The Advanced Torah Institute: “The Sparks of Perfect Torah” on Paradoxes and the Logical Completeness of Halakha — Rabbi Dr. Michael Abraham — Bar-Ilan University

[Speaker A] Rabbi Dr. Michael Abraham, who chose the title “The Torah of the Lord is perfect,” will try a bit to smooth things out and speak about the practical implications in learning and in Jewish law of Gödel’s theorems. Please, Rabbi.

[Rabbi Michael Abraham] Okay, maybe I’ll start by saying that I don’t know whether I’m allowed to be more logical than Nadav, but actually I don’t like his formulation, with all due respect. I think I wouldn’t give up so quickly on theoretical completeness and settle for practical completeness. And I’ll try to show, through examples, that there are places where practical completeness can’t be achieved when we’re stuck in a theoretical problem. So maybe the direction is even the opposite, although it seems to me that we’re not really saying very different things.

[Speaker C] Meaning, Gödel doesn’t care about contradictions.

[Rabbi Michael Abraham] Okay, fine. So whoever wrote that, never mind. So that’s the first basic principle. I’ll just really give a kind of introduction. We’ve been working in this group for something like, I don’t know, two or three years, something like that. It’s Uri, Shay, Dov Gabai and me, and maybe lately Belfer has joined us, perhaps. Our goal, in principle, is twofold. I call it import and export here; that’s what we call it. Import means bringing in, importing techniques from other fields in order to improve a bit our ability to deal with Talmudic-halakhic material. And export means taking certain insights, sometimes even tools, from the Talmudic field and trying to show their implications elsewhere. The warning against formalophilia—I spoke about that the last time I spoke here. I’ll just mention it briefly, also because it comes up a lot for us. The goal, of course, is not to do formalization for the sake of formalization. In other words, unless we succeed—at least that’s my goal, here we’re not completely unified—we try, at least from my perspective, to do things that can’t be done without formalization. Meaning, the fact that I translate the halakhic issue into mathematical language doesn’t help me much unless that language is a more convenient one for presenting the problems or handling them. So therefore I want to show that maybe here that really is the case. What has come out so far is five books: the logical hermeneutic principles, the textual ones, rules and particulars, deontic logic, logic of time, conflict resolution—which is the topic I’ll speak about today, part of what we dealt with. And now we’re dealing with questions of agency and representation in Jewish law. So let’s start with the topic of the completeness of Jewish law. And one more short introduction before I start with that issue. One could have spoken about this from above, and asked ourselves theological questions and principled questions about how Jewish law should be. Personally, I don’t believe in that kind of treatment; you have to see what it is, not what it ought to be. And many times it seems to me that this is a methodological flaw in many, many treatments of the nature of Jewish law, where people take all kinds of introductions to books, journalistic articles, and turn wishes—yes, the desirable—into the actual. One has to examine how Jewish law works, not how it is supposed to work. Facts are stronger than all sorts of theoretical and a priori statements. When I want to examine the completeness of Jewish law, then again, in taking Gödel’s theorem I’m careful to avoid a mistake that many, many people make, especially non-mathematicians: immediately applying it and saying, “This is complete, this isn’t complete,” and drawing all sorts of broad philosophical conclusions from Gödel’s theorem. Arnon warned us about that earlier, I think. And beyond the question of whether the halakhic system satisfies the conditions required for Gödel’s theorem to apply to it or not—that’s a question in itself. But I think that even the concepts of completeness and consistency with respect to Jewish law are concepts that have to be defined somewhat carefully. Meaning, even before the question of how I characterize the halakhic system mathematically or logically. So how can one approach this—not from above and not through general conclusions, but through the terrain itself? To see how this thing works. So the first thing that perhaps would naturally suggest itself is: what do we do with contradictions? How do contradictions appear, how do disputes appear actually, not contradictions? How do disputes appear? How can it be that from the same system two different people emerge with different conclusions? Ostensibly that means the system is inconsistent, in the usual language. But I don’t think—first, I don’t think that is necessarily true; the concept of consistency needs to be defined a bit more precisely. And second, the question is: what is the system? Meaning, it’s definitely possible that Reuven and Shimon, who both interpret the halakhic system, perhaps because it is not mathematical in character, are actually working within different systems. The different interpretations create different systems, and it is entirely possible that Reuven’s system is consistent and complete and has no problem with it, and likewise Shimon’s. They’re just different. The fact that there are disputes emerging from the same sources does not mean they are dealing with the same system, and therefore one has to be very careful in applying the concepts—and certainly the techniques—to this issue of Jewish law. Therefore I think it’s preferable to choose a slightly narrower direction, and I’ll try to speak a bit about loops. Loops—paradoxes, if you like—in Jewish law. If such things exist, that somewhat undermines the concepts, at least in the usual sense of completeness and consistency, because here I can no longer attribute it to different opinions, since I’m speaking from the perspective of a particular sage. From the perspective of a particular sage, if he himself gets caught in a loop, then clearly there is some problem here of completeness and consistency. And through these examples of loops I’ll try to show that I think practical problems can’t be solved if we are in a theoretical problem, at least in these cases. The theoretical problem has to be solved, because without that you can’t solve the practical problem. And therefore I think mathematics does have something to say here. And one last sentence: I will use inspiration from Gödel’s theorem—not its conclusions and not an interpretation of Gödel’s theorem, but inspiration. Gödel’s theorem tells us that in order to prove—the G statement was a correct statement, it was just not provable in T. Meaning, within T it cannot be proved. So how do we know it’s correct? Gödel’s theorem proves it. So it proves it, but apparently not in T. So where, then? You have to step out of the formal system you’re dealing with to a stronger language, as Dov said earlier, and in that stronger language perhaps you can solve the problems that cannot be solved within language T. That’s the only use I intend to make here of Gödel’s theorem. Meaning, this very intuitive insight that in order to solve internal problems sometimes you have to step outside, outside the box. Beyond that, I’m not committing to any conclusion from Gödel’s theorem about Jewish law; I have no such conclusion. So, fine, I’ve put the cards on the table. Let’s begin. We begin with a well-known Tosafot; there’s a parallel in Kiddushin. This is Tosafot in Bava Metzia dealing with a dilemma—or actually a three-way dilemma—of three possible moves, and the dilemma, or the three-way dilemma, is which of the three to choose. His own lost object takes precedence. So Tosafot says: “And if you say: his own lost object, his rabbi’s lost object, and the honor of his father—which comes first?” He has three, three actions before him, yes. His own lost object is lying before him—he needs to save it or wants to save it. His rabbi’s lost object, which he presumably also needs to save, perhaps also wants to. And the honor of his father, where he also has to honor his father. So there are three tasks. The assumption is that he can’t do all three; he can only do one. So he has to choose which has priority. The problem is that we can’t decide which has priority, because Tosafot says this: “If it is his own lost object,” if we decide that his own lost object comes first, then surely the honor of his father is preferable according to the view in the first chapter of Kiddushin that it comes from the son’s own resources. Meaning, there is an opinion in the Talmud that my obligation to honor my father applies even at financial cost. Meaning, I have to spend money in order to honor my father. Tosafot assumes that if so, then my father’s honor takes precedence over my own lost object. I will lose money, I will lose my own lost object, but for the sake of my father’s honor I am obligated to spend money. Meaning that my own lost object is not preferable to my father’s honor; my father’s honor is preferable to it according to that opinion. And if it’s my father’s honor—fine, then let’s do my father’s honor. He says no: if it’s my father’s honor, then his rabbi’s lost object comes first. His rabbi’s lost object comes before his father’s lost object, as was said above: “I am the Lord.” The Talmud derives from a verse that his rabbi’s lost object comes before his father’s lost object. Tosafot doesn’t mention here, because it’s the very Talmudic passage he is discussing, that of course he himself takes precedence over his rabbi, in order to close the loop. Meaning, he takes precedence over his rabbi. That’s the third principle, and I summarize it below. His father’s honor takes precedence over his own lost object—that’s what Tosafot says. His rabbi’s lost object takes precedence over his father’s honor—that too Tosafot says. And third, the Talmud says: his own lost object takes precedence over his rabbi’s. What do we do? There’s no way to decide the question, actually. So if we return to the question of completeness, in this case I think it’s more completeness than consistency, then basically there’s a question here that we cannot answer within Jewish law. That’s what is usually accepted as a problem of completeness. Another example: “They raised a question: a bird sin-offering, an animal burnt-offering, and a tithe—which comes first?” Tractate Zevahim. “Should the bird sin-offering come first?” What shall we say—regarding the bird sin-offering, which should be offered first, yes? If the bird sin-offering comes first, there is the tithe, which precedes it; the tithe is from an animal, so it comes before a bird—that’s the rule. “Let the tithe come first”—there is the animal burnt-offering, which precedes it; the burnt-offering takes precedence over the tithe in terms of certain criteria. “Let the animal burnt-offering come first”—there is the bird sin-offering, which precedes it. Again, three principles: the tithe comes before the bird sin-offering, the animal burnt-offering comes before the tithe, and the bird sin-offering comes before the animal burnt-offering. Three principles—what in mathematics is called non-transitive. Meaning, it is not the case that if x is greater than y and y is greater than z, then x is necessarily greater than z. In other words, there is a non-transitive relation here, and we cannot actually reach a decision. If we talk about non-halakhic examples, incidentally it’s quite hard to find such examples. It’s very hard to find something that is non-transitive, or a kind of normative loop—I tried to think of examples, it wasn’t easy. All kinds of dilemmas I thought of are basically dilemmas where you can say this or you can say that, but it’s not a loop. Rather, it’s something for which I don’t have one clear answer, or I don’t know what the clear answer is. Here the problem is—or may be—that I cannot justify any answer, not that I don’t know which answer is correct. I know that all of them are not correct. That’s something else. And that—I know—a soldier can fire at an enemy soldier with ammunition forbidden by international law—it’s gas, all right?—and hit only the enemy soldier, just him. He can drop a bomb that is permitted by international law—maybe he won’t get the Jonah award at the UN, but it’s permitted—but then civilians will also be harmed, because a bomb has a wider effect. And he can do nothing, in which case he himself will of course be harmed. All right? Now the question is: what is he supposed to do? So you see the pictures above—what happens to someone who doesn’t decide correctly, then that Jew Goldstone will come settle accounts with him. Okay, so now we have two concepts that I want to come back to, and my two predecessors also spoke about them. The concept of completeness—and I’m using an Escher-like image. Now the book Gödel, Escher, Bach has come out in Hebrew, so I’m using an Escher image. The image shown here is not by Escher. It’s a paraphrase of an Escher image; in the book we wrote that we wished there were such an image, and afterward Ruli showed me that there is. Because the problem of completeness can be presented, say, through a statement like the one already mentioned: “Statement A: Statement A is false.” Yes, it speaks about itself. Or “I am a liar,” as was mentioned earlier. Meaning, this is a problematic statement because I can’t arrive at an answer regarding it. I can’t arrive at an answer regarding it—neither true nor false. If it’s true, then it’s false; if it’s false, then it’s true. The source—from the New Testament, about the inhabitants of Crete—is actually not paradoxical, because if you say all Cretans are liars, then that means you too are a liar; and if you are a liar, that only means not all Cretans are liars—there could be someone else who isn’t a liar, not you, your cousin. Okay, so that’s not a paradox. So if you want to make it sharper, you do it with a single self-referential statement that refers to itself. The illustration is two hands erasing each other, not drawing each other, from an advertisement for shirts, by the way. And that is the illustration for this statement; it presents the problem of completeness because a system cannot give an answer to this statement or determine whether it is true or false.

[Speaker C] So it’s not complete in the conventional sense.

[Rabbi Michael Abraham] The question of consistency is represented by Escher’s original image, and here there is what I called in another context an anti-paradox. “Statement B: Statement B is true.” This is an anti-paradox because if we decide that this statement is true, that is consistent. If it is true, then that’s fine—it really is true. If we decide it is false, that too is also consistent, because it indeed comes out false, since it says about itself that it is true, so if it is false then it is lying. Right? So both the claim that it is true and the claim that it is false are consistent. Or in other words, this represents a situation in which one can derive from the system both A and not-A, both a positive answer and a negative answer, or both truth and falsity for the same proposition. That’s the problem of consistency. All right? Now when we speak in the logical context, then the problems of consistency and completeness are connected to each other, because the statement with respect to which the logical system is incomplete is the statement that asserts the consistency. There is some connection between completeness and consistency simply because the statement Gödel constructed in order to show incompleteness is a statement about the consistency of the system. In principle, these are two different properties. Meaning, in principle there could be—I don’t know, the mathematicians here will say—but I assume there could be systems with only one of the two properties. So these are not two overlapping concepts. Fine. So what do we do with the problem of completeness, with problems of completeness of the kind I presented earlier? A problem like that, non-halakhic, or these halakhic problems. These are basically problems of completeness, yes, not of consistency. Because in fact the dilemma presents me with a situation regarding which I cannot derive an answer from Jewish law. This is a problem for which Jewish law gives no answer, and therefore there is a problem of completeness here. What do I do with such a thing? So let’s start moving toward the Gödelian insight I mentioned earlier, a soft Gödelianism. Let’s start with a fateful normative dilemma, as I write here. Reuven wants to eat chocolate and at the same time also does not want to.

[Speaker A] He’s in tremendous turmoil over this.

[Rabbi Michael Abraham] A really not-so-simple dilemma. Compared to this, the soldiers are a joke; here there’s something existential. What do you do with such a dilemma? On the theoretical level, there’s a contradiction here. What do you mean—how can you say, at one and the same time, I want to eat chocolate and I don’t want to eat chocolate? Both A and not-A—that’s a contradiction.

[Speaker A] On the practical level it’s a dilemma.

[Rabbi Michael Abraham] Meaning, beyond the theoretical question of how to understand what he’s saying, practically speaking, what do I do now? I have two contradictory principles—what am I supposed to do? So Dov says: leave the contradiction alone; what difference does it make? The main thing is that we manage. Flip a coin. Flip a coin, and either you’ll get fat or enjoy yourself, or get fat and enjoy yourself, or get thin and not enjoy yourself, more precisely. Everything is fine. I think that in order to make a halakhic decision you also have to solve the theoretical problem. And therefore I don’t want to distinguish here between the contradiction and the dilemma. You have to solve the contradiction, and then maybe—and even then not necessarily—but maybe you can decide the practical dilemma. So how do you do it? Here, of course, it’s simple; this has already been said. These two statements actually express two different axes. From the standpoint of health, I want not to eat chocolate; from the standpoint of taste, I do want to eat chocolate. So therefore, what Rabbi Chaim of Brisk called two laws, or what Rabbi Breuer called two aspects. From the standpoint of taste I want the chocolate, and from the standpoint of health I don’t want it. That basically solves the theoretical problem, right? The contradiction. There’s no contradiction. It’s from this perspective and from that perspective, or Dov’s blue-and-black darkness. But that still doesn’t tell me what I do practically. Because on the practical level I’m still in trouble. What about deciding the dilemma? Meaning, okay, I understand—there’s no contradiction between these two desires—but how do you make the decision? So here too the decision isn’t so hard to make. If we’ve solved the theoretical problem, then we’ve identified two values involved in this dilemma. One value is health and the other is taste. So if that’s the case, now we can step outside the system—we already did that when we spoke about taste and health—and try to quantify, to translate in some way into a numerical translation, for example, or determine some measure for taste and health. Everyone can choose the measure he likes. Here it doesn’t matter at all what we decide; I’m only trying to show that there is a way to reach a decision. There is nothing problematic here in principle. That’s the point. I’m not trying to convince you right now what the decision will be, only that there is no problem. Once we solve the theoretical problem, the practical problem is also settled—settled at the level of principle. Meaning, there’s no longer a principled problem; now you just have to decide. So units of importance for taste, let’s call that x; importance of health, y. The value-weight of eating chocolate—I’m now comparing eating and not eating. So eating chocolate is n units of taste minus m units of health. Okay? And of course the opposite for not eating. Meaning, the weight of not eating is of course the reverse. And now each person will place for himself how much one unit of taste is worth relative to health, how much taste there is in chocolate, how much health damage there is in chocolate. Mix well, and whichever comes out bigger is what you should do. Okay? So the decision here is actually simple under two assumptions—or not two assumptions, really one. One assumption is what is called commensurability. Meaning that I can really place taste and health on the same scale, or measure both in the same units. Because notice: the units of importance of x and y, I call them by different names, but they are both measured in the same unit. In this case I represent it with a number, because otherwise this expression is meaningless. What does m times x minus m times y mean? Five apples minus eight oranges is not a meaningful expression. I have to translate everything into the same unit of measurement. So therefore my assumption is that there is commensurability here—a common measure, yes? That’s commensurability, meaning there is a shared measure for both. Beyond that, why did I split it into two? I could have said x minus y, y minus x, and that’s it. Why do I also need the full units, the n and the m? That’s because not every case is a two-way dilemma; sometimes it will be a three-way dilemma, and then we’ll need that separation. In any case, the reward is to eat salad, of course. Okay. Solving a non-binary dilemma. We go back to one of the dilemmas I presented earlier. His father’s honor comes before his own lost object—his father is z, his own lost object is x. His rabbi’s lost object comes before his father’s honor. His own lost object comes before his rabbi’s. So the solution to the contradiction, first of all, has to begin in theory. As I said, I’m not willing to enter the practical level before solving the theoretical problem. How is it theoretically possible that I hold three such principles that do not preserve transitivity? So first I have to look for a theoretical explanation before moving to the question. And notice—maybe I’ll sharpen this a bit more. Any such dilemma—his father’s honor and his own lost object—can be decided. There’s no problem. On the practical plane, in principle, there’s no problem. Any dilemma—a dilemma means two and not three—is decidable, yes, a decision can be made about it. The problem with these three principles together does not arise as long as in practice we are dealing only with dilemmas of two options. But it seems to me that already here, even before we get to the three-way dilemma, there’s a problem. The problem is in the theory, not in the practice. There’s only a problem in the theory here. Now Dov tells us, fine, why do you care? We know what to do. Why should I care about theoretical problems? So I say it’s not exactly like that, because when I now move to the three-way dilemma, and now I ask: I have my father’s honor, my own lost object, and my rabbi’s lost object all together. Now there are already three things, and now I want to make a decision—now I no longer know how to make the decision. So now I have a practical problem too. Then he’ll say, fine, fine, find a practical solution. I think the practical solution is connected to going back here and solving the theoretical problem. If I solve the contradiction, I have a chance—like with the chocolate—I have a chance of also deciding the practical dilemma. So therefore it seems to me that I really want to start from the theory, though of course the goal is to show what I do when I’m faced with a three-way dilemma. Because with each of these separately there is no practical problem. The practical problem is—the theoretical problem is, of course—how can it be that in one system there are three such principles? We return to the completeness of Jewish law, yes? How can it be that in one system there are three principles that do not satisfy the expected logical transitive relation among them? Okay. So now I begin from the theory. Exactly like with the chocolate, I ask myself not whether I want or don’t want, but why I want and why I don’t want. Like the blue and black—why was it black for one person and blue for another? Okay? That’s really stepping outside the box, stepping outside the system. So I ask myself which axes are playing a role here. In these three rules—what is at work here? Like taste and health in the case of chocolate, only this time it’s a bit more complex. So one axis is the red axis: financial interest. I want to save my own lost object, I want not to lose money. I mark the units with A, also in red accordingly. The second interest or value is the status of the commanding authority. His rabbi’s lost object is, as it were, a reference to the Holy One, blessed be He—“I am the Lord,” it says there in the Talmud—and the Holy One, blessed be He, takes precedence over his father. At least that’s how the Talmud treats it, without entering now into the content of the matter. So the status of the authority underlying the obligation also plays a role here. The Holy One, blessed be He, stands above his father. So therefore I mark this with the letter B. That’s a second parameter. A third parameter is the commandment to honor parents, to honor one’s father. All right? So that too has some weight; I mark it with the letter C. Up to this point, what have I actually done? Ostensibly I’ve solved the theoretical problem. I haven’t yet reached the three-way dilemma. I had a theoretical problem: I don’t know how I can have three contradictory or non-transitive principles. So I say, what’s the problem? This one is from the standpoint of A, this one from the standpoint of B, and this one from the standpoint of C. Like with the chocolate. I want to eat chocolate because of taste; I don’t want to because of health. So if I found the theoretical axes, the theoretical parameters, then ostensibly I solved the theoretical problem. In a moment we’ll see that I didn’t—but in a moment. Okay? So what do I do in order to decide the three-way dilemma? In order to decide the three-way dilemma I now have to give a weight, find some kind of measure function, yes, for each of the sides x, y, and z. And I want to evaluate each of the sides x, y, and z—and maybe just to remind you, these are the three sides, you see? His father’s honor is z, his own lost object is x, and his rabbi is y. I want to know what value-weight each one has—interest-value weight, yes?—in terms of A, B, and C. So I say: his own lost object has some weight—this is the previous n and m, only now we have three sides among which we must choose—so there is some weight, certain units of interest, units of commanding authority, and units of commandment, the commandment of honoring parents. And similarly, saving his own lost object, and his father’s lost object, and his rabbi’s lost object—each of them has some weight in terms of the three axes that I identified. Okay? Now, from the three previous principles—these three previous principles here, his own lost object precedes his rabbi’s, his rabbi’s lost object precedes his father’s—I can extract this. Meaning, these relations among the constants. For example, see what this means: A represents his financial interest, so his financial interest is more significant in saving his own lost object than in saving his father’s lost object, right? He gains more money. So therefore it is obvious that n11 is greater than n21. And so on. These three principles are translated into three relations. I can also bring in substantive considerations, and this definitely is not empty formalism—I need to rely on content and arrive at this set of values, this matrix: n11, and there are nine coefficients here, these are the nine coefficients. Okay, for example, there is no honoring of one’s father in terms of—whatever—there is no honoring of one’s father except in saving his father’s lost object. In saving my own lost object, from the standpoint of honoring one’s father, that’s zero. In saving the rabbi’s lost object, likewise, the honoring-father value there is zero. And so on. So I find such a matrix. I think it can be justified fairly well, but I don’t want to go into too much detail. What do I now have to do? This isn’t enough, because now I’ve found these coefficients, but now I have to determine A, B, and C. Remember the n times x minus m times y that I also did with the chocolate? Okay, so here too, same thing, I want to arrive at numbers. When I arrive at numbers, I know—I’ve determined the coefficients, now I want to determine these. Now here suppose I take something that comes from an interpretive judgment within Jewish law. Let’s say my own interest, for example, is not of very great weight, but it isn’t negligible. If my interest exists and no value stands opposite it, then I am certainly entitled to realize my interest or act for the sake of my interest. Therefore I give it a one. That’s the unit of measure. Okay? Obedience to an authority—the Holy One, blessed be He, or his father or something like that—let’s say that already has greater force than mere interest. That’s something of value.

[Speaker A] And honoring parents outweighs everything, so five.

[Rabbi Michael Abraham] Okay? Just so you can see the form of the reasoning. The moment I fixed these numbers and fixed these coefficients, those three become numbers. F of x comes out 1, f of y is 4, and f of z is 7. So what do we choose? z has the highest value, so the result is that one must save his rabbi’s lost object. z is his rabbi. Okay? So apparently, after solving the theoretical problem, because we found that we’re dealing with three different axes, like the taste and healthiness of chocolate, I can also decide the practical dilemma. But that’s not… that’s not true. Because if we now translate the halakhic instructions of the laws of precedence, these are the precedence rules that are supposed to hold, only this one holds. Meaning, only this one—that his rabbi’s lost object takes precedence over the lost object… what is that there? F z probably got flipped on me, I think. His father, yes. Fine—his father’s lost object takes precedence over his own, that holds. The other two do not hold. Look there at the 1, 4, and 7 that I arrived at—it doesn’t satisfy that. Now of course what does this actually mean? It basically means that the numbers I arrived at—1, 4, and 7—do not reflect the relationship between the halakhic values. I didn’t succeed. Now why didn’t I succeed? It’s no surprise. Why didn’t I succeed? Because if I translate f of x, f of y, and f of z into numbers, by definition I won’t succeed. There are no three numbers that satisfy all three of those. None, simply none. Any three numbers you want—it can’t be, because the relation is non-transitive, and numbers are always transitive. So in practice there’s some kind of problem here. I thought I had solved the theoretical problem, but no, I hadn’t. This measure I found—the measure is the evaluative weight that each side x, y, or z has—doesn’t really represent the instructions of Jewish law, so my model isn’t good; it doesn’t represent Jewish law. So maybe it will tell me what to do, but that won’t help. It’ll tell me… the wrong things to do. Therefore, in practice, here we have to fall into a certain despair, because the conclusion is that such a measure function cannot exist. There is no such measure function, because every numerical result must always be transitive. It’s impossible to have three such numbers, or three such results, that satisfy those three inequalities, so this shouldn’t surprise us. The second thing, of course, is that our problem is not in practice but in theory. With divine inspiration I already foresaw what Dov wanted to do, so the problem is in theory. We really do not succeed in representing the instructions of Jewish law with this device, in actually building a measure function that will measure or weigh each of the steps and give us the ability to decide. And of course you can’t do that by definition, because every such function maps the steps to numbers—but there are no three numbers that satisfy those three inequalities. So what do you do in such a situation? What has to be done, and there is no other choice, is to set situation-dependent weights. What does that mean? I’ll give you an example before I get into that. And again, this is to solve the theoretical problem, but once we solve the theoretical problem, the practical problem is also solved—solved in principle. There is a way to solve it. The halakhic interpreter still has to decide what weights to attach, but we’re no longer in the paradoxical realm, okay? Permitted versus overridden. For example, we know there is a prohibition against violating prohibitions, right? There are negative commandments in the Torah; you’re forbidden to violate them. But if there is a positive commandment against a negative commandment, then a positive commandment overrides a negative one. Yes, if someone has nothing to eat—meaning, he has no flour for matzah on Passover—and there is only flour from the new crop, which one may not eat until the day after Passover, then since there is a positive commandment to eat matzah, that overrides the prohibition against eating new grain. And therefore he may take flour from the new crop and make matzah from it. A positive commandment overrides a negative commandment. There is a dispute among medieval authorities (Rishonim) about how to understand a positive commandment overriding a negative one. Some make it into a kind of inequality, that the positive commandment is more important than the negative one. But others say that in a place where there is a positive commandment, the negative commandment does not exist at all. In halakhic language that is called “permitted.” It is not overridden by the positive commandment; it is permitted. It has no negative weight at all when you eat matzah from new grain. What is the meaning of “permitted”? The meaning of “permitted” is that the weight of eating new grain is not a universal weight; it is not a weight that is valid in every situation. When I have nothing against it, it is forbidden; it has a negative value. But when a positive commandment stands against it, then it is not because the positive value of the positive commandment overrides the negative value of eating new grain; rather, the negative value of eating new grain itself is already different—it does not exist; it becomes zero. What does that mean? That I cannot attach one number to every halakhic act; rather, I need to take into account the situation in which that act is found. If I now want to assign the value of violating a prohibition, eating matzah from the new crop, I say: if a positive commandment stands against it—zero, there is no problem, there is no cost when I ate it. If a positive commandment does not stand against it—minus a thousand. Meaning, it is very severe. Okay, so therefore the mistake we made earlier was when we defined the functions in this way: f of x, f of y, and f of z. Meaning, x, y, and z each give a number. That is not right. It should be f of z in one situation, and f of z in another situation will be a different function. In fact it’s a function of more than one variable; it has an index.

[Speaker A] So we’re back to, “Why did you make your rulings variable?” What? So why did you make your rulings variable?

[Rabbi Michael Abraham] Fine, there are places where you do make your rulings variable.

[Speaker A] Meaning the Sages don’t accept this principle?

[Rabbi Michael Abraham] Not always—you see that they don’t. There are examples; in the book we bring quite a few examples of this matter. Now what this means for us, basically—I won’t go into the details here—but what it means is that the function of act A depends on the question of what stands opposite it. There is f sub b of a and f sub c of a—where is it? f sub c of a. And it is not the same number. f sub b of a is the cost of performing act A when act B stands opposite it. And this is the cost or benefit of performing act A when act C stands opposite it. And the assumption is that they are not the same. There is dependence on the lower index. And that solves the transitivity problem, and there is no other choice. Meaning, without that, you cannot find a measure function and decide the dilemma. Now regarding the soldier’s dilemma, of course one can get into the substantive issues here, because everything here depends on the substantive issues, but what matters to us logically is only the fact that there has to be dependence on the situation. Now exactly how the dependence works—how to assess the worth, the value, of A in the presence of B or in the presence of C—that is already a question for the halakhic interpreter. That is not a matter for logical work. The schema—how to get to a situation in which there will be no contradiction—that does not replace the halakhic decisor. The halakhic decisor will have to decide what the size of this thing is. That is a question of interpretation, a question of evaluation; it’s something else. But he will have to make the decision in this way, because otherwise he will not be able to make a decision. Okay. Now what does this mean in the soldier’s dilemma? So yes, for example international law has a different weight—as I think everyone understands—when my life is threatened and when my life is not threatened. It is not true that the weight of obeying international law is a fixed and universal weight in every situation. There will be situations in which I will violate international law, if there is another value, or another interest, that is very, very important and in my eyes justifies that. Therefore it is not correct to assume that the value of violating international law is a universal value, and that is exactly the same way to solve this dilemma too. So let’s do an example, an example with which I’ll finish. What time is it now? Quarter to…

[Speaker A] That went fast. It makes it prettier.

[Rabbi Michael Abraham] Another minute or two and we have to finish? Okay. Look, this example works like this. I mentioned earlier matzah from the new crop; this is a paradox that several later authorities raise. And the paradox is built as follows. We are on Passover eve, and one has to eat matzah. Now grain from the new crop is still forbidden until the day after Passover, still forbidden. There is grain from the old crop, but it’s very expensive. Now there is a person who, in order to buy grain from the old crop, would have to spend all his money. Okay? To buy grain from the new crop—there is some at a reasonable price, negligible let’s say for the sake of discussion. Okay? What is he supposed to do? So we have three principles. One principle is that we do not spend more than a fifth on a positive commandment. A fifth of my assets I spend on a positive commandment. The second principle is that for a negative commandment one must spend all one’s assets in order not to violate a negative commandment. And the third principle is that a positive commandment overrides a negative commandment. Now when I try to think how to quantify these values, and actually represent the problem, then I say as follows. Suppose the value of this is a positive commandment and this is a prohibition, a and l, right? a is the positive commandment and l is the prohibition. And here, not violating a prohibition is minus l, and neglecting a positive commandment is minus a. So there are certain values for performing the positive commandment, violating the positive commandment, not violating the prohibition, and violating the prohibition. Okay? Now when I decide to eat old grain, what is the value? It’s exactly the same form as I said earlier, but I’m abbreviating. So eating old grain—what does that mean? I fulfilled the positive commandment of matzah, so I gained f of a, right? That is the value of performing a positive commandment. I gained the fact that I did not violate a prohibition—minus l—this is not violating a prohibition, so I gained that too, and of course I lost all my money. Right? So that is the total value of eating old grain. That is the option of—this is the cost that I gain or pay, really gain, for eating old grain. What happens with eating new grain? I fulfilled the positive commandment, but I also violated a prohibition. That is a negative number, f of l, yes? It is a cost; I violated a prohibition. Okay? Not eating at all—what does that mean? I did not perform the positive commandment, but I also did not violate the prohibition. Sitting on the fence. Not violating the prohibition, not doing the positive commandment, playing it safe. Let’s see if that really is the preferable option. Those are the three measures. It is obviously clear that this won’t work, right? Because of the consideration I mentioned earlier—because it is not transitive. Let’s see how it works. There is a principle that one spends a fifth of one’s assets. What does it mean to spend a fifth of one’s assets? That if I perform a positive commandment and I need to lose a fifth of my assets for it, that is the same as not performing a positive commandment, so there is equality here. That means one must spend a fifth on a positive commandment. This means one must spend all one’s assets on a prohibition. Here there is an inequality, and the value of the prohibition is at least all my assets; it is more than all my assets. Okay? So f of minus l is less than f of l—f of l is violating the prohibition; violating the prohibition is less good than the value of not violating the prohibition and spending all my assets. Okay? Now “a positive commandment overrides a negative commandment”—that means if I perform the positive commandment and violate the prohibition, that is this, you see? And if I do not perform the act and do not violate the prohibition, then that is this. So this is greater than that, because a positive commandment overrides a negative commandment. Okay? Now let’s substitute one into three and we get this. From equation two we get this. You see that there is a contradiction here. There is a contradiction. This quantity has to be greater than minus one-fifth x, but also less than minus x. All right? If x is ten, then it has to be greater than minus two and less than minus ten. There is no such thing. Now there is no such thing exactly because of the previous problem of transitivity. How do you solve it? This is an example of what I said earlier. Clearly one has to find here a situation-dependent weight. What does that mean? So look. Let’s take a moment to look at this equation—a non-obligatory positive commandment. A non-obligatory positive commandment is a positive commandment such that if I do it, I get reward, I gained something; if I don’t do it, nothing happened. There is no issue here of neglecting a positive commandment. Right? How much money does one have to spend on a non-obligatory positive commandment? Nothing. After all, I don’t have to do it at all. So clearly one does not have to spend money on it, right? Okay, so in a non-obligatory positive commandment, f of minus a is zero, so what is f of a? f of—sorry, f of a, sorry—the value is zero. f of minus a does not exist because it is non-obligatory; there is no problem of neglecting a positive commandment. It is zero because one does not have to spend money on it. That means that f of a is zero. The value I gained by fulfilling the positive commandment is zero. But there is no difference between a non-obligatory positive commandment and an obligatory positive commandment on the fulfillment side. What do I gain when I fulfill the positive commandment? It is the same in an obligatory positive commandment and a non-obligatory positive commandment. The difference is only that in a non-obligatory positive commandment there is no neglect, so f of minus a is zero. So that means I made a mistake here. In other words, money is not spent on fulfilling a positive commandment. Money is spent on avoiding the neglect of a positive commandment. And therefore in this case f of a has to be deleted from this equation, because it does not take part in the calculation of how much money one has to spend. For fulfilling a positive commandment, it has no value with respect to the question of how much money to spend. But where will it appear? Where is it? Where will it appear? In “a positive commandment overrides a negative commandment”—that is the whole idea. That the fulfillment of a positive commandment overrides the prohibition. So you see that f of a is situation-dependent. Here it is zero and here it is something. Now when you erase that thing, you’ll see that the contradiction disappears. The contradiction disappears, and you can find how much a positive commandment is worth, how much a prohibition is worth, and how much the Holy One, blessed be He, is worth if you want to buy Him. Okay. Summary, a brief summary. Let’s go back to the general questions. The big question is whether such a decision is a halakhic decision. In the end, the logic I did is logic. The decision is obtained when I plug numbers into these variables or these functions. Those numbers are supplied by a halakhic decisor according to his evaluation. Sometimes there are such considerations, other considerations. But these are considerations that I think are involved in very many acts of halakhic ruling. An act of halakhic ruling is not a positivistic step—it is not, yes in the positivist sense—meaning, it is not something that can be done by mechanical calculation. Clearly values enter here from the world of the halakhic decisor—his assumptions, his conceptions, and so on. That is why I said that in disputes it will be difficult to discuss completeness and consistency. Now, therefore I tend to think that such a decision is a halakhic decision. Because every halakhic decision is like that. But then that raises the question: what is the halakhic system? So the halakhic system actually includes all kinds of assumptions and insights of the halakhic decisor himself, not only what is found in the sources. And only then can you perhaps arrive at completeness—that is, find an answer to each of the questions. The incomplete system is the system without the halakhic decisor. The halakhic system pure and simple is the M-T system. Within that system there is no answer. There is no answer to the question. But if we go outside that system, begin to measure it, ask why—why does this cost this much and that cost that much, and this overrides that and that overrides this—we use a stronger language. In that language one can find a solution; the system can indeed be complete. Now it may be that in that complete system there will be some other Gödel sentence, never mind, but that is another matter. What there is here, essentially, is a kind of ongoing giving of the Torah, or an addition to Torah of all sorts of insights and intuitions that exist within the halakhic decisor. The decisions of the halakhic decisor themselves also enter into the halakhic system, and then one can relate to it as complete. It is a semantic question. The halakhic system in itself is certainly not complete. But then one also cannot make decisions, not even practical ones. You can’t. When we make practical decisions, that means we are bringing into the system—our universe expands, okay? The halakhic universe. We need to bring into it also the conceptions of the halakhic decisor himself, and only then does this thing become calculable—or, have an answer. Calculable, I’m not sure that’s the word. That’s it, thank you.

[Speaker A] Thank you very much to all the speakers and to everyone who managed to keep their head through all of Gödel’s sentences and their implications. We’ll meet again at the next Nitzotzot session. Thank you very much.