Positivism in Halakha and in General, Lesson 2

[Rabbi Michael Abraham] This is an experiment in the method of refutation, but I’m talking about logic. Logic generally has a destructive role, not a constructive one. Meaning, proving a conclusion from a set of premises—fine, that’s a logical argument—but in the end everything begins and ends with the premises. So usually that’s not something very significant. In other words, from the premises I assume I’ll reach my conclusions; that’s not what someone talking to me is going to newly teach me. Usually a logical argument will show me, will bring me, to a contradiction. Meaning, it will show me that something in my premises contradicts itself, or that my conclusion doesn’t really follow from them. Therefore its role—logic’s role—is a destructive one, and that’s what I mean by destructive: a demolishing role, not a poisonous one; as opposed to a constructive role. Its job is to destroy, not to build. And why? Exactly because of what I said before. Because logic is a necessary condition, but not a sufficient one. Meaning, if I show you that something doesn’t pass the logical test, then it’s not correct—I’ve killed your position. If I show you that it does pass the logical test, that still doesn’t mean it’s correct; it only means that now we can start talking. Therefore, in arguments, logic naturally tends to be useful in the negative direction, the destructive direction, not the constructive one. And by the way, what serves in the constructive direction is what’s called rhetoric. In our circles rhetoric has a bad name, because people see it as a synonym for demagoguery. But that’s not true. Demagoguery really is a derogatory term; that means a misleading argument, someone trying to confuse you. Rhetoric is not that. Rhetoric is someone who has a good power of explanation. It’s allowed to have a good power of explanation; that’s not something disqualifying—on the contrary, a good explanatory ability is wonderful. The Chafetz Chaim said that someone who lacks explanatory ability lacks understanding. Meaning, someone who has a good power of explanation probably also understands well; he probably also has a good position, or a correct one, or a persuasive one. Okay, so therefore, what is a good power of explanation? What does it mean to explain well? This is a very interesting point, because good explanatory ability is basically your ability to persuade a person of a conclusion in a positive way, as distinct from logical ability, which is your ability to find a contradiction in someone’s doctrine. That’s logical ability, okay? Good explanatory ability, or rhetorical ability, is basically something that combines logic and literature. Meaning, if you’re a good writer, if you know how to describe things well, to look at them from an interesting angle, then you have good rhetorical ability. And what does that mean? It means that by these means you can take a person and persuade him of something, to adopt something positively. That is, persuade him of a claim he doesn’t agree with. So the positive tools are rhetoric; the negative tools are logic. And logic is a necessary but not sufficient condition. Meaning, logical consistency says: okay, this is on the table, this can be discussed. What is not logically consistent is not on the table at all. Among the things that are on the table, how do I know what’s true? That I do by rhetorical means. Let’s put it in the language of a logical argument: a logical argument is built on premises. So the conclusion I can derive from the premises by logical tools—but how do I adopt the premises themselves? How do I know the premises are true? On what basis do I adopt them? For those who don’t know how to formulate this for themselves, premises are something arbitrary. Because there is—and again this is a reflection of the view that says that whatever isn’t proven is arbitrary. But if that’s so, then of course everything is arbitrary. We talked about that last time, because if the premises are arbitrary, then whatever you derive from them is arbitrary too. So what if you derived it by logical tools? You derived it from something arbitrary. So what does that help? Rather, the only option for not being either a skeptic or a fundamentalist is to understand that I have some ability to recognize things by some kind of observation. Meaning, I simply see that it is true. What’s called common sense, or intuition, or whatever you want to call it—but it’s some ability to see that a certain thing is true. I have no proof and nothing else, and it sounds right to me. Not because it came from on high, and not because I have a proof. I have nothing; first premises, by definition, have no proof. Rather, it seems reasonable to me. Okay? So I think this is a kind of observation and not of thought, but that’s another discussion; I won’t get into it here.

[Speaker G] It’s like an axiom.

[Rabbi Michael Abraham] Yes—how do you get to axioms? Yes, or to first premises. But axiom not in the arbitrary sense. Sometimes people say “axiom” meaning arbitrary—just like that. Not because it’s true, but because that’s what I assume. I want to argue that an axiom has a standing; it’s not something arbitrary. True, I have no proof for it, but by means of my intuition I understand that this thing is reasonable, that this thing is true. That intuition actually uses rhetorical tools. Meaning, I see things from a certain angle and I understand that it’s true. Now, for example, a good writer puts you into a situation and causes you to look at things from some angle that suddenly brings you into an insight that this is right or this is not right. That is the power of a writer. Now if I want to persuade someone—we have some dispute about some claim and I want to persuade him—then I can show him that it contradicts another claim of his. That’s persuading him by negative means, meaning showing him that his position is not correct. I can persuade him by positive means. I can say: look, I don’t know how to persuade you that what you’re saying is wrong, but let me show you why I think what I’m saying is right. How? Imagine a situation like this—I bring him a parable. A parable is a rhetorical tool, not a demagogic one. It’s a rhetorical tool; that is, a parable helps clarify things—a good parable, yes? So I use some parable, or I write him a story, and I say: come read the story, try to enter into the situation and try to see things from the angle I see them. Doesn’t that persuade you? And sometimes I’ll succeed—sometimes he’ll suddenly see that it persuades him. These are rhetorical tools. And here I’m showing him not that there is a contradiction in his doctrine—that I demolished what he thinks—but rather I brought him to see things from my angle, and by that I persuaded him that what I think is correct, not that what he thinks is mistaken. All right? That’s a rhetorical route, as opposed to a logical one.

[Speaker D] But rhetoric works in both directions, meaning also negatively. Sure. It also exists for refuting something.

[Rabbi Michael Abraham] Look, once he’s persuaded that what I’m saying is right, it follows automatically that what he said is not right. But the question is what follows from what. Do you persuade him that what he said is wrong, and it then follows that what I say is right? Or do you persuade him that what I say is right, and then he understands that what he thought was wrong? The first is logic, and the second is rhetoric.

[Speaker E] Right, right—and also regarding logic, when people sometimes answer you that not everything is logic, after all even you sometimes say that with paradoxes you have two first premises that are apparently in logical contradiction, but still you don’t give up either premise despite the logical contradiction because you’re very convinced of both of them.

[Rabbi Michael Abraham] No, no, that’s what I’m going to talk about.

[Speaker E] We talked about it in weakness of will and in God’s knowledge and free choice.

[Rabbi Michael Abraham] I’ll talk about it, I’ll talk about it, because the question is that you need to define the situation well. So indeed, the claim I want to make is that although my claim is that logic is not the whole story, logic is still a necessary condition. Meaning, that’s the framework of the discussion. Inside it a lot of other things still happen—rhetoric. But that’s the framework of the discussion. Whatever doesn’t meet logical standards is not in the game. And therefore in the end there is such a thing as proof by negation. That is, if I show you that what you say leads to a contradiction, then I’ve refuted what you claim, right? Assuming that—again, when I show you that what you say leads to a contradiction, it almost always involves what you say plus several other things, because it’s supposed to contradict something else. So you have to give up one of them. It’s not always giving up the thing we’re arguing about, but you have to give up something in your doctrine. Proof by negation is always problematic, because it doesn’t tell you which of your premises to give up. You know that this whole set of premises leads you to a contradiction, so clearly at least one of them you have to give up—but it gives you no clue which one to give up. You have to decide that for yourself, and sometimes you’ll give up not the premise we’re arguing about but a different one. Fine—but at least it’s a route one can try to use.

Now at this point I really want to get into what Oren was talking about before, and that is the question: what does it mean to live with contradictions, or what do we do with contradictions for which we have no solution? In principle, when I find a contradiction between two principles, that is proof by negation of the claim that both of them are true. Meaning, clearly it is not true that both are true. At least one of them I’m supposed to give up, okay? That’s what proof by negation is built on. Proof by negation says that if you assume this, you arrive at a contradiction, so you cannot maintain the whole set you assumed; something in it you have to give up. But very often there is a situation—what Oren described before, if I understood him correctly—very often there is a situation where my feeling is that there isn’t really a genuine contradiction here, but I can’t explain why. Okay? Meaning, I think both sides are true even though someone showed me there is a contradiction between them. In my opinion there is no real contradiction between them, but I can’t formulate, even to myself and not only to others, why there really isn’t a contradiction, because on the face of it they’ve persuaded me that they do contradict one another. Here we have a very problematic situation. Because on the one hand, it’s always possible that I’m not smart enough and someone smarter will come and show me that there is no contradiction between the two things—that certainly can happen. On the other hand, as long as I am convinced that there is a contradiction between the two things, I cannot live with both of them. Meaning, I cannot accept X and accept Y when from my point of view Y is equivalent to not-X, okay? Because then what I’m really saying is that both X is true and not-X is true from my point of view. Again, it may be that I’m mistaken and after someone explains it to me that’s really not so and Y is not equivalent to not-X. But for now, as long as for me it is equivalent, I can’t say that I accept both sides. All I can do is suspend judgment. That is, I can say: I don’t know what to do with this; both things seem true to me, I’m not willing to give up either of them. My sense of smell tells me there isn’t a real contradiction here, but I don’t know how to formulate it. But you still have to understand that the conclusions I draw from those premises I have to put in question. I can’t go on living with both and say, okay, I don’t understand why, but there’s no—I don’t think this is inconsistent. Because just as you say maybe someone will come and discover for you that there’s no contradiction, by exactly the same token maybe no one like that will come. That is, you are mistaken and there really is a contradiction here as you were shown, and you yourself have no answer to it.

So to say—for example, take divine foreknowledge and free choice, yes? What is usually thought to involve a logical contradiction: that God knows everything in advance and together with that we have free choice. Okay, so many people in this area say: yes, we live with this contradiction. Meaning, both things are true, because both of them—say—are given by the Torah, so I trust what the Torah says—

[Speaker A] And God’s knowledge is something else, maybe yes.

[Rabbi Michael Abraham] Yes, never mind, all kinds of formulations of that type. But the claim is that since I have some reliable source telling me both principles, I tend to think that both are true. There’s a contradiction between them, so maybe I’m not smart enough, but someone smarter will come and explain to me why there’s no contradiction here, okay? So what do you do in a situation like that? I say that even in such a situation—which can happen—I don’t think you can deny the fact that not each of us is so smart that he always catches every contradiction. A person can make a mistake or fail to understand or not have the ability to resolve contradictions; this is complicated. But still I think you have to suspend your judgment, and you cannot continue with it positively. It’s not the same as something for which I have no proof, what I was talking about earlier. I have no proof for it, but it seems reasonable to me; with that I proceed, and from that I also draw conclusions, and everything is fine. In a place where they showed me that there is a contradiction between things, my feeling that I’m missing something and maybe that’s the answer cannot allow me to continue living with both sides yet. It can only say: look, for the moment I’m not deciding to give up one of them, because I also don’t know which one to choose; both seem persuasive to me, and precisely because of that—they’re two sides of the same coin. Since both seem persuasive to me and I’m not willing to give up either of them, then it’s pretty clear to me that they also don’t contradict each other. To me it’s clear—but I don’t know how to answer what was presented to me, the argument presented to me. So then I say: fine, then I suspend judgment. But the conclusions—I really can’t draw a conclusion from a system that contains two principles that contradict one another, because on the logical level too you can derive any conclusion whatsoever from a system that contains a contradiction. So it seems to me that even in such a situation you have to understand that I cannot simply continue to live with both sides; that’s not an option. I can say, I suspend judgment—that yes. But I don’t see in this a proof by negation that would force me to give up one of those two premises. I remain in the laws of doubt; I don’t know, I don’t know how.

[Speaker E] Why shouldn’t it have a presumption of prior status? For every conclusion you already had, even after this doubt arose, you should remain with it.

[Rabbi Michael Abraham] Because you remain with contradictory conclusions all the time. Logically, as long as it’s a contradiction from your point of view, you can’t use logical tools to derive conclusions from it.

[Speaker E] Yes, even after you suspend judgment, but you still keep the same practical ways of acting, as if—because you have no reason to change them.

[Speaker D] That’s just conservatism.

[Rabbi Michael Abraham] No, not conservatism—or a rule of conduct, as it’s called in the halakhic context. Fine. But if I ask you what is really the case, not what you will do—the question of what you will do is a question I don’t know how to answer. But the question of what is really the case is, for example, how you judge others who think differently, or something like that. There I think you need to suspend judgment. You can’t continue with both sides as long as you haven’t clarified that it’s not contradictory.

Now here too we need to distinguish between two things. Very often you see two things as contradictory, but the contradiction is not a logical contradiction. Rather, it looks to you not compatible, yes—not sitting well together—but it’s not a logical contradiction. For example, I’ll give you an example: “Three things are too wondrous for me, and four I do not know.” So the Sages say that one of them is the red heifer, which King Solomon said. One of them is the red heifer—what about it? That it purifies the impure and renders the pure impure. That’s a paradox. There is no paradox there whatsoever in the logical sense. What’s the problem? The priest who deals with the ashes of the heifer becomes impure—“it rendered his garments impure”—and the ashes themselves, after they’re prepared, purify the impure. Where is the logical contradiction here? There’s something perhaps not understood here, and I once spoke about this—I even think there’s something here one can understand, as much as one can understand impurity and purity. The priest draws the impurity into himself, and therefore the ash is left with the ability to purify, because the impurity went to the priest who prepared the ash. Maybe, I don’t know, just a thought.

[Speaker F] Like washing the hands. If I’m feeding someone, his hands are tied behind his back, he has to wash his hands, not I. It’s not touching the bread that’s connected to purity.

[Rabbi Michael Abraham] Yes, although there it may be that it’s really just a non-distinction rule, because in the end the assumption is that when you eat, you eat with your own hands. So even when you do it in some pathological case not that way, they still tell you it’s the hands.

[Speaker F] But then someone who handles the bread—why doesn’t he have to wash his hands?

[Rabbi Michael Abraham] Also a non-distinction consideration.

[Speaker F] Someone making a sandwich and touching the bread doesn’t have to wash his hands.

[Rabbi Michael Abraham] Again, non-distinction, because usually the one who prepares is the one who eats—that’s the assumption. Maybe, I don’t know. Interesting question.

[Speaker D] By the way, apropos, this is a bit off topic—knowledge and choice. Choice is written in the Torah; where is knowledge written?

[Rabbi Michael Abraham] No, that God is all-powerful—“Is anything too wondrous for the Lord?” or something like that. So people assume that this means He’s supposed to know everything. I agree—I think He does not know, in my opinion—but when I bring it here it’s only as an example. People say: I live with this contradiction. What does it mean that you live with this contradiction? If one thing contradicts the other and you have no answer, what does it mean that you live with this contradiction? Then meanwhile one of the two is not true. That is, unless someone explains to you why it’s not a contradiction.

[Speaker A] You told him yes, but choice also isn’t written in the Torah—

[Speaker D] But—

[Rabbi Michael Abraham] You can say maybe give up foreknowledge. No, I agree, I really think that it isn’t—there isn’t—

[Speaker A] any meaning to knowing something that you can choose.

[Rabbi Michael Abraham] So maybe we’ll talk about that later; I spoke about it in another group. Yes.

[Speaker E] Free choice also isn’t written in the Torah?

[Rabbi Michael Abraham] No—“choose life,” “Behold, I set before you life and good, death and evil—and choose life.”

[Speaker E] Fine, but there’s also “and He hardened Pharaoh’s heart.”

[Rabbi Michael Abraham] No, “and He hardened Pharaoh’s heart” is the opposite—that’s the proof that there is choice, because if we know that choice was taken from him, that means that ordinarily there is choice and in his case it was taken away. And even there there are commentators who say it wasn’t taken away completely, only made harder. Well, again, I’m saying—with the Torah I’d manage. Meaning, I think there is choice because I think there is choice. As Maimonides said about the eternity of the world and corporeality. Meaning, if I were persuaded scientifically and philosophically that I don’t have choice, with the verses of the Torah I’d manage. That’s part of my claim that you can’t learn anything from Scripture. Here I want to distinguish between two types of contradiction. There is a logical contradiction, and there is a contradiction of the sort we have with the red heifer. There the reasoning is somewhat troubled by how it can be that it performs two opposite actions, eh? But there is no logical contradiction here. There isn’t. Anyone who tries to formalize this won’t succeed in formalizing it as something that is logically contradictory, as two opposite claims on the logical level. Therefore, a lot of the time when people talk about contradictions—in almost all cases, by the way, when people talk about contradictions—they mean contradictions of this kind, not contradictions of the logical kind. With contradictions of this kind there is no problem living with both sides. Meaning, if I think both sides are true, only I don’t understand how they fit with one another—but not that they logically contradict one another, only that I don’t understand how they fit—so what? I also don’t understand why we put on tefillin. There’s no problem there whatsoever. The problem of living with contradiction exists only where the contradiction is a logical contradiction. And why? Because the problem of living with contradiction is not the psychological dissonance that I feel—some tension, how can I live with this? I’m talking about a philosophical problem, that is, of living with contradiction. Living with contradiction means I live with claim X and claim not-X simultaneously. So do you believe in X or do you believe in not-X? You have to decide. Simply at the level of words, the phrase “living with contradiction,” when you are talking about a logical contradiction, is simply a meaningless expression. There is no such thing as living with contradiction. Either you believe in X or you believe in not-X. You cannot believe in both X and not-X at the same time. It’s simply impossible, even if you want to. This is not a problem of psychological tension or how you do it; it’s simply undefined. There is no such thing. If you believe in X, that means you do not believe in not-X, and vice versa.

So very often—and look at “the unity of opposites” and all this nonsense—very often it is simply applied in places where the contradiction is not a logical contradiction. The contradiction is simply not a logical contradiction. And then the example that made the coin drop for me was—I don’t even remember what he brought it in relation to—Benny Shlev has a book on Rabbi Kook, Between Rationalism and Mysticism, if I’m not mistaken. And there, in two places, he talks about the unity of opposites in notes in the book. He talks about the unity of opposites and brings the three-valued logic of Lukasiewicz—Lukasiewicz was a Polish logician. Poland is a logical superpower. Really. There’s a Polish school in logic; Poland is a logical superpower. One of the well-known logicians in Poland was Lukasiewicz, I think somewhere in the first half of the twentieth century. And he formulated three-valued logic, which basically talks about evaluating propositions in a logic that has three truth values, not two. Not true or false, but true or false or P—true or false or P. The P is paradox, never mind—but leave aside for the moment that it’s paradox: three truth values. Fine? Then of course there is no law of the excluded middle there. Meaning, either a thing is true or a thing is false—because there’s a third option. After all, something can be neither true nor false; it can be P, okay? Something else. Never mind—he built an entire logic with truth tables of three-valued logic. And Benny Shlev brought Lukasiewicz’s three-valued logic there to explain that when Rabbi Kook spoke about the unity of opposites in some context, then that’s perfectly fine; there’s a logic that describes it, everything is fine, and we’re all set. This is of course nonsense. It’s nonsense because three-valued logic is not really an alternative logic. It’s only a formal tool for describing certain domains of thought. There are places where I can describe things as if I have a three-valued logic—there’s no problem with that, no big deal. But it’s clear that the logic with which I think is always binary logic. Meaning, when I build Lukasiewicz’s tables, I build them with binary logic. The proofs I build for why here it says P, here T, and here F—those are proofs, some of them by negation. Once you’ve defined the three-valued logic, inside it you play the three-valued game, and that’s fine; there are domains where this model can be a useful model. Meaning, if you treat it as though there’s a logic here and as though there are three values and you can—like in quantum theory, by the way, same issue. Also in quantum theory there are people who think that if they formulate a logic that is not ordinary logic—there some people, for example, give up transitivity of logic, never mind, each one gives up something else in order to explain this strange logic of quantum theory—then they think they’ve solved the problem. They haven’t solved any problem. You cannot deviate from ordinary binary logic. It’s just nonsense. The fact that you gave it a formal formulation doesn’t solve anything; all you did was formulate formally what I’m saying in words. So what? The problem remains. After all, the proofs you do in quantum theory are also proofs by negation. Meaning, you use ordinary logic in quantum theory in order to formulate the theory itself. Quantum theory does not undermine our logic, because if it did you could also throw quantum theory itself in the trash, since it is built on our logic. You cannot say that you believe in quantum theory and also do not believe in quantum theory. That’s nonsense. Okay? Therefore there too people don’t understand that quantum logic is not an interpretation. People relate to quantum logic as one of the possible interpretations of quantum theory. One of the proposed interpretations of quantum theory is that binary logic really is not correct; quantum theory taught us that logic really needs to be a different logic. That’s nonsense. It’s not an interpretation. It’s merely a formal model that can help. You can use this mathematics of three-valued logic to display the calculations in quantum theory. Okay, if that helps you, that’s perfectly fine; as a model, fine. As long as you don’t give it the meaning that we’ve found logic here. In short, logic is not measured in a laboratory. You come to the laboratory with logic; logic does not emerge from the findings we measure in the laboratory.

So the claim, basically, what I want to say is that very often when people use the concept of the unity of opposites, this is what Rudolf Otto writes—a well-known Christian philosopher who wrote a book, and his famous book is called The Idea of the Holy. In the introduction to the English edition of the book he writes that the unity of opposites is the lazy man’s monster. Meaning, someone too lazy to think says: I’m unity of opposites, I live with contradictions—instead of really getting into the thickness of the beam and explaining why there is no contradiction here. No, no, I live in— So the point is that I think that where they have shown you a formal, frontal logical contradiction, usually you should understand that apparently you were mistaken. Meaning, there won’t be a solution to it. This feeling that I can live with the contradiction and only I’m not smart enough to explain why it’s not contradictory appears in places where the contradiction is not a logical contradiction but a contradiction of the kind I spoke about before—of

[Speaker F] the red heifer.

[Rabbi Michael Abraham] Meaning, this contradiction of two things that seem to me not to fit together, but yes—there are many examples of this. Most contradictions—I’ll give you another example. This too was a coin that dropped for me once when Galia Nadel died. So I read an obituary article in Haaretz written, I think, not by Yehudit Rotem—she has a daughter, I think, who also writes there, I don’t remember her name. She wrote an obituary. That is, she’s basically formerly Haredi, the writer Yehudit Rotem, well known. Her daughter too, I think, writes there; I forgot the name. It seems to me it was the daughter, and she wrote a bit about this figure, an interesting figure, and she tried to describe this figure to the readers of Haaretz. And among other things she was surprised by the strange combination in Galia Nadel’s personality: on the one hand he was very open, and on the other hand he was very extreme. And she said this doesn’t fit together—how can it be that a person who is so

[Speaker F] open…

[Rabbi Michael Abraham] No, no, absolutely not. Rabbi Shach was not open at all. Galia Nadel was both open and extreme. And she didn’t understand how those two things appeared in the same person. Now when I read that it was very interesting, because many times I’ve also sensed this within myself, that I think there are certain areas in which I am extreme, but I think all in all I’m an open person. So the question is why that isn’t contradictory. And then I started thinking, and basically I came to the conclusion that it doesn’t contradict at all. In the end my conclusion was not only that it doesn’t contradict—it usually comes together. Because when can you allow yourself to be extreme? When you are open and you know the other options and you’ve come to the conclusion that this is the right one and the others are wrong, and then you have enough confidence to say, okay, I checked the other options and I’m telling you that you’re talking nonsense; this is what’s right and that’s it. Whereas if I’m not open and not willing to examine the other paths, I can never know—maybe this is right and maybe that is right.

[Speaker F] But fundamentalists are also extreme. What? Fundamentalists are also extreme, even though they don’t examine anything.

[Rabbi Michael Abraham] No, I’m not claiming it must go together, but I’m saying that openness naturally, in my opinion, can definitely lead to extremism.

[Speaker H] And not so naturally, because usually if someone—after all, if someone goes out to check, he is willing to hear the other opinions, that probably means—maybe it’s because he says, what do I care if I hear them, it won’t hurt.

[Rabbi Michael Abraham] No, because he wants to know whether they might be right.

[Speaker H] So I’m saying, that probably means he assumes that his starting assumption is that what he says is not self-evident.

[Speaker A] Right. And after he checks, he may arrive at the conclusion—

[Speaker H] One second—and therefore I say, someone who starts from such an assumption, then usually when there are arguments that don’t depend on clear facts—that is, I’m not talking about something where you really check what’s there and then you see that it’s so, but arguments in ideas, in spirit—then usually even after you check, you may still remain in some doubt.

[Rabbi Michael Abraham] Meaning—not maybe. I’m saying, the fact that fundamentalists too are extreme—I’m not claiming the connection is a necessary one, but I’m saying—

[Speaker H] No, I’m saying it’s not a necessary connection; the connection is the opposite. No, the reverse connection is not necessary—

[Rabbi Michael Abraham] It can be; that I don’t agree with. On that I don’t agree. I do think that at least as a tendency I would expect that someone who is open—precisely that would lead him to greater extremism in general. With all the—I agree there are other possibilities too. Because it means he checked, and if he reached a conclusion, then he reached a conclusion. Meaning, on the contrary, then he no longer expects anything from the other side and he can dismiss it and say he disagrees, that he sees nothing in it—because I checked. But if I didn’t check, how can I say that I see nothing in it? Only if I’m just some fanatic fundamentalist who isn’t willing even to believe that someone else has—

[Speaker H] But then that’s extremism coming from being a fanatic.

[Rabbi Michael Abraham] No, that’s what I’m saying: there can be extremism from having arrived. There can be extremism from truly arriving at the conclusion that I am right and the others are talking nonsense. Not because I’m fanatical by temperament. And philosophical extremism, not psychological extremism. Philosophical extremism means that after I checked, I came to the conclusion that I’m right and the others are wrong. Fanaticism is a character trait. It’s not philosophy. I’m talking about philosophical extremism, not psychological extremism.

[Speaker E] There’s that story in the book of the Yavetz about Spain in the Inquisition, where the scholars gave up their lives less, and the ignorant masses were more pious—

[Rabbi Michael Abraham] Yavetz.

[Speaker E] So there you see actually that the more, let’s call it, extreme you are, then there is—

[Rabbi Michael Abraham] No, and there it’s wisdom, not necessarily openness. Intelligence, education, or something like that—I don’t think it’s openness. There it’s something—I once wrote about this in one of my columns—that the Bnei Brak ethos, after all, is the Yavetz; he is very central there. About simple faith and simple people who go with their truth without all the philosophizing of the Torah scholars. There’s something in ordinary householders that is much more stable, much stronger, than the Torah scholars. And that’s what the Yavetz says: that in the expulsion from Spain, those who were prepared to go with their truth to the very end were usually the householders. Torah scholars found all sorts of solutions for themselves. And why? It’s very logical. Because from the householder’s point of view, everything came down from Mount Sinai. He doesn’t know tricks. For him, what is forbidden is forbidden, what is permitted is permitted. He doesn’t— But a Torah scholar knows: okay, on this issue there is a dispute among the medieval authorities, I know I can introduce a doubt here, after all, and it’s life-threatening, so that’s—and I’ll do it indirectly with my left hand. If I’m a Torah scholar then I know you can play with

[Speaker A] the system, exactly, there’s—

[Rabbi Michael Abraham] You can play with this system. In this context I always think that people think there is oppression of women in Bnei Brak, that the men oppress the women. They don’t understand that the opposite is true. Usually the truly religious ones in Bnei Brak are the women. Meaning, if the husband wants not to go to kollel, he’ll get kicked by his wife who’ll throw him into kollel. That is usually—of course there are always examples—but usually. Why? Because the woman didn’t study. Since she didn’t study, then anything the husband does that doesn’t match what she, I don’t know, heard in seminary or from her parents—then he’s denying the fundamentals. What do you mean? Moses at Sinai said you have to wear forty-denier stockings—how are you buying me twenty-denier stockings? You’re a heretic, an apikores. Now it never even occurs to her that the stockings are some kind of dubious custom of one sort or another, and that the whole question is why one has to follow it. But someone who studied knows. So the ignoramus, when he looks at the Torah scholar, thinks that everything he does is tricks. Now by the way, sometimes that’s true. Sometimes the Torah scholar, because he has an excess of tools and excess possibilities and all that, often also uses them—when you’re in distress, like in the expulsion from Spain, I can understand that a Torah scholar finds for himself some route that in a calm view I’d say doesn’t hold water. But he persuades himself, because you know, it’s a very heavy price to keep struggling here. And the ignoramus—he has nothing, he knows no tricks. It’s forbidden, so that’s it. God said it, God said it—what do I care if they all burn? Meaning, he’s genuinely God-fearing. So in that sense there is something very strong in the ignorant masses. The ignorant masses—I just don’t think that because of this I should turn ignorance into an ethos, which is what they try to do as a result. I say: okay, one should be wise and try not to fall into the traps that a wise person can fall into. But I’m not willing to turn stupidity or ignorance into an ideal just because there indeed the person will go with his own thing to the very end. He’ll go with his falsehood to the very end. Okay—but who said that what he goes with—who said an ignoramus is pious? Who said that what he thinks is really the right thing? So this point is exactly the difference: the ignoramus is a psychological fanatic, not an ideological fanatic, because he doesn’t know ideology at all. He is a psychological fanatic. He understood that it is like this, and he’s not willing to consider other possibilities. The Torah scholar is not a fanatic. Why not? Because he considers possibilities, he understands there are different directions, and so on. He needs to be careful not to make considerations that are wrong just because of constraints. And in that sense he needs to be careful to be an ideological fanatic even if he is not a psychological fanatic—to be an ideological fanatic, to go all the way with what he thinks is right. Okay, so now I want to move on to the next stage. So now what I basically want to talk about a bit is really the concept of logical contradiction,

[Speaker A] meaning that—

[Rabbi Michael Abraham] And here we enter the paradoxical questions, because very often when we—sorry, I didn’t finish the example—the example of openness and extremism. Once I understood that those two things actually don’t contradict, or at least don’t necessarily contradict, then suddenly I understood that there is a certain kind of contradiction that is not a logical contradiction, but rather just seems to people not to fit. Now maybe you don’t understand why it doesn’t contradict, but you have a strong sense that it doesn’t. And if such a thing exists, it usually appears in a place where you can’t formalize and show there is a contradiction. If you can formalize it and show there is a contradiction, then you understand that no one is going to come and show you that this is not true. Though even there you could be mistaken—but that feeling won’t appear there. That feeling appears in those contradictions of this type: extremism versus openness, or the red heifer that purifies the impure and renders the pure impure, or all sorts of things like that that seem to us on their face incomprehensible, but are not really logical contradictions. And there, I would say, it is more a question than a difficulty. Meaning, then I have a question of how this fits together, not a hard objection. A contradiction is a hard objection. All right? Therefore I think that in the end, living with a logical contradiction is almost—if the contradiction is a logical contradiction, I almost cannot imagine a situation where I would find myself saying: okay, but nevertheless both sides are true, maybe some smart person will come and show me that there is no contradiction. In principle, even with a logical contradiction, it could be that I made a mistake in the formalization, it could be that I made a mistake in the logical calculation, and that can happen. But there it is already, in a certain sense, self-deception. The place where I am willing to live with contradictions is when the contradictions do not logically contradict each other. And therefore, for example, with God’s knowledge and free choice, the big question underlying the whole issue is whether this is a logical contradiction or not a logical contradiction, or a contradiction of the softer type I spoke about before. Because if it is a logical contradiction, then spare me all the formulations—you have to give up one of the two options. If you can argue that there is no logical contradiction here on the conceptual level, that they do not contradict one another, then we can discuss it. It may be that I don’t understand how it fits together, but some smart person will come and explain it to me. I can continue to live with both sides of the contradiction. In short, this brings me to the topic of paradox.

[Speaker E] Still, with will you did an analysis—you threw in some paradox there, between a person’s ability,

[Rabbi Michael Abraham] Yes, but there I think I tried to show why I don’t think it’s ultimately a paradox. I found a solution, because really if I didn’t have a solution, then you’d have to give up one of the two possibilities. So very often it’s clear that the more you identify with both sides, the more deeply you’ll dig in order to find the solution for why it isn’t a contradiction. But if in the end I dug and didn’t find one, and it remains a contradiction on the logical level, then that’s what’s called a proof by contradiction. Otherwise you’re not supposed to accept any proof by contradiction. Someone will show you there’s a contradiction in your position, and you’ll tell him, okay, but someone smarter will come and show it. You can’t talk like that. So how do we prove things to one another? Now you’ve dismantled not only the rhetoric but the logic too; meaning, there’s nothing left here. Meaning, you can no longer be convinced in any way, because you can always say okay—which is already fundamentalism. It basically means that what I think is not open to examination; even if someone proves to me that it contains a logical contradiction, I don’t accept that claim. And that seems problematic to me, a problematic stubbornness. Okay, so what is a paradox really? A paradox is a situation where two things—or not necessarily two things, but rather a system of assumptions—leads me to two contradictory conclusions, to a thing and its opposite. And then basically that means that something in my system of assumptions is not correct. And often a paradox, by the way, is some kind of experiment; a paradox is a proof by contradiction. If a system of assumptions leads you to a contradiction, that is a proof by contradiction that something in your system of assumptions is not correct. Or in scientific contexts, say, sometimes people do what’s called a thought experiment. A thought experiment is not an experiment done in a lab. They say: let’s try to imagine a situation, analyze it according to the criteria or principles I believe in, and I’ll see that I get a contradictory result. And that means that in the system of principles I believe in there’s some problem; I need to change it, give something up, or something like that. That’s the role of a thought experiment. A thought experiment is really just to produce, out of the system I hold, a kind of paradox. Because a real experiment—a real experiment doesn’t work like that. A real experiment simply shows you factually that what you’re saying is wrong. That’s all; it just contradicts the facts. You observe something and see that what your theory predicts does not come out in the experiment. A thought experiment brings you no new facts at all. So why should I give something up because of a thought experiment? Because a thought experiment isn’t really an experiment. It just says: let’s apply what you think to a hypothetical situation, and I’ll show you that a contradiction comes out there. So it’s just another formulation; a thought experiment is a paradox. A thought experiment is constructing a paradox out of a system of principles I believe in.

[Speaker E] Sometimes in the Talmudic sages a difficulty comes up and they remain with it unresolved. To say, okay, fine, not a disaster.

[Rabbi Michael Abraham] Right, that’s what’s called living with it, but that’s only where the difficulty is not logical. By the way, that’s what the medieval authorities (Rishonim) say: what’s the difference between a difficulty and a refutation? A refutation rejects what you challenged; a difficulty does not. What’s the difference? It’s exactly the distinction I’m talking about here. If the contradiction is a logical contradiction, then I do not rely on the idea that someone smarter will come and solve it for me, and therefore—someone has proved to me by contradiction that I’m wrong. “Difficulty” means exactly that there is something here that I don’t understand how it fits together, but it’s not a logical contradiction. I have no explanation for why it doesn’t fit together; someone smarter will come and explain. As long as I’m convinced of what I’m saying, I don’t throw it out because of a difficulty. That is exactly the difference. The fact that there is such a thing as a refutation shows that a difficulty is not that, because otherwise there wouldn’t be refutation. Otherwise about everything I could say maybe someone smarter will come and explain it to me. So why is there such a thing as refutation in the world? It means there are situations that are a refutation and situations that are a difficulty. And that’s actually a nice illustration of what I’m saying here. By the way, there are places where the medieval authorities (Rishonim) actually rule like the view against which the Talmud ends with an unresolved difficulty. The Talmud concludes with a difficulty against one of the views, and Maimonides rules it as Jewish law. And the rule-writers discuss this, saying that a difficulty does not necessarily reject the matter.

[Speaker F] Surely whole books were written of difficulties. Surely books were written with…

[Rabbi Michael Abraham] Wow. There’s a book called Chazot Kashot. It’s a book—there are difficulties; Rabbi Akiva Eiger, for example, is someone who very often leaves things as requiring further analysis in many places. There’s Rabbi Akiva Eiger on the Talmud, there’s Rabbi Akiva Eiger on the Mishnah, and on the Shulchan Arukh, I think. And there’s a book—three books in one volume—where each book resolves the difficulties of Rabbi Akiva Eiger: one on the Talmudic discussions, one on the Mishnahs, and one on the Shulchan Arukh. Simply a book that goes through Rabbi Akiva Eiger’s “requires further analysis, requires further analysis” and resolves them. One after the other. Who wrote it? Those are three different books. I don’t think it was the same person. But they bound them together, collected them, I think. It’s called Chazot Kashot. On the Talmud it’s Chazot Kashot, I think.

[Speaker D] Sometimes it says “and examine carefully”—what does that mean?

[Rabbi Michael Abraham] No, it means: look carefully into this and see that I’m right. It has nothing to do with a difficulty. When I say something and I’m afraid the reader won’t understand, that he should think about it again in order to understand that I’m right, I write—

[Speaker A] “And examine carefully,” look into it well.

[Rabbi Michael Abraham] If you look carefully, you’ll see that I’m right. Maybe at first glance you don’t see it; think again. “Examine carefully” means to be precise, that’s all. In any case, I don’t know why sometimes they write it with quotation marks.

[Speaker D] Some say it means “examine carefully and you’ll find a difficulty,” and some say “examine carefully and you’ll find the truth.”

[Rabbi Michael Abraham] Yes, I think originally it means “be precise”; maybe the quotation marks aren’t quotation marks at all, maybe it’s even two yods of “and be precise”—who knows. All kinds of corruptions like that developed over time. Let’s get back to our matter. In any case, as I said before, the paradox is basically an attempt to show a contradiction within a certain system of assumptions, and that itself constitutes a proof by contradiction against that system of assumptions. Meaning, one of those assumptions you have to throw out. That is the meaning of a paradox. And therefore I say: if the paradox is a logical paradox, then you can’t go on living with it. It is a proof by contradiction that you have to choose a side.

[Speaker G] On the side of the assumptions or on the side of the conclusions?

[Rabbi Michael Abraham] The conclusions are derived from the assumptions, so you can’t remain with the conclusions.

[Speaker G] Unless you didn’t derive them correctly.

[Rabbi Michael Abraham] No, that’s something else. That’s why I say: someone smarter can come and show that I made a mistake in the derivation—fine, that’s an error. But I’m saying, usually if I derived it and I know how to do that, then it won’t happen. Usually. And therefore I wouldn’t worry about that. You need to give up one of the assumptions. Now what is the paradox? I’ll bring a few examples so things will be more concrete.

[Speaker A] There’s that famous experiment in physics, Einstein-Podolsky-Rosen.

[Rabbi Michael Abraham] No, that’s EPR.

[Speaker A] No, that’s the thought experiment—

[Rabbi Michael Abraham] Which in the end was also carried out. They later did that experiment.

[Speaker A] I didn’t get to finish learning that Rosen one before the end of the degree.

[Rabbi Michael Abraham] Nathan Rosen. Yes. And now there’s Joe Rosen, his son; he’s a physicist too. In any case, the paradox—I’ll bring a few paradoxes just to make things more concrete. There’s the paradox of Achilles and the tortoise, the paradoxes of Zeno. Zeno presented all sorts of paradoxes that basically came to show that the concept of motion is a concept that leads to paradoxes—or in other words, the paradox proves by contradiction that there is no motion in the world. Okay? What we think, that things move—that’s an illusion. There is no motion in the world. I don’t know exactly what then there is instead, what exactly he meant—I have no idea. But the paradoxes he formulated are amusing and well-known paradoxes, and one of them, for example, is the paradox of Achilles and the tortoise. Achilles and the tortoise is basically—he says, suppose Achilles runs ten times faster than the tortoise, and they have a race. Now Achilles gives the tortoise a head start of ten meters. Okay? So Achilles runs ten meters per second and the tortoise goes one meter per second, for the sake of discussion. Okay? So when Achilles passes the first ten meters, the tortoise has already advanced another meter, right? It moves at one-tenth Achilles’ speed. By the time he passes that meter, the tortoise has already advanced another ten centimeters. By the time he passes those ten centimeters to where the tortoise was, the tortoise is already a centimeter ahead. And so on. Meaning, Achilles will never catch the tortoise.

[Speaker A] He didn’t know there was a convergent infinite series.

[Rabbi Michael Abraham] Yes. And the claim is that Achilles never catches the tortoise. That is basically Zeno’s claim. That’s one of his paradoxes. Another paradox is the paradox of the arrow in flight. He says: look at an arrow flying. At every moment it stands in a different place. So when does it pass between the places? At what moment does it change location? He says: at every moment, if you look at it at that very moment, as a mathematical point in time, then it’s standing here. A moment later it’s standing there. So at what moment does it pass? When does it make the transition? So that too is a paradox, subtler than the first. And so on; he has several paradoxes that come to undermine the concept of motion. These paradoxes are only apparent paradoxes, because today we know they have solutions. Meaning, they can be solved. The paradox of Achilles and the tortoise is, as Shmuel said, a geometric series.

[Speaker A] A convergent infinite series.

[Rabbi Michael Abraham] Yes. A geometric series, a convergent infinite series, because if you do the calculation you’ll see: after one second Achilles has passed the ten meters and the tortoise has gone one meter, right? When Achilles passes the next meter, the tortoise has gone another ten centimeters. How long did it take him to pass that next meter? A tenth of a second, right? In one second he goes ten meters. Meanwhile 1.1 seconds have passed. Now he passes the ten centimeters by which the tortoise is ahead, and the tortoise advances another centimeter. How long did it take him to pass those ten centimeters? A hundredth of a second, right? So now 1.11 seconds have passed since the race began. He passes the next centimeter—that’s a thousandth of a second. Meaning, in the end, when you add up all these steps, it describes the first 1.111111 seconds of the race—the first one and one-ninth seconds of the race. Okay? And that’s true. During the first one and one-ninth seconds of the race, Achilles does not catch the tortoise. It’s just that you are describing the first one and one-ninth seconds of the race and breaking them into infinitely many stages. But you broke into infinitely many stages a segment that is altogether only one and one-ninth seconds. Now when you do the calculation, you’ll see that it takes Achilles exactly that amount of time to catch the tortoise—

[Speaker A] It comes from that simple initial intuition that when you add infinity—

[Rabbi Michael Abraham] Things—infinite things means infinite time. Yes, exactly. But in the end what you described was the first one and one-ninth seconds of the race, and that’s true. When you do the calculation you’ll see that Achilles catches the tortoise after one and one-ninth seconds. Exactly then he catches the tortoise. Now in this description you’ll never reach one and one-ninth, because it’s point one plus one plus one—you’ll never get to the ninth; after infinitely many steps you’ll never arrive. So therefore, in that sense, Achilles never catches the tortoise. In the logical process you never arrive at the description in which Achilles catches the tortoise. But in reality, one and one-ninth seconds eventually pass, and when that one and one-ninth seconds passes, he catches—

[Speaker A] It exactly after one and one-ninth seconds.

[Rabbi Michael Abraham] Yes, exactly. So the claim is that Zeno’s paradox is only an apparent paradox. He simply made a mistake. Here, by the way, is an example of something that on the face of it seems paradoxical on the mathematical level—I have a mathematical proof, right? And then someone smarter comes and shows there’s nothing there at all. Someone smarter—someone arose who knew mathematics that people then didn’t know, okay?—and showed that it wasn’t correct. So there are—even examples of contradiction right on the mathematical-logical level where I’m missing something. That can happen; you can’t rule it out. But I’m saying one has to be careful and straightforward in this context, because otherwise you can entrench yourself in any position whatsoever. The same with the arrow in flight; about that I wrote an article explaining why this thing is not a paradox, but I won’t get into it here—maybe we’ll talk about it sometime. In any case, that too has a resolution. They also tie that one, by the way, to infinity, and in my opinion you don’t need to get to infinity to resolve it. So that’s one type of paradox. Another type of paradox is, for example, the liar paradox. Okay? The liar paradox is: a resident of Crete says, “All the residents of Crete are liars.” This is a sentence in the New Testament, where one resident of Crete claims something there; he said all the residents of Crete are liars, don’t believe anything they say. So from there they took this idea, and it’s a loop paradox. Zeno is not a loop paradox. Zeno is simply—you arrive in two ways—or in other words, Zeno is basically casuistic pilpul in the map of pilpul versus homiletics. Because here there is an argument built correctly, even though it’s clear to you that its conclusion is not true, that Achilles does not catch the tortoise. And what I said was that pilpul is an intellectual challenge—meaning, find what is wrong in the argument, because it’s clear to you the conclusion is wrong. But there are paradoxes that really are paradoxes. So Zeno isn’t exactly a paradox; it’s pilpul. But the liar paradox is a paradox. Why is it a paradox? Because it is basically a claim regarding which you cannot determine whether it is true or false. If it’s true, then it’s false; if it’s false, then it’s true; and basically it’s some kind of circle. Why? Because if all the residents of Crete are liars, then he himself, as a resident of Crete, is also a liar. But if he is a liar, then it’s not true that all the residents of Crete are liars, so he is telling the truth, so it is true, and so on. Of course that isn’t precise, as we already discussed once, because the contradiction of “all the residents of Crete are liars,” if that statement is false, means that there is a resident of Crete who is not a liar—but that doesn’t have to be the speaker; it could be someone else, and then it stops, it’s not a loop. The liar paradox really exists where there is no universal quantifier—meaning, where there is no “all” in the sentence. For example, to say, “I am lying now,” or “This sentence is false.” Okay? That is the liar paradox. Because then, if this sentence is indeed false, then it is true, because the sentence says that the sentence is false, so if it is false then it is true, and if it is true then it is false. Meaning, it is a paradox of self-reference, a loop paradox of self-reference. There are other paradoxes of self-reference, like the barber of Seville, yes? The barber who shaves all the people who do not shave themselves. So the question is whether he shaves himself or not. If he shaves himself, then he belongs to the group of people whom he does not shave, because he does shave himself, so he does not shave himself. And if he does not shave himself, then he belongs to that group that is shaved by him, so he does shave himself. So again, this is some sort of logical loop. There are more such paradoxes, for example the paradox of Protagoras, yes? The law teacher who stipulates with his student that he will pay tuition if he wins his first court case. The student—if he finishes his studies and wins his first case, then it turns out he studied well, so he should pay the tuition. And if not, it’s like success-based learning, meaning if you don’t win your first case you’re exempt from paying; if you do win your first case you have to pay. Fine. The student finished his studies and went on his way, went off to become a fisherman. So the teacher sues him, demanding that he pay the tuition. So now the judge has to decide whether the student is right or the judge—sorry, the teacher—is right. Now if the judge rules in favor of the student, then the student has won his first court case, so if that’s so he will have to pay tuition. And if the judge exempts him from the tuition, then he is obligated to pay the tuition, and so on. There are various others—the crocodile that makes conditions with a woman, yes? Whether it will devour her or not devour her—there are all kinds of amusing paradoxes of this type. These are paradoxes of self-reference. Ronnie Aharoni is fond of them, yes. There are other paradoxes that are not exactly self-reference—for example, there is the heap paradox, which I have already mentioned more than once; I won’t weary you with it again. There is, for example, the Swedish army paradox. The paradox of the army—is it familiar to you? The paradox of the surprise exam or surprise drill. Meaning, the commander comes to the soldiers and says: look, there will be a 24-hour drill, from midnight to midnight the next day, this week there will be a surprise drill—

[Speaker A] On one of the days of the week.

[Rabbi Michael Abraham] Okay? Now the soldiers calculate and say this: on the final Sabbath—suppose he is now at midnight on Friday night. They say to themselves: will the drill begin on the coming Sabbath night, on the coming Friday night? Obviously not. Why? Because then we’ll know, since that’s the only day left, so if the drill hasn’t been until now, then it won’t be a surprise—we’ll know. Right? So it can’t be on the coming Sabbath. Now it also can’t be on Friday, because if it can’t be on Sabbath, and we’ve reached Thursday and there has been no surprise drill, then obviously it will be on Friday, because after all it can’t be on Sabbath, it hasn’t happened until now, only Friday remains—so it can’t be on Friday either. And likewise not Thursday, not Wednesday, and so on, so it can’t be on any day. There are no surprise drills. That’s basically the point. But there are. Meaning, this too, by the way, is pilpul, not a loop. Because it proves to you by a logical route—and seemingly it’s a good proof; it’s very hard to put your finger on what is problematic in that proof. It’s a beautiful paradox. But it’s obvious to you that it isn’t correct: there are surprise drills; the fact is that I’m surprised. Meaning, when they do surprise drills, I’m surprised. So this too is a kind of pilpul paradox. Meaning, the argument is an argument that leads to one result, not to two contradictory results, except that this result is obviously not correct—obviously because of common sense, because of experience, whatever; so it is a contradiction with common sense. It’s not that you prove a thing and its opposite from the same system of rules. But this too presumably ought to have some kind of solution, yes? It’s clear to all of us that it should. In the loop arguments, or the liar paradox, there it’s not a contradiction with my experience; rather, there it really is something that leads to a conclusion and its opposite from the same system of assumptions. And then the big question really is: can there be such a paradox with no solution? Or is a paradox always just our oversight—if we were smart enough, we would always solve it? In other words, let’s formulate it differently: what does it mean to solve a paradox? Another question: what does it mean to solve a paradox? To solve a paradox is to show where in my calculation or logical process an error occurred, and then—

[Speaker A] There really is no paradox.

[Rabbi Michael Abraham] Because to solve a paradox—very often you do the calculation of the surprise exam, or the calculation of Achilles and the tortoise, which has a similar logical structure. Someone will come and say: okay, but obviously Achilles catches the tortoise, so this is nonsense, I’ve solved the paradox. You haven’t solved anything; you’ve shown why it’s a paradox. You need to show me, in the calculation, what in the calculation is incorrect. I know it’s a paradox because we know Achilles catches the tortoise. But a solution to a paradox is not to show that the conclusion is incorrect, but to point out what is defective in the argument that leads to the incorrect conclusion, to discover where the problem is. That’s why I said pilpul is an intellectual challenge; it’s a riddle. Okay? That is what it means to solve the paradox. Now the interesting question is whether there is a paradox that has no solution. Can such a thing exist? Because apparently not—if it’s a thing and its opposite, then if this one is true, the other one is not true. How can both a thing and its opposite be true? And if they are not true, then obviously something in my logical reasoning was flawed, even if I don’t discover what it was. But there must be a solution to the paradox, because a thing and its opposite cannot both be. Unless what I am saying is simply meaningless, just a linguistic illusion. There isn’t really a paradox here, just the appearance of a paradox; it’s not really a paradox, it’s simply a linguistic problem. What you’re saying is meaningless—that is basically what Aharoni claimed, that what you are saying is meaningless, and therefore this thing is not a paradox. So, okay, we’ll continue with these topics in—.