Paradoxes and Contradictions in Jewish Law, Lesson 1

This is an English translation (via GPT-5.4). Read the original Hebrew version.

This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.

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🔗 Link to the transcript on Sofer.AI

[0:00] Introduction: dealing with contradictions and paradoxes
[2:15] The deception of dichotomous logic
[4:33] The law of the excluded middle and contradiction
[12:58] Light and darkness: privative opposition
[14:03] Cold and heat: contrary opposition
[19:01] Organ donation and the ethical discussion
[20:47] Brain death versus cardiac death
[23:27] The third possibility: thinking outside the box
[28:14] The dichotomy and the third logical possibility
[30:29] Organ donation as an example of dichotomous analysis
[33:05] Brain death versus cardiac death — urgency of decision
[34:27] Choosing the safer option in a tie
[45:30] Translating into logic and distinguishing the field of argument
[47:20] Systematic problem-solving — the S.I.T. methodology
[51:14] The heap paradox and fuzzy logic

Summary

General Overview

The text sets itself the goal of understanding how to deal with contradictions, paradoxes, and dichotomies, and how to find a “third possibility” even when logic seems to force a choice of yes or no. It presents the law of the excluded middle and the way a mistaken application of it creates artificial debates in which the speaker agrees with neither side, and it proposes developing a systematic toolbox for identifying hidden assumptions and breaking out of discussion frameworks. It demonstrates this through a Zen koan about the answer “mu” as neither yes nor no, through a distinction between negation and opposite, through a halakhic example of nullification by majority, through the organ donation dilemma, and through fuzzy logic and the heap paradox, all the way to the idea of fractional dimensions and fractals.

Zen Koan, Hofstadter, and the Answer “Mu”

The text opens with a koan brought in Hofstadter’s book Gödel, Escher, Bach about the Zen teacher Joshu, who is asked whether a farmer has Buddha-nature or not, and he answers “mu.” Following Hofstadter, it explains that “mu” means neither yes nor no — that is, a third kind of answer that neutralizes the dichotomy. This is presented as an introduction to the idea that one can escape situations in which only two options are presented as though they exhaust the entire space of possibilities.

The Law of the Excluded Middle, the Law of Non-Contradiction, and the Feeling of “Logical Distress”

The text describes how a contradiction is usually presented as a pattern of “X or not-X,” in which there is no third possibility, because “both X and not-X” contradicts the law of non-contradiction, and “neither X nor not-X” contradicts the law of the excluded middle. It claims that logic itself is dear to him, but that many times people “do logic incorrectly,” and so logic can often mislead. He describes the recurring experience of debates in which “this one argues this way and that one argues that way,” and he finds himself agreeing with neither side, and tries to clarify how one can formulate a third possibility without disputing the law of the excluded middle.

Opposites, Logical Negation, and Two Kinds of Opposition (Aristotle)

The text cites a point in the name of the Maharal that two opposites always have something in common, and that things with nothing in common are not “opposites” at all but simply different, like “a clock” and “a bird.” He recounts a conversation with a friend about whether there is anything that has no opposite, and arrives at the conclusion that objects do not have opposites, only at most logical negations, and logical negation is not the same as an opposite. He presents an Aristotelian distinction between “contrary opposition,” which is a positive opposite, and “privative opposition,” which is a negation that nullifies the thing, illustrating it with numbers: zero is the nullification of one, while minus one is the positive opposite of one. He connects this to the philosophical question of light and darkness versus cold and heat, and argues that light and darkness are a privative opposition, whereas cold and heat mix by canceling each other out, and so they are a contrary opposition.

Halakhic Example: Nullification by Majority, Oneg Yom Tov, and Tzitzit “for their own sake”

The text cites a responsum of Oneg Yom Tov, siman 4, on the question whether a minority that is nullified by the majority “takes on the name of the majority” or “only loses its own name.” He illustrates this with tzitzit strings, where most of the strings were spun for the sake of the commandment and one string was not, and argues, following Oneg Yom Tov, that nullification does not turn the defective string into one spun properly, but only removes its status as “not for its own sake,” so that an intermediate, “neutral” state is created that is neither this nor that. He presents a practical difference in the case of a further mixture with strings that were not spun properly, and the discussion of “piece by piece becomes nullified,” and the consequences for invalidation or nullification, connecting it to a dispute about whether the minority “changes essence,” similar to the idea that “a piece becomes carrion” (hatikha na’aseit neveilah).

The Liar Paradox and Correct Negation versus Opposition

The text brings the popular formulation of the liar paradox about a Cretan who says that all Cretans are liars, and argues that this is “not a paradox,” because the negation of “all Cretans are liars” is not “all Cretans tell the truth,” but rather “there exists at least one who is not a liar,” or “not all of them are liars.” He concludes that very often people make an incorrect negation because they replace negation with opposition, and in this way they create what seems like a paradox or a dichotomy that is not actually necessary.

Organ Donation: Brain Death versus Cardiac Death and Questioning the Framework of the Discussion

The text presents the dilemma of organ harvesting between brain death and cardiac death, and formulates the debate as a dichotomy surrounding the definition of the moment of death: if brain death is death, organs may be taken without killing; if cardiac death is death, taking the heart kills and is therefore forbidden. He argues that one can find a “third option” not by denying the dichotomy of the moment of death, but by stepping outside the framework that assumes as a starting point that “it is forbidden to kill one person in order to save another.” He cites the Talmudic reasoning, “Who says your blood is redder? Perhaps that man’s blood is redder,” and presents the possibility of challenging its application in a situation where one person can live a full life while the other cannot return to life according to medical consensus, while acknowledging that “slippery slope” arguments are a different discussion. He emphasizes that he is not proposing any violation of the law of the excluded middle, but rather examining the assumptions built into the process of logical formalization that leads to a dichotomous presentation, and illustrates this with the metaphor of “choose any color you want, as long as it’s blue,” as against choosing “green.”

Defining the “Playing Field” as a Dichotomy: The Eagle and the Duck and the Public Relations of Concepts

The text brings an anecdote written by Baruch Kahane about “the eagle and the duck,” in which the eagle invites the duck to an aviation contest and the duck moves the contest to the pond for swimming, in order to show that whoever defines the playing field defines the dichotomy in advance. He continues with anecdotes about a hamster versus a mouse and a donkey versus a lion to argue that public relations and names alter perception, and that even “eagle and duck” are sometimes just image. He uses this to reinforce the principle of refusing to accept a discussion framework that creates an artificial dichotomy.

Breaking Dichotomies as a Method: SIT, Altshuller, and a Systematic Toolbox

The text describes a friend who founded a company called S.I.T. (Systematic Inventive Thinking), which proposes a systematic solution to problems that require creativity, and connects this to a Russian Jew named Altshuller, who tried to build such a toolbox. He describes a method in which one defines the problem, lists the parameters that create the deadlock, and then systematically tries giving up different parameters and checking what results, so that new possibilities emerge without relying on a random “creative burst.” He argues that this is exactly parallel to searching for the underlying assumptions at the basis of a dichotomy and examining whether one of them can be discarded in order to generate a third path.

Fuzzy Logic and the Heap Paradox: Continua Instead of Yes/No

The text presents as a second tool the move away from “binary” logic when the dichotomy itself covers concepts that are continuous, and gives as a central example the heap paradox, in which “a heap” is not yes/no but a scale of “heap-ness” between zero and one. He gives the example of the claim “there’s no point in giving tests,” which divides students into lazy and diligent, and breaks this down by showing that diligence exists on a spectrum and not in only two classes, and he gives a similar example in the name of Leibowitz about sports. He demonstrates that everyday language requires artificial cutoffs for practical decisions such as “when is it afternoon,” but the cutoff does not solve the philosophical problem of the meaning of the concept, and extends this to the claim that all everyday concepts are exposed to similar continua, such as colors, “bald,” and continuous transitions between a “prayer book” and “tissue paper.”

Fractals and Fractional Dimensions: Breaking Dichotomies That Seem Fundamental

The text presents Mandelbrot’s idea of fractals and fractional dimensions, and shows how apparently there are only whole-number dimensions of 1, 2, 3, but in practice there are forms whose dimension is fractional, like “one and a bit.” He demonstrates this through a coastline, where increasing the resolution does not increase the measured length linearly but adds inlets that were not seen before, so that scaling reveals a “dimension” that is not whole, at least over certain ranges of scale. He weaves into the discussion remarks about increased surface area and practical and halakhic possibilities that arise along the way, and concludes that this too is another case in which a dichotomy that seems unbreakable turns out to depend on assumptions and on the mode of measurement.

Summary of the Tools and Planned Continuation

The text summarizes two central tools for breaking dichotomies: examining the framework and assumptions that lead to the dichotomous presentation, and identifying cases in which there is a continuum and shades of gray, so that yes/no logic is not appropriate. He states that there are additional tools he will present later, and that further examples will illustrate how an orderly toolbox makes it possible to move tool by tool in order to find a way out of situations that seem stuck.

Full Transcript

[Rabbi Michael Abraham] Yes, this topic — how to deal with contradictions, with paradoxes, with how to neutralize dichotomies. I mean, obviously this can spread out into a huge number of topics, each one of which is a topic in itself. We’ll encounter a few examples. So I don’t know how far to go with this subject; you could talk about it for quite a few months. We’ll see, we’ll see how it develops. Maybe, maybe I’ll start with a little piece of what’s called a koan. A koan — in Zen there are these short stories called koans. What? That’s what they call them there, koans. They’re like stories that provoke thought, little stories that provoke thought. So one such story appears in Hofstadter’s book Gödel, Escher, Bach. And this was centuries ago: there was a Zen teacher named Joshu who lived a long life and reached the age of one hundred and nineteen. One day, when Joshu and a certain monk were standing together in the monastery, a farmer passed nearby. The monk asked Joshu: does that farmer have Buddha-nature or not? Joshu answered him: Mu. And when Hofstadter tries to explain there what that means, he says that mu means neither yes nor no. In other words, there is some third kind of answer, which is not — don’t get trapped in the dichotomy of either he has Buddha-nature or he doesn’t; there is also a third option. So that’s a good introduction to the matter, because I too often feel, despite the fact that I’m very fond of logic and I usually persecute deviations from logic to the bitter end — but, but many times the feeling is that people are just doing the logic wrong. And very often those statements that there are only two possibilities, because logic forbids any deviation — either yes or no, what’s called the law of the excluded middle — are an incorrect application of logic. And therefore logic can very often, very often mislead. Whoever has already heard me mention more than once that in a great many debates, where this one claims this and that one claims that, I very quickly find myself disagreeing with both sides. And then I ask myself: how can I disagree with both sides? They always present us with a picture: either you think this way, and if not, then that way. I mean, what else could there be? The law of the excluded middle. And when I realized that this happens many times, I tried to examine for myself how this really fits with the rules of logic — that is, how do you get out of dichotomies? How do you formulate a third possibility when only two are presented to you, two that supposedly cover the whole space of possibilities? And somehow several patterns developed for me, I managed to identify several patterns that can be used — these are kind of master-patterns — that can be used in a great many problems to try to see that there is a third possibility. Meaning, that the dichotomy does not cover all the possibilities. So I want first of all to say a little about the concept of contradiction and deviation and the law of the excluded middle and so on, and then get a bit into those techniques, that toolbox. Usually, when a contradiction between two things is presented to us, that contradiction is supposedly presented as X or not-X. In other words, those are the two possibilities, and beyond X or not-X there is no third possibility. Both X and not-X is a contradiction — that is, it doesn’t say anything. Neither X nor not-X contradicts the law of contradiction. Neither X nor not-X contradicts the law of the excluded middle. In other words, there is no third possibility; that’s why it’s called the law of the excluded middle. Meaning, the third possibility is excluded. That is, there is either X or not-X, there isn’t something else. So therefore it seems that you have to place yourself either on one side or on the other. And when you feel that you don’t identify with either of the two sides, you’re actually in logical distress. How can that be? What, are you violating the law of the excluded middle or the law of contradiction? So first of all, there is a point — and the Maharal says this in several places — that usually, when we talk about opposites, they have something in common. Two things that have nothing in common are never opposites of one another. A clock is not the opposite of a bird; they’re just two different things. In order to be opposites, there has to be something shared, and within that there is somehow a division into yes or no. Say, you can be a living creature that is a bird or not a bird, but both are living creatures. But a bird and a table are not perceived as two opposites. A table is not a bird, but that is not usually what is perceived as two opposites. Meaning, whenever we talk about two opposites, there is something common that binds them. And that’s what I started saying earlier before I told you — I once said that I have some good friend with whom I always used to enjoy thinking about all kinds of interesting philosophical problems. And one time I asked him whether there is anything that has no opposite. I thought about it for quite a while and couldn’t find anything that has no opposite. So he says, what do you mean? A cloud has no opposite, a table has no opposite, lots of things have no opposite.

[Speaker D] No, those are objects, right? Objects don’t have opposites.

[Rabbi Michael Abraham] Exactly. The insight in the end was that all matter in the universe that is not an object…

[Speaker C] What? All matter in the universe…

[Rabbi Michael Abraham] Yes. No — there is logical negation but not an opposite. In other words, logical negation is not the same thing as an opposite. To be not-this is not to be this thing’s opposite. An opposite is something positive. We once talked about two kinds of opposition: there is contrary opposition and privative opposition. That’s already an Aristotelian distinction. There is, say, in numerical terms, in the language of numbers — then zero can be seen as the opposite of one, and minus one can also be seen as the opposite of one. What is the difference between those two kinds of opposition? Zero is a privative opposite in relation to one. You nullify the one and you are left with zero. What?

[Speaker D] That’s negation.

[Rabbi Michael Abraham] Yes, exactly — which is a negation that is not necessarily an opposite. And minus one is a positive opposite. Meaning, it’s not just not-one, but something positive that has an opposite property. That is really contrary opposition. Just this morning we studied regarding nullification by majority. There is a responsum of Oneg Yom Tov, siman 4, where he discusses the question whether the minority that is nullified by the majority takes on the name of the majority or only loses its own name. It’s exactly the same thing. That is, if for example he says there are tzitzit strings that were spun for their own sake. They were spun for their own sake — the string has to be spun for the sake of the commandment of tzitzit, or for the sake of tzitzit; there is room to discuss that a bit, but it doesn’t matter, it has to be done properly for its own sake. There is one string that was not spun for its own sake. Now it got mixed in among the other strings; there is nullification by majority. If there is nullification by majority, then supposedly you can use all of them. He says no, such a thing is not nullified by majority. Why? Because this conception, as though the majority changes the essence of the minority and turns it into something like itself — yes, takes it over — is not correct. The majority removes the minority’s name from it; the minority loses its name. So now, regarding that string, it is no longer correct to say that it was not spun for its own sake, but that still does not mean that it was spun for its own sake. In other words, it doesn’t turn into something positive like the majority; it only loses the designation of something that was not spun for its own sake. Okay?

[Speaker D] There’s some intermediate state — here’s the third possibility.

[Rabbi Michael Abraham] Yes, that’s why I’m bringing it up, we’ll talk about that more.

[Speaker D] The practical difference is if you now mix it with seven that were not for their own sake, whether that means it’s now nullified by majority or not.

[Rabbi Michael Abraham] Yes. The opposite nullification would apply. According to the one who says — if we nullify permitted material in forbidden material, and that too is a question, whether permitted in forbidden is nullified, that’s a dispute. But let’s say that permitted in forbidden is also nullified, then you have seven strings that were not spun for their own sake and one that was, so the one that was spun for its own sake loses its name. Now it isn’t correct to say that it was spun for its own sake; that’s enough to invalidate it. It doesn’t need to have been spun not for its own sake; it’s enough that it was not spun for its own sake.

[Speaker D] Seven for their own sake plus one not for its own sake will be nullified, and then if you add another seven not for their own sake, you get to a situation where there are definitely now eight not for their own sake and seven yes.

[Rabbi Michael Abraham] Now if it’s piece by piece, is it nullified?

[Speaker D] What? There are seven for their own sake plus one not for its own sake.

[Rabbi Michael Abraham] Okay, so it was nullified.

[Speaker D] Meaning all eight are for their own sake?

[Rabbi Michael Abraham] Right, it was nullified. No, not exactly, because that’s exactly the point: that string did not become a string spun for its own sake — after all, that’s what Rabbeinu Yom Tov said — so there aren’t eight strings that were spun for their own sake; there are seven strings spun for their own sake and one neutral one. Now another seven come, and that’s why I said not the seven all at once but one by one — another seven come that were not spun for their own sake. You have seven against seven, so it is not nullified. If you hold that the eighth string itself becomes a string spun for its own sake, and that’s how some later authorities (Acharonim) understand it — it’s a dispute — there are those who hold that it really does become properly spun, then it will nullify the others. There is the idea of piece by piece becoming nullified, that if they fall in one by one, then they will all become nullified, so they would indeed be nullified. But if all seven come together, then not.

[Speaker F] An infinite number of not-for-their-own-sake would become nullified within seven if they cross one by one.

[Rabbi Michael Abraham] Right, like in a mikveh. What?

[Speaker F] And that opinion fits with the conception, with the ruling, that a piece becomes carrion — is that what it’s called? That if I mix meat and milk, then it can…

[Rabbi Michael Abraham] Meaning it itself becomes meat-and-milk, so the minority changes the quantity. Okay, fine, the dispute touches on a piece becoming carrion, yes, hatikha na’aseit neveilah. Okay, to bring that down here, as it were? What? He says that this is apparently yes, but there is the law of that, it’s written in the Talmud. So the question is how he explains that law — that’s another story, we don’t want to get into that now, and I’m not in that topic right now. In any event, that is really the difference. Meaning, either you say it loses its name, and then this is privative opposition: it does not become the opposite, it only loses its name, it goes from one to zero. That’s not zero. What?

[Speaker C] That’s not zero.

[Rabbi Michael Abraham] Yes, so I said — that’s what I said earlier — this is negation, this is negation, it’s not zero. Okay? The opposite of zero in that sense could be either one or minus one; you don’t go back to one, it’s another number. Yes, never mind, let’s say there are only three things in a binary world — or actually a ternary one — one, minus one, and zero. So those are two kinds of opposition. That’s just an interesting note; you know this appears in Aristotle, and in Giv’at HaMoreh Shlomo Maimon also talks about it a bit. There is, for example, a relation of opposition between light and darkness, and the perennial question is whether light is the absence of darkness or darkness is the absence of light. Today, in the physical conception, it’s quite clear that darkness is the absence of light, but philosophers puzzled over this matter. And likewise with cold and heat and the like — with these opposites there is always a question: which of them is the real thing and which of them is merely an absence? Now when you think about it in this mathematical language that I presented earlier, you see that it simply depends on which pair of opposites you are talking about. If we are talking about light and darkness, then it is pretty clear that the opposition there is privative, not contrary. Why? Because if you add light to darkness, you get light. There is no cancellation between them; it’s not one and minus one canceling each other out, it’s one plus zero. In other words, light plus darkness — yes, there is no flashlight that is a source of darkness; there is a flashlight that is a source of light, and you can stop it and then darkness is created.

[Speaker G] But there are silence speakers. What? There are silence speakers, by broadcasting in reverse phase that zeros it out, yes, destructive interference.

[Rabbi Michael Abraham] If there are silence speakers, if it’s…

[Speaker D] Slices of it, then in any event it would just be reduced light. I didn’t understand — light and darkness, light and not-light.

[Rabbi Michael Abraham] It’s not a mixture of light and darkness, it’s stopping the light. It’s not the same thing. It’s stopping the light. No, that’s exactly the point, but there are no sources of darkness.

[Speaker F] So if there is such a thing as a speaker that creates silence, then why can’t you stop…?

[Rabbi Michael Abraham] The speaker doesn’t produce…

[Speaker F] Silence — it nullifies the sound.

[Rabbi Michael Abraham] Yes, it produces sound whose frequency is such that it cancels the other frequency through destructive interference. It broadcasts opposite sound, right, true. So basically, basically, that’s light and darkness. By contrast, with cold and heat — when you want lukewarm bathwater, you mix hot water and cold water, right? They cancel each other out. So clearly cold and heat — and again, cold and heat are of course our sensations, not temperature; everything is relative to us — but cold and heat, whatever their subjective definition may be, mix in a way that cancels the other. Meaning that the relation between them is contrary opposition and not privative opposition.

[Speaker F] Why isn’t it cold and heat? It’s the cold object and the hot object.

[Rabbi Michael Abraham] Yes, fine, so its coldness and its heat mix together and an object is created that is lukewarm. The way to mix cold and heat is of course through objects that carry the cold and the heat; I don’t know how you mix cold and heat detached from objects. But still, what is mixing here is not the objects but the cold and the heat too — the cold and the heat, not only the objects. So that is an example of contrary opposition. Okay, in any event, for our purposes: when I am really talking now about contrary opposition, not privative opposition, then in contrary opposition there has to be something shared. One and minus one are both on the number line, both of magnitude one, only the sign is opposite. So here is the first hint. What about two? Or zero. Zero too. Here is a third option. In other words, either one or minus one. You understand that the moment I go beyond the thing shared by the two opposites, after all the two opposites are defined within some framework that is common to them, if I manage to go outside the framework, additional possibilities can open up for me. That’s one way of defining additional possibilities — to get out of the dichotomy. Okay. When we talk about a dichotomy, the first tool worth checking if we are looking for a way to escape it is to look for what the two opposite sides have in common and to try to go beyond that. Now maybe I’ll give a few examples.

[Speaker H] Sorry — with one and minus one and zero, then your privative thing is the negation of it? Again? What is privative? Is it its negation?

[Rabbi Michael Abraham] Yes, logical negation is basically nullification, it’s not opposition.

[Speaker H] Fine, so specifically the negation turns it into… it’s either this or its negation? It’s not either one or minus one, but either one or zero.

[Rabbi Michael Abraham] Either one or not-one. Logical negation takes me from one to zero. Opposition takes me from one to minus one. But by not-one you mean zero. Yes, right, something that is not one. That’s it — I’m marking it as zero just as a symbol.

[Speaker D] Meaning I’m not standing opposite it, I’m standing more in the middle of the…

[Rabbi Michael Abraham] The point is, I nullify the one, I don’t turn it into something else. Exact logical negation, of course, is everything that is not one, not necessarily zero. It’s just that in a world where there is only one and zero, there is no opposition, only negation. Let’s put it that way, okay? There is no opposition. The opposite of one is minus one. In a binary world in which there is only one or zero, there is negation but no opposition. Maybe that’s a more precise formulation. In any event.

[Speaker H] And maybe that’s why you can talk about not-one even outside the domain of numbers.

[Rabbi Michael Abraham] Yes, of course. A cloud is also not-one in that sense. That’s exactly what I’m saying. Negation does not assume something shared between the thing being negated and the result of the negation. Opposition does. Meaning, one and minus one have something in common: both have absolute value one, both are numbers, only their sign is opposite. So they have a lot in common, and within that shared genus there are two species, and those species have opposite properties. Okay? But when I talk about negation, not opposition, there doesn’t have to be anything shared between the thing being negated and the result of the negation. The negation of one is not-one. I marked it as zero, but it doesn’t matter — anything that is not one, okay? That’s negation. Maybe this is also somewhat connected to what happens in the liar paradox. The popular formulation of the liar paradox is when a Cretan says — yes, the source is in the New Testament — a Cretan says that all Cretans are liars. Usually people see that as a paradox, but it’s not a paradox, because the negation of “all Cretans are liars” is not “all Cretans are not liars” or “all Cretans tell the truth,” but rather that there is at least one who is not a liar. That is the negation. And is it himself?

[Speaker B] It could be — no, it’s not necessarily himself.

[Rabbi Michael Abraham] He himself could be telling the truth about someone else who is the liar, but that is enough for the statement itself to be false. In other words, many times when we make an incorrect negation, what we are really doing is opposition, not negation. The opposite of “all Cretans are liars” is “all Cretans tell the truth.” The negation of “all Cretans are liars” is every other situation. For example, that at least one is not a liar, or two, or three, or all of them. What? That there exists at least one who is not a liar, yes, that not all of them are liars. Maybe that’s a more precise formulation. In any event, maybe I’ll bring one example — we’ve also talked here about many things in various contexts, so right now I’m basically just connecting them. The example, for instance, of organ donation. I once wrote about this in Techumin, and with organ donation, basically how is the debate conducted? The debate is conducted under the following two assumptions — or not two assumptions, really one assumption. It is forbidden to kill one person in order to save another. That is the basic assumption. Now when we are in a situation where a person is between brain death and cardiac death — I’m talking about organ transplantation in that specific state, not all organ transplantation, or organ harvesting for transplantation — so I say: a person is between brain death and cardiac death, the brain stem is no longer functioning, let’s say, but the heart is still functioning. Okay? Biologically he is still — the pump is still working. I think something in the brain also still has to be there; it can’t be that the brain is completely gone, otherwise I don’t think the heart can work without that. But apparently some part doesn’t. So now the question is whether it is permissible, say, from a patient who is between brain death and cardiac death — whether it is permissible to take an organ from him, like the heart for example, in order to transplant it into someone else. Once the heart finishes, you can’t take it and transplant it into someone else anymore, and before the person reaches brain death we don’t kill him and take his heart in order to save someone else. Therefore the dilemma arises only in that specific segment between brain death and cardiac death. The question is whether you can take the person’s heart and in doing so basically kill him in order to transplant it into someone else who can gain life from it — a whole life. So the debate surrounding this matter basically revolves around the point of when the moment of death is defined. Is the moment of death the death of the brain, in which case what? Then the person, in the state I described earlier, is already considered dead because we are after brain death. If the person is already considered dead, then taking the organ from him is not killing him, so you can take the organ from him in order to transplant it into someone else and save him. But if we define the moment of death as the death of the heart and not the death of the brain — let’s pause a second on what brain death means; it’s not enough to say brain death, what is brain death? Okay, there is some medical definition called brain death, the details don’t matter right now, and I myself am not expert in the medical details.

[Speaker J] Maybe the brain stem, I assume that’s also disputed.

[Rabbi Michael Abraham] No, what difference does it make — let there be a dispute about that too, but I’m not dealing with that dispute, I’m dealing with this one. So in this dispute there are two possibilities, either brain death or heart death; whatever it is, the medical definition called brain death. Okay? Now if I am considered dead once the brain dies, then you can take my heart — that is not called murder — and save someone else; on the contrary, it is a great commandment to do so. But if brain death is not the moment of death and heart death is, then when they take my heart they are actually killing me: I was alive and now they are killing me. And since that is so, transferring it to the second person is essentially killing me in order to save him, and that is forbidden. That is the debate. And at a certain stage I thought about this and something here seemed very absurd to me. Supposedly there are two possibilities: either this is death or that is death. Fine, so of course there is always the route of saying, okay, maybe the moment of death is somewhere in the middle. Fine, that’s the banal option, but here it won’t help. I’m speaking now about the principled question, okay? So after that middle moment — however you determine it.

[Speaker J] There are also other considerations. What? There are also other considerations: maybe in this case it’s forbidden even if he is indeed considered dead, because I don’t want to open the door to who knows what.

[Rabbi Michael Abraham] For example, I want to raise the opposite consideration: even if he is considered alive, maybe you still can take it. Okay, but according to the same logic — the same logic. Meaning, when I look at these two opposing definitions, heart death or brain death, let’s say for the sake of discussion that there are only two stages — let’s turn this into a binary discussion. Okay, there are only two stages. So I ask myself whether it is possible not to agree with either side. Supposedly you have to be either here or there. At some point the person dies; somewhere between the stage of brain death and the stage of heart death the person dies, so you have to decide. The moment I say — again, the model we are discussing right now is binary: there is brain death, immediately after that comes heart death, there are no other stages, just for logical simplicity, okay? So you have to choose a position, meaning either this or that. You don’t know? Fine, so you don’t know — that’s a doubt, but it’s not a third position. The question is whether there is a third option. And my answer is yes, there is a third option. The third option is to step outside the box. You put me inside the box; you are basically saying, look, as long as the person is alive, taking his heart is killing him in order to save someone else, and that is forbidden. Both sides agree on that. The whole question is only whether the person is considered alive or not, but both sides agree that if he is alive, then it is forbidden to kill him in order to save someone else. That is the point common to both of them. And now, when I look for a third option, it may be possible to attack that point, because that is the point that defines the opposition here, the contrast between the two possibilities. And indeed, what I argued there — and I really think this is true even according to Jewish law — is that it is not true that it is forbidden to kill one person to save another. And when the Talmud talks about the fact that it is forbidden to kill one person to save another, it brings the reasoning: “Who says your blood is redder? Perhaps that man’s blood is redder.” Yes — your blood is not redder than his, so you cannot kill him in order to save yourself. So if I’m speaking about the case of organ donation, I say to the patient: listen, your blood is not redder than the blood of this brain-dead person, and therefore you cannot kill him in order to save yourself — or the doctor cannot kill him in order to save someone else — because no one is preferable to the other. That is basically the Talmud’s assumption. Now the question is whether I am willing to accept that in the case of someone who has suffered brain death versus a person who can live a full life. Even in such a situation, do I say that the blood of the one who can live a full life is not redder than the blood of someone who is essentially already dead — not according to the formal halakhic definition, but dead in terms of personal functioning?

[Speaker D] No, now you’re…

[Rabbi Michael Abraham] Raising slippery-slope arguments. You’re now saying, okay, where do you stop?

[Speaker D] Good question.

[Rabbi Michael Abraham] I’m only talking about the logic of a third way. I’m saying that in logic I can find a third way. You can say that you don’t want this third way because it’s all a slippery slope — no problem, that’s another argument. I also don’t think I agree here, but never mind, I’m willing to accept that.

[Speaker H] Let’s suggest a different assumption.

[Rabbi Michael Abraham] What? That he’s not dead?

[Speaker H] Yes, he’s not dead. So why say — if he’s not dead, why say that his blood isn’t the same color?

[Rabbi Michael Abraham] Because no, it’s not about the color. We’re talking about defining the equality of life, the value of life. I’m saying that the value of life…

[Speaker H] If he’s not dead, then as long as he’s not dead he has life. Why? Because it could be… Why is he considered not dead? Because it could be that he’ll live, and then he’ll be in excellent shape.

[Rabbi Michael Abraham] No, he can’t come back to life. There is no return to life. He can’t come back to life. That is wall-to-wall medical consensus. Nobody disputes that.

[Speaker D] Maybe the dead person can come back to life, and that would be the department’s margin of error.

[Rabbi Michael Abraham] Maybe the dead person can come back to life? Fine. We’re beyond the margin of error. As long as he is brain-dead, the doctors say: that’s it, the person is not coming back to life. There is no such example. There was once — I brought it up — there was a case like that in Denmark, it was published about a year ago or something like that, and everyone’s assumption was simply that there had been a mistake in the initial determination. Again, but assuming the initial determination was correct, there is no return. No return. Meaning, again, I’m not a doctor, I’m saying this is what it…

[Speaker D] The diagnosis — a person being taken to burial gets up.

[Rabbi Michael Abraham] No, that proves nothing. It could be that they didn’t diagnose correctly, so he wasn’t brain-dead. Fine.

[Speaker D] At that moment you really thought he was dead.

[Rabbi Michael Abraham] So an error in diagnosis is a technical matter.

[Speaker D] It’s a matter…

[Rabbi Michael Abraham] It’s a technical matter, an error in diagnosis.

[Speaker D] Also when there is…

[Speaker B] Cardiac death, they can also make a mistake…

[Speaker D] In the diagnosis.

[Rabbi Michael Abraham] Exactly. Why not? Cardiac death too — yes, maybe afterward it’ll suddenly jump again, maybe you give him some electrical impulse and maybe that’ll bring him back, who knows?

[Speaker D] At first they wait an hour in the ward, he remains attached, and in every…

[Rabbi Michael Abraham] In other words, you said that people being led to burial come back. Right, exactly. So that means we’re not talking about technical problems. I’m talking about the basic logical question; leave practice aside for a moment. And on the basic logical question I’m saying: I have a model. I’m now describing a different situation, okay? A situation medicine doesn’t know. There are two states, and we know how to formulate both of them exactly. There are no measurement errors. It’s a hypothetical question, okay? There is brain death and cardiac death. We know how to define them well, there are no mistakes, everything is fine, and there are only two possibilities, no more. Okay? Let’s talk under those assumptions, because what interests me here is the logic, not the practice. So logically, in such a situation, apparently there is a complete dichotomy here. Meaning: either here or there. What other possibilities could there be? Now, that’s true on the level of the moment of death—you have to choose either here or there. What I’m claiming is only that when you talk about the question of whether to take organs, it’s not certain that this is determined automatically by deciding the moment of death. That’s what I’m challenging—not the dichotomy between whether death is brain death or cardiac death; there, in the model I described, there are only two possibilities. But the question whether that determines the result in the issue I’m wrestling with—that’s a different question. Because that dichotomy is presented under the assumption that someone who is alive may in any case not be killed in order to save someone else. That itself is worth examining. Now again—even if I don’t agree with it, it doesn’t matter—I’m talking about this logical option, that this logical option needs to be examined. Afterwards you can say we don’t agree with this option, slippery slope, or we just don’t agree, whatever, that’s not the point right now. But I’m saying that on the principled level, I’m showing you how logically one can produce a third option where it seems there isn’t one.

[Speaker H] No, but that doesn’t change it… not an option… not logically.

[Rabbi Michael Abraham] Of course it does, because if I’m checking what “logical” means—

[Speaker K] I’m not disputing…

[Rabbi Michael Abraham] What is the law of the excluded middle? I’m not disputing the law of the excluded middle. This is not a claim against the law of the excluded middle. Rather, when I conduct a philosophical discussion and map it onto a logical dichotomy, then suddenly it’s worth checking whether there are some additional assumptions along the way before I got to the dichotomous presentation. So from the dichotomy I can’t depart: either death is brain death or death is cardiac death. You can’t go beyond that, again, in the binary model.

[Speaker L] But you’re only arranging one side.

[Speaker E] You’re doing a trick here… what do you mean, a trick in philosophy? What does that mean? You’re telling us, listen, you have either brainstem death or heart death, so someone will tell you, listen, there’s also an intermediate state where the brain is working at half power, say. You tell him, I’m not talking about that. The question I’m deciding right now is… Then they come to you and say, listen, what do you rule—this or that? And you say, that’s not the question at all, my question is different.

[Rabbi Michael Abraham] Of course. I was talking about a hypothetical question, okay? I was talking about a hypothetical question. We have a binary situation: either brain death or cardiac death.

[Speaker E] But you’re not… they’re asking you the question.

[Rabbi Michael Abraham] Fine, and I’m asking that question. What’s forbidden? I want to discuss a hypothetical question; that’s allowed. It’s a way of showing how an orderly analysis of the problem can generate additional possibilities beyond the two—that’s all. Now, you can disagree, you can say it’s not practical, everything is fine. I think it’s correct and also practical. But that’s a different discussion; we’ll talk about that when we talk about organ donation. Organ donation here serves me only as an example, to show how we do an analysis that yields a third possibility where it seems hopeless—where it seems there is no third possibility.

[Speaker L] Yes, but you’re arranging only one side of the equation. Because true, the hypothetical question is like that, but then you can’t use the idea that one person’s blood is less red than another’s. Because what the Talmud is talking about, for our purposes—that one person’s blood is redder, and therefore it is forbidden to kill—speaks about a reality that is not dichotomous. It speaks about ordinary reality. You’re making some kind of forced interpretation here that sets things up and creates a third option with respect to the law.

[Rabbi Michael Abraham] Even in a non-dichotomous situation, if my blood were less red than yours, would you be allowed to kill me in order to save yourself? Apparently that’s what is written in the Talmud. Not exactly—almost all the halakhic decisors say no, because of slippery slopes, doesn’t matter. But on the principled level, that’s what is written in the Talmud. No—even in the non-dichotomous case. The Talmud speaks only in the non-dichotomous case.

[Speaker L] Precisely because the Talmud takes all that reality into account, that’s why it enters in. Not the other way around—the opposite.

[Rabbi Michael Abraham] If they say it in a non-dichotomous situation, then in a dichotomous situation they certainly would say it. If in a situation where there’s a continuum, you still say that if the value of Reuven’s life is slightly less than the value of Shimon’s life, then it is permitted to kill Reuven in order to save Shimon, then in a case where it’s defined binary, obviously it would be permitted.

[Speaker D] The opposite. But there’s no life-value here. The Talmud told you blood-value. It said blood because it’s not value. Right.

[Rabbi Michael Abraham] First of all—

[Speaker D] It’s someone who is a little less oxygenated, which is nonsense.

[Rabbi Michael Abraham] Value—forget blood. The blood represents the color of the person, but not literally the color of the person.

[Speaker D] Yes, I’m saying, it’s a metaphor. But the metaphor is coming to say: don’t find a difference.

[Rabbi Michael Abraham] And who said that? There are halakhic decisors who say this in Jewish law too. There are decisors who claim that it is permitted to kill a terminally doomed person in order to save a living person. And his blood is less red. He’ll get to that.

[Speaker E] The very fact that the Talmud—by the way, not against his will.

[Rabbi Michael Abraham] I’m talking if he agrees. Because a living person—even if he agrees—it is forbidden to do this. I’m not talking about doing it against someone’s will. I’m talking about when he agreed. He signed: I agree. It is still forbidden for me to do it. It is forbidden for me to kill one in order to save the other.

[Speaker D] Even if he’s half-and-half, almost a terminally doomed person for sure?

[Rabbi Michael Abraham] What? I didn’t understand. Someone with brain death is more than a terminally doomed person. I’m saying: the terminally doomed person still functions completely; it’s just that within a few months he’ll die. Okay? That’s what I’m saying. So a person who has died by brain death—here I’m already really getting into Jewish law, it’s not important.

[Speaker D] But surely there’s no one hundred percent. Obviously not. And still they define it by probability.

[Rabbi Michael Abraham] If he doesn’t—

[Speaker D] If he is unlikely to live a year, he’s terminally doomed.

[Rabbi Michael Abraham] I’m saying more than that: the halakhic conception is that it is one hundred percent, not probability. But still he is alive, and only within a few months will he die. That’s much more. After all, a person who is brain dead usually dies within a few hours—he also dies a cardiac death. It’s just a matter of a few hours. That’s why these dilemmas are usually urgent, because within a few hours you have to make a decision what to preserve and how to do the surgery, and the whole story is complicated. So I’m saying: exactly at this point there are those who say this even about a terminally doomed person. Most halakhic decisors are not inclined to say this, but there are decisors who say this about a terminally doomed person. By the way, what about a non-Jew? There are decisors who want to say it is like this there too—that his blood too is less red than the blood of a Jew.

[Speaker J] A man and a woman—why go there?

[Rabbi Michael Abraham] Or a man and a woman—no decisor says that. That’s precedence in Horayot. Right, on that too no one says it, at least no one I know. That’s precedence. There are precedence laws in tractate Horayot. The precedence laws in Horayot say that a man takes precedence over a woman, and a Torah scholar takes precedence over an ignoramus, and all kinds of things like that, because everything is determined by the question of how many commandments one is obligated in. Whoever is obligated in more commandments—his blood is redder in that sense. But that is only with regard to returning a lost object, livelihood through charity, priest and Israelite—

[Speaker F] What? Isn’t it also about whom to save in the source?

[Rabbi Michael Abraham] If you need to save—but not to kill one in order to save the other. Rather, if I have two people whom I need to save—say they’re drowning in a river, or in captivity, or something like that, where there is mortal danger—then who takes precedence over whom? There one has priority over the other. That’s the topic I’m referring to. And still the halakhic decisors say that regarding killing, all the distinctions made there in tractate Horayot are irrelevant. Why not? That’s what they say. Because with regard to murder, it doesn’t justify it. Because for murder, it doesn’t justify it. If you have two equivalent options and you need to choose, then any slight tilt in favor of one option means you should choose the better option. After all, in any case you can only save one. So you have to decide to save one and not the other—that’s not your decision. Otherwise both will die. There is no logic in letting both die. Siamese twins—we talked about that last time.

[Speaker L] Why, if there are two—

[Rabbi Michael Abraham] On the road, and one person has a flask of water—should both die? What? Yes, that’s the flask of water case. It’s connected to Siamese twins. Maybe we’ll talk about that sometime. I don’t think it’s right here, but fine, that’s another discussion.

[Speaker B] But I don’t understand how you started the discussion. In the distinction you’re making, you say: theoretically there’s a dichotomy, and people think there’s nothing to do. I think there’s a third way. And then when you bring the third way, you completely depart from theoretical thinking—you bring Jewish law that changes the whole situation. Of course. So all the time you say, “I’m not on the practical side, I’m on the theoretical side,” but it’s exactly the opposite—this is the most practical side there is!

[Rabbi Michael Abraham] No, no, one leads to the other. I’m on the theoretical side in the sense that I’m talking about a binary picture of either brain death or cardiac death.

[Speaker B] But the theory doesn’t stay that way.

[Rabbi Michael Abraham] Of course it stays that way. I said it stays that way.

[Speaker B] No, you’re bringing a third thing.

[Rabbi Michael Abraham] Right, because usually—and I’ll get to this in a moment—usually when we conduct a philosophical discussion, even if not practical, even if theoretical, even a philosophical discussion—one time I even argued about this a bit with Yehudit—even a philosophical discussion, once we’ve formalized it logically, we have embedded in that formalization all kinds of assumptions. And therefore, after we’ve formalized it logically, it usually looks as if it’s either X or not-X; there is no third possibility. And that is true at that stage. But check very carefully whether presenting the problem as X or not-X really presents the original philosophical problem. Along the way, in the formalization, when you transfer it into the form of either X or not-X, all kinds of assumptions are built in. And when you try to give up one of them, suddenly you discover that more possibilities grow. So of course, once I’m already there, there is no departing from either X or not-X. There isn’t. I don’t mean to claim there is a logical third way such that the law of the excluded middle is false. Of course it’s true. I fully believe in it. And I also oppose all those who claimed there are exceptions to it. There are people who argued that, and I don’t agree with them; I think their claim is mistaken. On the contrary, I go all the way with the law of the excluded middle. I’m not trying here to challenge the law of the excluded middle in the logical sense. What I mean to say is that logic does not always reflect precisely the problem that interests you. By the time we get to the logical stage, that is after a process called formalization. We basically translate the philosophical problem into some logical schema. And now I’m saying: fine, within that logic, yes or no, the calculation is simple. I’m saying: check the translation stage carefully, because at the translation stage it may be that you escaped several additional options that exist. So again, I’m not trying to propose a logical third option. I’m trying to propose an escape from the logical straitjacket—that is, that the logic is not really describing the true problem.

[Speaker K] No, and by “escape” you mean an escape that also doesn’t contradict either of the two ends of the dichotomy. Certainly.

[Rabbi Michael Abraham] The law of the excluded middle remains in force. There is no third option. I’m not contradicting that. I’m saying there is no third option once you’ve entered into that framework. But who says—why am I obligated to enter that framework?

[Speaker E] Because reality gives you that specific thing, and you enter into what… it can’t be… this is first-rate halakhic ruling…

[Speaker L] We have here a side… usually… by necessity.

[Speaker E] Okay, it’s connected. That’s the main thing here.

[Rabbi Michael Abraham] Because if you go straight to the Talmud—but if all the medieval authorities and all that—

[Speaker E] Suppose for argument’s sake…

[Rabbi Michael Abraham] But no—why? I’m claiming it’s not only the Talmud. I’m saying: who said anyone disagrees with this at all? So the sages of our time decided otherwise—so what? I don’t agree with them. But that’s not… I don’t think this is returning to the Talmud. I’m claiming it is compatible also with what the medieval authorities say. I don’t think I’ve gone back to the Talmud here, okay? This ruling is not against the medieval authorities. It is completely with the medieval authorities.

[Speaker K] Fine, I support this point, so I just want to say one more thing. After all, every legislation or every halakhic ruling also has reasons behind it. Right? If we’re talking about something logical, then there are reasons behind it. If you can deal with it and show that your third way still fits the logic of the two ends of the dichotomy, then you really can go with the third way.

[Rabbi Michael Abraham] Correct. It doesn’t contradict the dichotomy.

[Speaker L] Yes, but we’re talking about the translation stage, Rabbi—that’s like saying, “I’ll let you choose any color as long as it’s blue.” Exactly!

[Rabbi Michael Abraham] That’s what someone is telling me who presents me with a dichotomy, and I tell him, “Thank you very much, I choose green.”

[Speaker L] One hundred percent. Those aren’t the rules, and that’s why I said that it’s—

[Rabbi Michael Abraham] No, you think that’s outside the rules. Many times that is exactly the point. Very often—understand—this class is not a class in logic. Meaning, in logic there is either X or not-X. That’s it, period, end of story. Completely simple. There isn’t another word to add. We finished the logical chapter. Now I move on. I’m talking about the “move on.” Many times logic captivates us; it takes us prisoner.

[Speaker D] What? There you have to take it in terms of survivability. Meaning, someone with brain death has no chance of living more than a month. Obviously. And therefore he is a candidate. Because otherwise you couldn’t get brain transplants—for example in Parkinson’s they sometimes transplant pieces. There are people who have cardiac death without brain death. For example, someone who is on bypass, on the heart-lung machine—they switch him over, and the heart is out, and they can’t get it back working, in any way. Whatever they do, it won’t restart, and usually on bypass… and the brain is still working? The brain is perfect! Yes? You can take from him two other rights… other organs too. Now, is the person dead-alive? Is he dead dead? Or is he alive? Ten minutes—they tell him, do ten minutes on bypass, if it doesn’t work kill him? What can you do?

[Rabbi Michael Abraham] Also. No, okay, so that’s—

[Speaker D] Transplant the brain—the person is talking to you.

[Rabbi Michael Abraham] I’m talking—I’m talking about the process. No, why?

[Speaker D] You can keep him on the machine.

[Rabbi Michael Abraham] No—if there’s no one to pay.

[Speaker D] Who will pay for that? You? Me?

[Rabbi Michael Abraham] Fine, so that’s another discussion. Again, that’s a completely different discussion.

[Speaker D] Why? We don’t kill people like that because of money.

[Rabbi Michael Abraham] No, obviously, obviously. But to keep them… okay, Yossi, but that’s the practical question. I’m not talking about practice right now. That’s another question. The question is what to do with the money and all that. But in the normal process, when a person dies, it goes from the brain to the heart. That’s the normal course. There may be opposite situations, I don’t know. But the normal case is this case, and therefore whenever there is deliberation about harvesting organs like the heart or lungs or something like that, the deliberation is at those moments. And at those moments it’s usually a matter of hours, not months.

[Speaker D] Yes, but here she is one hundred percent alive right now. The kidney you’re going to take—

[Rabbi Michael Abraham] Fine, so it’s one hundred percent alive, but the person is dead—or dead or not dead. I’m saying that is exactly the question.

[Speaker D] I’m saying the Jewish definition should be when the soul leaves him. Now define for me when that happens and with what sensor? I don’t know. You don’t have it. You can’t derive any—

[Rabbi Michael Abraham] No, why not?

[Speaker D] If the person is dead and the soul is still inside him—

[Rabbi Michael Abraham] What are you talking about? Doubt does not override certainty. But Yossi, doubt does not override certainty. You’re assuming a stringent doubt, and I’m saying doubt does not override certainty. You are coming to save a living person, and this one is in doubt. So what? So the one who is doubtful—so what? It’s not simple at all. Leave it—I don’t want to get into the halakhic question. I’m using this as an example. Many times we face a problem like this, where the translation of it leads us to a dichotomous picture. And in order to get out of that picture, we need to go back and examine the translation and say: we are not willing to play on the field you are setting before us. Yes, someone once wrote that in an article. He reviewed one of my books—Baruch Kahane, the psychologist. And he brought there a story about the duck and the eagle. That’s what he wrote; that was the title of the review too. So he says: the eagle invites the duck to a contest—a flying contest, who flies faster? So the duck says, no problem, come into the pool and let’s have a contest, let’s see who swims faster. Meaning, you decided that the arena of competition is the air. There you definitely win; there’s no contest. But why should I accept that arena? Come to my arena. Meaning, many times the one who sets the arena defines the dichotomy. But if I say I’m not playing in that arena—why do I have to play in that arena?

[Speaker C] And how was that connected to the book?

[Rabbi Michael Abraham] What? And how was that connected to the book? He claimed that I’m not willing to play in the standard arena, and therefore I propose a third option. It’s very simple there, with the analytic and the synthetic, doesn’t matter.

[Speaker C] And you’re the duck?

[Rabbi Michael Abraham] What? I’m the duck. Yes, I’m proud to be the duck. In any case, you know, the eagle has a good reputation—but what’s wrong with a duck? Do you know what the difference is? There’s a hamster—that’s a very nice animal, right? And a mouse—when people see it they immediately jump onto chairs and scream, “Mouse! Mouse!” What’s the difference? It’s almost the same thing. What’s the difference? It’s only public relations. Meaning, if you had decided the mouse is a pet, then it would become a hamster and everything would be fine. But a mouse is very scary, and a hamster is very cute.

[Speaker N] A donkey is a more useful animal than a bear or a lion…

[Rabbi Michael Abraham] And it has a worse name, exactly. It doesn’t have good public relations. Right. Okay, so the same with the eagle and the duck. I’m a duck, fine. The eagle has good public relations, so that’s why I’m a duck. Okay. In any case, this option is one option. Meaning: to examine the translation process on the way to the dichotomy—do we really agree with it? Another way—and again I’m saying, it will almost always be an option of this type, that’s the important point here, which is why it’s important, not just an anecdote—in almost every dichotomy you present me with, I can hardly imagine something where it wouldn’t work.

[Speaker H] What do you mean “almost”?

[Rabbi Michael Abraham] Maybe all of them. I only say it for caution. I don’t know, because I haven’t gone over them all. So for caution I say: almost everything I can think of.

[Speaker J] Since there are no foundational assumptions that are one hundred—

[Rabbi Michael Abraham] percent, then every foundational assumption can look different. And there is always some foundational assumption—that’s what I mean. There is always some framework sitting at the base of the dichotomy. That’s not a claim. No, it’s not a claim—I’m saying, a dichotomy, you present a dichotomy.

[Speaker J] That is made up of claims.

[Rabbi Michael Abraham] Whether a certain thing exists or does not exist? Just—I’m thinking now. Okay, is there a third possibility?

[Speaker J] I assume an assumption of existence, I don’t know.

[Rabbi Michael Abraham] What do you mean by an assumption of existence? Yes—either it exists or it… I’m not assuming anything. Either it exists or it doesn’t exist. I’m asking you now: is there a third option? I don’t know—here it’s a bit hard for me to think. Maybe. We need to think about it a little. I don’t think I would immediately say that everything can be done this way. But it’s true that most of the interesting things—the vast majority of the interesting things—are dichotomies that are created because of some conceptual framework, within some conceptual framework. These are not dichotomies born in the Six Days of Creation. So that’s why I’m saying: always pay attention to the conceptual framework. Now, that sounds very simple, but many times we fall into this and get taken captive by it. They present us with: are you Religious Zionist or Haredi? If you’re not Zionist, then you’re Haredi, no? What other religious option is there? Well, there’s the option of being secular, but say within the religious option, right? Is there another option? I say yes, there is another option. Now, many times in an argument they say, okay, so you’re Haredi. I’ve often been hit over the head with being called Haredi because I said something that went against the accepted view in Religious Zionism, so that means I’m Haredi. Not true. I’m neither this nor that.

[Speaker B] And the reverse too, yes.

[Rabbi Michael Abraham] And the reverse too. Same thing. Doesn’t matter.

[Speaker B] I’m saying, there is a third option.

[Rabbi Michael Abraham] I usually find myself in the third direction. I don’t know why. But usually when I see an argument—maybe it’s the evil inclination, I don’t know—but almost always when I

[Speaker B] see an argument, I don’t agree with either side. But Gidi, here probably it’s important that most dichotomies do exist. That’s just how it is. There’s no way—

[Rabbi Michael Abraham] My experience is different. There are a great many dichotomies where there is a way. It’s very hard to justify your claim or mine. I’m saying, from my experience, many—let’s say, I don’t know, not necessarily most—a great many of the dichotomies are dichotomies that can be broken. Okay? That’s a minimal claim and enough for me. I don’t know whether it’s most or not.

[Speaker B] But every dichotomy has a practical side, like you just said about—what does “practical side” mean?

[Rabbi Michael Abraham] If they have some background. Okay, and I’m saying that very many dichotomies do have some background—that’s what I’m saying. Since that is so, it is worth examining it. And again I say: so many times they put us in front of some Sophie’s choice, either you go right or you go left. I want to fly upward. Who said I have to stay on the ground? Now, many times you simply don’t think of that. So I’m saying, it’s not that there is some earthshaking logical innovation here, but it’s worth noticing because many times you don’t think of it. I have a friend—that same friend I mentioned earlier, that we like to think about philosophical things—he founded a company called SIT, Systematic Inventive Thinking. Meaning, a systematic way of solving problems that require creativity. Usually we think systematicity and creativity are opposites.

[Speaker J] There are rules for that? Something like that.

[Rabbi Michael Abraham] There was some Russian—a Russian Jew named Altshuller—who basically first thought about this idea and tried to define some sort of toolbox. Two Israelis studied with him; they brought it to Israel when they finished a doctorate—or actually started a doctorate—they brought it here and did a doctorate on it. I met with one of them because I also wanted to get into this area, but it doesn’t matter—in the end I didn’t get into it, and afterwards he founded a company and now he gives consulting.

[Speaker B] If you had gotten into it you’d be a millionaire. You lost a lot of money.

[Rabbi Michael Abraham] Not sure, not sure—he’s not such a huge millionaire, but yes, he founded a company, and that company tries to offer solutions. He talks with giant corporations—IBM, Philips—meaning, they invite him, the whole board of directors sits with him for study days, and he solves problems for them systematically. So it’s interesting.

[Speaker D] And is there really a method to these things?

[Rabbi Michael Abraham] Okay, governments invite him. There are all kinds of things. Giant companies are not easily impressed. Okay, maybe. But I’m saying the fact is that apparently people—maybe—

[Speaker J] maybe he’s creative. The fact is that he—

[Rabbi Michael Abraham] solves problems for people. I mean, I know—he’s creative because he’s creative. No, no. He shows them—no, he shows them how, systematically, he takes them to solve the problem. He doesn’t solve it himself.

[Speaker L] Yes, but he comes and knows how to bring them to an invention.

[Rabbi Michael Abraham] And how does it work, by the way? It works the way I just said. Usually it works like this: his techniques—most of his techniques, he has several techniques, but most of the—actually they’re not even his, he just collected them—most of his techniques are basically built like this: define the problem, write down the set of parameters that underlie and lead us to being stuck, and now try giving up each one of them without any logic—just throw one away and see what comes out. Does something sensible come out? Okay, that’s an option. Doesn’t something sensible come out? Okay, throw away the next one. There are ten things here that define the playing field. Throw away one—it doesn’t work. Throw away two, throw away three, throw away one and four—it doesn’t matter—all the possible combinations. Once you do this systematically, you arrive at all the options there are. So basically you can systematically reach everything there is. Fine, there are also options of adding, true, beyond the options that depend on subtraction. So I’m saying: usually people expect you to be creative in order to get there, but in fact you can do it systematically. What I’m trying to do here is exactly the same thing. And many times solutions to dichotomies like these look like some burst of creativity, when in fact one can do it quite systematically. Meaning: try to check what assumptions stand at the basis of the dichotomy and examine each one—can it be given up? Can it be disputed? In that sense it is exactly the same thing. Okay? Only in the… I don’t make money from it here like my friend does, and that’s fine, but it’s still interesting in the philosophical realm.

[Speaker H] And is there a situation where you can say there is validity to different positions, and then there’s no chaos?

[Rabbi Michael Abraham] That’s what I’m getting to now.

[Speaker H] No, because you said you don’t know…

[Rabbi Michael Abraham] No, no, that’s what I’m getting to now. I’m saying that in the previous sense, I don’t—well, I know the solution, but there are still more techniques for deviating from a dichotomy. The next technique—here I talked about attacking the shared foundation, okay? That’s the first technique. The second technique is really to leave binary logic. And here it is already almost on the logical plane—almost—even though it’s still not exactly. Meaning, the sorites paradox is of course the example you’ve heard from me many times. The heap paradox? The heap paradox, yes, exactly. What is “a lot”? That’s the big question. And this paradox basically says that when I ask whether this collection is a heap or not a heap, I assume that the answer can only be yes or no. Whereas in fact it may be that the logic more suited to answering this kind of question is fuzzy logic, or vague logic, and not binary logic of yes or no. Some kind of continuum: something that is not a heap, something that is a bit of a heap, something that is fairly much a heap, something that is very much a heap, something that is completely a heap and really really a heap—I don’t know, there aren’t enough words in Hebrew. So say between zero and one, heapness. Its degree of heapness is zero, 0.1, 0.34, I don’t know, pi over four heapness—whatever you want, yes, it should be between zero and one. So there is a continuum of levels of heapness. Okay? Now, many times when something is presented to us as dichotomous, binary logic is assumed, and one has to check whether binary logic is really the right way to handle this kind of question. Now it turns out that with concepts that are not abstract mathematical concepts, it is almost never correct to handle them with binary logic. Almost never. Meaning yes, it starts with dilemma arguments. I don’t know—there’s no point in giving exams. Why? Because the lazy students won’t study even if there is an exam, and the diligent students study even without an exam, so why give an exam? Either way, the exam is unnecessary. What’s wrong here? There are people who study somewhat. Yes—the division between lazy and diligent is not dichotomous. It’s not pathologically lazy versus ultimately diligent. There is a whole continuum of diligence. Zero diligence is lazy; 0.1 is a little lazy; 0.3 is fairly lazy; 0.5 is so-so; 0.7 is already fairly diligent, and so on. It’s a continuum between the lazy person and the diligent person. Therefore, you’re right that for the pathologically diligent and the pathologically lazy there is no point in exams, but for the whole range in the middle—or not the whole range, but for part of the range in the middle—there is a point in doing it. So what was the mistake? The mistake was that we assume the relevant logic here is binary, and that’s not true. This is vague logic, continuous logic. Okay?

[Speaker L] Leibowitz once said that about sports. He said sick people can’t do sports because they’re forbidden to, and healthy people don’t need it, so…

[Rabbi Michael Abraham] Yes, right.

[Speaker J] This is basically the same kind of solution to a problem, because you changed the space of possibilities. Or—well, I don’t know exactly—it’s not exactly non-binary. You simply said there are many many more possibilities, not just those two possibilities.

[Rabbi Michael Abraham] Not clear. But I’m saying: here I didn’t go to some other assumptions. Rather, I attacked it on the logical plane itself. Meaning, you assume it’s either one or zero. Not true. It’s all the options between zero and one.

[Speaker J] The assumption is that it’s either one kind of person or another kind of person, so you—

[Rabbi Michael Abraham] say no, that assumption is… fine, okay. But that’s not an external assumption. Meaning, it’s not the framework that set the dichotomy. Rather, it’s the dichotomy itself. Meaning, I’m saying the logic itself is not dichotomous.

[Speaker H] You could call that a private example. The word “binary,” theoretically binary, because “bi” is two. Binary is exactly—

[Rabbi Michael Abraham] Okay, binary, right. I should have said binary. Yes, but why does everyone always say “binaric”?

[Speaker H] But now it comes out nicely, because it means “between.”

[Rabbi Michael Abraham] It’s like something I saw today—you know there’s an “Uncyclopedia”? Uncyclopedia. On the internet I saw today an entry on Agnon, yes, an entry on Agnon in the Uncyclopedia. It’s a humorous encyclopedia; they make fun of the whole thing. It’s actually pretty nice. There were some very nice bits there. Yes, maybe I’m already captive to that. In any case, so that is the second option. Meaning, there is another option, the option that says: you are presenting me with a dichotomy, but who said binary logic is the right logic? Maybe binary logic is not the right one. Okay? And by the way, in a great many questions—almost anything that touches ordinary concepts—it is actually more correct to handle it with this kind of logic and not with binary logic. For example—yes, examples I already gave—when is it afternoon? Children ask when they can go out to play in the yard. Right? So twelve o’clock is not afternoon. Adding one second does not change the status from noon to afternoon. Like the heap? Yes, right—but five o’clock is afternoon. So what’s going on here?

[Speaker J] In the end, as a human being, you still have to relate to it, you still have to cut somewhere—you cut—

[Rabbi Michael Abraham] turn—

[Speaker J] things into as if they really are.

[Rabbi Michael Abraham] But the cut is made artificially. The cut basically says: the law determines that from four o’clock it is afternoon. No problem, no problem. But I’m saying: one must be careful not to be captive to the law on the philosophical plane. Because on the philosophical plane there is a continuum between noon and afternoon. The law has to cut, because it has to tell people what is permitted and what is forbidden. There is no “a little permitted and a little forbidden.” Okay? By the way, in Jewish law there is, but in law generally there isn’t. So because of that, you need to say, okay, between two and four that’s called noon. Fine? Before that it isn’t, and after that it also isn’t. But that is an artificial cut. When you ask a question in a paradox, when you ask a question of this kind of paradox, the artificial cut won’t help. Because I’m asking what I really call “afternoon,” not where it is defined in the law. In our everyday language, the concept of afternoon is coherent? It’s a concept that doesn’t withstand criticism. Here, I’m showing you an ontological analysis—it doesn’t withstand criticism. So it won’t help to cut it and say that from four it’s afternoon. I’m asking what I really mean in everyday language. Like the heap paradox, the bald man paradox—all of these are the same thing.

[Speaker B] Why do you call this logic? It really jars me. It’s not binary logic or multi-valued logic—it’s not logic at all. These are vague concepts.

[Rabbi Michael Abraham] It’s conventional. It’s conventional to call it logic.

[Speaker B] For example, by the way, in statistics they cut it. They said a significant result is this, an insignificant result is that, and that too is on a continuum.

[Speaker D] Right, it’s on a continuum, but in statistics… what? I didn’t understand. Receiver operating characteristic. Meaning what? You draw a line, as if—what? Why don’t you let them go out in the afternoon? Because the neighbors are still sleeping and so on, and then from a certain hour most have already woken up and so on. That’s how you place the line.

[Rabbi Michael Abraham] No, but you’re still making some cutoff—

[Speaker D] some sort of cutoff, and that’s how you set a criterion, and that’s how you determine this criterion. Fine, no problem. What is signal and what is noise? What is signal?

[Rabbi Michael Abraham] Fine, but that again is an artificial determination—

[Speaker D] and it goes for every such test. What is called pathological and what is—

[Rabbi Michael Abraham] Fine, but that’s true. But it’s an artificial decision. From when is speed dangerous on the road? There too you make a cut.

[Speaker D] No, where it’s best, meaning optimal—

[Rabbi Michael Abraham] But it’s still an artificial cut. Because I’m asking you from when, conceptually, it is afternoon. Not the question of when children are allowed out into the yard, but when is it afternoon. Why does that interest you? It interests me. It’s a philosophical question. Nothing more—a philosophical question. Practical implication? I’ll give you a practical implication. I’m telling you that today—

[Speaker D] Do you want to offer a sacrifice from plag hamincha?

[Rabbi Michael Abraham] No, nothing. You ask me whether it’s afternoon now. You made a bet with your friend whether it’s afternoon now or not. You ask me whether it’s afternoon now. Practical implication for betrothing a woman—yes, exactly. You say—it doesn’t matter. No, exactly, that’s the point. I’m talking about a philosophical question. On the practical level, we always cut somewhere, and that’s fine. An artificial cut solves the practical problem. But I want to solve the philosophical problem, not the practical one. For the philosophical problem, an artificial cut won’t help. I need to understand what afternoon really is. Now, there is no everyday concept that is not exposed to this attack. No everyday concept. If you take, I don’t know, this prayer book and this tissue package, okay? Then I say: put the simplest computer program in the world. It takes a pixel, say of the same-sized picture, takes the pixel from here and the corresponding pixel from there—0.9 times this plus 0.1 times that—and draws the resulting image. The image will be very close to this, but not completely. A little tissue, but already less. Right? And you can make a continuum like Escher’s metamorphoses there. You can make transitions on a graph.

[Speaker D] Yes, exactly—between fish and birds in Escher, or all kinds of metamorphoses like that.

[Rabbi Michael Abraham] Yes, exactly. So in fact you see that even the concept “prayer book” is not fully defined. It can turn into tissue continuously. So from when is it called a prayer book? From 0.8 prayer book and 0.2 tissue? From 0.9 prayer book and 0.1 tissue? From when? There is no everyday concept that is not like this.

[Speaker D] You’re talking about the image of it, not the thing itself.

[Rabbi Michael Abraham] Yes, yes, the picture. No, you can also make objects like that. Produce this picture now—produce the picture, make it a real object.

[Speaker D] Take the—

[Speaker J] the prayer book and parts—

[Rabbi Michael Abraham] of it, start removing and adding from the prayer book to the tissue. Produce it— theoretically, yes, 3D printing. A 3D printer, okay? Give it this input, and a 3D printer produces this thing for you, okay? So now what is it? Is it a prayer book or not a prayer book? Fine. By the way, I’m not sure you can produce every such thing. What?

[Speaker I] I wanted to make some kind of language of gray in everyday life. He’s talking about the level. Fine, I’m saying there’s no such thing.

[Rabbi Michael Abraham] What do you mean, gray? I also— nobody is talking about that thing. Right, exactly, that’s what I’m saying. What is red? It’s a whole continuum of wavelengths that we cut up. Our straight line is built in a certain way, and we cut it: from here to here is called red. And why is something a little farther not red, or something a little before that yes red? You could cut it a little earlier too. There’s no sharp line here. We somehow impose a sharp line one way or another, but in fact it’s a continuum. Now, our whole world is actually full of continua, and we cut it up for convenience—for ease of use, and so that we can somehow communicate and make decisions. But a lot of the philosophical problems arise—and sometimes they’re practical too—when you use philosophy as some kind of model for practice, then we also have a problem on the practical level. They present it as, say: is this red or not red? It’s a little red. And we don’t think to answer that way. That’s also an option: it’s a little red and a little not. Fine.

[Speaker J] When does a fetus become a human being?

[Rabbi Michael Abraham] Yes, when does a fetus become a human being? We once talked about decision-making and the rabbi and reality and all kinds of things like that, and there really was a whole collection of questions, all of which are basically questions whose answer is a continuous answer. It’s like the nicest example in mathematics—you know what are called fractals, yes, broken dimensions. We’re used to one dimension, two dimensions, three dimensions—an integer number of dimensions, that’s what we’re used to. Then there was someone named Mandelbrot, a mathematician at IBM, yes, from IBM’s research institute, originally Polish, I think, and he invented a broken dimension. There are shapes whose dimension is one and a third, or pi. Okay? Not one, not two, not three, but not an integer—rather a fractional number.

[Speaker B] In nature there are a lot of those.

[Rabbi Michael Abraham] More than that—he says that in nature there are only such things, that’s his claim. There are no integer dimensions at all; that’s our abstraction. In nature there are no integer dimensions, only fractional dimensions. That’s his claim. I don’t think that’s exactly right, but never mind, that’s the claim. What are you saying? So he breaks something that, on the face of it, seems really dichotomous: either you’re one-dimensional, or two-dimensional, or three-dimensional. And who could even think about length—

[Speaker D] width, three-dimensional, two-dimensional, one-dimensional.

[Rabbi Michael Abraham] Yes, those are the integer dimensions. But now, for example, take a coastline, okay? That’s one of Mandelbrot’s famous examples. A coastline. Take a drawing of a coastline, okay? What characterizes one dimension? One dimension is: if you increase the resolution by a factor of two, the length increases by a factor of two, right? If you increase the resolution by a factor of four, it increases by a factor of four, and so on. What characterizes two dimensions? Look at area. Increase the resolution by a factor of two, the area becomes four times as big, not two times as big, right? The index—in other words, the exponent by which the size of the thing you’re talking about increases—that defines what the dimension of the thing is, right? Now he says that with a coastline like this, when you increase the resolution by a factor of two, the length doesn’t grow by a factor of two but by a factor of three. By a factor of three—or by 1.7 if you want, that’s less than three. More than one.

[Speaker E] Because of the ups and downs.

[Rabbi Michael Abraham] Yes, I’m saying it’s more than one, so let’s say it grows by a factor of three, okay? Then three divided by two is one and a half; log base 2 of 3 is actually the dimension. You see that this thing is actually a shape whose dimension is not an integer. It’s not one, it’s not two—it’s one and something. Could it also be infinity? No, not infinity. Infinity is just, say, whole infinity, so to speak; infinity is a limit. But an integer as large as you like—that’s fine, that’s just a simple abstraction, a simple generalization. I’m talking about a fractional dimension, a dimension of one and a third. Okay? What about a complex dimension? I don’t know. A negative dimension? I don’t know—maybe there are such things too; I’m not expert there in what people do with it. But in principle, anything is possible. Meaning, you’re talking about some magnitude such that if you look at it through binoculars that magnify by a factor of two, you’ll see a length that’s three times as big. The length of the coastline will be three times as big. Why? Do you understand why? Because if we think about a map—a map on a scale of 1:50,000—draw the map now on a scale of 1:10,000, then it won’t increase by a factor of five, it will increase by more. Why? Because you’ll see inlets that on the larger-scale map you didn’t see. More inlets will be added, and so it will actually increase the length of the line that you see and add size; it won’t be by a factor of five—five is just the basic one—but by six, say, or seven, or something like that.

[Speaker D] But that’s on condition that the more you magnify, the more inlets you always find.

[Rabbi Michael Abraham] Right—that you have more, that you can keep magnifying, because otherwise the dimension is an integer. If now you magnify by twenty-five, it will grow by the same factor. Okay, that’s the assumption, and the claim is that at least over part of the way, that happens.

[Speaker D] No, you can draw it; he has models for how to draw such a thing.

[Rabbi Michael Abraham] After all—

[Speaker C] It’s obvious that once you get to a scale of less than a meter, there’s nothing fractal at all in a coastline. All of this happens on scales of kilometers.

[Rabbi Michael Abraham] No, I’m not sure. Why? There are millimeter-sized inlets. What do you mean? Millimeter-sized inlets in the rocks that the shore touches—millimeters.

[Speaker C] Never mind, but it’s not important.

[Rabbi Michael Abraham] Take—what I’m saying is, on the principled level, the claim there is that at least over some range, between scale one and, say, magnification by a hundred, there’s some point where each time you get a fixed dimension. Meaning, doubling will give you the same increase as quadrupling and as multiplying by eight—in other words, each time it will preserve the same factor. So you can define that as the dimension of the coastline. Later it will break down; at a hundredfold it’ll break down, at half it’ll also break down, doesn’t matter.

[Speaker B] But a drawing does have an integer dimension. The fact that we have measuring devices that create distortions—why does that make it a dimension?

[Rabbi Michael Abraham] Who says reality has an integer dimension? What? Because in reality you look at it through glasses.

[Speaker B] Reality is some kind of truth. It could be that you’re distorting the—

[Rabbi Michael Abraham] You’re looking at it through glasses of a kind that don’t reveal to you something extra when you magnify. Okay? But if you were looking through glasses that do reveal it— no, so I’m saying, in reality you don’t know what reality is. You don’t know what reality is; what you know is what you see. No, not existing reality—the reality you see. The reality you see, not existing reality.

[Speaker J] But even there, no matter how far down you go, to the atomic level, I don’t know, quantum, there will still be—

[Rabbi Michael Abraham] I said: there is a certain range within which the dimension is defined. It’s not defined from zero to infinity.

[Speaker J] It’s a range, right.

[Rabbi Michael Abraham] I’m saying it’s a range between, say, magnification by one—that is, normal size—and by a hundred. Doesn’t matter. In that range you’ll see that when you magnify by two it grows by three; when you magnify by four it grows by six. No need to define—what?

[Speaker M] No need to define it at all. No need to define it at all—why call that a dimension? I’ll buy a quarter-acre lot, bring a tractor, make pits and hills, and now I’ll measure all the rises and hills—suddenly I have three quarter-acres. There, a start-up, right?

[Rabbi Michael Abraham] There, a nice start-up. A nice start-up. A terrific start-up. Wow, I hadn’t thought of that. That’s a nice start-up, right? Every contractor does that. Yes. He takes an area and checks it not by height. No, it’s simple. No, but what he’s saying is actually nice: turn your field into a field on which you can grow more crops. Right. Dig a pit, shape it like this, and then ostensibly it has a much larger area.

[Speaker L] Yes. There is—Rashi says this; there’s a Talmudic passage like that. Rashi says that when they checked the Land… so he checked that on the top of the mountain there’s more area you can cultivate than the area on the slope. Okay, so they did bring that up. But in any case, you’d have to sell it that way; I don’t think people make money from that. What?

[Speaker C] An oil filter in a car—filters in a car. An oil filter. They’re built with a membrane of paper folded many times over, and that way you create a very large surface area in a small volume.

[Rabbi Michael Abraham] Of course, I know that they increase the surface area of things in such ways. But it’s nice to define the field that way; that’s nice.

[Speaker D] Let’s say fractal: someone ordered X amount of ink to draw a map at a certain scale, and suddenly he makes the scale larger, and the ink isn’t enough for him.

[Rabbi Michael Abraham] He has to know it’s six times the ink, not five times the ink. If you go from 50,000 to 10,000, it’s six times the ink and not five times the ink. Yes. Exactly. Inlets. So I’m saying this is an example of breaking a dichotomy that seems to us completely unbreakable—meaning, either one or two or three. Who would ever have imagined that there are fractional dimensions, and that even this can be broken?

[Speaker L] Whoever ordered ink and didn’t have enough.

[Rabbi Michael Abraham] Yes, whoever ordered ink and didn’t have enough. Or he was bored and sitting at IBM and had to invent new mathematical models. So it doesn’t matter, but give some respect to ambition. Yes, right. So in the end, what I want to say is: so far we’ve presented two tools for breaking dichotomies. One tool is to try to examine the framework within which the dichotomy is formulated—whether it’s possible to argue about something from within the assumptions of that framework. And the second tool is that the dichotomy itself very often paints in black and white things that in fact have many shades of gray in the middle. And you have to notice that there are also shades of gray in the middle. We’ll see examples of these things, and I’ll bring a few more such tools—there are quite a few more such tools for breaking a dichotomy—and then we’ll move on to examples and see how this is applied. And maybe then I’ll convince you that these things are not entirely trivial; meaning, that there are places where it seems to us that we’re stuck, and if we have an organized toolbox, you can simply go through it. Meaning, to see whether you can use this tool, this tool, this tool, and in the end we’ll get answers. Okay.

Paradoxes and Contradictions in Jewish Law, Lesson 1

Table of Contents

Summary

General Overview

Zen Koan, Hofstadter, and the Answer “Mu”

The Law of the Excluded Middle, the Law of Non-Contradiction, and the Feeling of “Logical Distress”

Opposites, Logical Negation, and Two Kinds of Opposition (Aristotle)

Halakhic Example: Nullification by Majority, Oneg Yom Tov, and Tzitzit “for their own sake”

The Liar Paradox and Correct Negation versus Opposition

Organ Donation: Brain Death versus Cardiac Death and Questioning the Framework of the Discussion

Defining the “Playing Field” as a Dichotomy: The Eagle and the Duck and the Public Relations of Concepts

Breaking Dichotomies as a Method: SIT, Altshuller, and a Systematic Toolbox

Fuzzy Logic and the Heap Paradox: Continua Instead of Yes/No

Fractals and Fractional Dimensions: Breaking Dichotomies That Seem Fundamental

Summary of the Tools and Planned Continuation

Full Transcript

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Table of Contents

Summary

General Overview

Zen Koan, Hofstadter, and the Answer “Mu”

The Law of the Excluded Middle, the Law of Non-Contradiction, and the Feeling of “Logical Distress”

Opposites, Logical Negation, and Two Kinds of Opposition (Aristotle)

Halakhic Example: Nullification by Majority, Oneg Yom Tov, and Tzitzit “for their own sake”

The Liar Paradox and Correct Negation versus Opposition

Organ Donation: Brain Death versus Cardiac Death and Questioning the Framework of the Discussion

Defining the “Playing Field” as a Dichotomy: The Eagle and the Duck and the Public Relations of Concepts

Breaking Dichotomies as a Method: SIT, Altshuller, and a Systematic Toolbox

Fuzzy Logic and the Heap Paradox: Continua Instead of Yes/No

Fractals and Fractional Dimensions: Breaking Dichotomies That Seem Fundamental

Summary of the Tools and Planned Continuation

Full Transcript

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