Fuzzy Logic and Data Mining in the Talmud and Beyond – Lesson 2 – Rabbi Michael Avraham

[Speaker A] Mm mm mm mm. Okay, we’re recorded, just like that.

[Speaker B] That’s it, everything is now on the record.

[Speaker A] Oh, hello, Rabbi. Peace be upon you. All right, I hope the meeting will join, but notice that when the meeting joins you’ll have to mute and all that. But don’t do it yet, otherwise we won’t hear you.

[Rabbi Michael Abraham] No, right now I’m… I understand. There are various people joining, I gather.

[Speaker A] No, are you, are you joining as… like, are you joining the Zoom?

[Rabbi Michael Abraham] Wait, so I’ll mute.

[Speaker A] Okay, let’s see when they arrive, God willing.

[Speaker F] Now we’re in suspense. Go ahead.

[Speaker A] Okay, we’re just waiting for you to join from the main computer, because right now…

[Speaker F] Yes, so there’ll be a meeting we can record. You didn’t think of that?

[Speaker A] You didn’t think of that? You didn’t think. No, don’t let Nimrod run this, it doesn’t work. As long as he doesn’t run it, everything else is fine, but Hananel is there, and I trust Hananel.

[Speaker B] Remihi, we can’t hear you.

[Speaker A] I think now we need to do some abduction from this, right, and then we’ll understand, we’ll make some kind of theory about…

[Rabbi Michael Abraham] Wow, maybe I shouldn’t disconnect my thing.

[Speaker B] I’ve got a whole collection of theories, what? A whole collection.

[Rabbi Michael Abraham] Zoom isn’t high-tech; we don’t manage with Zoom.

[Speaker B] Fifty academic degrees in one room and Zoom beats you all. Unbelievable.

[Speaker E] So should we do it like this?

[Rabbi Michael Abraham] How is it now? Is it being recorded?

[Speaker A] Yes, it’s being recorded. In principle we can start, since the most interesting figure is here and we can hear him and he’s being recorded, and if during the meeting someone manages to put you up on the main screen so we can also see you and all that, that’s fine too. But beyond that, I think we can start.

[Rabbi Michael Abraham] Okay, I’ll sing over this catastrophe. Okay, I’ll just briefly remind you what happened last time. I tried to present, in a nutshell, the motivation for this whole matter and the context in which I place it. Basically I spoke about the relationship between logic and mathematics on the one hand and science on the other: science deals with accumulating new information, while logic and mathematics, in principle, extract more and more information out of some set of assumptions. Without going into details, you can define it in different ways, but that was the direction. So what I said was that my motivation was to try to define in a more systematic way how we gather information, because logic usually doesn’t deal with that. Logic usually deals with how we analyze existing information, how we get more and more information out of the assumptions, and therefore in science we’re actually dealing with a softer kind of logic, let’s call it that: analogy and induction. I made a distinction there between deduction and analogy and induction. I said that Francis Bacon tried to present certain schemas of scientific inference, but that doesn’t really reach the point of formalization; it’s some kind of elimination of factors. And what I’m trying to do here is something that already comes closer to formalization. Of course it doesn’t thereby become deduction, otherwise I’d have thrown out the baby with the bathwater, but I do think it gives some more orderly tool for carrying out this process of accumulating information, or gathering information. This process lies at the basis of scientific work, legal work, everyday thinking, halakhic thinking—basically all non-mathematical fields. All non-mathematical fields in fact also require tools of a different sort in order to get from assumptions to conclusion. And therefore, in a certain sense, it seems to me that this formalism sheds light on—or even makes it possible to locate—fundamental processes of thought in all those fields. What the practical level is—maybe I’ll comment on that later. They mentioned Solomonoff, and I said it’s somewhat similar. Maybe at the end, if possible, we can discuss that. I’m really not an expert in the matter, but it seems to me to be something like Solomonoff induction. My basis is actually that I begin with an a fortiori argument. An a fortiori argument, what’s called in Latin a fortiori.

[Speaker G] Can you zoom out and then… what? Maybe shrink the document, zoom out.

[Speaker C] I can drag the bar.

[Rabbi Michael Abraham] I’m not getting the hang of this thing.

[Speaker C] You can drag the bar from the side…

[Rabbi Michael Abraham] Ah, that’s what I tried to do, but

[Speaker C] without

[Rabbi Michael Abraham] much success.

[Speaker C] Oh, nice. Okay.

[Rabbi Michael Abraham] So I start with an a fortiori argument, as it’s called, a fortiori reasoning, which is basically an inference from the lighter to the stricter case, from what is less clear to what is more clear. So there are examples. Say, a biblical example: הן בני ישראל לא שמעו אלי ואיך ישמעני פרעה (“Behold, the children of Israel did not listen to me; how then will Pharaoh listen to me?”). Right? If the children of Israel didn’t listen to me, then Pharaoh certainly won’t listen to me. That’s a kind of a fortiori reasoning. Or if a person can’t lift one kilogram, then obviously he won’t manage to lift fifty kilograms. Or say if a spacecraft with a certain horsepower succeeds in escaping Earth’s gravitational field, then with greater engine power it will certainly escape Earth. The same thing—there are all kinds of a fortiori arguments in all kinds of fields. There’s an interesting legal a fortiori argument that maybe I’ll come back to later. I saw it in a book by Perelman, Chaim Perelman, who was a philosopher of law and a logician of law. He brings the Belgian law, the Vandervelde law, which says that you’re not allowed to sell two liters of wine in a pub to workers, so that they won’t spend their whole weekly salary on wine. Now somebody came and wanted to buy ten liters of wine. So they told him, impossible, the law forbids it. He said, “No, the law forbids two; I want to buy ten.” So that’s a fortiori, right? It turns out that when it reached court, the court said the buyer was right, not the seller. Very interesting. In any case, maybe we’ll get to that later. So there are all kinds of types of, or all kinds of contexts for, a fortiori arguments. In the Talmud, a fortiori reasoning usually appears in a different structure, because the a fortiori arguments I’ve brought until now are based on one datum and infer from it a conclusion. “Pharaoh won’t listen to me”—sorry—”The children of Israel didn’t listen to me,” conclusion: “Pharaoh also won’t listen.” One datum, one conclusion. In the Talmud, usually the a fortiori argument is based on three data points, not one. So I call this Talmudic a fortiori, as opposed to biblical or everyday a fortiori. And it works like this. Say, also in ordinary life—I brought an example here: Reuven passed the history exam and failed physics. Shimon passed the physics exam. Will Shimon pass the history exam? Right? You can say a fortiori—that is, if he passed physics, which is harder, because we see that Reuven failed physics, then if he passed physics, which is harder, he certainly should pass history. Everybody understands that this isn’t necessary, but it’s an a fortiori argument. Talmudic a fortiori is built in a very similar way. Say we want to know how betrothal is effected—betrothal with a “b,” right? Kiddushin or marriage. Kiddushin is the act that precedes marriage, what is done under the wedding canopy with the ring and so on. So there is a discussion about how kiddushin is actually effected: by money, document, and intercourse. Those are the three options for effecting kiddushin. The Talmud discusses what about the wedding canopy. The Talmud says: money can effect kiddushin, but cannot effect marriage. So the wedding canopy—the wedding canopy, which can even effect marriage, certainly it can also effect kiddushin. In the end this is rejected, but that’s an example of a fortiori reasoning that appears in the Talmud, and again this is a fortiori from three data points. And still the feeling is that these are similar arguments. Why are they similar arguments? Well, in the simple sense, it seems to me that in the ordinary one-datum a fortiori—הן בני ישראל לא שמעו אלי ואיך ישמעני פרעה—there is a hidden assumption, which is the hierarchy rule. Right? That obviously Pharaoh will listen to me less than the children of Israel. So if the children of Israel didn’t listen, then Pharaoh certainly will listen even less. So in fact there isn’t just one assumption here; there are two assumptions. One assumption is the datum, and the second assumption is the hierarchy rule. Okay? Now where does the hierarchy rule there come from? How do you know that Pharaoh will listen less? Common sense says so, reasoning. Right? It seems pretty clear to us that Pharaoh will obey Moses less than the people of Israel. That reasoning doesn’t always work, but that was the assumption there. In the three-datum a fortiori, the same thing happens. It’s just that the hierarchy rule is extracted from two of the data points. In other words, think, for example, about the case of Reuven with history and physics. From Reuven’s achievements, we reach the conclusion that history is easier than physics, right? So I used the two data points about Reuven and extracted from them a hierarchy rule. What now remains is the a fortiori argument we saw: one datum and a hierarchy rule, right? Shimon passed physics; the hierarchy rule is that history is easier to pass than physics; therefore he certainly—or not certainly, but probably—will also pass history.

[Speaker H] You’re saying that you’re defining a dimension that’s actually implicit and saying where it falls on the spectrum. And there could be more than one dimension. I didn’t understand. The level of difficulty of some material is a spectrum of learning difficulty, but there are probably additional dimensions too.

[Rabbi Michael Abraham] And that’s the reason why it’s not necessary. Right? Yes, of course. So here we’re assuming this is not an argument from the deductive world. That’s why I say: I’ll try to formalize it, but it isn’t from the deductive world. So obviously there are some hidden assumptions here that can be refuted or challenged, or—

[Speaker H] discovered, exposed.

[Rabbi Michael Abraham] Yes, we’ll see that later. So therefore, basically we’re dealing here with two types of arguments that are of the same kind. The difference between them is only the question of where the hierarchy rule comes from. Does the hierarchy rule come from a priori reasoning, say—or it doesn’t matter, even a posteriori, but not from the data in front of us—or is the hierarchy rule derived from two out of the three data points? Then I have a hierarchy rule and a datum, and like with the mathematicians’ kettle, I already know what to do with this thing.

[Speaker C] Now there’s another assumption here too—that you can derive the rule… the hierarchy from the two.

[Rabbi Michael Abraham] Right. That very derivation is, of course, one I assume can be done. It’s basically a kind of generalization. I say: if this degree of severity, or this relation between history and physics, exists for Reuven, then it probably exists for all human beings. And you can say that there is some general hierarchy here. Yes, of course, there is… I talked about this last time, that behind analogy there sits a hidden induction. So here’s the example. That is, a fortiori is some kind of analogy, a somewhat different kind of analogy, and behind it there actually sits a hidden induction. And that brings me to the note at the end of the page. Right, I ask whether this argument is deduction. Now when you ask ordinary people, you often get the answer that yes, it looks terribly necessary, as if it were self-evident. But you’ve already seen for yourselves that it isn’t. There are all kinds of hidden assumptions here that I need to add in order to reach the conclusion, and therefore this is not deduction. By the way, Adolf Schwarz, one of the earliest scholars of the hermeneutic rules in the Rabbinical Seminary in Vienna, for example argued that a fortiori is full deduction. But that’s simply not true, it’s not…

[Speaker E] What you’ve brought here is a process of partial deduction, and in order to complete it you activate abduction.

[Rabbi Michael Abraham] Exactly. Induction. We’ll get to abduction in a moment. For now, induction. To say that if for Reuven it is true that physics is harder than history, then it is true for all human beings—that is not abduction, it’s induction.

[Speaker E] Fine, okay. I’m saying there was some implicit assumption here.

[Rabbi Michael Abraham] The assumption that a generalization can be made.

[Speaker H] The assumption is—

[Rabbi Michael Abraham] that one can make the—

[Speaker H] this generalization, that there is this hierarchical relation. That’s the assumption. No.

[Rabbi Michael Abraham] The issue is that there is this hierarchical relation. To say: I have this hierarchical relation for Reuven. Now, okay, now I assume that Reuven is a representative sample. In other words, that I can learn from him to all other people, that it will be true for everyone. That assumption is, of course, one you can argue with. Fine. That’s actually the induction assumption. It gives… I’m saying just as it’s true for Reuven, so it’s true for all human beings. You’re saying this would be deduction if enough necessary conditions were present? If everything were present. Look, any non-deductive argument can be completed and turned into a deductive one, right? Make an analogy: this frog is green, this one is a frog, so this one is also green. Right?

[Speaker H] That’s analogy, not deduction.

[Rabbi Michael Abraham] But obviously if I add another assumption, that every color true of

[Speaker E] this frog is true of all frogs—once I state the assumption explicitly. That completely loses the distinction between deduction and something else if we make that completion. I don’t know what problem you’re trying to solve. I mean, if we have a set of facts and a law, okay? You say: I want to understand what follows from it—that’s the problem of what follows from it. That’s one thing. If you come and say: the problem I’m trying to solve is that I see here a process of a fortiori reasoning. That is, I look here at a linear process where I already understand its feature, but I can’t say what law was applied—that’s the problem I’m trying to solve, what the hierarchy relation is, say.

[Rabbi Michael Abraham] So that’s what I’m trying to understand.

[Speaker E] So you’re trying to expose the implicit assumption?

[Rabbi Michael Abraham] Yes, later on. Yes. So this isn’t abduction?

[Rabbi Michael Abraham] No.

[Speaker E] Why? Because here, what is the assumption you have here? Go back to this example of history and physics.

[Speaker E] Why here the a fortiori rule of “if the children of Israel listened”—right? A person that Pharaoh? One explanation is that the children of Israel listen more than… there could be another explanation.

[Rabbi Michael Abraham] An explanation of what? You have no explanation here, it’s a rule. You have one datum; that datum is not explained by the rule. You have two assumptions: the datum and the rule. The rule explains nothing. It’s simply two assumptions, one of which is a datum and the second is a rule. When I have three data points, then you can talk about induction or abduction. Because then you take two data points and create from them a rule. Okay? In the first case the rule is just a datum, it is not… that’s it. In the second case we’ll still talk about abduction; right now it’s only induction because you take the two data points about Reuven.

[Speaker E] The activation itself is inductive, and from it you’re basically trying…

[Rabbi Michael Abraham] Okay, that I’ll do later. That I’ll do in a minute. So therefore it’s clear that this isn’t deduction; we need to use further assumptions. When we refute a fortiori reasoning, for example, if someone wants to challenge this inference about history and so on, he’ll bring an example of someone else who succeeded in physics and didn’t succeed in history. Right? That would be a refutation of the a fortiori. What does he refute? He doesn’t reject any of the data; the data remain exactly as they are. This one succeeded, this one did not succeed. Those are data. What he refutes is that hidden assumption that if it holds for Reuven then it holds for everyone. Not true. It may hold for Reuven, but for others it may be that they do better in physics than in history. Okay? So therefore, it is precisely the hidden assumptions that very often we attack with refuting arguments, with objections. The explicit assumptions are often easy to state, because those can’t really be attacked. But often we hide the more sensitive assumptions, and those are the ones at the center of the discussion. Okay, so let’s look at the general structure of a fortiori reasoning. The general structure of a fortiori reasoning—now I also move to analogy—the general structure of a fortiori reasoning. Now let’s go back to the Talmudic example I brought earlier. This is money and wedding canopy, and this is marriage and betrothal. And this is a typical structure; every a fortiori argument is built like this. Every a fortiori argument. Assuming for now that it is binary. In a moment we’ll see what happens when it isn’t binary. Assuming it’s binary, it’s zero or one, meaning either it effects kiddushin or it doesn’t. So this is the a fortiori table; there isn’t another one. A table of three data points. You have money doesn’t effect marriage but does effect betrothal. The wedding canopy effects marriage, and now the question is whether it also effects betrothal—question mark. And the a fortiori reasoning says yes. Okay? In other words, the data, the three data points, are here, and the question mark—in the a fortiori argument I fill the question mark with a one. I give the answer one. Now while we’re at it, I’ll also present analogy. Analogy is basically the same table, except that here there appears a one. Why? Because it’s basically analogy. I say: just as intercourse effects both marriage and betrothal, I assume that the wedding canopy is the same thing—it too effects both marriage and betrothal. I make an analogy between intercourse and the wedding canopy. Okay? Here it isn’t exactly analogy; a fortiori is a kind of analogy, but it’s not exactly analogy because you see that it isn’t similar. Here it’s zero-one, and here the claim is that it’s one-one. But there is something stronger here. It’s not similar, it’s stronger, but it still is.

[Speaker H] Over a binary field for now.

[Rabbi Michael Abraham] What? Yes, for now I’m talking only over a binary field. So those are basically the two tables, of a fortiori and analogy. Now the question is, what we want to know is how this thing works. And it isn’t all that simple. Why isn’t it all that simple? Because let’s look for a moment at such a table—say look above. You can present this table in the way it appears here. Now, how would I formulate the a fortiori? Suppose I look at the top row, the row of money. Jews always start with money, right? So let’s look at the row of money. Betrothal is stronger than marriage, right? You can see that in this row. Now I go down and say: fine, if so, then here too betrothal will be stronger than marriage. So if marriage is one, then betrothal is also one. That’s basically the argument, what stands behind the a fortiori reasoning. But of course there is also a vertical argument. I rotated the matrix for convenience, but you could leave it as it is and go by the columns. Look at this column. In this column I see that the wedding canopy is stronger than money, right? Teaching you, as Jews, that there are things stronger than money. So if with regard to marriage we saw that the wedding canopy is stronger than money, then with regard to betrothal too the wedding canopy should be stronger than money, and therefore it works. Right? So here the assumption is that betrothal is easier to effect than marriage. And here the claim is that the wedding canopy is a stronger instrument than money; it is able to effect more things. Okay? That’s the meaning of “stronger,” right? Is that the same argument? Are these two different arguments or two different formulations of the same argument? Now, this appears in every a fortiori argument, because a fortiori is generic; this is the a fortiori table. In every a fortiori argument, you can go by the columns or by the rows. Two different arguments. Agreed?

[Speaker I] Meaning, in certain situations you can infer about one from the other, but for present purposes it’s not the same set that contains the same set.

[Rabbi Michael Abraham] Basically, if we formulate it like this—remember the induction assumption that was in the background? Notice: what is the induction assumption in the upper formulation? That betrothal is easier to effect than marriage, right? By contrast, what is the assumption here? That the wedding canopy is stronger than money. There’s no connection—these are two completely different assumptions.

[Speaker G] Stronger in what sense? What do you mean, in what sense? In the sense of what it effects. Okay. I think because the a and the n enter a relation of order all at once, and the h and the m enter a relation of order all at once.

[Rabbi Michael Abraham] I fully agree, I’ll get to that in a moment. But on the face of it, when we look at this table, apparently there are two different arguments here, each of which passes through a different generalization, right? Therefore, in fact, these are two different arguments.

[Speaker H] Right, though seemingly they’re intertwined.

[Rabbi Michael Abraham] In this case I take these two data points, build an induction from them, and use this third datum to reach the result. Right? In this case I take these two data points of the column, build a hierarchy rule, and from this third datum I reach the result. Seemingly two different arguments. Okay? Let’s see an indication, for example, of this matter. Before the indication. Let’s go on; I’ll enlarge the table, a more complex argument. What happens here? Sorry, yes. What happens here? I have a refutation of the a fortiori. Look at the a fortiori here. This is the a fortiori; these four cells are the basic table, the first matrix. Now here there’s redemption. Money can effect redemption, and the wedding canopy cannot effect redemption. Fine? Never mind redemption of second tithe; it’s not important. In Jewish law all sorts of things are redeemed, and you can do that only with money and not with a wedding canopy. This is considered a refutation of the a fortiori. Remember the grades in history and all that? We simply found someone whose grades are different. Okay? Which a fortiori does this refute—the horizontal one or the vertical one? Seemingly only the column a fortiori. Because what does the column a fortiori say? That the wedding canopy is stronger than money, right? And here we see that’s not true. Here is a counterexample. That money… But if the assumption is that betrothal is easier to effect than marriage, what has that got to do with it? This p doesn’t touch that. Right? It doesn’t refute the assumption that a is greater than n, that a is stronger than n.

[Speaker I] You used that at the beginning to prove the second.

[Rabbi Michael Abraham] No, no—each of them is a separate argument. I didn’t use either of them to show the other.

[Speaker I] You used betrothal and money to prove that the wedding canopy is stronger.

[Rabbi Michael Abraham] Right? No, I used only marriage. To prove that the wedding canopy is stronger than money I used these two cells, in the marriage column. That’s all. Now I transfer it to the betrothal column. So apparently there is no assumption whatsoever about the relation between betrothal and marriage, right? Here I make an internal comparison within marriage, between zero and one. I’m not assuming that marriage is stronger than betrothal or weaker than betrothal—nothing. It’s an internal comparison within the marriage column. Therefore, this refutation, which casts doubt on the relation between money and wedding canopy, refutes this a fortiori but doesn’t touch this a fortiori. In order to refute the a fortiori of the rows, I need a row refutation. Add here, say, intercourse, or something else, another act that succeeds in effecting marriage but doesn’t succeed in effecting betrothal. That would refute the a fortiori of the rows. This is an indication that we’re dealing with two different arguments, right? If one refutation refutes A and leaves B intact, and another refutation refutes B and leaves A intact, then that means they are independent arguments.

[Speaker H] But that’s like you’re talking about other separate dimensions. Fine, go on.

[Rabbi Michael Abraham] Okay. Now I make a first generalization, which is basically a scientific generalization. In the Talmudic context this is called analogy from two verses. What is analogy from two verses? I take money and intercourse and try to learn from them about the wedding canopy. Right? That’s my question mark. I want to know whether the wedding canopy effects betrothal. That’s the unknown. Fine. Now I have lots of data. So I begin with a fortiori from money. You see these four cells? Those are the four cells of the first a fortiori, right? I say—but there is a refutation. The refutation is this one, marked in green. I have a counterexample. It isn’t true that the wedding canopy is stronger than money, right? So I say, okay, then let’s learn differently. Look now at the shaded cells. I learn from intercourse to wedding canopy. Okay? This is already analogy. You see here there is a one, not a zero.

[Speaker J] Rabbi Michi, sorry for interrupting—can I say one little thing? Yes. I have a question. I was here in the first few minutes, but I’m not sure everyone is clear on what the difference is, what analogy means. Analogy from two verses is already more advanced.

[Rabbi Michael Abraham] No, no, analogy is just analogy. I showed before the table of analogy. Okay, fine. So I say, here I made an analogy. If there were only these four cells, I could also fill in one, from analogy. Okay? But there is a refutation. You see? Here I see that intercourse is not similar to the wedding canopy.

[Speaker G] Can you just explain what that y is there in column b?

[Rabbi Michael Abraham] Wait, where was I… yes, where are you? Ah, that’s a yevama. These are simply halakhic parameters of a yevama; say intercourse works there, but the wedding canopy does not. And neither does money. Okay? I’m just bringing counterexamples. So basically, notice what I did here. The Talmud says that once I have this table, even though the a fortiori has been refuted—you see from the green markings—and the analogy has been refuted from the shaded markings, each one separately, still the two together, without adding any extra datum, do the job anyway. Even though each one separately doesn’t do the job. And what we actually have here is scientific generalization. Because what is scientific generalization? Say I take this bottle, I let go of it, and it falls to Earth. Right? Fine. Maybe this bottle is made of plastic. Okay? So I take this plate, let go of it, it too falls to Earth. And okay, maybe this plate is made of porcelain or earthenware, I don’t know, and it too falls to Earth. So okay, maybe you can’t learn anything? No, you can. Because I basically say: wait, then probably—Francis Bacon’s induction—the material itself isn’t what matters here, because if you take different materials and both fall, then probably the fact that both have mass is what causes them to fall to Earth, and I make a generalization. And that’s exactly what happens here. Notice.

[Speaker I] Is this the common factor here?

[Rabbi Michael Abraham] Yes. That’s why it’s called the common denominator. I basically look for what these two have in common, and I say that that is probably the relevant factor. Notice that this is, of course, striving for the simplest explanation, right? Because when I look for the common denominator, that is the simplest explanation. Wait, we’ll get to abduction in a moment. So now what comes out here, notice, is that what is written in this table is exactly scientific generalization. Scientific generalization from only two examples. Of course, scientific generalization usually relies on more examples, but the logic is pairwise logic—take all the examples of scientific generalization, pairwise it works like this. In other words, if you take them in pairs. What do I mean? I have two things—you see—both money effects betrothal and intercourse effects betrothal, right? So I want to say: good, then let’s generalize that the wedding canopy will also effect betrothal. Let’s write one here. And no—intercourse has a unique feature, you see, it works for a yevama. Money doesn’t have that unique feature; you see, money doesn’t work for a yevama. Right? And I say money also has a unique feature: it effects redemption. You see, intercourse doesn’t effect redemption. You see? This is porcelain and this is plastic, right? And we said: fine, if so, then probably those unique properties aren’t what do the work. Rather, what they share is that both are actions that effect marriage, and therefore they also effect betrothal. Okay? Therefore this is really an inference completely equivalent to scientific generalization. In pairs. Of course scientific generalization needs more examples, never mind, but in pairs this is how it works. Okay? Now, that’s regarding the presentation… you can generalize. Look here: I took the sugya of the wedding canopy in tractate Kiddushin on page 5, and I got this table. This is a four-by-seven table, okay? One of the more complex ones in the Talmud; there may be more. But this is one of the more complex ones in the Talmud, four by seven, already more complicated. And again there is a question mark here, and I want to know what to fill in there: is it one or zero? Now here there are lots and lots of data. Simple intuitions won’t work anymore here. Because simple intuitions—we know how to fill in a fortiori. We know how to fill in analogy. But here, if I give you this table, what do you do with it? Okay? Here formalism will help us. Okay. Now, one more remark before I move to the formalism. Look here. I’m now doing—you mentioned, Darian, that this is over a binary field. Fine, what happens when it isn’t binary? When it isn’t binary, then look at the grades. Say Moshe got a grade of eighty here and a grade of forty there. And Hannah got a seventy there, and here we don’t know what she’ll get. Okay? Now look: if I do the a fortiori by columns, then look what happens here. Hannah is stronger than Moshe, right? Therefore if Moshe got eighty here, Hannah will get at least eighty or more—but at least eighty, right? But if I do the a fortiori by rows, then this exam is easier than that exam. In this subject it’s easier to succeed than in that exam, right? So if Hannah got seventy there, how much will she get here? At least seventy. And here it comes out at least eighty. These are different lower bounds. That’s another indication that these are two different arguments, because you see that the conclusions of the two arguments lead to different conclusions, right? Seemingly it’s the same thing here—this is an example…

[Speaker H] But do you have, like, a guaranteed minimum?

[Rabbi Michael Abraham] In each of them there is a different guaranteed minimum. In one it’s seventy and in the other it’s eighty.

[Speaker H] No, but if you do a full union then maybe you have a guaranteed minimum?

[Rabbi Michael Abraham] If you say that both of them are true, then you can take the eighty, because in particular the second argument is also true, so you have an eighty bound and a seventy bound and both are true, so the bound is eighty. Fine? You can prove that it’s above eighty and you can prove that it’s above seventy, so obviously you’ve proved that it’s above eighty. Fine. So in the Talmudic example too there is a similar example to this, and it’s an example with half, one, and zero. There too, again, if we go by the rows, the result here is at least half. If we go by the columns, the result here is at least one. Okay? Now why am I saying this? Because then from here and from the refutations, it comes out that these are two different arguments. But there is a puzzle. Nowhere in the Talmud does anyone rotate an a fortiori argument. Nowhere do you bring an a fortiori, someone refutes it, and then say, okay, let’s try the columns instead of the rows and escape the refutation. A refutation does not refute the vertical a fortiori. Nobody ever does this. And the question is why. That bothered me for some time; Amit saw it faster than I did. So the answer to this, in my opinion, lies here. Let’s look for a moment at the example of the a fortiori here. Now I say as follows: I want to understand, to find a model—now I come to abduction, right? I want to find a model that will explain the three data points. The data are what is written in black. What is written in red are two possibilities, either one or zero. I want to know which possibility is correct, whether it’s one or zero. Okay? How will I choose the explanation? The simpler one, yes? Solomonoff in one form or another. I’ll choose the simpler one. Then I say as follows: let’s say I look at… I assume they have some feature, let’s call it alpha, okay? that allows it to effect betrothal but is not enough to effect marriage. Okay? That means that to effect betrothal, one alpha is enough, right? But to effect marriage you need two alphas. Just for convenience I make it discrete, okay? So two alphas. Okay. Now in h, when I try to guess what my model is for describing the features of h—after all, it succeeds in effecting marriage, and in order to effect marriage, I remind you, you need two alphas. So clearly it has at least two alphas, right? If so, then certainly it will also effect betrothal, because one alpha is enough for that. Therefore this is one. Okay? But notice there is another option. I could say that m has a feature alpha. It succeeds in effecting a because a requires alpha in order to be effected, but it does not succeed in effecting n because n requires beta—not two alphas, but something else. Not that it requires a greater quantity of alpha; it requires beta. Then the model would be that m has alpha, h has beta, to effect a you need alpha, to effect n you need beta. Then the answer here would be zero, right? Someone who has beta won’t succeed in effecting a, because in order to effect a you need alpha. And now the question is: which of the models is correct? Each model gives a different prediction, right? This model gives one prediction, and this model gives zero. Here I strive for the simplest solution, and I say this is a model with one parameter explaining it—alpha, at two levels—whereas here it’s alpha and beta. Occam’s razor tells me that a one-parameter model is more plausible, simpler, and therefore I adopt it. Okay? Therefore the answer is one. This means that what I actually did here is an abductive process—not complete, because I still haven’t identified what alpha is. But I built some abstract model: there needs to be some parameter in the background. I also know at what levels it has to be present within money, within the wedding canopy, for example. One could speak about degree of benefit, and then say how much benefit one gets from money, how much from intercourse, from the wedding canopy not so much. Fine? But we can talk about that—but that is already identification, and I can’t derive that from the model here. What I can derive from the model here is that there is some such parameter alpha; afterwards we’ll think how to identify it. So I’ve done half the road of abduction, and I’m basically finding some model with theoretical entities, as philosophers of science say—those alpha and beta things. I don’t yet know what they are, but there are some such alpha and beta things that explain the three data points. And after I reach this model, I can complete the missing datum and give a prediction for a future experiment in science, or in this context, I can say what the law is in a lacuna, in a case where I don’t know the law. Notice, though, what I gained. Now you can see that the vertical argument and the horizontal argument are exactly the same argument—what Amit said earlier. Because do you understand what happens here? In order to produce the hierarchy, say, of the one, what did I assume? I assumed that m has a strength of alpha, and that a requires enough alpha in order to effect it, while n requires two alphas in order to effect it, right? But to complete this, that isn’t enough. I also need to explain why h succeeds with n and with a. It has to be that h too has more—has two alphas. That means that the hierarchy between m and h has to be expressed in the same units as the hierarchy between a and n. The hierarchy of the columns and the hierarchy of the rows must be on the same parameter. That’s basically what you said earlier: in what sense is h stronger than m? With respect to the kinds of actions a and n. Understand? That’s the formal formulation of what Amit said earlier. You were saying that we thought we were speaking only at the level of who is stronger—h is stronger than m. No: stronger in what respect? And if you assume that to be stronger in this respect is like being stronger in that respect, then you are assuming that both the rows and the columns speak the same language. In other words, they are all located on the alpha scale. Each has a certain level of alpha characterizing it, which is required in order to effect it or which it has as a power to effect things. Yes, in the rows it’s how much power you have; in the columns it’s how much power is required to effect the columns. But it all has to be expressed in terms of alpha. Because if it isn’t expressed in terms of alpha, then there is no horizontal a fortiori, and therefore there will also be no vertical a fortiori. What happens when there is a refutation? Look here at a column refutation. I apply the same technique. In a column refutation, now I have another column in the table, p. This is the refutation, you see? It refutes. I said before that it refutes this hierarchy; it doesn’t touch that hierarchy, right? That’s what I said earlier. Because this p is an example that h is not stronger than m. But it doesn’t depend on the question of what the relation is between a and n. Okay? But now look—you see that it does depend on it. And therefore it isn’t true that it refutes only one side and not the other. It’s the same argument. Whoever refutes the first refutes the second too. Why? Because now let’s try to look for a model that will explain the five black data points. Those are my hard data. If you try to look for an explanation—yes, what we’re really doing mathematically, say, is filling in one here and zero here and looking for the simplest model that explains this table. The simplest model that explains this table—I check which model is simpler, and that is the winner. If this model is simpler, the answer is one. If this model is simpler, the answer is zero. Okay? That’s the algorithm we’re activating here.

[Speaker K] Simpler in what sense?

[Rabbi Michael Abraham] So in a moment I’ll talk about that. Earlier we saw number of parameters, for example; that’s one criterion of simplicity, but it isn’t enough. I’ll complete that in a second. So what I’m actually saying is this: in order to do this, we began—I may not have mentioned it—we did this work with Dov Gabay and Uri Schild from King’s College and Bar-Ilan. In any case, what we do in order to build a model heuristically—it’s not mathematical, by the way there is a master’s thesis where someone proved various theorems about these models, because once the tables start getting larger you can no longer do it heuristically, and then there are all kinds of theorems—not a complete set, there is no complete algorithm, we don’t even know the computational complexity of the algorithm, nor whether there is uniqueness. But uniqueness isn’t all that important in the simple sense; I’ll comment on that in a moment. So what I’m saying is this: I now take the columns of the table, p, a, and m—you see, p, a, and m are the different circles. Fine. Now here I make the model of filling in one. I say as follows: the strongest row is one-one. So it sits here. The others enter into it with arrows, so they are basically—you see—less than or equal to it. Fine. So a enters into it with an arrow, and p enters into it with an arrow, right? But between a and p there is no connection. They are just drawn together here, but really they should be separate. There is no arrow between them because there is no hierarchy, right? Those are two independent columns in the matrix, the first and the third. Okay, so what happened here? Right, so basically we get a table like this. Now why is that good? Because this gives me, heuristically, a way to find the model from the table. Given a table, I now want to draw the minimal model of m, h, a, p, and m. Which alphas and betas are here and how are they distributed among them? So look how it’s done. I assume that the simplest thing is alpha, that’s a, into which all the arrows enter. Now I go backward: so it has to contain alpha. That can be either two alphas or alpha and beta. If there were only this, then I would say two alphas, because that’s simpler than alpha and beta. But if I have two and they’re independent, then it has to be that one of them is two alphas and the other is alpha and beta, because otherwise there would be a hierarchy relation between these two as well—but there isn’t. I see in the matrix that there isn’t. These two are independent columns, right? So the picture must be this. So here I found the model. a is alpha, and this is heuristically the simplest model. a is alpha, m is two alphas, p is alpha and beta. That’s when the filling is one. What happens when the filling is zero? Then it no longer becomes the strongest. You see, the relation is no longer so simple. a and p are exactly identical columns, identical vectors, right? So a and p are both in this circle, m is in that circle, and they are independent; there are no arrows between them. Right? This is zero-one, this is one-zero. Okay. Now the question is which is simpler? Number of parameters won’t help here, because in both there are alpha and beta. So we came to the conclusion that we need to add more elements to the criterion of simplicity, not just number of parameters. Then we said: okay, we also went to more complex graphs and said that in graph theory, when you make a graph, there are three features that, a priori—we decided this—it wasn’t ad hoc, but rather a priori we said there are three parameters in graph theory that could be relevant. These are connectivity—that is, how many disconnected subgraphs there are in the graph; the number of independent vertices—say here there are two and here there are three. You see how many vertices, meaning how many circles. Fine. And the number of changes of direction. A change of direction is basically a measure of how simple the hierarchy is. Right? If there are no direction changes, if there is a simple hierarchy—this is more severe than that, that more severe than that, all transitive—then that means the model is simple. Clearly there is alpha, two alphas, three alphas, right? That’s simple. If there are changes of direction, that means something here breaks the hierarchy; in effect an additional parameter has been added. Therefore we said a priori that direction changes too could be a relevant parameter. Then we proposed a suggestion, and the suggestion says this: the criterion of simplicity is, first, the number of parameters in the model. We solve the model heuristically and reach the conclusion whether in the one-filling there are more… I compare filling in one and filling in zero. Now in the one-filling I check how many parameters there are in the model, what the connectivity is, how many independent vertices there are, and how many direction changes there are in the longest path—the path with the most direction changes. Okay? And I do the same for the zero-filling. Now I say: if one of them has an advantage on all the parameters, then it is preferable. But if one has an advantage in one and the other has an advantage in something else or in two other things, there are no tradeoffs. That means there is independence between them. Because think of a fortiori: say I bring a counterexample. Fine, it may be that I’m better than him in history but worse than him in physics. That’s already enough so that I can no longer infer a conclusion. I don’t know what the right answer is, but it already refutes the a fortiori. A refutation doesn’t mean that the correct filling is zero. A refutation means there is no correct filling for me. I can’t infer a conclusion. That’s what a refutation means. Okay? Therefore I say that in terms of the criterion of preference, superiority, for me if it is not univocal—not one-directional—meaning if there are advantages here and advantages there, I don’t care how many, but if it’s not clear, the direction isn’t clear, then that is a refutation.

[Speaker G] Fine? Can we go back to the definition of filling?

[Rabbi Michael Abraham] What?

[Speaker G] What do you mean by filling?

[Rabbi Michael Abraham] One and zero, the red ones. I fill the table with one, fill it with zero, and make a model. I look for the graph that is created, build the model. The filling that gives me the preferable model is the correct filling. Okay? Now, apparently this sounds a bit ad hoc, right? Okay, number of parameters didn’t work out for us, so we added graph theory; and you could pull out other points from graph theory, so why exactly these three? Although I think these three criteria are pretty natural as criteria for graph simplicity. The fact that we said this a priori—historically, in fact, we said it a priori—so it isn’t… More than that, one second—more than that, there are two interesting indications that we noticed. One indication is that with this large table I showed you earlier—with this table we got stuck. That’s the end of the sugya. We followed the whole sugya, solved every stage, and saw that it really matches the model. At this table we got stuck; it didn’t work out. According to the sugya, the filling had to be one, but we didn’t get one. We tried to play with the preference criteria; here we already did ad hoc games, and we tried to find preference criteria that wouldn’t destroy everything that had come before and would give us the result. We didn’t find any; we couldn’t do it. We sat over this for three days, sweating over the thing. Okay? What did we discover in the end? That we had made a mistake in one of the data points—we had simply typed a zero instead of a one. And suddenly it became a great confirmation. Because that basically means that you can’t just make things up ad hoc. The fact is that when it doesn’t work out, it doesn’t work out. We weren’t able to ad hoc our way out of it. And when we worked correctly, it simply worked out. That gave us good confirmation that this isn’t just ad hoc. In other words, the fact that we got stuck and couldn’t get out of it because of that mistake—it’s an indication, not of course a proof of anything.

[Speaker E] Look, first of all I want to say I find this model very hard to understand all the way through, because there are dimensions here that are hard to read in this and that, but I’ll tell you what it reminds me of a little: a decision tree. In a decision tree, the operators there—a decision tree receives data, I have features, and now I need to divide into groups by the features. And there, depending on the kind of decision tree, you can place the operators at the nodes where you say okay, if you’re like this then you go down that path, and at the leaves in the end I have the classification, where I say okay, you fell here, so I build the world, and then if I have a new case I can understand where it falls. And in a decision tree there is exactly the same problem, because you can say that the decision tree may just… it may not be perfect, meaning there is contradiction in the data as you say, and then you need impurity and there are measures of purity. You say how far am I producing a separation here in my leaves that is a good separation between, say, positives and negatives. There are also aspects of pruning, because you can get overfitting. What’s the question? What… what did you say, what do you want to say? In a decision tree too you’re trying to say which tree…

[Rabbi Michael Abraham] Maybe you can do it with a decision tree; maybe. I don’t know. No, this is not a decision-tree method.

[Speaker E] Because… in those contexts too, in the world of learning from a decision tree, people also tried to propose criteria that they talked about—you talked about simplicity and number of rules, there it’s pruning, you say I’m trimming my tree when it gets too…

[Rabbi Michael Abraham] And why will we go by criteria of simplicity? Because that’s Occam’s razor; that’s generally what guides us in choosing between hypotheses. Okay, but I assume that appears everywhere in one form or another. But here it isn’t a decision tree. The tree I’m describing here is only a representation of the data in the table. It’s just an aid in order to analyze the table and extract from it a model. What I’m really supposed to do is this: this is the table given to me; these are the data I have. Now I’m told: bring me the theory. What explains these data? Do you understand? It’s really scientific work. In other words, you have a collection of data, you’ve made measurements—what happens here, what happens there—you have fever. An apple lowers your fever, soup doesn’t lower your fever, but if you have a fever of thirty-nine then this lowers it and that doesn’t lower it, but if you run then it will lower your fever—fill the whole table with all the kinds of experiments, and now I say okay, let’s build a theory.

[Speaker E] What… what features do the apple, the running,

[Rabbi Michael Abraham] and the soup have that succeed in lowering fever this way but don’t succeed in lowering it that way, and I build a

[Speaker E] medical or physiological theory in that case. It’s exactly the same thing. I think the table is just what makes it very challenging to analyze this too.

[Rabbi Michael Abraham] Fine, I said—

[Speaker E] about the complexity of how you do this practically. A hard question. I don’t know how to answer it. No—even theoretically. I think one of the things that makes it hard to understand is that there is something very unnatural about reading a derivation tree off a table.

[Rabbi Michael Abraham] It’s not a derivation tree.

[Speaker E] But it’s a kind of… okay, I have objects and rules, and I’m trying…

[Rabbi Michael Abraham] But this tree describes a hierarchy. You don’t run on it and get decisions. It’s a pictorial description of a hierarchy among halakhic concepts. What hierarchy?

[Speaker E] It’s a derivation because there is something in common.

[Rabbi Michael Abraham] No, it’s not a derivation. It’s stronger than that, because it can do this and it can’t do that. You haven’t derived anything yet. The derivation happens after the tree exists and I analyze it and extract various parameters from it and use that for derivation. The tree itself is not a derivation tree. It’s only a representation of the data in the table. That’s all. A graphical representation of the data in the table. It’s just convenient to handle them through the representation of a graph. And it doesn’t describe something essential here, at least not in the decision-making process. Okay? So that’s first. Second, remember the table of the common denominator? Here you can see it. How many kinds of such tables are there? In principle, three. No more. What do I mean? There can be a case where here there is zero and here there is one, right? That’s either analogy or a fortiori, and I join the two into a three-by-four table. Okay? Again, this is in pairs. You can extend it to as many examples as you like, but the basic logic works in pairs. Okay? So we have either zero-one, or one-one, or zero-zero. If you make zero here and one here, just switch the rows; it’s not a different table. It’s the same table. Okay? So three tables. Now when you make the tree of these three tables—I don’t think I brought it here—that when you make the tree of these three tables, we discovered to our amazement that in each of them, each of them, the filling is one, and in the common denominator the result is one. Each of them was decided by one of the three graph-theoretic parameters. One by connectivity, one by number of vertices, and one by direction changes. And again, this came out for us—it wasn’t ad hoc. In other words, we thought these were the three parameters that sounded reasonable to us, and they simply spread out exactly over these three cases. Each one—meaning that these three preference criteria are indeed essential criteria. That is, it looks like something correct to use. Right? Our three basic processes of generalization—and there aren’t any more. Three basic processes of generalization, one decided by connectivity, one by number of vertices, and one by number of direction changes. So this means that a criterion built from these three, plus the claim arising from number of parameters, sounds like an extremely reasonable criterion, and it works. You can apply it to every stage of the sugya; at every stage of the sugya it works. Now it explains—and this is quite a long sugya, it goes all the way to this big four-by-seven table. Look for example here: I brought the… you see the drawing? This is with filling one and this is with filling zero. It already starts getting more complicated. Now to do this manually, and on large tables—it isn’t big data, of course, these are small tables. But actually with this table you can still do it, and these are the fillings we get, and you see the result comes out correctly. I compare in terms of connectivity, in terms of number of vertices and so on, and the result comes out correctly. So what does this story really mean? What it basically means is that when we make analogies, generalizations, a fortiori arguments, all sorts of things like that, plus all the ways of combining them with each other—refuting one, bringing another one in to support it, refuting the complex structure, bringing in yet another thing—there can be an endless composition of these things. Every scientific process, in fact, is composed of these building blocks. Every process of thought in general, every non-deductive process of thought, it seems to me. Here I’m being a bit ambitious. As far as I understand it, every process of thought that is not deductive, of soft logic, can be built from these building blocks. And therefore, in the end, you can make some giant table, obviously a huge one, containing all sorts of data. So in principle, if this is right, then this thing is actually an algorithm that gives me the answer for any set of data I got from an experiment. You haven’t reduced the theory—that basically means this is a mechanization of the process of abduction. Because in fact I take the data and derive from them the model—what alpha and beta, what each thing has. In fact, I’ve built a model here with theoretical entities not yet identified, but with theoretical entities such that I can tell you what theory would explain these data.

[Speaker E] Now in logic you can, with two quantifiers, actually describe every kind of formula, right?

[Rabbi Michael Abraham] Right?

[Speaker E] And also and, not, and if… three.

[Rabbi Michael Abraham] I don’t think so, why?

[Speaker K] By the way, those are operators. He’s talking about operators.

[Rabbi Michael Abraham] You’re catching the operators; that’s something else.

[Speaker E] There are all kinds of simple operators, and also…

[Rabbi Michael Abraham] That just means you can build all the tables from one table. All the two-by-two tables, the four, yes, exactly. Or Sheffer stroke, or nand, or something like that.

[Speaker E] And is that somehow connected to what we’re seeing here?

[Rabbi Michael Abraham] Yes, why? I don’t… okay. Basically, what I want to say is this. We begin with a tool whose role is to give me a datum or a prediction for a case I don’t yet know, a future experiment in the scientific context. But I get there through a process of abduction. I basically build a theory out of the existing data, and the theory gives me a prediction for what is going to happen in the next case. So this process actually does both things. It both constructs the abduction, the theory, and also uses the abduction in order to give a prediction for a case that I don’t yet know.

[Speaker G] But only partially, still.

[Rabbi Michael Abraham] It isn’t necessary; in a moment I’ll comment on that, yes, of course.

[Speaker G] There is a really cool perspective here and I’m really enjoying it, I understand how a fortiori here has something that could be a very interesting tool. But why, why do you say this might be a model for any abduction whatever? Why should all data gathering resemble a fortiori?

[Rabbi Michael Abraham] Of course you have to build in here that it isn’t binary, because the results of measurement in physics, for example, can be continuous. And indeed, the table can be: in this experiment the velocity came out this way, the energy that way; in this experiment it came out… fill all the data from all kinds of experiments, make a table out of it. In principle it’s the same thing. At a very high level of principle, obviously, yes?

[Speaker G] But… but if, say, suppose instead of measuring velocity I decide to measure shm-velocity, some gerrymandered feature of velocity?

[Rabbi Michael Abraham] Then build a theory about shm-velocity.

[Speaker G] But then you won’t get a set… your best explanation will be a funny explanation.

[Rabbi Michael Abraham] That’s already a question of units. It may indeed be that I won’t get a correct and productive explanation.

[Speaker G] That’s okay.

[Rabbi Michael Abraham] So the claim is basically that we have some systematic way to locate abduction, and I said it’s a bit similar to Solomonoff—again, those who understand can correct me—but of course it’s Solomonoff for the poor. In other words, I’m not doing two to the minus Kolmogorov of the hypothesis, but rather a delta function. I take the simplest one. Fine? But still, that’s basically it. And once you’ve taken the simplest one, then of course there is no longer any statistics, because its prediction has to be the correct prediction, and then it has to be probability one.

[Speaker L] What justifies taking the simplest?

[Rabbi Michael Abraham] Occam’s razor. On the contrary, Solomonoff wants to argue that there is some chance that even the not-simplest one is correct—a smaller chance—and you sum over all possibilities. I say: let’s take a toy model, let’s take the simplest as if it were completely correct. You can generalize this to Solomonoff. Now if I take all the possible models and assign them a criterion, yes, Kolmogorov complexity of H of each hypothesis, and do a sum over two to the minus Kolmogorov of H, then fine, I can also make this into Solomonoff. That is, in principle. I made it into a delta function because it’s simpler, but that Kolmogorov computation is Solomonoff and I don’t know how one could actually do it.

[Speaker I] But the difficulty is really identifying alpha. What is alpha?

[Rabbi Michael Abraham] Okay, now that’s already a question to which I have no answer.

[Speaker I] That’s a question that goes all the way back to the beginning. It’s not in logic; it’s a question

[Rabbi Michael Abraham] for an expert, it’s

[Speaker I] not syntactic. It’s not only after you’ve found some unknown and now you want later to identify what it is. Even in order to find it in the first place you somehow need to try to understand what the parameters are of…

[Rabbi Michael Abraham] No, no—that’s the advantage, that I don’t. I can draw the model without resorting at all to the semantics of these alphas and betas, not at all. Purely syntactically I can extract it. That’s exactly the great advantage of this. And more than that—on the contrary—after I’ve found the models, suddenly I see, for example, that in money I got alpha and in intercourse I got two alphas, so I think alpha is probably benefit.

[Speaker I] And what you said earlier, that it still isn’t the same thing—I still don’t agree that experiments and money and intercourse are the same thing.

[Rabbi Michael Abraham] It’s the same argument, but it’s not the same thing.

[Speaker I] Between them there is a

[Rabbi Michael Abraham] relation that if you refuted one, you refuted the other.

[Speaker I] Right. But one is the cause of the other, which makes them not identical arguments.

[Rabbi Michael Abraham] No, no, the arguments are identical. Fundamentally, when you refute one you refute the other. When one is true the other is also true.

[Speaker I] But that doesn’t make the arguments identical. Conceptually, for present purposes, kiddushin, okay, they are what cause money…

[Rabbi Michael Abraham] That doesn’t matter. Again, I’m not talking about causality.

[Speaker I] The arguments

[Rabbi Michael Abraham] are identical in the sense that when this is true, that is true, and when this isn’t true, that isn’t true.

[Speaker I] Money or kiddushin cause… rather betrothal and marriage, and not the other way around. The fact that they have the same structure by which you can separate them, I

[Rabbi Michael Abraham] understand, but that’s not connected to…

[Speaker I] the arguments themselves are not identical.

[Rabbi Michael Abraham] You know, like nand, for example, that’s not A and B. You can call it not A or not B. That’s also the same thing, right? Is it the same argument or not the same argument? It’s the same—not an argument, a proposition. Is it the same proposition or not? In logic it’s accepted that it’s the same proposition. They both have the same… when this is true, that is true, and when that is true, this is true.

[Speaker G] There are different ways of individuating arguments. What Erez is getting at here is that… maybe a philosophical concept called ground: there may be two things that are, let’s call it, logically equivalent, but we think one is the ground of the other. For example—I think this is Kit Fine’s counterexample—we can think of Socrates and the singleton containing Socrates, and there are all kinds of relations between them such that whatever you say about one is true of the other.

[Rabbi Michael Abraham] In set theory, the empty set and the set containing the empty set are not the same thing at all.

[Speaker G] Right, but everything we say about Socrates we can say that the singleton of Socrates is a set containing an element that… and then all these properties hold. But it seems to us that Socrates is in some way prior, grounds it.

[Rabbi Michael Abraham] Fine, okay. In terms of equivalence between arguments, it’s equivalent. Whether to call it the same thing or not is another question.

[Speaker G] There are all kinds of… in the end both of you are right in some sense.

[Rabbi Michael Abraham] Okay. Loosely speaking there is—

[Speaker E] here an equivalence relation, I think, like the one you’re trying to find. Huh? There is an equivalence relation here. You’re basically looking for an equivalence relation such that there is no contradiction with the hierarchy, so that once you apply this equivalence relation you’ll be able to fold the graph down to a minimum that expresses the… what graph?

[Rabbi Michael Abraham] Are you talking about my graph or the graph… a knowledge graph?

[Speaker E] The graph I think of in my head about this thing is distributed differently. And in my opinion—but you’re right that it connects to graphs from the direction you’re looking at, I think simply in that…

[Rabbi Michael Abraham] I need to see how it connects to a graph in the sense I’m talking about here. Right now I don’t see it. I don’t know, maybe I’m missing something. In any case, the point is that this abduction is a kind of data mining. I take data and in effect extract from them some information. Yes, once for example I sat with a friend of mine trying to build a model like this that would try to predict corporate collapse based on various data. This company had such-and-such data scope, collapsed in such-and-such way or earned so-and-so. We’ll fill the data into a table, I extract from it parameters that I can’t identify. Parameters. And I can predict what the chance is—or not what the chance is, but whether it will collapse in that year or not. Now of course the prediction is deterministic, and it won’t really be deterministic, but it will still give you some kind of measure for this. In other words, or how a judge in a certain place will rule. Give me all the cases. That guy drove the wrong way without a license on the sidewalk, as in the old comedy sketch, and got so many years. That one drove like this and got… I put everything into a table with different judges. Now I say: I’ve come before this judge and I drove in such-and-such a way, what will I get? So this method can give you a tool to predict that. In other words, you can really do data mining in this way. And for our purposes, actually—I probably need to finish, right?

[Speaker I] How do you think this works in neural networks? I mean, these networks, in their manifold somewhere, expose the parameters in this way, only not in this kind of form.

[Rabbi Michael Abraham] You could perhaps say that it’s equivalent to this, if at all. But whether it works this way or doesn’t work this way is another question. I’m not sure how well-defined that question even is.

[Speaker K] Models that make predictions, because in the end they extract parameters.

[Rabbi Michael Abraham] It’s equivalent. Maybe it will be equivalent to this—meaning it will give the same prediction as this. Whether it works this way or doesn’t work this way returns us to the earlier question.

[Speaker I] You asked some question—he knows how to expose alpha, beta, and all those things.

[Speaker K] But there it relies on statistical theory and proofs of regression converging to…

[Rabbi Michael Abraham] This is a completely deterministic method. Why? Completely deterministic.

[Speaker K] No, but here it’s something else. Here it’s logical inference; it doesn’t rely on regression of variables that in the end have to converge statistically somewhere, and that’s what neural networks are based on in the end. Here it’s something else.

[Speaker H] What do you mean by converge?

[Speaker K] In the end, when you have a statistical variable, regression converges. There is a proof that it converges to some value. Never mind, we can take this offline. But here it isn’t statistical; here it’s deterministic logical inference. So in that respect it’s different, but the idea is very similar.

[Rabbi Michael Abraham] I’ll maybe finish with a few sentences, if I’m allowed a bit more.

[Speaker J] Just before you finish, Rabbi Michael, I want to sharpen for everyone that what you’re saying is… basically when you come now to a wedding there won’t be a ring there, there won’t be money for kiddushin, there’ll only be some kind of wedding canopy and everything else is canceled. That’s what Rabbi Michi wants. The whole world of weddings changes, the whole world of relationships looks different. Just putting it on the table, that’s all.

[Rabbi Michael Abraham] Okay. In any case, first, regarding existence theorems, uniqueness, level of complexity of how I get to the model—I don’t have answers. I said there is a master’s student who wrote a master’s thesis on it, and he has several theorems on the issue, but he’s far from closing the matter. Second, what is a little troubling here, apparently, is that I’ve completely mechanized a process of non-deductive thought. I have a calculation; I tell you what the answer is. Where is the catch here? I mean, what is still non-deductive here? Because after all, this is not deductive. And the answer is that when I build the table, assumptions lie behind it. The assumptions are that the same set of parameters really stands behind all the entries of this matrix, that is, behind both the rows and the columns. Exactly. That is, I’ll give you an example that appears in the literature on hermeneutic rules: I can prove that one has to put fringes on a doorpost. Fringes are put on a garment.

[Speaker K] Right, but there too it’s not unambiguous. There the conclusion is always statistical; the conclusion is always with some probability.

[Speaker I] When you don’t have the assumptions except in a kind of implied way, yes? But without that assumption, it turns into something statistical, I think, in some sense.

[Rabbi Michael Abraham] Just give me a second to finish because I’m already running late; I just want to finish. First of all, existence theorems, uniqueness, complexity—all these I don’t know much about. A little, a little I have some information, not much. Second, what is troubling here, as I said, is how there can be a completely mechanical, completely deterministic algorithm that reaches the result of analogy or induction. That doesn’t sound plausible. So I say: it lies in constructing the table. When you constructed the table, you assumed that the same parameters that govern marriage and betrothal, wedding canopy and intercourse and so on, are the same parameters. There is an assumption here. And this assumption is brought to absurdity by the example, for instance, that I can prove that a doorpost is obligated in fringes. We put fringes on a four-cornered garment.

[Speaker K] True.

[Rabbi Michael Abraham] But a four-cornered garment, which is exempt from mezuzah, is obligated in fringes; a doorpost, which is obligated in mezuzah, all the more so should be obligated in fringes. What does that mean? That is an a fortiori argument which, if you look at its table, is basically built as one-zero, zero-one. Two independent rows, which basically means that these two columns cannot appear in the same table. This one is governed by alpha and that one is governed by beta. Therefore the fact that you attached them to the same table already means that you made some decision there. And in this sense, when I ask whether there can be a decisor, preacher, judge, philosopher, scientist, that is mechanical, as I write here, I say the point at which you will not be able to do it mechanically is the construction of the tables. That is, when you construct the tables, you decide which entries belong in the matrix. That decision is one for which we have no algorithm. For the time being it’s apparently a decision for an expert or for a statistical large language model, whatever—but it is not a deterministic model.

[Speaker K] And that basically brings it to this: building the table creates causality out of correlation. It basically creates—whoever builds it creates causality, and that’s the pitfall, because correlation is not causality.

[Speaker H] You can check that after the fact, if it created an assumption.

[Speaker K] So before that, when you build the table, you can build—

[Rabbi Michael Abraham] Not to check whether it came out right—that’s fine. But I’m giving predictions now; the question is how seriously to take such a prediction at all. You won’t do an experiment at CERN on a prediction you don’t give any…

[Speaker H] When you check all the permutations of these tables that exist, you need to know how to narrow them down correctly before you start running lots of things.

[Rabbi Michael Abraham] Now one needs—

[Speaker E] to check some additional aspect looking ahead. Are analogy and induction in deduction? Those are the two rules.

[Rabbi Michael Abraham] Analogy, induction, a fortiori if you like—that’s analogy, yes, and analogy from one verse and two verses is analogy and induction, a fortiori, refutations of them—

[Speaker E] these are the three building blocks.

[Rabbi Michael Abraham] From here on, you can make whatever extension you like. It can be tables of a thousand by a billion. In other words, it’s just the expansion into more and more, but these are the building blocks from which you can build the rest.

[Speaker M] Yes, and this is the first heuristic reasoning component that the system uses. Analogy will probably be the fourth, and maybe there’ll be a possibility to get into very, very interesting things here, because a large part of what we’ve done quite… This is also the first reasoning and heuristic that the concept uses here. The next thing will probably be the fourth. And we’ll see—maybe there’ll be a way into very, very interesting things, because a large part of what we’ve done is quite reflected. And that’s something that from this moment on, what’s nice is that it builds the… it builds a schema there, and in the previous slide we said that the schema of the data is what builds the V, and on that slide there, the V.

[Speaker G] If the parameters themselves aren’t continuous—say if there are quantum jumps there and so on—does what you’re saying still work?

[Rabbi Michael Abraham] In principle—look, I started with binary, and binary also works.

[Speaker G] Because okay, suppose we have certain behavior up to some threshold wavelength, where it’s continuous, and then you have a quantum jump, and then continuous behavior up to another length…

[Rabbi Michael Abraham] If that quantum jump introduces some element that wasn’t in the model until now, I guess there’ll be a problem there. Or you’ll somehow have to put that in too in some form, create additional implications of it or something like that. If it’s a jump because that’s the functional dependence, then I don’t see a difference.

[Speaker I] Thank you very much, Rabbi, that’s all for now. Thank you all. Thank you very, very much. With our laptops today.

[Speaker G] Tell me for a second whether this is a fitting title for what this should be. There’s a discussion about a more entropic explanation—what makes an explanation more entropic.

[Speaker E] There are all kinds of—

[Speaker G] properties that people talk about as desirable.

[Speaker E] One of them is simplicity of the lecture—

[Speaker G] and what you’re basically saying is: here’s

[Speaker E] a way to understand simplicity—

[Speaker G] understand it through graph properties.

[Rabbi Michael Abraham] So maybe through properties of the data in the table. The graph only represents the table. Properties of the data in the table.

[Speaker G] You’re basically telling me: understand simplicity in graph-theoretic terms and you’ll discover wonders. That’s basically it.

[Rabbi Michael Abraham] Again, graph theory here is not essential; it’s a representation. In other words, it’s not a graph the way Nimrod, say, meant, where the graph doesn’t really represent something in his decision process, but it does represent, in some way, the data in the table in a form that’s easier to work with.

[Speaker E] Yes, there is some unusual flexibility here in these things, and there is an optimization you’re basically looking for in order to arrive at some possible interpretation. And then you’re trying to compare between these models.

[Rabbi Michael Abraham] That’s the problem of

[Speaker E] abduction—that is, getting from facts to a hypothesis. The problem of the model in the end is a bit more just formatting the problem. What do we know? You come and say: I know the facts,

[Rabbi Michael Abraham] and I also know the relation between them, because I put them in one table. That’s it. That’s all I know.

[Speaker N] And in the domain of Judaism, in that sort of domain of Judaism, can you do an option tomorrow at the same time—tomorrow at 4:30 instead of 4?

[Rabbi Michael Abraham] Tomorrow no, I have a problem Tuesday afternoon.

[Speaker N] Okay, fine.

[Speaker I] Tell me for a second, is there any chance in the morning? It surely creates much greater complexity once you have a number of parameters that are supposed to explain the…

[Rabbi Michael Abraham] I didn’t get into it here. Yes, you can do a rotation in parameter space. Say if you take the parameter alpha and beta and call it gamma, okay? Or alpha and two betas and call it gamma—it doesn’t matter. As long as alpha and two betas appear everywhere in the same form, then it’s not two parameters, it’s one. And you can rotate parameter space, and that’s equivalent to rotating a matrix, by the way. And therefore, therefore, in fact, you can—what?

[Speaker G] Basically the dimension of the…

[Rabbi Michael Abraham] Exactly. You’re basically looking for the dimension. And therefore it doesn’t really matter. That’s why, for example, with identifying the parameters, it’s a bit tricky. Because if you formulated the parameters as alpha, beta, gamma, say, and you don’t find an identification for them, it may be that the real identification is alpha and two betas—that’s the intuitive thing you do know how to identify. In terms of the process, identification really is something trickier.

[Speaker E] One more table? Which one? In the table, the problem… I want to understand for a moment.

[Speaker I] The problem, as you say, is to put in the right variables. And to create a system that knows how to generate these things from the users for whom there really is similarity between them. I think there has to be some…

[Speaker E] That’s what he’s saying—the problem with this thing, with abduction. I just want to understand in this table, what

[Speaker I] is the V, this V?

[Rabbi Michael Abraham] Exactly, a vector of entries.

[Speaker I] Why does it matter?

[Speaker E] Great, I was worried if it wasn’t… it wasn’t clear to me here where this is a feature and where it’s a value. Where is this…

[Rabbi Michael Abraham] These are concepts, these are concepts. Or legal outcomes, and these are legal actions. Everything else doesn’t matter. The actions are giving money, standing under the wedding canopy, intercourse, and giving a document. And the outcomes are marriage, betrothal, redemption, yevama. What is there? I don’t even remember anymore. Three more outcomes.

[Speaker E] Okay. Wait—and this won’t always be actions and outcomes, right?

[Rabbi Michael Abraham] In principle no, but I think in most of the cases I know it’s actions that lead to a halakhic outcome. You can do it through entries of actions and outcomes. There may also be other examples; I’m not sure it’s always like that.

[Speaker E] What are these actions acting on?

[Speaker I] It’s introducing noise, sort of.

[Rabbi Michael Abraham] Say you give money to a woman; the result is that she is betrothed to you. You give money to a woman; the result is that she is married to me—not betrothed. Where is the woman here?

[Speaker I] Where is the woman modeled?

[Rabbi Michael Abraham] The woman doesn’t appear here, and neither does the man. Giving money is an action. The giving of money goes from the man to the woman. It doesn’t matter; for me, an action creates a result.

[Speaker E] Okay, I understand. So basically this is the missing part. It’s like you gave the signature of a function but didn’t give the arguments it

[Speaker G] acts on.

[Speaker E] Why?

[Rabbi Michael Abraham] It’s a function, it’s an action.

[Speaker E] Redemption by money leads to the result that a sacred thing becomes ordinary.

[Rabbi Michael Abraham] It is redeemed. Its sanctity goes onto the money, and now it becomes ordinary.

[Speaker J] If people want, maybe on that day or another day we can study the sugya from the Talmud itself, and then what Rabbi Michi taught us here, which is really the higher level above it, will be even clearer.

[Speaker G] Because this is just a field I’m not really…

[Speaker E] Fine, but that would be overkill. I’m just trying, after all, to understand action and result in a systematic form. The results here—what are they actually talking about?

[Rabbi Michael Abraham] A new halakhic state. A certain action creates a new legal state. There was some legal state, I did an action, a new legal state was created. For example, that the woman is betrothed to me. For example, that something sacred becomes ordinary. Things like that. A woman, my yevama, some woman—say if her husband died childless and she needs levirate marriage. It doesn’t matter; at the level of this formalism it’s not important. You do an action and the legal state changes from x to y.

[Speaker I] He didn’t

[Speaker E] say it, but it’s different. First of all you gave examples about students and grades, where one one’s grade is higher than another’s and therefore the other one

[Speaker I] took

[Speaker E] a certain exam, and then on a harder exam the second one, accordingly.

[Rabbi Michael Abraham] There indeed the entries here are people and subjects. Okay. Fine? People and subjects. Let’s say the person who takes the exam—you can look at that as an action, though that’s a bit artificial. The person takes the exam, and the result is the grade he got.

[Speaker A] Okay, but think of it like this: you have some kind of status of an outstanding student, okay? And then you have things that help, kind of events that help the student be outstanding, and things that refute that, and you need to see some kind of status, and there are events that affect it,

[Speaker G] yes-no, yes-no.

[Speaker E] That’s what you’re trying to put in, and it isn’t this. No, I’m trying to put into a database of… to model the form you thought that what you’re trying to do. Here you put zero and one. You also do that. Right. But the modeling here is because of

[Speaker G] this whole question of specificity.

[Rabbi Michael Abraham] No, I look at actions and results.

[Speaker E] You want to take it into a model of people and… no no, you also did it here. Here, look, this I don’t understand. The graph? A and B? Okay. Alpha and beta? Alpha and beta? Right. Here I say that I have, like,

[Rabbi Michael Abraham] p has two features. It has alpha; it has an alpha component and a beta component. Okay. And that’s the result of the calculation. The data… in the table are data. From this table I extract the model. I know that p has alpha and beta, n has only two alphas, and so on. Fine? And a has alpha.

[Speaker I] And now, Rabbi, now the penny dropped for me as to why you called it data mining.

[Rabbi Michael Abraham] Yes, my claim is that you can extract from the data.

[Speaker I] Now the penny dropped for me. Okay. Can we switch tractates? Yes, of course. Well…

[Speaker E] You can put it, if it suits you, in the folder

[Rabbi Michael Abraham] called “Judaism”

[Speaker E] or I don’t know, ontologies, or something like that there?

[Speaker H] If you send it to me I’ll look at it tonight. I’ll put it in the WhatsApp group so that anyone who wants can. Excellent. Thank you very much.

[Rabbi Michael Abraham] Goodbye.

[Speaker E] Thank you. The Talmud in tractate Sabbath, page 21b.

[Rabbi Michael Abraham] What is Hanukkah? As the Rabbis taught: בכ”ה בכסלו יומי דחנוכה תמניא אינון, דלא למספד בהון ודלא להתענות בהון (“On the twenty-fifth of Kislev begin the days of Hanukkah, eight they are, on which eulogizing is forbidden and fasting is forbidden”).

[Speaker E] And this is what we ask: Our Father, our King, be gracious to us and answer us, for we have no deeds; deal with us in charity and kindness. What is charity and kindness? The greatest act of charity that the Holy One, blessed be He, does with us is that He gave us the holy Torah, through which we can cleave to Him. And that is the essence of the giving of the Torah: to receive the Torah with joy and awe. And especially in these days before the festival of Shavuot, one has to prepare to receive the Torah and separate from all idle talk, and strengthen oneself in Torah study, in analysis and in diligence. And may it be His will that we merit the crown of Torah and merit to see the consolation of Zion and Jerusalem speedily in our days, amen.