Halachic Positivism, Lesson 1

Table of Contents

• Proposition, truth values, and logical positivism
• Paradoxes, anti-paradoxes, and nonsense
• Halakhic examples: the anti-paradox of Beit Shammai and Beit Hillel
• A halakhic example of a paradox: Rav and Shmuel on monetary law and prohibitions
• Self-reference and analytic solutions to paradoxes
• Tosafot in Eilu Metziot: non-transitivity between halakhic values
• The paradox of matzah from new grain, the halting problem, and stepping outside the rules

Summary

General Overview

The speaker seeks to define positivism and examine the possibility that Jewish law is positivistic, while stating from the outset that this “cannot be the character of Jewish law.” He builds the discussion “from the ground up,” beginning with the concept of a proposition in Aristotle, moving through paradoxes and exceptions to truth values, and from there arriving at halakhic examples of decision loops in which a rule-system may fail to “halt.” He raises the fundamental question whether Jewish law must be a complete system that gives guidance for every situation, and argues that if so, there is no escaping the need to step outside the system of rules. Therefore, Jewish law is not merely “a collection of rules” in the positivist sense.

Proposition, Truth Values, and Logical Positivism

Aristotle defines a “proposition” as a sentence to which one can attach a truth value of true or false, as opposed to sentences that do not assert anything and therefore are neither true nor false. The speaker brings Zermelo’s theorem about chess to show that propositions that seem trivial are not always so, and from there he moves to the idea that there are apparently exceptions to the notion that everything is simply true or false. He describes logical positivism as an extreme approach according to which a proposition is only something that can be empirically verified or logically proven. Therefore, metaphysical claims like “God exists” and moral claims are not propositions at all, but nonsense or a collection of meaningless words. The speaker agrees with positivism only in the physical context, but rejects it in the general sense, arguing that “God exists” is a proposition with content even if the dispute over its truth is not empirical.

Paradoxes, Anti-Paradoxes, and Nonsense

He defines a paradox as a situation in which one presents a proposition that appears to be a proposition, but it is impossible to determine whether it is true or false. He analyzes the formulation “All Cretans are liars” as a logical mistake rather than a paradox, because the negation of a universal statement is a particular statement, so the loop stops. He connects this to Maimonides in Words of Logic and to Boethius’s square of opposition. He argues that “I am a liar” also includes a generalization (“All the statements I make are false”), and therefore the loop can be stopped by the same mechanism. By contrast, the formulation that preserves the liar paradox is direct self-reference without generalization, such as “This sentence is false,” or a pair of sentences that refer to one another. He distinguishes between a paradox, where there is no truth value, and an “anti-paradox,” such as “Sentence A: Sentence A is true,” where one can assume truth and remain consistent, and can also assume falsehood and remain consistent. He distinguishes paradox/anti-paradox, which have meaning but a problematic truth value, from nonsense such as “Can the Almighty create a stone that He cannot lift?” or sentences like “Virtue is triangular,” where the question of true or false does not even get started because there is no coherent meaning to the concept.

Halakhic Examples: the Anti-Paradox of Beit Shammai and Beit Hillel

The speaker presents the Talmudic passage in Eruvin about three years of dispute until a heavenly voice declared, “These and those are both the words of the living God, but the Jewish law follows Beit Hillel,” along with Tosafot’s question based on “It is not in heaven” and their answers. He cites Tosafot elsewhere, explaining that the decision got stuck because of a methodological dispute over what counts as a “majority”: Beit Hillel count “the majority of feet,” while Beit Shammai count “the majority of heads” or “the majority of wisdom.” So voting cannot decide the matter, because it just circles back to the same dispute. He argues that where the halakhic rules “collapse” and a loop is created that threatens communal wholeness, the heavenly voice “rescues” the system. Therefore, “It is not in heaven” does not apply when no decision is possible from within the rules themselves. He notes the Talmud’s reason that Beit Hillel prevailed “because they were pleasant and humble, and they would cite the words of Beit Shammai before their own,” and leaves open the question whether this is a moral reward or a revelation of truth. He suggests interpreting it as a decision that really settles the rules of the game in favor of the majority of people, even though the heavenly voice itself did not state that reasoning explicitly.

A Halakhic Example of a Paradox: Rav and Shmuel on Monetary Law and Prohibitions

In the dispute between Rav and Shmuel over “one who stipulates against what is written in the Torah,” the speaker describes a structure in which the question is how to classify the dispute itself: Rav says it belongs to monetary law, and Shmuel says it belongs to prohibition and permission. Since the rule is “the Jewish law follows Shmuel in monetary matters and follows Rav in matters of prohibition,” a loop results: if you follow Rav, it turns out the Jewish law should follow Shmuel; and if you follow Shmuel, it turns out the Jewish law should follow Rav. He defines this as a paradox, as opposed to an anti-paradox, because the decision rule flips back on itself and does not allow a stable assignment of a ruling.

Self-Reference and Analytic Solutions to Paradoxes

The speaker argues that a large share of paradoxes are self-reference paradoxes, and brings the barber paradox—the barber who shaves everyone who does not shave himself—as a classic example. He describes Russell, in Principia Mathematica by Russell and Whitehead, as proposing the construction of a hierarchical language (“type theory”) in which statements can refer only to lower levels, thereby forbidding self-reference. He objects to this and sees it as “not a solution” but the creation of a new language in which the paradox cannot be formulated, similar to a “Stalinist solution” that eliminates whoever formulates the problem. He adds that this approach narrows things too much, because it also forbids harmless self-reference such as “This sentence is made of words,” and so it “throws the baby out with the bathwater.” He also presents the heap paradox and argues that the proposal to define a heap as beginning only from 13 stones does not solve the paradox of ordinary language, but merely replaces the language. Only an analysis that reveals what we really mean may show that the paradox was only apparent.

Tosafot in Eilu Metziot: Non-Transitivity Between Halakhic Values

Tosafot in Eilu Metziot presents the dilemma of “his own lost item and his father’s lost item—his own takes precedence,” “his own lost item and his rabbi’s lost item—his own comes first,” and “his father’s lost item and his rabbi’s lost item—his rabbi’s comes first.” From this arises the question of a situation involving his own lost item, his rabbi’s lost item, and his father’s honor, where honoring father and mother is “at the son’s expense” according to one opinion in Kiddushin. The speaker formulates a non-transitive loop in which his father’s honor overrides his own lost item, his rabbi’s lost item overrides his father’s honor, and his own lost item overrides his rabbi’s lost item—so there is no decision. He suggests reading this as a legitimate halakhic question that perhaps has no answer, or as a difficulty meant to refute the position that honoring parents is “at the son’s expense,” based on the assumption that “the Torah of God is perfect” and that Jewish law is supposed to be complete. He frames the fundamental dispute: is it permissible for there to be questions in Jewish law that have no answer, or is the lack of an answer a failure that collapses the view?

The Paradox of Matzah from New Grain, the Halting Problem, and Stepping Outside the Rules

The speaker brings the “paradox of matzah from new grain,” attributed by later authorities (Acharonim) to figures like Rabbi Elchanan and Rabbi David DeZehav: on Passover eve there is a positive commandment to eat matzah, but the flour is from new grain, which is prohibited until the day of the Omer offering; on the other hand, flour from old grain is so expensive that it would cost “all his money.” From the rules “a positive commandment overrides a prohibition,” “one must spend all his money in order not to violate a prohibition,” and “one must spend up to one-fifth of his wealth to fulfill a positive commandment,” a loop emerges: eating new grain is rejected because he can buy old grain; buying old grain is rejected because one is not obligated to lose all his money for the sake of a positive commandment, so it is preferable not to eat; not eating is rejected because a positive commandment overrides a prohibition, so he may eat the new grain—and so the cycle repeats. He describes this as a situation where, if you program a “computer” according to the set of rules, the program “will not halt” and will not produce output, connecting this to Turing’s halting problem. He argues that if the assumption is that Jewish law must provide guidance for every situation, then there is no escape except to “do something a computer does not know how to do”—that is, to step outside the system of rules and examine the problem from an external perspective. This is a non-positivist act, leading to the conclusion that Jewish law is not merely a collection of rules, to the point of a kind of “halakhic Gödel theorem.”

Hello everyone. Today I want to begin dealing a bit with halakhic positivism, and positivism in general. And of course, as I do that, I’ll also want to define the concept of positivism and ask whether we can really say that the character of Jewish law is in fact like that. Very often we assume that it is, but it seems to me that that’s not correct. Meaning, that cannot be the character of Jewish law. But I’m not going to start from the top; I’m going to start from below. I’ll begin maybe with a few remarks about paradoxes, and from there maybe I’ll move out to the more general question that I’ve touched on in the past in a few contexts. I spoke a bit, for example, about the relationship of Jewish law to rules, and that too is connected to this issue. All right, so let me begin a little with paradoxes. Aristotle has a definition for certain kinds of sentences, called a proposition. What’s the difference between a sentence that is a proposition and one that isn’t? A sentence that is a proposition is a sentence to which you can attach a truth value, true or false. Meaning, if I ask someone what time it is, that’s a sentence. But that sentence is not true and it’s not false. It doesn’t say anything, it doesn’t assert anything, so it’s not a proposition. Okay? In other words, there are sentences that don’t claim anything, so I can’t say of them either that they’re true or that they’re false. Sentences that do claim something are called propositions. For example: it’s dark outside now. That’s a proposition. It happens to be an incorrect proposition, but it is a proposition. Since I can say about it that it’s false, I can also say about it that it’s true. What characterizes propositions is that they have one of two truth values: either they are true or they are false. That sounds like something trivial. But it’s like what we once discussed about Zermelo, about games, where he says—I mentioned this once—that in every game of a certain type, say chess, in every chess game either White wins, or Black wins, or it’s a draw. That’s a mathematical theorem. I think it takes a semester or so to prove it, and by the way it’s not a trivial theorem. It’s not trivial at all, because the claim is that this is a characteristic of the game. If you’re not talking about the game itself, but about two people sitting at a board, then of course either this one wins or that one wins or there’s a draw. The claim is that the game of chess as such, by virtue of its own character, dictates one of three outcomes, and only one. We just don’t know which one, but either the outcome is always that White wins under optimal play, or always that Black wins, or always a draw. Now, we don’t know—we don’t have enough computational power to know which of the three is correct—but in principle only one of the three is, and that’s a very nontrivial theorem. Anyway, regarding propositions, in a moment we’ll see that even there it’s not all that trivial. Meaning, not everything can only be either true or false. Sometimes there are exceptions to that, at least apparently. And I want to touch a little on what’s called paradox, which is one of these exceptions. Before that, though, one more remark: when I say there are approaches—what’s called logical positivism is an extreme approach within positivism, and it too has a few variants. It’s an approach that says that a proposition is only something that can be empirically verified or logically proved. Meaning, for example, every fairy has three wings—that’s not a proposition. It’s not a proposition because there’s no way to check whether it’s so or not. Again, I’m saying: even an incorrect proposition is still a proposition, but there has to be some way for us to check and show that the proposition is incorrect. By empirical means, for example: that tree is green—that’s an empirical proposition. I look. If it’s green, then it’s true, and if it isn’t green, then it isn’t true. So I have an empirical way to check the proposition. But propositions that apparently look like ordinary propositions—you can say “propositions” in quotation marks—if there’s no real way to check them, then, says the positivist, that’s not a proposition. For example, positivists will say that the claim “God exists” is not a proposition. There’s no way to check it, no way to reach some compelled conclusion that everyone would have to admit once confronted with it. They basically say it doesn’t mean anything. In other words, it’s not that the proposition is false; it simply doesn’t say anything, it’s just a collection of words. Because as long as you have no ability to verify it, to test it, it has no content. I’m saying this is an extreme approach; I don’t agree with it. But okay. It doesn’t say anything. It seems to us as though it says something, but really it says nothing. Quite a few atheists claim, for various reasons—not always out of positivism, but for various reasons—that basically the claim “God exists” is nonsense. Not that the proposition is incorrect, but that it’s nonsense, that it means nothing, because if you haven’t defined what God is, then it has no content. So metaphysical propositions, the positivists say, are not propositions. The same goes for moral propositions—they are not propositions either. Meaning, that one must do such-and-such or one must not do such-and-such—those are not propositions. How do you test whether something like that is correct or incorrect, true or untrue? They don’t deny that we feel that murder is forbidden or theft is forbidden and that one ought to help others. That’s how we feel. But it’s not a proposition that can be discussed in terms of true or false. Someone else doesn’t feel that way—then he doesn’t feel that way. It’s a completely subjective matter. There are also what are called propositions in physics, like that we have eleven dimensions. No—that can be tested; it has implications. It has experimental implications. Someone thought of a way to test it. No—why? It has experimental implications. String theory is something that in principle is testable. Again, and if not, then it’s only a question of technology, but in principle a proposition in physics—specifically in physics, by the way—I am a positivist. In physics, something that cannot be empirically tested really is not a proposition in the physical context, because physicists have nothing to deal with there. Physicists are supposed to decide propositions, whether they are correct or incorrect, by means of experiment or proof based on other theories, it doesn’t matter—but it has to be testable. Where I disagree with positivism is in the broader context, not the scientific one. In the non-scientific context, in other kinds of propositions, I do think the claim “God exists” is a proposition. You can say it’s not correct—I’m not discussing right now whether it is correct or not—but I am saying that it is a proposition, it means something. Now we have to decide whether it is correct or incorrect. So that’s why, regarding—I already got ahead of myself a little on positivism just to give some insight—but what I actually want at the moment is to define a proposition as something about which one can say true or false. But if I’m not a positivist—and I’m not—that doesn’t have to be done by empirical means. Meaning, I can say that in my view it’s implausible and therefore it’s not correct. That’s fine by me. Now you can argue with me; that doesn’t mean you necessarily have to be convinced if I say something like that, but for me that is enough of a tool to determine that a certain proposition is true or false. What happens with paradoxes? In a paradox, what happens is that we are presented with a proposition and it is impossible to determine whether it is true or false. Fine, you’ll say: then it’s not a proposition. Okay, but it’s something that appears to be a proposition and yet cannot be determined to be either true or false. The proposition—the most famous paradox is the liar paradox, I think I already mentioned it. Yes, it has its source in the New Testament. In the New Testament there’s some inhabitant of Crete, and he says: all the inhabitants of Crete are liars. And then of course, if he himself is an inhabitant of Crete, then it follows that he too is a liar, and then they aren’t liars, and it becomes some kind of logical circle. And I think I already remarked that this is not correct—that this sentence is not a paradox. It’s a logical mistake. Because if someone says all the inhabitants of Crete are liars, let’s try to continue the loop and see whether it really keeps going. If all the inhabitants of Crete are liars, and he too is an inhabitant of Crete, then he too is a liar. If he’s a liar, what does that mean? That what he said is not true—that all the inhabitants of Crete are liars—but what is true? That not all of them are truth-tellers, but that there is at least one truth-teller. Meaning, not all of them are liars, right? And who is the truth-teller? Maybe his cousin. He himself remains a liar; his cousin is the truth-teller, or his friend. And that’s it—it stops there. The loop doesn’t continue. So if I now say I am a liar—not all the inhabitants of Crete. Whenever there is a universal statement, the problem here is that the negation of a universal proposition is an existential proposition. When I say every X is Y, that’s a proposition. Now what is its negation? That there is an X that is not Y, right? Not that every X is not Y. Not every X is Y, or there exists an X that is not Y. Okay, that is the negation of a universal proposition. A universal affirmative proposition. Maimonides in Milot HaHigayon already talks about this, and before him of course this is Boethius’s square of opposition. Affirmative universal—if you have a universal affirmative proposition, every X is Y, that’s universal affirmative. Universal negative is every X is not Y. Existential affirmative is there exists an X that is Y, and existential negative is there exists an X that is not Y. Okay? So the negation of a universal affirmative proposition is an existential negative proposition. If every X is Y, its negation is there exists an X that is not Y. Okay, so the negation of all the inhabitants of Crete are liars is that there is an inhabitant of Crete who is not a liar. Now if I say I am a liar, then apparently there is no universal statement here. If there is no universal statement, then this is a good loop, right? Then there’s no way to stop the loop. But that too is not so. When I say I am a liar, I am really saying all the sentences I utter are false. And here too I can stop the loop in exactly the same way. When I say all the sentences I utter are false, including this one, why is it false? Because there is one sentence I uttered that is not false. Not this sentence—another sentence. And here too it stops again. Meaning, whenever there is a universal statement, you won’t succeed in reaching the liar paradox. The formulation that leaves the paradox intact is one that does not include a universal statement. For example, when I say this sentence is false. Only about this sentence itself. Sentence A: sentence A is false. That’s already a paradox. Here there is no universal statement, it will never stop, and here it really is a big question whether this sentence means anything. There is broad room here for positivism in this context. But this is what is called the liar paradox, or in another formulation: sentence A: sentence B is true. Sentence B: sentence A is false. Here too there’s a loop; it’s exactly the same thing. What? You broke it. No, if you break it into two, then you broke it—but it’s still the same. Anyway, that’s the liar paradox. What’s the problem? What is paradoxical about the liar paradox? Meaning, why is it called a paradox? What characterizes it? What characterizes it is that you cannot assign it a truth value. Neither truth nor falsehood. Right? If it’s true, then it’s false; if it’s false, then it’s true. Meaning, there’s no way to say this sentence is true or false and stop there. So that, basically, is the paradox. Some will say it isn’t a proposition. Meaning, it’s not something to which one can assign a truth value, so it’s basically not a proposition—or in the language of the positivist, it doesn’t say anything. There’s—if I remember right, I don’t know whether we once discussed this too—another example, not exactly the same thing. I’ll maybe give you an example. Sentence A: sentence A is true. Not false—false is the liar paradox. Here, no: sentence A: sentence A is true. What does that mean? Is that a proposition or not? Is it true? Is it false? What would you say about that? It adds nothing. Doesn’t matter. So it doesn’t add anything. There are lots of sentences that add nothing. I’m still asking: is it true or false? There isn’t enough information to refute it, to prove it false. False? Why? There isn’t enough information to refute it, so you can’t say it’s false. No, I’m not speaking in the empirical sense. In the empirical sense, what you’re telling me is that it’s like the claim all ravens are black. There too, I don’t have enough information either to refute or confirm it, but that doesn’t mean the sentence has no truth value. It does—I just don’t know it. Fine? But I’m asking about the sentence itself. There’s no content in it. What? There’s no content in the sentence. Why? This sentence is true. Why is there no content in it? A sentence that states that it is true is indeed true. Fine? It has no empirical content. It doesn’t say something about the world, only about itself. Fine, it says those things. The problem with it is the following. If I assume it is true, then it will be true. Right? If sentence A is true, then the proposition sentence A is true is true. Right? Meaning, that’s consistent. If I assume it is false, then it will be false. Right? Because if I say sentence A is false, then the proposition sentence A is true is false. Right? But then it comes out that this sentence can be true and can be false. That’s not a paradox, and I once saw in some article that this was called an anti-paradox. It’s an anti-paradox, because a paradox can be neither true nor false, whereas an anti-paradox is something about which you can say it is true, but you can also say it is false, and in both cases you remain consistent. Okay? And both of these are anomalous propositions. Take the proposition there is light outside now—you can’t say about it that it is true and then also say whatever you like but it is true. There is light outside now. Okay? Meaning, it is not something that is both true and false. Fine? It could be false if I said it at night, in which case it would only be false. Meaning, an ordinary proposition is either true or false, but not both together. Fine? And here I’m talking about something… something that can be both true and false. The proposition whether God can create a stone He cannot lift, or something like that. Yes, that is a paradox—or regarding God, if you define Him as omnipotent, then in my opinion this is not a paradox but nonsense. Nonsense because there is a difference between a paradox and—not—a paradox has meaning. I understand perfectly well the proposition sentence A is true, sentence A: sentence A is true—I understand it completely. The question whether it is true or false is another matter. That sentence has no meaning, because basically what you are saying is whether the omnipotent can create a stone that He cannot lift. But a stone that the omnipotent cannot lift is like a round triangle. It’s just nonsense, it doesn’t say anything. But that is different from “I say that this sentence is false.” No, it’s not false—it’s nonsense, it says nothing. It’s like saying virtue is triangular. Is that sentence true or false? Or: there is more water in the ocean than kindness in human beings—I once gave that example. Is that sentence true or false? It’s not false and it’s not true; it doesn’t say anything, it has no meaning. You can’t—yes, you can’t compare those two things. So I’m saying the point here is not that it is also true. When I speak about a paradox or anti-paradox, the sentence has meaning. The question arises when I try to examine whether it is true or false. In the claim about God and the stone—yes, that’s the omnipotence paradox—when you try to examine that proposition, you won’t even be able to ask whether it’s true or false because you don’t understand what it means. After I understand what it means, I’ll ask myself whether it is true or false. And this says nothing. A stone that the omnipotent cannot lift is like a round triangle. Is there such a thing as a round triangle? Explain what that is, and I’ll tell you whether there is or isn’t. In fact it isn’t even correct to say there isn’t one. Not that there isn’t one in the world—there is no such thing, the concept doesn’t exist, it is contradictory, it has no meaning. Okay? So in practice we really have four kinds of what you might call propositions, if we call them propositions. Ordinary propositions can be true. Ordinary propositions can be false—that’s the second kind. Propositions of a third kind are paradoxical: they are neither true nor false. And anti-paradoxical propositions can be both true and false. I’ll maybe give two examples, two halakhic examples. First example: there is a dispute—we discussed this too once—there is a dispute between Beit Shammai and Beit Hillel, and the Talmud in Eruvin says that for three years they could not decide, until a heavenly voice came forth and said: These and these are the words of the living God, but the Jewish law follows Beit Hillel. Now what actually—yes, Tosafot there asks: but why? Isn’t it “it is not in heaven”? How can one decide Jewish law by a heavenly voice? So he gives three answers there. I once mentioned that I think there is a much simpler answer; you don’t need to get that far. Tosafot elsewhere says that the dispute between Beit Shammai and Beit Hillel couldn’t be decided because they held a vote, and the majority decides. So he says that was because they had a dispute about what “majority” means. Beit Hillel argued that the majority means the majority of people—what I called the majority of feet. And Beit Shammai say no, you don’t count feet, you count heads. Meaning, the majority of heads decides, not the majority of feet. The minority—the majority of wisdom, okay? Now once there was this methodological dispute, there was no way to decide it. What should we do, hold a vote? We’ll stay with the same problem, right? So Tosafot says that’s why they couldn’t decide it. And then what happens? If you can’t decide it, you can’t tell me “it is not in heaven.” When you say “it is not in heaven,” what you really mean is: listen, you were given rules of Jewish law; if a heavenly voice comes out of heaven, you’re not interested. You have to work with the rules you were given. You have to make a decision according to the halakhic rules, not by way of a heavenly voice or Elijah the prophet or whatever it may be. But in a place where the halakhic rules collapse, you find yourself in an undecidable loop, and then a heavenly voice comes out to rescue you. What will you say—that we pay no attention to a heavenly voice because we only accept decisions by our own tools? But we can’t reach a decision with our own tools. If there had been no heavenly voice—if there had been no heavenly voice—we would have been stuck, and that is why it came forth. That’s exactly the point. If there had been no heavenly voice for those three years, yes, then that’s it—they were stuck until the heavenly voice came out. This is the only place where there were problems, so to speak, of methodology and things like that—problems in the sense that they threatened the integrity of the community. I once spoke about this, about the historical background of the matter. This was an acute enough problem because it looked as though the community was splitting in two, between Beit Shammai and Beit Hillel, and therefore the heavenly voice decided after all to solve the problem. But the heavenly voice—what did it rely on? The Talmud says—basically the Talmud says—because they were pleasant and humble, and they would state the words of Beit Shammai before their own. That’s the rationale the Talmud gives us: the Jewish law was decided like Beit Hillel. And I spoke about whether that rationale is a reward for good behavior or whether that rationale actually shows that Beit Hillel are really right. The Jewish law follows Beit Hillel, and not Beit Shammai? That’s some tradition attributed to the Vilna Gaon, but that’s written after they said it. I don’t know. I write all kinds of things too, and then they become written. I don’t know. It’s written there—it’s in a book of the Raavad—trying to satisfy both sides. Anyway, the point is that what happens there is that we are, say, before the heavenly voice came out, in some kind of loop. In what kind of loop? There are halakhic disputes between Beit Shammai and Beit Hillel. Apparently we were supposed to hold a vote. Right? So if we held a vote, what would happen? Beit Hillel would of course support their own position, and Beit Shammai would support their own. And then Beit Hillel say fine, that proves we are right because we are the majority, and Beit Shammai say that proves we are right because we are the majority—the majority of wisdom, not the majority of people. And then what happens? That is what is called an anti-paradox. Why? Because if we were to decide that the majority of people is what counts, then the majority in number would count, because that was the dispute. After all, this dispute is about itself. Fine? And if we were to decide that the majority of wisdom is what counts, then the Jewish law would follow Beit Shammai because the majority of wisdom is what counts. How do you measure wisdom? That’s not a simple question, but the Talmud assumes—the Talmud assumes in a few places—that it can be measured. What does it mean when a court is greater in wisdom and number, when one has to repeal enactments? Then the later court has to be greater in wisdom and in number. Greater in number, fine, you count—although that too is more problematic, because the question is that it’s always seventy-one. So what is there to count? Maimonides himself in the Laws of Rebels says the meaning is all the sages attached to the Great Court. You have to see how many there are. That is greatness in number. But what is greatness in wisdom? You can see, yes? Sometimes you can see that one is a greater sage than another. It may be that when they are close in level it will be harder to decide. And there are situations where you understand that this fellow is wiser than that fellow; I think everyone understands that. I didn’t understand the earlier point. If you just assume that in terms of Jewish law one of them really is right, how we conduct ourselves is another matter. No, I’m not asking who is right in terms of the substantive dispute—whether the rival wife of a daughter is permitted or forbidden—the halakhic dispute. I’m asking who is right in the sense of whether we follow the majority of people or the majority of wisdom. But even then, according to the Torah, one of them is really right. Maybe only one is right. That’s perfectly fine in terms of truth, but I’m asking how I am supposed to decide now when I have two such opinions. After all, I don’t know what the truth is. And there are such opinions. And if I rule like Beit Hillel, it comes out that I rule like Beit Hillel because the majority of people say one should go by the majority of people. And if I rule like Beit Shammai, then I go like Beit Shammai because the majority of wisdom says one should go by the majority of wisdom. But that’s not wisdom. The majority of wisdom says one should go by wisdom, and the majority of people says one should go by the majority of people, so that says nothing. Did Beit Shammai themselves say that in a court of seventy-one one follows the majority of wisdom? Again. In a court of seventy-one… it doesn’t seem they were sitting there inside the Sanhedrin. No, I mean, did they make that claim there too, in a court of seventy-one? Apparently yes. More than that: there are medieval authorities and halakhic decisors who say this as practical Jewish law even today. When there is a dispute in a court of three, or twenty-three, no matter what court, and there is a dispute between a minority of wiser sages and a majority of less wise ones, then one follows the minority of the wiser sages. You count wisdom, not number. There are halakhic decisors like that. In practice it is generally accepted not to rule that way, but the reason is—and also, most people rule otherwise. But I assume the accepted practice not to rule that way is for considerations of peace. How are you supposed to know now who is wiser? You stir up quarrels, so it’s better to cut it off that way, not really because… According to what you’re saying, the heavenly voice should apparently decide the rules of the game, not this specific dispute. It should say: the majority is counted this way. That’s what it did: the Jewish law follows Beit Hillel. It didn’t say why. What? It said the Jewish law follows Beit Hillel; it didn’t say why. So apparently one can explain that as setting the rules of the game. Yes. That’s an explanation I’m giving for what the heavenly voice said. The heavenly voice itself said nothing more, it gave no reasoning, so I don’t know. But I’m saying: if the problem that arose there really was the problem Tosafot describes—that there was a meta-halakhic discussion whether what decides is the majority of wisdom or the majority in number—then the ruling is probably on that level too, and that ruling said that one follows the majority of people. Now the question is how the medieval authorities and later authorities and halakhic decisors—how the medieval and later authorities explain that heavenly voice, those who say that one follows the majority of wisdom. They’ll say it was only specifically for Beit Hillel and Beit Shammai. Exactly. Always—we once discussed… But the Talmud does explain why the heavenly voice came forth. In the Talmud? Yes. That they were pleasant and humble and would state the words of Beit Shammai before their own. There is a rationale there. What? So there is a rationale there. Yes, correct, there is a rationale. Again, the question is whether that is a reward for good behavior or a rationale showing that that is the truth, that the truth is with them. So that is an example of an anti-paradox. There is a parallel example of a paradox, and that is a Talmudic topic. There is a dispute between Rav and Shmuel regarding stipulating against what is written in the Torah. And whenever there is no overreaching claim against me in a monetary matter—or what appears to be a monetary matter—there is a dispute between Rav and Shmuel, and the rule in disputes between Rav and Shmuel is that the Jewish law follows Shmuel in civil law and follows Rav in ritual prohibitions. Meaning, in monetary law we rule like Shmuel, and in prohibitions we rule like Rav. Now in that dispute, there is a question what happens with stipulating against what is written in the Torah in a monetary matter, so that his stipulation stands. But the question is whether the question itself—let’s formulate it this way—there is a dispute between Rabbi Yehuda and Rabbi Meir whether in a monetary matter, if one stipulates against what is written in the Torah, his stipulation stands or not. Rabbi Yehuda says the stipulation stands, and Rabbi Meir says the stipulation is void. We rule like Rabbi Yehuda. The question over which Rav and Shmuel disagree is whether that dispute is a dispute in monetary law or not. Meaning, whether that dispute itself is itself a dispute that belongs to the laws of monetary matters. This comes out of the reasons they raise there in the discussion. I won’t get into the details of the sugya; it’s a bit complicated, but the structure is like this. Rav says that this dispute is a dispute in monetary law. And Shmuel says it is in the area of prohibition and permission. Is one allowed to stipulate against what is written in the Torah? That’s a question of prohibition. Fine? And Rav says this is a matter of money—whether I waived it for you or didn’t waive it for you—it’s a matter of waiver, yes, so it is basically a monetary matter. What happens in such a situation? Rav says this is a monetary matter, which means that the Jewish law follows Shmuel, because in monetary matters, in disputes between Rav and Shmuel, the Jewish law follows Shmuel. And Shmuel says this is a matter of prohibition, and therefore the Jewish law follows Rav. So what do you do there? That is already a paradox, not an anti-paradox, right? It is exactly the opposite of Beit Hillel and Beit Shammai. In the case of Beit Hillel, if you go with them, then the Jewish law really does follow them. If you go with Beit Shammai, the Jewish law follows them. Here, with Rav and Shmuel, if you go with Rav, the Jewish law follows Shmuel. If you go with Shmuel, the Jewish law follows Rav. The question is—or we need to define the question more precisely. Meaning? The definition of the question. That is their dispute—what is the definition of the question? Is this a question in monetary law, or a question in prohibition and permission? Is that what they are disagreeing about? Yes, that is what they are disagreeing about. That is what they are disagreeing about: whether this question is a question in the laws of prohibition and permission, or a question in monetary law. Fine? So this is basically an example of a paradox, because you cannot assign it either the value true or the value false, unlike an anti-paradox where you can assign both. Okay. Fine, let’s return for a moment to paradoxes in general, without connection to Jewish law. Quite a large number of paradoxes are paradoxes of self-reference. Meaning, when something refers to itself, that often creates paradoxes. For example, yes, like what I said earlier: sentence A: sentence A is false. That is basically a sentence that refers to itself, right? That’s classic self-reference. There’s the Seville barber paradox. The barber who shaves all the people who do not shave themselves. Fine? So he has a sign on the shop: do not come in here if you shave yourself. Meaning, I shave only those who do not shave themselves. I want commercial potential. Someone who shaves himself comes to me only once; it’s not worth my investment. Now the question is whether he shaves himself or not. If he shaves himself, then he belongs to the group of people who do not get shaved by him, right? So he does not shave himself. And if he does not shave himself, then he belongs to the group of people who do get shaved by him, so he does shave himself. Right? That’s a paradox. And again, this is a paradox of self-reference. Okay, that is, a sentence that refers to itself. There are all sorts of paradoxes of this type. There is—yes, I may have mentioned this once, I don’t remember whether in this context or not—the Principia Mathematica of Russell and Whitehead, the book that I suspect no one has read. Correct me if I’m wrong. Yes? Yes, all the way through, let’s say, if you want to be generous—but no one read it all the way through. Okay, I at least got stuck somewhere in the introduction. There is some such ambition there, until Gödel came along and said: throw the book in the trash, because there is no way—you can’t do it. Anyway, he proved that you can’t do it without reading the book. Meaning, I think, in principle it’s impossible; he proved that the project cannot be carried out. In any case. So in the introduction to the book, Bertrand Russell discusses paradoxes of self-reference. And I say Bertrand Russell—I don’t know why, I’m always fixed on this being basically Russell’s book. I don’t know why. There’s Landau and Lifshitz—that too is a mythological series in physics—and there it’s well known that there is not a single idea of Lifshitz and not a single word of Landau. All the ideas are Landau’s and the wording is Lifshitz’s. I actually met Lifshitz later; he even came to Israel, was at the Technion. I don’t know what happened with him today. Landau was in Russia, yes. Anyway, there in the introduction Bertrand Russell—I think it’s him—proposes a solution to self-reference paradoxes. And he basically argues that one needs to build the language hierarchically. This is what he calls the theory of types. Meaning, to build—to divide the sentences into different types, one above the other, and to determine that every sentence can refer only to types lower than itself. Meaning, a sentence cannot refer to sentences that belong to its own type or above. Okay? And of course that prevents self-reference, because a sentence that refers to itself is basically an illegal sentence in that language, a sentence referring to something belonging to its own type. The problem with this solution is that it solves nothing—typical of Russell. By the way, he was a very smart man. I think he founded analytic philosophy, and in that sense everyone’s burden hangs on his neck. Meaning, this solution is Stalin’s solution to paradoxes. Stalin would solve paradoxes by cutting off the head of whoever stated them. That’s basically what Russell did, only in a more refined way. He basically says: I will now define a language in which the paradox cannot be formulated. That’s all. Now there is no paradox, right? Everything is fine. He invented a new language, and in that language the paradox cannot be stated, so now everything is all right. But in our language there are paradoxes and all of that. Fine, then don’t use this language, use that language. And all of this because of the paradox of the set of all sets. In the end he wanted to get there, okay? But by the way, there too it is the same problem. And in practice, analytic solutions to paradoxes are usually of this kind—always of this kind. This is what is called an analytic solution to a paradox. Meaning, it is basically some sort of linguistic analysis, a correction of the language, and after the correction of the language, now the paradox can no longer be said. They say the paradox has been solved. I claim no: after correcting the language you created a new language in which the paradox cannot be expressed. So now everything is fine—but what do you mean it cannot be expressed? It still exists. The fact that you forbade me to express it, or that you’ll put me in jail if I express it, does not solve the paradox. You simply invented a new language in which it won’t exist. By the way, this was Leibniz’s dream—to create some formal language in which paradoxes would not appear. He didn’t succeed, but that was his dream. I dispute even the dream, not only the ability to succeed. I think it helps with nothing. If you produce a language with no paradoxes, it will be a terribly constricted language. Meaning, you simply won’t be able to express many things that in fact you would want to express, and some of them would even be legitimate to express. For example, there are self-reference sentences that do not create paradoxes. This sentence is made of words. That too is self-reference, right? And no paradox comes from that. Why can’t I say it? To say every sentence has to be made of words—that is a sentence that is illegal in Bertrand Russell’s language, because it is itself a sentence, meaning it refers to itself, right? Now what’s the problem? It’s an entirely innocent sentence; it raises no paradoxes at all. Why forbid saying it? That basically means you’re throwing out the baby with the bathwater—you’re throwing out a lot of water just because there might be a baby in it. Meaning, you are basically doing something formal that doesn’t really solve the problem, but only creates a language in which the problem cannot be expressed. Therefore, the theory of types is one kind of what’s called a solution to paradoxes that doesn’t really solve them, but forbids expressing them. I’ll give an example we’ve discussed more than once in the past: the heap paradox. In the heap paradox too, there is basically the same phenomenon. If I say that one pebble is not a heap, and adding one pebble does not change the status of the pile—yes, adding one more pebble does not change it—but a million pebbles is a heap, then those three propositions contradict one another, right? It’s impossible to affirm all three. And then the question is what the solution is. Some people say: what’s the problem? Then define a heap as 13 and up, and that’s that. Everything is fine. Once you define it that way, there’s no problem, the paradox really—no, that’s not a solution to the paradox. Why not? Because you basically formulated another language or another conceptual system in which the paradox does not appear. Fine, I wasn’t asking about that language. I was asking about our language. We use a different language. It’s not true that we define a heap as 13 stones and up; that doesn’t really represent our actual definition of the concept “heap.” Okay, so if that is the case, then what did you really do? You presented me with another definition in which there is no paradox. Thank you very much, but I asked about the paradox in this definition. And in this definition it really does raise a paradox. And now I ask: what do we do with that? Don’t offer me an alternative definition in which the paradox doesn’t appear. That won’t help me. When can this kind of thing help me—linguistic analysis? I mean analytic analysis—analysis is a very useful thing. But they took it too far. Analysis, if it reveals to me what I really think, is a solution. Meaning, if it shows me that in truth we do treat a heap only as 13 stones and up, and suddenly we see that that really is true, then that would be a solution to the paradox. We hadn’t noticed; we did analysis, and the analysis revealed to us what we really think, so there’s no problem because it was really only a pseudo-paradox. But if the analysis artificially defines some definition in which there won’t be a paradox, then it solves nothing. I ask a question in ordinary language and you answer me in another language. So basically the analytic solution to paradoxes does not really solve them, generally speaking. And then the question arises: how can one nevertheless deal with them? Must one deal with them? Maybe paradoxes are fine? I’ll give you one example. There is a Tosafot in Eilu Metziot. Tosafot says there: his lost object and his father’s lost object—his own lost object takes precedence. His lost object and his rabbi’s lost object—his own comes first. His father’s lost object and his rabbi’s lost object—his rabbi’s comes first, because his father brought him into this world, while his rabbi, who taught him wisdom, brings him to life in the world to come. So Tosafot says there: and if you say, his lost object and his rabbi’s lost object and honoring his father. Fine, we have a dilemma. We have his lost object in the river, his rabbi’s lost object in the river, and some task of honoring his father. Now the question is what to do—which takes precedence? If his own lost object, then honoring his father takes precedence, according to the view in the first chapter of Kiddushin that honoring one’s father comes from the son’s own resources. There is a dispute in Kiddushin whether the obligation of honoring father and mother comes from the son’s own money—that from my own money I have to honor them—or whether I only have to invest effort, but the money is theirs. The well-known story about Rabbi Chaim—you know it? Rabbi Chaim of Brisk, of course. A student came to Rabbi Chaim and said to him: look, I have vacation from the yeshiva and I need to travel to visit my parents, but the train costs a lot of money. So since in practice we rule that one has to honor one’s parents from their resources and not from one’s own, am I obligated to travel to visit them? So Rabbi Chaim says: certainly not, go on foot. You are paying money to make things easier for yourself, not in order to visit your parents. What do you want? Anyway, there is a dispute in Kiddushin—is that right? There is a dispute in Kiddushin whether one honors one’s parents from their resources or from one’s own. So according to the one who says it is from the son’s own resources, meaning from his own, a problem arises here. Why? Because again we have: his lost object, his rabbi’s lost object, and honoring his father. So come on, there are three possibilities: either he saves his own lost object, or he saves his rabbi’s lost object, or he attends to honoring his father and does not save the two lost objects. He can’t do two actions, only one. Okay? He’s not a woman—women can do two things, but a man can only do one thing. That’s why women don’t need to study Torah; men need all these hairsplittings, women understand on their own. So he says: and if it’s honoring his father, then his rabbi’s lost object takes precedence, as we said above, because his rabbi brings him to life in the world to come—so his rabbi’s lost object takes precedence over his father’s lost object. So it turns out you have no way to decide. Yes? Honoring his father versus his own lost object—honoring his father takes precedence. His rabbi’s lost object versus honoring his father—his rabbi’s lost object takes precedence. His rabbi’s lost object versus his own lost object—his own lost object takes precedence. So it’s not transitive. Right? Meaning, basically no decision can be made here. Now the interesting question is: Tosafot answers something—he answers, though it’s not completely clear what he means there—but what? Rock-paper-scissors. Rock-paper-scissors, yes. So the question is how to understand Tosafot’s question here. Is it a difficulty or a question? It’s a question, because those are two different proofs, no? A difficulty—okay, not always, because a difficulty can also be an internal contradiction, not only a contradiction with something else. A contradiction is a difficulty, not a question. Meaning, look, one could say that Tosafot is asking a question: what would the law be in a case like that? And that’s all. True, it’s not practical law, because in practice it’s not from the son’s own resources; it’s from the father’s. But he’s asking: according to that view, here is an interesting case—what is the law? And then it’s a question. But one could also say that Tosafot is raising a difficulty. Tosafot is saying: wait a minute, here is a situation about which Jewish law has nothing to say, there is no halakhic answer. But the Torah of God is perfect; the Torah is supposed to give an answer to every halakhic question. “Perfect” in the sense of completeness, in logic. Meaning, there is an answer to every question in the halakhic context; it is a complete system. So if that’s so, here we have a halakhic question about which we have no halakhic answer. Now this is a difficulty, and what should its solution be? Very simple. It contradicts the opinion that honoring parents comes from the son’s own resources. Because if you affirm that honoring parents comes from the son’s own resources, you enter a loop, a contradiction. This is a proof by contradiction. So that means he is apparently not right, and that’s it. In other words, suppose I had no answer to this question. Would that be a contradiction of the opinion that honoring parents comes from the son’s own resources? It would be refuted, because it brings us to a contradiction. That’s how one constructs a difficulty, right? I show that your position leads me to a contradiction, and with that I have proved you wrong. Or maybe not. Maybe it’s only a halakhic question, and if I don’t know the answer then I don’t know the answer, but fine—there is still such an opinion, that it comes from the son’s own resources. In principle you don’t have to solve paradoxes. It’s entertaining to deal with them, but you don’t have to solve them. Who says? There are paradoxes; that’s part of life. There’s no answer. Yes, there’s no answer. But is it allowed for there to be no answer, or is Jewish law complete? Must Jewish law give an answer to every halakhic question? Or not? Fine, there are questions for which it doesn’t, and that doesn’t contradict anything. If my opinion leads to a situation in which there is a halakhic question with no answer, that still doesn’t mean I’m wrong. It means that within Jewish law, according to my conception, there isn’t an answer to every question. Fine, so what happened? Then it’s a question. “Teiku” means there is a correct answer, I just don’t know which one. Here they’re saying there is no answer. Not that I don’t know what it is—there is no answer. It’s exactly like the anti-paradox versus the paradox. There is a view under which there is no answer; then you go to the second view that gives an answer to all these things. No, no—that’s exactly the question. Why? Who says? You are assuming that Jewish law has to be complete, and then you say: if your thesis proves to us that Jewish law is not complete, then your thesis falls. But who says your assumption is correct? That Jewish law really is a complete system, that it is supposed to give an answer to every question? It may be that reality presents us with a case regarding which we have no halakhic answer. And then what? I don’t know, I’ll drink wine, I don’t know what we’ll do. Whatever you do will be legitimate. Fine, then choose what you want to do. I don’t know. Fine, passive inaction is preferable—I won’t do anything. I don’t know. But passive inaction is not even an option here, because if it were an option, fine, that would be an answer. “Passive inaction is preferable” is certainly not correct. That’s the worst option, because then you fulfilled neither your own commandment, nor your rabbi’s, nor did you engage in honoring your father. Fine, that’s Buridan’s donkey. Right, Buridan’s donkey standing between two identical troughs and dying of hunger in the middle. Why? Because he has no way to decide to which of the two troughs to go. So that’s the solution of passive inaction in such a case. You have three values and you don’t know which is preferable to which, so therefore you do none of them. Then choose—start with one of them, and then you’re performing the commandment. But then you’ll have the question: why did you start with that one? Why did you start with that one? In short, a seemingly insoluble question. It’s like what people always say: someone fights for vegetarianism, and they say to him, tell me, have you already solved all the problems of human beings? As though the person criticizing him has already solved all the problems of humanity and vegetarianism too. Very often this is really an attempt to do nothing. When you say, wait, this value doesn’t override that value and vice versa, so therefore I do nothing—that’s certainly not a solution. Meaning, do the more important value, the less important value—do something, do what you know how to do. So yes, it really is insoluble, but on the other hand, who promised you that there would be a way out of every question? Who said such a thing has to exist? Now that’s an interesting question, but then I read Tosafot as a question, not as a difficulty. The question is what the law is in such a case according to that view. Either there is an answer or there isn’t. Tosafot afterward proposes an answer. Either there is an answer or there isn’t, and I can read this as a difficulty. And the difficulty basically says: wait, this view leads me into a loop; therefore it is apparently incorrect, and we throw it out and rule that honoring parents comes from the father’s resources, and that is indeed how we rule. In any case it’s always preferable for you to go with the second view. What do you mean “preferable”? The question is who is right, not what is preferable. That is exactly the point. Halakhic ruling—that’s what we once discussed when we talked about halakhic truth—halakhic ruling is not what is preferable, but who is right. At least according to my view that there is halakhic truth. Meaning, that we cannot simply play according to the rules of the game. Not everything that the rules of the game say is really what one ought to do. So where would you place, for example, “his hand and his hand are on his shoulders”? Apparently that too is a logical loop that the Jewish law comes and decides. No, I’m not sure. It may very well be that this is indeed a solution to a loop. Because before you said they come simultaneously, you assumed that one can’t be simultaneous, that each of them has to come first and the other afterward, and then you really can’t decide. But when you say that they really come together, then the paradox is—fine, then you solved it, they come together. At the same point in time, yes. Logically it applies together; it’s not so much a question of time, it’s not a physical event. Anyway, the question in the background here is whether Jewish law is complete. Meaning, if there is a paradox in Jewish law, or there is a case in Jewish law about which Jewish law doesn’t know what to say—is that possible on the one hand, or problematic on the other? It can happen; that’s part of life. So the question is: can paradoxes appear in Jewish law, and if so, what do you do? These are not independent questions, because once Jewish law tells us what to do, then it doesn’t really remain a paradox, okay? So these are questions that are connected. I’ll maybe give one example that will begin moving us toward the topic. There is a paradox brought by several halakhic decisors and commentators—Rabbi Elchanan brings it, and Rabbi David DeZehav, and several later authorities. Maybe I mentioned it: the paradox of matzah from new grain. Suppose we are on the eve of Passover, okay? Now on Passover night one has to eat matzah. I want to bake matzah, but for that I need flour, and on that date the flour is still from the new grain, because the day of the Omer offering is the day after Passover, and the offering permits the new grain. Until we get there—meaning, before Passover we are still before the day of the offering, and therefore the new grain is forbidden, okay? New grain is forbidden by the Torah, as is stated. Today this is already caught by the phrase “new grain is forbidden by the Torah” from the Talmud—it’s a halakhic paraphrase of the Hatam Sofer. So the dilemma arises when I basically have produce from the old crop and produce from the new crop. The produce from the old crop is at a very high price and is running out, and the produce from the new crop is cheap, okay? Now the assumption is that a person has to spend all his money in order not to violate a prohibition, and has to spend up to one-fifth of his money in order to fulfill a positive commandment, okay? That is the halakhic rule. What happens now? So let’s see. This is basically the same triangle we saw now in Tosafot. Why? Because he has three options. He has the option of buying flour from the old crop, baking matzah, and eating it. A second option: to buy flour from the new crop, bake, and eat. A third option: not to eat at all, okay? In this case, by the way, not eating at all is an option. Why? Because there is the prohibition of new grain. But this is not Buridan; this is not a Buridan case. In this case, not eating at all is one of the three options. In Tosafot’s case there were three options: deal with honoring his father, his own lost object, or his rabbi’s lost object. To do nothing is a fourth option—that is, to do none of those three. Here, doing nothing is one of the three options inside the loop. But how can that be? Isn’t there a commandment to eat matzah! Fine, but there is the prohibition of new grain. The prohibition is not just that I say I’m not doing anything because I don’t know what to do. I’m not doing it because there is a prohibition, not because I don’t know what to do. Meaning, that’s one of the options, one of the three. Okay? Now how does this work? I say as follows: let us assume I eat matzah made from new grain. Why? Because a positive commandment overrides a prohibition. The positive commandment of eating matzah overrides the prohibition of new grain, okay? So I eat matzah made from new grain. One second—what do you mean, eat matzah made from new grain? But to avoid violating a prohibition you have to spend all your money, so buy flour from the old crop. It costs all your money, but it will save you from violating the prohibition of new grain. So eat matzah from the old crop. Fine, so the solution is to eat matzah from the old crop. One second—not so fast. If I eat matzah from the old crop, then what am I doing? I’m spending all my money for what? To fulfill the positive commandment of eating matzah. But to fulfill a positive commandment, it’s only one-fifth of my money; I don’t have to spend all my money. So the solution is not to eat at all, right? So not to eat at all, because this is a commandment that costs me all my money, and I’m not required to fulfill a commandment at the price of all my money. No, but that isn’t the solution either. Not to eat at all? Then eat from the new grain! Because of the prohibition you won’t eat at all? But a positive commandment overrides a prohibition. The positive commandment of eating matzah overrides the prohibition of new grain. So what’s the problem? Eat from the new grain—and round and round again. Okay? There’s a loop—never mind “round and round,” just a loop. So now there’s a loop here, similar to what we saw in Tosafot there. And again, the question is this: if I really view Jewish law as some set of rules, and what I have before my eyes is only the set of rules I just described—A: a positive commandment overrides a prohibition. B: there is a positive commandment to eat matzah. C: there is a prohibition against eating new grain. Plus the financial data—how much each flour costs and how much one has to spend in order not to violate a prohibition or in order to fulfill a positive commandment. Those are the halakhic data. I give this to a computer and now I say to it: tell me the answer. It won’t stop. It won’t stop. Not that it gets stuck—it will keep spinning, yes. And this is very connected to our topic, this halting problem of Turing. We’ll get to that in a moment because that is exactly what we are talking about here. But is there no prohibition against spending more than one-fifth? Is there a prohibition to spend more than one-fifth? No, it’s just not obligatory. I don’t want to. I want to go to the beach; the entrance there costs money. I don’t want to spend more than one-fifth, I don’t want to spend more than one-fifth. So if a computer is the illustration here—if a computer looks at this data set, this set of rules, if I program it according to this set of rules, it won’t give me output. It won’t give me an answer; the program won’t halt. Okay, so now the question, as I asked earlier, is whether this is a question or a difficulty. Someone who says it’s a question will say: fine, the computer doesn’t halt, and there is no halakhic answer to this question. What should we do? I don’t know what we’ll do—go to sleep and let Passover pass without our noticing, I don’t know. I don’t know what to do, but there isn’t one. Who says you always have to know what to do? I don’t know. That’s if it’s a question. Because if it’s a question—fine, then I have no answer. Is it a difficulty? Because the assumption is that in Jewish law there must be an answer to every question. And here there is really a difficulty, so there has to be a solution. Now here there isn’t any dispute I can hang onto like in Tosafot—say, fine, I honor my parents from their resources and not my own, and that solves the loop for me. Here, no. All the principles I’ve said until now are agreed upon. Everything is agreed upon. So now the question is what do you do in such a case? In such a case there is no way out except to do something a computer does not know how to do. And that means to go outside the system of rules and try to look at the problem not from inside the rules but from outside the rules. Okay? And if I go outside the rules, that is basically a non-positivist act. I’m already giving away the insert, but later on I’ll define more clearly what positivism is and why this is not positivist. But it requires going outside the rules, because within the rules there is no solution. So again, I say: someone who says this is a question does not have to go outside the rules. He says, I work only within the rules, I have no answer, fine, there is no answer. Such situations exist. There is in principle no answer. Yes, there is in principle no answer. So there are situations regarding which Jewish law has nothing to say; it does not know what to say. If someone comes and asks you this question, within the rules—but we learned to say “I do not know.” What can I do? I don’t know. No, you’re not saying you don’t know; you’re saying there is no answer. No, fine, no—he asks me what to do. What to do? I don’t know. Does the fact that I don’t know mean there is no answer? No—I don’t know what to do in the absence of an answer. Not that I don’t know what the answer is, but what to do in the absence of an answer—I don’t know. They tell a story that in New York many years ago, in a shared building, the neighbors complained that they built a sukkah below. So there came a ruling from a rosh yeshiva that one has to take down the sukkah all eight days. All eight days. I think that was a story in Canada and the judge was called Goldstein. Fine, but maybe within Jewish law this is coercion, and then you enter under coercion. But still, even under coercion, what am I supposed to do? Not eat, yes eat? Am I coerced regarding new grain or coerced regarding the positive commandment? If I go this way it pushes me off, if I go that way it pushes me off—so what is coercion here? I could have done it; it just comes at another price. Fine, but I’m saying that too—if you could do it but it only comes at another price, the question is whether that counts as coercion. I don’t know, maybe. And therefore there is Jewish guilt: I’m guilty. I wasn’t a positivist. Okay, so the claim I want to make is that if Jewish law really is supposed to give guidance for every situation—meaning, this is a difficulty, not a question—then there is no escape but to conclude that one must go outside the system, or in other words that Jewish law is not a set of rules. Because within a set of rules, many times a situation can arise which, like a Turing machine that does not halt, means there is a set of rules and I try to compute the solution and it won’t halt—there is no solution. And if I claim that there must be a solution in every case, by that I have said—and again, let everyone decide what he thinks—if I claim this, “the Torah of God is perfect,” it has to be complete, to give me guidance in every situation, then it has to be that it is not a set of rules. That is basically Gödel’s theorem in Jewish law. Okay? So the next thing we want to look at.