Doubt and Probability—in Halakha, Thought, and in General—Lesson 21—Rabbi Michael Abraham
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- Majority that is before us and majority that is not before us
- Parallel weaknesses and the difficulty of a categorical hierarchy
- Statistics as two steps and the distinction between “distribution” and the way it is obtained
- Court majority, Sefer HaChinukh, and Rabbi Shimon Shkop
- Objection to idealizing the minority and clarifying the concept of rationality
- A probabilistic model for justifying majority rule and a priori assumptions
- Conditional probability and the confusion between directions of the question
- The lecture schedule and continuation after Sukkot
- Repentance, why a penitent is preferable to a completely righteous person, and the value of the process
Summary
General Overview
The speaker summarizes the distinction between a majority that is before us and a majority that is not before us, and shows that both types have different weaknesses, so there is no way to determine categorically which is stronger in every situation. He formulates the use of majority as a two-step process: determining a distribution and then applying it to a particular case, and argues that the basic dispute is not whether “there is statistics” but how one arrives at the distribution: by generalizing from a sample or by a priori reasoning. From there he explains why the majority of a religious court belongs to the category of a majority that is before us, and develops a probabilistic model that illustrates how a priori reasoning can justify preferring the majority. Later he points to a common confusion between conditional probabilities in different directions (P of B given A versus P of A given B), demonstrates this through statistical errors in law and medicine, and concludes with words of encouragement for the Days of Repentance, in which the value of repentance lies in the process itself and not only in the result.
Majority that is before us and majority that is not before us
The speaker presents two sides among the medieval authorities (Rishonim) on the question of which is stronger, where the plain sense of the Talmud seems to lean toward a majority that is before us being stronger, while Nachmanides seems to lean toward a majority that is not before us being stronger. He defines a majority that is not before us as a majority based on generalization from a sample, scientific induction, and a majority that is before us as a majority based on a priori reasoning, such as the majority of stores. He connects the problem of generalizing from a sample to Hume’s problem of induction and to the dispute between actualism and informativism, and presents a principled difficulty in accepting scientific generalizations as the weak point of a majority that is not before us. He explains that a possible solution to the difficulty is the ability to “see the generalization” with the mind’s eye, as a result of cognition arising from interaction with the world and not only from internal thought, but he emphasizes that this cognition is not certain in the way sensory perception is, and therefore it is natural that there is more disagreement about it.
Parallel weaknesses and the difficulty of a categorical hierarchy
The speaker argues that the weakness of a majority that is not before us lies in the step of generalizing from the sample to the general rule, even though the sample itself is clear because it was observed. He argues that the weakness of a majority that is before us lies in determining the basic datum itself, such as the assumption that the chance that a piece of meat that was found came from any given store is equal, and from that comes the ninety-ten distribution, which is not the result of observation but of reasoning. He explains that statistics comes only after accepting the distribution, whereas the way of arriving at the distribution is logic of various kinds, and therefore it is hard to determine a priori which is stronger; it depends on the situation and on the strength of the reasoning or the generalization. He concludes that one cannot derive one from the other, because each has a side of strength and a side of weakness, and it is enough to show a weakness in each in order to reject an absolute hierarchy.
Statistics as two steps and the distinction between “distribution” and the way it is obtained
The speaker divides the use of statistics into two stages: determining the distribution and making a decision based on the distribution in the case before me. He emphasizes that in both kinds of majority one ultimately arrives at a numerical distribution, and therefore in a certain sense both can be called probability or statistics, but the difference is in the way one arrives at the distribution. He states that in a majority that is not before us the distribution is built from generalization from a sample, while in a majority that is before us the distribution is built from a priori reasoning, and only after there is a distribution do the statistical calculations begin. He interprets Rabbi Shimon Shkop’s statement that a majority that is before us “is not statistics” as a claim about the stage of building the distribution, not about the calculation stage after the distribution has already been determined.
Court majority, Sefer HaChinukh, and Rabbi Shimon Shkop
The speaker raises the question that comes from Sefer HaChinukh, which explains that one follows the majority in a religious court because usually the majority is correct, and emphasizes that on the face of it this sounds like a majority that is not before us, because it relies on the “majority of panels” over many cases. He explains that in practice it is impossible to gather a sample in court disputes in order to check empirically whether the majority was right, because there is no independent source of information that could determine in a particular case whether the majority or the minority got the truth right. He concludes that the majority in a religious court cannot be a majority that is not before us in the sense of generalization from a sample, and therefore he classifies it as a majority that is before us, where the distribution is built from a priori reasoning. He adds that the a priori reasoning applies directly to the case before me, so even if this were the first legal discussion ever, the ruling by majority would still be accepted on the strength of that same reasoning, without needing to go through the “class of cases” in history.
Objection to idealizing the minority and clarifying the concept of rationality
The speaker rejects the claim that the minority deserves extra weight because it is “brave” or “thinks outside the box,” and distinguishes between originality and correctness. He argues that a minority can also be completely foolish, so there is no rule that says the minority is right, and he brings the example of “flat-earthers” to show that courage against consensus is not a measure of truth. He clarifies that disagreement is not a contradiction to rationality, because rational is not equivalent to certain, and different conclusions can be reached even when the same facts are before everyone. He also adds an example of a paradox in capital cases, where a judge is not permitted to change his vote in order to achieve a desired final result, because his role is to say what he thinks and not to manage the outcome.
A probabilistic model for justifying majority rule and a priori assumptions
The speaker constructs a model in which a perfect judge gets the truth right one hundred percent of the time, and therefore there is no value in adding judges and there is no disagreement, whereas a “complete idiot judge” gets the truth right in half the cases, and therefore there is no advantage to two over one. He argues that the advantage of majority decision begins when the judges are “good but not perfect,” with a correctness rate between fifty and one hundred percent, and he demonstrates a calculation in which three judges of ninety-percent quality produce a reliability of 0.972 for the majority ruling. He emphasizes that the model rests on assumptions such as independence between the judges’ opinions and some knowledge about the “quality” of the judge, and these assumptions cannot be measured empirically, so this is a mathematical expression of a priori reasoning. He mentions an article by Nadav Shnerb in the book Keren Zavit, in the chapter “What Do We Gain from Adding More Fools,” and describes the argument there that one can sometimes also benefit from adding judges of lower quality, while qualifying that this is not always true.
Conditional probability and the confusion between directions of the question
The speaker presents a distinction between two different questions: given that the truth is A, what is the probability that the judge will rule B, as opposed to given that the judge ruled B, what is the probability that the truth is A. He illustrates the difference with the case of Munchausen syndrome by proxy, where it was claimed that the probability of two crib deaths is one in sixty-four million and therefore the mother probably murdered them, and he explains that this inference is mistaken, among other reasons because the events may be dependent and because the number of cases in the population is large, so even rare events will occur. He gives the example of a test for a rare disease with ninety-nine percent reliability and a prevalence of one in a million, and shows that a positive result can indicate a very low chance of disease because of the high rate of false positives in a large population. He argues that reliability calculations of one kind do not answer the question of the “quality of the result” in the opposite direction, and warns that switching the direction of the probability causes smart people and experts to make mistakes. He says he will return to the topic later, including the effect of prior indications that led to sending the patient for testing, and notes that this is especially important in medicine when patients panic over a positive test for a rare disease that was administered without justification.
The lecture schedule and continuation after Sukkot
The speaker concludes the theoretical part by saying that the next lecture will only be after Sukkot, because next Thursday night is the end of Yom Kippur. He promises to try to present the subject of conditional probability more clearly and even to write it up in a more efficient form for next time.
Repentance, why a penitent is preferable to a completely righteous person, and the value of the process
The speaker is asked for an idea for Yom Kippur and replies from the Talmud that a penitent is preferable to a completely righteous person, explaining that the meaning is that the value of repentance lies not only in the result but in the process itself. He formulates this as a distinction between the value of the function and the value of the derivative, and compares it to changes in a factory that refresh and help by the very dynamism, even if the final state is not “more efficient” than the initial state. He brings Zeno’s paradox and the concept of velocity at a point in time to emphasize that dynamism can be significant even when there is no change of state at the point itself. He interprets Maimonides’ words, “until the Knower of hidden things testifies of him that he will never return to this sin again,” as testimony about his state at the moment of repentance, even if later he returns and fails, and argues that a person’s purpose is to repent and improve, not to be perfect. He concludes with the blessing: May you be sealed for good, and have a good year.
Full Transcript
[Rabbi Michael Abraham] Okay, let’s get things a bit organized and see where we are. We talked about a majority that is before us and a majority that is not before us; we only discussed the last part. A majority that is before us and a majority that is not before us—we saw that there are two sides among the medieval authorities (Rishonim) as to which of them is stronger. The plain sense of the Talmud seems to be that a majority that is before us is stronger; in Nachmanides it seems that a majority that is not before us is stronger. I talked about the difference between them, and the claim was that a majority that is not before us is basically a majority based on generalization from a sample, scientific induction if you like, while a majority that is before us is a majority based on some kind of a priori reasoning, like the majority of stores. In the last few lectures we talked about the problematic nature of generalizing from a sample, because with generalizations from a sample, the problem Hume posed regarding scientific induction comes up. We saw the problem, and from it comes the dispute between actualism and informativism, and basically I tried through that to point to the difficulty of accepting scientific generalizations, which is really the weakness of a majority that is not before us. On the other hand, the claim was that we do trust science, and science is based precisely on generalizations of this kind, so my claim was basically that this is the problem of Kant’s synthetic a priori and so on. In the end the problem was—or not the problem, the solution to the problem was—that apparently we have some kind of ability to see the generalization, to see with the eyes of the mind, not with our ordinary eyes, to see the generalization. Meaning, this is a result of cognition and not of thought. But of course this is cognition not on the level of seeing with the eyes, and therefore even if it is cognition and not thought, that still doesn’t mean that the degree of certainty here is like the degree of certainty of something I see with my eyes. The fact is that there are far more arguments about generalizations and theories and things of that sort than about things I see directly. Therefore I didn’t mean to say, when I said that this is cognition and not thought, that it is certain like sensory cognition. All I meant to say was that this is the result of interaction with the world and not of a thinking process that takes place solely inside my mind, without interaction with the world, and therefore it’s no wonder that it also yields insights about the world. What?
[Speaker B] So this is basically—
[Rabbi Michael Abraham] The solution to the problem of the synthetic a priori. That’s roughly where we stand. Now I want you to notice where we’ve really arrived. We’ve arrived at the fact that we have two kinds of majority in Jewish law: a majority that is before us and a majority that is not before us. In each of them there is a weakness. In a majority that is not before us, the problem is the generalization. The sample itself is fairly clear, because I saw it; the examples I saw, I saw. But from those examples I generalize and arrive at a general law. So the problem in a majority that is not before us is in the step of generalization. By contrast, in a majority that is before us, the problem is in determining the basic datum itself. The basic datum itself is the result of speculation. Meaning, if I have ten stores, nine kosher stores and one non-kosher, I assume that the chance that the piece of meat I found is kosher is ninety percent. That assumption is not a generalization from a sample, and it is not the result of observation, and not anything of the sort; it is simply a priori reasoning. I assume that the chance of being lost from each of the stores is the same, and therefore I assume that the probability here is ninety percent. So here, the problem or weakness of a majority that is before us is not in the step of generalizing from the sample to the general rule, but in the basic determination itself—in saying that there is in fact a ninety percent chance for this piece of meat. And this is an important point, because I really think it’s hard to determine categorically which of the two types of majority is stronger.
[Speaker C] What’s the problem with the second one? It’s just simple statistics.
[Rabbi Michael Abraham] In a majority that is before us? Yes.
[Speaker C] Intuition, okay—
[Rabbi Michael Abraham] Intuition can be right and it can be wrong.
[Speaker C] What do you mean, probability could be wrong? Probability theory could be wrong?
[Rabbi Michael Abraham] No, this isn’t probability—that’s exactly the point. Probability is the result of the intuition. It’s not the basis for the intuition. Once I have the intuition that each store has equal weight in the loss of pieces of meat, then it comes out that the statistics is ninety percent. But that determination itself, of the distribution, is the result of reasoning. Just like in a majority that is before us—well, I checked—they are completely identical.
[Speaker C] All the stores are identical, there’s no difference at all.
[Rabbi Michael Abraham] And therefore you have a reason to think they are probably the same.
[Speaker C] No, I checked. I checked, I investigated.
[Rabbi Michael Abraham] You can’t check such a thing.
[Speaker C] I went to all the stores, ten stores, checked how they produce, what cuts they sell, they all look the same.
[Rabbi Michael Abraham] You can’t know. What’s the route to the store, how strong the knife is—you can never know.
[Speaker C] True, but obviously the more I check, the more—it’s obvious that the reasoning gets stronger. If I spend a hundred years checking, then it becomes much stronger.
[Rabbi Michael Abraham] And that’s exactly the last sentence I said: it’s really hard to determine categorically which of the two types of majority is stronger, because it depends on the situation. There are situations in which my a priori reasoning is very strong; there are situations in which it’s less strong. And the same goes for generalization. There are generalizations that seem more persuasive to us—again, never certain—but more persuasive; and there are generalizations that seem less persuasive. Therefore I’m pointing to a weakness in a majority that is before us and a weakness in a majority that is not before us. From there to some categorical determination of which one is stronger—the distance is great. So it seems to me that a more accurate way to present this is basically to say: there really is no way to determine a priori which is stronger. But it is true that you can’t derive one from the other. You can’t derive one from the other because each has aspects this way and that way, and therefore you cannot derive the present one from the absent one or the absent one from the present one. In other words, there is no way to derive. And for that it’s enough for me to show that there is a weak side here and a weak side there. It doesn’t have to be that the bottom line establishes a clear hierarchy that one of them, absolutely, after all the calculations, is stronger.
Now, another important point that arises from the picture I described—and this is what I basically also said to Shmuel a moment ago—notice that when we talk about using statistics, we are basically talking about two consecutive steps. The first step is to determine the distribution, and the second step is to make decisions on the basis of that distribution about a case that comes before me. Right? These are really two different things. Now, both in a majority that is before us and in a majority that is not before us, at the end of the day I arrive at a distribution. As for the piece of meat, there is a ninety percent chance that this piece of meat is from the kosher stores. So the distribution is ninety percent. Now I find a piece of meat, so I apply the distribution to the case before me and I make decisions. That’s the second stage. But I always have some statistical distribution that tells me what percentage things go one way and what percentage they go another way. That exists both in a majority that is before us and in a majority that is not before us. Therefore you can perhaps call both of them probability or statistics. The difference between them is the question of how I arrive at the distribution. In a majority that is not before us, I arrive at the distribution by generalizing from a sample. I observe a sample and within that I generalize. The generalization gives me the distribution. Once I have the distribution, a case comes before me and I apply the distribution to that case. In a majority that is before us, I also have a distribution—ninety percent. The only question is how I got there. And I got there not from generalization from a sample, but by a priori reasoning. Therefore, if you ask which of these two things is statistics, it’s hard to decide. In both cases there is a distribution in the end. The question of how to arrive at the distribution is what distinguishes a majority that is before us from one that is not before us. The difference is in how you arrive there, but the how-you-arrive-there is not statistics. The how-you-arrive-there is reasoning. Here I make a generalization, here I make a priori reasoning. Only after I use the reasoning, the generalizations, all these tools that I have, a distribution is created, and now I begin to work with statistics. Once there is a distribution, I can do statistical calculations. Statistics starts from that point. How I got to the distribution—that can happen in all sorts of ways that, I don’t think it’s correct to connect them specifically to statistics. Those are the data that build the statistics. And the difference between a majority that is before us and one that is not before us is found there. But after I’ve gone through those initial stages—this one that way and that one that way—I have a distribution, and now I begin to work with statistics. Once there is a distribution, I can make calculations. Statistics starts here.
Now that basically means that when I ask the question whether this is statistics or not statistics—yes, a question that the later authorities (Acharonim) also ask in one way or another, whether this is statistics or not statistics. Rabbi Shimon Shkop—we saw him, yes—he says that a majority that is before us is not statistics. What does it mean that it’s not statistics? It is statistics, after you’ve reached the conclusion that it’s ninety percent. The only question is how you got to ninety percent. In other words, the question whether something is statistical or not statistical, I think, usually refers to the final stage. Meaning, if I have a distribution, let’s do the calculation regarding the case before me and see what the chance is that the result is this or that. That’s statistics. But how did I arrive at the distribution? That’s never statistics, neither in a majority that is before us nor in one that is not before us. So the difference between them is only in the question whether this is science. A majority that is not before us is science. A majority that is before us is local reasoning; it’s not connected to science. I have a situation before me and I assess it through reasoning; I assess what distribution really describes this situation. Therefore, for example, with a majority that is before us, one of the two examples brought in the Talmud is stores—a piece of meat and the distribution of stores—and the second is majority in a religious court. And I said, and we already asked this, yes, what Rabbi Shimon Shkop asks on Sefer HaChinukh. Sefer HaChinukh says that—why do we follow the majority in a religious court? Because generally the majority is right. And then, if you look at it according to the simple definitions, it turns out that majority in a religious court is actually a majority that is not before us, because you’re looking at the majority of panels in which there was a dispute between two judges and one, and the claim is that in most cases the two were right and not the one. Okay? So you are actually talking about a majority that is not before us, the majority of all cases that existed in the world. The case before me is one particular case. Meaning, I am not following two judges; the majority being discussed here is not the majority of two judges against one, but rather the majority of panels out of all the panels in which there were two against one—in most of those panels, the two were correct. In a minority of those panels, the one was correct. So the majority and minority here are the majority of panels versus the minority of panels, not the two judges in this panel versus the one in this panel. Therefore this is basically a majority that is not before us. And that’s Rabbi Shimon Shkop’s question.
And the claim I basically explained—I’m just reviewing it to anchor it in what we saw—the claim that in a religious court this is a majority that is before us, I explained by elimination. The claim was that in a majority that is not before us, I accumulate data on a sample and then make a generalization to the general group. In the case of courts, I have no way to accumulate sample data, because I have no way at all of knowing in a particular case whether the majority was right or the minority was right. There are two judges who say Reuven is liable and one says Reuven is exempt. Fine. How do I know whether Reuven really was liable in that case? I have no way. The evidence that was before the judges is also before me. So I can think about the case, but I have no independent source of information outside the court against which I can check the result that came out of the court and ask whether it hit the truth or not. Therefore, when I now examine all the cases that took place in split panels where two judges disagreed with one judge, and I now want to examine a sample—let’s take a sample of, say, fifty such cases, and see how the results are distributed in those fifty cases: in how many of the cases the two were right and in how many the one was right—I have no way to do that, because I have no way at all to determine in a given case that I’m looking at whether the two were right or the two were wrong. I have no way to do that. And since that is the case, majority in a court cannot be a majority that is not before us, even though the majority being referred to—as Sefer HaChinukh says—is not the majority of judges but the majority of panels. So by that definition it should really have been a majority that is not before us. But it does not satisfy the logic of a majority that is not before us, because there is no generalization here based on a sample. I have no way to find the sample. And therefore, therefore—so what is it based on? So it’s a majority that is before us, right? Since it isn’t a majority that is not before us, it’s a majority that is before us. But why is it a majority that is before us? In what way is it similar to pieces of meat? It’s true that in both cases I cannot accumulate data on a sample and then make a generalization. That is true both for pieces of meat and for court majority. Therefore it isn’t a majority that is not before us. But what makes them similar? Why do they both belong to the same category of majority that is before us? And the answer is that in both cases I am basically constructing the distribution—the ninety percent, say, or whatever it may be—on the basis of a priori reasoning. And every place where I build the distribution on the basis of a priori reasoning, that is called a majority that is before us. And therefore it also isn’t science, because a priori reasoning is just reasoning. There is no observational dimension here. Observation plays no part in this matter. So basically what we have here is some kind of a priori reasoning.
[Speaker C] Yes. Maybe you could say that in a religious court, when there’s a majority, we’re not looking for the truth. There’s no objective truth at all; the truth is what the majority says, what the victorious narrative of the people in power says. And since a religious court, you could say, is political—
[Rabbi Michael Abraham] But that’s not true. We are looking for the truth. What do you mean?
[Speaker C] No, maybe in a religious court we’re not looking for the truth because they know there’s no truth, but rather once the majority—
[Rabbi Michael Abraham] Why is there no truth? Of course there is truth. Either he borrowed—
[Speaker C] Or he didn’t borrow, what do you mean?
[Rabbi Michael Abraham] Go to the court of your own generation—they tell you, go to the court of your own days. Obviously, go to the court of your own days—
[Speaker C] That still doesn’t mean there is no truth; it means—
[Rabbi Michael Abraham] It means that you have to obey the court of your own generation, and that’s the truth at that time. It could be that in another court there would be a different truth, because—because—but again that’s not relevant. According to—
[Speaker C] By that logic, even a majority that is not before us has no truth. You follow the majority because that’s what the majority determines, and the law says to follow the majority. No, no, that’s something else. The majority is an indicator of the question of what the truth is. I’m speaking right now specifically about a religious court—about when we are looking for what the law says.
[Rabbi Michael Abraham] A religious court is no different from any other majority. You can take that same logic and say: why are most women not incapable of bearing children? Because I’m not interested in the truth; after all, Jewish law says follow the majority, so therefore one hundred percent of women are incapable of bearing children. No. When I ask, for example—well, I ask whether slavery is permitted or forbidden—the correctness of my ruling does not depend on whether the woman before me is in fact incapable of bearing children or not, but on what the law says. The law says to follow the majority, so I am one hundred percent right.
[Speaker C] No, what I’m saying is, I ask myself whether slavery is permitted or forbidden. So at the level of Abraham our forefather, if we were to gather all the courts of the Abrahams and they would say that this—since what determines it is what that—
[Rabbi Michael Abraham] What does that have to do with statistics? No, it’s not relevant to us. The question of whether values—whether there are objective values and whether values change over the generations—we’re not going back to that argument again. Why is that—
[Speaker C] Fine, no, since the court comes to determine the binding value in light of the specific case.
[Rabbi Michael Abraham] The court determines no value. In most cases the court has to determine facts, not value. Did Reuven borrow or not borrow? Did he repay or not repay? A factual question. So in the end, what these cases that are called a majority that is before us have in common is that at the end of the day there is a distribution. There is some distribution—ninety percent, eighty percent, whatever it may be. The way I arrived at that distribution is a priori reasoning. I have no way to accumulate information on a sample and make a generalization and produce the general law. I produce the general law directly through reasoning. That is called a majority that is before us.
Now, regarding the stores, it’s fairly clear to us what the reasoning actually says. The reasoning says that if I have no other specific datum, then the chance that it came from each store is equal, and if there are nine kosher stores and one non-kosher, then the distribution is ninety-ten. Right? That’s basically the reasoning. I have no way to measure this, I have no way to check it through a sample, but it is reasoning—very sensible reasoning—and therefore I don’t agree with Rabbi Shimon Shkop that it isn’t logical. I do agree with him that it cannot be measured scientifically, it can’t be checked, but it is logical. How is the reasoning constructed regarding a religious court? Why do I determine in a religious court—so people say, because in most cases the majority was right against the minority. Let’s say a hundred cases where there were two judges against one; I assume that in most of those hundred cases the two were right and not the one. But as I said before, that’s nonsense. I have no way of knowing that in most cases the majority was right. Meaning, if I knew something to say about the general statistics and I said, the case before me is one of the cases in that group, I know the distribution in the group, so I apply the group’s distribution to the case before me. But I don’t know the group either—just as I don’t know the case before me, I also don’t know the group. So how did I decide that really, in most cases too, the majority was right? Even for a majority that is not before us, the question is how you decided. How do I know that in most cases where there were two against one judges throughout history, in different places and times, how do I know that in most cases the two really were right and not the one? It’s just reasoning. But if it’s reasoning, then let’s come back to the court before me—forget all the panels that ever were. The court before me has two judges against one, and the reasoning says that most likely the two are right and not the one. I no longer need to get to that general group of all cases in which there were proceedings with two judges against one, because I don’t have any more information about the general group than I do about the case before me. Just like with pieces of meat in stores: I can’t say that most pieces that were lost came from the kosher stores, therefore this piece probably also came from the kosher store. That’s nonsense. Because how do I know that most pieces came from the kosher stores? From a priori reasoning. And that same a priori reasoning applies also to this specific piece. I don’t have to go through all pieces and then apply it to the piece before me. I can say it directly about the piece before me, because it’s the same reasoning itself that speaks about all pieces. There is no difference at all. I don’t have some statistical information telling me something about the group as a whole. Therefore, in the formulation of Sefer HaChinukh—and here I close the circle—therefore in Sefer HaChinukh’s formulation, I said that this is not a majority that is not before us, because although it is true that I am relying on the majority of panels that ever existed, I have no way of making a generalization in order to arrive at the majority of panels.
Now I’ll go one step further. Since I have no way of making that generalization, then how did I arrive at it? Through a priori reasoning. But that very same a priori reasoning tells me the same thing about the case before me. I don’t need to go through all the cases in which there was a dispute between two judges and one. Let’s say there had never been any legal proceedings at all. This is the first legal proceeding ever, and now it stands before me. And there are two judges against one. Will I decide that the two are right and not the one? The answer is certainly yes. Why? I still don’t have any statistics or information about the general group of proceedings in which there were disputes between two judges and one. I don’t need it. Because the information I have about all cases is nothing more than the result of a priori reasoning that I have about each one of them, and in particular also about the case before me. I don’t need to go through the general group and then apply it to the case before me. The reasoning speaks directly to the case before me. The same thing I say about all cases—that the two are usually right and not the one—that also applies to the case before me. If there are two judges against one here, my reasoning says that most likely the two are right, about this case, even if there were no other cases at all. So—
[Speaker C] So what’s the problem?
[Rabbi Michael Abraham] There’s no problem at all. I’m only saying that the resemblance to a majority that is not before us now becomes even more remote. I’m simply explaining even more why, and even better resolving Sefer HaChinukh against Rabbi Shimon Shkop’s question. Before, I said: it’s true that we refer to the general set of panels, but I had no way of arriving at information about the general set of panels by generalization from a sample, because I have no way of finding the relevant sample. Now I’m saying: forget it—if that is really so, then I don’t need to refer to all the cases at all. What is the relevance of all cases? I’m speaking about this particular case. In this particular case, if there are two against one, the reasoning gives that the two are right. In most cases the two—in all likelihood the two are right. Rabbi, Rabbi—yes?
[Speaker C] Rabbi, every time I react against this, because specifically with panels it would seem that there should be more weight given to the lone opinion. Because again, that first one stated some position out of the three, or out of the seventy-one. Then the second one already heard the first and says the same thing. The tenth, the fifteenth says the same thing as the previous ones. Then one person comes along and says: I heard all of you and it doesn’t seem right to me. It doesn’t seem right to me and I’m thinking independently. I would think that intuitively the Rabbi would actually take him more seriously. No?
[Rabbi Michael Abraham] He gives his arguments, they give their arguments. He didn’t persuade them any more than they persuaded him. So now I’m left with fourteen against one.
[Speaker C] No, but who was braver? Who had to think outside the box?
[Rabbi Michael Abraham] What do you mean, braver? Thinking outside the box? So what? Outside the box doesn’t mean it’s right. Outside the box means, at most, that he’s original.
[Speaker C] Original and correct are not—
[Rabbi Michael Abraham] Synonyms.
[Speaker C] Again, everyone heard everyone’s arguments before deciding. Nobody has some factual novelty about reality. There was a value judgment, only a value judgment, and therefore—no—
[Rabbi Michael Abraham] A value judgment? No, no, no.
[Speaker C] We said that earlier, no.
[Rabbi Michael Abraham] First of all, in most cases the discussion is factual, not evaluative.
[Speaker C] But everyone knows all the facts, and everyone knows all the facts. Right. No one persuaded any—
[Rabbi Michael Abraham] Anyone. They reached different conclusions, but as for the weight of the evidence, they have to think independently. So to decide in that way is somehow irrational? Is it irrational? It’s completely rational.
[Speaker C] Then how can there be a dispute if all the facts are before everyone?
[Rabbi Michael Abraham] We think—
[Speaker C] Rationally, so apparently we define the concept of rational differently.
[Rabbi Michael Abraham] You define rational as certain. I don’t. I define rational as reasonable. So why is one person’s reasoning different from another’s? He has different reasoning.
[Speaker C] So this is something beyond rational, already in the realm of intuition.
[Rabbi Michael Abraham] These are just words. Call rational whatever you want—it’s words. So not rational, yes rational—bottom line, the majority is more likely to be right. That’s all. It doesn’t matter whether it’s rational-majority or not.
[Speaker C] I’m saying, here my explanation helps a bit, because we’re really not looking for the factual truth.
[Rabbi Michael Abraham] No, no, no, we are looking for the truth. If the question is whether Reuven murdered, if we’re looking at whether Reuven borrowed, yes? Definitely yes. We’re absolutely looking for the truth, and usually it’s factual truth too—not always, but in many cases, in most cases. We have no direct access to it; we have indications toward it, and one has to decide in light of the indications what it was. Right, that’s what happens in a religious court.
[Speaker D] What about the fact—
[Rabbi Michael Abraham] That basically—
[Speaker D] That the wise are usually a minority?
[Rabbi Michael Abraham] So specifically—fine—I already wrote a column about that, I don’t know if you read it. There’s a note there at the end of the column, worth reading. I added it afterward in a different color. There I explain. What’s the name of the column? There are few wise people, but there are also few fools. Extremely foolish people are few, and extremely wise people are few. So “the few”—it’s not true that the few are right. That’s a joke of mine, it’s not—
[Speaker D] What do you mean? If they had set up in the Sanhedrin, for example, if the Sanhedrin had been debating science and they had put Einstein’s theory against others and argued about it, then apparently his opinion would have been rejected because he would have been in the minority.
[Rabbi Michael Abraham] By the same token, you could have put forward a completely foolish theory and it too would have been rejected.
[Speaker D] Yes, that’s true.
[Rabbi Michael Abraham] So would it also have been correct? Just because only a few hold it?
[Speaker D] True, definitely, but specifically the majority is the—
[Rabbi Michael Abraham] No, no, what you’re saying is correct, and then there is no “but.” No, there’s no “but.” Correct, without a but. A completely foolish theory that the minority supports, a completely intelligent theory that the minority supports. Now tell me, what’s the rule? Is the minority always right? No. That’s it. Okay, anyway, so the claim is that… But I argued something a bit different. I said: what happens in a place where you have only half of the Gaussian? Just as an aside I’m adding this. What happens in a place where there’s only half of the Gaussian? Meaning, say, think for example about faculty members in academia, okay? Faculty members in academia are not all great geniuses, as is well known, but I think there are few absolute idiots there. Let’s talk about reasonable departments, not gender studies and things like that. Let’s talk about departments that deal with things with some substance. So there are no complete fools there, okay? So basically what you have there is only half of the Gaussian of the population. In that case there’s more room for the idea that the minority is right. You can argue about that too, but that’s already more complicated.
Okay, anyway, how do we really arrive at this—what is this reasoning by virtue of which we reach the conclusion that if there’s a dispute between two judges and one, then the two are right and not the one? So here there’s actually a kind of statistical calculation, but I’m describing it so you’ll see what the statistics are here and what the assumptions really are. So think, for example, we have three judges, and two say Reuven is liable and one says Reuven is exempt, okay? Now I ask myself: how can I measure the advantage of the two over the one? So let’s take a model as an example. Fine? Suppose that if I have a judge who is a perfect judge, he hits the truth in all cases. Okay? In that case, a two-against-one situation simply won’t happen. Right? Because when there are two against one, it’s obvious that either the two were wrong or the one was wrong. A judge who never errs—no matter how many judges you put there, a two-against-one situation will never happen. Therefore, with a judge who never errs, it’s enough to appoint a single judge; there’s no value in adding more judges to the panel. That judge will deliver a true judgment in the fullest sense even without any addition beyond him. That’s the perfect judge.
And the perfect idiot judge—what is the perfect idiot? Yes, the judge with zero ability. A judge with zero ability is a judge who hits the truth in half the cases and misses it in half the cases. It’s not a judge who hits the truth in 0% of the cases, but one who gets it right in half, because that means it’s a random draw; there is no correlation between the truth and what the judge says. When there’s no correlation, that means that in half the cases he is right. And a complete idiot judge is right in half the cases. If the judge manages to be right in less than half the cases, then he’s such a huge idiot that this is actually worse than flipping a coin. And of course there is value here: once you know that this is the judge, that actually helps, because then you always listen to what he says and do the opposite, and then you have a pretty good chance of being right. Yes, that’s the Sema in section 3, that “the opinion of laymen is the opposite of the Torah view.” Meaning, what Sheinfeld, the Haredi writer, when he came to the Brisker Rav, and the Brisker Rav said to him, tell me, how do you always… So that’s more or less the situation.
In any case, that’s the complete idiot judge and the perfect judge. With a perfect judge, a single judge will always hit the truth; there’s absolutely no need to add judges to him, right? And there will also never be a disagreement among the judges. With a perfect idiot judge there will be disagreements among the judges—on the contrary, it’s very likely there will be, because he’s making fifty-fifty random choices, right? So some judges will say yes, some judges will say no, but there is no advantage of two over one. The chance that the two are right and the chance that the one is right are half and half. It’s still fifty-fifty. There’s no advantage of two over one.
When does the game begin? The game begins when the judge is a good judge but not perfect. What does that mean? That he is right in some percentage between fifty and one hundred percent. Okay? Say sixty percent, something like that. Now the game begins. Why? Because if I take one judge, in sixty percent of the cases he’ll say the truth and in forty percent he won’t. What happens if I add two more judges and then I decide to follow the majority? In that situation, it could be that my chances of being right improve—by the way, not always—but there is a chance. You can do the calculation; I don’t want to get into statistical calculations here, but in such a situation there is a chance that adding judges improves the situation.
If, for example, let’s say I have three judges whose quality is ninety percent. In ninety percent of the cases they say what really happened—yes, they’re right. So now if I have one judge, then in ten percent of the cases he is wrong, and in ninety percent of the cases he is right. What happens if I have three judges? With three judges, what is the chance that the panel will issue a correct ruling? That’s one question. What is the chance that the panel will issue a correct ruling? You can do calculations, right? What is the chance that all three will be wrong? One thousandth, right? One tenth cubed is one thousandth. What is the chance that two are right and one is wrong? Sorry, that two are wrong and one is right? Because that too is an error of the court, since we follow the majority. In short, if I have three judges, in how many cases will the judges be wrong? In how many cases? And each judge is of ninety-percent quality. And I’m asking: in how many cases will this court be wrong? When will this court be wrong? Either when all three judges were wrong, or when two were wrong and one was right. Right? In all other cases, the ruling comes out correct. Okay?
So let’s do the calculation. If all three were wrong, that’s a probability of one thousandth, because each one has a one-tenth chance of being wrong, one-tenth times one-tenth times one-tenth is one thousandth. That’s the probability that all three were wrong. What is the chance that two were wrong and one was right? That’s one hundredth times 0.9, which is 0.09. 0.9 times 0.1 times 0.1. But there are three such possibilities: either Reuven was wrong, or Shimon was wrong, or Levi was wrong. Therefore that is actually twenty-seven thousandths, plus one thousandth, so twenty-eight thousandths that this court will be wrong. Okay? And accordingly, nine hundred seventy-two thousandths that it is right. Which is much better than nine hundred thousandths, which is a single judge. Meaning, if we added two more judges, then we gained a ruling of better quality. Okay? That’s the calculation.
There’s some article by Nadav—Yossi, you once sent it, Yossi Laor. You once sent it; maybe we’ll send it on WhatsApp to whoever still doesn’t have it.
[Speaker E] An article by Nadav Shnerb in his book Keren Zavit, and the chapter title—I don’t remember which Torah portion it is—“What Do We Gain by Adding Fools.” And there he says, I don’t remember the exact numerical values, if there’s one judge with ninety percent accuracy and two judges who are fools with only seventy percent, it still comes out better, no?
[Rabbi Michael Abraham] Not always, not always, no no.
[Speaker E] He says that still, you still gain from the addition of complete fools.
[Rabbi Michael Abraham] Not always. Sometimes.
[Speaker E] You gain, not always. The probability of an erroneous ruling when one judge is at 0.9 and two additional ones are at 0.7—still, the court’s ruling… I don’t remember the calculation for 0.7, but it’s not always true.
[Rabbi Michael Abraham] I don’t remember the calculation for 0.7 either, but it’s not always true. Meaning, if you have two judges worse than 0.9, you can lose from adding two judges. But in the calculation where all three are of the same quality, usually you gain. So you gained from adding two judges. So in other situations you don’t always gain, but you can see that by adding judges you do gain.
Now notice what this calculation actually did. This calculation is what created its distribution, so to speak. In 972 out of 1000 cases this panel will be right, and therefore this panel will be the rule of truth. And so the way of arriving at this distribution is not actually the result of measurement and generalization from a sample. It is the result of a priori reasoning. For example, this reasoning says that the judges’ opinions are independent. This reasoning also says that the quality of a judge is 0.9—you somehow know that, even though I really have no way of knowing what a judge’s quality is. And therefore, basically, this is just some mathematical model that expresses an a priori intuition. And that a priori intuition says that the majority generally has a greater chance of being right than the minority. Therefore all these models are models that can’t really be measured empirically; they are not the result of generalization, not the result of sampling, but they help me demonstrate this a priori intuition that leads me to the conclusion.
[Speaker C] Rabbi, I have a question. The Rabbi said earlier that if I say the minority is right because it went against the majority, then by the same token a complete fool also goes against the majority. But that’s not…
[Speaker E] No, I didn’t understand.
[Speaker C] The Rabbi said earlier that when I said that apparently you should listen to the minority because the minority did something the majority didn’t do—the majority followed the herd, they were sheep, to a certain extent they felt good because they were going along with everyone—then the Rabbi said no, my explanation isn’t good because it could also be a perfect idiot, and that also doesn’t have to mean he’s the wise one.
[Rabbi Michael Abraham] No, I gave that answer to someone else’s question, not to your question. To your question I gave the answer that everyone took all the considerations into account and reached different conclusions, and the fact that you are original doesn’t mean you’re right.
[Speaker C] No, but there’s an addition here. After all, the minority also heard the opinions and decided something. But here there’s something special: that minority has some special quality—it was brave enough to go against everyone. You can’t say that about the others.
[Rabbi Michael Abraham] Why do I care that he’s brave? I don’t understand. The question is whether he’s right.
[Speaker C] No, but when you have to decide, it requires all kinds of abilities, intuition and all that—courage, no?
[Rabbi Michael Abraham] Not bad at all—he can form an opinion. In any case, when they start counting the opinions, they start with the junior one, so you still don’t hear the opinion leaders first. In short, I don’t accept this regardless of minority and majority; that’s not the answer I gave. It’s an answer to another question, not to yours. To your question, I simply don’t accept it. It’s not because of the question whether the minority is more right or less right.
[Speaker C] No, I also…
[Rabbi Michael Abraham] I’m saying, the question is why not say that the minority is always right, because there is a minority of wise people. So to that I said there is also a minority of fools. But that’s not your question.
[Speaker C] I didn’t say a minority of wise people. I didn’t say a minority of wise people. I’m assuming everyone is wise because, like—I don’t remember who—the Rabbi brought the Sefer HaChinukh, that we’re talking about people equal in wisdom. But still, in order to decide you need some special courage, and this idea of equal wisdom—there’s no such thing as equal wisdom. Specifically in the present ruling, you showed some kind of courage.
[Rabbi Michael Abraham] By that logic, the earth is flat. No, no. Yes. Why? The majority thought the earth is flat; they’re very brave, going against the accepted approach.
[Speaker C] But in order to be a perfect idiot in the current state of science and come say the earth is flat, you really have to be a special kind of idiot.
[Rabbi Michael Abraham] And you decided that the earth isn’t flat. But I’m talking now—suppose I have a court, the minority says the earth is flat; look what a brave minority.
[Speaker C] But there aren’t such people, amazingly enough there aren’t.
[Rabbi Michael Abraham] Of course there are. Are there flat-earthers? There are.
[Speaker C] Fine, okay, but I don’t know, I haven’t investigated the matter.
[Rabbi Michael Abraham] That’s not what you’re saying, it’s not… After all, there are considerations, they do the… In the Sanhedrin they followed the majority. And if you have twenty-two judges who say Reuven murdered and one says he didn’t, then I’d bet the twenty-two are right, and the one is very brave and thinks outside the box—and is wrong.
[Speaker C] Okay, now I’ll ask it another way, another different question: if they accepted my position that we should decide in favor of the minority, how could it function at all? Because people sit, hear one another—after all, they deliberate before determining… each one says his opinion, they discuss it among themselves. That matters, the preliminary discussion before they determine innocent or guilty. There’s a discussion, and they sense that the majority is moving in a certain direction. If they accepted my view, then they could never decide, because the moment they leaned in a certain direction and one person said otherwise, they’d understand that the truth is with him, because there—he’s right, the minority is right. So they’d go with that minority, and again…
[Rabbi Michael Abraham] They wouldn’t do anything. Each person has to state his opinion. Even if you know that the other person is right, that’s irrelevant; if this is your opinion, say your opinion. Okay, you’re bringing me to those paradoxes of—you know that in capital cases… yes, what happens if there are twenty-two judges who found him guilty? So if you are the twenty-third judge and you think he is guilty, if you say he’s guilty, he’ll go free, after all. Because if the entire court said he’s guilty, then he goes free. And if you say he’s not guilty, then he’ll be convicted. So if you think he is guilty, are you allowed to say he is not guilty so that his true verdict will actually come out? The answer is obviously not. You have to say what you think. It doesn’t matter what others say or whether they are right or not right; weigh it in your own mind, say what you think. That is a judge’s role. And in the end we count.
Okay, anyway, back to our issue. So what is the claim, really? The claim is that I have some sort of a priori intuition that says that if there is a dispute between a majority and a minority, then generally the majority is right. Meaning, most likely the majority is the one that is right. Now here, this is the a priori intuition, and therefore because of this intuition, the majority in a court is an “available majority” just like the majority of stores. Just like with stores, where majority is a priori reasoning—the majority of the stores—so too here, the majority of judges is a priori reasoning, and not generalization from a sample.
Now I’m using this as a springboard to the next point. When you read Nadav’s article that I’ll send you—actually, I didn’t send him this remark; I think I should send him this remark—there is a problem there. There is a problem with the calculations he makes. Why? There are two questions we can ask about a judge’s quality. They look very similar, but they are really not similar. Completely different questions.
When I say, for example, that a judge’s quality is ninety percent, what do I mean? Do I mean that given that Reuven murdered, there is a ninety percent chance that the judge will also determine that Reuven murdered—that is, that he will determine the truth? That means a judge of ninety-percent quality. Meaning that in nine… in ninety percent of the cases he will give the correct ruling. Okay? That’s one measure. And there is another measure, the reverse: assuming that the judge said Reuven murdered, what is the probability that Reuven really did murder? What I described before is a conditional probability that says this: given that Reuven murdered, what is the probability that the judge will rule that Reuven murdered? If that is ninety percent, then he is a judge of ninety-percent quality. But when I ask the question: the judge ruled that Reuven murdered; now, should I accept his ruling or not? That is the reverse question, because that question depends on the opposite probability. Given that the judge ruled that Reuven murdered, what is the probability that the truth is that he murdered?
[Speaker C] So that depends on the prevalence of murderers? If ninety percent are murderers and he rules for every…
[Rabbi Michael Abraham] Correct. Correct. Correct. In a moment I’ll explain it, but that’s right. Wait, once again I’m getting tangled up here with the… one second. Okay, look. Suppose event A… wait. No, you know what? Why do this here? I’ll do it in Word; it’s more comfortable for me. There, can you see? Let’s denote the claim that Reuven murdered by A. And the claim that a judge ruled that Reuven murdered by B. Okay? Now the judge’s quality actually derives from the relation between these two claims, right? A perfect judge is a judge who rules every time that Reuven murdered, he also rules that Reuven murdered. In one hundred percent of the cases he is right. But regarding a perfect judge it is also true to say the reverse: that if he ruled… that Reuven murdered, then clearly Reuven murdered, one hundred percent. And you understand that that is not the same claim. It is not the same claim if the judge is not perfect.
Let’s see, for example: when I ask what the quality of the judge is—wait—I ask what the quality of the judge is. So for anyone who doesn’t know the notation, I’ll explain: P is probability, of event B given event A. Yes, that slash means “given.” This is called conditional probability. Conditional probability means: what is the probability of B given A? Okay? I say this: given that Reuven murdered, what is the probability that the judge will also rule that he murdered? That is the quality of the judge, right? Okay, that basically tells me: if I have a hundred murderers, how many of those murderers will the judge convict? Right, suppose he’s a ninety-percent judge, then he will convict ninety out of the hundred murderers; he’ll miss ten. Okay?
But now notice: suppose the judge ruled that Reuven murdered. Now I ask another question: what is the probability that Reuven really murdered? You understand that this question—oy, with the Hebrew and English this is a plague. Okay. What is this question? This question is the quality of the ruling. Right? What does that mean? Given that the judge ruled that Reuven murdered, what is the probability that Reuven really murdered? Is this ruling correct or not? Now it sounds like the same question, but in the mathematical notation you see that it’s not the same question.
And to explain this, I’ll illustrate it with another example I talked about in the past. Let’s suppose that you—I’ll tell the case of Munchausen syndrome, Munchausen syndrome by proxy. What does that mean? There was a woman in Britain whose two small children died—babies died. She was brought to trial on charges of murder, and some British doctor came there, a psychiatrist I think, British, Sir Roy Meadow, and he argued that this woman is a murderer because the chance of crib death for a child is one in eight thousand. So the probability that two children died of crib death is one in sixty-four million. An infinitesimal probability. Therefore she probably didn’t experience that; rather, she murdered them.
This poor woman was sent to prison and sat there for quite a significant period until some statistician came and shook up the idiot judge and the idiot doctor and explained to them that they had probably put an innocent woman in prison. Why? You can challenge this in many ways. First challenge: since we’re dealing with statistical problems, I’ll also address the other challenges, although my topic here is not those. First challenge: the events are not independent. Meaning, who says that if crib death for one child is one in eight thousand, then for two children it’s one in sixty-four million? It could be—after all, we just saw the Talmudic passage that says “the spring causes it,” right, the Talmud in tractate Yevamot, which says that if her husbands died—three husbands died—then she is forbidden to marry a fourth time, or two, there’s a dispute among the tannaim. Why? Because there is some concern; one of the explanations in the Talmud is that “her spring causes it.” Meaning, marital relations with her cause the man’s death. She has some kind of, I don’t know, problematic fluid structure, her bodily fluids, and therefore that kills the husbands.
Now if that really is the woman’s trait, then you can’t measure the probability that two husbands die as the product of the probability that each one separately dies. The probability that two die is the same probability, more or less the same probability, that one dies, because the cause of death is the woman; it’s not something statistical. There is something shared between the death of the first husband and the death of the second husband; they are not independent. When the events are independent, I can multiply the probability of each one and that will give me the probability of both. But if the events are dependent, then it’s not correct to multiply them.
Say, think for instance of a biased coin, okay? And that coin has a ninety-percent chance of landing heads, not fifty percent; it’s not a fair coin. No, actually that’s not good, not a good example. I’m saying: in a place where we have a common cause for two events, there is dependence between the events, and you can’t multiply the probabilities. Say, a woman who definitely kills the… suppose a cannon is aimed at a certain place, and suppose its hit is perfect at that place, then I can’t now calculate what the probability is that a shell will fall twice in the same place; it will fall in the same place because for the same reason it fell twice in the same place. Okay? There’s nothing to multiply there. It’s not exact, which is why I keep retreating, because you can multiply probabilities—it’s one times one, which gives one. But you understand the idea. The point is that when the events are dependent, you can’t multiply the probabilities.
So in our case, if there really is something that causes the children to die in this woman’s case—I don’t know, something genetic or whatever—then you can’t say the probability is one in eight thousand and then ask how two children would happen to die in the same home. They would die in the same home because this woman’s genetics are what caused their deaths. So it’s very likely there would be another child who would die in that same home. If the first child already died in that home, then the second probably would also die there, because there is something in the genetics of that woman—or the husband, doesn’t matter—that causes this child to die. Therefore you can’t multiply the probabilities.
But that’s the less problematic challenge. Why? Because leave that aside, let’s talk about one dead baby. What is the probability that one baby dies? One in eight thousand, right? One in eight thousand is also a very small probability. So why don’t they put in prison all those whose child died? Why do you need two? What if one died? After all, the probability that he died of crib death is one in eight thousand, right? And that means you probably murdered him and should go to prison. Why are you only talking about two dead babies? Why not one baby who died?
And the answer is of course because… think, say, that the probability is one in eight thousand. How many mothers are there in Britain? Or how many children were born in Britain? I don’t know, in the last year, okay? In the last year how many children were born—say a million, okay? I don’t know how many; let’s assume that. Fine? So if a million children were born, and the probability is one in eight thousand, that means that around a hundred of them will die, right? Roughly—even a bit more, a hundred and twenty. A hundred and twenty children will die in a year in Britain of crib death.
Now we caught one of the mothers whose child died. Can I send her to prison because the probability that the child dies is one in eight thousand? Do you understand? This is nonsense, complete nonsense. There are a hundred and twenty mothers whose child will die; we know that with certainty. So how can you put someone in prison? Now if I have two children who died, then one in sixty-four million. Suppose I’m allowed to multiply the probabilities, leave it aside, say they are independent, okay? One in sixty-four million, fine. Excellent. So if there are several—I don’t know how many families there are in Britain—say twenty million, I don’t know, fifteen million, fifteen million households—then it is absolutely possible that in one of them two children died of crib death. The probability is one in sixty-four million. That’s not enough for a criminal conviction. Why? Because the number of households in Britain is so large that even if the probability is small, in one of them it will happen.
If you ask, for a given child, what is the probability that tomorrow he will die of crib death? One in eight thousand. But if the child has already died, and you ask what is the probability that he died of crib death as against the probability that his mother murdered him, clearly it is far more likely that he died of crib death than that his mother murdered him. Doubt and probability.
So maybe I’ll illustrate this with something easier to explain. Think about—I may have mentioned this already, I don’t remember—a problem with doctors. Suppose a patient comes to you and you send him for a test, a test for some rare disease. And let’s say the reliability of the test is one percent—meaning ninety-nine percent, one percent error rate. Both false negative and false positive, let’s say symmetric. Meaning you can get a sick person reported healthy in one percent, or a healthy person reported sick in one percent. So the test is ninety-nine percent reliable. You sent the person for the test and he came out positive, meaning he came out sick. Now I ask: what is the probability that he is sick? Anyone who doesn’t know—want to try?
The reliability of the test is ninety-nine percent. The person took the test and came out sick. Now I ask what is the probability that he is sick? The obvious answer, I assume a large part of you want to say, is ninety-nine percent, right? The test is ninety-nine percent reliable, apparently the probability is ninety-nine percent. Well, not so. Why not? Let’s assume that this rare disease is a disease that occurs in one in a million in the population. Say one in a million. So now suppose I run the whole population, all one million, through the test. How many will come out sick? Ten thousand. Ten thousand will come out sick. Why? Because the test is wrong in one percent of cases, right? Now I have one million healthy people and one sick one—one million minus one, one million healthy and one sick one, right? Now among the million healthy, ten thousand will come out sick, because the test is wrong in one percent. One percent of a million is ten thousand. Right? So ten thousand out of the million will come out sick. How many are really sick? One. So if the test identified me as sick, what is the probability that I am sick? One in ten thousand. Because only one out of the ten thousand whom the test identified as sick is actually sick.
So notice: the test is ninety-nine percent reliable, it identified me as sick, and the probability that I am sick is one in ten thousand. In short, there is practically no chance that I am sick. I can go home happy and cheerful. How does that square with the fact that the test is ninety-nine percent reliable? So I’ll explain. Let’s go back to the previous file. Look, when I talk about the reliability of the test, what question am I asking? If the person is sick, what is the probability that the test will identify him as sick, right? That is called the reliability of the test. If the person is sick, what is the probability that the test will identify him as sick? That’s ninety-nine percent. But if I ask the question: the test identified him as sick; what is the probability that he is actually sick? One in ten thousand. Because the question about the quality of the test is this question, and the question about the quality of the result is this second question. And you see there isn’t much connection between these two questions, even though they look terribly similar.
So if we now speak about a judge, not a test: the judge’s quality is ninety-nine percent, a judge of the highest caliber. The judge’s quality is ninety-nine percent. What does that mean? That out of a hundred murderers he will identify ninety-nine murderers. He’ll only miss one. Now I ask the reverse question: a person comes before me and the judge ruled that he is a murderer. Now I ask: what is the probability that he is a murderer? You understand that this question is this second question. So if, say, there is one murderer per million in the population, and the judge is of ninety-nine-percent quality, then if he ruled that someone is a murderer, the probability that he really is a murderer is one in ten thousand; there is basically no chance that the judge was right. And that is a ninety-nine-percent judge. And why is that? Because the percentage of murderers in the population is tiny; it is one in a million in the example I gave. So what? A large number of innocent people will come out as murderers because the judge errs one percent of the time. Out of a million people he will identify ten thousand as murderers, even though he is an excellent judge, because ten thousand out of a million is actually a very good judge—he is right in ninety-nine percent of cases. It’s just that one percent is ten thousand people.
[Speaker D] But you can’t test this based on the general population; you have to test it based on, among those who are under suspicion, how many of them are murderers.
[Rabbi Michael Abraham] Okay, we’re already getting closer to the solution. First of all, I’m presenting the question. In the end, the problem is that when I ask—say, when I test the quality of the judge—I ask the question: given that he is a murderer, what is the probability that the judge will identify him as a murderer? Ninety-nine percent, wonderful judge. But when I ask the reverse question: the judge ruled that so-and-so is a murderer, that ninety-nine-percent judge ruled that so-and-so is a murderer. Now I ask what is the probability that so-and-so really is a murderer? Zero, meaning one in ten thousand, almost no chance. Meaning the questions look terribly similar, but the answers to them can be completely different from each other. Why? Or when does this happen? In a place where the number of murderers or the percentage of murderers in the population is very small.
I often compare it to a fishing net. When you want to catch small fish, you need a net with even smaller holes, otherwise the fish will slip out, right? When you want to capture a phenomenon that is very rare—yes, one in a million in the population—you need a net whose reliability is on the order of one in a million. A net whose reliability is one in a hundred—meaning one percent—which is an excellent net, but not for fish the size of one in a million. When you want to catch one murderer out of a million people, you can’t take a judge whose reliability is such that he errs one in a hundred times. You understand that if he errs one in a hundred times, his skill has no value in catching a murderer who is one in a million.
And therefore, checking the reliability of the judge—what all of Nadav’s calculations, which you’ll read when I send you the chapter, deal with—is the question of in how many cases the court will be right. But I ask the opposite question: assuming the court said this, in how many cases is it really so? Not in how many cases the judge will rule correctly, but in how many cases what he ruled will turn out to be correct. It sounds terribly similar, it’s terribly confusing, and these are completely different questions.
I’ll give you an example. Sorry, Rabbi—
[Speaker F] Excuse me, Rabbi.
[Rabbi Michael Abraham] Suppose I have a judge of ninety-percent quality. Now three judges are sitting in judgment; two say Reuven borrowed and one says Reuven did not borrow. What is the probability that Reuven borrowed? So according to Nadav, twenty-eight out of a thousand, right? Sorry, nine hundred seventy-two out of a thousand. We did the calculation earlier. A ninety-percent judge—again, a ninety-percent judge—I ask, what is the chance that the court will issue, say, a correct ruling that matches the facts? So we said: what is the chance that it erred? Either all three are wrong, that’s one out of a thousand, or two are wrong and one is right, that’s twenty-seven out of a thousand. So all told, the probability of error is twenty-eight out of a thousand. In nine hundred seventy-two out of a thousand cases the court will issue the correct ruling. This is a court of quality 0.972. That is the reliability of the court.
But what does that reliability represent? It represents the question: given that this is the case, what is the probability that the court will also rule this way? The answer is 0.972. Now let’s do the reverse calculation and I’ll show you that it’s not the same thing. Let’s do the calculation: the court ruled that Reuven borrowed. What is the probability that Reuven really borrowed? Okay, it ruled two against one that Reuven borrowed. Very simple calculation.
Notice: to match a value of 0.9—say, Reuven murdered, okay? A: Reuven murdered, and the judge, the judge is of quality p. Fine? This is the quality of the judge. What does the quality of the judge mean? Notice that, assuming Reuven murdered, the judge will indeed rule that Reuven murdered, right? That is the quality of the judge. Now I ask: two judges ruled that Reuven murdered, and one ruled that he didn’t. What is the probability that Reuven murdered?
So look, if Reuven murdered, yes? Then I have two judges—I have two judges who rule that he murdered, right? The probability of that is 0.9. Oy, with the Hebrew, it’s killing me here. Fine, I’ve given up. I won’t manage with the Hebrew. I’ll say it in words. Look, there are two possibilities: either Reuven murdered or he didn’t murder. If Reuven murdered, yes, then the probability that two judges were right and one erred is 0.9 times 0.9 times 0.1, right? 0.9 means they were right, two were right and one was wrong, and the probability that he is wrong is 0.1. What is the probability that Reuven did not murder? That means two judges were wrong and one was right, correct? That is 0.1 times 0.1 times 0.9.
So one of them—now if you do the calculation, what you actually need to calculate is as follows. This, this is p times p times… right? That’s on the side where two were right, so the probability that this one was right and that one was right, times the probability that the third erred, okay? The probability… that’s if Reuven really murdered and two judges ruled that he murdered. What happens if the truth is that he did not murder? Then that’s one minus p times one minus p, right? That’s the error, the probability that two were wrong and one was right, times p, okay? Meaning, if Reuven really did murder, then the two who ruled that he murdered were right, and the one who ruled that he did not murder was wrong. That probability is 0.1, 0.9, 0.9. One minus p is the chance of error, p is the chance of being right. If Reuven really did not murder, that means that the two who ruled that he murdered were wrong. That is one minus p times one minus p, and the probability that the third—and the third was right in saying that he did not murder—is p, okay?
Now I need to measure this against that. That is actually what I need to measure. If I want to know what is the probability that, given that they ruled he murdered, he really murdered—meaning I am now asking about the quality of the ruling, not the quality of the judge. The quality of the judge is p, that’s 0.9, but I want to test the quality of the ruling. Meaning, assuming they ruled that he murdered, what is the probability that he murdered? The answer is this divided by the sum of these two. Right? Because this is the probability that he murdered divided by the probability that he murdered plus the probability that he did not murder, which is the total—that is basically one. Okay? So this is actually this divided by the sum of these two.
When you do the calculation, you’ll discover that it is p. The result is p. That is to say, ninety percent, and not as Nadav says, ninety-seven point two. Nadav is measuring the question—it’s not that he is wrong—he is simply measuring a different question. He measures the question of how many of the murderers the court will catch. How many of the murderers the court will hang. Nine hundred seventy-six out of a thousand. I ask: assuming the court ruled that he is a murderer, how many of those whom the court convicted are actually murderers? In how many of them did it err? The answer is: ninety percent are actually murderers, and not 972, not ninety-seven point two percent. It’s a different question.
[Speaker B] And that has nothing to do with how many murderers there are? What? It has nothing to do with how many murderers there are?
[Rabbi Michael Abraham] I already swallowed that probability here, of how many murderers there are. Fine, I decided not to get into formalism here, so I can’t explain it with Bayes’ formula, but I got around it through the ratio I described here. Anyway, in short, the details of the calculation don’t matter at the moment. It’s really a bit hard to explain without the formulas, and I’m not even trying to write it properly here.
But what I basically want to say is that it’s very important—and I’ll maybe talk about this next time, I’ll try to think how to explain it in a clearer way—the direction of the question is very important. P of B given A and P of A given B are two different questions. Both in medical diagnosis and in legal evidence, and we need to pay very close attention to which of the two questions we are asking. And I’ll show you all kinds of examples where we mix up the two kinds of questions and reach incorrect conclusions. Very often even smart people, and experts in the field, fall into this.
Okay, so we’ll stop here. In fact, I think we’re not going to meet again until after Sukkot. Right? Next Thursday is already Yom Kippur.
[Speaker F] Next Thursday is the evening after Yom Kippur.
[Rabbi Michael Abraham] Yes, so only after Sukkot will we meet next time. So for next time I’ll try to think how to present this in a way that will be clearer, and maybe also write it on the file in a more efficient way.
[Speaker C] Maybe the Rabbi would really help us doctors a lot with this, because it’s genuinely hard to explain to patients. Sometimes patients come with some positive test for a very rare disease. And they’re under pressure. They’re terrified, not sleeping for a month. And I’m trying to explain to them that they can go home cheerful and happy. Do something, fine, we’ll do something further, but…
[Rabbi Michael Abraham] You can relax.
[Speaker C] It just doesn’t sink in.
[Rabbi Michael Abraham] One of the problems is first of all explaining it to doctors, and only afterward to patients, because the answer is not so simple. And you sent him to the test because you had indications, and there is some suspicion that he actually has this rare disease.
[Speaker C] No, not—I’m not talking about that case. Usually he comes to me with a test from someone else, and I don’t know why they sent him.
[Rabbi Michael Abraham] There was no reason he took the test, just like that?
[Speaker C] No, just like that. The doctor says, check a marker, do it just to mark it off, like—
[Rabbi Michael Abraham] Like during COVID, and those who came out sick—
[Speaker C] During COVID, because they had some suspicion? “Give me a general checkup.” They come to the family doctor: “Give me a general checkup.” And if he’s a little weak-willed, then he sends him for all sorts of markers for extremely rare diseases, and it comes back positive and then he doesn’t sleep anymore. And then the doctors start making a living off him. He says to him, fine, we’ll do something, because otherwise you won’t sleep at night, but the truth is it’s unnecessary. Not only is it unnecessary, he—
[Rabbi Michael Abraham] He won’t sleep at night—even less after that. Exactly.
[Speaker C] No, so that’s clear.
[Rabbi Michael Abraham] Meaning, if there are no indications, then it’s simply nonsense. But usually when you send a particular patient for a test for a rare disease, it’s because you have indications. Once there are indications, it changes the whole picture. And I’ll talk about that next time too.
[Speaker H] Rabbi, some short idea for Yom Kippur? Last year it was very successful.
[Rabbi Michael Abraham] I don’t remember what it was last year.
[Speaker H] Labor for a higher purpose and the haftarah of Jonah?
[Rabbi Michael Abraham] Ah, that. The truth is, I have a lot of long things; I’m not finding something short.
[Speaker H] We’re buying.
[Speaker C] Maybe we can ask something? Maybe the Rabbi will answer it and that might spark some idea for him. We’ve now begun the Ten Days of Repentance, right? It’s this project we do every year and we know it well. Seemingly a difficult project in every possible sense. After all, if we were really rising every year, we would expect that after three thousand years of Judaism, four thousand years, and all the time people are repenting, they should already have reached the heavens in the intensity of their holiness and all that. Not only is that far from reality, but the Sages tell us: if the medieval authorities (Rishonim) were like angels, then you are like donkeys. That’s basically what they’re saying.
[Rabbi Michael Abraham] And I’m going to talk about that, not about what you had in mind, okay? I’ll talk about that. So I’ll say it like this: look, it says in the Talmud that a penitent is preferable to a completely righteous person, right? And they disagree—yes, it’s a dispute among Amoraim—but in the end everyone cites this conclusion, that a penitent is preferable to a completely righteous person. And the question is why. Seemingly, a penitent is someone who should be, at most, a completely righteous person. Meaning, he committed sins; let’s say he did complete repentance, cleaned away all the sins—then he becomes, at most, a completely righteous person. How can it be that a penitent is preferable to a completely righteous person? Clearly what that actually means is that the value of repentance is not in its result, or not only in its result, but the value of repentance is in the process of repentance itself. In other words, the penitent is measured not only by the question of what spiritual state he reached after repenting, because that really cannot be more than a completely righteous person. But if the penitent underwent a process of repentance, then beyond the fact that he reached the state of a completely righteous person, he also repented. And the act of repenting itself has some value beyond the state to which it leads him. And therefore a penitent can be preferable to a completely righteous person, because beyond having become a completely righteous person and his spiritual state being like that of a completely righteous person, he has the added advantage that he also repented on the way. And the act of repenting—in short, what I’m saying is that the value of repentance is not the value of the function but of the derivative. Meaning, it’s the value of the path you take and not only of the state you arrive at. Yes, think about a factory where a consultant comes in and says, listen, make some changes, it’ll refresh things and so on. He can recommend changes that won’t improve the condition of the factory. And still, the very fact that there is dynamism in the factory refreshes the people. Meaning, that itself can improve the outcome even though, in terms of calculating the state you changed into, it’s not supposed to be more efficient than the previous state. But the very fact that you went through a change did something to you. Meaning, there can be a situation in which the change is measured not by—not through the difference between the initial state and the final state, but by the very fact that you made a change, even if the final state has the same value as the initial state. Yes, think about—yes, I once wrote about this and also spoke about it in the past—about the definition of velocity. Velocity is defined as difference in position divided by difference in time, right? How much place I changed and how much time it took me to change that place. And from here arises Zeno’s paradox, because Zeno asks, the paradox of the arrow in flight—Zeno asks: so basically at every moment the arrow, when you look at an arrow that is moving, at every moment you see it standing in a different place. So when does it move? When does it pass between the places? At every moment it stands in a different place, so when does it pass between the places? When is it moving? Because Zeno assumes that the arrow cannot move at a point in time. If you look at it at a point in time, it is not moving. It is standing in one place; you can’t move in a single point in time. So at every point in time it stands in a different place. So when does it move, or how does it move? And the answer is that there is motion even at a point in time. The arrow has velocity at a point in time. It will not change its place in a single point of time. To change place you need an interval of time. At a point in time you can’t change place. But velocity it has even at a point in time. Yes, we know that velocity is the derivative of the position function. And the result, after differentiating the position function, is that the velocity at a point in time is defined. At every point in time there is a value for the velocity function. A body has velocity at a point in time. So usually if you ask physicists, they’ll tell you no, no—velocity exists only over a time interval. But it’s convenient for us to speak in terms of velocity at a point in time, when really what we mean is velocity over a small interval around that point. Yes, in an interval as small as we want around that point in time. But that’s not true. A body has velocity at a point in time. The way to calculate it is by taking a certain interval, seeing how much distance it covered, dividing that by the time—that gives me the result of the velocity. But that’s only the way to calculate. Velocity exists at a point in time. And what this actually means is that dynamism is not measured only by the question of where you were before and where you got afterward. Change in itself can take place at a point in time where there is no—meaning, the dynamism exists at a point in time even though there is no change in state, because at a point in time there cannot be a change of state, but dynamism exists even at a point. His spiritual state is like it was before. And on this Maimonides says: until the Knower of secrets will testify about him that he will never return to that sin again. What does that mean? After all, we all know that in the end yes, we do return—what can you do? So how can the Knower of secrets testify about him that he will never return to that sin again? The Knower of secrets testifies about him that right now he will not commit that sin. In another moment maybe yes, but right now he truly repents, he truly regrets, he truly wants to improve, so he has improved. The desire to improve has improved. True, in the end the state I arrive at will not be better than what it was before, but that does not mean my repentance has no value. Since a penitent is preferable to a completely righteous person, the process of repentance has value—not only in the sense of the state I arrive at as a result of the process of repentance, but my very being in a process of repentance has value, even if the state I arrive at in the end is the same state I was in last year. The very fact that I’m troubled by it, that I’m trying to improve, that at that moment the Knower of secrets testifies about me that I will never return to that sin again from the standpoint of that moment—that “never” will end in another second; in the next moment I’ll do it, but at this moment, if you look at me, I am truly completely clean. From the standpoint of this moment, it is “I will never return to that sin again.” In the next moment it will come back; it will pass over me. So I am a penitent. And this is a little connected to what you mentioned with Jonah—that basically we were born here not to be perfect but to perfect ourselves. So it does connect to that. We were born in order to perfect ourselves, not in order to be perfect. And many times the process of self-perfection does not lead me to a state in which I am more complete than I was before. But if I underwent a process of self-perfection, then that has value. And our goal is basically to repent, not to become more righteous. Even if we don’t become more—we need to try to become more righteous—but the value of repentance does not depend on whether we succeeded in becoming more righteous. If we did the dynamic process that is supposed to make us more righteous, even if in the end we did not become more righteous, we repented. That is the value of repentance, and in fact it is its main value, because what is demanded of us is to do repentance and not to be righteous. That’s probably what I talked about last year. Okay, that’s it.
[Speaker H] Thank you very much.
[Rabbi Michael Abraham] May you be inscribed and sealed for good—or really, may your final sealing be for good, a good final sealing—and may we have a better year than the one we had, which is basically saying nothing. May it be a good year, not a better one than the one we had.
[Speaker B] Thank you very much, success,
[Rabbi Michael Abraham] Goodbye.