Doubt and Probability—in Halakha, Jewish Thought, and in General—Lecture 3

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[Rabbi Michael Abraham] No, the doubt can depend on desire, but you have to avoid that. That’s exactly the claim. I agree, people have psychology, nobody is perfect, but you need to try to avoid it, because it’s a distortion. Those psychological influences are a distortion. They exist—I agree that factually they exist. I just think that the fact that they exist is a failure, it’s a distortion that has to be overcome, and that there’s no connection between the question of whether I despise him and the question of whether he’s my father. Even wicked people have children. And therefore the claim is that my argument is an argument on the philosophical plane, not the psychological one. On the philosophical plane, if I’ve reached a clear conclusion that it is X, and it is established as being X, if someone raises a hypothetical claim of maybe not-X, then I won’t necessarily relate to that, because from my perspective that’s not an option I have reason to be doubtful about. The Talmud in Kiddushin says that we execute and stone on the basis of established presumptions. What does it mean that we execute and stone on the basis of presumptions? Simply speaking, the meaning is probably this very issue. Let’s say someone—it says, “One who strikes his father or his mother shall surely be put to death.” Now, you know that fatherhood is only a matter of presumption. The presumption is that most sexual relations are with the husband. The mother is known—we know from whose womb you came—but the father is a presumption: most sexual relations are with the husband. On the basis of such a presumption I can kill a person? Meaning, who says you struck your father—maybe he’s not your father? So the answer is that we execute and stone on the basis of presumptions. What does that mean? We’re not talking about presumption in the ordinary evidentiary sense—like the presumption that a person does not repay before the due date, or things of that sort. We’re talking about presumption in the sense of something established as true, something that is clear to us as true. If it is clear to us that it is true, even though there is a hypothetical possibility that it isn’t. Meaning, there are situations—there are illegitimate children—meaning, there are children whose father is established as their father and he really isn’t; he’s not actually your father. That can happen. But if he is established as such, then I need a good reason before I put myself into a situation of doubt. Before I say, okay, actually I’m not sure—although you can always not be sure, there is some hypothetical chance that it’s not true. But that is exactly the meaning of presumptions: that something is established to be so, and therefore for us it is so. And in order to argue that there is doubt, you also need reasons—not only to argue that it isn’t true, for that of course you need evidence, but even to argue that I should be in doubt between several possibilities—for that too you need reasons. Meaning, doubt is not something that arises without a reason. The Talmud at the beginning of Bava Metzia discusses whether there is or is not a direct monetary connection—whether you have some connection to this money, and that’s a different kind of doubt from doubt that arises when you have no connection at all to the money. For example, that case of the boat—in the Talmud in the chapter Chazkat HaBatim, they speak about some boat sailing on a river, and two people are arguing about whose boat it is. Now, there is no indication at all that it belongs to either of them. Maybe it belongs to someone else altogether. That’s called a doubt without a direct monetary connection. There is no connection at all, according to the standard explanation. There’s a dispute there between Rashi and Tosafot, but that’s the accepted explanation of direct monetary connection: that basically the doubt arises only because of their claims. The situation itself does not arouse doubt. The doubt is created only because of their claims. That’s called a doubt without a direct monetary connection. By contrast, if something happened before our eyes—for example, someone threw a bill of divorce, and it is uncertain whether it landed closer to him or closer to her. We don’t know whether it fell closer to her, in which case she is divorced, or closer to him, in which case it never really left his possession, and so it is not effectively a bill of divorce. Now here the doubt is not because of their claims—I see it. I see that the bill of divorce fell somewhere, and I am uncertain whether that place is closer to her or closer to him. So the doubt is created by the situation itself, not by the claims. That’s called a doubt that does have a direct connection. Or whatever, all sorts of things of that kind. It’s not a monetary connection because this isn’t money, but it’s that kind of doubt. So that distinction is exactly the point. If you have doubt without a direct monetary connection, and now someone comes and says this boat was his, he says “it stood up”—in that kind of situation I don’t even see it as a situation of doubt. The fact that someone claims something—he claims it, he can keep claiming it forever. Why should I take it into account at all? The whole world could also come and say that this boat is theirs. So you say it—so what? In other words, when the doubt is created by your claims, that doesn’t really succeed in arousing doubt if there is indeed an established state of affairs. Meaning, if there really is a situation that I know, I have clear knowledge, and with your claims you come to remove me from that clear state, that won’t do. A mere claim does not remove us from a clear state. You have to bring some evidence that connects you to that boat. I don’t know—someone once saw you sailing that boat somewhere, or whatever exactly—then some doubt starts to arise, a doubt with a real connection. And then we have to get into the laws of evidence. In a doubt without a direct monetary connection, you don’t enter the laws of evidence; the presumption remains as it was, and nobody has succeeded in undermining it. But if someone threw a bill of divorce, and it is uncertain whether it is closer to him or closer to her, here too there was a prior presumption: first of all, she was married to him. Now the only question is whether an act of divorce occurred or not. So here too I am trying to remove from an established state—she was married to him, that is the known fact. Now the question is whether there was… but here there is a reason for doubt, because I saw that he threw a bill of divorce. I don’t know whether it fell closer to him or closer to her, so I am in doubt. But you can’t say that this doubt fails to arise at all—that this situation does not succeed in arousing doubt. Of course it succeeds in arousing doubt. Therefore here, even though there is a prior presumption—meaning, the woman was married to him and that is clear, and now the question is whether she has left that situation or not—apparently this is similar to… to that boat, to the discussions about the boat. But no, it is not similar. Because in the case of the boat this is a doubt without a direct monetary connection, a doubt born only from their claims. You claimed something, and therefore you think that because you made a claim, that should arouse doubt in me. No. So you claimed it. So what if you claimed it? I know this is the state of affairs, and a claim alone does not remove us from it. By contrast, if the doubt is created by objective reality, I simply see that a bill of divorce was given here, and I’m uncertain whether it’s closer to him or closer to her, then the doubt arose against my will. It’s not because someone claimed something. And once the doubt arises, the fact that there was a prior presumption—fine, there was. So what? But the doubt—I can’t ignore it. It has to be discussed. It could be that I will still follow the prior presumption, but this is already called a state of doubt. And the same is true regarding Bertrand Russell’s celestial teapot, and the same is true with “your mother is my wife and you are my son,” and all claims of that type. There is a situation that is clear; a mere claim does not succeed in arousing doubt. In other words, a mere claim is not something that arouses doubt. If there is a good reason to doubt, then maybe yes. And again, it’s not because I’m certain that I’m right—there is no certainty in anything. We talked about that. There is certainty in nothing. But if the state of affairs is established as such, then in order to dislodge it you need a reason. That’s the point, even if it’s only ninety-five percent and not one hundred percent. Not because it is one hundred percent true, but because that is the correct methodology of discussion. So one—another aspect perhaps of the same issue—is a situation called a major doubt. What is a major doubt? I have a dispute among decisors or among the medieval authorities regarding a certain halakhic question. Usually, in the halakhic world, people are accustomed to relating to such a situation as a situation of doubt. And they basically distinguish between two types of doubt, or even three types of doubt. One type is factual doubt, a second type is legal doubt, and a third type is a major doubt. Meaning, if there is a dispute among halakhic decisors, that too supposedly arouses in us some kind of doubt. And in all these types of doubt, people generally apply the laws of doubt—Torah-level doubt is ruled stringently, rabbinic-level doubt leniently, or presumptions, or all the rules for deciding doubtful cases. But for example, the Mishnah near the beginning of the chapter Klal Gadol—not at the very beginning actually, later on—the Mishnah distinguishes there between someone who doesn’t know, between two kinds of unintentional sinner: someone who doesn’t know that today is the Sabbath, and someone who doesn’t know that it’s forbidden to do this on the Sabbath. Let’s say someone is selecting on the Sabbath. So he could be unintentional because he didn’t know today is the Sabbath—that is an error in facts. Or someone could know that today is the Sabbath, but not know that selecting is forbidden on the Sabbath. So that is an error regarding the law, not regarding the facts. Just as there are two types of unintentional error, there are also two types of doubt. There is factual doubt and there is doubt in the law. There is the well-known Maharik, cited by the Rema in Even HaEzer: what happens if a woman commits adultery? She becomes forbidden to her husband. But if it was unintentional, then for a priest she becomes forbidden, but for an Israelite she does not. But says the Maharik: there is a difference between two types of unintentionality. If she erred regarding the prohibition of adultery—meaning, she understood that she was a married woman, and she understood that the man with whom she was having relations was not her husband, but she did not know that it was forbidden—says the Maharik, that is not the type of unintentionality under discussion. In such a situation she does become forbidden to her husband. Only if she does not know that this is not her husband—she thought she was having relations here with her husband, and in that she erred, she erred in the facts and not in the law—then indeed she does not become forbidden to her husband. What is the idea? The idea is that in the first case, where she errs in the law and not in the facts, she still had relations with someone who is not her husband. Meaning, she damaged the marital unit not in its halakhic dimension but in its factual dimension. The marital unit is supposed to be exclusive in terms of sexual relations; polyamory is not recommended by Jewish law. Therefore, once she allowed herself to do something that destroyed the marital unit—not because of the prohibition involved, but because of what it factually is. But if she didn’t know at all that this wasn’t her husband, then she did not dismantle the marital unit; she thought it was her husband, she simply made a mistake. So here, if she is unintentional, that truly does not forbid her to her husband; the marital unit did not break apart. But for our purposes, again, this is basically a distinction between error in facts, factual error, and error in the law. And similarly in other halakhic contexts they generally don’t distinguish. Meaning, both types of unintentional sinner are called unintentional. And likewise regarding doubts: there is doubt that is factual doubt, and there is… doubt in the law. For example, the doubt I described earlier regarding the bill of divorce—uncertain whether it is closer to him or closer to her—that is factual doubt. If I have a bill of divorce written in a way such that it is doubtful whether it is valid or not—I don’t know. Or if I have doubt regarding the validity of the witnesses on the bill of divorce, or things of that kind—then that is… no, that’s basically factual doubt, doubt about the validity of the witnesses. If I have a halakhic doubt regarding validity—I don’t know whether such a witness is halakhically valid or invalid—then that is legal doubt, not factual doubt. Usually legal doubt and factual doubt are viewed very similarly. Here and there some distinctions are made between the different types of doubt. Rabbi Akiva Eiger has a point regarding double doubt, that in legal doubt a double doubt might perhaps still be treated stringently, whereas in factual doubt not. But the prevailing approach is that there is no difference between the two types of doubt. But there is a third type of doubt, as I said earlier, and that is major doubt. Major doubt is a situation in which there is a dispute among halakhic decisors, and now the question is what to do. The accepted halakhic approach is that if there is a dispute among halakhic decisors, then I act according to the laws of doubt. This is basically a kind of legal doubt. There is legal doubt and factual doubt. Legal doubt means doubt in the law. So here too this is basically doubt in the law. Doubt whether the law follows Maimonides or follows the Rashba. Therefore this is basically a type of legal doubt. And I have written and argued—and I’ve said this in many lectures before as well—that I don’t think such a state is even defined as a state of doubt. In other words, I don’t accept the category of major doubt at all. There’s no such animal. Why? Because in principle I am supposed to—at least if I am competent, if I am a person able to issue halakhic rulings, at least for myself—then if I look at this halakhic topic and I see that Maimonides and the Rashba disagree about it, if I have my own formed position, then I am basically supposed to do what I think. Why should I care that Maimonides and the Rashba disagree with one another? If I do not have a formed position of my own, then I am in doubt not because Maimonides and the Rashba disagreed, but because I myself do not have a formed position. So again, this is not major doubt but ordinary doubt—my doubt. In other words, the fact that Maimonides and the Rashba disagree is not a reason to be in doubt. It may be that the fact that Maimonides and the Rashba disagree is what aroused doubt in me; otherwise maybe I would not have noticed that there is another side. But once that aroused the doubt in me, now the situation is doubtful because I am in doubt, not because Maimonides and the Rashba disagreed. And the practical difference is that if I am not in doubt—if they did not succeed in arousing doubt in me, if I have a clear position—then this is not a state of doubt, and I am supposed to do what I think, regardless of the fact that Maimonides and the Rashba disagreed. In this context, when I wrote this in an article many years ago, I brought the well-known story about Rabbi Yonatan Eybeschutz. A priest said to him—came to him and said, why don’t you convert to Christianity? Why don’t you follow us? After all, your Torah says, “Incline after the majority,” and we are the majority. You can discuss this for many reasons, but Rabbi Yonatan Eybeschutz says to him: I follow the majority when I am in doubt. If I am not in doubt, I do not follow the majority. Now this is often brought as a kind of joke, but it is not a joke; it is a completely serious argument. It is a serious argument that says something very similar, by the way, to Rabbi Kook, whom I mentioned earlier, or “we stone on the basis of presumptions.” He says: if I arrive at the conclusion that I am right, that Judaism is true and Christianity is not true, then the fact that the majority thinks Christianity is true—so what? They think that. But I have a clear position of my own. The fact that someone says something does not arouse doubt in me. So what if he says it? If I am in a state of doubt and I do not have a position of my own, then there is room for the claim: let’s follow the majority. But in order to follow the majority I first have to decide that I am in a state of doubt, because only in states of doubt do we follow the majority. Think, for example, about the case the Talmud discusses of someone who finds a piece of meat in the market. In a city, somewhere, and there are ten shops in the city, nine of them kosher and one non-kosher, then by strict law you may eat that piece of meat because we follow the majority. If there were nine non-kosher shops and one kosher shop, then you should be stringent and not eat the piece of meat, since most of the shops sell non-kosher meat. What happens in a city that has nine non-kosher shops and one kosher one, but I found a piece of meat with a high-level kosher seal on it? In such a situation would I follow the majority and say: no, the piece is probably non-kosher? The answer is of course not. Why? Because if it has a high-level kosher seal, then I am not in doubt; I know the piece is kosher. The laws of majority apply only to situations defined as situations of doubt. If I am not in a state of doubt, why should I care whether the majority points one way or the other? A majority one way or the other is only a rule telling me what to do if I am in doubt. If you’re in doubt, follow the majority. But if I’m not in doubt, then I am not interested in what the majority says. Exactly the same thing as with the argument with the priest—which of course never happened, but all the good stories are told about Rabbi Yonatan Eybeschutz. The claim is that if I am confident in my way, then the fact that someone argues otherwise, or even that the majority argues otherwise, should not move me from it. In other words, in order to doubt you need a reason. Therefore, if I am not in doubt, then no—there is no reason to doubt. The fact that someone says something is not enough in order to doubt. But this really sharpens for us the next important point: in order to activate rules for resolving doubts, rules of conduct or rules of decision in Jewish law, I first of all have to take step one and declare the situation to be a situation of doubt. That is the first decision I am supposed to make. And if I have decided that this is a state of doubt, now I go to my toolbox and begin discussing which tool to use in order to resolve the doubt. But it is very important first to go through stage one: first determine that this is indeed a doubtful situation. Because if not, there is no point—all the toolbox is irrelevant. Meaning, it is relevant only where I have decided that I am in a state of doubt. And this brings me back to what I said earlier: in order to doubt, you need a reason. And therefore before we discuss which tools to use and what the significance of the tools for resolving doubt is, first we need to discuss what is even called a state of doubt. And then we can discuss which tools to use and what each such tool means. Now, there is a similar question—what’s called peer disagreement. A disagreement among peers. There is a dispute, I don’t know, regarding any issue whatsoever, philosophical or scientific or whatever. It always comes up: who says you’re right? There are others, no less intelligent, who think differently. Regarding faith this comes up a lot, but it doesn’t matter; it applies to everything under the sun. Who says you’re right in the struggle against the Palestinians? There are many intelligent people in the world who think not. Why do you think yes? So I say: that is an argument of the kind… in other words, it is not an argument about the issue itself, but what I would perhaps call an argument about the person. Meaning, they try to arouse doubt in me not through arguments on the merits, but by pointing out that there are many intelligent people who think differently. That is the issue in philosophy called peer disagreement. Meaning, the fact that there are peers who disagree with me supposedly ought to arouse doubt in me—maybe I’m not right? What, am I the greatest genius? Now this is an argument that confuses a great many people, and my answer to it is really not simple. I won’t get into it; I wrote a column about it on my site. But I do not accept that argument. From my point of view, if you have substantive arguments, then raise them. But the fact that someone thinks differently does not help me in any way. If he is intelligent, at most it can cause me to reconsider my considerations. It may be that I missed something because he is an intelligent person, and if he thinks differently, then I should reconsider my considerations. But after I have reconsidered my considerations and still reached the same conclusion, then the fact that others disagree with me should not move me in any way. In other words, it is not something that arouses doubt in me. Again, not because of arrogance or because I am convinced that I am right, but because I have no better way to clarify the matter than to weigh the best considerations available to me. And that I have done. And if I reached some conclusion, then that is my conclusion. The fact that someone thinks otherwise—what am I supposed to do with that? Very often things like this are presented as if they are some kind of knockout argument. This argument is absolutely not a knockout argument. I’m not even talking about all sorts of claims based on the wisdom of crowds, like that priest and Rabbi Yonatan: we are the majority, therefore the truth is with us. Or if most people don’t believe in God, then the truth is with them. Or if—very often various versions of this—the more educated people generally tend to be more atheistic or less believing, so that proves there is no God, or things of that kind, which are arguments about people and not about the issue itself. When I want to decide questions of this type, I need to weigh the arguments themselves. Bring me the arguments of those intelligent people and I will need to think about them. But the fact that intelligent people think something—fine, they think something. I think otherwise. In other words, that is not supposed to change much for me. As for the wisdom of crowds, I’ve also discussed that in the past. The wisdom of crowds, to my mind, is a very important thing. Meaning: take what the crowd says and do the opposite. Usually the majority is wrong unless proven otherwise—that is my approach to the matter. So if you tell me that the majority thinks something, then apparently that thing is not true—unless by chance the majority hit the truth, because there are good arguments, and then I’m open to being convinced. But the default is that the majority is probably wrong. That is of course somewhat rhetorical, but not entirely. You have to check and distinguish between situations. What people often bring in connection with the wisdom of crowds is, for example, when they try to estimate how many leaves there are on a tree. Or how many animals are grazing in a meadow, cows grazing there in the meadow. Let’s say there are hundreds of cows, and you can’t count them. So: how many cows are grazing here? There may be some person who has some experience in estimating the number of cows in a given area, and he’ll throw out some estimate that is better than the estimate of the amateur, of someone with no experience in such a thing. But if a thousand people come and give estimates, and I average their estimates, the claim of the wisdom of crowds is that they will get closer than the expert. That there is power in the quantity of people, and that’s where the whole idea of the wisdom of crowds comes from. But the wisdom of crowds in that context is based simply on the law of large numbers. And the claim—the claim involves the law of large numbers and an assumption about the distribution. There are two assumptions here. One assumption is that the distribution of the errors of the amateur—not of the expert. The amateur errs, and can err completely, can be very far from the correct answer, but the assumption is that the distribution of the errors is a distribution around the correct mean, and you can err in both directions in the same way, a kind of Gaussian. Okay? The center is the correct answer. Assuming that is how the errors are distributed, then if you take many people, each of whom is at another place on the Gaussian, and you average them, you’ll arrive at the correct answer. And the law of large numbers says that the larger the number of people, the more the average approaches the correct answer, or the answer expected a priori, depending on the context. And in that sense, indeed, the law of large numbers gives a tool that is better than the opinion of an expert. But all that is only in cases where you have some access to the truth, and you can make mistakes because you are an amateur, and the mistakes are distributed as I described before around the mean symmetrically. Then you say: good, if all the people err in that kind of way, then the law of large numbers works. But in theoretical questions or questions of thought—not questions of estimating reality, where you can speak about errors distributed in one way or another, but questions where you have to use your head and think about what is true and what is not true—this is not a matter of distributions of errors or number estimates, like how many leaves are on a tree or how many cows are grazing in the meadow, but some theoretical question or another. In that case, I think that what the majority says is usually mistaken. Unless proven otherwise. Therefore the wisdom of crowds is also not a tool I hold in especially high regard. In any event, all these arguments and things basically say that before I approach solving a problem and using probabilistic tools or halakhic tools to resolve doubt, I first have to pass through the stage where I declare this situation to be a state of doubt. In other words, the decision that this situation is a state of doubt is a very important decision, and very often our mistake falls exactly there. Because the situation is not even a state of doubt, and therefore there is no point discussing probabilities and questions of whether we follow the majority or not, because we are not in a state of doubt at all. If someone tells me, look, I rolled a die and the result was even. Fine? Now someone says to me, look, I think it came up five. Why? Because the probability is one-sixth; five is no worse than any other result. If I know that an even result came up, then I do not care that the theoretical probability on such a die of getting five is one-sixth. Because I have information that tells me those distributions are irrelevant. When I am in a state where I have information, I am not in doubt. If I have partial information, I am in partial doubt. Partial doubt too removes some of the possibilities from the discussion—all the odd results. If I know it came up two, then that removes all the possibilities from the doubt because I know the truth. The result is two. So there is no point saying maybe it came up three, after all the probability of two is one-sixth and of three is one-sixth—who says it’s two? What tells me is my eyes; I saw it come up two. So don’t talk to me about probability questions. If I know, then I am not in a state of doubt. If I am in a state of doubt between the possibilities, then among the doubtful ones I can use probabilistic tools. But if I am not in doubt, or if the possibility is not one of the possibilities among which I am doubtful, then there is no point talking about probabilistic tools. Exactly as in Jewish law, so too in probability. We will talk about conditional probabilities; conditional probabilities are basically just that. A conditional probability means that the probability has to be calculated in light of the information that I know, and that information must be taken into account. Probabilities that do not take that information into account lead to error, lead to mistake. And that is exactly the idea. When you use conditional probability, you are basically saying: I am doing the probabilistic calculation on the basis of partial information I already have. That information rules out some of the possibilities but leaves others. Among the possibilities that remain, I try to do a statistical calculation to see which is more likely or which is true. But I will not use the general distribution over all the possibilities, because I have removed some of them; I have partial information. If I have complete information, then of course I’ve removed all of them. In Rabbi’s case, he had complete information; he knew that he was the son of his parents. So the fact that someone comes and says no, no, no, you are my son—that doesn’t interest me. Yes, there is such a possibility, there are illegitimate children, I don’t know, maybe half a percent of people are illegitimate, I’m just throwing out a number—so maybe that’s the case? How can you rule it out? The answer is: I can rule it out because I have other information, and I do not take those probabilities into account. In other words, all these things lead to the same point: that you have to set up the alternatives properly, decide that I am in doubt, and decide that I am in doubt between these possibilities and not others. And only then does our discussion begin as to how to resolve the doubt. Probability, or other halakhic tools, or other conceptual tools—that is already the final stage. But it is very important to set up the doubt correctly. First, to decide that there is doubt, and second, to decide what the possibilities are between which I am in doubt, and which possibilities are not. Because there are also possibilities—even if I am in a state of doubt—regarding which I am not in doubt. Therefore it is very important to define the possibilities before I start making calculations here. Now, when I speak about defining the possibilities—and that is why I said today I would speak about defining the state of doubt itself, not yet how to resolve it—when I define the possibilities themselves, there are of course two basic logical tools I have to take into account. There is the law of contradiction and the law of the excluded middle. As a foundation—they usually do not cover the whole issue, but they are the necessary infrastructure. The law of the excluded middle says that either X or not-X; there is no third possibility. Okay? So once you have two such possibilities, don’t look for a third one; there isn’t one. And the law of contradiction says that both X and not-X cannot be true at the same time. In other words, you cannot hold both such possibilities simultaneously. But in the probabilistic world you can, because in the probabilistic world you are often in doubt between X and not-X and you want to know which of them is true. So you are not claiming that both are true together; you are claiming that both stand before you awaiting decision. That is fine; there is no problem with that. On the contrary, very often that is the situation—you are deciding between contradictory possibilities. Therefore, when you define a situation as one of doubt, you should not be put off by the fact that the doubt is between contradictory possibilities. On the contrary, very often, most of the time, the doubt is between contradictory possibilities. I cannot say both X and not-X, but I certainly can say that I am in doubt whether X or not-X, two possibilities stand before me and I now need to decide which of them is correct. But it is true that if both stand before me, there is no third standing before me. In that sense, the law of the excluded middle does indeed speak also about states of doubt. The law of contradiction speaks only about the result of the calculation, not about the state of doubt itself. As for the state of doubt itself, two contradictory possibilities can definitely stand before me. But the law of the excluded middle does speak about the state of doubt itself. There are not three possibilities standing before me—either X or not-X, there is no third possibility. Whoever knows hypothesis testing in statistics knows that hypothesis testing always sets up a null hypothesis and an alternative hypothesis. And the relation between the null hypothesis and the alternative hypothesis must always be one of contradiction. Because if the relation is not one of contradiction, then who says that the null and alternative hypotheses are the only possibilities? Maybe there is also a second alternative, a third alternative, a fourth alternative. And therefore, the fact that you refuted the null hypothesis does not mean that you proved the alternative hypothesis, and vice versa. Because there may be other possibilities. Therefore in statistics—and this is a very common mistake in statistics—when you set up two possibilities, you have to set up two possibilities that exclude one another. Meaning, these are the only two possibilities there are. Either this or that; there is a contradiction between them, and there cannot be a third. These two rules must both hold. And then you can start doing hypothesis testing—that is a whole doctrine in statistics, how to test hypotheses with all kinds of probabilistic and statistical calculations and so on. But setting up the possibilities—and anyone who has dealt a bit with statistics knows that very often, in many cases, the failure lies in setting up the possibilities, not in calculating the probability that the null hypothesis is true or that the alternative is true, or the significance of this determination. Rather, the question is whether the null and alternative really are two possibilities that exclude one another. Are these two contradictory possibilities? Is there no third possibility? And here there can be a great many mistakes when we come to make probabilistic calculations or decide doubts. Therefore, when we discuss how to set up the possibilities before us when we need to make a decision between them, it is very important to take into account the law of contradiction and the law of the excluded middle. Now on this matter I want to make a comment—comments on the law of contradiction and comments on the law of the excluded middle. Regarding the law of contradiction: it is true that two contradictory possibilities cannot both be true together, but they can stand before me for decision as a state of doubt. I am in doubt between X and not-X. But not always what we see as a contradiction really is a contradiction. Even Maimonides in the Guide and the Rashba in a responsum following Maimonides distinguish between two types of contradictions. I’ll call them in more modern language physical contradictions and logical contradictions, or you could say physical impossibilities and logical impossibilities. What does that mean? If I say there is a contradiction between the law of gravity and the claim that a certain body with mass remains suspended in the air, I cannot accept both claims; they contradict one another. Does that rule out the possibility of miracles? The Holy One, blessed be He, can leave an object with mass hanging in the air. The answer is no. Why not? Because the contradiction is a physical contradiction. In other words, the laws of physics forbid this object from remaining in the air. But if the Holy One, blessed be He, knows how to suspend the laws of physics, or is not subject to the laws of physics, then He can leave the stone in the air. In other words, when I speak about a physical contradiction, I assume that the subject of the contradiction is subject to the laws of physics. But if something is involved here that is not subject to the laws of physics, then there is no contradiction. In a somewhat similar context, people often speak about human free choice, and one of the common arguments against the libertarian conception—the conception that man has free will—is the view that says: look, for every event either there is a cause or there is no cause. There is no third possibility. If there is a cause, then it is deterministic; if there is no cause, then it is random. Either way, it is not free choice. And again, regarding inanimate objects or animals or something like that, this is an argument I accept in principle, because the two possibilities are either randomness or causality. But with respect to human beings, I argue that this dichotomy is not correct, because our will can bring things about even if they have no cause, and that is not a random event. My claim is that the human will, like the Holy One, blessed be He, in the previous example, is an exceptional function within nature. In other words, other things in nature cannot perform this kind of action, but I as a libertarian think that human beings do have that capacity. Human beings can perform such an action. And again, you see that this dichotomy between whether there is a cause or there is no cause breaks down. It breaks down because the contradiction between these two things is context-dependent. In the context of animals or inanimate objects, it is true: either there is a cause or there is no cause. In the context of human beings too it is true: either there is a cause or there is no cause—but no-cause can be purposive and no-cause can be purposeless, and therefore there are three possibilities, not two. Therefore, when we speak about contradiction between two possibilities, contradiction is very often context-dependent. And this is even more prominent when we speak about the law of the excluded middle. When we speak about the law of the excluded middle, we are basically saying that either X is true or not-X is true, but there is no third possibility. Now it turns out that not infrequently there is a third possibility. And that brings me as well to situations—I gave lectures on this in the past—about states that look dichotomous, but when you take a second look you discover that the dichotomy is not all that convincing or not necessary. Often I’ll give examples: when they ask a religious person, tell me, are you Religious Zionist or Haredi? Supposedly they are putting before him a dichotomous dilemma: either you are here or you are there, there is no third possibility. If you are not Haredi, that means you are Religious Zionist, and vice versa. This example speaks to me very strongly, because many times people asked me that question—which of the two sides I belong to—and somehow I often felt that I belong to neither of them. But how can that be? There is the law of the excluded middle. In other words, either you are here or you are there; there is no third possibility. But no, there is a third possibility, even in situations like these. Or regarding evolution: do you believe in God or do you believe in evolution? Again, supposedly a dichotomous dilemma. And notice that both believers and neo-Darwinians share in this dichotomous perception of the dilemma. They both agree that it’s either this or that—only these choose one and those choose the other. And I claim that it is not either this or that; I reject the dichotomy. In other words, I think there is a third possibility. Therefore very often when they present us with a dilemma between two possibilities, there is some implicit assumption that these two possibilities are dichotomous. Dichotomous means, first, they contradict one another; second, they cover all the possibilities—there is no third possibility, a complete partition in the language of mathematicians. So the assumption that there is some dichotomous picture here is an assumption that needs to be examined carefully. And therefore, again, when I discuss here how I set before myself the possibilities between which I need to make a decision, by probabilistic tools or otherwise, I am still dealing with the question of how we set up the possibilities before us, how we know that we have set them up correctly, that there are no additional possibilities, or that there are superfluous possibilities, or things of that kind. In this case I’m talking about what I would call dissolving dichotomies, breaking dichotomies. What does that mean? There are several… I collected a toolbox for dissolving dichotomies because I encountered this need in very many contexts. Usually when I see a dispute, I find myself agreeing with neither side. When I agree with neither side, that basically means that apparently there is some third possibility that neither side noticed—or at least neither side agrees with; it doesn’t matter—but there is another possibility, although they are fighting with one another as if those are the only two possibilities. And then when I ask myself, wait, but how can that be? It’s either yes or no—what else could there be? Either you are Zionist or you are not Zionist. Is there another possibility? What else could there be? So I developed a kind of arsenal of tools, a toolbox for dissolving dichotomies. There is of course the first tool, and it is very relevant to us: the tool of vagueness. What does that mean? The sorites paradox, which I’m fond of. One pebble is not a heap. If there is a collection that is not a heap, the addition of one pebble will not change its status—that’s the second premise. And the third premise is that a thousand pebbles are a heap. Now each of these premises by itself sounds reasonable, but all three together cannot be accepted; they contradict one another. Right? If one pebble is not a heap, and adding one pebble does not change the status, then two are also not a heap. But if two are not a heap, then adding a pebble to make three also doesn’t make a heap, and not four, and not five, and not six—so then a thousand also isn’t. So how is a thousand yes? In other words, all three of these premises each by itself sound reasonable to us, but all three together contain an internal contradiction. And there is an entire family of such paradoxes—the bald man paradox. Someone who has one hair on his head is bald. Someone who is bald and gets one more hair is still bald; that does not change his status. But someone who has ten thousand hairs on his head is not bald. Again, these are three premises each of which sounds reasonable, but all three together contain a contradiction. You cannot accept all three together. And the solution to all these paradoxes—and there is an infinite family of such paradoxes—is that concepts like “bald” or “heap” are not binary concepts. In other words, they are not concepts in which either you are a heap or you are not a heap. Rather, there are different degrees of heap-ness. Meaning, there is a pile of stones whose degree of heap-ness is 0.01. And there is another pile whose degree of heap-ness is 0.2, and 0.4, and 0.7, 0.93, up to 1, which is completely a heap. Okay? So if I speak in that language, in the language of different degrees of heap-ness, then you see that this dichotomy breaks down, the one that says either you are a heap or you are not a heap. No—because there is “I am a bit of a heap,” “I am quite a heap,” “I am very much a heap,” and things like that. I refuse to enter that yes-or-no language. And very often problems or contradictions that are presented to us, dichotomies for decision that are presented to us, suffer from assuming binary thinking. In philosophy they call all the dilemma arguments in the world “dilemma arguments.” There is no point in training in sports. Why? Because if a person has talent, then he doesn’t need training—he has talent. And if he has no talent, all the training in the world won’t help—he has no talent. Here training means competitive sports. So what is wrong with that argument? Its dichotomy. There are very many people who have talent, but training can still improve their achievements further. There are people for whom it really won’t. Let’s say I assume that even if I trained a lot for marathon running, my achievements in that area would not be especially impressive—let’s say at least after the first two hundred meters. So the claim is that the fact that I have talent does not mean I’m already at the top, such that training cannot improve the situation. In short, “has talent” or “has no talent” is not dichotomous; there are different levels of talent. Or: there is no point in giving an exam, because lazy people don’t study even with an exam, and diligent people study even without one, so what’s the point of giving an exam? Again, the same issue: there are different levels of diligence. At intermediate levels of diligence there is definitely a point in giving an exam, because it will motivate people with that level of diligence to study more. People with optimal diligence will indeed study even without the exam, and the completely lazy will not study even with the exam. But the world is not made up only of perfectly diligent people and perfectly lazy people; it is made up of the whole range of levels of diligence in between. Therefore all arguments of that type are dilemma arguments; they assume some dichotomy in the background that it is either this or that. And this consideration shows us that very often the concepts are not binary concepts. The concepts in everyday language are vague concepts. Vague means they are not zero or one, not black or white; there are also many shades of gray. Fifty shades of gray. Therefore the dichotomy does not exist to begin with. The dichotomy presented to us is simply incorrect. We’ll talk more about fuzzy logic and statistics and so on—I’ll come back to this example—but I’m bringing it here to show that now if someone asks me: okay, now I need to decide whether this person is diligent or not diligent. I have a statistical question, let’s check what he did, see his achievements, and decide whether he is diligent or not diligent. These are questions I’m asked all the time, and I try to explain to people: look, the question is not well-defined. What does “diligent” mean? There are different levels of diligence. The question is not well-defined—what do you mean, diligent or not diligent? And so too with almost every everyday concept in our language, as opposed to mathematical concepts. The concepts are not binary, and therefore you cannot ask whether war is preferable to—I don’t know what—or whether some agreement is good or not good. What does good or not good mean? There are considerations in its favor and considerations against it; in the end you need to weigh them somehow and form a position. But good and not good is a very bad way to ask the question. When you ask the question in a dichotomous form, you are cooking up, or creating for yourself, a superficial answer, an incorrect answer, an answer that misses nuances. In other words, when you set up the possibilities before you for decision, you need to be aware of the whole range of possibilities and not set up only two when there are more than two. That is with respect to the law of the excluded middle. When they set before you two possibilities—either he is diligent or he is not diligent, there is no third possibility—that is simply not correct. There are many third possibilities, different levels of diligence between fully diligent and not diligent at all. There are different levels. Another method is the method of “who says I even have a sister?” Meaning, when someone asks me whether my sister is a prostitute or not, I can say: I don’t have a sister. I don’t have to enter the dichotomy of whether she is or is not. Or someone asks me: is the current king of France bald or not bald? Look in the group of bald people, you won’t find him; look in the group of hairy people, you also won’t find him. Aside from the fact that hairiness comes in degrees—there are different levels of hairiness—but even suppose I define a binary level. Why won’t you find him? Because France has no king. So very often the dichotomy put before me assumes some assumption that I can attack. The moment I attack the assumption shared by both possibilities, suddenly other possibilities are born and the dichotomy breaks down. When people ask me: are you Religious Zionist or Haredi? I say: I’m not. The answer is that there is another possibility in the background, another nuance that they are not putting on the table. For example, what Avrum Burg called the hyphen. Burg said that the main thing in Religious Zionism is neither the Zionism nor the religiosity but the hyphen between them. And I think that is very true. Religious Zionism is an outlook that says my Zionism is religious. It’s not that I’m Zionist and religious. I’m Religious-Zionist, meaning my Zionism is religious and my religiosity is Zionist. They perceive it as one fabric. By contrast, I can say: I’m Zionist and I’m religious, but there is no hyphen between them. My Zionism is not religious; I’m Zionist like Ben-Gurion. Like the joke about the rabbi of Ponevezh, who did not say Hallel and did not say supplication prayers on Independence Day. So his students asked him: tell us, if you are Religious Zionist, then say Hallel; if you are Haredi, then say supplication prayers—how can it be that you say neither Hallel nor supplication prayers? So he answered: I’m Zionist like Ben-Gurion. Ben-Gurion also did not say Hallel and did not say supplication prayers on Independence Day, because he didn’t pray, of course. But beyond the joke, there is in my view a very serious statement here: the Zionism of the rabbi of Ponevezh—he was Zionist—but his Zionism was not religious. Therefore he did not think one should say Hallel in such a situation. I don’t agree with him on the matter, but that’s what he thought. Meaning, he thought that if it doesn’t have a religious dimension, then you don’t say Hallel over it. And he was still Zionist. He was a secular Zionist, like Ben-Gurion was a secular Zionist. That is what Leibowitz—I once defined myself this way, and then someone told me I stole it from Leibowitz; apparently I had seen it and forgotten. I said that I am Zionist, I am a religious person and a secular Zionist. My Zionism is secular, but I am both Zionist and religious. But there is no hyphen between the Zionism and the religiosity. I am a religious person and I am a Zionist person, but my Zionism is not religious and my religiosity is not Zionist. There is no connection between them. I am Zionist like Ben-Gurion—that is exactly the same thing. So you see that suddenly a third possibility is born here, although ostensibly both sides in the debate agree that there are only two possibilities. And therefore they constantly try to force me to say: are you with us or against us, are you on this side of the equation or on that side of the equation? I’m on neither side of the equation, because I do not recognize this equation as a dichotomy. There is a third possibility, and perhaps even more than that. And the third possibility is always created when you attack the assumption shared by both sides. In this case of Zionism, what is the shared assumption? That there must be some religious dimension to Zionism and to the state. Those who think the religious dimension is positive are Religious Zionists with a hyphen; those who think the dimension is negative are anti-Zionist Haredim. But I claim no—there can be things that are secular, that have no religious significance, neither positive nor negative. They belong to the sphere of the secular. You see that by that I attacked an assumption shared by both sides in this dichotomy. And in exactly this way one can produce third and fourth possibilities and so on by attacking a hidden assumption shared by both sides, and that assumption is what creates the dichotomy. And there is always such an assumption. When a dichotomy is presented before you—say, a bird is not the opposite of a table. Why not? Because there is nothing shared by a bird and a table. But a bird, in a certain sense, is the opposite of a land animal, because both are animals—this one is on the ground and that one is in the air. So you can say that a bird is in some sense the opposite of a land animal. There has to be something shared by the two things in order for them to be opposites of one another. They need to belong to the same framework, the same conceptual framework, intellectual framework. And the moment you attack the shared framework, you break the dichotomy. You discover that there are additional possibilities. A bird is not the opposite of a table. Why? Either it’s in the air or it’s on the ground? No, because it’s not an animal. An animal in the air is a bird; an animal on the ground is a land animal. But a chair is not an animal—it is not in the game. Like the current king of France: there is no such king, so there is no point saying either he is hairy or he is bald. No, there is no such king at all. So very often breaking the dichotomy comes by attacking the assumption that constructs the dichotomy. And that assumption is generally shared by the two opposing sides. They fight one another over opposite or contradictory conceptions, but they assume in the background something shared, and that is what creates the dichotomy. When you attack that, suddenly additional possibilities are revealed.

[Speaker D] You tend to say that within the framework of, say, bringing people into a prayer quorum, you exclude Reform Jews. You say that from your perspective they can’t come and be part of the quorum. And I’m saying that if we go your way, then it doesn’t have to be a dichotomy between faith or religion. Rather, a person can come and say to me: I take some parts from Reform, I take some parts from religiosity, and you can’t really remove him from the picture. Right? He takes a little from everything.

[Rabbi Michael Abraham] Right, and that’s exactly why I really don’t say that Reform Jews don’t join a prayer quorum. What I do say is that a Reform conception, on the essential level, does not join a prayer quorum. But a person—a person is a complex creature. So with a person, the question is: you have to examine what he actually believes, and based on that decide. If this person believes in God, then in my opinion he joins a prayer quorum even if he is not committed to Jewish law. He joins a prayer quorum, because he believes in God and his prayer has meaning. But a person who does not believe in God, or at least not in a God toward whom we have some kind of obligation, or to whom prayer has meaning—then he’s a potted plant. There’s no point including him in a prayer quorum. So that’s why, many times when I talk about these things, I say: when I classify these kinds of categories, I’m classifying arguments, not people. People can hold all kinds of arguments that belong to all kinds of categories. It’s not right to classify people in a dichotomous way. I completely accept that comment. I’ve even said it myself more than once. I think it’s absolutely correct. Okay, that’s it. If there’s another comment or question? All right then, Sabbath peace, and may we hear good news.

[Speaker B] Sabbath peace,

[Rabbi Michael Abraham] Goodbye.