Positivism in Halakha and in General, Lesson 4

Table of Contents

• Russell and Whitehead’s theory of types and self-reference paradoxes
• Analytic philosophy, problems of language, and the example of the flying arrow
• Paradox, thought and language, and the analogy to physics of divergent sums
• Paradoxes in Jewish law and in law: normativity, contradiction, and theology
• “Two verses that contradict one another”: Rabbi Akiva, Rabbi Yishmael, and the Raavad
• Rules, loops, and stepping outside the system: Gödel’s theorem and understanding “from outside the system”
• Matzah from new grain: a positive commandment overriding a prohibition, one-fifth, and all of one’s wealth
• A solution through weighing and hierarchy: giving weights to rules and qualifying the rule of “all of one’s wealth”
• Positivism in Jewish law and in law: a closed system versus an open and casuistic system
• Tosafot: one’s own lost item, the lost item of one’s rabbi, and honoring one’s father as a triple loop
• This is true in management, it’s true in interpersonal relationships, and it’s doubly true in the service of God.

Summary

General overview

The text presents a critical approach to proposed solutions for paradoxes, especially self-reference paradoxes, and argues that a real solution has to be reasoned and principled, not just an ad hoc prohibition against formulating the problem. It distinguishes between paradoxes that arise from language and are therefore solved by conceptual analysis, and situations in which the problem may lie in the principles of thought themselves or in normative structures such as Jewish law and legal systems, where genuine contradictions may exist. It develops a direction of solution based on “stepping outside the system” and examining the rules themselves, including creating a hierarchy or assigning weights to rules, and connects this to a critique of positivism as viewing Jewish law as a closed computational system.

Russell and Whitehead’s theory of types and self-reference paradoxes

The text describes Russell and Whitehead’s theory of types in the introduction to Principia Mathematica as a solution to self-reference paradoxes such as the liar paradox, by constructing a hierarchical language in which a statement can refer only to statements below it and therefore cannot refer to itself. It argues that this move does not solve the problem but merely forbids presenting it, and is therefore an ad hoc solution that replaces the language instead of answering the paradox in the language in which it is formulated. It adds that the prohibition is too broad, because not every self-reference is paradoxical, and so the fix unnecessarily harms valid sentences as well, which highlights the lack of seriousness in a solution of the form “declare paradoxical sentences illegal.”

Analytic philosophy, problems of language, and the example of the flying arrow

The text attributes to the analytic school, of which Russell was one of the founders, the claim that many philosophical problems arise from disruptions in language and conceptual imprecision, against the background of the assumption that reality itself contains no paradoxes, because nothing can both be and not be in the same respect. It accepts that some problems really are linguistic in origin, but rejects the generalization that all problems are of that kind, and insists that the justification for a solution must be its internal logic and not merely the fact that it makes the paradox disappear. As a successful analytic solution, it presents the “paradox of the flying arrow” through the distinction between “is located” and “is standing still,” and argues that the paradox is created by confusing location with zero velocity, so conceptual analysis dissolves the difficulty without any need for infinite mathematical tools.

Paradox, thought and language, and the analogy to physics of divergent sums

The text tells of a lecturer from the Technion, Boris Shapiro, who justified computational manipulations in physics that lead from an infinite result to a finite one on the grounds that in reality there are no infinities, and therefore a method that restores finitude is “probably right.” It compares this to the analytic motivation according to which there are no paradoxes in reality, so their source must be in us, but argues that the failure may lie in the principles of thought and not only in language. It suggests that one could view the theory of types as a restriction on thought, but emphasizes that it is presented mainly as a syntactic restriction on sentence structure rather than guidance about what to think, and therefore in his view it is still not a solution.

Paradoxes in Jewish law and in law: normativity, contradiction, and theology

The text argues that the assumption that “there are no paradoxes in reality” is less applicable in normative systems, because Jewish law and legal systems are not physical facts but products of thought, and therefore genuine paradoxes may exist within them. It brings the story of Gödel, who found a contradiction in the American Constitution and concluded that it could be used to lead to dictatorship, and mentions the logical rule that from a contradictory system one can derive any conclusion. It adds a common theological claim that in divine Torah there can be no contradictions, and presents this as incorrect, because Jewish law is in practice formed through human interpretation and even includes rabbinic tools such as “two verses that contradict one another.”

“Two verses that contradict one another”: Rabbi Akiva, Rabbi Yishmael, and the Raavad

The text describes a systematic difference between Rabbi Akiva and Rabbi Yishmael in the way they apply the interpretive rule of “two verses that contradict one another.” It attributes to Rabbi Akiva a model in which one seeks “a third verse that decides between them” in the sense of joining one side and deciding like “two against one,” without engaging in a full reconciliation of the rejected verse. It attributes to Rabbi Yishmael a model in which the third verse helps reconcile the two verses and interpret each as speaking in a different context, so that there is no contradiction, and it cites the Raavad at the beginning of the Sifra, according to whom a contradiction can be reconciled through reasoning as well, and not necessarily by a third verse. It suggests the possibility that the gap reflects an ideology regarding contradictions, and raises the question whether there is a place where Rabbi Yishmael offers a reconciliation and Rabbi Akiva rejects it and prefers a different form of decision.

Rules, loops, and stepping outside the system: Gödel’s theorem and understanding “from outside the system”

The text presents Gödel’s theorem as the claim that in certain axiomatic systems there exists a sentence that is true but cannot be proven within the system, and emphasizes that Gödel “broke” the project of Principia Mathematica, which tried to ground all of mathematics in a complete and consistent way. It describes the puzzlement over how one can prove the truth of an unprovable sentence, and resolves this by saying that the proof exists but outside the system, that is, by means of tools not included in the axioms of that framework. It presents this as a general model for solving entanglements: when the mechanistic application of rules closes in on a loop, one has to move to a meta-systemic level and examine the rules themselves.

Matzah from new grain: a positive commandment overriding a prohibition, one-fifth, and all of one’s wealth

The text presents a halakhic dilemma called “matzah from new grain,” based on three rules: a positive commandment overrides a prohibition, for a positive commandment one spends up to one-fifth of one’s wealth, and for a prohibition one spends all of one’s wealth. It constructs a case in which flour from new grain is cheap but forbidden as new grain, while flour from old grain is permitted but so expensive that it costs half of one’s assets or more, and shows how trying to apply the rules leads to a loop in which each decision is pushed aside by another decision. It distinguishes between a logical contradiction and practical inapplicability, and suggests calling this a “conflict” or a “loop” rather than necessarily a paradox, comparing it to dilemmas like chocolate that is “fattening but tasty,” and to the example of saving a life on the Sabbath.

A solution through weighing and hierarchy: giving weights to rules and qualifying the rule of “all of one’s wealth”

The text argues that if there is a solution to loops of this kind, it must come from stepping outside the system of rules and creating a hierarchy or “weights” that evaluate the cost and value of infringing one rule as against fulfilling another. It suggests that in the case of “matzah from new grain,” the breaking point is the assumption that one is obligated to spend “all of one’s wealth” in order not to violate a prohibition even when that prohibition is overridden by a positive commandment, and argues that a prohibition overridden by a positive commandment does not “weigh” the same as an ordinary prohibition, so there is no necessity to pay a great deal of money to avoid it. It concludes that on this approach the solution is to eat matzah from new grain, and emphasizes that the move is not a mechanical calculation but a judgment about the rules themselves, an action that a computer does not perform when the rules are given to it as axioms.

Positivism in Jewish law and in law: a closed system versus an open and casuistic system

The text identifies this difficulty as especially problematic for positivist thinking, which sees Jewish law as a closed computational system of rules from which answers can be derived deductively in a mechanical way. It argues that the proposed solution requires the opposite assumption: that Jewish law is not a closed system but an open one that allows meta-rule discretion and judgment of the rules themselves. It compares German law as reflecting a positivist tendency with British law as casuistic, based on precedents and induction, and formulates the point that the very willingness “to step outside” already undermines positivism even before a solution is found.

Tosafot: one’s own lost item, the lost item of one’s rabbi, and honoring one’s father as a triple loop

The text cites Tosafot describing a triple case involving one’s own lost item, the lost item of one’s rabbi, and honoring one’s father according to the view that honoring one’s father comes “from the son’s resources,” and shows how different priority orders create a loop in which each option is displaced by another priority. It argues that, in principle, situations like these can be solved only if one stops applying the priority rules as absolute givens and begins weighing the rationales and costs behind them in order to rank the various infringements and fulfillments. It states that the ability to solve complex problems depends on one’s point of view, and that in such cases the solution becomes possible only by rising to a level where the rules themselves become the object of judgment and not merely tools of calculation.

This is true in management, it’s true in interpersonal relationships, and it’s doubly true in the service of God.

The text states that the ability to solve complex problems depends on whether one remains inside the system of rules or manages to rise above it and discover new possibilities. It argues that this is true in management and in interpersonal relationships, and adds that it is doubly true in the service of God, where one strives to reach what is “beyond reason and understanding,” not out of contempt for reason but out of recognition of its limits. It concludes with the claim that every problem has a solution if you look from the right place, and that this is “the secret of faith”: that there is a ruler of the palace who determines the real rules.

Someone who got a flat tire said that from there it was impossible to move. Crazy. But we stood at the intersection for a quarter of an hour. By the way, today at around five, something like that, there was a traffic jam there—someone I was talking to on the phone was stuck there for two hours. Wow. It was at the exit from Israel Aerospace Industries. There was apparently an accident, I don’t know exactly; he told me he saw all kinds of ambulances. He stood there for two hours. Wow. In that same place. I thought maybe it was that other accident. I don’t know. Okay. Last time I spoke about paradoxes, and I want to discuss a bit what one does with them. If there are any—did I mention Russell’s theory of types or not? In the introduction to Principia Mathematica—the monstrous book by Russell and Whitehead. Nobody read it. Right? Nobody read it, or almost nobody. It’s a book with no words in it. They said that the person who wrote it didn’t read it. That guy was… really? There’s not a single word in the book, only formulas. Three volumes like that, of formulas. In any case, in the introduction there, they propose a solution to paradoxes of self-reference, like the liar paradox that I mentioned—I brought several examples of this—and the claim is that they build a language based on the principle of the theory of types. Meaning, they divide propositions into different types, build a hierarchy of types, and say that every proposition can refer only to propositions lower than it in the hierarchy. Of course, that means it can’t refer to itself. This comes to solve the problem in set theory, the paradox of set theory. Right? That’s basically the motivation. In any case, Russell and Whitehead’s claim is that if we adopt the theory of types, then paradoxes of self-reference won’t appear. I never understood the logic behind this solution, because of course it doesn’t solve the problem, it only forbids presenting it. In other words, you build a language in which it will be impossible to formulate the problem, and everything is fine. That is, it’s simply forbidden; it’s illegal to formulate a sentence that refers to itself. Beyond that, this “solution,” in quotation marks, is an ad hoc solution. What do I mean? If there were some internal logic, beyond the fact that it leads to paradoxes, some logic as to why one ought to use such a language and not another language, I’d understand. But if you produce this language in order to solve the paradox, then it’s not really a solution to the paradox. Because it only says: okay, let’s adopt another language in which it won’t appear. Fine—but you haven’t answered it. So I’m now using the old language, and I can formulate it. What’s your answer? “Let’s adopt a new language and not say it”? That’s Stalin’s solution to paradoxes. Whoever says it gets his head chopped off, and then there are no paradoxes and everything is fine. But it’s more than that, because this principle also forbids lots of sentences or claims that contain no problem at all. Not every self-referential claim leads to a paradox. There are many self-referential claims that don’t lead to paradoxes, and they’re perfectly fine claims. I don’t know—for example: all sentences are made of words. Okay? A claim, right? A claim that also refers to itself, because it too is a sentence. Okay? Does that lead to some paradox? It’s a claim that also refers to itself. So there’s nothing essentially problematic about self-reference. Why establish this principle? Because there’s an entire class of paradoxes that arise from self-reference, like the liar paradox: “This sentence is false,” so a sentence referring to itself. Okay? Or the barber who shaves all the people who do not shave themselves. I brought several examples of that. So it’s true that there are some paradoxes whose basis is self-reference. Self-reference is something prone to paradoxicality, but it’s not true that self-reference is essentially problematic. There are self-referential claims that raise no problem—why forbid those too? So even as an ad hoc solution, it’s too broad an ad hoc solution. In other words, you forbid things there’s really no reason to forbid. That only sharpens even more the previous problem I mentioned—that this doesn’t solve the paradox, it simply forbids expressing it. And along with that, it forbids expressing other things for which there’s no problem at all, so why forbid expressing them? You could just as well solve paradoxes in this way by saying that every paradoxical sentence is illegal. Why limit yourself to solving paradoxes of self-reference? Every sentence you see is paradoxical, declare it illegal, and everything is fine—we’ve immediately solved all paradoxes that were born, that will be born, that ever were, and that ever will be. You understand that this isn’t serious. It’s not really something that can solve the paradox. I think this is part of the so-called analytic approach. Bertrand Russell was one of the founders of analytic philosophy. And analytic philosophy advocates the claim that the problems in philosophy are rooted in distortions in language, in incorrect use of language. The claim is basically—and by the way, there is logic behind this—that in the background there stands some assumption that reality itself contains no paradoxes. Reality itself is either like this or like that. The paradoxes are somehow our artifact, something artificial born in us. Now where is it born in us if it doesn’t exist in reality itself? So they say: in language. In other words, apparently our language contains inaccuracies or something of that sort, and the use of language can somehow, because of this, lead to paradoxes. Therefore analytic philosophy, when dealing with philosophical issues, basically solves problems by linguistic means, by linguistic analysis. They analyze the concepts, redefine them, and thereby eliminate the problems. Now this in itself isn’t illegitimate, because there are problems whose source really is language. But I don’t think it’s true that all problems are like that. Self-reference is an example. And therefore, what will be the criterion? How will we know when yes and when no? When there is logic in the analysis—not merely the fact that it solves the paradox. That’s not the justification for adopting it. The justification for adopting it is that it is genuinely logical independently of this, and after that I also show you that it solves the paradox. That’s called a solution to a paradox, because it means you used something in an illogical way and therefore the paradox arose—that’s fine. Then I say: use it in a logical way and see that the paradoxes don’t arise, okay? I’ll give you an example. I spoke about the paradox of the flying arrow, right? We talked about it not long ago, the paradox of the flying arrow. My claim basically was this: the paradox says that at every single moment, if you look at an arrow flying, at every single moment you see it standing in a different place. So when does it move? When does it pass between the places where it stands? At no moment does this happen, because at every moment it stands—just stands in a different place. In short, this is a challenge to the concept of motion. My claim was: some people solved it with infinity, with all kinds of things like that. I don’t think you need to go there. A conceptual analysis solves it. That is, my claim is that at every moment the arrow is in a different place, not standing in a different place. “Is in” means it has a location; it is located in that place. “Stands” means not only that it is located in that place, but that it is located in that place and its velocity is zero. Right? Now who said its velocity is zero? It is in that place; it is not standing in that place. Now here, when I explain the difference between “is in” and “stands,” everyone understands it and would accept it even apart from the context of paradoxes. In other words, there is logic to this conceptual distinction not only because it happens to solve the paradox, but because it makes sense in its own right. It’s obvious that there is a confusion here between “is in” and “stands.” It’s obvious. I’m not saying it ad hoc just because it solves the paradox for me. Rather, everyone understands that it’s really true. Consequently, now I tell you that the paradox also disappears. That’s a paradox that is distinctly a paradox whose source really is in language—a linguistic problem. You’re confusing the concept “is in” with the concept “stands.” Once we analyze those concepts, we’ll see that they are not the same thing. Everyone who stands is also in some place, but not everyone who is in some place is standing, right? The identity is not bidirectional. And therefore this is an analytic solution to a paradox that is genuinely a correct solution, because the problem really is in the language. But there are things—or at least there can be things; I don’t know whether there are—maybe they really are right and all problems are linguistic problems. But I don’t know analytic solutions to all problems. Maybe I’m not smart enough; if I really searched, maybe I’d find that they are right, that all paradoxes can be solved with analytic solutions. But for example, the theory of types, in my view, is not a solution. Maybe there is another analytic solution there, but the theory of types is not a solution. Why? Because it’s ad hoc. Okay? So the motivation to create analytic solutions is understandable to me. The motivation is that there can’t be real paradoxes. In reality it’s either this way or that way. It can’t be both this way and not this way. In other words, in reality either something is such-and-such, or the barber shaves the people, or he doesn’t shave the people. So what can it be? What creates the problem here? Maybe something in the concepts I’m using—“shaves,” “all the people”—maybe there’s something in the concepts that, if we examine it well, we’ll see that it leads to problematicity. Because in reality itself, either he shaves or he doesn’t shave. There can’t be paradoxes in reality. And this reminds me of something very nice. Once, when I was at the university doing my doctorate, a lecturer came from the Technion, Uri Shapira. He came and gave us a course in field theory techniques, various things like that. In short, one of the problems in field theory—Feynman diagrams and all sorts of things like that—one of the problems there in those techniques is that when you sum the diagrams, the sum diverges, meaning it is infinite. Fine. So he’s standing at the board and says: okay, we’ve arrived at an infinite sum, let’s switch the order of summation, or swap the integral with the sum—we swap them, and now everything is fine, the result is three. Okay, that’s it, we calculated it, let’s move on. I said to him: one second. I studied mathematics with mathematicians, not physicists—that’s my advantage—and the mathematicians taught me that if a series does not converge, even if it doesn’t converge absolutely, you may not swap the sum with the integral. You may not do that. It’s illegal. So it cannot be that a divergent series, when you swap the sum and the integral, suddenly converges and now everything is fine. So he thinks for a moment and says to me: look, in reality the fields are not infinite, right? Clearly the phenomenon we’re calculating, in the end its result is something—three, I don’t know what—but it’s not infinity. There is no infinity in reality. Okay? So if I reach a finite result by this route, then that is probably the correct route. That’s an interesting claim. It’s not a ridiculous claim. It’s an interesting claim. I don’t think it’s correct, but it’s not ridiculous, because he’s saying: look, in reality there are no infinities. The infinities are a problem in our method of calculation. So if I found another method of calculation that brings me to a finite solution, then that is probably the correct one—assuming there isn’t another method that also brings me to a finite solution. Never mind. I’m saying something similar here about the analytic solution. The analytic solution basically says: look, in reality itself there are no paradoxes. Paradoxes exist in our hallucinations but not in reality. In reality, either something is true or something is not true, not both. Okay? So what does that mean? Apparently the problem is somehow in us, we said. Fine, so maybe it’s a problem in language, and we’ll do a verbal or linguistic analysis and thereby solve the problem. That’s the motivation, and there is logic in the motivation. It’s not absurd. These people were geniuses. Bertrand Russell was an accomplished genius. He wasn’t a stupid man. The point I think they miss is that maybe there are no paradoxes in reality, but the paradoxes can be in our logic, not in our terminology. That’s not the same thing. In other words, our mode of thought may be problematic. It’s true that in the world itself I too tend to think there are no real paradoxes. A paradox is always a mental phenomenon; it’s not a phenomenon of reality. Okay. But thought is not only language. Thought also contains principles of thinking, and it may be that our problem lies in our principles of thought about reality and not in language. In that sense, I’m willing to accept their motivation, I just think they took it too far. They say it’s only in language, because in reality it can’t be. Okay, that it can’t be in reality I accept, but that doesn’t mean it’s only in language. It could be that there is something in our thinking. And similarly, the theory of types, for that matter, could be viewed not as a restriction of language but as a restriction of thought. Meaning, our thinking contains some unresolvable holes. But I’ll try to think about this a bit. I think the theory of types speaks about language, not thought, because it speaks to you about the structure of sentences; it doesn’t speak to you about what to think. I think—well, one has to think about it, because apparently one reflects the other, a sentence reflects some thought—but I don’t think so. Here, in my view, it’s really syntactic. That is, it’s entirely formal. So therefore I say that I’m willing to accept that in reality itself there are no paradoxes, but I don’t think all the solutions have to be found in language. One has to find something in the problem in our logic—what in our logic is not okay, what in our thinking is not okay. But yes, I do agree that in the end there are no real paradoxes. There cannot be real paradoxes. The paradoxes are something of ours, at some linguistic or mental level—but it’s something that is in us. Now this raises a very interesting question, for example regarding halakhic paradoxes, because halakhic paradoxes—Jewish law does not deal with reality. Jewish law is a normative system. It’s not a question in physics, of what the physical fact is such that it equals both three and five. That can’t be—that’s a paradox in reality. In reality it either equals three or it equals five. But when I speak about Jewish law, I say: this woman is both permitted and forbidden. What does it mean that she is both permitted and forbidden? That’s a paradox. But no, because Jewish law is a normative system. It could be that our normative system really is defective. There is no solution to the paradox. Our normative system contains paradoxes. Because a normative system is a product of thought, let’s call it that. It is not a physical fact or some scientific fact. Therefore there I am not sure it is correct to say that in the system itself there cannot be paradoxes, and that it is always just a problem in our thinking about the system, because the system itself is a system of thought and not a system of facts. Okay? So therefore the same is true in law, of course, not only Jewish law but law in general. Is it possible that within a legal system of some state there lies a paradox? I think I told you about Gödel. On the website, on my site, I wrote in one of the columns—I have a series of columns on what philosophy is. In the last column I brought a logical proof that philosophy exists. And that proof is built on a proof Gödel gave, a formalization that Gödel made of the ontological proof. In other words—never mind, that’s another formal pilpul. Now Gödel was a somewhat crazy type, of course, like all geniuses, or many geniuses. And the story goes—and by the way it is probably true—that when he arrived in Princeton in the United States, at the Institute for Advanced Study with Einstein, he came there as a new immigrant from Germany. So they took him to the American immigration office. He had to be examined on the American Constitution and American history, and Einstein and von Neumann accompanied him. Yes, exactly, and von Neumann. That trio—no creature can stand in their company. I mean, the intelligence level of those three people equals roughly the intelligence of five generations in the entire universe, it seems to me. But they went there to the American immigration office, and you can imagine whom they might find there. Okay? Now, when Gödel was preparing for the exam, he found a contradiction within the American Constitution. That is, he claimed that the American Constitution contains a contradiction that could lead to dictatorship, and therefore it is contradictory, because the basic principle is democracy. So if that is so, there is a contradiction in the American Constitution. If so, then one can derive any conclusion from it. Because that is a rule in logic: from a contradictory system one can derive any conclusion. So if that is so, there is no point in being examined on the American Constitution. The American Constitution says whatever I want. There’s no material. Okay? Now try explaining that to a clerk, you understand? I always think of that when a policeman stops me. For example, a policeman once stopped me. He says to me: listen, I was without a seatbelt—back in the days when it wasn’t as serious as it is today. Today it’s already almost absurd to drive without a seatbelt. Back then it was forbidden, but people did it. I drove without a seatbelt. A policeman stops me and says: tell me, aren’t you ashamed? Don’t you understand it’s dangerous? Aren’t you afraid for your life? I wanted to explain to him: look, driving is also dangerous. It’s only a matter of risk level. I mean, there is a level of… I wanted to explain to him—I didn’t say a word because I knew I’d double the fine if I… but I wanted to say to him: look, if I drive a car, then I have some chance—I don’t know what percentage—of getting hurt. With a seatbelt I have half a percent chance of getting hurt. Now where exactly is the line of risk that one is not allowed to take? Why do you decide that it passes at one percent and not half a percent? Now try explaining that to the average policeman. Who knows whether you’ve run into the policeman who knows how to write or the one who knows how to read. What, there weren’t even two? Yes, exactly. So in short, it’s terribly amusing. In any case, when we were in basic training we had a preparatory class before the swearing-in ceremony, and there they explained the text of the oath, its meaning, all kinds of things like that. Then it started—it was a platoon of hesder yeshiva guys—so people started asking all kinds of questions: what does this mean and what does that mean, and how does this fit with that, and how can you swear, and one guy said that when he enlists he doesn’t want to sign an obligation, so it’s coercion, so maybe it isn’t valid—all sorts of pilpulim. Then at some point the instructor said: enough. So someone said to him: I have a question. I have one thing that can answer your questions. So he said: okay, fine. So he said to him: according to Wittgenstein’s philosophy there is an inherent gap between the sentence and what it represents, and therefore the questions are necessary. Doron Avital, commander of Sayeret Matkal—his doctoral dissertation was on Wittgenstein. Not that I’m sure the dissertation is any good. No, no, who? Very interesting. Doron Avital, commander of Sayeret Matkal at the time of the Tze’elim Bet disaster. A book came out of his dissertation. A book came out of his dissertation? I once met him on the beach; we had a long discussion about his book, and I gave him criticism of the book. No, he’s a very interesting person. Fine. In any case, the claim basically is that paradoxes—yes, I do agree that there ought to be a solution. Or alternatively, that it’s some kind of bias, and that too is a kind of solution: simply to show some sort of solution, or to show that there is some kind of bias here, meaning the paradox doesn’t really exist, it’s a misleading formulation or something like that, but there isn’t really a paradox here. But there ought to be some sort of solution. Yet in the halakhic context, as I started to say, it isn’t so simple. Because in the halakhic context, since this is not a physical fact but all of Jewish law—or any legal context, it doesn’t matter—a normative rather than factual context, there it is not entirely clear that I can assume there is really no contradiction in the material itself and that the whole problem is only in how I relate to the material, in my logic or in my language or whatever it may be. There could be a contradiction in the thing itself. Now in the context of legal systems—this is how I got to Gödel—in the context of legal systems, here Gödel found a contradiction in the American Constitution. Okay? Now in the context of the Torah, some people will tell you: fine, but the Torah is a divine creation. The Holy One, blessed be He, does not stumble where an immigration clerk or the American Congress stumbles, or even the Founding Fathers. Meaning, those were human beings. Not all Americans will accept this, but yes. The Torah is divine; it can’t be. There can’t be… “The Torah of God is perfect,” right? “Perfect” in the sense of whole. There can’t be paradoxes there. Therefore a paradox in Jewish law raises a kind of problematic feeling in people, a theological problematicity. But of course this is not true, because it is obvious that Jewish law is a human creation, not a creation of the Holy One, blessed be He. The Holy One gave some sort of initial infrastructure—I don’t know what—commandments. But the commandments don’t say anything until they are interpreted, and the interpretations are human interpretations, not to mention rabbinic commandments. Even Torah-level commandments are human interpretations. And in human interpretations, contradictions can of course always arise. There is even a hermeneutic principle in Torah interpretation: two verses that contradict one another. That exists even in the Torah itself. Exactly. By the way, that’s very interesting, but it depends, because Rabbi Akiva and Rabbi Ishmael interpret that principle differently. I also wrote about that once. Rabbi Akiva and Rabbi Ishmael interpret that principle differently, and the claim is that it doesn’t appear very often in the literature of the Sages—“verses that contradict one another.” It appears maybe five times or something like that. Three times with Rabbi Akiva and twice with Rabbi Ishmael. Because in Rabbi Ishmael’s thirteen principles it appears as “two verses that contradict one another”; it’s the last principle on his list. But Rabbi Akiva also uses it. But if you check the expositions they make, you’ll see they use this principle differently. Rabbi Akiva uses precisely the formulation that appears with Rabbi Ishmael, but that formulation actually belongs to Rabbi Akiva. Meaning, Rabbi Akiva says: two verses that contradict one another—you have one verse that says this, one verse that says that—look for a third verse that will decide between them. What does it mean, “decide”? Join one of the sides, and then it will be two against one, and we will rule like the two. But what about the one? Is the one just talking nonsense? It’s a verse! There’s a verse saying the opposite! You’re not ruling here between sages. This is a verse that the Holy One, blessed be He, gave us in the Torah. It contains two verses: one says X, the other says not-X. Now there is another verse that says X. Okay, so now I know what to do: I need to do X. But how do I explain the verse not-X? There is a problematic verse here. Rabbi Akiva isn’t interested. He says: the third verse decides. “Decides” means like a balance scale. Two against one; therefore that is the Jewish law. Rabbi Ishmael, when he looks at it—look at his expositions, how he does it—he reconciles the two verses. The third verse helps me show what this verse is talking about, what that verse is talking about, and that there is no contradiction. And sometimes, by the way, the Ra’avad in his commentary to the beginning of the Sifra, on the baraita of the thirteen principles, says there that basically every reconciliation we make of a contradiction between verses is the principle of two verses that contradict one another. You don’t need a third verse to decide between them. Sometimes it’s a third verse, sometimes it’s just reasoning. The reasoning says that this one speaks about this and that one speaks about that. We make a reconciliation, and by that we solve the problem. In other words, Rabbi Ishmael is not willing to accept that there is a contradiction in the Torah; there must be a reconciliation to the contradiction. Rabbi Akiva says no, there are contradictions in the Torah. Therefore what I am looking for is only what to do. What to do? If there are two verses in one direction and one verse in the other direction, I go with the two, because you follow two against one. But I don’t really have a reconciliation—so what do I do with that verse that says the opposite? No, because contradiction comes from our understanding, not from the divine understanding. And if it’s from our understanding, then how? So let’s look for a reconciliation and see where our understanding is mistaken. It could be that we don’t know how to find a reconciliation, but if we do know, then all the better. And if we don’t know, then there is a contradiction. You’re saying that we wouldn’t even call it that. With Rabbi Akiva, the principle that Rabbi Ishmael gives starts the process. We look for a reconciliation. What Rabbi Akiva says is: okay, but if we didn’t find one, what do we do? Exactly. Now we are in the case of two verses that contradict one another, and then we bring the third verse. Could be. Interesting comment. In any case, the truth is one has to check this, because it seems to me there is one place where Rabbi Akiva and Rabbi Ishmael disagree, and then it would mean that Rabbi Ishmael found a reconciliation and Rabbi Akiva doesn’t accept it but instead takes the third verse. And if Rabbi Ishmael found a reconciliation and you have no alternative reconciliation, then adopt his reconciliation. If that’s so, one has to check, and if that’s so, then we see that for Rabbi Akiva this is an ideology, not only for a case where one fails to find a reconciliation. But I don’t know. For example, Rabbi Akiva—when everyone was crying, he laughed when he saw a fox coming out of the Holy of Holies. Everyone was crying and he was laughing. He was always the antithesis. Exactly. Rabbi Akiva stands before contradictions and lives with them in peace. He doesn’t look for reconciliations. Fine, okay, I understand—so there’s a contradiction. The question is what to do. One has to solve the practical problem of what to do, but the fact that there is a contradiction doesn’t upset me. You said, so there is a contradiction. In any case, the claim is that in Jewish law there really can be contradictions that have no solution. That’s possible. On the contrary, precisely because it doesn’t deal with facts in their simple sense, and because it is a human creation. Then indeed the question arises: okay, so can paradoxes appear in Jewish law, and if so, what does one do with them? And I say again: it could be that paradoxes appear and they have no solution, and maybe I also don’t know what to do and that creates a practical problem for me, because I don’t know what to do—but there’s no principled problem here, because on the principled level it could be that there really is a genuine paradox there. In other words, one of the principles in Jewish law is simply not correct; it contradicts another principle, or two other principles, or I don’t know exactly how the contradiction is structured. Therefore one of the—but I don’t know which one—so I’m in a real problem because I don’t know which one to throw out and what to do. But this doesn’t raise a theological problem. There’s no principled problem here. Fine, there is something I don’t know. Now what to do—I have no idea. You can always interpret. What? You can always interpret. No—if you can always interpret, then we’ve solved it. The problem is what one does when one cannot interpret. I mean, that’s the… I’ll bring examples. I’ll bring examples where we are in trouble. Maybe before I get into that, I want to speak a bit about the way one ought to look for solutions to paradoxes. Usually—and I’ll get to this later because it touches on the question of positivism—usually paradoxes are created by the use of rules. Usually. Why? Because if you say something specific, usually there won’t be something specific that is exactly its opposite and creates a contradiction. Usually a rule that says something general can run into another rule at some point because it says many things. So contradictions are often the result of using rules. That’s not a law—it itself is not a rule—but I think it’s true in most cases. In most cases the problem is the use of rules. Why am I saying this? Because the way to try to solve contradictions is to try to get out of the rules. That is, to move outside the rules. When we are inside the system of rules and using them, we are in a loop and cannot get out. But if we go outside and think about the rules themselves, try to see whether the rules are correct or how correct they are or whether there is a hierarchy among them—all these possibilities—then maybe we’ll find the paradox. Often this feeling of paradoxicality, this feeling that there is no way out, stems from the fact that we are trapped inside the rules. The rules seem to us like something certainly correct, and now the only question is how to apply them. But the applications lead you into a loop—there’s no way to apply them. So are we stuck? No, we are not stuck. It could be that one has to go outside, beyond the rules, and think about the rules themselves. A parable for the matter—Gödel’s theorem, I mentioned Gödel earlier. Gödel’s theorem is a theorem about axiomatic systems of a certain type, but many of the interesting systems we know belong to this type—roughly equivalent to arithmetic, never mind. And systems of this type, Gödel claims, contain a statement that is true but cannot be proved within the framework of the system. Meaning, you have to assume it is true, but it cannot be proved within the framework of the system. That is Gödel’s theorem, considered one of the most important theorems in logic in general—in the twentieth century certainly, but in general. Its proof is very interesting, and by the way it turns out even to be applicable—his proof. Codings, encryptions, are basically founded on Gödel’s theorem. What he did says that what Russell tried to do cannot be done. Correct. He shattered Principia Mathematica with that claim. He basically dismantled from the ground up the project of Principia Mathematica. Today the book has museum value. There’s no point even reading it, so there’s no need to make an effort. In any case, because Russell tried to present a complete picture of all mathematics, to base it on set theory and build all of mathematics from it in a consistent and complete way. And Gödel showed that this cannot be done. He did this—I don’t know—twenty years after Russell wrote the book, or twenty, thirty years after. In any case, this claim that there is a statement that is true but unprovable. Now what is beautiful about this business—and this drove me crazy for years—there are two ways in mathematics to prove a theorem. Suppose there is an existence theorem: a differential equation of a certain type has a solution. That is a theorem in mathematics. By the way, mathematicians usually don’t know how to find the solution; they know how to prove there is a solution. If you want the actual solution, go to a physicist. The mathematicians prove that there is a solution to such an equation, and even to such a class of equations, not only to one particular equation. Now that can be proved in two principal ways. One can prove by mathematical techniques the existence of a solution, and one can simply find it. If I find a solution, that also proves there is a solution, right? Obviously. Okay, so that is what is called a constructive proof. A constructive proof basically proves that there is a solution by presenting a solution. I show you there is a solution—here, look. Now what Gödel did: he had to build a statement within his axiomatic system, show that it is true, and show that it cannot be proved. Okay? So he showed that it cannot be proved. But how does he show that it is true? After all, you need a proof to show that it is true. So how do you show it is true if you claim it has no proof? There is something problematic here. Again, “true” for a mathematician—you have to understand. In philosophy something can be true even without proof, or in everyday life something can be true even without proof. But for a mathematician there is no such creature. “True” means “proved.” No games there. So how can it be that he proves that this statement is true and proves that it has no proof? If he proves it is true, then that proof is the proof of the statement, so it has a proof. This drove me crazy for years until I understood—I no longer remember how, where the coin dropped—that it has a proof, but outside the system. In other words, the axiomatic system is built, say, out of ten axioms. Take geometry. In geometry there are five axioms or four, I don’t remember anymore—five, in Euclidean geometry. There is a certain set of axioms. Okay? Now I can prove all sorts of theorems on the basis of those axioms. Now I can construct some theorem in geometry—and by the way one can do this—a theorem in geometry that will be true and have no proof in geometry. Meaning, you won’t succeed in proving it on the basis of the four or five axioms of geometry. But I can prove that it is true. So how did I prove that it is true? I proved it by non-geometric techniques. In other words, by techniques that are outside that axiomatic system, by general intellectual techniques I proved that it is true. Meaning, if it isn’t true, that leads me to a contradiction in some sense. But those are techniques that do not belong within the system. Or in other words, to prove this theorem I need to go outside this system of rules, this axiomatic system, and think about the rules themselves, and by means of that perhaps prove the truth of the statement. Okay? So in that system the statement is… meaning, in the broader system, right, but in that broader system there will also be another statement that is true and unprovable. And then again one has to go outside, correct. Until you get to systems sufficiently complex that Gödel’s theorem does not apply to them. For example, systems whose number of assumptions is not countable. An infinite continuum—the number of assumptions, yes, the number is always countable; the cardinality of the assumptions, yes, exactly. So yes, correct, it’s a kind of regress. Why am I saying this? Because this is exactly the point: you’re stuck in some tangle. And you’re sure there’s no way out, because you live inside a well-defined system of rules, you apply it, and you cannot get out. This leads to that and that leads to this, and suddenly you find your tail in your mouth. Something here doesn’t work. But you don’t notice that in fact you are operating within a very particular, rigid framework of rules. Let’s try to examine the rules themselves. Let’s see whether this works or doesn’t work. I’ll give you an example. Once—a halakhic example. Rabbi Elchanan Wasserman brings this, also Rabbi David Tevel; he’s not the first, there were others before him. They discuss a nice little paradox beloved by the later authorities: matzah made from new grain, the topic of matzah from the new grain. What does that mean? A few preliminaries. First preliminary: there is a rule that a positive commandment overrides a prohibition. If there is a clash between a positive commandment and a prohibition, the positive commandment prevails. I fulfill a positive commandment, even though the price is violating a prohibition. For example, I need to eat matzah on Passover. Okay? And I have no grain, only grain from the new crop. The new crop becomes permitted the day after Passover, right? The waving day. So on Passover itself, on the eve of Passover when there is a commandment to eat an olive’s bulk of matzah, you cannot eat new grain; it is forbidden. But I have no other flour. So they make matzah from new grain and eat matzah, despite violating the prohibition of eating new grain. Why? Because the positive commandment of eating matzah overrides the prohibition of eating new grain. Okay? How does this relate to the rule that a commandment does not come through a transgression? On that the later authorities have written at great length, to their delight. What is the relation between a commandment that comes through a transgression and a positive commandment overriding a prohibition? There are many answers. One of the answers—the Maharam Chalava in tractate Pesachim, and many others discuss this—one of the answers is that a commandment that comes through a transgression applies only to theft and not to other transgressions, for example. Or that a commandment that comes through a transgression applies only to commandments that come for appeasement, like the lulav or prayer or a sacrifice. But not in other things. Or all sorts of… in short, there are many explanations. Or that it’s not simultaneous. If it’s simultaneous, then a positive commandment overrides a prohibition. If it’s not simultaneous, then the positive commandment does not override the prohibition, and then the opposite happens—the prohibition invalidates the commandment. Okay? I think that’s the Ritva, but what? No, it’s a dispute among the medieval authorities. All of this already comes up among the medieval authorities. In any case, the first rule is: a positive commandment overrides a prohibition. The second rule is that in order to fulfill a commandment a person has to spend up to one-fifth of his wealth. The ordinance of Usha: one who gives should not give more than one-fifth. The Talmud even writes, according to the Jerusalem Talmud, that this is Torah-level, not an ordinance, but in the plain sense in the Babylonian Talmud it seems to be an ordinance. In order not to violate a prohibition, one must spend all his wealth. Okay? For a positive commandment, one-fifth; for a prohibition, all his wealth. Now then, we said: a positive commandment overrides a prohibition, one-fifth for a positive commandment, all one’s wealth for a prohibition. Let’s see whether that’s enough for me. Let’s start the calculation and see whether those are the premises I need. So now the situation is this: I am before Passover, I need to bake matzot. Now in the market there are two kinds of flour. There is flour from the new crop at a reasonable price, the price of normal flour, but I may not use it until the day after tomorrow, until after Passover, okay? And there is flour from the old crop, but the flour from the old crop is very expensive. In order to buy the flour for my matzah I would have to spend all my wealth—three quarters of my wealth, okay, it doesn’t matter—half of my wealth. What should be done in such a situation? Let’s start making the calculation. Stage one, I say: fine, then I’ll take old flour and fulfill the commandment. Absolutely not. To spend half my wealth on old flour, I don’t have to; I only spend up to one-fifth of my wealth on a positive commandment. Therefore I won’t make matzah from the old crop. Maybe I won’t eat matzah at all. What do you mean, not eat matzah? Eat matzah from the new crop. After all, a positive commandment overrides a prohibition. So take matzah from the new crop, which is reasonably priced, no problem, bake the matzah and eat it. A positive commandment overrides a prohibition. What do you mean? But I’m violating the prohibition of new grain, and to avoid violating a prohibition I have to spend all my wealth. So let’s buy flour from the old crop, which costs me half my wealth, in order not to violate the prohibition of new grain. Not in order to fulfill the positive commandment of matzah, but in order not to violate the prohibition of new grain. And in order not to violate a prohibition I must spend all my wealth, so certainly half my wealth. So if that’s the case, then to make matzah from the new crop—what do you mean? Sorry, then to eat matzah from the old crop—what do you mean? And so on and so forth. In short, this is a never-ending loop. Okay. Now the question is what to do. But is this a paradox, what’s happening here, or is it apparently a loop? Paradox or not paradox—there is room to discuss it—but it is a loop. Maybe that’s a good point. Let me note something about it now. A paradox is usually defined as something where there are two principles that contradict one another on the logical level. Here no principle contradicts the other. The fact that one must spend one-fifth of one’s wealth on a positive commandment, all one’s wealth on a prohibition, and that a positive commandment overrides a prohibition—there is no contradiction among these principles. Rather, their application—that is, they are not applicable in a certain situation. In most situations there is no problem. But in this particular situation I have no way to apply them, because each path is rejected by another path and there is no transitivity. Path A is preferable to path B, path B is preferable to path C, and path C is preferable to path A. In other words, there is no hierarchy among the preferences of the paths. What does that mean? The fact that something is non-transitive is not a contradiction. There are many relations that are non-transitive. For example, being the father of. That is a non-transitive relation. Right? I am the father of my son, my father is my father, but my father is not the father of my son. In other words, if A is the father of B, and B is the father of C, it does not follow that A is the father of C. He is the grandfather of C. Okay? Therefore being the father of is not transitive. Many relations are not transitive. Okay? So the fact that there is a non-transitive relation among these principles is not in itself problematic. So where is the problem? There is no paradox here in the logical sense, a contradiction among the principles. There is no contradiction in the content of the principles. Rather, what happened is that I succeeded in producing a practical situation in which the combination of these three principles prevents me from applying this axiomatic system, from reaching a conclusion. But that does not mean there is a contradiction among the principles. Okay? It only means they are not applicable. Therefore there is room to hesitate. You could say that if you add an additional assumption that these principles are also supposed to be fully applicable in every situation, then of course the absence of such a situation is a refutation of the principles. But don’t assume that, and there is no refutation. But still, of course, on the practical plane I need to know what to do. And I have no answer what to do, so I would call this more a conflict than a paradox. Essentially it is a conflict: I don’t know what to do. “Conflict” is a statement about me. “Paradox” is a claim that there is a contradiction among the principles themselves. When I am in conflict, I am in a dilemma; I don’t know what to do, but there is no contradiction. Yes, as I’ll return again to my beloved chocolate example: if I say that it’s not good to eat chocolate because it makes you fat, but on the other hand it’s good to eat chocolate because it tastes good, then what does that mean? There is no contradiction between the statement that it is unhealthy and fattening and the statement that it tastes good. Both are true and they do not contradict each other. Rather, practically, now when I ask myself whether to eat or not, I’m in conflict. Because I don’t know which principle to use now, because both are true. Precisely because both are true. And precisely because they do not contradict one another, I don’t need to choose one and throw out the other. I can remain with both. And precisely because of that, a conflict is created in me. I don’t know what to do now—to eat or not to eat. Okay? So that’s an example of a situation that really isn’t paradoxical but conflictual. That is, I don’t know what to do practically, but there is no principled contradiction between the… Here with the matzah it doesn’t look like a contradiction… it’s as though… it doesn’t look like just a conflict between two principles; there is something paradoxical here. You’re saying it looks like a contradiction, not like a conflict. Yes, it does look like a contradiction. A contradiction between the principles, not a conflict on the practical plane like with the chocolate. Why? Because here there isn’t… it’s not that the statement is, on the one hand, according to the principle that a positive commandment overrides a prohibition one needs to do something specific, and according to the principle of… in this particular situation. Situation. If there were no such situation in the world, say there were no such situation in the world, then there would be no problem. Why would there be no problem? After all, if there is a contradiction among the principles, then it ought not to depend on the situation. The existence of a situation means that you only have a practical problem—what to do in that situation. Because if there is a contradiction among the principles, the contradiction exists independently of the existence of the situation. But even in, say, the liar paradox, it also exists in a certain situation where there is such a sentence that says… No, because that is the whole paradox. It isn’t produced by the situation. The paradox is only there. It’s not maybe… No, but here the principles are general principles that exist in other contexts, whereas the principle that one must spend up to one-fifth of one’s wealth on a positive commandment is true in every situation for every positive commandment, or spending on a prohibition. There the liar paradox says something only about this sentence itself, and in that very sentence it says nothing. It’s not something, it’s not a chance collision—it is the sentence itself. Like danger to life on the Sabbath. Danger to life on the Sabbath—to preserve life—that is a principle that is perfectly fine; one can preserve it, one can apply it everywhere. Keeping the Sabbath is also a principle that one can apply everywhere. But when I get a situation of danger to life on the Sabbath, I’m in conflict because I don’t know there what to do: save and desecrate the Sabbath, or not save and give up the value of life. So that is a conflict, but there is no contradiction between the value of life and the value of Sabbath observance. In their content they don’t touch one another at all. There is no contradiction between them. So the claim basically is that there is some sort of conflict here. Now what does that mean? Before I offer a solution, I want to show in what direction one should look for the solution. In principle the solution arises from the fact that I have three rules: a positive commandment overrides a prohibition; for a positive commandment one spends up to one-fifth; and for a prohibition one spends all one’s wealth. Those are the three rules that stand at the base and create the conflict, paradox, conflict, loop—whatever you want to call it. Now if I stay within this system of rules, I will never find a solution. That’s the proof—I gave you the proof. There’s no… I don’t think there is a mistake in the calculation I made, meaning in the application of these three rules. That is the correct application, and therefore I am condemned to spin in an eternal loop if I remain within the rules. Unless it is possible that there is no solution—there really is a problem in Jewish law, a halakhic problem. I claim there is a solution, and the solution must be found by going outside the axiomatic system, outside the rules, exactly like with Gödel. That is, not to play inside the rules, but to go outside the rules and try to think about the rules. For example: establish a hierarchy among the rules. For example, say when I eat… that was my first attempt, say, to solve this conflict. Suppose when I eat matzah from the old crop. Fine? Then I violated the rule that one need not spend more than one-fifth of one’s wealth on a positive commandment, but I fulfilled the rule that one must fulfill a positive commandment. Correct? Now if I say that the rule that one must fulfill a positive commandment and that other rule not to spend more than one-fifth of one’s wealth each have some price, or value, or weight—you understand that I have gone outside the system of rules, because now I am weighing them. I am no longer assuming them and simply acting within their framework. I go outside and try to evaluate how much this rule weighs, how important it is. Then I create a hierarchy among the rules, and I say: look, if there are three moves here—or to eat matzah from the old crop, or to eat matzah from the new crop, or not to eat matzah at all—those are the three possibilities. Each one of these possibilities violates one of the rules and fulfills another. One option violates rule A and fulfills rule B, a second option violates rule B and fulfills rule C, a third option, and so on. Okay? Now if I assign weights to the rules—rule A, B, C—I say: let’s see what the cost is of violating rule A and fulfilling rule B. The cost of violating rule A, say, is X, and the benefit of fulfilling rule B is Y. So the value of taking this step—for example, eating matzah from the old crop—is Y minus X. And now if I weigh the rules, I can try to calculate the value of each of the actions, and then I’ll choose the action whose value is highest, or least negative, however you want to put it. Okay? In principle there should be no problem with that. In principle that is a solution to the paradox. How did this miracle suddenly happen? When you present it from the inside, it looks like a dead end, like the liar paradox. How did I suddenly find a solution? I found a solution because I didn’t play inside the rules. That is, if I assume the rules, I am basically saying I do not judge the rules at all. For me the rules guide me what to do, and I only apply them. Now I say: wait, wait. Before applying the rules, I want to think about the rules themselves. Let’s see how much each rule weighs, how important it is, what benefit—halakhic benefit, I mean—it gives me, or what halakhic damage it gives me, and then I say: I assign a price to each step and rank the steps according to the price. I’ll take the step whose price is the highest or whose damage is the lowest. Okay? And that is a solution. It is a solution to the paradox. The question, of course, is how to do this ranking of the principles. Now this is a problematic step for people who think in a positivistic way, and here I begin touching the topic of this series—positivism—because people who think in a positivistic way do not allow themselves at all to make this kind of consideration. Positivistic thought is thought that does calculation. For it, the legal system or the halakhic system is a computational system. There is a set of rules. Give them to the computer, feed them into the computer, and in principle the computer will give you the result. If I could feed into a computer everything I need to determine from a halakhic standpoint, then that is what would come out. Okay, only we don’t know how to do that, and therefore it can’t be done with a computer. But in principle it could be done. If that is really so, what are we actually assuming here? We are assuming that the entire halakhic system is nothing but a collection of rules. That’s all. By contrast, what I presented earlier—the mechanism I presented—notice that it denies this. Or even before what I do—what is the subtext? Why do I allow myself, or not allow myself, to go outside the system? Because going outside the system means that I have some way of judging and weighing the rules. Now from what vantage point am I doing that? From within the system of rules I cannot do it, because there the rules are given—the axioms of the system. I cannot judge them. I need to judge the axioms themselves. There must be some broader system, exactly like with Gödel. There must be some broader system within whose framework I look at that system and weigh the rules. So by definition I have said that Jewish law is not a closed system. If I take this step, even before I’ve taken it—before I’ve given a solution—the very fact that I am willing to take such a step means that I’ve said Jewish law is not an axiomatic system. It is not a closed system. Or in other words, Jewish law is not positivistic. Positivism is a conception that says basically that the system is some kind of mechanical calculation. Give me the system of rules and the rules of calculation, and the results are the outcome of a calculation. There is no discretion here, nothing outside the set of rules. Okay? That, by the way, is how the Germans understand, say, the legal system. German law is generally based on the positivistic conception. It is more positivistic. That has changed a bit recently. By contrast, British law is exactly the opposite. British law is casuistic. It is based on precedents, on cases, on analogies between cases. Positivism works with deductions. There are rules, and from them I derive the result for the case before me. Among the English, by contrast, it works with inductions. There are cases, and from them I try to generalize, or make an analogy and generalize, and determine what the answer is to the case before me. Okay? Therefore the claim I am making here is that when I set out to solve the paradox, even before I solve it, I am basically saying that Jewish law is not a positivistic system but an open system, a casuistic system. I’ll return to that. But let’s still focus on this paradox. Basically the claim is that one must look at the system of rules—I return to matzah from the new crop—one must look at the system of rules, try to weigh them, and those weights will pave some way out of this infinite loop. Now when one thinks about it that way, it seems to me that the answer is almost obvious in this context. Because think for a moment. Suppose I eat matzah from the new crop. So how is it built? Let’s recap. Say: eat matzah from the old crop—that’s the ideal path, right? The ideal path, very nice. But it’s expensive to travel on the ideal path, and in order to fulfill the positive commandment of eating matzah I am not obligated to spend half my wealth. Maybe I’m even forbidden to spend it. There are those who say one may not spend more than one-fifth. “One who gives should not give more than one-fifth.” The simple wording means it’s a prohibition to spend more than one-fifth. Now, that’s eating matzah from the old crop. So that can’t be. So what then? Then let’s not eat matzah at all. Because in order to pay for the positive commandment, to fulfill the positive commandment, I have to pay a lot of money, so I won’t eat. Wait, but there is matzah from the new crop. Matzah from the new crop is a prohibition. What do you mean, a prohibition? But the positive commandment overrides the prohibition. So because of the positive commandment to eat matzah, eat it from the new crop and that will override the prohibition of eating new grain. But what do you mean? In order not to violate a prohibition you have to spend all your wealth. And you have an option not to violate the prohibition. Spend half your wealth, buy grain from the old crop, and you won’t have to violate the prohibition of new grain. Okay? What’s the problem? So eat from the old crop, and we’ve returned to the starting point. Where does it get stuck? It seems to me it gets stuck at the last consideration I mentioned. When I need to eat matzah from the new crop, I claim that the solution is to eat matzah from the new crop. That is what must be done in this situation. Is that the only option left? No. No option remains in the calculation I presented earlier. Why does this one remain? Because what is the challenge to it? What do you mean—after all, you’re violating a prohibition. To avoid violating a prohibition one must spend all one’s wealth. Buy grain from the old crop, which is expensive, make the matzah from it, and avoid violating the prohibition of new grain. Not true. In order not to violate an ordinary prohibition, I have to spend all my wealth. But in order not to violate a prohibition that is overridden by a positive commandment—who said I have to spend all my wealth? A prohibition that is overridden by a positive commandment is not really a prohibition. The proof is that the positive commandment overrides it. Maybe of course it would have been preferable to fulfill the positive commandment without the prohibition. But it is no longer obvious that in order to fulfill the positive commandment without the prohibition, as opposed to fulfilling the positive commandment with the prohibition, I need to spend all my wealth. I need to spend all my wealth in order not to violate the prohibition, period. But in order not to violate the prohibition in a way that… also for that I need to spend all my wealth? Who said? Now understand that what I basically did here was go outside the system of rules and judge it, qualify it. I said: here it applies and here it does not apply, or here it weighs such-and-such. Here, when it comes to violating a prohibition, it weighs as much as all my wealth—not violating a prohibition. But not violating a prohibition that is overridden by a positive commandment, which a positive commandment overrides—that no longer weighs so much. Therefore I claim that this is what must be done. Now if so, then one must eat matzah from the new crop. That is basically the solution. But notice the mechanism I used. How did I arrive at the solution? I arrived at the solution by refusing to assume the rules and merely do the calculation mechanically like a computer. A computer can’t do what I just did, because the computer is told: listen, these are your three rules, and the computer doesn’t know how to say: wait, wait, let’s think about the rules, see how much each one weighs. Because if the computer knew how to do that, we would be in serious trouble. The computer doesn’t know how to do that, okay? I give it the rules and it works according to them. That is the advantage of the human being, in a certain sense. And this leads us into other dilemmas—whether there really is an advantage of the human being, whether a human being is not merely a sophisticated computer but something beyond that, something essentially beyond that. But there are debates about that. Still, the claim is that such a move is one a computer cannot make. In other words, it is a move in which I am given a set of rules, and instead of acting according to the rules I go outside, judge the rules, ask myself how much each one weighs, where it applies and where it does not apply, which is more important than which, and accordingly determine a hierarchy among the rules—a hierarchy that the system itself does not determine. Now you may ask me: according to what standard do I determine this hierarchy? According to my reason. But here the reasoning sounds very persuasive: if for a prohibition one must spend all one’s wealth, then a prohibition that a positive commandment overrides is obviously a lesser problem. The proof is that one is allowed to violate it, right? Clearly it is a lesser problem. So in order not to do that I need to spend all my wealth? Who said? Okay? Yes. Now perhaps in another direction: if I… that the obligation… well, maybe this is a question in itself, but is my obligation to spend all my wealth in order not to violate a prohibition something I understood passively? Meaning, if you can save all your wealth by violating a prohibition, don’t do that and instead lose it all. But suppose I know that tomorrow someone will come and force me to violate a prohibition. No, no, forget tomorrow. Tomorrow is another question, because it could be called putting oneself under duress. It could be forbidden to put yourself under duress. That is another topic, a dispute among the medieval authorities. But I’m talking now about simultaneity. Or in simultaneity, in an active way, not a passive way. Say I have pork, okay? And I need to eat meat; doctors told me I need to eat meat. Now the pork is reasonably priced, but I have—similar to the case, basically—and I have kosher meat but it costs a huge amount. Okay? It costs a lot of money. Do I have to spend all that money in order not to violate the prohibition of pork? The answer is yes. You have to actively spend all that money in order not to violate the prohibition of pork. Because here there is no commandment. What is missing as compared to matzah from the new crop is the commandment. Meaning, the assumption is that saving one’s life is not considered a commandment for this purpose… Okay, one could indeed hesitate here. Does the commandment of saving life override this because… yes, right. Because if I have kosher meat, then I don’t need to save my life; it’s not in danger. Fine, that’s a discussion. It could be that there’s a difference between if I am already sick, in which case perhaps there is a commandment to heal oneself or something like that, and then indeed it would be exactly like matzah, and if I’m only afraid that if I don’t eat meat I’ll become sick. Then it could be that meanwhile there isn’t yet any commandment on me, only that I will enter a state in which I will be sick. Fine, one can deliberate about that. But the principle is that even actively, meaning you need to spend all your wealth in order not to enter into a problem of violating a prohibition. Fine, but suppose a robber comes to me and says… what? Either you pay all your wealth or I will force you to commit a transgression. Assuming this would count as my transgressing, because coercion is the problem. There is a Mishnah Berurah in section 329 or 330, I think. The Mishnah Berurah discusses this. What happens when someone is forced to commit idolatry? No, no, not idolatry. Idolatry is… no, even idolatry. The question is whether it is one of the cases of be killed rather than transgress. Wait, how does it go there? You can spend all… you can spend all your wealth. There, some… I don’t remember. If they force you… no, they tell you: if you don’t give me all your wealth, I’ll force you to worship idols, something like that. That’s his novelty there. I’ll force you to worship idols. Now you have to spend all your wealth to avoid idolatry, but on the other hand idolatry is one of the cases of be killed rather than transgress, so take all the money from me in order that I not worship idols. Fine. In any case, the point is that if we want to solve paradoxes, or loops, or conflicts of this type, where there is some sequence of rules or some axiomatic system that creates a conflict or a contradiction for me, it is impossible to solve this within the system. Therefore it becomes an ideological question. That is, if you think Jewish law is this set of rules, period, and anything you do beyond interpretations of the rules is illegitimate—who are you to make rules? The Jewish law is from the Holy One, blessed be He, and it is imposed on you, and these are the rules, this is what you have—you can’t make calculations about the rules themselves—then indeed there is no solution. But if you are willing to accept that Jewish law is not a closed set of rules, not positivism, then in such a case you can go outside, judge the rules, and determine a hierarchy among them, and then find some kind of solution. Maybe another example—I have two or three in mind. There is an example in Tosafot in two places, both in Pesachim I think and in tractate Bava Metzia, in “These Are Found Articles.” Tosafot there discusses his own lost article and his father’s lost article and his rabbi’s lost article, I think his father’s honor and his rabbi’s lost article—here. The Talmud says as follows: the Mishnah says: if it’s his own lost article and his father’s lost article, his own takes precedence. Fine? His own lost article and his rabbi’s—his own takes precedence. His father’s lost article and his rabbi’s—his rabbi’s takes precedence, because his father brought him into this world but his rabbi who taught him wisdom brings him to life in the world to come. And if his father is a sage, then his father’s takes precedence, because his father is both things together. Now Tosafot asks this: his own lost article—and if you say, his own lost article and his rabbi’s lost article and the honor of his father. Again, a three-way situation. Fine? Which takes precedence? He has three things, right? His own lost article, his rabbi’s lost article floating down the river—he can save only one of them, okay? And on top of that his father also wants a cup of tea, so he can’t jump into the river right now; he has to make his father a cup of tea. Fine? Now the question is: he has honoring his father, his own lost article, and his rabbi’s lost article—what does one do? So Tosafot says: it’s a loop. Why? Which takes precedence? If it’s his own lost article, then honoring his father should take precedence, according to the one who says in the first chapter of tractate Kiddushin that the cost comes from the son. Meaning, you have to spend your own money to honor your father. This is a dispute in the Talmud, and in practice we rule that no, but there is an opinion in the Talmud that my obligation to honor my father applies even if it costs me money—not only from the father’s funds but from my own. The well-known story about Rabbi Chaim, yes, where a yeshiva student came to ask him whether to travel by train to visit his parents since it costs money—and in practice we rule that honoring parents is only at their expense; I don’t have to spend money to honor my parents—so why travel by train to honor my parents? It costs me money. Rabbi Chaim told him: correct—go by foot. Or stay in the yeshiva; that is, don’t travel to visit your parents. But yes, in practice we rule that it’s not from one’s own funds but only from the parents’ funds—you are not obligated to do it from your own. As opposed to the laws of charity: if he has charity money, then yes, parents take precedence over someone else. But beyond the laws of charity, you are not obligated. But here he is speaking according to the opinion that yes, I am obligated to do it even from my own funds. So if my lost article stands against honoring my parents, clearly honoring my parents takes precedence, because I will lose the article and that’s fine—at the expense of my own money I need to honor my parents. So he says: and if it’s honoring his father, then what will you do? Honor the parents, so leave your own lost article. But you said his rabbi’s lost article takes precedence over honoring his father—so save his rabbi’s lost article. But wait, his rabbi’s lost article—his own lost article takes precedence over his rabbi’s lost article. His own lost article takes precedence. But honoring his father takes precedence over his own lost article, so there is some sort of loop here. Okay? That’s what Tosafot claims. Fine. He claims that here there is no issue of his father’s honor or some technical matter, but on the principled level there is basically an insoluble paradox here according to Tosafot. Now how can one solve such a thing? Again, one has to go outside, outside the system of rules that determines priorities here. Why does his own lost article take precedence over his rabbi’s lost article? What principle stands here, and how much does it weigh, how important is it? Why is honoring one’s father at the expense of one’s money? In what sense? Why is it important? How significant is it? Once we begin weighing and judging the rules instead of merely applying them technically, perhaps we’ll find a solution here. We’ll do the same calculation I mentioned before, say. When I save my own lost article, then I harmed honoring my father—that’s minus X, the price, minus X—but I saved my own lost article, that’s my financial gain, so plus Y. So the value of that step is Y minus X. Fine? Now save his rabbi’s lost article—and my own lost article takes precedence, so I lose my own lost article, that’s minus Y. Fine? But on the other hand, true, his rabbi’s lost article takes precedence over his father’s lost article, because his rabbi made him a Torah scholar, okay? So there is Z here. And so on. Weigh each such step—what is the gain and what is the loss in it? Create a hierarchy of what the benefit is in each of the sides, and take the side whose benefit is greatest or whose damage is smallest. So on the principled level one can solve any paradox if one is not willing to surrender to the game inside the box. Go outside the box, outside this set of rules that tells me what to do. I go outside, and I think about the rules themselves. Then I can try to solve the paradox. Okay.