular and general, and general and particular and general, and presents them as an intuitive definition of sets as opposed to formal definitions of sets; it was done together with Yossi Maorbach and Gabi Hazut. The third book deals with deontic logic, that is, the logic of norms: obligation, prohibition, and permission, and the relation between them; he argues that paradoxes in deontic logic can dissolve when viewed through a Talmudic lens. The fourth book, now in preparation, deals with the logic of time, including alternatives and conditions, going backward in time and forward in time.
The a fortiori Inference as a General Pattern of Thought and Non-halakhic Examples
The speaker presents the a fortiori inference as a common inference that appears in many contexts, and illustrates it with an everyday example about history and physics exams, noting that the conclusion is not necessary because different skills may be required. He presents a scientific a fortiori inference about spacecraft with engines of different strengths, and notes that there too the inference is not certain, because factors like the spacecraft’s weight or air density can refute it. He also gives a legal a fortiori example involving the “Vandervelde law” in Belgium, which prohibited selling two-liter bottles of wine, and shows that three-liter bottles were permitted because the purpose of the law was to prevent workers from spending their entire weekly salary, whereas a three-liter bottle cost more than a worker’s weekly wage.
The a fortiori Inference in the Topic of the Bridal Canopy in Kiddushin 5 and a Matrix Structure
The speaker returns to the halakhic a fortiori inference and presents the topic in Kiddushin 5, whose goal is to examine whether a bridal canopy can effect betrothal or kiddushin, since it does effect marriage and the question is whether it can also effect kiddushin or betrothal. He presents the a fortiori inference as follows: money does not effect marriage but does effect betrothal, and a bridal canopy does effect marriage, so the question is whether a bridal canopy will also effect betrothal. He formulates this as a structure of two actions and two outcomes: giving money and bridal canopy, versus marriage and betrothal, and the whole question is which action can bring about which result.
Symmetry, Rotations, *Dayo*, and Refutations in the Talmud’s View
The speaker asks whether an a fortiori inference is symmetric under rotation and presents two different formulations: one in which a relation is established between betrothal and marriage as easier and harder, and a “rotated” formulation in which the assumption is that bridal canopy is stronger than money. He presents these as two different assumptions and would expect, in principle, that one could be attacked without knocking down the other, yet he argues that the Talmud sees the whole thing as symmetric.
He gives an example of *dayo*—“it is enough that what is derived from the law be like the original case”—and shows that when grades are quantified, one can get a different result depending on the formulation, yet “in the Talmud they never reverse an a fortiori inference,” and *dayo* “works in both directions or in no direction at all.” He gives another example, the redemption refutation in the topic of the bridal canopy: “What is unique about money is that it redeems consecrated property and second tithe,” and stresses that when this refutation is raised, the a fortiori inference is not rotated; the Talmud treats it as though both versions of the inference have fallen. He adds that a refutation does not mean that the cell is zero, but rather that the result remains a question mark and could be either zero or one.
Tosafot in Tractate Shabbat and the Confusion That Leads to the Need for a Technique
The speaker tells a story about his son, who came back from yeshiva with an a fortiori argument in Tosafot in tractate Shabbat that “nobody knows how to explain,” and presents the progression: an a fortiori inference, “the vessels prove it” as a refutation, an attempt to add another distinction, and then “even a three-by-three piece does not become a tent.” He presents the question of why a characteristic of the vessels relative to the others refutes the inference if the original assumption is a different hierarchy, and the difficulty that when one tries to formalize it, “something that is very easy to understand—once you try to formalize it, you lose your head,” whereas when reading Tosafot it seems obvious.
“Logical” Hermeneutic Rules, Matrix Language, and the Common Denominator
The speaker defines the “logical hermeneutic rules” as a fortiori inferences and refutations of them and refutations of the refutations, an inference from a prototype and refutations of it, and the common denominator as an inference from two verses together with refutations and combinations, while “two verses that contradict one another” may perhaps belong here too but not within this scheme, and the rest are more textual hermeneutic rules. He says that the matrix language was adopted from Avi Livshitz and presents everything in a table.
He describes how the Talmud, after an a fortiori inference with a refutation and a prototype inference with a refutation, “joins the a fortiori inference and the prototype inference into a full table,” and this is called the common denominator, where “neither of them works on its own,” but together, without adding any further datum, the result is one. He says there are three types of common denominator, and that “the result is not sensitive” to the values in a certain column, and gives as an example the final stage of the bridal canopy topic as a complex picture that would be hard for a second-grade child to carry out.
Chemical Analysis of Concepts, Occam’s Razor, and Deciding by Simplicity
The speaker proposes a “chemical analysis of concepts,” according to which behind actions like money and bridal canopy there are active components that produce the results, and the issue is not only who is stronger or weaker but what theory explains the successes and failures. He presents the possibility of filling in the question mark with either one or zero, and argues that the correct filling is the one whose underlying theory is simpler. He explains that with one filling, the situation can be explained with a single parameter in different intensities, whereas with zero there is no one-parameter model and two are required, and therefore one chooses the filling of one by Occam’s razor.
A Refutation as Equivalence Between Fillings and a Graph Algorithm
The speaker presents an analysis of a refutation table using an algorithm that generates a diagram and represents an order relation between columns, and shows that in a certain case both the zero filling and the one filling require at least two parameters, and therefore “the fillings are equivalent,” and that is what defines a refutation. He explains that later in the topic the number of parameters alone is not enough to decide, and a prototype inference of 1,1,1 with a question mark remains equivalent if one looks only at parameters, so they add graphic simplicity criteria: connectedness, number of independent points, and number of direction changes. He presents a full preference criterion with four components: number of parameters, connectedness, number of independent points, and number of direction changes, and defines a decision as requiring an unambiguous advantage, whereas any ambiguity is a refutation.
Confirmations: Three Types of Common Denominator, Consistency Check, and Solving Tosafot
The speaker argues that the three graphic measures were first chosen from graph theory as measures of simplicity, and only afterward it was discovered that each of the three types of common denominator is decided by a different measure, and he presents this as confirmation that all three are needed. He describes another confirmation from the large table where “we sweated blood” and could not produce a result until it became clear that a zero had been entered in one place, and he presents this as a sign that the model is not ad hoc.
He returns to the Tosafot problem and presents a decision between fillings according to the number of points in the graph, such that one filling is preferable at one stage and the zero filling becomes preferable after a change, from which it follows that Tosafot’s argument was correct. He concludes that for this reason it also does not help to “rotate” an a fortiori inference in order to escape a refutation or *dayo*, because in the end one gets an equivalence that does not change under rotation, and therefore the balanced, standard a fortiori inference is one argument with two formulations.
Technical Summary: Filling a Missing Cell and Discovering a Parametric Theory
The speaker summarizes that the technique allows every table and every inference and every combination of refutations to fill in the empty cell by means of the same mechanism, without distinguishing between types of inference. He argues that the order in which the Talmudic discussion progresses is a didactic order for someone working with intuition and common sense, whereas in mathematics one can receive the data at the end and derive the result without going step by step. He says that the model also uncovers the theory behind the laws, that is, how many “components” or parameters are involved and which combinations of them are present in different actions, and from there one can ask how to identify the alphas and betas in halakhic terms such as benefit or other components.
Scientific Generalization, the Problem of Induction, and the Context of Discovery
The speaker argues that the model is a completely mechanical algorithm for making non-deductive inferences and serves as a supplement to Aristotelian logic. He illustrates scientific generalization in the form of a common-denominator structure concerning the falling of objects—a pencil and a ball—and identifies “mass” as the microscopic parameter that explains the conclusion. He argues that this provides, in a certain sense, a solution to the problem of induction by means of one assumption alone—Occam’s razor—and connects this to Hans Reichenbach’s distinction between the context of discovery and the context of justification, presenting the model as an algorithm for the context of discovery.
Mechanism, Artificial Intelligence, and the Distinction Between Imitating a Human Being and “the Right Answer”
The speaker presents a principled difficulty: if there is a mechanical mechanism for non-deductive inferences, then a computer could be a halakhic decisor, a scientist, or a judge. He says the realization struck him during a computer science lecture at Tel Aviv University, where a disagreement in artificial intelligence was described between the goal of reaching the best answer possible and the goal of reaching the answer that a perfectly thinking human being, as a human being, would reach. He states that the model does the second, and therefore it can make mistakes; that is why this is induction and not deduction, but even so one still gets full mechanization as an imitation of a human process.
Future Directions: Algorithm, Continuous Data, Data Mining, and the Space of Matrices
The speaker presents topics for future work, including finding an algorithm rather than merely proving it, and not merely proving the existence of a solution for every table and filling. He asks to develop a model for tables with continuous values such as grades, and to apply it to questions like predicting the collapse of companies on the stock market from continuous data. He presents this as data mining, where there is a missing item and the algorithm completes it, so this is not deduction but the accumulation of information, and he ends with the question of what mathematical space these matrices inhabit and how to represent it in order to solve the problems.
Conclusion and Handing Over to the Next Lecture
The speaker thanks “Rabbi Doctor Michael Abraham,” and it is said that the questions will be at the end and that there will be a round so as to get impressions from all directions and only then ask questions. The speaker invites “Rabbi Shapira, here Doctor, head of the Higher Institute for Torah” to deliver his lecture on “Does logic work?”
Full Transcript
[Rabbi Michael Abraham] What I want to do is briefly present what we did in the first project, in the first book. But before I start getting into the substance of the matter, just a short description. We—like I already said, the four of us, Roshi, Edgar, Babi, and I—the basic goal of the project is twofold: import and export, as Dובי mentioned earlier. Import means using newer techniques to decode things that appear in the Talmud, and export means finding all kinds of insights that emerge from the Talmud and seeing whether we can use them to help other communities or other fields of research. And at the bottom there’s just a kind of barbed remark reflecting some of the arguments among us, because there are people who suffer from formalophobia. Formalophobia is a fear of formalism, but there’s also the opposite disease, formalophilia. Formalophilia is the disease of people who think that if we translate this from Hebrew into x’s and y’s and mathematics, then we’ve brought redemption to Zion, we’ve fixed the world like in physics. And I just want to be cautious about that, because at least in my own personal view, that is not an end in itself. Formalization is worth doing if it helps us somehow understand the material we’re dealing with; the mere translation of it into formulas and mathematical mechanisms doesn’t really seem to me to have value in itself. So far, what we’ve done is four books—or three, with the fourth in preparation. The first deals with logical-deductive inferences, which I’m going to present; I’ll present its main points in a moment. The second dealt with textual hermeneutic rules, some of the textual hermeneutic rules—that is, general and particular, particular and general, general and particular and general—and there we saw that this is really an intuitive definition of sets, as opposed to formal definitions of sets. That was done with Yossi Maorbach and Gabi Hazut, the second book as well. The third book deals with deontic logic, meaning the logic of norms: obligation, prohibition, and permission, and the relation among them. There are all kinds of paradoxes in deontic logic that it seems to me can be dissolved through a Talmudic perspective—like the old joke, go find yourself a father and scatter. The fourth book, which is currently in preparation, deals with the logic of time, which was mentioned a bit earlier in connection with alternatives and conditions, going backward in time, forward in time, and so on. Okay, of course we’re dealing with the first book. So first of all, the first book deals with logical-deductive inferences, and as a point of departure I took one of them as an example—perhaps the most common one—and that is the a fortiori inference. I just want to show that an a fortiori inference appears in many, many contexts of thought that we use. An everyday a fortiori inference: Reuven passed the history exam and failed physics; Shimon passed the physics exam—he’s fixed the world—so will Shimon pass the history exam? I won’t forgive you, Horowitz. Will Shimon pass the history exam? So the a fortiori inference says yes. Of course it’s not necessary, right? Why isn’t it necessary? Because it could be that different skills are needed for physics and history, and indeed it turns out that even history requires skills. A scientific a fortiori inference: spacecraft A, with an engine of 1,000 horsepower, manages to escape Earth’s gravitational field; spacecraft B has an engine of 2,000 horsepower. The question is whether B will also manage to break free of the gravitational field. Here too you can make an a fortiori inference, and here too it’s not certain. If spacecraft B is heavier, then it won’t help that the engine is stronger, or if it arrives in a place where the air density is different—other data can refute this a fortiori inference. So it’s not a necessary consideration, but on the other hand it’s a consideration we do make, so it’s worth understanding it. A legal a fortiori inference, and an especially amusing one, which is why I chose it even though it’s not typical: there is a law called—or there was a law, I think it no longer exists today in Belgium—called the Vandervelde law, and that law prohibited selling two-liter bottles of wine. The question is whether it was also forbidden to sell three-liter bottles of wine. So apparently if you’ve sold three liters then all the more so you’ve sold two, right? So this is a simple a fortiori inference, and the Maharsha even says that there is no refutation of an a fortiori inference of “if it includes two hundred, it certainly includes one hundred.” But in fact even such an inference has a refutation. It’s not true. It was permitted to sell three-liter bottles, and the reason is that Belgian common sense, in this case, worked well, because the purpose of the law was to prevent workers from spending their entire weekly wages on a bottle of wine. But a three-liter bottle of wine costs more than a worker’s weekly wage, so that they didn’t prohibit. Okay. So those are three a fortiori inferences appearing in three different contexts. I’m returning to the halakhic a fortiori inference, because we’re going to work with it, but through it I want to show a completely general pattern of thought. In other words, everything I say in the Talmudic domain is also true in all the other domains—there’s nothing special here about the Talmudic context—and if we return to the export side, then the claim is that what we understand here actually explains parallel inferences in other areas too, not just in the Talmud. In this case, the very topic Dov spoke about was the topic through which we built the model—a topic in Kiddushin page 5. The purpose of the topic is to examine whether a bridal canopy can effect betrothal or kiddushin. It does effect marriage; the question is whether it can also effect kiddushin or betrothal. So the a fortiori inference is built like this: money does not effect marriage, but it does effect betrothal. A bridal canopy does effect marriage. So will a bridal canopy effect betrothal? Here’s marriage, here’s betrothal. So if money does not effect marriage but does effect betrothal, then a bridal canopy, which does effect marriage—one means yes, zero means no—then of course it will also effect betrothal. That’s the a fortiori inference. In fact every typical a fortiori inference is built in this way. And basically there are, if we formulate it just for what follows, two actions and two results. In other words, we have two actions: giving money and a bridal canopy, and we have two results: marriage and betrothal. And the whole question is who succeeds in doing what. So there is an a fortiori consideration here. Apparently a very simple consideration, right? Amichai asked me earlier, so why do you even need to formalize an a fortiori inference? Every child knows how to make that inference. And I want to show that there is still importance to formalizing the matter, so as to get out from under the suspicion that I suffer from formalophilia. So because of that, let’s try to see some of the problems that exist even in this simple a fortiori inference before I broaden out to even more complicated inferences. First of all I want to ask whether the a fortiori inference is symmetric under rotations. I came from physics; nothing to be done. So the original a fortiori inference is this one, but it can be formulated in two ways. It can be formulated as follows: money succeeds in effecting betrothal and does not succeed in effecting marriage. So clearly effecting betrothal is easier—sorry—easier than effecting marriage. So I have established a relation between betrothal and marriage, right? The relation is that A is easier—here larger means easier—easier to effect than marriage. Therefore, if a bridal canopy succeeds in doing the difficult thing, then certainly it will also succeed in doing the easier thing. This a fortiori inference is actually assuming a relation between these two. Now I make a reflection, okay? I rotate the thing. I have the same table, and clearly the a fortiori inference will work here too. Exactly the same thing, I’m just trying to switch the terms around, but it works the same way. Now what’s the difference? The difference is that in this a fortiori inference the basic assumption is different from the basic assumption in this one. Here we assume that betrothal is easier to effect than marriage. Here that assumption doesn’t exist—you don’t need it. Here the assumption is that the bridal canopy is stronger than money. Okay? So there are really two different assumptions here. So if I ask whether these are two different arguments or two formulations of the same argument, apparently the answer should be that these are two different arguments, two different formulations. What does it mean that they are two different arguments? It means that there could be something—Dov spoke earlier about attacks—there could be something that attacks this argument, while this argument remains valid, remains standing. One is attacked and the other remains valid. Okay? Or the reverse. Right. A rotation of an a fortiori inference. In other words, if I can prove that A is not easier to effect than marriage, for example by finding another action where the relation here is one and zero, the opposite relation, then I have broken the hierarchy between these two. But the hierarchy between these two has nothing to do with that. So I would expect that if, for example, there were a refutation of this a fortiori inference, one could always rotate it. But that turns out not to be correct.
[Speaker C] Because the hierarchy that the bridal canopy is stronger than money follows from your basic assumption that betrothal is weaker than marriage. Why?
[Rabbi Michael Abraham] How does it follow from there? I don’t see any connection. But let’s leave it, because I have a lot and I need to move quickly; afterward there’ll be discussions. The symmetry of an a fortiori inference under rotations—on the one hand it ought to be asymmetrical, as we saw before. Two different assumptions, each saying something else, and so each can also be attacked differently. Two examples to show that the Talmud at least sees it symmetrically. The first example is *dayo*: it is enough that what is derived from the law be like the original case. Let’s go back to the example of the grades in history and physics and quantify the argument, okay? One got 40 in history and 80 in physics. That can happen. 40 in history and 80 in physics. The second got 70 in history—notice, not 80, 70. The question is how much he’ll get in physics. Now, if I formulate the argument in the sense that A is stronger than N, then if in N he got 70, in A he will certainly get at least 70, right? But if I formulate the argument like this, meaning that H is stronger than M—or that he is more talented, not that the subject is easier but that the person is more talented—then the result will be 80, right? In other words, the result of applying these two a fortiori rules is a different result. That reflects the same asymmetry I was talking about before, right? But it turns out—and the same thing is represented by Rabbi Tarfon, who was mentioned earlier, it’s the same matrix—that in the Talmud they never reverse an a fortiori inference. Even when *dayo* is raised and someone tries to answer this *dayo* argument, where someone tries to infer 80 and the other says, hold on, hold on, hold on—you can’t get more than 70, because you learned it from 70. So apparently I would have said, fine, let’s reverse it into the opposite argument and then it really will be 80. No. *Dayo* works in both directions or in no direction. That’s what comes out from the Talmud. A second implication: column refutations or row refutations. Let’s go back to the original a fortiori inference. There is a refutation here. It’s redemption. The topic starts with this, right? Money does not effect marriage but it does effect betrothal. A bridal canopy effects marriage, so certainly it should also effect betrothal. That’s how the a fortiori inference is built. Then the Talmud comes and says: what is unique about money is that it redeems consecrated property and second tithe. Of course you can’t redeem consecrated property and second tithe with a bridal canopy—that would have been very cheap—but they didn’t give us that option. So because of that, what happens? What have we broken? The relation between these two, right? In other words, the a fortiori inference assumed that H is stronger than M. Not true—look, here it doesn’t happen. Has that broken the relation between these two? No. But it turns out that when this refutation is raised, they do not rotate the a fortiori inference. Both of them collapse. The Talmud treats it as though both versions of the a fortiori inference have fallen. Why? You could rotate the a fortiori inference, assume this assumption and not that one—sorry, this assumption and not that one—and this refutation does not break it. So again we see, both in *dayo* and the same with row refutations—it doesn’t matter, it’s exactly the same thing—that both *dayo* and refutation show that the Talmud treats an a fortiori inference as something symmetric, even though apparently these are two different things based on different assumptions. Okay? So that’s one more problem—or two more problems. Just parenthetically, I’ll add already: you have to understand that a refutation does not mean that the result here is zero. A refutation means that the result here remains with a question mark. It could be either zero or one. That will be important for us, of course, later on. Okay. Now I’ll bring in Kobi, and earlier Kobi said he wants us to convince the yeshiva students that they have something to do at the university too, so they can start fixing the world in physics. So here’s a story: my son came home from yeshiva last Sabbath, and he said they had an a fortiori inference in Tosafot in tractate Shabbat, and nobody knows how to explain it. It doesn’t work. What do you mean? Rabbi Yaakov of Corbeil asks: let us make the a fortiori inference this way. Whatever the details are—warp and woof, three kinds of threads, bones, not important right now—just as warp and woof, which are pure regarding creeping creatures, are impure regarding leprous afflictions—pure regarding creeping creatures, impure regarding leprous afflictions—so too a three-by-three piece, another kind of object, not important for us now, which is impure regarding creeping creatures, should it not all the more so be impure regarding leprous afflictions? Up to this point, that’s a basic a fortiori inference. Okay? And then Tosafot says: and one can say that vessels prove otherwise. Vessels prove otherwise, because they become impure regarding creeping creatures but do not become impure regarding leprous afflictions. So we have refuted this a fortiori inference. Okay? So far, a refutation like the one I showed on the previous slide, only in this case it’s a row rather than a column. Doesn’t matter. Okay? And if you say: what is unique about vessels is that they neither transmit nor become a tent over the dead while attached—now let’s rotate, now let’s add another column. Okay? Vessels are zero with respect to that property; they don’t do it. And the three-by-three piece and the warp and woof do do it. So one and one—we’ve refuted it. Why does that refute it? That was the question he asked. Why does that refute it? The property of vessels relative to the three-by-three piece and the warp and woof is irrelevant, because what we assumed—we assumed this hierarchy, not this hierarchy. In this case it’s a three-by-three table, not a two-by-two table, but it’s the same thing. What difference does it make whether vessels are more stringent or less stringent? Within the vessels themselves, this it succeeds in doing and this it doesn’t. So I have broken the asymmetry, the assumption of the a fortiori inference. What difference does it make now that I showed that vessels are less stringent than these two? And then comes the next stage in Tosafot: even the three-by-three piece also does not become a tent. In other words, here it should be zero, not one. So that’s interesting: so you succeeded in showing that this too is basically as bad as that one—so what? Why does that restore the a fortiori inference? One of the problems with formalization, by the way, is that something that seems very simple to understand—when you try to formalize it, you lose your mind. When you read the Tosafot you understand; none of this is understood. I mean, it seems obvious. But that was his question. Fine. So in a moment we’ll come back to this confusion at Mercaz HaRav. Just to sum up: we basically have the question why they don’t rotate an a fortiori inference, both with respect to *dayo* and with respect to refutations. The second question is that even in more complicated problems like the three-by-three one we just saw, where again it seems that the symmetry is operating, when you make that kind of refutation it also refutes that consideration. And there are still more problems that I haven’t presented here. As a result of all this, there are people who wanted to say—some people like saying this, I don’t know, I think Rabbi Shapira says this too, I don’t know if he likes saying it, but he does say it from time to time—that an a fortiori inference is some kind of rule that isn’t logical. I don’t know what will happen when he hears whether logic works or not, but I want to claim that this is indeed a logical rule, and therefore I move to the next slide. Okay, in order to understand the hermeneutic rules, we need a language. And the hermeneutic rules basically tell me this. I’m speaking now about the logical hermeneutic rules. To define what logical hermeneutic rules are, I say: an a fortiori inference, a refutation of it, refutations of the refutation, and so on; an inference from a prototype, a refutation of it, and refutations of the refutation, and so on; and the common denominator—that is, an inference from two verses, with refutations and so on, and all their combinations. For me, that is the world of the logical hermeneutic rules. Two verses that contradict one another is also something worth discussing, but it is not in this scheme. The rest are more textual rules and don’t belong to this family. Okay, so the language is as follows. We got this language from Avi Livshitz, someone from Jerusalem who was with us at the beginning and left us at some point. In any case, this language is his. He proposes presenting everything in a table. Look—for example, this is an inference from a prototype based on two verses, the common denominator. So here I have two sources and a target. We want to learn the target from both of them. Okay? So it starts like this: we learn by an a fortiori inference—this block of four. Okay? And now we refute the a fortiori inference; we already saw that. The refutation of the a fortiori inference. The whole darkened frame is an a fortiori inference with a refutation against it. Now we say, okay, the a fortiori inference didn’t work, close the store, let’s start over. Let’s look at this, the darkened part. Right? That is an inference from a prototype. We try a prototype inference, and now we raise a refutation. This darkened part is a refutation of the prototype inference, right? Because we are making some analogy between these two, and this tells us that it doesn’t work. Here too, of course, you break this analogy but not necessarily that one, exactly as with the a fortiori inference. But that’s the refutation. What does the Talmud do after it raises these two things? It joins the a fortiori inference and the prototype inference into a full table. That is what is called the common denominator. And lo and behold: neither of them works alone, but without adding any further datum, this set of data leads me to a result of one. Neither one works by itself—not the prototype inference and not the a fortiori inference. Make the full table with those same data, without adding any data, any refutation, or anything at all, and the result is one. Okay? That’s point one. By the way, a prototype inference can either be based on two a fortiori inferences, and then there will be zero here and zero here, or on two prototype inferences, and then one here and one here, or on a prototype inference and an a fortiori inference. There are three types of common denominator. Okay? The difference is simply in this column, which shows that the result is not sensitive to the values in that column, right? Because no matter what is in that column, the result is always one. Okay. Fine. One more example just to get an impression—I won’t explain it at all. The topic of the bridal canopy at its conclusion—this is ultimately its picture. After the common denominator and an a fortiori inference and refutations and a prototype inference and another refutation and an even larger common denominator and refutations of that and changes in the refutations and all sorts of things like that—this is what you get at the end. And now I’m not sure a child in second grade could do that. So what we propose doing is a chemical analysis of concepts. What does that mean? When we want to understand phenomena in the world in general, in a scientific world, what we really want to understand is what components generate the phenomenon. So if I see here money and a bridal canopy that succeed in effecting betrothal and marriage, usually we are used to looking at this on the phenomenological plane: who is strong, who is weak, who is a bully—who manages to overcome whom? But in truth there is a theory sitting behind it. There is a theory behind it saying that within money there is some component that succeeds in effecting betrothal. In other words, betrothal has the active ingredient that can effect betrothal, if we speak in medical language, and regarding marriage that active ingredient is not relevant. It will not effect marriage. And in a bridal canopy there are components that will succeed in effecting both. Okay? So that is basically the meaning of the chemical analysis.
[Speaker D] And now I’m saying, do you want to test that? Huh? That marriage contains the component that creates betrothal—do you want to test that?
[Rabbi Michael Abraham] So yes, I want to test it. So the fact that there are two possibilities—we’ll see in a moment. No, a question mark. A question mark, and I want to check what’s correct. So what am I saying? To check what’s correct, I do an analysis.
[Speaker D] What does a bridal canopy mean without prior kiddushin?
[Rabbi Michael Abraham] No, no, the Talmud itself says that—it brings that refutation itself at the end—but I’m ignoring it for now because I’m talking only about the scheme. Okay? So look: what I’m really saying is this. What we see is something like this. There are two ways to fill this in: either one or zero. Okay? We ask ourselves which is the correct filling. The answer we propose is that the correct filling is the one whose underlying theory is simpler. Okay? The theory underlying a filling of one is this theory: there is a single parameter, there is only one active component in this system, in this representation. It appears in different intensities. Money has it at one level of intensity, and the bridal canopy has it at intensity two. To effect marriage you need two, and therefore only the bridal canopy succeeds, and that is why marriage is two; and for betrothal one is enough to effect it. Okay? Notice that the greater the intensity, the easier it is to effect, and that means it has stronger power. But how did you rule out the possibility that it’s completely different? What?
[Speaker B] How did you rule out the possibility that the bridal canopy doesn’t have the component that creates betrothal because betrothal is something else?
[Rabbi Michael Abraham] No, wait, wait, wait, one second, I’ll explain in a moment. As for this—if I fill in a zero here, there’s no model with one parameter that will manage to explain it. We’ll need two in any case; the two columns are independent. Okay? So there has to be an alpha for these and a beta for those; alpha does this and beta does that. Now, Rabbi Shabtai asks, quite rightly, maybe here too there are alpha and beta? After all, there are alphas and betas that would explain this. We take the simplest model—that’s Occam’s razor. Occam’s razor says you always take the simplest model among the possible models. The simplest model here has one parameter; the simplest model there has two. There is no explanation of this table with one parameter, so the filling is one. What does that actually mean? Notice what this now says about a refutation. I’m going back over the exercise now in order to analyze a refutation table. The table of a refutation that I showed earlier can be either zero or one; we don’t know. I’m now checking what theory explains this and what theory explains that. Now, how do you do that? There is some algorithm—we won’t go into its details. We basically start drawing a diagram, because when you saw the big table I drew earlier, it already becomes very hard to see the alphas and betas, which theory is the one that explains it. You need to develop a mathematical model that takes me from the table to the solution: what components there are in N, in I, in N, in I, and in P. So there is such an algorithm, and this algorithm is built as follows. I basically say: I set an ordering relation between the columns. So look, for example, in this case, I is the strongest, right? It has the highest values. P is weaker, and N is weaker, while N and P are independent. So I represent it like this. This is the strongest; this one goes into it because it is weaker than it, and this too is weaker than it, and between these there is no connection. So that is the model. What happens here? Notice that these two suddenly become identical and independent of this one. So here there are two independent ones, where here it is both I and P because they are identical. Already once you draw it, there is a simple and clear way to get directly to the solution. You get directly to the solution, and the solution is that here you need two parameters and here you need two parameters. Here you need alpha and beta, and here you need alpha and beta; the solutions are written here. This is alpha and beta, this is two alphas, this is alpha, and here are the solutions for the operations. It doesn’t matter right now; we won’t go into the details. Since in both fillings, both zero and one, you need at least two parameters, that means the fillings are equivalent. If the fillings are equivalent, what does that mean? That’s a refutation. Right? A refutation is not that the result is zero, as I said earlier. A refutation is that the two fillings are equivalent.
Okay, now that wasn’t enough. We got stuck later in the passage, because later in the passage there were things that the parameter of number of parameters was not enough to decide. The simplest example is a paradigm case. The simplest there is. A paradigm case, of course, is a one-one-one table with a question mark. We saw it earlier. So if we fill in a one, then everything is one, if of course it’s only one point. Both N and M are the same point, and that is alpha. There is only one parameter involved here, but here too there is only one parameter involved, and it turns out that you can’t decide. A paradigm case comes out such that the result is not one—it’s a refutation; you can fill it in with either one or zero, however you like. Now someone could say that here there is alpha, and two alphas is still more complexity. We prove that this is not true. Meaning, in other arguments you can see that if there is a difference in the level of intensity of the parameter, that does not decide the issue. So what does that mean? It means that we need to introduce additional parameters into the decision criterion. Which filling is preferable? Not only how many parameters the theory requires, but also the properties of the graph. The properties of the graph basically mean what relations there are between the columns, how closely related they are to one another. The more they are related to one another and there is some ordering among them, then the answer is simpler. So because of that, this is of course simpler, because there is only one point here; this is a very simple graph. This graph is more complex, so it is less simple. So here we decide it on the basis of the number of points in the graph, even though in terms of number of parameters it is the same thing. By contrast, in other graphs it turns out that we need three independent indices. By the way, from the outset we thought that if topological indices were needed, these were the three candidates: connectedness—meaning, if it were independent then there would be two here that are not connected; that counts against the graph, it is less simple—the number of independent points, which is what decided here, and the number of direction changes in the graph, meaning the relation between the arrows; we won’t get into that right now. The full preference criterion is built like this: we check the two possibilities, a zero filling and a one filling, in terms of four things. First, the number of parameters there are in each theory, for the one-filling and for the zero-filling; the connectedness of the two graphs; the number of independent points; the number of direction changes. If there is a decisive advantage in one direction across all the parameters—it is at least the same or better—then the inference is decided. If there is ambiguity, meaning it has an advantage in this but a disadvantage in that, or vice versa, it doesn’t matter how many advantages there are—you know that for a refutation it is enough to raise some difference. Even if there are many differences in favor of one side, if there is just one difference in favor of the other side, that refutes the inference. Therefore, as far as we are concerned, the moment it is not univocal, that is a refutation.
Now, something very interesting turns out. We chose these three things more or less arbitrarily, in consultation with graph theory; it seemed to us that this measures the simplicity of a graph. After that we went on to examine the common denominator, and it turns out—as I said earlier—that there are three kinds of common denominator: a common denominator built on two a fortiori arguments, on two paradigm cases, and on an a fortiori argument plus a paradigm case. It turns out that one of them is decided by this, the second by this, and the third by this. And I’m saying: we found that only after we had defined the parameters, so for us that is confirmation. It means all three are necessary, because if any one of them were missing, it would be impossible to decide the common denominator, and that means these are probably the right three, since we found that only after defining the parameters. A second confirmation: in the big table I showed you earlier, we sweated blood over it for two or three days; we couldn’t figure out how the result was one or zero, whichever it was supposed to be. We said, okay, something is broken here; we need to throw out this model. And what turned out the next day? That we had simply gotten confused: in one place we put in a zero in one of the tables. It was impossible to produce a reasonable decision criterion that would give us the correct result, and in my opinion that is also excellent confirmation of the model, because it means this is not ad hoc. It means that if something here doesn’t work, we won’t manage to force it in ad hoc. Okay?
Okay, the next stage: we go back to Mercaz HaRav, solving the problem. Now we’re already equipped with a technique. So here we have a three-by-three. The one is the first stage there, and zero is the refutation of the refutation, and this is the refutation of the refutation of the refutation—the red one. Notice what happens: in black, these are the two fillings. Which is preferable? This one is preferable. Neither of them is valid, right? But this one has two points and this one has three. And when you replace the one with a zero—that’s the red—it is preferable. So this is the one-filling and this is the zero-filling. Okay, that’s all. Therefore it is clear that Tosafot made a correct argument. More than that—notice that already here, just as in an a fortiori argument, I asked why the Talmudic text, when it raises a refutation, does not reverse the a fortiori argument in order to escape the refutation or the limitation. It turns out that it would not help anyway, because whenever we insert a refutation, the model ultimately comes out equivalent for a one-filling and a zero-filling, whether you reverse it or not, because it does not change the model. Therefore it is clear that the a fortiori argument really is not two different arguments. Contrary to what people think, the balanced a fortiori argument and the straightforward one are not two different arguments; they are one argument, two formulations of the same argument.
Okay, just so you can get a sense of it, this is the big table I showed you earlier—this is the solution. Okay, summary: what we actually learned here is two things. First, we have a technique for every table, every kind of inference, every combination of refutations, a fortiori arguments, common denominator, whatever you want. Fill it in and that’s it. We do not distinguish at all between types of inference. Fill in ones and zeros in the matrix for me, and I’ll tell you what the result should be in the empty square. That’s all; it changes nothing. The order in which the passage proceeds is unimportant. The order is a didactic order, for someone working with intuition, with common sense. In mathematics, give me the data at the end and I’ll tell you the result; there’s no need to go step by step. But notice what this model does. It does two things. First, it fills in a missing square. Second, it reveals to us the theory that stands behind these laws. That is scientific generalization. In other words, this model basically tells me: look, in this problem of marriage, betrothal, redemption, various components are involved, parameters. I don’t know who they are; I don’t know how to identify them. But there are four of them—that I know from the previous slide. There are four of them. And I know that one of them exists in canopy, money, and intercourse, but not in document, and the second exists in a different combination, and so on with regard to the results. Now I can come and ask myself how to identify who these alphas and betas are. So I have advanced the process of understanding a very significant step forward. I not only made an a fortiori argument; I ask myself which parameter, in the eyes of the Torah, really succeeds in effecting marriage or betrothal. So if I see, for example, that it is only money or intercourse, it is clear to me that this is benefit, right? Which is absent in canopy and document, for example. Or all kinds of things of that sort. So I can decipher it. That is exactly the method of scientific generalization. So I can find the theoretical model and fill in the missing square.
Just one note, which I also wrote here: the interpretation of the parameters—who is alpha and who is beta—is a bit problematic; it is not quite as simple as I described earlier. It could be that one of the relevant parameters is half benefit plus three-quarters transfer into one’s domain, and that would be called alpha. You need to rotate this in parameter space, diagonalize it, in order for these to become meaningful parameters. That’s just for the connoisseurs. The general significance of this inference basically says the following. What we found here is a completely mechanical algorithm for making derivations, for making non-deductive inferences, which is almost a contradiction in terms. That is why logic does not deal with things like this, because it works with clear and mechanical answers. Here there is, in effect, a completion of Aristotelian logic. All the inferences—I gave earlier examples of a fortiori arguments from all kinds of fields. Scientific generalization, for example—what is scientific generalization? I see a tree falling to the earth and a ball falling to the earth, and then I ask: will this microphone also fall? Don’t worry, I’m not going to do it. Will this microphone fall? And then I ask: can the pencil teach me about the microphone or not? What applies to the pencil—namely, that it writes—this one doesn’t write. So I say, fine, the ball will prove it. The ball doesn’t write. Then I say, what applies to the ball—namely, that it is round? The tree will prove it. And the argument comes back around. The common denominator of the tree and the ball is that both have mass; therefore this too has mass. And what is the mass? It is the microscopic parameter of the problem; that is alpha. So that is how scientific generalization works. In other words, in every field where we make generalizations from data to theory—scientific generalization—there is a completely mechanical way here to do it.
And therefore, what this gives is, in a certain sense, a solution to the problem of induction. The problem of induction, which Horeliss presented earlier, basically says: how do we take data and turn them into some theory? You can generalize them in many ways. How can I build a theory from data, okay? In an inductive way, yes? Some kind of generalization. Here there is a process that maps it completely by means of only one assumption: Occam’s razor. That is, that you take the simplest model. Once you take the simplest model, you can explain exactly what in philosophy of science is called the context of discovery. Hans Reichenbach distinguishes there between the context of discovery and the context of justification. The context of discovery is how one discovers a theory, and the context of justification is how one justifies it, how one tests it in the laboratory. So the context of justification is entirely logical. But the context of discovery is usually treated by philosophers like divine revelation. What we did here is an algorithm for the context of discovery. That is how one discovers—without, of course, identifying the alphas and betas.
[Speaker B] You’re using your own tolerance rules, yes, obviously.
[Rabbi Michael Abraham] I’m saying: Occam’s razor. The big question that really troubled us in this context was: how can this be? If there is a completely mechanical mechanism that makes non-deductive inferences, that means that a non-deductive inference has a clear, predetermined, mechanical answer. So a computer can basically be a halakhic decisor, a computer can be a scientist, a computer can be a judge. All the inferences in all the fields I discussed earlier can be computerized. Basically, it’s a table; I can compute them. How can that be? So here the penny dropped—at least for me—in a lecture we gave at Tel Aviv University in computer science, and there they said that there is a dilemma for people in artificial intelligence. Artificial intelligence people are divided among themselves over the ideology of the field. Some argue that the goal of the machine, of artificial intelligence, is to reach the best answer possible—it sounds banal. Others say no, the goal of artificial intelligence is to reach the answer that a human being who thinks perfectly as a human being would reach. Sometimes a person makes a mistake, but I can do it as well as possible the way a human does. If a human would make a mistake, I can’t be better than him. What our model does is basically this and not that. Therefore it can make mistakes; that is why this is not deduction but induction. But the perfect mechanicality means that I can mechanically imitate what a human being does. Not that I can mechanically arrive at the correct answer. That is an entirely different question.
And just to finish, we have quite a few topics—only some of them I brought here—for further work. One is to find an algorithm, not to prove that this algorithm works. Not to prove the existence of a solution for every table and filling. To make a model for tables with continuous values, like grades in history and physics—there it wasn’t zero or one but seventy, eighty, and forty. So how do you make a model that basically predicts the collapse of companies on the stock exchange? So I need to take continuous data—their profit percentages, their asset percentages, all kinds of things like that—and ask the question: will it collapse next year or not? In principle, this model is supposed to do that. To do it to the extent that a human being can do it. Again, not to arrive at the correct answer, but to arrive at the answer that a rational human being arrives at. Okay? But for that, of course, you need a model that is not just zero-one but more than that. And what we are really talking about here is data mining. Basically, I have one missing item here and I use the algorithm to complete the missing item. That is why this is not deduction, because I accumulate information through this process. I am not merely exposing information that already exists with me, so this is basically a non-deductive algorithm. And of course, at the root of everything, the question is: these matrices lie within some mathematical space, and the question is how to represent it in order to solve all the problems here. So that is all.
[Speaker E] Thank you very much to Rabbi Dr. Michael Abraham. The questions will be at the end. We’ll do a round; it’s worth waiting in order to get an impression from all directions and then ask questions, sometimes to two people at once. I would like to invite Rabbi Shapira, here a doctor, head of the Institute for Advanced Torah Studies, to deliver his lecture on whether logic works.