Doubt and Probability—in Halakha, Thought, and in General—Lecture 4
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Table of Contents
- [59:32] The difference between negative and positive doubt
Full Transcript
[Rabbi Michael Abraham] Okay, let’s begin. Up to this point we’ve talked about the concept of certainty, epistemic certainty as opposed to ontic truth. Right, certainty is basically a description of my own state: whether something is certain in my eyes or not. Truth is a description of the state of affairs in the world, not of my state of knowledge about it. So I talked about the relation between truth and certainty, belief and certainty. I talked about practical certainty and philosophical certainty. Meaning, philosophical certainty—there’s no such thing; that is, there is no 100% certainty about anything. But there can be practical certainty. In other words, there are things that seem completely clear to us. There is always some possibility that we’re mistaken, but for practical purposes, that’s what’s called certainty. I talked about the different maturations: skepticism versus synthetism versus fundamentalism. After that I dealt with the question, or the point, that in order to doubt, you need a reason. Right, in Ein Ayah by Rabbi Kook, in the passage in tractate Shabbat on page 30, where someone comes to Rabbi and says to him, “Your mother is my wife and you are my son”—basically, you’re a mamzer, I had relations with your mother and you’re illegitimate. So Rabbi says to him, “Would you like to drink a cup of wine?” He drank and burst. Meaning, Rabbi is not willing to entertain doubt if there’s no good reason to doubt. We spoke a bit about autonomy—that I’m not required to doubt just because there’s a dispute among sages on a certain issue. On the other hand, what’s called peer disagreement—meaning, if there’s a disagreement among peers, if my peers think differently, is that supposed to make me doubt? Up to here, that was the general introduction. After that I moved into the question of how, before getting into ways of deciding in cases of doubt, do we lay out the options between which I’m supposed to decide? In other words, how do we formulate the problem before the way we deal with it or solve it? So here I said that we always need to take into account the law of non-contradiction and the law of the excluded middle. The law of non-contradiction says that a thing and its opposite cannot both be true at the same time—or in other words, don’t confuse me with pluralism. Meaning, pluralism is simply nonsense. There’s no such thing as both a thing and its opposite being true. And the law of the excluded middle basically says that either the thing is true or its opposite is true, but there’s no third possibility. Meaning, it’s true that they both can’t be, but on the other hand it has to be one of the two; there can’t be something third. When we map out the possibilities between which we’re hesitating, we need to take into account these two basic logical rules. I distinguished between—or before that, yes. In the context of contradictions, I distinguished between logical contradictions and philosophical contradictions. Meaning, there are things that don’t fit together on the philosophical level, but there is no logical contradiction there. And there are, say, things that go against the laws of nature. Things that go against the laws of nature—if you tell me that a certain stone remains suspended in the air and doesn’t fall to the earth, that contradicts the laws of nature, but it’s not a logical contradiction. Meaning, in principle there could be a world in which such a stone really would stay in the air. In our world there’s a law of nature that says no—it falls—so that contradicts the laws of nature. A logical contradiction is something that contradicts the laws of logic. So these are two kinds of contradiction, and sometimes it’s important to distinguish between them. For example, when we talk about the question of foreknowledge and free choice—when the Holy One, blessed be He, knows everything in advance, the question is whether we have free choice. It’s very important to decide in advance whether the contradiction that exists between God’s prior knowledge and our freedom of choice is a logical contradiction or a physical contradiction. Meaning, if it’s a logical contradiction, then even the Holy One, blessed be He, cannot overcome it. If it’s a physical contradiction, meaning that we cannot deal with going against the direction of time or knowing things in advance, fine—but the Holy One, blessed be He, is omnipotent, so He can. When we discuss the Holy One, blessed be He, physical contradictions trouble us less. He can keep a stone suspended in the air, but He cannot make a round triangle. Therefore the difference between logical contradictions and physical contradictions is important, because as part of what we call the law of non-contradiction, we need to ask: what kind of contradiction—logical or physical? After that I said that there are different ways to get outside the dichotomous framework. They present me with two dichotomous possibilities, either this or that, and many times—meaning, I said earlier how one should deal with them, so it’s one of the two and there is no third, and the law of non-contradiction, the law of the excluded middle, and so on—but many times, many times when we’re presented with two paths, they don’t really exhaust all the possibilities. Either they don’t really contradict one another, or they don’t cover all the possibilities. And there I talked about several ways to get out of dichotomies, to neutralize dichotomies, by basically stepping outside the framework of the discussion. So I gave examples such as evolution versus faith, creationism, what’s called creationism.
[Speaker B] What would you say—what would you say about Gödel, that there is no system that’s both consistent and in which you can prove everything—can the Holy One, blessed be He, yes, can He—does that mean something?
[Rabbi Michael Abraham] No, because that’s in mathematics, not in physics. It’s a theorem in mathematics, in logic. And the laws of logic bind even the Holy One, blessed be He.
[Speaker B] So even the Holy One, blessed be He, can’t prove everything.
[Rabbi Michael Abraham] Correct. Meaning, when we talk about logical impossibilities, not physical impossibilities, that includes even the Holy One, blessed be He. It’s like a round triangle. Even the Holy One, blessed be He, cannot make a round triangle, like I mentioned the example of the stone, right, that the Holy One, blessed be He, can’t lift—there too it’s the same thing. Meaning, a stone that the omnipotent cannot lift is like a round triangle. It’s obvious that there is no such stone. In other words, this is not an injury to God’s omnipotence. When I say that the Holy One, blessed be He, cannot prove all the theorems within an axiomatic system, as in Gödel’s theorem, that is not an injury to God’s omnipotence. There simply is no such proof. There cannot be, there does not exist such a proof. It’s like when I say that the Holy One, blessed be He, cannot make a round triangle. I didn’t say that the Holy One, blessed be He, is limited. Rather, a round triangle is simply an empty concept. There is no such concept. So explain that concept to me, and then I can try to think whether the Holy One, blessed be He, can make it or not. When you speak about that concept, you’re speaking about a collection of words that means nothing. Therefore there’s no reason to get excited that there is some injury here to God’s omnipotence. God’s omnipotence is the ability to do everything that is thinkable. But if there is something that isn’t defined, something that simply cannot exist, then the fact that the Holy One, blessed be He, can’t do it is not an injury to His omnipotence, simply because there is no such thing. That’s all. Just like the Holy One, blessed be He, also cannot kill Himself, meaning simply cease to exist. Because He is a necessary being. A necessary being cannot fail to exist. So is that a deficiency? Is that an injury to His omnipotence? No. That possibility simply does not exist. That’s all. Wait, what happened here? So I said that if I want to break dichotomies, meaning they present me with two possibilities, and I see that I don’t agree with either of them, I’m basically looking for a third possibility, or I understand that there is a third possibility. But how can that be? After all, they’re presenting me with two dichotomous possibilities. It’s either this or that and there is no third. We talked about it—the law of non-contradiction and the law of the excluded middle. So how can it be that one can disagree with both sides in a dispute? And I said that I usually find myself in exactly that situation. Meaning, I see disputes and usually I don’t agree with either side. And I say that there is a whole toolbox that can help me neutralize dichotomies, or find a third possibility. For example, Religious Zionism versus Haredi Judaism, the examples I brought. Religious Zionism versus Haredi Judaism—as if either you’re Religious Zionist or you’re not Religious Zionist. What else could there be? Seemingly those are two possibilities, there is no third and it also can’t be that you are both. So I say no, that’s not true, because in “Religious Zionism” there is an assumption embedded in these two options—that the meaning of Zionism has to be religiously significant, either positively or negatively. Meaning, if you’re Religious Zionist then it’s positive; if you’re anti-Zionist and religious then it has negative significance. I say no, it could be that Zionism has no religious significance. So I’m a Zionist and I’m religious, but there is no hyphen between my Zionism and my religiosity. So that’s an example of stepping outside the dichotomy. They present me with two possibilities: wait, are you Religious Zionist or are you Haredi? A lot of people say, wait, wait—if I’m not Religious Zionist then I’m probably Haredi, or vice versa. If I’m not Haredi then apparently—after all, there is no third possibility. Meaning, you can’t be both, and there is no third possibility. So no. Very often when dichotomies are presented to us, there is a third possibility. It’s a false dichotomy. The same with evolution, which I mentioned before—evolution and creationism. So either the Holy One, blessed be He, created the world or it came into being naturally. What else could there be? It could be that the Holy One, blessed be He, created the natural process through which the world came into being, or life came into being, and so on. All kinds of things like that, which seem terribly simple when I present them here, but in life you can see that lots of people get terribly tangled up in this. They feel they have to choose either this possibility or that possibility and there is no third. Free choice, for example—an argument. Great, about free choice. I don’t think I brought that example. The perhaps most common argument against the existence of free choice is a “whichever way you take it” argument. It goes like this: every event either has a cause or has no cause, right? There’s no third possibility. So if it has a cause, then that event occurred deterministically. The cause dictated the outcome. If it has no cause, then it’s indeterministic, so it’s just arbitrary, random, accidental. Okay? Either way, there’s no room for free choice, because free choice is neither this nor that. But there is no “neither this nor that.” Either an event has a cause or it doesn’t have a cause. Therefore, basically, there is no room at all for this mechanism we call free choice. Because it’s either randomness or determinism; there is no third thing. That’s the claim. A very common claim—in almost every discussion about free choice, this claim comes up in one form or another. Where is the mistake in this claim? The mistake in this claim is that once again this is a false dichotomy. Why? Because when I say there is no cause, that does not automatically mean indeterminism. Free choice too is something that happens without a cause. And therefore it’s true that either there is a cause or there isn’t, but under the possibility of “there isn’t a cause” there are two sub-possibilities hiding. Either there is no cause and it’s just arbitrary, or there is no cause but it is the result of deliberation, and that is free choice. In another context I said either there is no cause and there is a purpose, and then that is free choice, because you act toward a certain goal—not from a cause but toward a certain goal. Or there is no cause and there is also no purpose, and then it’s just randomness or arbitrariness. Okay? So here’s another example of a dilemma argument. Yes, in logic these are often called dilemma arguments. Dilemma arguments basically present you with two possibilities and I prove to you—either I prove by elimination that if I ruled this one out then it must be true, or vice versa, or I show you that a certain proposition is true whether you assume this or assume that, and therefore clearly it is true. That’s called a dilemma argument. For example, they tell me there is no point in giving exams. Why? Because someone who is lazy doesn’t study even with an exam, and someone who is diligent studies even without an exam. So whichever way you take it, there’s no point in giving an exam. That’s called a dilemma argument. Where is the mistake in this dilemma argument? The mistake is that those two possibilities—either he is lazy or he is diligent—are not the only two possibilities. Since there are different levels of laziness. Someone can be completely lazy, fairly lazy, not so lazy, a little lazy, and he can be completely diligent. There is a whole range or spectrum of possibilities between the absolutely diligent person and the absolutely lazy person. Therefore the view that there are only two possibilities is a false dichotomy. In fact there are more possibilities, and this thing is basically, yes, this is basically an example of the heap paradox. The heap paradox is basically a tool for neutralizing dichotomies. Right, they tell me one pebble is not a heap. If there is a collection of pebbles that is not a heap, then if I add one pebble that won’t change its status. Sounds reasonable. But a thousand pebbles are a heap. That too sounds reasonable. All three assumptions sound reasonable, but they don’t fit together. You can’t affirm all three. You can’t say that one pebble is not a heap, and also say that adding one pebble doesn’t change the status, and also say that a thousand pebbles are a heap. That can’t be. Why? Because if one pebble is not a heap and adding one doesn’t change anything, then two also aren’t; if two aren’t then three aren’t; if three aren’t then four aren’t; so a thousand also aren’t. So how can a thousand be? In other words, there is some contradiction here. And the answer to this is that the claim that adding one pebble doesn’t change the status is a mistaken claim. Adding one pebble changes the status a little bit. Not from not-a-heap to heap, because the concept of a heap is not dichotomous. It’s not that there are only two options, either yes-a-heap or not-a-heap. There are different degrees of heap-ness. And when I add one pebble, I increase the degree of heap-ness of the pile. Okay? So once again this is an example of a false dichotomy. Either yes-a-heap or not-a-heap. What else could there be? No, there’s a continuum. It’s fuzzy logic—we talked about that.
[Speaker C] Rabbi, but very often the additional option is very often a bit of a fiction—or a lot of a fiction. What fiction? Very often the extra options people create so that it won’t be dichotomous are a bit of a fiction.
[Rabbi Michael Abraham] If it’s a fiction, then don’t accept it. I’m not getting into the specific question right now.
[Speaker C] The question of heap-ness, for example. After all, once again, existentially speaking, you’re standing in front of a heap and you have to decide. You can’t say—and we’ve never heard of any person in the history of the world who said “sort of.”
[Rabbi Michael Abraham] I don’t completely agree, but I think we shouldn’t get into that argument because it’s a specific argument. So regarding a heap, you disagree. But I’m talking about the principled question.
[Speaker C] Also regarding free choice—the purpose. You ask yourself what caused you to choose that purpose. No? So again I come to a halt at the question of causality. So is it random? You just happened to choose it?
[Rabbi Michael Abraham] No, again, you’re begging the question. You assume that if there is no cause, then that’s randomness. No. No cause means either randomness or teleology or choice. No, that’s exactly the point. That’s the assumption I’m attacking.
[Speaker C] But how did we escape causality? The question was causality.
[Rabbi Michael Abraham] There’s no need to escape. You assume that we need to escape from something because you assume there are only two possibilities: either causality or randomness. And I’m saying, that’s not true—your assumption is wrong. There’s nothing to escape from.
[Speaker C] What, because we escaped to the future, that solved the problem of this option of…
[Rabbi Michael Abraham] We didn’t escape. There are three possibilities. You call it escaping because that already has a negative connotation. No—there are three possibilities.
[Speaker C] No, but I arrived at the two options because I understood intellectually that either you choose… I know from life that people either choose randomly—they toss a coin—or they choose. If we invent purpose… no, the opposite, if they choose, if—
[Rabbi Michael Abraham] If you know from life that they choose, then everything is fine, so what’s the problem?
[Speaker C] Why did we invent the word “purpose”? We could just as well have invented some other word and said that that’s a third option.
[Rabbi Michael Abraham] What does it mean that they choose? To choose is purpose. That’s what choosing is. You choose something because you want to achieve…
[Speaker C] But the question that troubles us at the root is why you chose—that’s the problem.
[Rabbi Michael Abraham] No, “why did you choose” is a mistaken question, because when you ask “why did you choose,” you’re basically expecting…
[Speaker C] No, what brought you to it? What brought you to take that side?
[Rabbi Michael Abraham] Let’s not play with words. “What brought you to it” is the same thing as “why did you choose.”
[Speaker C] Well, obviously.
[Rabbi Michael Abraham] And the answer to that cannot be… you’re asking an illegitimate question. Why? Because when you ask why I chose, you’re basically assuming there is some cause that made me choose, but the whole claim is that choice has no cause. So there is no point in asking why you chose.
[Speaker C] Right, right. So then when I say “purpose,” I could just say ABC and claim I have a third option and I solved this dilemma. You don’t understand ABC, so I also don’t understand what purpose without a cause means. The fact that I called it “purpose,” which sounds like a nice word, doesn’t solve the problem.
[Rabbi Michael Abraham] That solved the problem; you just don’t understand. Fine. I do understand. I see three possibilities here. It could be that you don’t see them, so you’ll be a determinist, fine. But again, that’s a specific argument. I’m talking at the principled level. At the principled level I want to claim that even when we are presented with a dichotomy, that dichotomy is not necessarily correct. It could be a false one. Now, it may be that about some dichotomies we won’t agree. You’ll think they’re real and I’ll think they’re false. I don’t care. Those are already arguments that have to be conducted separately in each case. I’m talking only about the logic of the matter, and the logic of the matter is that even when a dichotomy is presented to us, we shouldn’t take it for granted that that’s really how it is. Sometimes it’s a false dichotomy, and therefore it’s worth considering that when we lay out the options for discussion.
Okay, so up to this point, what I’ve really done is talk about setting up the alternatives. Defining a state of doubt is always doubt between several possibilities. And until now I discussed the question of how I set up the different possibilities. So now I’m in a state of doubt between several possibilities. What do we do in such a situation?
What you do in such a situation is, basically, you have to make decisions under conditions of uncertainty. Right? Meaning, again, if I have a way to determine which possibility is correct and simply eliminate the others because they contain an internal contradiction or something like that, fine, then there are no real alternatives, so the doubt was only an apparent doubt. I’m talking about real doubt, where even after I formulate a position and reach a conclusion, that doesn’t mean I’m certain. It could be that the other possibilities are correct. I think this is the correct possibility, but not that I have a logical proof or a mathematical proof that it’s correct.
That’s called decision-making under conditions of uncertainty. Right? When I decide how to invest in the stock market, it’s not that I reached the conclusion that this is the correct way and everyone else is talking nonsense. Otherwise that’s not really decision-making under conditions of uncertainty. Then I have certainty. Decision-making under conditions of uncertainty—where statistics is a central tool for this, probability or statistics—is basically decision-making in a situation where even after I decide, the other possibilities are not ruled out. They still exist; I just claim that they are less probable, and therefore I choose that option. In other words, I start speaking in the language of probability, plausibility, statistics, things of that sort.
Therefore, after I’ve set up the state of doubt, I need to make sure that the doubt is a real doubt. Meaning, if I have two possibilities, but I have a logical proof… Let’s say I have two possibilities: one possibility is that the sum of the angles in a triangle is 180 degrees; the second possibility is that the sum of the angles in a triangle is not 180 degrees. Two possibilities. Am I in doubt? The answer is no, because I have a proof that it’s 180 degrees. The possibility that it’s not 180 degrees is a hypothetical possibility; it’s not true—in Euclidean space and all that, leave me alone with other spaces. So it’s not really that two possibilities stand before me. Once I determine it, I arrive at the conclusion that there isn’t really any doubt here. The two possibilities don’t both exist.
When I’m talking about decision-making under conditions of uncertainty, it means I have no proof against one possibility or in favor of the other possibility. They all remain standing, and even after I decide, they all remain here on the board. The table. And I still have to make decisions about which of the possibilities I’m going with. And here too—that’s the great innovation of statistics and probability and so on—here too there is a rational way to make decisions. There’s no way to resolve it or say there’s no doubt here. There is doubt, and that doubt will continue to accompany us afterward as well. And still there is a way to make decisions under conditions of uncertainty. Meaning, rationalism doesn’t stop in areas of certainty. Rationalism also exists in the world—in worlds of doubt. And there too there are still ways in which we make decisions.
Now, among the possibilities for decision-making, I want to talk a bit about types of decision-making in such situations. First of all, I want to talk about two concepts that are often confused with one another: plausibility and probability. They look similar, but they’re not the same thing. Right? People ask me, what is the probability that God exists? How do you answer such a question?
Now, when people ask me what the probability of something is, you need to assume some distribution, a sample space, an event space. There has to be some sort of background map that tells me what the possibilities are, what the weight of each possibility is, and now I can answer the question of what the probability of that possibility is. Say I have a fair die. Then the chance that it lands on each face is one-sixth. So I know the distribution; I have the space of possibilities. I roll the die twice. They ask me what’s the chance that you’ll get five and six. Right? So I say: I have the space of possibilities, 36 possibilities. I can count how many of those possibilities are five and six. That number divided by 36 is the probability.
In other words, when people ask me what the probability is of getting something, it is always assumed that in the background there is some defined probability space. Right? I know what the possibilities before me are; that’s everything we’ve talked about until now. I laid out all the possibilities. I know the weight of each possibility. For example, with a die, I know I have six possible outcomes. More than that: I also know how to weight them. Meaning, I know how to assign each of the possibilities a weight of one-sixth. Now they ask me, what’s the chance you’ll get a two? The answer is one-sixth. What’s the chance you’ll get an even number? The answer is one-half, sorry. What’s the chance you’ll get a result smaller than three? One-third, right—one and two—one-third.
Why, how do I know how to answer these questions? Because when people ask me about chances or probability, I simply count possibilities. It’s basically combinatorics. I count possibilities, and when I count possibilities, that gives me the tool for calculating probabilities of things. Weighted possibilities—I don’t always have equal weights, not important; sometimes there’s a distribution—but still I count possibilities in a weighted way and arrive at the answer of what the probability will be.
But if they ask me what the probability is that God exists? I don’t know how to draw all the… The possibilities are either that God exists or that God doesn’t exist. So is the probability one-half? There are two possibilities? No. Why not? Because I don’t know the distribution. Meaning, I don’t know whether the weight of each of these two possibilities is equal. Since I have no weight for each of these two possibilities, I can’t answer that the answer is one-half. With a die that I don’t know is fair, if they ask me what the chance is that it lands on two, I can’t say one-sixth. I don’t know the distribution. Give me the data about the die and I can try to answer that question.
Therefore, it’s not enough to locate the possibilities; you need to understand what the distribution is and what the weight of each one is. So sometimes I don’t know what all the possibilities are. Sometimes I know what the possibilities are but I don’t know the weight of each one. And in all these situations I basically can’t use the tools of probability. I can’t calculate the result of how plausible something is.
But contrary to what people often think, I can still speak here about plausibility. Not probability, but plausibility. For example, someone can come and say: look, this world didn’t come into being by chance. Therefore, if there is such a complex world, then apparently there is a God. I have no way to make a probabilistic calculation and tell you what the chances are that God exists or doesn’t exist. But I can still tell you: listen, it seems highly plausible to me that God exists. And here I need to use plausibility, not probability. Plausibility rests on reasoning, not on calculation. Probability rests on calculation. Okay?
So it’s true that very often I don’t have a sample space, I don’t have a distribution, I don’t have things of that type, and then the statisticians say, forget it, then you’re just… you’re not allowed to talk here. You can’t talk in such situations, because it isn’t mathematically defined. For example, once I had some… there’s an interesting blog by Gadi, not exact, what’s his name. A math blog, Gadi something, not Alperovitz, I don’t remember what his name is. Alexandrovich maybe, Alexandrovich I think. So I argued with him a few times, and among other things there was one time I said, let’s say I’m talking about the fine-tuning argument, right, about the values of the constants in physics—what the speed of light is, what the gravitational constant is—and the claim is that if you change the values of the constants a little, then reality would collapse, meaning there would be a crash, there would be no life, nothing interesting would be formed here, nothing complex.
So that’s the fine-tuning argument. Now let’s ask, at the hypothetical level, suppose that for life to emerge you need exact values of the constants—not values that are almost precisely this number, but it has to be exact, meaning only this value gives life. Okay. Now I ask: what is the chance that the value of… let’s say there is one constant that is distributed uniformly between zero and one. What is the chance that the result will be 0.4? Zero, right? There are infinitely many possibilities, but actually that’s not really defined. When there is a continuum of possibilities, you need to talk about probability density, not probability. Meaning, probability density always talks about an interval, the probability of a certain interval, as small as you want, but an interval. You can’t talk about the probability that you’ll get exactly 0.4. Right? There isn’t any. If there are infinitely many possibilities, it isn’t defined probabilistically.
Why isn’t it defined? You could say it’s zero. No, that’s also not exactly zero, because if it were zero, then how do all the chances add up to one? You have to add up all the possibilities, the chances of all the possibilities, the probabilities of all the possibilities, and the result has to be one. You won’t be able to define it here in that way. You have to speak about intervals, as small as you like, yes, moving to a continuum—you can’t talk about discrete events.
Now I ask whether I am still allowed to ask the question: suppose that in this world the value of the constant is 0.4, and that’s what came out. Okay, what does “came out” mean? That’s reality, let’s say. And only 0.4 gives life; if it were 0.3999, then it wouldn’t give life—only exactly 0.4. Right, 9s forever is 0.4, but not forever. Okay, so only that gives life. Now I ask: what is the chance that this happened on its own? I think it’s only the handiwork of the Holy One, blessed be He, because it is not plausible that we would get this exact number, the one that exactly gives life. Okay? That’s the fine-tuning argument.
So the mathematician will tell me: what do you mean, what’s the chance? You can’t calculate the chance of such a thing; the problem is not mathematically defined. You can’t talk about the probability of a discrete result in a continuous space. It just isn’t mathematically defined. Okay, but you also can’t say “the Holy One” with the definite article. No, no, don’t take me there, I’m not getting into the Holy One. I’m talking about statistics here. The Holy One here is just an example; leave me alone with theology right now. I’m using the example only to illustrate the idea, okay?
So now I say: then that means that if we have exactly this precise value of 0.4, that means the Holy One probably made it; it didn’t happen by itself, it can’t be. Now I say: but when you do the probabilistic calculation—there is none. The probabilistic calculation is not defined. You can’t talk about the probability that it will be 0.4 if there is a continuum of possibilities, right, between zero and one. Now can I still claim: yes, but still, in such a case it is clear that the Holy One did it? I think so. I had an argument about this with that Gadi. Clearly yes. I won’t talk about probability; I’ll talk about plausibility. But still, I can’t make a probabilistic calculation—the tool of probability isn’t available—but it isn’t true that the statement is undefined. The statement is defined; I just don’t have a mathematical way to use the tools of probability here. Okay, fine. But clearly the philosophical logic—yes, to say that such a thing is not plausible as something that happened without a guiding hand—that I can say even without a probabilistic calculation.
And therefore it is very important to understand that when I talk about probability and when I talk about plausibility, I am talking about two different things. And very often it is precisely mathematicians who get stuck on this point, because if it isn’t mathematically defined, then from their perspective you can’t talk about it. And I say, that’s not true. At the philosophical level I can talk about it even when it isn’t mathematically defined. And very often the discussions, when we want to resolve doubts or deal with states of uncertainty, I need to use plausibility and not probability.
Maybe an example from the last two columns I wrote: I wrote about transgender identity and asked what the halakhic status of a transgender person is. Let’s say a woman who underwent surgery and turned herself into a man at the level of sex—yes, sex. Now the question is what her gender is, right? What is the gender? Okay? Or not; choose whichever you want. My claim there was this.
Maybe now I can do a conceptual analysis and ask whether, for example, the exemption from positive time-bound commandments for a woman—is that because of her sex or because of her gender? Now from the Talmud we won’t be able to prove it, because in the Talmud it always goes together. But still—yes, but what about an androgynos? There’s an androgynos in the Talmud, which is… No, no, no, I showed why the androgynos also doesn’t help here. Not important; you can see that in my columns. Okay, it doesn’t help here.
So the point is, again, this is only an example, so I don’t want to get into too many details. So now I say: I don’t know whether it’s sex or gender. Okay? So I’m in doubt, basically, right? What do you do in a state of doubt? On the face of it: with a Torah-level doubt we rule stringently; with a rabbinic-level doubt we rule leniently. So regarding positive time-bound commandments, I think that if we are in a state of doubt, then the person has to observe them. Let’s say a woman who became a man or a man who became a woman, it doesn’t matter—out of doubt, on the face of it, they should observe positive time-bound commandments, because a Torah-level doubt is ruled stringently. There are opinions that with a positive commandment we do not say that a Torah-level doubt is ruled stringently, only with prohibitions. The simple reading is that we also say it regarding positive commandments. So that is one possibility.
But someone can come and say: I have a reasoning, and the reasoning says, look, it is not plausible that the Sages made a person’s status with regard to Jewish law—what he is supposed to do halakhically, spiritually, ethically—depend on the question of what sexual organs he has. Why would that matter? That’s a reasoning. Again, I have no proofs; I’m talking about reasoning. My reasoning says that it makes much more sense that the laws, at least those laws that were stated regarding women, depend on gender and not on sex. I’m saying this as a matter of reasoning. And in his functioning—that when I come with certain halakhic claims or demands, or don’t come with certain halakhic demands, toward a person, it stems from the person’s personality and not from their sexual organs. Because personality is more relevant to the service of God, to spiritual matters, to observing commandments, and so on. That’s how it sounds to me from the standpoint of reasoning. Okay? Without getting into the argument now; it’s just an example.
Now people say: yes, but you’re in doubt—Torah-level doubt stringently, rabbinic-level doubt leniently. I say no: I’m in doubt because I have no probabilistic way to decide between the two possibilities. I don’t know. Either it depends on sex or it depends on gender. But I have considerations of plausibility—not probability, plausibility—that tell me that in terms of plausibility, in my opinion, it depends on gender and not on sex. Okay? Now as far as I’m concerned, that too is a consideration. I can also resolve doubts through plausibility, not only through probability. Okay? So that’s exactly an example of it. In other words, what we call reasoning—“why do I need a verse? It is reasoning”—what the Talmud calls reasoning is basically considerations of plausibility and not considerations of probability. Considerations of plausibility.
Can I know what “plausibility” is in English? Reasonable. Reasonable, yes, as opposed to probability, right? Meaning, as opposed to probability. Yes. Okay. But “reasonable” is the broadest and most subjective thing there is. Yes נכון. Talk about reasonability, yes? Meaning it’s reasonable and not reasonable—it sounds binary, either it’s reasonable or it isn’t. I say no, there are different levels of plausibility. But it’s not probability; I’m not quantifying it in numbers, I don’t know how to calculate it. Okay? And I still claim that one can talk about resolving a state of doubt or making decisions in a state of doubt by the force of considerations of reasonability, not of probability. Okay? And that’s what is called in the Talmud “why do I need a verse? It is reasoning.” The reasoning being discussed there is not probability; it is plausibility. In other words, this is what seems sensible to me. Why? I don’t know. I don’t know how to present a calculation for why it is more sensible than the opposite. I have no calculation; it’s not probability. But it is not true that the moment there is no probabilistic calculation, I’m supposed to keep quiet. No—I can still speak the language of plausibility.
On the other hand, of course, maybe we’ll talk about this later. Let’s say I reach the conclusion that it is highly implausible that God does not exist. Right? Because such a complex world could not have come into being without God. Okay, let’s say. Now there are two possibilities: either God exists or He does not exist. So is that a proof that God exists? Because if it is highly implausible that God does not exist, then apparently God exists.
Now you understand that what I did here is like what I do in probability. Because the sum of the probabilities of these two possibilities is one. Right? The probability that God exists plus the probability that God does not exist is one. Correct? There is no third possibility; this is the law of the excluded middle. Fine? And they are opposites. So therefore they have to sum to one. Okay? Now, if I could find a number that tells me what the chance is that God does not exist—say 0.0001—then I know that the chance God exists is 0.9999, because it is the complement to one.
But if I speak in the language of plausibility and not probability, then I don’t have a number for the chance that God does not exist. But I can still do one minus the plausibility. As if—one… it’s not really a mathematical operation, but I can still prove things by negation. I can still say that if it is very, very implausible that God does not exist, then that is a proof that He does exist. Even though I don’t have numbers here, only plausibilities. But the plausibilities also…
Rabbi, you’re using the word proof. What? Rabbi, you’re using the word proof. Yes. Really a proof? A proof in the philosophical sense, not the mathematical sense. Meaning, a proof in a philosophical sense. I’m saying it’s a good argument in favor of the existence of God—let’s put it that way. Fine, “proof” is a strong statement.
So I’m saying that even when I’m talking about plausibilities, I can still bring a proof by negation. If I ruled out the plausibility of option A, then I’ve shown that option B is much more plausible. Even though I’m not talking about numbers and they don’t sum to one, because these are not probabilities but plausibilities. But you see that the ways of thinking can still be relevant even when I’m talking about plausibilities rather than probabilities.
In the halakhic world, what marks plausibilities is what is called reasoning. Yes, what I said earlier—“why do I need a verse? It is reasoning.” What connects to the world of probability is the laws of majority. Right? If we follow the majority—nine stores, ten stores, all questions of that kind—there I really count possibilities, and then I can speak even in the language of probability, not only in the language of plausibility. I’ll qualify that a bit later, but for now, for purposes of the introduction, that’s good enough. Okay? So I have plausibility and I have probability, and one has to notice that these are two different things, both of which are legitimate even though they are not the same thing.
In other words, anyone who says that only probability is scientific is mistaken. Is plausibility always subjective? Depends what you mean by “subjective.” Meaning, plausibility can involve disagreements. That’s true. But the question is whether because there are disagreements, there is no one truth? That’s another question. Meaning, even when there are disagreements, it could be that I’m right and you’re wrong. The fact that we disagree—so what? It doesn’t mean we are both right or that there is no truth. So the question is what you call subjective.
If by subjective you mean that probably a lot of opinions will appear here—then probably yes, much more than in cases of probability. Because with probability there is a calculation, it’s easier to persuade. But the question is whether plausibility is an indication of truth. My answer is yes. In that sense it is not subjective. And if we have an argument—you think it’s plausible that God exists and I think, say, that it’s plausible that God does not exist—then we have an argument. One is right and the other is wrong. But the fact that there is an argument does not mean there is no truth here. Okay?
That is often an argument in favor of postmodernity, right, in favor of pluralism. Here, look, there are lots of possibilities, lots of opinions in the world. Therefore clearly there is no truth, and truth is… truth is absent, flocks upon flocks. So there are many truths and all kinds of things of that sort. But that is a mistake. It is a mistake at the logical level. Because the fact that there are many opinions is a descriptive fact. Right, factually there certainly are many opinions. There is nobody who is not a pluralist in that sense, in the descriptive sense. In the descriptive sense there are many opinions in the world, yes, obviously. You can’t deny that.
Substantive pluralism wants to say that there are many truths, not that there are many human opinions. And you can’t derive substantive pluralism from descriptive pluralism. Meaning, the fact that there are many opinions among people does not mean that there are also many truths. Because on the way from many opinions to many truths, you also assume that all the people are right. So if there are many opinions and all the people are right, then there are also many truths. But no, I do not assume that. I assume that if there are many opinions, then one is right and the others are wrong. If they are opposite opinions, for example. So one is right and the others are wrong. Therefore, the fact that there are many opinions does not mean there are many truths. Okay?
Are there halakhic implications, for example regarding a judge who makes a mistake, or regarding a rebellious elder, or something like that, if you disagree in terms of plausibility or in terms of… We’ll still talk about those things. It’s a bit tricky, but I’ll talk about those things. There will be examples, and we will have to discuss whether those examples are plausibility or probability. Because for example… majority… I’m just throwing out one example that we’ll deal with later. A majority that is before us. I have ten stores in the city, nine sell kosher and one sells non-kosher. I found a piece of meat in the market. Here people usually connect it to probability. You count possibilities. There are nine kosher stores and one non-kosher, so there’s a ninety percent chance it’s kosher. Okay? So people usually connect that to probability.
The same with a majority that is not before us. A majority that is not before us—for example, most women are not barren by nature. So I can do statistics and see that eighty percent of women are not barren by nature, or ninety percent, I don’t know, so I have a majority; I can even quantify it. I will argue later that a majority that is before us is plausibility and not probability. Yes, but plausibility is very problematic. Because I’m basically claiming something simple. To use a comparison—not wrestling, not wrestling, boxing. There’s a knockout and there’s winning on points. Always on points. Wait. With points there’s no problem, because there are judges, they know how to count the points, no tricks. Whoever amassed one point more is the winner. Okay. Here with plausibility we are basically in the same situation, except that we don’t know how to count the points. Right. Not only do we not know how to count the points—what is the decision in the end? What is the decision in the end? There is no objective factor that can make the decision, no judge. Exactly, so what is the decision in the end? The decision is what I think. That has no meaning, because in the end what basis will we decide on? If a person, say, for psychological reasons or for any social reason—you raise objections—then his plausibility will lean in that direction. They’ll always be irrelevant. You’re raising skeptical objections. You say, look, but the fact that you think this way—so what? Maybe you’re biased, maybe all kinds of things influenced you. Leave it, I’m not getting into skepticism. I’m not a skeptic. Someone else will be a skeptic. What do you mean maybe? Your decision ultimately falls because of an emotional factor.
That’s what you think. That’s your reasoning. But your reasoning is biased because I disagree with you. I think it is not true that everything we say is biased. No, the points are points, only since we don’t know how to count them, in the end we will incline toward psychological or social bias. No, we will decide that it is true. You call it “incline” because you think it’s a bias, and I don’t agree with you. Now in this argument itself I’ll say that you’re biased, and you will probably say that I’m biased, and we won’t get out of it. So what’s the point of conducting this argument? I’m simply saying: as far as I’m concerned, I think there is truth. I’m not a pluralist. I think there is truth. So the existence of a disagreement or the existence of different opinions changes nothing. I’m not troubled by peer disagreement. I also have a column on the site dealing with peer disagreement, yes, it’s an issue in philosophy.
Of course I will seriously consider the opinions of others, of course. No need to be arrogant and be certain that I’m the one who is right. So you have to listen to what others say. But after I’ve listened and weighed it all and in the end reached a conclusion, that is my position, and I don’t care that others think differently. What did you want to comment?
Rabbi, I wanted to ask: you said earlier that it’s an indication—you used the word indication of truth and not truth itself, which is something else. What is an indication of truth? I don’t remember. That’s a word you used two minutes ago. You said that since it is very plausible that the Holy One, blessed be He, exists, or that I don’t know, that we have a soul, or any claim about the metaphysical—so you used… you said because it is very plausible, or because the opposite is very implausible, and then you used softer language, you said that this is an indication of truth. I’m willing to give up “indication” here; I don’t think that’s necessary. I already don’t remember what wording I chose, but I can say no, as far as I’m concerned this is true. If it is plausible, then it is true. Leave indications aside. That’s a philosophical question. Is the sense of plausibility inside me an indication of truth, or is it my tool for determining truth? Fine, that’s a philosophical discussion that I don’t think is worth getting into here. As far as I’m concerned, this is the truth, okay? Not certainty—truth. I said certainty is epistemic; truth is at the ontic level.
So that’s regarding plausibility and probability. Now one small question. Yes. At the beginning of this topic, when I ask what the chance is, is that the same thing as asking what the probability is? Yes. Is there any difference here? Same question, same meaning, same answer. Yes.
So the last point I still wanted to touch on is positive doubt and negative doubt. I’ll mention again the Ein Ayah that we saw, yes, Rabbi Kook on Sabbath page 30, right, “You are my people, my wife, and you are my sons.” Right, someone comes to Rabbi, and he waves him off and says, leave it—for in order to doubt, you need a reason. Therefore, not every time I have two possibilities before me does that mean I am in doubt and it’s fifty-fifty. No. If I have no reason to doubt, then I don’t doubt. Okay? In order to doubt, you need a reason. By the way, that is also true in Jewish law, it’s true everywhere. In order to doubt, you need a reason.
I brought the example of Bertrand Russell’s celestial teapot, which basically says the same thing. If someone comes and tells me that orbiting the planet Jupiter there is a small teapot, I’ll tell him, but I don’t see it. Well yes, because it’s small; you can’t see it. Fine. Now I have two possibilities: either there is such a teapot there or there isn’t such a teapot there. Is it fifty-fifty? So Bertrand Russell says: of course not. Why assume there is such a teapot there? There are two possibilities, but the first possibility is completely implausible, and therefore I reject it. But in order to doubt, you need a reason.
If there were some indication that there is something orbiting the planet Jupiter, and someone came and told me that that something is a teapot—fine. I already saw that there is something orbiting there, some strange thing, and now someone comes and tells me it’s a teapot. Now I say, okay, I have a reason to doubt, because I saw something orbiting Jupiter; I just don’t know what it is. So here I have a reason to doubt, and then I will doubt whether it is a teapot or not. But if I have no reason at all to doubt, then the fact that there are two possibilities does not mean I am in a state of doubt. In order to declare a situation a doubtful situation, you need a reason. Right? That is basically Bertrand Russell’s claim.
And to that I say: correct, but if I have good arguments that there is a God, and now someone comes and tells me yes, and God also revealed Himself and said such-and-such, that is no longer a celestial teapot. It’s like the case where there is something orbiting Jupiter—I already know that. Then someone else comes and says yes, it’s a small teapot. Now I don’t know yes, I don’t know no, but I know there is something orbiting there. So here I already have a reason to doubt. Only now am I doubting whether it is a teapot or not. Okay? This is already real doubt. That’s called having a reason to doubt. The reason does not resolve the doubt. The reason makes the doubt possible. Meaning, if I have a reason, doubt is created. Okay? If I have no reason, then this is not called a state of doubt.
Meaning, it is not enough that several possibilities stand before me in order to determine that I am in a state of doubt. You also have to examine whether there is reason to assume that all those possibilities are real. What in the halakhic context is called an objective basis in the monetary item itself. A doubt that has an objective basis in the property itself, in the halakhic context, looks different from a doubt that has no such objective basis. What is a doubt without such an objective basis? That’s, for example, the case of that ship in tractate Bava Batra, where there is a boat sailing in the river and two people come and each says it is his. That is a doubt with no objective basis in the property itself. Why? Because there is nothing that links that boat specifically to the two of them; anyone in the world could say that the boat is his. There is no reason to assume it is specifically yours or specifically his, other than your own claim. But there is no objective something, objective evidence.
By contrast, if two people are each holding onto a cloak—two people are each holding onto a cloak, each says I found it, not that it’s mine. They both agree it was a lost item and I found it. Now it’s in their hands. The cloak is in their hands. So clearly the cloak belongs at least to one of the two of them, or if they picked it up together then it belongs to both of them. Here there is a reason to doubt. If there is a reason to doubt, this is a doubt with an objective basis in the property itself. Each one has a connection to the item even apart from his claim. The very fact that he is holding the cloak means that the cloak has some connection to him, regardless of his claim. And now he also claims it’s his. That is a doubt with an objective basis in the property itself. Okay? So here I have a reason to doubt; in the case of that ship I have no reason to doubt. But if nobody is arguing with them, then fight among yourselves. If you think the boat is yours, then fight it out between yourselves, and whoever takes it takes it.
So that is the point of a reason to doubt. I said, if someone had told me, I don’t know, twenty years ago, that the United States has a Black president—twenty-five years ago—I would have hospitalized him on the spot for temporary insanity. Okay? A few years later Barack Obama was elected. Now if someone had come and said in 1990, someone had come and said there is a Black president of the United States and he passed Obamacare, the healthcare reform, okay? I would say, and I met Tinker Bell this morning and she also fluttered her wings at me. In other words, don’t confuse my mind—in short, this is a celestial teapot. Okay?
And now, a few years later, Barack Obama was elected. I know there is a Black president of the United States. Now someone comes and says, look, his name is Barack Obama and he passed Obamacare, the healthcare reform. Okay? That’s already not something that… So I don’t know his name, I don’t know what actions he did. So someone tells me his name is such-and-such and he did this and that action. Fine, okay. So why not accept it? Meaning, here at least doubt has arisen, if not more than that, because I already know there is such a president. Exactly like the celestial teapot.
The same thing with the Holy One, blessed be He. If I know that there is a Holy One, blessed be He, from philosophical considerations of one kind or another, and now someone comes and says, and He also revealed Himself and said such-and-such, then this is no longer a celestial teapot, because here I have a reason to doubt. Meaning, there is already a Holy One, blessed be He; now either He revealed Himself or He didn’t—I don’t know—but there is already a Holy One, blessed be He, so there is reason to doubt. If I don’t raise at all the possibility that there is a God, and now someone comes and says that He revealed Himself to him, I have no reason at all to enter a state of doubt.
Therefore, in order to declare that the situation I’m in is a doubtful situation, you need reasons. Not every time several possibilities stand before me is it called that we are in a state of doubt. But now I’ll qualify this a bit. Jewish law distinguishes between a doubt where there certainly exists a prohibition and a doubt where there may not be any prohibition. Right? One piece out of two pieces, or one single piece. Let’s say there is a piece of fat here and I don’t know whether it is forbidden fat or permitted fat. Not exactly meat, yes, whatever, something from an animal, and I don’t know whether it is forbidden fat or permitted fat. This is called a doubt involving one piece. So out of doubt I have to be stringent: a Torah-level doubt is ruled stringently. I’m forbidden to eat it.
What happens if I ate it? Then I violated the prohibition because a Torah-level doubt is ruled stringently. Another case: a doubt involving one piece out of two pieces. I have two pieces. I know one is forbidden fat and one is permitted fat. I took one—I don’t know which of them—I took one of them and ate it. I still have a doubt, right? Maybe I ate forbidden fat and violated a prohibition. This is called a doubt where there certainly exists a prohibition. And its halakhic status is different. Meaning, someone who ate a piece in such a state has to bring a provisional guilt-offering, whereas someone who ate a single doubtful piece does not bring a provisional guilt-offering. A provisional guilt-offering comes only for a doubt where there certainly exists a prohibition.
What is the difference between the two doubts? Someone may say: what, this is fifty-fifty and that is fifty-fifty. What difference does it make whether it’s one piece that I don’t know what it is, or two pieces where I know one is this and one is that but I don’t know which is which? Either way I took one piece and ate it; there’s a fifty percent chance I ate forbidden fat. So what difference does it make? On the face of it, this is only a halakhic difference, not a probabilistic one. The probabilities are the same. Later I’ll argue that no, it is a probabilistic difference, not just a halakhic difference. But for now I’m just setting it up.
What is the difference between saying that in the case of one piece it’s fifty-fifty, and saying that in the case of one piece out of two pieces it’s fifty-fifty? The difference is that one is negative doubt and the other is positive doubt. What does that mean? I have one piece here and I don’t know what it is. There is a possibility that it is forbidden fat; there is a possibility that it is permitted fat. I have no way—I have no reason—to doubt, right? I have no way to know why to assume it is forbidden fat or why to assume it is permitted fat. I have no positive reason to doubt; I simply know nothing. The two theoretical possibilities that exist here are either that it is forbidden fat or that it is permitted fat. I don’t know. I declare that such a thing is fifty-fifty even though I don’t really have a distribution; I know nothing. That I call negative doubt.
What happens in positive doubt? In the doubt of one piece out of two pieces I have good reasons to assume this was forbidden fat and I have good reasons to assume this was permitted fat, because I know that among these two pieces one is forbidden fat and one is permitted fat. Now I took one and I don’t know which one. But I have a positive reason to assume it was forbidden fat, and I have a positive reason to assume it was permitted fat. Okay? This is positive doubt, not negative doubt, even though here too I’m talking about fifty-fifty and there too I’m talking about fifty-fifty.
I’ll give you another example. Let’s say I roll a die. Okay? Now if they tell me the die is fair and ask me to bet on the chance that it lands on five—how much do you stake? Well, one against six, right? Meaning, that’s the chance. Any offer that is more than one against six, I’ll take it, okay? Because it’s worthwhile; that’s the probabilistic calculation, the chance is one-sixth. Fine? Now another case. I have a die. I have no idea whether it is fair or not. I have no idea how it is built. Nothing. I have no information. Now they tell me: you have to bet on the chance that it lands on five. Okay? You have to bet, meaning tell me how much money you put down—you have to bet.
Now I think most people would still bet the same thing, at a chance of one-sixth, even though they have no idea whether the die is fair. An unfair die could always land on five; it could never land on five and always land on two. Okay? You don’t know, you have no clue. But since there are many, many possible dice, right? Fair and unfair and all kinds of varieties, you have no way of knowing. You have no way to prefer one face over another, and therefore you would still assume that the probability of getting a five is one-sixth.
You understand that the calculation in the first case is essentially different from the calculation in the second case. The calculation in the second case speaks of one-sixth because of lack of knowledge. The calculation in the first case speaks of one-sixth because of knowledge. I know the die is fair. In the first case I have a positive reason to be in a state of doubt, and the probabilistic calculation is based on positive knowledge. In the second case I have no positive reason to declare the situation doubtful. I simply have a lack of knowledge, so therefore I am in doubt. I have no reasons to doubt, but I simply know nothing, so I’m in doubt. And the probabilistic calculation also—on the face of it it looks the same, but no. The probabilistic calculation there speaks of a calculation from total lack of knowledge.
Therefore, true, I will probably bet on a chance of one-sixth because I don’t have anything better. All the faces of the die are in equal standing for me; I know nothing about any of them. But the one-sixth chance I spoke of here is a chance that is the result of complete lack of knowledge. So I’m talking about chances, I’m talking about doubts, but actually the situations are situations where there are doubts and calculations that are a result of positive knowledge, and there are doubts and calculations that are a result of lack of knowledge. Negative doubt, or a negative majority, okay, negative probability. Fine? This is another distinction that we need to keep in mind for the coming discussions.
Okay, I’ll stop here. If there are comments or questions. But even in negative doubt all the possibilities still converge to one? Yes. A probability function always sums all the possibilities to one; otherwise it’s not a probability function. After all, practically speaking, one of the six faces will land, right? Even an unfair die will land on one of the six faces. So whatever number or whatever plausibility or whatever probability you assign to each of the faces, clearly the sum has to come out to one, because one of the results will certainly occur.
If you ask me what the status of the following claim is—when a die is rolled, one of the results from one to six will occur—what is the chance that this claim is correct? One hundred percent. One, right? One hundred percent. Okay? That means the sum of the possibilities adds up to one here too. We just don’t know the weight of each one? Correct. We don’t know the relative weight, and we assign equal weight because of lack of knowledge. But it doesn’t matter; even if the weight is not equal or something else, the sum always has to be one.
Let’s say that in the end it turns out that this die is so unfair that it always lands on five. Five is so heavy that it cannot land on any other face, only on five. Okay? In that case too it still sums to one. The chance of getting a five is one, and the chance of getting all the others is zero. Altogether the chances are one. Because in the final analysis, one of the results from one to six will always occur.
Yes, so in the case with the meat, where we don’t know anything about it, which is defined as negative doubt. It’s an interesting question whether the meat is negative doubt or positive doubt; we’ll talk about that more. On the face of it, the meat actually resembles a doubt where there certainly exists a prohibition, because there are ten stores before me, nine kosher and one non-kosher, and the meat came from one of them, because it’s a city whose doors are locked. I meant in the comparison to the provisional guilt-offering, in the case where you have a piece and you don’t know whether it is forbidden fat or permitted fat. Okay, yes. So there it’s as if in truth it’s not fifty-fifty; you can’t know. Maybe it is fifty-fifty, but you attach the fifty-fifty probabilities because you have no ability to prefer one possibility over the other. But that is out of lack of knowledge. It is a probability that is the result of ignorance, not probability as a result of knowledge. Probability too is based on knowledge—not complete knowledge, because if I had complete knowledge I wouldn’t need probability, I would know what will happen. But determining the relative weights of the possibilities can be done on the basis of knowledge. If I know that the die is fair, then I have knowledge in hand, and that assigns a one-sixth chance to each face on the basis of knowledge. The knowledge doesn’t tell me what will happen, but the knowledge determines the distribution. But sometimes I determine distributions on the basis of lack of knowledge. Fine? Goodbye, Sabbath peace. Sabbath peace.