Doubt and Probability—in Halakha, in Thought, and in General—Lecture 9
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
- The structure of the series: defining doubt and rules of decision
- Truth, facts, claims, certainty, and doubt
- Epistemology and the sources of knowledge
- Mathematical certainty versus factual uncertainty
- Logic, axioms, Platonism, and the “if-then”
- Practical certainty and practical doubt in Jewish law
- Monetary connection, possession status, and monetary law
- “To be in doubt you need a reason” versus personal anxieties
- Majority, kosher examples, and the mistake of assigning probabilities
- Probability, prior information, and conditional probability
- Types of doubt: negative doubt and positive doubt
- Doubt without a specific indication: an unknown die
- Rules of decision: deciding versus acting under doubt
- Checking whether the possibilities are complete and undoing false dichotomies
- Epistemic doubt versus ontic doubt (ambiguity in halakhic reality)
- An analogy to quantum theory and clarification of the absence of logical contradiction
Summary
General overview
The lecture continues a series on doubt and probability in Jewish law and more generally, and sums up the previous lectures before moving on to the section on rules of decision. The speaker distinguishes between truth as belonging to claims and certainty as belonging to the person, and defines doubt as a person’s epistemic state in relation to a claim, not a property of the facts or of the claim itself. He argues that full certainty is possible mainly within a logical-mathematical framework, whereas claims about the world are never absolutely certain but at most practically certain; therefore, the laws of doubt deal with practical doubt that requires a real reason for it to arise. He then defines how one identifies a state of doubt, how one weighs possibilities and probabilities based on the information given, and adds a distinction between epistemic doubt and states of ambiguity in the halakhic reality itself.
The structure of the series: defining doubt and rules of decision
The series is divided into two parts: first, defining the concept of doubt and the different kinds of doubtful situations, and then examining the forms of deciding in cases of doubt. The speaker sets up in advance the connection between halakhic rules of decision, especially majority rules, and probabilistic calculations. He notes that in the previous lectures he had already reached the second part, but he returns to summarize the foundations of the first part so that those who were not there can follow along.
Truth, facts, claims, certainty, and doubt
Truth and falsehood are properties only of claims, assessed by the correspondence between the content of the claim and the state of affairs in the world, whereas a fact is by definition a fact, and there is no meaning to an “incorrect fact.” Certainty is a judgment about the person’s state in relation to a claim, and so we speak about levels of confidence from zero to one hundred with respect to claims, not with respect to facts. Doubt belongs to a third plane: the person is doubtful regarding the claim, even though the claim itself is either true or false and reality itself is not “in doubt.” A doubtful situation is defined as a person’s lack of certainty as to which possibility is correct among several possibilities, such as “the meat is kosher or not kosher.”
Epistemology and the sources of knowledge
Factual claims are made by human beings, and they raise the question “How do you know?” through the senses, testimony, reading, scientific research, or other ways of knowing. The speaker assigns the question of knowledge to epistemology and clarifies that the assessment of certainty is a function of the knower’s condition. He emphasizes that even a true claim can be said as a mere guess, without knowledge, because truth does not depend on the route by which one arrived at the claim.
Mathematical certainty versus factual uncertainty
The speaker distinguishes between analytic-mathematical claims, which have absolute certainty, and claims about the world, which are not certain. He argues that mathematical claims do not say anything about the world, and the moment one applies them to reality, one must assume that the world is described by the mathematical model, and that assumption is not certain. He gives examples from Euclidean geometry versus the curvature of space according to Einstein, and emphasizes that empirical refutation does not undermine a mathematical theorem but rather the physical assumption that the theorem applies to the given situation. He uses the example of adding forces as vectors to show that the laws of nature determine how one “adds” physical magnitudes, not mathematics alone.
Logic, axioms, Platonism, and the “if-then”
In mathematics, proofs rest on assumptions, and certainty does not come from proof but from confidence in the assumptions and in logic itself. The speaker formulates mathematical truth mainly as an “if the assumptions, then the conclusion” claim, and says that asking “How do you know the inference?” leads to the claim of an immediate grasp of logic. He presents a Platonist position according to which there is an abstract world in which the axioms of geometry are true claims about that world, while others would see them as definitions only. He concludes that factual claims are never fully certain, but a claim about claims can belong to logic and therefore can be certain, such as the statement “No factual claim is certain.”
Practical certainty and practical doubt in Jewish law
The speaker distinguishes between general philosophical doubt and practical doubt that activates the laws of doubt, and argues that only when there is no practical certainty do we treat the case as one of doubt. He gives the example of the phrase “To be in doubt you need a reason” through the story in Sabbath 30 about someone who says to Rabbi, “Your mother is my wife and you are my son,” and Rabbi rejects him because there is no practical justification for raising a doubt. He defines a state of practical doubt as requiring a rationale that justifies hesitation, and not merely the hypothetical fact that there are two possibilities in the world.
Monetary connection, possession status, and monetary law
The speaker uses the distinction between “there is a monetary connection” and “there is no monetary connection” to show when a religious court recognizes a doubt. In the case of a boat on the river that two people claim, there is no independent indication connecting the object to the claimants, and therefore “whoever is stronger prevails”; this is not a doubt that requires a court decision. In the case of two people holding a cloak, possession creates a legal link, and therefore there is a monetary connection and a state of doubt that leads to the rule “they divide it.” He emphasizes that signs or indications can turn a situation with no monetary connection into one that does have such a connection.
“To be in doubt you need a reason” versus personal anxieties
The speaker gives the example of a boy asked whether he is bar mitzvah age for a prayer quorum and argues that there is no need for discussions of credibility without a real reason for suspicion, because in practice one believes the claim so long as no relevant suspicion has arisen. He sharpens the point that Jewish law is not determined by anxieties and “obsessions” but by what it is philosophically correct to think in the given situation, and someone who raises groundless concerns does not create a halakhic doubt. He cites the principle “we execute and stone on the basis of presumptions” and shows that practical certainty is established even in capital cases so long as there is no reason to arouse doubt, despite the theoretical existence of exceptional possibilities.
Majority, kosher examples, and the mistake of assigning probabilities
The speaker explains that “follow the majority” operates only where there is doubt, and cites Rabbi Yonatan Eybeschutz’s answer to a priest that the Christian majority is not decisive where there was no doubt to begin with. He illustrates that when there is a piece of meat with a kosher seal, one should not apply the majority of non-kosher butcher shops, because there is no practical doubt here, only the hypothetical possibility of forgery without any indication. He also adds a philosophical example about probabilities regarding the existence of God and argues that if there are only two possibilities, the probabilities must add up to 1, and therefore one cannot claim that both possibilities are “unlikely” to different degrees without explaining what absorbs the rest of the probability.
Probability, prior information, and conditional probability
Probability is described as a tool that measures a person’s level of certainty regarding different possibilities, and therefore depends on the information available to him. The speaker demonstrates this with a fair die: before additional information, the probability of a four is one-sixth, and if it is known that the outcome is even, the probability of a four is one-third. He shows that practical decisions depend on information, and connects this to problems of information, such as the advantage of someone who holds excess information within economic systems.
Types of doubt: negative doubt and positive doubt
The speaker distinguishes between a negative doubt of absence of knowledge, such as one piece of meat where we do not know whether it is pig or cow, and a positive doubt of “one piece from among two pieces,” where it is known that one is forbidden and one permitted and they became mixed up. He notes that this distinction appears in the law of the provisional guilt-offering, which is brought specifically for “a doubt where prohibition had a fixed presence,” in the case of one piece from among two pieces. He suggests that even if the probability appears to be the same, there may be either a probabilistic difference or at least an intuitive-legal difference in the significance of failure where there are positive reasons for the mistake.
Doubt without a specific indication: an unknown die
The speaker describes a situation of a die about which we have no information as to whether it is fair, and argues that one still prices the bet as though there are six equal possibilities in the absence of any reason to prefer one side. He uses this to show that negative doubt can still be practical doubt, because the known structure of the possibilities defines the doubt even without a specific indication. He distinguishes this from the case of an arbitrary claim such as “You are a mamzer,” which has no anchor that generates practical doubt.
Rules of decision: deciding versus acting under doubt
The speaker notes that there are rules that decide a doubt and remove it, such as accepting testimony, as opposed to rules that leave the doubt in place but determine how to act despite it, such as “the burden of proof is on the claimant.” He defines “dealing with doubt” as either deciding the legal reality or determining permitted/forbidden action while remaining in a state of not knowing. He postpones the detailed classification of rules of decision to later in the series.
Checking whether the possibilities are complete and undoing false dichotomies
To define a doubt correctly, one must make sure that the possibilities presented really cover the whole space and that a third possibility has not been omitted. The speaker describes how people are presented with false dichotomies and gives a social example of “Religious Zionist or Haredi” as against additional options. He argues that identifying missing possibilities is a necessary step before applying rules of decision or probabilistic considerations.
Epistemic doubt versus ontic doubt (ambiguity in halakhic reality)
The speaker distinguishes between epistemic doubt, where there is a defined truth but the person does not know it, and ontic doubt, where the legal state itself is not sharply defined. He gives the example “One of your two daughters is betrothed to me” and argues that this is not a situation where there is some specific married woman and only information is lacking, but rather an ambiguous state in which there is no unequivocal answer to “which one” is betrothed, even from a comprehensive perspective. He gives another example of “I consecrated one of five coins” without defining which one, and describes this as a state in which a status change takes effect on one unit but without the possibility of pointing to it. He parallels this to the concept of “superposition” and emphasizes that the correct formalization is a combination of coherent alternative states, not a simple contradictory claim of “both and both.”
An analogy to quantum theory and clarification of the absence of logical contradiction
The speaker argues that the layman’s interpretation of “the particle went through both slits” creates an apparent contradiction, whereas the formal description is a sum of states, not the simultaneous attribution of contradictory properties. He says that a question such as “Where did the particle pass?” is not a legitimate question within the proper framework of description, and emphasizes that a genuine logical contradiction would collapse any ability to infer conclusions from the theory. He concludes that intuitive understanding is sharpened by formalization, and that foundational questions such as “Why is logic certain?” remain, in his view, at the level of something “self-evident” that admits of no further breakdown.
Full Transcript
[Rabbi Michael Abraham] Okay, as I wrote on WhatsApp, instead of starting a new series today, I’m simply going to continue a series that began on Fridays. I already gave eight lectures in that series, and now we’re going to continue it on Thursdays. Now, since I assume some of the people here weren’t there on Fridays, I’ll first give some kind of summary of the eight lectures that were already given, and after that we’ll continue onward. So, the topic of the series is basically doubt and probability in Jewish law and in general. So the structure is supposed to be that first of all we want to talk about the concept of doubt itself. Meaning, what is doubt? To define different states of doubt, and then to ask what we do when we find ourselves in states of doubt. In other words, the first part is simply to define what a state of doubt is, or how I decide that I’m in a state of doubt. That itself is a body of teaching that needs to be learned or gone through. And after I’ve decided that, we enter the world of the different forms of decision in cases of doubt. So the series is basically divided into two parts. I’ve actually already reached the second part, but since we’re in summary mode, I’ll go back over the first part as well. Of course, in the rules of deciding doubts, we’ll have to see what the relationship is between them and probabilistic calculations, especially the decision rule of majority. What is the connection between that and probability? So we’ll talk about those things too, about different rules of decision and so on, which is really what I started at the end of the last few lectures in the series. So, I began with the relationship between truth and certainty. When we talk about some truth, truth is a property of claims. A certain claim says something, and the claim can be true or false. And that’s a property of the claim. How do I check whether the claim is true or false? Usually the accepted view — and again, philosophers make a whole big fuss out of this — but the accepted view is that we make a comparison. A comparison between the content of the claim and the state of affairs in the world that the claim describes. Let’s say the claim, “It’s dark outside now,” is a true claim if I check, observe the world, and I see that it really is dark outside. Then there is correspondence, a fit, between the claim and the state of affairs in the world that it describes, and then the claim is true. If there is no correspondence, then the claim is false. These concepts of truth and falsehood are properties of claims. There’s no such thing as true facts and untrue facts; there’s no such thing. Truth or falsehood is a property of a claim, not of facts. A fact is by definition a true fact. An untrue fact simply doesn’t exist, such a thing. There can be a factual claim that is not correct. An incorrect claim does exist; an incorrect fact cannot exist. So a claim can be true and can be false; facts are facts. A claim whose content describes a fact is a true claim. A claim that describes something that is not factually correct, that is not a fact, that does not exist in the world, is false. So truth and falsehood are properties of claims and not of facts. Now the question, of course, is how do we know?
[Speaker B] Rabbi, but facts can also be subjective and interpreted differently by two people. No, no, that’s not facts,
[Rabbi Michael Abraham] that’s already what you think about the fact. Facts have no subjectivity; facts are facts. There can be different people who think the fact is such-and-such, and you think the fact is something else. So we raise two claims, not two facts. We have a claim about what the fact is. I claim the fact is this, and you claim the fact is that. With claims, one of us is right and the other is wrong; one is true and one is false. Or both of us could be wrong. Let’s say it’s zero-sum, let’s say there are only two possibilities. Then one is right and one is wrong, one true and one false — that’s with claims. But there is only one fact, the fact itself. The fact is just the fact. There is no true fact and untrue fact. So that’s regarding facts and the claims about them. And of course these claims are claims usually made by people. Okay? Or by AI. But in principle, claims are made by people. Now, the moment a person makes a claim, he is basically relating to the world. Right? He claims that there is some fact, a factual claim at the moment. So he claims that there is some fact in the world. Okay? That is basically his claim. And then the question arises: how do you know? How do you know that’s the fact? He can say: because I thought so, because I observed, because I listened, I don’t know, senses, something, I read it in a book, I don’t know exactly what. There can be all sorts of different ways by which we become aware of facts. Okay? That’s called cognition. Or epistemology. Epistemology is the theory of knowledge. Yes, cognition is the way people become aware of facts. And there are different ways of knowing things: through the senses, maybe through reading books from other people, through testimony, scientific research of one kind or another, doesn’t matter, scientific generalizations, this kind of reasoning, that kind of reasoning — all sorts of ways of knowing facts. The facts exist in the world. The claims are made by us. But the claims are about facts. So when I make a claim about some fact, the question always comes up: how do I know that? How did this thing, this fact, become known to me, such that I make a claim that this fact is true? That is basically connected to, belongs to, epistemology. Now we have different ways of knowing things. And the result of these ways of knowing things can come at various levels. It can be that I’m sure it’s true, certain, a certain claim. It can be that I’m not sure it’s true, I think it’s true. It can be that I’m sure it’s not true. Of course, the levels of certainty can be however you like, between zero and a hundred, between zero and one. So my knowledge of the world is very often judged in terms of certainty. How sure are you that this is the claim? Okay? One hundred percent means sure, certain. Eighty percent means very likely, sixty percent less so, forty percent already much less, and so on. So that already measures my level of certainty in the claim. Notice, I’m not saying that the claim is — that the fact is — forty percent true. The fact is a fact. The level of certainty relates to claims. There’s no such thing as certain facts. There are claims; maybe if there is such a thing, there are certain claims. About this claim I am convinced that it is true. So after I establish that a claim can be true and can be false, the question is what degree of certainty I have in the claim. The claim is true or false; that has nothing to do with me at all. The claim is true if its content matches the state of affairs in the world. Regardless of that, even if, let’s say I say that it’s night outside now, I close my eyes, I haven’t looked anywhere, and I say that it’s night outside now — I’m guessing. The claim that I said is a true claim, even though I had no way to know it because my eyes were closed. Why is the claim true? Because the content of the claim matches the state of affairs in the world; it really is dark outside now. Therefore it has nothing whatsoever to do with the question of whether the claim is true or not true; it has nothing to do with the question of how I arrived at it. It simply has to do with the content of the claim in relation to the fact in the world. If there is correspondence, it is true, and if not, then not. When I talk about level of certainty, that is already a claim — that is already a judgment that concerns the person. Not facts and not claims, but the person. The question is: the claim can be true, but how convinced are you that the claim is true? So if I say that I’m sure of it, that’s certainty. The statement that this claim is certain is a statement about me, not about the claim. Do you understand? That is, there are facts, there are claims made about the facts, and there is the person making the claim, and the question is how sure he is of the claim. These are three different levels. The question of the truth of the claim has nothing to do with the person at all. It has to do with the match between the claim and the fact. If there is a match, it’s a true claim; if there isn’t a match, it isn’t a true claim. The question of certainty is a question that doesn’t concern the claim at all. It is a question that concerns the person. The question is how sure are you that this claim is correct. What is your level of certainty in this claim. Okay? Therefore certainty is a judgment about a person’s state. It is not a judgment about a claim. Okay? So certainty or uncertainty, meaning the different levels of certainty. Now when we talk about states of doubt, which is the topic of the series, to which of these planes do states of doubt belong? Most simply, to the third. Right? The facts aren’t doubtful. The claims also aren’t doubtful; they are either true or false. Doubt doesn’t belong to the claim. The doubt is the doubt in the person who doesn’t know whether the claim is correct or not. Or the question is how much he knows. Maybe he thinks eighty percent, ninety percent, twenty percent, doesn’t matter, one hundred percent. That describes the person’s state in relation to the claim. So when I say that I am in a state of doubt, that is not a statement about the world. It is also not a statement about language. It is not a statement about the claim. It is a statement about me in relation to the claim. I don’t know if this claim is correct. Let’s say I don’t know whether a certain piece of meat is kosher or non-kosher. Okay? Now, the fact of whether it is kosher or non-kosher is a fact. I don’t know, the Holy One, blessed be He, knows. Okay? That is a fact. The claim “this meat is kosher” — since I don’t know, then I don’t know — but the claim is either true or false. Whoever checks it against the state of affairs in the world will discover a true claim or a false claim. A state of doubt is created when we reach the third layer, the layer of the person’s relation to the claim. The person does not know whether this claim is true or false. On the epistemic plane, yes, on the cognitive plane. I’m not sure this claim is a true claim. So I am in a state of doubt. It’s not that the claim is doubtful and it’s not that the fact is doubtful. The person is doubtful regarding the claim. The claim is either true or false. There is no… facts are only true, or only facts, they are simply facts. A claim can be either true or false, and the person’s relation to the claim can be anywhere from zero to one. Sure that it’s true, not sure that it’s true, sure that it’s not true. Yes? The whole range between zero and one. Therefore states of doubt deal with a person’s knowledge of reality or with his level of trust in claims about reality. Okay? That is basically a definition of a state of doubt. Now the next question is whether certainty is possible at all. So here we have to divide between two kinds of claims. There are claims that are true in themselves, analytic claims, in Kant’s language, let’s say. Two plus three equals five, or the sum of the angles in a triangle is one hundred and eighty degrees. These are mathematical claims. Mathematical claims are certain. Meaning, here there is absolute certainty. The claims… but these claims say nothing about the world. Mathematical claims say nothing about the world. The moment I apply that mathematical claim to the world, it already ceases to be certain. Meaning, if I claim that on the sheet of paper in front of me, when I draw a triangle, the sum of its angles is one hundred and eighty degrees, that claim is not certain. The claim that the sum of the angles in a triangle is one hundred and eighty degrees is a claim about the Platonic world in which space is straight Euclidean space, and there the sum of the angles in a triangle is one hundred and eighty degrees, in that abstract triangle. When I draw a triangle on the board or on paper, I need to assume that the paper is Euclidean space. Now of that I’m not certain. Maybe yes, maybe no. I think yes, almost certain yes. But maybe not. Right? Even our ordinary space, we tend to think it’s straight space. Right? If you draw a triangle here, the sum of the angles is one hundred and eighty degrees. But Einstein tells us that this is not so. No. Space is curved, and the sum will not be exactly one hundred and eighty degrees, depending on how close you are to a very, very large mass that curves space. And when space is curved, the sum of the angles is not one hundred and eighty degrees. Therefore, when I make the claim that the sum of the angles is one hundred and eighty degrees as a claim about the world, that is not a certain claim. When I make the mathematical claim that the sum of the angles in a triangle is one hundred and eighty degrees, that is a certain claim. It deals with some abstract Euclidean space that I know how to define, and in that abstract world it is a certain claim. But every time I go out to apply it to some reality in the world, I assume that the world is a model of that mathematical theory, and that assumption is not certain.
[Speaker B] And if we take something not connected to geometry and mass, just plain one plus one equals two? Same thing.
[Rabbi Michael Abraham] Why? If I say that one plus one equals two, that is a certain claim. But if I say that I added an orange into… I put an orange into a basket and added another orange. Now I ask, what is my level of certainty that there are two oranges there? Not one hundred percent. Ninety-nine.
[Speaker B] Why?
[Rabbi Michael Abraham] Because in the world itself it could be that adding oranges is not described by arithmetic addition. I’m not… that’s a law of nature. That’s no longer connected to mathematics. And that has to be decided. I’ll give you an example, yes, this is the example I always use, not in this series, but I’ve spoken about it before. It was always in the context of when I taught recitation — not always — I taught mechanics at the university, I think it was one year. I taught mechanics, and in the first recitation I opened with the question: is the statement two plus three equals five a scientific statement, a scientific claim? Or in other words, is it falsifiable? And that’s Popper’s definition of a scientific claim: that it can be subjected to an empirical falsification test. And describe to me an experiment such that if it succeeds, it will confirm the claim, and if it doesn’t succeed, it will refute the claim. If there is such an experiment, then the claim is a scientific claim. Okay? That’s Popper’s minimal definition, of course there’s a lot to discuss there, but it’s the most accepted one, at least the minimal definition of what science is. Okay? Now, so I said to people: don’t you think this is a scientific claim, that two plus three equals five? So suggest to me an experiment. What, what kind of experiment? For example, like with the oranges. You have two oranges in a basket, add another three oranges and count. If it comes out five, then excellent, and if it comes out four and a half, then no, that’s not good. Or minus seventeen, then not good. So apparently the theory has collapsed. That means the theory is a scientific theory. Okay? Of course we’re sure it won’t be refuted. We have some clear conviction because we know how oranges and baskets behave, and when you add oranges into baskets, we understand — it works like mathematics. But that is an insight about the world. It could be that we’re wrong here. In my opinion the chance is very small, but it could be that we’re wrong. And why did I say this at the beginning of a mechanics course? Because I said to them, let me show you an example. Think of a body that is here. A force of ten newtons acts on it northward. North is upward. Ten newtons northward. Another force of ten newtons acts on it eastward. Okay, you see it westward, never mind, I’m assuming it’s eastward because it’s a mirror image. What is the resultant force? Seemingly there’s a force of ten plus a force of ten, ten plus ten equals twenty. The resultant force is twenty newtons. And that is of course not correct. The resultant force is fourteen point something. That’s the parallelogram law, and it basically goes in a diagonal direction, and the magnitude is ten root two. Fourteen point something. Did we do an experiment, apply a force of ten newtons plus a force of ten newtons, and the result is fourteen point something and not twenty — so did we refute the law that ten plus ten equals twenty? No. We did not refute the law that ten plus ten equals twenty. What did we refute? We refuted the law in physics, not the law in mathematics, which says that the addition of forces is done by, is described by arithmetic addition. And that in order to know the resultant force, you need to do arithmetic addition between the values of the forces you’re adding, ten plus ten. Not true. The theory that is responsible for, or represents, the addition of forces is vector addition, not arithmetic addition. Okay? Vector addition will give you fourteen point something. What does that actually mean? It means that when I apply the rule ten plus ten equals twenty to reality, ten plus ten certainly equals twenty. That cannot be refuted. If it doesn’t work in some situation in reality, what does that mean? Not that the mathematical law is wrong, but that it was wrong to apply that mathematical law to the reality in question. And that is an assumption in physics, not in mathematics, and therefore it can be refuted by experiment. Now, adding oranges into a basket — we are pretty convinced, I’m pretty convinced, that if we do the experiment it won’t be refuted. Okay? But how do you know? There are surprises in the world. The world sometimes behaves in very strange ways. And if you add electrons instead of oranges, then indeed you won’t always get two electrons when you add one and one. Okay? Therefore the world is more complicated. And therefore we cannot have full certainty in any factual claim at all. Even if I attach some mathematical model to it and on the level of the mathematical model the claim can be proven. And therefore when I want to talk about truth… wait. Yes.
[Speaker C] Where did you get the idea that… not where did you get it — why do you say there is absolute certainty in mathematics, in ten plus ten equals twenty? What is the proof? Isn’t there some axiom we assume? In mathematics, in arithmetic there is an axiom. So why is that certainty? Axioms aren’t certain; it’s an axiom, it’s not certain.
[Rabbi Michael Abraham] An axiom is completely certain. What do you mean not certain? I know it, it is clear to me, I grasp the axiom and it is true.
[Speaker C] I couldn’t understand, I couldn’t understand the difference between reality and mathematics. Why with the basket is that… different from mathematics?
[Rabbi Michael Abraham] Because to say that one plus one equals three is a logical contradiction. To say that two oranges plus three oranges will give four oranges is not a logical contradiction. It’s a problem in physics.
[Speaker C] And in fact that doesn’t happen, so there is certainty that two oranges plus three oranges is five oranges and not four.
[Rabbi Michael Abraham] Until now it hasn’t happened to you, so what?
[Speaker C] Also one plus one — my child writes one plus one and writes that it equals three. Until now in mathematics too, reality has also proved itself, right?
[Rabbi Michael Abraham] Have you ever encountered a body that doesn’t fall toward the earth? The law of gravity has proved itself, right?
[Speaker C] On earth, yes.
[Rabbi Michael Abraham] Right, yes, we’re talking here at the moment. Also outside the earth, we haven’t found any exception to the law of gravity.
[Speaker C] Maybe on the moon it doesn’t fall toward the earth, but never mind.
[Rabbi Michael Abraham] It matters; that doesn’t contradict the law of gravity, it simply revolves around it. Nothing contradicts the law of gravity. Now the question is what happens — are you sure that the next mass you observe will also fall toward the earth?
[Speaker C] From my experience, yes, but I…
[Rabbi Michael Abraham] Not sure? There are many things about which we accumulated experience but in the end it turned out we were mistaken, right?
[Speaker C] Our generalization wasn’t correct.
[Rabbi Michael Abraham] What does that mean? It means that many times things seem very, very certain to us, but because… yes, but I didn’t ask about
[Speaker C] the generalization, I asked about two plus three with oranges.
[Rabbi Michael Abraham] That’s what I’m saying, this is about facts in the world, because facts in the world are based on generalization. You learn from experience, and generalization is an act that is not certain.
[Speaker C] But oranges — let’s talk about oranges — I’m not generalizing anything, an orange.
[Rabbi Michael Abraham] But the truth of mathematics you are generalizing. If you ask what will happen when you add two oranges plus two oranges — if I saw that it was four, then that’s certain, I saw it. But I’m asking what the law is: when you add two oranges plus two oranges, what do you get? Until now I always got four, but how do you know tomorrow it won’t be five, or four and a half, or minus thirteen? I don’t know.
[Speaker C] How do you know tomorrow in mathematics that two plus two will be five? I can’t understand the difference.
[Rabbi Michael Abraham] I’m trying to answer. So what I want to say is that laws in physics are based on generalizations, and laws in mathematics are based on immediate apprehension, not on generalizations. And since I know in an immediate way that two parallel lines do not meet, then it is clear to me, that is certain to me; it cannot be refuted. And not only because even if I succeed in doing an experiment that refutes it, it would refute a physical assumption, not the mathematical theorem. There is no way to refute it.
[Speaker D] So it’s only intuition. It’s true that two lines, but intuition
[Rabbi Michael Abraham] How many of our intuitions
[Speaker D] have turned out to be mistaken?
[Rabbi Michael Abraham] I completely agree that it is intuition, I just wouldn’t add the word “only.”
[Speaker D] No, but we have endless intuitions that turned out to be wrong. Right, but not mathematical intuition.
[Rabbi Michael Abraham] Where does the Rabbi get the confidence that mathematical intuition too is correct?
[Speaker D] Why? He’s asking why, the fellow here. Why? Why is logic certain? Why is logic certain?
[Rabbi Michael Abraham] It comes out of the definitions, what do you mean? You have Euclidean space. Euclidean space not in the world, some Platonic Euclidean space like that, not something that really exists, something imaginary. In that space there are certain definitions; from those definitions some result follows. That does not come by way of generalization, it comes by way of analysis. It is simply a consequence of the definitions.
[Speaker D] Why doesn’t the Rabbi answer him — I’m not claiming that it’s true, I’m only claiming that if you accept the assumptions, the axioms, then the theorems are true. If you don’t accept them, then not.
[Rabbi Michael Abraham] Mathematics does not say that the theorems are true. Mathematics says that if the assumptions are true…
[Speaker D] Exactly, that’s what I said, that the Rabbi could have answered the previous fellow that you’re not really claiming it’s true; you’re claiming that if you accept the axioms, the assumptions, then the conclusion, the theorems, follows from there with certainty.
[Rabbi Michael Abraham] Let’s go back again, I’ll go back again. Mathematical truth is not the conclusion. Mathematical truth is “if the assumptions, then the conclusion.” Now when someone asks me, “And how do you know that?”, that is the question for mathematics. The question is not about the results, the question is not that the sum of the angles in a triangle is 180 degrees; the question is who told you that under Euclidean assumptions the sum of the angles in a triangle is 180 degrees? To that I say: because it is self-evident. Not the conclusion — the inference.
[Speaker D] I rely on logic with certainty.
[Rabbi Michael Abraham] Yes, exactly. Meaning, when a mathematician comes and I ask him what the sum of the angles in a triangle is, if he’s a responsible person he ought to tell me: I have no idea. I have no idea — tell me what your assumptions are, what space you’re talking about. Once you tell me that, he’ll be able to calculate the sum of the angles in the triangle for me. Okay?
[Speaker D] And if we ask about logic itself, who says our inference is always…
[Rabbi Michael Abraham] Exactly, ah, so the previous question — why didn’t I answer him that I’m speaking only about the inference? Because the question was about the inference. The question you’re asking now is the question he asked before: where… where do you know the inference from? Because that you cannot observe in the world. Because if you observe the world and discover that the result isn’t correct, you simply discovered that the assumptions aren’t correct. Then you have no way to check or refute the “if-then.”
[Speaker C] But maybe our axioms in mathematics come from our world? Because we know that two oranges plus three oranges equal five oranges, we reached the conclusion in mathematics that 2+3=5.
[Rabbi Michael Abraham] So Saul Kripke wants to make that claim. I don’t agree with that claim. I argue that maybe didactically it helped me become convinced of the matter, but after I became convinced of the matter, I understand it from itself. Meaning, there’s a difference between using an experiment as a didactic aid — which can also work in mathematics — when you teach children arithmetic, how to count, how to add, so you teach them through various beads and adding things to one another, but that’s only a didactic aid. After you’re done with that, it’s a ladder you used and then you throw the ladder away. Now I understand — once I understand the concepts 2, 3, and 5, I understand from the concepts themselves that 2+3=5. It doesn’t depend on experiment. The experiment helped sharpen it.
[Speaker D] If someone had raised the Rabbi from childhood,
[Speaker C] How do you prove this thing in mathematics? What — I didn’t hear?
[Rabbi Michael Abraham] How do you prove in mathematics that 2+3=5? No, that can be proven, but there are assumptions on which that proof is based, and the assumptions of course cannot be proven. I claim that the assumptions are certain not because they have a proof. Proof is a tool by which you cannot arrive at certainty, because proof is always built on assumptions. Therefore proof is really not a tool for arriving at certainty. The only question is what certainty you have in the assumptions, and I claim that the assumptions…
[Speaker C] What is the basic assumption that we assume in order to arrive at 2+3=5?
[Rabbi Michael Abraham] That two parallels never meet in Euclidean space.
[Speaker C] Not in our space, I mean that 2+3=5.
[Rabbi Michael Abraham] So in number theory there are assumptions like Zermelo, Zermelo-Fraenkel, and all kinds of things like that. There are many axiomatic systems that lay out number theory. Take three assumptions, five assumptions, and that can lay out all of number theory. There are formulations in mathematics; they teach this in math courses on number theory.
[Speaker D] And if we took a child — the Rabbi’s father was a physicist — and educated him from age zero in vector addition, constantly showing him vectors in the world, showing him, look, 10+5, you apply this parallelogram law and you see the result. And that’s what he was exposed to from the beginning, all the time, only that. Then when he gets to age 10, 12, 15, I expose him to simple arithmetic. Doesn’t the Rabbi think he might say, wait a second, where did you pull this weird thing from, that 2+2=4?
[Rabbi Michael Abraham] It could absolutely be hard to teach him that, but after I taught him, he would understand that it’s self-evident. So that’s a didactic difficulty; again, I repeat.
[Speaker D] I actually even support the previous fellow’s claim that maybe, after all, our basic intuition does indeed come from the world.
[Rabbi Michael Abraham] Anything is possible, but since the claim that mathematical intuitions are not correct leads to contradictions, then I don’t know what to do with such a claim. It’s not something we can deal with, at least. I’m claiming there’s nothing to deal with at all. In any case, we have no way to deal with such a thing. At least on that level, it’s certain that from our point of view we have no option not to hold by it. Okay? At least that much.
[Speaker C] Have there ever been mathematical proofs that were refuted? I can’t hear. If there were ever mathematical proofs that people were sure of and then they were refuted?
[Rabbi Michael Abraham] Of course not, what do you mean? You can find a mistake in a proof, yes. You can’t refute it. Refuting is not finding a mistake. If it’s simply a mistake, then it was never true; that’s not a refutation. Meaning, it could be that someone wrote an incorrect proof, fine, obviously.
[Speaker C] So why are you so sure about 1+1, that it’s certain? Maybe one day that’ll be refuted too.
[Rabbi Michael Abraham] How would they refute it? What would they refute?
[Speaker C] Just like they haven’t refuted two oranges.
[Rabbi Michael Abraham] There’s no proof — what proof would they refute? Or would they find that it’s a mistake? Refute — impossible altogether, because no experiment…
[Speaker C] That’s why I also don’t understand — you could say the same thing about oranges in our world, the exact same sentence about oranges. Why is it specifically mathematics where you say high certainty, while in reality you tell me this could contradict the laws of physics? Okay, maybe it could contradict both the laws of physics and the laws of mathematics; I don’t know what will happen.
[Rabbi Michael Abraham] I’ll answer again, because I have nothing more to say than what I already said. In the scientific context, our conclusion can be refuted, and that has happened more than once. And because of that, factual conclusions are never certain, even when I think they’re true. That’s science. As for mathematics, it depends what we’re talking about. There are axioms in mathematics, and there is the logic that leads from the axioms to the conclusions. Okay? There are certain axioms in mathematics that say nothing; they are simply definitions. You can define, for example, a group or a vector space; it’s a collection of definitions. There’s no true or false here; these are the definitions. Now, from these definitions I can derive various claims, and then the claims are basically — the mathematical claim is that from these definitions or these foundational assumptions you can derive those theorems. Just the “if… then.” That’s mathematics. And here this is logic; there’s no way to refute it. Okay? You can find a mistake in the proof, but there’s no way to refute it.
And regarding assumptions like, for example, the assumptions of geometry — there, there is some kind of… at least I, as a Platonist, claim that the assumptions of geometry are not definitions. They are theoretical claims about a hypothetical abstract Platonic world, which is a Euclidean world. There is such a world. And in that world there are certain characteristics, which are the axioms of Euclidean geometry. And those are the characteristics of that world. I simply know that this is so; I have an immediate apprehension that this is so. And from there I can also derive various conclusions. That’s different, for example, from the case of assumptions about the definition of a group. Because there I’m not sure there exists such an entity somewhere that I’m merely describing. It could be that I created it by the very definition. Regarding geometry, for example, I do think that a Platonic space of that sort exists, and in that sense I claim that the axiom of geometry that two parallels do not meet is a true statement. Others will say: that’s simply the definition of Euclidean space. Just a definition. A definition cannot be true or false; it’s just a definition. And then, according to them, all that mathematics says is only: “If these assumptions are true, then these are the conclusions.” Just the “if… then.” I want to make a Platonist claim that says the assumptions too are true — with respect to a Platonic world, not our world. In our world it’s a matter of observation, and you can sometimes be mistaken. In the Platonic world it’s clear to us; that is the concept of Euclidean space. Therefore there is some kind of immediate certainty here. You can say you don’t accept that because you’re not a Platonist. Then you’ll accept it as a definition. But I don’t think it makes sense to treat such a thing as a statement that is not true.
[Speaker C] Meaning, you can say either that it’s a definitely true statement — that’s my Platonist view — or you say there is no such thing as true or false here because those concepts don’t exist; this is not a factual claim but a definition.
[Rabbi Michael Abraham] And definitions, again, cannot be true or false. A definition is a definition.
[Speaker C] That’s the definition of… those are the two options. To say that it’s a claim, but a claim that isn’t true or certain, that could be false — meaning not certain — that’s not an option that I… That’s acceptable to me. I’m only saying: in mathematics isn’t there an initial definition of “one”? So those are also definitions. That too is like a definition. Obviously. I didn’t understand. I said that a group, for example — the basic definitions of a group are the definition of that concept. No, mathematics in 1+1. Is the number one, this unit, I don’t know what to call it, I don’t understand mathematics.
[Rabbi Michael Abraham] A group is mathematics; you mean arithmetic.
[Speaker C] Okay. One. One plus — I mean the one to which you add one. Is that a definition? Is “one” some sort of definition? No.
[Rabbi Michael Abraham] I claim it’s a statement. Again, Platonists will say it’s a definition. Meaning that the foundational assumptions of number theory are definitions. And from there on, these are derived theorems. Meaning: if those definitions are true, then those are the results. I claim those assumptions are statements, not definitions.
[Speaker C] What is the statement of one? That 1+1=2. It is—
[Rabbi Michael Abraham] A statement. What is this “one”? “One” is not some object in the world.
[Speaker C] I define the word “one” with the concept “one.”
[Rabbi Michael Abraham] I’m not talking about any object in the world, again.
[Speaker C] Yes, yes, I’m not talking about the world, I’m talking about the abstract Platonic one. Is that one a definition or a statement? It’s not a statement because it’s nothing.
[Rabbi Michael Abraham] One is not a statement. One is a Platonic object. 1+1=2 is, in my opinion, a statement. Others will say it’s a definition. Fine? One is not a statement. One is an object. A Platonic object. Again, for whoever accepts that.
[Speaker C] And is “two” defined as 1+1, or is “two” an entity in its own right?
[Rabbi Michael Abraham] No, two is defined as two. 1+1=2 is, in my view, a statement; it’s not a definition. Others will say, “No, that’s the definition of two.”
[Speaker C] So maybe that’s the difference between mathematics and the world of — I mean arithmetic and our world. In arithmetic there’s the “two,” which is — like, an apple plus an apple is not two. It’s always an apple plus an apple. It’s never “two apples.” Because there is no entity called “two apples.” It’s convenient for me to abstract it as “two apples.” I’m not — you can define “two apples” as an entity.
[Rabbi Michael Abraham] I don’t see why not. The entity can be — after all, the apple itself is also composed of parts. So maybe there’s no entity called an apple either?
[Speaker C] No, I defined an apple as the thing that comes out of—
[Rabbi Michael Abraham] What, you can define for it—
[Speaker C] But it’s always an orange plus an orange; it’s not two oranges, that’s not the definition.
[Rabbi Michael Abraham] Fine, let’s stop, because this isn’t our topic, so it’s a shame. Okay, so regarding the subject of certainty, I’m saying: mathematical certainty, at least in my view, does exist — logical-mathematical certainty. But in every factual claim, claims about the world, there is never certainty. Never certainty. People always attack that, right? I’ve said that many times I add that I’m not sure of anything — except this very claim, that I’m not sure of anything. About that I’m completely sure. Okay? Meaning that nothing is certain.
Now, that very claim itself — that nothing is certain — the question is whether that is a claim about the world or a claim in logic. I claim that it is a claim in logic, and therefore it can be certain. And therefore there is no contradiction. People often attack this skepticism that says nothing is certain: and how can you be sure of that itself? So first of all I can say: no, I’m not sure even of that itself. That’s also an option. I claim no — I can be sure of that itself. I can be sure of that itself because it is a claim in logic, and logic can indeed be certain. Factual claims are not certain. And a claim that deals with claims is not a claim that deals with facts. All the other claims are claims that deal with facts, and there there is no certainty; and the claim that says no claim is certain — that claim, what is its subject? What does it deal with? It deals with claims, not facts. So in fact it belongs to the domain of logic, not the domain of physics. And a claim that belongs to logic can be certain; this does not contradict the claim that no claim in physics is certain. Okay? Fine, that was a philosophical pilpul.
In any case, for our purposes, there are of course situations where on the practical level we have certainty. Practically, like gravity, right? So practically I assume that all of us are convinced that the law of gravity is true — meaning that masses attract one another with a certain force. Okay? Even though I call this practical certainty, because it is a claim about the world, and a claim about the world is never truly certain. There is no absolute certainty here, but for all practical purposes, yes, for every practical purpose, I treat this claim as a certain claim. Or the claim that what I see indeed exists — in my eyes that is a claim certain on the practical level, even though I could be wrong; there are mirages sometimes too. But practically, for me, that is a certain claim.
Therefore, when we come down to the practical world, there can also be certainty in factual claims. Now, once we do not have certainty on the practical level — meaning because on the philosophical level there is certainty about nothing; we are always in some kind of uncertainty — but even on the practical level, in the legal rules of doubt and in questions of doubt, we deal only with places where we have practical uncertainty, not philosophical uncertainty. If there is practical certainty, then we do not call it a situation of doubt. If we have practical uncertainty, meaning on the practical level I do not know whether this is true or not true, then we define that situation as a situation of doubt. Once we have doubt, that means that before us there is more than one alternative as to what the correct claim is. Okay? Or what the fact in the world is, or what the correct claim is — it’s either this or that, and I’m in doubt between the two possibilities. Is the meat kosher or is the meat non-kosher, impure or not impure, and so on. Okay? These are several possibilities.
Therefore, when we talk about situations of doubt, first of all we need to decide that there are several possibilities, and second, we need to decide that we have no certainty regarding one of them as the correct possibility. Then we need to map out what possibilities are before us, and if we can, we assign some weight — called probability — to each of the possibilities. Okay? You can’t always do that. But sometimes we assign a weight to each possibility, and then we arrive at probability. Probability basically measures my level of certainty in each claim, and therefore the total of the probabilities must add up to one hundred — meaning, all the possibilities ultimately add up to one hundred percent. So if I have thirty percent certainty in this direction, then apparently there is seventy percent in the other direction, if there are only two possibilities.
Very often there is a claim — say when people speak about the existence of God, right? So I say that to say that the world exists without anyone having created it is an obviously unreasonable claim. The world is complex and so on; it can’t be that nobody created it — an obviously unreasonable claim. So what is the probability of the claim that… that there is someone who created it? One minus the probability of that other claim, right? Meaning, if the probability of the claim that nobody created it is ten to the minus ten — I’m just quantifying it, right? — then the probability that God exists, assuming there is a world, is one minus ten to the minus ten. A great many atheistic claims collapse because of this banal calculation. All kinds of atheists argue: wait, wait, true, it’s unreasonable, but the existence of God is even less reasonable, so I still prefer this unreasonable possibility over the possibility that God exists. That’s nonsense. Since either there is such a factor or there isn’t such a factor — there is no third possibility. So if the probability that there isn’t such a factor is ten to the minus ten, then that means the probability that there is such a factor is one minus ten to the minus ten. You can’t decide that the probability of this is ten to the minus ten and the probability of that is ten to the minus one hundred, even smaller. So what, then what takes up all the rest of the probability? What possibility accumulates all the rest of the probability? There is none. Either there is a Creator or there is no Creator; there is no third possibility.
Okay, so it sounds terribly simple, but I’m laying these introductions because so, so many things in our lives, and even in philosophical discussions, so very many things fall apart over these simple understandings. Very simple understandings, but if you put the conceptual world in order, it solves a great many problems. We’ll encounter some of them later on.
I said that in order to be in doubt now, we are in doubt among several possibilities. That is the state of doubt. So we’ve defined the state of doubt. Now still, in order to decide that there is practical doubt between possibilities, the rule is — in Jewish law too, but not only in Jewish law — that in order to doubt, you need a reason. This is the story in the Talmud, again, I’ll try to shorten it a bit because I’ve gone on too long already — the story in tractate Shabbat 30a, about someone who came to Rabbi and said to him: “Your mother is my wife, and you are my son.” Meaning: you are a mamzer. I had relations with your mother and you are a mamzer. You are not your father’s son. So he said to him: “Would you like to drink a cup of wine?” He drank and burst. Meaning, Rabbi waved him off. Okay? What does that mean? Don’t waste my time. Why don’t waste my time? It could be true. After all, there are mamzerim in the world. It happens. Right? So how are you so sure it isn’t true? I have no doubt on the practical level. True, theoretically there is such a possibility. On the practical level, I’m certain. And in order to doubt, you need a reason.
What does that mean? That doubt means practical uncertainty, because philosophical uncertainty is not enough to create doubt. Philosophical uncertainty only means there is also another option. Right? Maybe not — all those questionings, right? Who says? Why? Maybe not? Maybe the opposite? Right? That is not a reason to doubt. Give me a reason why there is actually a side to say that perhaps the opposite claim is true, and then I have reason to doubt. Why? Because a doubt has arisen in me on the practical level, not only on the hypothetical level. So if someone comes to Rabbi and says that he is a mamzer, the possibility exists on the hypothetical level. There is such a possibility; there are people in the world who are mamzerim. It could have happened. But I have no reason at all to suspect that this is the case. And therefore on the practical level I am convinced. If on the practical level I am convinced, then the laws of doubt do not apply. Okay?
Therefore this is often called, in the language of Jewish law, that in order to doubt you need a reason. A person who has not brought a reason to arouse the doubt will not create doubt. Very often in the context of monetary law, for example, we talk about situations where there is a concrete monetary basis to the claim and where there is no such basis. For example, the topic of “let it remain” in the third chapter of Chazkat HaBatim, the third chapter in tractate Bava Batra: there is a boat floating along the river, and two people claim it is theirs. I claim it is mine and he claims it is his. Okay? That is a situation where there is no concrete monetary basis. What does it mean that there is no concrete monetary basis? I have no reason to assume that it’s yours; anyone can come and say that it’s his. There is no indication independent of your claim that ties the boat to you. Anyone in the world could come and say this boat is his. That is a situation with no concrete monetary basis; I do not have a doubt about who is right. I do not define such a situation as doubt. I define such a situation as one that the religious court does not deal with. Whoever is stronger prevails. Do whatever you want. The religious court has nothing to say about this matter.
By contrast, “two are holding a garment.” If two are holding a garment, each one physically holds the garment in his hand. Possession is a reason to doubt. Now the religious court is in doubt whether the garment is yours or yours. The ruling is: they divide it, because here there is a concrete monetary basis. Here there is a connection beyond the fact that I say it is mine. In the world itself there are facts in the world that lean toward or support this side, that this garment is mine, this cloak is mine. And he too has such factors. Since that is so, this is a situation of doubt. There are reasons that arouse this doubt. Okay?
[Speaker B] What exactly is the difference, Rabbi, between the two cases — that in Bava Metzia there is actually pulling or holding of the garment?
[Rabbi Michael Abraham] I’m holding it in my hand, and possession, at least on the legal level, creates a connection between me and the object. It’s not like if now someone from the outside world came and said, no, no, that shirt doesn’t belong to either of you, it’s mine — we would wave him away; he wouldn’t even enter the religious court. The court would rule only between the two of them; it would not rule with some third person. In the case of the boat, two people come to the religious court and say the boat is ours; any other person could also come join the discussion — no, that boat is mine — because there is no thing, no fact in the world itself, that supports their claim. They’re just saying it. Anybody can say it. That’s called a situation without a concrete monetary basis.
[Speaker B] But what if one of them brought identifying signs, say?
[Rabbi Michael Abraham] Then that’s something else; then there already is a concrete monetary basis. Fine, that’s exactly the point. Okay, so you need reasons in order to doubt. For example, I think I mentioned the Torah talk that happened when I was in Bnei Brak. Someone’s son had reached bar mitzvah, so he gave some little Torah talk and said: tell me, we’re about to pray the evening service, we have nine men and one boy. They ask the boy: tell me, are you already bar mitzvah? He says yes, and then “He, being compassionate, atones for iniquity,” and they go ahead and pray the evening service. Maybe he’s lying? Does he have credibility regarding whether he is bar mitzvah? If he’s a minor, then maybe he also cannot testify about himself, and this is something that builds itself. Okay, so who says such a minor is believed to testify about himself?
Now the discussion there was a long and winding one, and I was tearing my hair out over that discussion. I don’t need proof or his credibility about the fact that he is thirteen years old. I have no reason to assume he isn’t. I have no reason. A person comes and says he is thirteen, then apparently he is thirteen. If you bring me a reason, something that raises suspicion that maybe he is lying, maybe — I don’t know — things like that, or that he has an interest, or I don’t know exactly what, something like that — fine, then I have reason to doubt, then perhaps he is not thirteen, and I’ll begin entering the discussion about the laws of evidence: who is believed, who is not believed, how we verify it, how we don’t verify it. But if I have no doubt, I don’t enter the discussion at all, because there is no reason that arouses doubt. In order to define a state as doubt, it is not enough that there are two paths before me; that is hypothetical doubt. A state of doubt has to be practical doubt. Practical doubt means there are reasons in favor of this path and reasons in favor of that path. Maybe there are more reasons here and fewer there, so it’s seventy-thirty — it’s still a state of doubt, just with different weights. But a state of doubt always has to be uncertainty on the practical level, and that means there has to be a reason that tells me there is room to doubt.
This is Bertrand Russell’s “celestial teapot” argument; it’s exactly the same thing. The celestial teapot, right? Someone comes and tells you: look, there is a tiny teapot orbiting the planet Jupiter. No, I have no idea how you know that. Where did that come from? Who said there is such a thing? Bertrand Russell says: what do you mean — you also don’t know it’s not true. Fifty-fifty, no? Either there is or there isn’t; you don’t know. Bertrand Russell says: what? Not at all. Zero-one hundred, not fifty-fifty. Obviously there isn’t. Why obviously there isn’t? He says theoretically maybe there could be one? Yes, maybe, but I have no reason that raises in me a doubt that maybe there is such a teapot; after all, you can’t know that any more than I can. The fact that someone is babbling fantasies proves nothing. I have no reason to doubt that there is such a teapot. Therefore for me it’s not fifty-fifty, even though there are two possibilities. There are two possibilities, but only hypothetically. On the practical level I need there to be doubt on the practical level, not on the philosophical level, and in order for there to be doubt on the practical level, there have to be reasons that arouse doubt — a concrete monetary basis.
[Speaker C] Okay, does the doubt depend on the obligation to clarify? I didn’t understand — does the obligation to clarify depend on doubt?
[Rabbi Michael Abraham] Yes. If you have no doubt, then you don’t need evidence.
[Speaker C] So if a child comes and he could ask his father whether he is thirteen, he could call his father and ask if he is thirteen — do I have to do that, or can I believe the child?
[Rabbi Michael Abraham] No, if he tells you he is thirteen, then I certainly believe him. This is what happens every day, after all; this is really what people do, right? Any normal place, that’s what they do, right? Only the hair-splitters start discussing what his credibility is and what this is and what that is. As the Talmud says, in Kiddushin: we execute and stone based on presumptions.
[Speaker C] But it’s not a presumption that he is thirteen?
[Rabbi Michael Abraham] No, wait — what does “presumptions” mean? It is a presumption. What does presumption mean? It means what is established in the world. For example, if I am presumed to be the son of my parents, then now if I struck my father, yes, they execute me — “one who strikes his father or mother shall surely be put to death.” Maybe he’s not my father? Maybe I’m a mamzer? How do they execute? After all, to execute you need absolute certainty, right? Do you have absolute certainty? There are mamzerim in the world; not always is the one presumed to be my father really my father, right? No — as long as I have no reason to cast doubt on the fact that he is my father, even though theoretically there is such a possibility, I do not doubt it — to the extent that in capital law we execute on that basis. On that certainty. It’s exactly the same thing. If you have no reason to cast doubt, then there is no doubt. The situation is not defined as a situation of doubt.
It’s like the question they ask, right? The priest asks Rabbi Yonatan Eybeschutz: why don’t you go after us? We are the majority, the Christians. Right? It says, “incline after the majority,” so follow us. So what does Rabbi Yonatan Eybeschutz answer him? That I follow the majority in a place where I am in doubt. If I’m not in doubt, why should I follow the majority? Which is exactly the same principle. What is he saying? If I am in a state of doubt, then I begin looking for evidence and decision rules and majority rules and all kinds of things like that. But first of all I need to decide that this is a doubtful situation. If this situation is not doubtful, then I do not look for decision rules; I know what the truth is. There is no doubt; I know what the correct path is. So why do I now need evidence that it’s this rather than that?
[Speaker C] Is this a psychological matter? Meaning, if a person does think — I think this child might be lying.
[Rabbi Michael Abraham] It’s philosophical, not psychological.
[Speaker C] No, not psychological. A person has anxiety issues. He comes to a store to buy something, there’s kosher certification, and he says maybe they made it at home, maybe they didn’t really make it…
[Rabbi Michael Abraham] I said: if he has problems, let him take a pill. Problems don’t define Jewish law. What defines Jewish law is philosophy, not psychology. The question is what it is correct to think in such a situation, not what a person’s personal neuroses lead him to fear.
[Speaker C] But I can suspect this child is lying because he wants to go home quickly now. He’s being delayed now…
[Rabbi Michael Abraham] If there are good reasons, then it’s not because you’re neurotic, but because there really is reason to cast doubt. Fine.
[Speaker C] But because I’m neurotic, I worry about these worries.
[Rabbi Michael Abraham] No, no. If there are such concerns, then it has nothing to do with neurosis. If there are such concerns, then there are such concerns, philosophically. Neuroses play no role here. If there really is such a side, then it’s a state of doubt. If there is no such side, but with your neurosis you worry about it, then it still is not doubt. So take a pill. It’s like someone who finds a piece of meat.
[Speaker C] But who decides? In my opinion there is doubt; in your opinion there isn’t.
[Rabbi Michael Abraham] The truth decides. Reality decides. What — who decides that someone isn’t sane? He says, I’m sane, I’m Napoleon and I’m sane, and all of you are crazy. Who decides he isn’t sane? The truth decides he isn’t sane, because he isn’t sane. That’s all.
[Speaker C] But aren’t there such things as children who lie?
[Rabbi Michael Abraham] There are doubtful questions from here until forever. That’s not… there’s no point dealing with that. There are… I find a piece of meat in the street, and there are nine non-kosher butcher shops there and one kosher shop. But this piece of meat has a premium kosher seal on it. Should I follow the majority of stores? After all, it says “incline after the majority”; I follow the majority. The majority of stores are non-kosher, so this meat is forbidden to eat? No, of course not. Why not? After all, we follow the majority? In situations of doubt we follow the majority. And in order to follow the majority, first of all I have to decide that I’m in a state of doubt. If there is a kosher seal, then I’m not in doubt. Ah, maybe someone forged the seal and… yes, maybe. If there is some indication of that, or some concern, some basis for that concern, talk to me. If not, the mere fact that such a possibility exists does not make this a state of doubt. That is philosophical doubt, not practical doubt. We deal with practice.
[Speaker D] Rabbi, why can’t the Rabbi make an analogy from what the Rabbi just said to the issue of logical positivism? When that priest came to Rabbi Yonatan Eybeschutz and said “the majority,” he told him, “you haven’t said anything.” You simply haven’t said anything because you didn’t arouse any doubt in me. So when a person comes and says, “there are a billion and a half ants,” you haven’t changed my epistemology at all. It didn’t even bring me to the tiniest fraction of a doubt that you’re right, because you have no ability to verify it — and then it’s as if you said nothing. Exactly like Rabbi Yonatan Eybeschutz told him, “you said nothing,” so the logical positivist says to that claim, “you said nothing.” No — that the Rabbi… after all, the Rabbi does not accept logical positivism.
[Rabbi Michael Abraham] Logical positivism does not say that. Logical positivism says that the claim that there are a billion and a half ants is not a claim at all. It’s not a matter of truth and falsehood; it’s a pseudo-claim. It doesn’t accept his claim. If someone tells me there are a billion and a half ants in the world, I won’t accept it. But that has nothing to do with whether this claim is true or not. It could be true and it could be untrue. You have no way to know and I also have no way to know, so I don’t deal with it and I also don’t believe you. Fine. But that does not mean the claim is not subject to a judgment of truth or falsehood. And the positivist claims that you can’t say the claim is true or false. That is certainly not correct.
So in order to doubt you need a reason. Now, as I said earlier with the story of Rabbi Yonatan Eybeschutz, that actually brings me to what is called — maybe first — there is negative doubt and positive doubt. Negative doubt is basically, say I have a piece of meat and I don’t know whether it is kosher or not kosher. I don’t know whether, say, it could be pig or cow, fine? And I don’t know whether it is pig or cow. So I’m in doubt, right? By contrast, there could be a situation where I have one piece out of two pieces. There are two pieces, one of which I know is pig and one of which is cow, and they got mixed up. I don’t know which is the pig and which is the cow. That is called a doubt where the prohibition was already established, or a doubt of one piece out of two pieces. For the second kind of doubt one brings a provisional guilt offering; for the first kind of doubt one does not bring a provisional guilt offering. There is a difference between these two kinds of doubts.
This is negative doubt, a doubt of absence of knowledge. I don’t know whether it is pig or cow — fifty-fifty, doubt. The second doubt, of one piece out of two pieces: I know there is one pig and one cow here; I just don’t know which is which. Meaning, I have good reasons to assume a certain piece is pig, but I also have no less good reasons to assume it is cow. I have positive reasons to doubt. But in that case, also in one piece — not one out of two pieces — I am in doubt even though apparently I have no reasons to doubt. So why am I in doubt? Didn’t I say earlier that only where there are reasons do we define a situation as doubt? The answer is that I have reasons to doubt there too. Why? Because here is a piece whose nature I do not know. Now, what can it be? It can be meat, it can be pig, it can be cow. That’s what it can be, and there are pigs and cows in the world. Okay? So therefore it is clear that here there is reason to doubt. I have no positive indications in favor of pig or in favor of cow, as in one piece out of two pieces, but there is here… Why does that matter?
[Speaker D] Why does that matter?
[Rabbi Michael Abraham] Ostensibly, probabilistically, it doesn’t matter, although I’ll get to that too later on, and I’ll argue that maybe it does matter. But for now I’m saying probabilistically it doesn’t matter, and still these are two types of doubts. Okay?
[Speaker B] And for which one do I bring the offering?
[Rabbi Michael Abraham] For one piece out of two pieces. There need to be reasons to doubt — positive reasons to doubt.
[Speaker B] Meaning if I ate from one of the two pieces, or what?
[Rabbi Michael Abraham] Yes. If you ate both of them, then you certainly ate a prohibition. You ate one, and you don’t know whether you ate the kosher one or the non-kosher one.
[Speaker B] And in the first doubt, if I ate, I don’t bring a guilt offering?
[Rabbi Michael Abraham] No. A provisional guilt offering is brought for doubts.
[Speaker B] But in the first one I also had doubt.
[Rabbi Michael Abraham] True, but only for a doubt where the prohibition was established do you bring a provisional guilt offering; for the doubt of a single piece you do not bring a provisional guilt offering. What’s the logic of this? In the end, what’s the logic?
[Speaker E] It’s the same thing; statistically it’s the same probability.
[Rabbi Michael Abraham] That you are assuming. I’m not sure you’re right, but let’s leave that, because we’ll get to it. I claim there is even probabilistic logic to this, although people generally think this isn’t a probabilistic claim. It’s only at the legal level, I don’t know what to call it — when there are positive reasons, maybe the experience of the transgression is different; you can try to suggest explanations for it. You ate something for which there was a positive reason to think it was pig, so that’s a bit different from eating something whose nature you have no clue about. True, it could be pig, but you have no positive reason. So again, even if there is no probabilistic difference here, there is room to understand that with doubt of the second kind there is more reason to be stringent — not because the chance that you failed is greater, but because maybe the impact of failing is greater or something like that.
[Speaker D] It really felt here that that was the motto of the difference.
[Rabbi Michael Abraham] Not feeling — intuition.
[Speaker D] If we assume statistically it’s the same statistics.
[Rabbi Michael Abraham] Not feeling — intuition, legal intuition. But let’s leave it, because we’ll talk about it. I’ll give another example where it’s a bit trickier, although it’s still similar. Think of a die. We roll a die, and now I need to bet whether it will land on four. Fine? So the chance is one-sixth, if the die is fair. The chance is one-sixth, right? So I would bet one in six that it lands on four. That’s the reasonable probability, right? Usually people would bet one against six — actually one against five. Okay? What happens if I have a die about which I have no information at all — whether it is fair or not fair? It could be loaded so that it always lands on one; maybe it always lands on even numbers; I don’t know how it is built. I have no information about the die, and now they tell me I need to bet on the chance that it will land on four. What do you bet? You have to bet; you have no choice. They force you to bet. Also one against five.
[Speaker E] Also.
[Rabbi Michael Abraham] Right? Why? Because I have no reason to prefer one face over another, even though I have no information at all. This doubt is a negative doubt, and still I judge it in terms of doubt; I even apply probabilistic rules, that the chance is one-sixth, even though I have no information about the distribution. For a fair die I have positive information that the distribution is uniform; there is a one-sixth chance for each face. I know that because the die is really symmetric; it is fair. So there, when I bet one against five, I do it on the basis of information. With a die about which I have no information, I bet on the basis of absence of information.
[Speaker B] And here again — the difference will only come after several throws.
[Rabbi Michael Abraham] Then I get information, exactly. It will come or it won’t come, but I don’t have information; I don’t know. By the way, throwing it won’t help you — say the first few throws definitely won’t help. Even if in the first few throws it keeps landing on four, that could still be coincidence; that can happen with a fair die too, theoretically. Obviously if it’s a hundred times in a row, that already raises suspicion, but if it landed that way two or three times, you still don’t really know the nature of the die. Okay?
So here we see that this is negative doubt. Ostensibly there is no reason to doubt, but if you think about it, it’s exactly the same as one piece not from two pieces — the doubt of one piece. Because there is reason to doubt. After all, I know it will land on one of the faces, and I know the die has six faces. So each face is a reason why there is some chance it will land on that face. In that sense it’s like: there are pigs and cows in the world; I have a piece of meat before me; so obviously it is either pig or cow, because those exist in the world. So true, I have no concrete information about this particular piece — it’s not like one piece out of two pieces — but still, this is definitely a state of doubt. It’s not the same as Rafi’s case we brought earlier, where someone comes and says, “You are a mamzer.” Why assume I am a mamzer? So you said it — so what if you said it? There’s no reason. There are mamzerim in the world, fine, but I have no reason to assume I am a mamzer. Okay? There I really do not doubt.
So this is a bit delicate; notice, these are cases that look similar on the surface, but no — you have to distinguish between them. It’s very important. A great many paradoxes and problems in the laws of doubt are solved simply by this conceptual analysis.
[Speaker B] Rabbi, maybe I’m jumping too far ahead, but what about a case of agunot, for example?
[Rabbi Michael Abraham] What do you mean?
[Speaker B] Agunot is a case of doubt that is sometimes without evidence, without anything, and they decide this way or that, meaning—
[Rabbi Michael Abraham] There too it is a doubt where the prohibition was not established, but still there is doubt here. Either the husband died or the husband is alive; there are no other possibilities. And clearly he is either dead or alive.
[Speaker B] Right, but like—
[Rabbi Michael Abraham] Like a piece of meat that is either pig or cow, but I don’t know which — but clearly it is either pig or cow. That is a state of doubt.
[Speaker B] Even though it’s without any, yes, without any evidence.
[Rabbi Michael Abraham] Right. And still there are two possibilities here, and clearly only one of them is correct, so you have to deal with it. This is indeed a state of doubt.
[Speaker B] But I mean: to deal with it when I’ll never receive the evidence.
[Rabbi Michael Abraham] That’s what the laws of doubt are for — no matter — but I will still treat it as a state of doubt. Fine? The laws of doubt are also a way of dealing with things.
[Speaker B] Meaning yes, to deal with it and remain in doubt.
[Rabbi Michael Abraham] I’m saying we need to deal with a situation of doubt. We’ll get to that in the second part of the series, which is coming up soon; I just want to finish the summary. But “dealing with it” doesn’t necessarily mean remaining in doubt. There are rules—I talked about this in the previous Friday lecture—there are rules that tell you, “You remain in a state of doubt, but this is what you should do despite the doubt,” and there are rules that resolve the doubt. Okay? If two witnesses come, then you’re no longer in doubt—you’ve resolved it, right? On the other hand, “the burden of proof rests on the one who seeks to extract from another”—you’re in doubt, you remain in doubt, but fine, the rule is that despite the doubt, it stays with the current possessor. Okay? So there are two—actually more than two, there are three—we talked there about three types of decision rules, and I’ll get to that. So that’s regarding the reason for doubting and the types of doubts.
Now of course, the state of doubt is defined by the information I have. Say I have a die, and I know that the result came up—I rolled the die, I already rolled it yesterday—but I don’t know the result. So the chance that the result was four is one-sixth, assuming it’s a fair die. A one-sixth chance, right? Now if they tell me, “Look, the result was even,” and now I ask what the chance is that it was four—the answer is one-third, right? Meaning that probability or likelihood—probability depends on the information I have about the situation, on the prior information. Right? Since probability is a claim about the person, it’s a judgment about the person’s state, how confident I am in something—remember? There are the facts, there are the propositions, and there is the person who knows the facts, meaning who holds the propositions. Okay? States of doubt always concern the person. Right? States of doubt always concern the person.
Wait, I don’t even remember what I started with. Right—so since that’s the case, it’s clear that probability, which is the tool with which we deal with states of doubt—not in law, but in the scientific or factual world—we deal with states of doubt by means of probability, okay? In the halakhic and legal world there are additional rules, but in the ordinary world we deal with it through probability. Now, probability is a tool that deals with states of doubt, meaning with the person’s relation to propositions. And therefore it’s obviously very dependent on the question of what information the person has. If he knows the result was even, then the probability that it was four is one-third, because there are only three even results. If he doesn’t know it was even, then the probability that a four came up is one-sixth.
Now either it was four or it wasn’t; from the perspective of the propositions, it doesn’t matter what I know. It does matter from the standpoint of my decision. Because for my decision, the information I have is very important. All the issues of insider information on the stock market and things like that are exactly because of this. You’re supposed to make decisions based on the information available to everyone; you’re not allowed to use information that only you have, because it changes your probability calculations, it gives you an advantage. Okay? So someone who has extra information isn’t allowed to participate in the game. Meaning the information definitely affects decisions, probability calculations, and the decisions I’ll make based on those probability calculations. And that brings us also to conditional probability and all sorts of things like that, but we’ll get to those matters too.
Now basically I’m summarizing here. I’m skipping over lots of other things because these are the important things. All the examples—paradox, contradiction, the law of the excluded middle, and so on—whoever wants can listen to the recordings; it’s all there. But what matters to me are just these principles, because we’ll need them later.
So until now, what we’ve done is basically set out—or provide—criteria that have to be met in order for me to declare a situation a doubtful one. If the situation is doubtful, I need to understand what the options are between which I’m uncertain, and I need to check all the information I have about those options. Because then I can weight each such option, attach a certain probability to it. Those are basically the things that define the state of doubt. I need to decide first of all that there is a reason to doubt—that this is a practical doubt and not just a hypothetical doubt. I need to define what the possibilities are between which I’m uncertain, and I need to define whether I have any additional information that can help me decide the weight of each possibility, the likelihood that this is the correct possibility. And that’s usually a probability distribution. So that’s basically what defines the state of doubt.
Given that, we now come to the second part. And the second part is basically: how do we deal with states of doubt? There are rules—assuming I’ve reached a state of doubt. That’s what we really started in the last lecture or the one before, meaning I more or less summarized the first six lectures. So in the last lecture or two I already started discussing the decision rules for doubts. And I divided those rules into a few kinds, but I’ll leave that for next time; I’ll start that next time.
I just want to complete one more point that I wanted to complete here. When we set the possibilities against one another, we have to check very carefully that we’ve really covered all of them. Because very often a crossroads is presented to us with two possibilities: you have to decide, either you’re here or you’re there; if you’re not here you must be there, and vice versa. But if you think carefully, you’ll discover that at that crossroads there’s another path—the so-called third path. Okay? Sometimes you can discover that what was presented to us as a crossroads is only a partial crossroads. And I spoke a bit about ways of dissolving dichotomies. A dichotomy is presented to us—either this or that, two possibilities—and I gave a few ways to discover or expose third, fourth, fifth alternatives, and so on. But here I won’t go back into all of that again. I’ll just say that you have to check very carefully that the possibilities presented to us really are all the possibilities. First, that they exist. Second, what their likelihood is. And third, that we haven’t missed possibilities we failed to take into account. And very often, when you think carefully, you suddenly discover there are more possibilities that simply weren’t presented to us.
They say: are you Religious Zionist or are you Haredi? If you’re a religious person, then you’re either Religious Zionist or Haredi—what else could you be? No, there could be something else. I could be a religious person and a secular Zionist. So I’m not Haredi and I’m not Religious-Zionist with the hyphen. But when the dilemma is presented to me, it’s not a trilemma, it’s a dilemma—there are only two options, either Haredi or Religious Zionist. That’s why I know this, because every time some people accuse me of being Haredi, and others accuse me of not being Haredi, and there you have it: one master says one thing, another master says another thing, and they do not disagree. Meaning, that’s because there’s also a third possibility. In other words, it’s simply not true that there are only two possibilities. There are also accusations that I’m Reform; there are all kinds of accusations. People see a binary picture, and that’s a mistake. In the laws of doubt especially, and in thinking generally, you need to know to examine the crossroads very carefully before you go solve the problem. First check whether the problem was presented correctly and completely—are these all the paths or not.
Now one last point I wanted to complete today. I made a distinction: until now I presented the state of doubt as a state of epistemic doubt. Meaning, I don’t know which proposition is true or which option is correct. That’s something that speaks about my knowledge. Okay? But I gave examples of this—quantum theory, there may be examples in physics, the accepted interpretations—but in Jewish law there are much simpler examples; you don’t need to work hard to understand it, of a doubt that I called ontic doubt, not epistemic doubt. Ontic means the doctrine of being—that is, a doubt in reality. Not a doubt in my knowledge of reality. What does it mean, a doubt in reality? After all, I said a fact is not something doubtful. A fact is a fact; either it is a fact or it isn’t, but a fact is not something doubtful.
So here, think about someone who betroths by giving the father a perutah and saying, “One of your two daughters is betrothed to me.” Right? The topic of betrothal not fit for consummation. “One of your two daughters is betrothed to me.” Now usually the way the medieval authorities (Rishonim) and later authorities (Acharonim) phrase it is that this is a state of doubt: I don’t know which of the two daughters is betrothed. But a doubt of this type is not a doubt like any other. Say I sent an agent to betroth a woman and the agent died. He betrothed a woman and died. I have no idea who the woman is that he betrothed. But if I asked the Holy One, blessed be He, He would tell me which woman he betrothed, right? It’s just that I lack the information. That’s an epistemic doubt. It’s cognitive doubt: I don’t know what the truth is; the doubt is in me, in the person.
But there are doubts that are doubts in the object, not in the person. What does that mean? For example, someone betroths one of a man’s two daughters. He gave him a perutah and said, “One of your two daughters is betrothed to me.” Now here, it’s not a doubt in the sense that there is one who is betrothed, I just don’t know who she is. There is no specific one who is betrothed to me. Even the Holy One, blessed be He, cannot point to which of these two daughters is really betrothed to me. There isn’t one real answer that I just don’t know. The state of the world itself is vague.
[Speaker B] Wait, so the situation the Rabbi is describing is before the father gives the perutah to one of the daughters?
[Rabbi Michael Abraham] No, no, no, the father received the perutah. He doesn’t have to give it to the daughters.
[Speaker B] The father betroths his daughters, he takes the perutah.
[Rabbi Michael Abraham] No, but he has to transfer it—
[Speaker B] to one of the daughters—
[Rabbi Michael Abraham] he doesn’t keep it for himself.
[Speaker B] No, no, he—
[Rabbi Michael Abraham] he keeps it entirely for himself. He buys himself—
[Speaker B] a Bazooka chewing gum. So how can there be betrothal like that?
[Rabbi Michael Abraham] A father can betroth his daughter; he receives the money. If I want to betroth a woman, can I give the perutah to the father? Yes, if she’s a minor or an adolescent girl. If she’s an adult, then she accepts betrothal herself.
[Speaker D] Rabbi, doesn’t all this cheapen the whole discussion?
[Rabbi Michael Abraham] No, no, don’t get me into that problem, I don’t want to get into that. We’re dealing now with the laws of doubt. What I want to say is that doubt of this kind is a doubt in the world itself; it’s a doubt in the object, not in the person. In the people themselves, there isn’t one who is betrothed and one who isn’t, with me just not knowing who is who. Neither of them is really betrothed nor really not betrothed; they are in a state of quantum superposition.
[Speaker B] No, but either there is betrothal or there isn’t; there’s no such doubt.
[Rabbi Michael Abraham] Of course there is betrothal. This is called betrothal not fit for consummation. It’s the dispute between Abaye and Rava whether such betrothal counts as betrothal or not. Okay? But there is an opinion that it does count as betrothal. And the opinion that it doesn’t is only because it’s not fit for consummation, but in principle there’s no problem at all. It’s a doubt. If I betrothed one of two women by means of an agent—both of them sent me an agent, they’re not sisters, they sent me an agent and I give him a perutah and betroth one of the two—this is a situation where according to everyone there is betrothal. There’s no dispute about that. The dispute of Abaye and Rava is only if these two are two sisters. If they are two sisters, then if Rachel is betrothed to me, Leah is forbidden to me because she is my wife’s sister. And if Leah is betrothed to me, then Rachel is forbidden to me because she is my wife’s sister. So it comes out that this is betrothal not fit for consummation. That’s the whole problem. But undefined betrothal as such is no problem at all. You can give a perutah to an agent to betroth one of two women for you without specifying which, and here, according to everyone, the betrothal takes effect.
[Speaker B] Takes effect on which one of them?
[Rabbi Michael Abraham] On both of them in a superposed way. On one of the two, without specifying which. It’s like the Rogatchover—Or Sameach once said about the Rogatchover that when he traveled by train he would run through half of the Talmud in his head by heart. They asked him, which half? And he said, whichever half you want. Meaning, same thing. Only one woman is betrothed. Which one? Either one of the two. And it’s like quantum superposition. Meaning, this woman is betrothed and that woman is betrothed—but only one. And to define this state logically, you need to do a bit of work. But I won’t get into that. But yes, it’s a real state.
For example, I have five coins in my pocket and I consecrate one of them to Temple maintenance, without specifying which one—I consecrate one shekel. I have five one-shekel coins in my pocket, and I consecrate one shekel from my pocket to Temple maintenance. Which of the five shekels is consecrated property? There’s no way to know. One of them—but I can’t point to one of them. Even the Holy One, blessed be He, won’t be able to point to one. It’s not that there is one here and I just don’t know which one it is. There is no concrete one, but still it’s only one—I consecrated only one shekel, not five shekels. Okay? So these are pathological situations in which the doubt is a doubt in reality itself, not in my knowledge of reality. Reality itself is doubtful. I called this vagueness, a state of vagueness, as opposed to ambiguity—yes?—as opposed to a state of doubt. Doubt is in my knowledge of reality; ambiguity is vagueness in reality itself. Meaning, reality itself is not defined in an unequivocal way.
[Speaker B] But again, in the case of the agent with the two women, there too doesn’t he give the money to one of them?
[Rabbi Michael Abraham] I didn’t understand.
[Speaker B] In the case of the agent who takes the perutah for the two women…
[Rabbi Michael Abraham] No, the agent—of course he gives it to one of them.
[Speaker B] Right, so in reality I’m married to one of them, not to both of them superposition-ly, I don’t know what to call it.
[Rabbi Michael Abraham] He’s asking… no, but she is betrothed even before she receives the perutah. She is betrothed the moment the perutah reaches the agent. Now he can come to a religious court and ask: “Tell me, to which of the two should I give the perutah?” It doesn’t matter. It is not a condition of betrothal that the perutah reach the woman. The moment the agent acquires it, he acquires it for her. She is betrothed.
[Speaker B] So the practical difference is only during that time when the money is still in his hand?
[Rabbi Michael Abraham] No, no, the practical difference is permanent. Right now there is a question to whom to give the perutah—a different question, I don’t know.
[Speaker B] No, but there’s no question because he said, “I want either this one or that one,” so there’s no such question.
[Rabbi Michael Abraham] Why is there no such question? I now have a perutah—whom do I give it to?
[Speaker B] It doesn’t matter. No, I can’t decide.
[Rabbi Michael Abraham] What do you mean, “it doesn’t matter”? I need to give it to the right woman. What do you mean, “it doesn’t matter”?
[Speaker B] Can I draw lots? No, but if the man says that both are acceptable to him… not that both are acceptable to him, it doesn’t matter who is acceptable to him.
[Rabbi Michael Abraham] The question is whom he betrothed, not who is acceptable to him. He betrothed only one.
[Speaker B] Right, but he said, “one of the two.”
[Rabbi Michael Abraham] If he betroths one of the two and tells the agent, “You choose which one,” no problem—that’s ordinary betrothal. But no. I betroth one of the two, I don’t care which. It doesn’t matter which.
[Speaker B] He doesn’t determine which one.
[Rabbi Michael Abraham] Who doesn’t determine which one? I, the man doing the betrothing.
[Speaker B] Fine, obviously, I understand. So who does determine it?
[Rabbi Michael Abraham] No one. There is no one. It remains in that state, a vague state. No one can determine it. There is a Ritva who wants to argue that here the law of retroactive clarification would apply, and according to the view that there is retroactive clarification it would be possible to determine it, but that’s puzzling. The simple understanding is no—there’s no way out, it remains stuck like that. It remains vague.
[Speaker C] If the agent gives it only to one of them?
[Speaker F] What? If the agent gives it only to one of them, then that’s it, it’s over.
[Rabbi Michael Abraham] If the agent does what? I didn’t understand.
[Speaker F] Gives the perutah only to one of them.
[Rabbi Michael Abraham] Then he’s a thief.
[Speaker F] If he gives it to neither of them, then neither of them is betrothed.
[Rabbi Michael Abraham] No, then they’ll have to go to a religious court and litigate over the perutah, maybe they’ll rule “divide it.” But that’s just monetary law. That’s just monetary law; the betrothal took effect.
[Speaker F] How did it take effect? They haven’t received the perutah yet.
[Rabbi Michael Abraham] No, they don’t need to receive the perutah. The agent acquired it on behalf of the woman. No, but—
[Speaker F] that’s not really—
[Speaker C] a doubt in reality, Rabbi. Because you introduced… it’s not a doubt in reality, it’s a doubt because you put yourself… it’s because of what?
[Rabbi Michael Abraham] It’s a doubt in legal reality.
[Speaker C] It’s a doubt because you put yourself—
[Rabbi Michael Abraham] in that state.
[Speaker C] —in such a state that you’re saying that once I have such a doubt then I choose that there should be here…
[Rabbi Michael Abraham] I created this situation, but now this is the situation. Why do I care that I created it?
[Speaker C] In the same way, you could also create a situation in which once a person—
[Rabbi Michael Abraham] says such a thing… I did that.
[Speaker C] No, I mean by setting intent in Jewish law. Jewish law could have determined that there is a state—
[Rabbi Michael Abraham] It could have determined anything, but it didn’t. That’s it—this is the legal situation.
[Speaker C] But this isn’t a doubt in reality.
[Rabbi Michael Abraham] It is a doubt in reality. A doubt in halakhic reality. Who is the woman who is betrothed to me? In legal reality, yes.
[Speaker C] Okay.
[Rabbi Michael Abraham] There are things in physics, similar doubts in quantum theory, but there it’s really complicated.
[Speaker B] Rabbi, but why, if the agent gives the money to one of them, is he a thief?
[Rabbi Michael Abraham] Obviously, because the money is possibly hers and possibly the other one’s, and there’s no way to resolve that doubt. So how can you give that perutah to someone? This is “two people grasping the same garment” with regard to the perutah. They’ll have to go to a religious court; maybe they’ll tell them “divide it,” maybe “whoever is stronger prevails,” doesn’t matter, they’ll tell them something. It’s simply still unresolved. Whatever it is—those are monetary laws.
[Speaker B] So why does that make him a thief? Why does that make him a thief? Why does that make him a thief?
[Rabbi Michael Abraham] You gave a perutah to someone when it isn’t exclusively hers. If I have money that may be yours and may be someone else’s, can I give you that money? Of course not. Maybe it’s his.
[Speaker B] No, but when the man said this, he didn’t mean that half goes to this one and half goes to that one.
[Rabbi Michael Abraham] He says that one… no, it’s not half-and-half, it’s superposition. It’s both this and that. But again, for that you need to do a little logic to define it fully. We won’t do that here. I can’t hear.
[Speaker B] Where is this Talmudic passage?
[Rabbi Michael Abraham] In Kiddushin 56, I think—I don’t remember exactly. Look up “betrothal not fit for consummation.” This is Yael Kagan’s kuf.
[Speaker G] Rabbi, according to that view that the betrothal takes effect when it’s not fit for consummation, and you explain that it’s this kind of superposition, that it’s both and both, but neither one is any one of them—
[Rabbi Michael Abraham] Practically speaking, both my wife and my wife’s sister.
[Speaker G] And neither one is your wife.
[Rabbi Michael Abraham] No, both are both my wives and my wives’ sisters. Okay, which is a logical contradiction.
[Speaker G] No, why is that a logical contradiction? Both my wife and… no, because that points to two—
[Rabbi Michael Abraham] Wait a second, wait a second. Because it’s forbidden to marry two sisters. But there is no logical contradiction here. More than that—for this you need the logic I mentioned earlier. Look, I’ll say it in one sentence because you’re all asking. In one sentence I’d say this: the more correct definition is not really “possibly this one is betrothed to me and possibly that one is betrothed to me.” Rather, there is here a superposition between two states. One state is that Rachel is betrothed to me and Leah is not. The second state is that Leah is betrothed to me and Rachel is not. Each of those two states is coherent. Right? And I am in both of them. That is not the same thing as saying both are my wives. Saying both are my wives is a contradiction. Because it cannot be that she is both my wife and my wife’s sister, because if she is my wife’s sister then betrothal cannot take effect with her, so she cannot be my wife. But here it is a superposition between two states, each of which is coherent. One state is that Rachel is my wife and Leah is not; the second state is that Leah is my wife and Rachel is not.
[Speaker B] Yes, but not both together.
[Speaker G] No, but excuse me, why don’t you say the same thing about God’s foreknowledge and free choice? That is also possible and that is also possible. It’s possible that the Holy One, blessed be He, knows in advance what I’ll do, and it’s also possible that I have free choice and that He doesn’t know. Each one by itself is possible.
[Rabbi Michael Abraham] Yes, but I want to claim both propositions. Not that I’m in doubt between the two. I claim both of them with certainty. Both this exists and this exists… that’s just a contradiction. It’s like saying both are my wives, not in superposition. This one is my wife and that one is my wife. That you can’t say. The whole idea of the logic of superposition is that it is not a logical contradiction. If it were a logical contradiction, then quantum theory couldn’t be a scientific theory, because you wouldn’t be able to infer any conclusions from it.
[Speaker F] But why?
[Rabbi Michael Abraham] I found a word called—
[Speaker F] —superposition, and now I have a unity of opposites, everything’s fine.
[Rabbi Michael Abraham] But you don’t understand how it doesn’t amount to a logical contradiction. It’s not just a word; it’s quantum. It describes a different logical state; it’s not just a word.
[Speaker F] But our intuition is still that same intuition of an absolute contradiction.
[Rabbi Michael Abraham] No, no. Why?
[Speaker F] What changed in the intuition once we gave it that name?
[Rabbi Michael Abraham] Once you knew the logical formalization of the matter, you would discover that even intuitively you understand that it’s not a contradiction. That’s exactly the point. The logical formalization doesn’t float in the air. It helps me sharpen things that intuitively I can also grasp.
[Speaker F] I’m asking the Rabbi: how can it be that the electron is both here and here? The Rabbi has to admit that our intuition doesn’t grasp that.
[Rabbi Michael Abraham] No, that’s the mistake. The mistake is that the electron is not both here and here. Wrong—that is an incorrect description of the quantum state. Rather? Rather, there are two states. Two slits, right? One state is that the electron passed through slit A and did not pass through slit B. A coherent state, right? Yes. Good. The second state is that it passed through slit B and did not pass through slit A. That too is coherent.
[Speaker F] Which negates the first.
[Rabbi Michael Abraham] And we are in super—wait. And we are in super—
[Speaker F] Wait, Rabbi, when I say it passed through slit A, there’s an asterisk: and not through B. Right, yes. And when you say B and not A, that’s a plain logical contradiction, and everyone is happy because they say “superposition” and everything’s fine.
[Rabbi Michael Abraham] Wait, I’m trying to say something. I have one state where the particle passed through slit A and did not pass through slit B. Coherent? Yes. Good. Second state: the particle passed through slit B and did not pass through slit A. A coherent state? Yes? Good. Now the state that describes the particle is not that the particle is in both slit A and slit B. Rather, the state that describes the particle is the sum of those two states. And that is not the same thing as saying that the particle passed through both slit A and slit B, the way laymen say.
[Speaker F] But where did the particle pass?
[Rabbi Michael Abraham] There is no such thing; the question is illegitimate.
[Speaker F] “Particle” is a legitimate word, yes. “Where” is a legitimate question. I ask where the particle passed, and then they say the question is illegitimate.
[Rabbi Michael Abraham] It’s an illegitimate question. Physics says you can’t ask it. Can you answer me the question whether a good quality is triangular? Or what the difference is between—
[Speaker F] But Rabbi, where does the intuition come from? I’m not coming from… I’m coming from the intuition of an eight-year-old child, and the Rabbi says that it isn’t—
[Rabbi Michael Abraham] I understand that, because my question is mistaken. We should move on a bit. So I’m saying that when you look at this logical formalization, you understand that this formalization does not describe a particle that passed through both slits. It describes that the particle’s state is made up of the sum of two states. What is a particle anyway? A particle is not a little ball like you think; it is some kind of wave distributed in space. This whole layman’s way of looking at it… no, that’s exactly what I’m claiming… is it both here and there?
[Speaker F] No, that’s not right. We describe it as a particle; that’s a misleading description.
[Rabbi Michael Abraham] It may be that in another hundred years there’ll be something else—I don’t know whether it’s waves or something else. Your concept of particle is the concept… your concept of particle is the concept of an eight-year-old child; it’s distorted. That’s exactly the claim. The claim is that we are dealing with imprecise language that creates a logical contradiction. If it were a logical contradiction, you couldn’t infer any conclusions from it. There is a theorem in logic that if you have a system containing a logical contradiction, you can infer from it any conclusion you want.
[Speaker F] Does the Rabbi have an intuition about the thing? I can’t hear.
[Speaker F] Does the Rabbi actually have an intuition to understand what the quantum state is?
[Rabbi Michael Abraham] All those who get tangled up with the quantum state simply don’t understand. They don’t understand that a particle is not a little ball. It’s a layman’s perspective. It’s a wave function described as a sum of functions, each of which is a function of the particle. And where—
[Speaker F] What would its ontology be? What is its entity?
[Rabbi Michael Abraham] Its ontology is that it is a wave spread out in space in some fashion.
[Speaker F] Ah, with that I agree, so it’s not a particle. Okay, that’s exactly what I’m claiming.
[Rabbi Michael Abraham] It is a particle. That’s what’s called an electron.
[Speaker F] Yes, but that’s only our description of the thing. There isn’t really such a thing. It’s a wave.
[Rabbi Michael Abraham] It’s a wave.
[Speaker F] With that I completely agree. Fine, so…
[Rabbi Michael Abraham] Good.
[Speaker G] But if it is… why does it refrain from going through both slits?
[Rabbi Michael Abraham] What? I didn’t understand.
[Speaker G] If so, no… if that’s not how we define it, if it’s this kind of wave, then we don’t understand why it refrains from going through both slits.
[Rabbi Michael Abraham] It doesn’t refrain; it goes through both slits. Okay. It’s just that if you put a detector in one of them, then it will turn out that it passed through only one. Those are already the deceptions of quantum theory. But again, that’s not a logical contradiction. It’s just hard to grasp physically. There is no logical contradiction here at all. And there cannot be a logical contradiction. All those people—including physicists—who get confused about these things, get confused in the philosophical interpretation. They may be good physicists. In the philosophical interpretation of the matter, they don’t understand it correctly. Meaning…
[Speaker G] So then you can believe in Kabbalah. Huh? Then you can believe in Kabbalah. Why? Because it’s the same kind of… intuitively it’s the same kind of contradictions, things you don’t understand…
[Rabbi Michael Abraham] In Kabbalah there are no contradictions at all, nothing of the sort; these are simply abstract concepts. In quantum mechanics there are things that look like contradictions. You need to work on it to understand why there’s no contradiction there. In Kabbalah it’s not contradictions but abstract things. Okay, friends, let’s stop here—it’s already late. Rabbi, a question…
[Speaker F] Rabbi, just one last question. Why in the end… I still didn’t fully understand. Why is logic necessary and certain? Why, if for example some postmodernist—some postmodernist, some Michel Foucault—had been a child and someone had raised him on the knees of non-logic…
[Rabbi Michael Abraham] I’ll save you the time—I have no answer. That’s just how it is; it’s self-evident. If you don’t understand that, I have nothing to explain. No, I mean that seriously. I’m not saying it cynically. It’s simply… it’s like someone asking me why do you trust your eyes? Maybe they deceive you. I trust my eyes because they show me. That’s it. I know that they—
[Speaker F] That’s why I asked the Rabbi: if, say, some postmodernist had raised his child on the knees of contradictions and—
[Rabbi Michael Abraham] There would be a didactic problem—I answered. He—
[Speaker F] No, but the Rabbi understands that he would feel what the Rabbi feels—that it’s simple? He would understand that the opposite is simple.
[Rabbi Michael Abraham] No. There would be a didactic problem, but I’d overcome it if I were a good enough teacher. Even people whose minds have been messed up by their education can be corrected; it’s not deterministic. The fact that people make mistakes proves nothing; people can make mistakes.
[Speaker F] No, I meant intuitions.
[Rabbi Michael Abraham] There can be opposite intuitions. An intuition can be mistaken, yes, I agree. Okay. But once you clarify it, I—
[Speaker F] think you can arrive at the correct intuition. When the Rabbi argues with him, like about faith / belief, about ethics / morality. When the Rabbi argues with him he’ll say, but my intuition is the opposite. I answered Shmuel that I have nothing to say to him.
[Rabbi Michael Abraham] But even to the child, he—
[Speaker F] he’ll also tell you, I have nothing to say to you. Right, right.
[Rabbi Michael Abraham] Right, right. True, even in ethics / morality I don’t know what to tell him. If he asks why murder is forbidden, I don’t know—because it’s immoral to murder. Why not? I don’t know, that’s just how it is. What can I tell him? If you don’t understand that on your own, I have nothing to tell you. Yes. Okay, Sabbath peace, goodbye, may we hear good news.
[Speaker E] Thank you very much, Sabbath peace.