Paradoxes and Contradictions in Halakha, Lecture 5

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[Rabbi Michael Abraham] They can’t justify it in the sense of… no, fine, I’m saying that according to their own approach, it can’t be justified in terms of morality. Of course it’s moral, and they can be very moral, because they also believe in the Holy One, blessed be He—they just don’t admit it. Fine, those are all the calculations I make on their behalf. But I’m saying that according to their own approach, there can’t be morality, because morality has to stand in relation to something external, something against which I compare the claim “it is forbidden to murder,” and determine whether it is true or false. Otherwise I arrive at the relativity of that claim. If only I am here, then all I can say is that I think it is forbidden to murder because that’s how I feel. So I say, I think it’s forbidden to murder; you say it’s permitted—you’re simply mistaken. You’re not perceiving the Idea of the Good correctly. Okay, so this objective something has to be somewhere—something abstract, I don’t know exactly what—out there, against which we place our ethical claims, our moral claims. That is the standard by which we test them. And therefore, in some sense there has to be something here—I don’t know, call it God or whatever you want to call it, because every proof defines a different God—but some kind of abstract entity standing out there, and opposite it, or from it, comes the source of the moral standard, the source of the validity of moral determinations. In any case, here too I tend to think that this is indeed a claim. Because the price of saying it is not a claim is basically to say that there is no truth and no falsehood here, everybody is right, and really I’m saying nothing except reporting subjective feelings. So this too is a claim, even though the way to make the comparison here is really—I think many people feel that there is a way to make such a comparison. If they give themselves an accounting, they also demand of others to understand that such-and-such really is moral or really is not moral. I don’t think they see it as some subjective experience of theirs. But when you push someone into a corner, he says, no, no, actually yes, you’re right, it’s only my feelings and there’s nothing beyond that—and yes, I’ve heard that many times. So it’s hard here to point to what exactly this comparative process is that lies at the root of the matter. Fine, so this is just an example of what a claim is. A claim has to be judged in terms of truth or falsehood. Positivism, which sees claims only in terms of how you verify them—verify with an aleph, yes—also has a legal position, not surprisingly. It views the legal system too that way; there is positivism in the legal world as well, and legal positivism basically sees the legal system as a kind of axiomatic system. Meaning that the system starts from certain basic assumptions—if you like, Kelsen’s basic norm, the Grundnorm, what is called the basic norm—and from it secondary norms are derived, and from them further norms are derived, and so a logical structure is built, which is basically the legal system or the law code. Okay? That’s at least how Hayek sees it. And basically the positivist conception says that morality—or law, sorry—the judge, basically what the judge has to do is one of two things: either observe the law code, simply read the law code and apply what is written there, or deduce conclusions from it deductively. Even if something is not written explicitly, if you can infer it deductively from what is written, that’s also fine. Nothing beyond that. That is the positivist conception. And again, it’s a very demanding conception, asks a lot, sets a high bar, and therefore greatly narrows the system. The positivist conception in philosophy is like that, and the positivist conception in law is like that too. Later on I’ll return to law, which is why I’m mentioning it here as well. I once spoke about how at the Nuremberg trials this positivism was shattered precisely there, because they understood that legal positivism had basically led to a situation in which whether something was legal or illegal was measured only through the law code or what could be derived from it. But in Nazi Germany the law code—or what was derived from it—was to kill Jews, or kill Roma, or establish concentration camps. That was the law code, that was the law, that was the command of the authorized authorities. And if that’s the case, then you actually cannot prosecute anyone for carrying it out, because that is the law, if you hold a positivist conception. So at the Nuremberg trials they had to deal with the question of how it is possible to judge someone for carrying out the law. And that’s where positivism actually broke down. Some still insist on it, but it’s no longer what it was—certainly not what it was in the first half of the twentieth century, that’s what I mean. There’s still Englard and a few dinosaurs who are still positivists, but it’s already an extinct species. So the positivist claim basically says—and I’m defining it because I’ll use it later on—it says that everything, and this is also philosophical positivism, is basically either an observation or a logical derivative of an observation. That’s all. Only those things have meaning. About everything else there is nothing to talk about; they are meaningless. Okay? Both in law and in philosophy. Of course there is no room, for example, for what is called judicial legislation in a positivist conception, yes? When a judge encounters a particular case for which he does not find a simple answer in the law code, and he still has to make a decision, then sometimes he makes a judgment that has an element of legislation in it. He expands the law code a bit in order to apply it to the new case, the case before him. And there is a lot of criticism of that—whether a judge may do this—because ostensibly that is an act of legislation, and he was not appointed to be a legislator, he was not elected, he is the judicial branch, not the legislative branch. But on the other hand, there is no interpretation that is pure deduction. Today we already understand that; such a thing is impossible. A judge cannot help but legislate. The only question is one of degree. Aharon Barak got a lot of criticism for doing too much of it. But there is no judge who doesn’t do it. It’s impossible not to do it. Anyone who thinks interpretation can be done only by deduction will not be able to interpret anything. It’s simply a ridiculous illusion, this illusion of doing interpretation only by deduction. Nonsense. There is not a single conclusion you can extract deductively from a law code.

[Speaker C] Maybe on a computer you can? What? In a computer program.

[Rabbi Michael Abraham] Yes, but run the computer program on the law code and you’ll see that it won’t produce an answer to any question by purely deductive means. Not a single question.

[Speaker C] No, I mean interpretation of a text in the context of literature, poetry.

[Rabbi Michael Abraham] Yes, but not only text. Interpretation of anything is not deductive. Meaning, it’s clear that there is an expansive dimension here and not only an analytic one. In any case, that’s philosophical and legal positivism. And basically it sees the legal system as a closed system, an axiomatic system. It has assumptions, rules of inference, and you use the rules of inference to derive conclusions from the assumptions, and that’s it. Anything outside that—the legal system has nothing to say about it. Meaning, if by chance the Israeli law code says, “A thief shall be punished by such-and-such,” and it doesn’t say that it is forbidden to steal—I saw this in a book by Haim Cohen, and I think we also spoke about it once—that it doesn’t say it is forbidden to steal, it says the thief shall be punished by such-and-such. So therefore there is no prohibition against stealing. There is only an obligation upon the courts to impose a punishment on someone who is caught and convicted of theft, but there is no prohibition on stealing. That’s his claim, at least. And again, there’s a kind of positivist element here. It is completely obvious—what does “a thief shall be punished by such-and-such” mean? A thief shall be punished by such-and-such because it is forbidden to steal. Fine, but the law code says “a thief shall be punished”; it does not say it is forbidden to steal. So someone who sticks to the law code or to what can be derived from it deductively—that is the conclusion he comes away with.

[Speaker D] So the court can’t, for example—a judge can’t, say, order… for example, to return stolen property. If that isn’t stated explicitly, say, then the judge decides that the stolen property should be returned. Let’s say, yes, say.

[Rabbi Michael Abraham] I’m saying, nobody actually behaves that way, but that is the positivist conception in its pure form. Yes. Okay, now, I’ve also defined what a claim is and what positivism is and how it relates to claims. With those two concepts I want to continue. There are sentences that are claim-like, but are not really claims. Up to now we’ve encountered two types of sentences that are claims: a false claim and a true claim. It is now dark outside, or it is now light outside. Okay? But there are sentences that can receive a value—maybe let’s phrase it differently, let’s start from the fourth type. There are sentences that cannot receive any value at all, neither truth nor falsehood. That is what is called a paradox. Yes, the liar paradox. “All Cretans are liars”—I think we once discussed that. When a Cretan—yes, it appears in the New Testament—when a Cretan says, “All Cretans are liars.” So this is known as the liar paradox, but of course it isn’t paradoxical. Because the paradox is built in such a way that if he says all Cretans are liars, then he too is a liar because he is a Cretan. If so, if he is a liar then the sentence he said is false. So if it is false, then they are not liars. If they are not liars, then he too is not a liar, so he is telling the truth. But he said they are liars. If he is telling the truth and they are liars… that’s the paradox. But it isn’t a paradox. Why? Because if he says that all Cretans are liars, then it follows that he too is a liar. Since he is a liar, the sentence he said is false. When I determine that the sentence “All Cretans are liars” is false, that does not mean that everyone tells the truth. It means that there is at least one truth-teller. And that truth-teller could be not him but his cousin or his neighbor. Okay? He indeed is a liar and the sentence he said is false, and that’s where it stops. The loop does not continue. Wherever there are quantifiers—yes, generalization, a sentence that begins with “all” or “there exists” or things like that—you usually won’t be able to create paradoxes. That is, these loopy paradoxes always have to sit on sentences without quantifiers, without generalizations, yes, without reference to groups of things. For example, if I say, “I am a liar,” is that the liar paradox? Also not. Why not? Where is the quantifier here?

[Speaker D] “I am a liar” meaning…

[Rabbi Michael Abraham] All my claims—I always lie—and all my claims are false. So if I say that, it means that this claim too is false; the meaning is that there are claims that are not false, but not necessarily this one—maybe another one. And here it stops. Whenever there is some kind of generalization, you won’t be able to continue this loop. The loop stops.

[Speaker D] Wait, but according to that logic I can maybe infer a conclusion from the fact that you said you are a liar.

[Rabbi Michael Abraham] That’s a different question.

[Speaker D] The question is whether there is at least one other true claim.

[Rabbi Michael Abraham] The question is what I meant. The question of what I meant is a different question. I’m speaking at the level of parsing the sentence, regardless of what he intended.

[Speaker D] So if I understand you literally. If not literally, but the way the paradox understands it. Say, you didn’t say that you are a liar, but rather you said—

[Speaker E] —that every sentence you say is false.

[Rabbi Michael Abraham] My claim—translation—my claim means: there is at least one sentence you say that is true.

[Speaker D] But that’s an incorrect translation.

[Rabbi Michael Abraham] It’s a correct translation. That is the logical meaning of the sentence. That’s why I say: if you want to ask me what he meant to say, then clearly—

[Speaker D] —he didn’t mean to say that.

[Rabbi Michael Abraham] No, I’m asking what he intended to say.

[Speaker D] That’s the translation of the sentence. But ostensibly it’s not only the translation of the sentence. Ostensibly, from the very fact that he asserted it, it follows that this is so, right? What do you mean, yes?

[Rabbi Michael Abraham] I didn’t understand. But after all, that’s not correct.

[Speaker D] It could be that all the other sentences he says are false, and this sentence—

[Rabbi Michael Abraham] And then what will happen with this sentence?

[Speaker D] Well, that’s it, then that’s exactly the paradox, isn’t it?

[Rabbi Michael Abraham] Fine, that’s why I say: then I prove it by contradiction. Since the paradox is a contradiction, I say: if the alternative leads me to contradiction, then I have to assume the opposite. And assuming the opposite means that there is one sentence that is true. And again, I am not talking about the speaker’s intention.

[Speaker D] Fine, but obviously the speaker can say it and also intend the sentence—

[Rabbi Michael Abraham] He can intend it and not know what he intends, because when you do the logical calculation that intention does not come out consistent. Fine, all contradictions are like that. Fine, I’m not talking about human beings, I’m talking about claims. It’s not… it’s always the question, what’s the connection, I… There’s the principle of charity, of Ronald Dworkin, also a philosopher of law, who says that we interpret what a person says in the way most favorable to him. But that interpretation is not always really his intention. Sometimes a person really made a mistake. But in order to conduct the debate, there is generally such a rule that says we interpret him in the most sympathetic way, and then we begin to quarrel or argue. Okay, in any case, this possibility—a paradox—for example if I say: sentence A: sentence A is false. That’s a paradox, right? Here there are no quantifiers. “Sentence A is false” speaks only about itself without anything additional, so that’s a loop, and it really is a paradox. Or: sentence A: sentence B is false; sentence B: sentence A is true. Same thing. That’s a paradox. There is a fourth type of claim. I said a true sentence, a false sentence—both are claims. There is a sentence that is neither true nor false; it is generally accepted to think that it is not a claim. It basically claims nothing. A sentence that claims nothing cannot even be neither true nor false. It cannot be neither true nor false because it claims nothing. Okay? And there is a fourth type of claims—or sentences, sorry—namely sentences of the following sort. Let’s say: sentence A: sentence A is true. What do you say about that? Not a paradox, just—

[Speaker D] Meaningless.

[Rabbi Michael Abraham] Meaningless? Is it true or false? Is it true? It can be both. Right. Even if you assume it is true, it comes out consistent, right? If it is true, then that indeed fits its content and so it really is true. And if you assume it is false, that also fits. If it is false and it claims—

[Speaker D] —that it is true, then it really is false. I would define any sentence all of whose reference is to its own truth value. Fine, never mind.

[Rabbi Michael Abraham] Before it is paradoxical, it is simply meaningless, and because of that… those are debates I’m not going to enter into here. One can—many do define it that way. I don’t want to get into that right now. Now if indeed this sentence can receive both the value true and the value false, and both are consistent, then that is a fourth type of sentence, right? There is a true sentence, a false sentence, a sentence that cannot be either true or false, and a sentence that can be both true and false at the same time. Because “it is now dark” can be true now and tomorrow morning it will be false. Fine, I am talking about being true and false simultaneously under the same circumstances. What is that? So that I call an anti-paradox. It’s an anti-paradox, something that can be true and can be false with no contradiction. The law of excluded middle, as it were, doesn’t apply to it, right? Both true and false. Or the law of non-contradiction, if you like, never mind. There are halakhic examples of paradox and anti-paradox. I’ll give one of each. I’ll give an example of an anti-paradox. An anti-paradox: the dispute between Beit Shammai and Beit Hillel, which I also once discussed. The dispute between Beit Shammai and Beit Hillel was not resolved for several years until a heavenly voice emerged and said: the Jewish law follows Beit Hillel. Tosafot explains in Eruvin why the dispute was not resolved: because Beit Shammai were sharper. They were wiser, greater Torah scholars, I don’t know exactly. Beit Hillel were more numerous. And the question they were actually arguing about, besides all the halakhic disputes, was a methodological question. The question was: when you take a vote, you should follow the majority. It says, “follow the majority.” Which majority? We once talked about majority of heads or majority of feet—yes? Whether you count heads or you count feet. Beit Hillel say you count feet, and Beit Shammai say you count heads. The determining majority is the majority of wisdom, not the majority of people. And Beit Hillel say it is the majority of people. Now what is the status of such an argument? I won’t get into it—it’s an interesting topic, and we discussed it. But what is the logical status of such a situation? In such a case, if I decide that we count heads, then the Jewish law will follow Beit Shammai—even on the issue of counting heads itself, the Jewish law will follow Beit Shammai. Right? Again: Beit Shammai, sorry, yes.

[Speaker E] Beit Shammai, yes.

[Rabbi Michael Abraham] Yes, because where does it get stuck? Why couldn’t they resolve this dispute? Let them vote. Because even the vote on that very question—

[Speaker B] Yes,

[Rabbi Michael Abraham] —the dispute about that very question itself lands us in the same tangle. Beit Shammai are wiser and Beit Hillel are more numerous, and again I’ll have the question of how to decide. I have no way to decide. Right? So now what comes out is that in such a case, if I decide that we follow the majority of wisdom, then the Jewish law will follow Beit Shammai. Beit Shammai are the ones who in fact claim that one follows the majority of wisdom. Do you see the analogy to “this sentence is true”? If I decide that we follow the majority of people, then the Jewish law will follow Beit Hillel, who say you count feet, who indeed say to follow the majority of people. So every such assumption remains consistent with the situation. And in fact this is an example of an anti-paradox in the world of halakhic ruling. Okay? Basically, if you assume this assumption it remains consistent, and if you assume its opposite it also remains consistent. So in fact you can assume either one of them; there isn’t one that is true and one that is false.

[Speaker D] Why? Just because it remains consistent doesn’t mean it’s true or false. Certainly.

[Rabbi Michael Abraham] You have no other data at all. That is the argument. There is no external datum that decides. All you can do is try to see what remains consistent. Whatever remains consistent, you can’t—

[Speaker D] —rule it out.

[Rabbi Michael Abraham] But that doesn’t mean it’s true.

[Speaker E] This is halakhic ruling—what I can know.

[Rabbi Michael Abraham] This goes back to Tosafot’s question elsewhere. What? It goes back to Tosafot’s question. Maybe, could be. But I’m saying here: in the end, if I assume that in Jewish law what I can know is the law, I’m not interested in what the Holy One, blessed be He, says—“it is not in heaven.” By the way, this comes up there. After all, in the Talmud there a heavenly voice comes out and says the Jewish law follows Beit Hillel. And Tosafot asks there—this is another Tosafot, by the way, on a different page, that’s page 13 and this is page 6—but Tosafot asks there: after all, we do not pay attention to a heavenly voice, or “it is not in heaven,” as in the Oven of Akhnai, so how did they pay attention to a heavenly voice here? He gives all kinds of answers: this was before “it is not in heaven,” I don’t know, all sorts of answers. Now within page 6 it’s simple; you don’t need any answers. Because the dispute was over which majority is counted, the majority of wisdom or the majority of people. That’s another Tosafot. But then I say, according to that Tosafot, it’s simple. “It is not in heaven” is said when I have a way to decide using halakhic tools. Then we say, leave it alone, we don’t heed a heavenly voice; decide according to what the rules of Jewish law tell you. “It is not in heaven”—do it yourself. But if I’m stuck and have no halakhic way to decide because I’m in an anti-paradox, then in that case you do indeed go by heaven. What can you do? The Holy One, blessed be He, has to get us out of the tangle that we can’t get ourselves out of. And therefore there they do follow the heavenly voice.

[Speaker D] And there’s another stage here of—

[Rabbi Michael Abraham] Once the heavenly voice came out, there is no truth in the sense of a decision—

[Speaker D] The truth was revealed only by—

[Rabbi Michael Abraham] Fine, never mind. By our halakhic criteria we are in a state of anti-paradox. We cannot decide it either way. It can be decided either this way or that way, and it will remain consistent. If you were to give control to Beit Shammai, everything would be fine in terms of the system’s logical consistency; and if you were to give control to Beit Hillel, that too would be consistent. And is this a regular dispute?

[Speaker D] No, because usually in a dispute… not a halakhic dispute. No, an ordinary halakhic dispute has nothing to do with such a thing.

[Rabbi Michael Abraham] In an ordinary halakhic dispute, when I decide whether the rival wife of a daughter is permitted or forbidden, then if I decide that she is forbidden, that doesn’t mean she is forbidden; it means that I decided she is forbidden. But where does it remain consistent? Here the dispute is a halakhic dispute: if I say the Jewish law follows the majority of wisdom, then it indeed comes out that the Jewish law follows the majority of wisdom. And it remains consistent. Fine. So that’s the example of an anti-paradox. What is the example of a paradox? There is a dispute between Rav and Shmuel regarding stipulating contrary to what is written in the Torah. The details don’t matter right now, but the rule is that the Jewish law follows Shmuel in monetary law and Rav in matters of prohibition. In civil law the Jewish law follows Shmuel, and in matters of forbidden and permitted it follows Rav. Now there is a topic in the Talmud where there is a dispute between Rav and Shmuel over how to classify it—

[Speaker H] Whether it is a prohibition.

[Rabbi Michael Abraham] According to Shmuel, the classification of this matter is that it’s a prohibition, and according to Rav, the classification of this matter is that it’s monetary law. They are arguing how to explain the fact that in monetary matters a person can stipulate contrary to what is written in the Torah—whether this is a permission or whether it’s a monetary issue, that I waive my right and therefore it overrides what is written in the Torah. It’s waiver; that’s basically the dispute. But never mind—the logical structure means that Shmuel claims this is a question of prohibition, and Rav says it is a monetary question. So the calculation I make—but that is what comes out. Okay? So in such a case, according to Shmuel it follows that the Jewish law is like Rav, and according to Rav it follows that the Jewish law is like Shmuel. Now here it’s a somewhat interesting question. Ostensibly it’s a paradox. If Shmuel then Rav; if Rav then Shmuel, and so on. But on second thought I’m not sure. It may be that this too is an anti-paradox. Because according to Shmuel, he says this is a matter of prohibition, and therefore he too says: rule like Rav. Fine. Rav indeed says his position—that’s his position—there’s no further wheel spinning here. Only Rav says, besides that being his position, Rav also says that it is a monetary question—sorry, not a question of prohibition. Therefore he proposes: rule like Shmuel. But that’s not a wheel; it’s basically two possibilities, each of which remains consistent with itself, and then maybe it’s an anti-paradox. Fine, one can discuss it. The question is whether Rav’s conclusion follows from its being a prohibition, and then it’s not two consistencies but one and the same assertion. Fine. So this is also a structure in the halakhic world in which there can be a paradoxical situation, a situation where if the Jewish law is like Shmuel then it is like Rav, and if it is like Rav then it is like Shmuel. So that is basically parallel to what is called a paradox. Now what happens—or before what happens—basically we know many paradoxes, in Jewish law too there are several, but also generally in our thinking. There are paradoxes that actually have a solution, and then they are only apparent paradoxes. Yes, Achilles and the tortoise of Zeno, Zeno’s paradoxes against motion—I think maybe they are all merely apparent, not real paradoxes. Because you can analyze them and show that what happened there was a mistake. It’s not really a loop, it’s not really a contradiction. Maybe it’s a loop, but not a contradiction. But there are paradoxes that, at least as far as I know—at least it seems—they have no answer. Maybe I’m not clever enough and there is an answer, but on the face of it, it seems to be a loop with no answer. For example, the surprise examination paradox. Do you know the surprise quiz or surprise drill? The teacher comes to class and says: there will be a surprise quiz this coming week. Okay? Or a surprise drill in the army—that’s why it started in the Swedish army. Now what happens? Let’s calculate. Suppose the surprise quiz will be on Friday, the last school day of the week. But if the surprise quiz did not happen by Thursday, then on Thursday night we already know that on Friday there will be a quiz. So it won’t be a surprise; we can prepare. Okay? So it won’t be then. Therefore it can’t be Friday. Fine, so let’s go backwards. If Friday is already off the table, let’s see—if it were on Thursday, then on Wednesday night—exactly—then on Wednesday night I already know it will be on Thursday, because Friday is impossible, and until now it hasn’t happened, so all that remains is Thursday. And again it’s not a surprise quiz. And so too Wednesday, Tuesday, and so on. So basically there cannot be a surprise quiz. There is no such thing. And we all know there is such a thing as a surprise quiz; we can be surprised by a quiz, right? This, for example, is a paradox for which at least I don’t know a solution. It’s very—

[Speaker G] —very—

[Rabbi Michael Abraham] —interesting, because it ought to have some solution. It’s so frustrating. The initial analysis is not natural. Fine, but it’s a correct analysis. What do you mean, not natural? Here, I’ve just presented the analysis to you, and now you understand it. Now if I tell you there will be a surprise quiz next week, will it surprise you when I give the quiz? Of course it will surprise you. But why? After all, there is a very clear calculation.

[Speaker B] Ah, that calculation works against time. So what? The surprise is when the quiz lands on you and you didn’t anticipate it.

[Rabbi Michael Abraham] The explanation is only about the form of the analysis. What difference does it make whether I—

[Speaker B] —analyze from the front or from the back?

[Rabbi Michael Abraham] It’s only a question of how to do the analysis, not how to carry out the analysis. Never mind how. I’m not analyzing how I’m going to carry out the surprise. Besides, this itself yields such a paradox, and therefore I’m surprised. And that was the answer I once thought of, but it too has problems.

[Speaker D] Problems.

[Rabbi Michael Abraham] The very fact that you can’t determine any day—that itself creates the surprise. But on the other hand the analysis becomes valid again. Fine, so what do we do? But after all, on Friday it really can’t be. It cannot be on Friday, because if it’s on Friday then on Thursday night I prepare for the quiz. It can’t be. Well then, if Friday is impossible, Thursday is impossible too. The analysis really does continue to be correct. Yes, of course, his vote does not determine anything, because the chance that a party lands exactly on a round number of mandates, a round number of voters that divides evenly into the value of a mandate, is zero. I’d bet it has never happened. The probability—thirty thousand is a mandate, something like that—what is the chance it lands exactly on the thirty thousand and eighty-seven, or whatever it takes for a mandate? There’s no chance of that.

[Speaker H] But that’s not really a paradox.

[Rabbi Michael Abraham] It’s somewhat similar, because the mechanism of analysis resembles the previous mechanism very much. I’ll now show why. Because then people immediately say: yes, but if everyone did as you do, and you are one person, then obviously there would be an effect. True—but I am deciding only for myself and not telling anyone else anything. They will make their decisions as they make their decisions. Even if they don’t go vote, then maybe it will have an effect, but still my addition changes nothing. My addition still changes nothing. So it doesn’t matter that if everyone doesn’t vote it will have an effect—true—but when I decide not to vote, I am making a decision only for myself; I am not influencing others. So in fact it seemingly finds a solution, but after the solution it still doesn’t work. Or the income tax paradox, yes? Why not hide a thousand shekels from the tax authorities? No one will notice. It’s not just that they won’t notice or won’t catch me; it won’t affect anything. For the state budget, a thousand shekels is nothing. No service will be cut, no debt will be reduced, nothing. It’s simply nonsense.

[Speaker E] In elections it has no effect at all, and in a thousand shekels—

[Rabbi Michael Abraham] It has an extremely tiny effect probabilistically.

[Speaker E] But in the end—

[Rabbi Michael Abraham] There is a difference here: there the probability is tiny, and here the effect—one thousand shekels are removed from the fund. Yes, fine, but after all you know it’s all just papers; it isn’t actual money. It’s papers, it’s numbers written in a computer. So what difference does it make if it says one thousand less there? Nothing in the world will change. Nothing. Nothing in the world will change. Everyone will receive his allowance, his salary, everything will be the same; nobody will feel anything anywhere. Fine, okay.

[Speaker D] In any case, with the same logic with income tax, you can also take it further. If I don’t come to work one day, even if that doesn’t affect anything, right? And that’s the question.

[Rabbi Michael Abraham] Of course. And then people always ask, wait, and what if everyone shuts down the tax authority? Fine, but whatever everyone does, they do anyway. What I decide does not affect them. So still what I do affects nothing.

[Speaker G] And the financial systems also weren’t built on the basis of anything.

[Rabbi Michael Abraham] Yes, but if they don’t know about me, then how will that cascade be created? They would have to be affected by me for it to happen. There there is a mechanism of influence. The differential equation shows what the relation is at the next moment based on the state at the current moment. Suddenly it creates some sort of divergence. But here there really is no connection, because what I do or don’t do has no connection to what others do. I don’t tell them that I didn’t vote or that I hid income tax.

[Speaker G] But because of you the Ministry of Finance will discover that tax collection wasn’t as expected.

[Rabbi Michael Abraham] How will it discover? A thousand shekels it can’t discover. Nonsense. It appears as a number so far after the decimal point that it’s impossible to detect.

[Speaker E] It isn’t felt at all. It will show up as some zero point zero zero something eight hundred places after the decimal. More and more.

[Rabbi Michael Abraham] There are paradoxes that are closer to the liar paradox—self-reference paradoxes. So there is, for example, the barber paradox, yes? The barber who shaves all the people who do not shave themselves. The question is whether he shaves himself. She’s a woman, the barber. So what?

[Speaker G] Then she shaves all those who don’t shave themselves. The barber can be a woman. So what? Then she shaves all the people who don’t shave themselves. Human beings.

[Rabbi Michael Abraham] Fine, I’m talking about people, never mind. So the barber—if I ask myself whether he shaves himself, then look: if he shaves himself, then he belongs to the group of people who shave themselves, but that is exactly the group he does not shave. And if he does not shave himself, then he belongs to the group of people whom he does shave, so he does shave himself. And so on. Or the paradox of the law lecturer, Protagoras, yes? The Protagoras paradox: a law teacher agreed with his student that if the student wins his first case, then he pays tuition, and if not, then not. Something like: if I trained you properly, pay me; if not, then not. Fine. The student finishes his studies, everything is okay, each goes his own way. And what does the fellow do? He doesn’t pay. So his teacher sues him in court to pay tuition. Now the judge has to decide.

[Speaker B] If he—there hadn’t been any trial.

[Rabbi Michael Abraham] This is the first trial. Now, if he acquits the student, then the student won his first trial, so he must pay tuition. And if he rules against the student, then the student lost his first trial, so he doesn’t have to pay tuition. Okay, all kinds of tricks of that sort. In any case, all these self-reference paradoxes are also basically some kind of loop where you can’t—if the answer is yes then it’s no, and if it’s no then it’s yes. Okay? The set of all sets that do not contain themselves as members—Russell’s set theory paradox—it’s all the same thing. Or the paradox—do you know the paradox of numbers? I forgot what it’s called; that one also has some name. Take any number that can be described verbally in various ways. For all I care, the number three is the smallest odd prime number. Just a description of the number three, okay? Or the smallest number divisible by three—never mind. Of course using it itself to define itself is problematic, yes. Okay, then define six as the second number divisible by three. Never mind; one can define a number by means of a set of words, by means of a description, okay? Now let’s take the set of all numbers that have some description—there are many ways to describe every number—that have at least one description that fits them and requires fewer than, or up to, one thousand letters, okay? There is a set of such numbers. Now let’s take the smallest number not included in that set. Okay? Here, I just described it in fewer than a thousand letters: the smallest number that cannot be described using a thousand letters, using up to a thousand letters. Okay? So here too there is this sort of paradoxicality. Okay, so in short, all these paradoxes of this type, beyond the fact that they are amusing, create a problem. And the problem stems from this—and this is the problematic thing about a paradox: the problem stems from the fact that if there is a paradox inside a system, that means the system is inconsistent. If it is inconsistent, that means one can derive from it any conclusion one wants. This is a well-known logical issue: if you have a system with an internal contradiction, you can derive from it any conclusion you want. You can derive from it that it is permitted to murder and that it is forbidden to murder. It is inconsistent, it contains a contradiction, so I can somehow construct a valid logical inference that takes the assumptions of the system—say, a legal system—takes the assumptions of the legal system, what is written in the law code, and assuming there is some contradiction within the system, it doesn’t matter what, it doesn’t matter between what and what the contradiction is, I can always derive from it whatever legal conclusion I want.

[Speaker G] You’ll ask what—you’ll say what, that there are sentences dependent on other sentences that you can—

[Rabbi Michael Abraham] No, no, the claim is stronger. If there is a contradiction between two sentences, I can derive a conclusion that has nothing whatsoever to do with those two sentences.

[Speaker E] And that’s the problem? You use them as premises and assume by contradiction? Yes, easily.

[Rabbi Michael Abraham] So there is a way to derive any conclusion we want. Now a legal system from which any conclusion can be derived and defended in court is worthless—as a legal system, I mean. Therefore if there is a paradox sitting inside a legal system, we are in trouble not only intellectually. We are in real trouble—personal, practical, and legal. Only not intellectual? Why? Because what—

[Speaker D] —will we make of this inference? You understand that the inference is not correct, even though it is orderly.

[Rabbi Michael Abraham] So it’s not correct—but there are two laws written in the law code. So show me what is wrong with the inference I built. It is permitted to use logic in court. The premises are taken from the law code.

[Speaker D] And then what?

[Rabbi Michael Abraham] What is the answer? A positivist.

[Speaker G] A positivist, yes, but wait, here too—you got every… I don’t know. Say you’re the Minister of Education and you have to bus children to a school that has, I don’t know, endless safety defects. Fine, okay, now if you don’t send them to that school and you have no other option, then you’re a criminal because you didn’t send children under the compulsory education law. And if you do send them to that school, then you’re a criminal because you sent them to a school that is dangerous for them. Okay, and you have to solve it, and you don’t have any—and in the end you choose something… so there is a paradox, an excellent example of a paradox, okay, I understand the paradox. And still there is some solution.

[Rabbi Michael Abraham] That solution—and this is basically the conclusion I ultimately want to reach—the solutions you are talking about, in this sense, are not logical solutions. For a positivist, there is no solution to this situation. No solution. Meaning, in positivist thinking, if there is—and now you understand how naive positivism is, because there is no legal system without problems of this sort—then it cannot be. Rather, obviously the judge sitting there has already done, will do, this himself. You don’t need the judge for that. He already makes his own judgment, and it is a judgment of common sense: what, so he’ll stop classes for a few days, fix the school, and send them back there, or find them some other solution of this kind or that. But wait a second, there is a compulsory education law; they have to go to school every day. Fine, okay, so the compulsory education law means except in situations where… fine, but that’s not written. So if you are a positivist and only recognize the possibility of deriving logically from the law code—

[Speaker G] —you end up with the rabbi’s response to “Why was a pin found where a sword should have been?” And we said: fine, great, you’re right—but I have to run a state here.

[Rabbi Michael Abraham] The Talmud asks this seriously. The Talmud brings it as an argument for why not to execute any murderer. Rabbi Akiva and Rabbi Tarfon basically say: if we had been in the Sanhedrin, we wouldn’t have executed anyone. Maybe there was a puncture where the sword entered. You didn’t see, so you can’t know. But these are those Rabbi Yirmiyah-type questions about fifty cubits and things like that. In any case, the existence of paradoxes creates a certain problem in a legal system, because from them you can derive any conclusion. That’s really the difficulty in the existence of paradoxes. Now what this basically means is that clearly positivism cannot be correct. It’s a proof against positivism, you could say. And that positivism can perhaps be true, but then there won’t be any legal system that can function under positivism. Not as a legal approach. It may be true in some abstract theoretical sense, but no concrete legal system can conduct itself in a positivist way. It’s simply nonsense to think you can run a legal system that way. So this isn’t just some difference of opinion in a debate; it’s not the kind of disagreement we really have. There’s no such thing. Today what’s called positivism is of course an approach that very much wants to adhere to the law; it’s not pure logic. Yes, exactly. Pure logic is nonsense.

[Speaker G] Nothing here can be pure,

[Speaker E] nothing pure.

[Rabbi Michael Abraham] The legislator can—fine, he’ll legislate for a month—but until the legislator does that, what will you do in the meantime with the children in school? It’s a technical problem, but a substantive one. After you add that qualification, something else will arise. Something else will arise. There’s no way out of it.

[Speaker E] That’s what judges do today too, they’re constantly adding. Right. So I’ll do it in the form of laws—that’s exactly the point.

[Rabbi Michael Abraham] But the system itself, before you added that law, do you agree that this system was worth nothing? It had a problem; it was worth nothing. A positivist says: no, that system simply wasn’t legal. Now I’m telling you: because if it contained a contradiction, then any conclusion could be derived from it.

[Speaker E] No, but the moment I find,

[Rabbi Michael Abraham] a positivist, still a positivist,

[Speaker E] the moment I find a problem, I fix it.

[Rabbi Michael Abraham] Yes, no, so fix it. But if there’s a problem in it, even if you didn’t find it—once there’s a problem in it, it may be that the conclusion you reached is nonsense. I may be able to prove—maybe I’m not clever enough, but I may be able to prove—the opposite conclusion too. I can throw this system in the trash.

[Speaker B] Your induction won’t help.

[Rabbi Michael Abraham] Now after all, I also know that even when you fix the system, there will still be contradictions. There is no system without problems.

[Speaker B] Contradictions again?

[Rabbi Michael Abraham] No, contradictions. Contradictions. There will always be contradictions. And paradoxes and arguments of that sort, yes—you won’t be able to solve them. And because of that you won’t succeed. No, the chance is zero. The chance is zero. Certainly don’t build on that. You’ll never be able to know that this is really so.

[Speaker G] Look how every time we go to the extreme. The Rabbi always takes it…

[Rabbi Michael Abraham] No, I’m speaking only from a purely logical point of view, obviously, obviously.

[Speaker G] Obviously. In anything you put into reality, you’ll have a problem.

[Rabbi Michael Abraham] Right. And therefore I really say that the conclusion that follows from here is that you simply can’t relate to a legal system—or really to anything, in my opinion—in a positivist way. Maybe in mathematics that has some meaning.

[Speaker G] Even there it’s implicit in mathematics, in the numerical solution of problems; there’s always the endpoint problem at the beginning, zero, for example in time. There is no delta t minus. So there are dynamic systems that you can’t even start. Why? Because when you want to solve them numerically, it isn’t solvable.

[Rabbi Michael Abraham] Not solvable isn’t a problem.

[Speaker G] No, then you can’t—it’s not defined.

[Rabbi Michael Abraham] You can’t put

[Speaker G] it

[Rabbi Michael Abraham] into a computer.

[Speaker G] No, obviously, you can’t put it into a computer.

[Rabbi Michael Abraham] But the computer can’t solve it either. No, it’s not defined in the sense that there is no… you… there is no moment before it.

[Speaker G] When you solve it, you always take a moment before and a moment after.

[Rabbi Michael Abraham] Exactly, because you can’t do a numerical solution for that system. But that doesn’t mean it has no solution. It means the computer is helpless in relation to it. No, it comes to describe a real problem. Because time is a real problem and it is continuous, not discrete. But even there, there’s no delta t minus. Why? Of course there is. You don’t need delta t minus. There are differentials. You don’t need to get to minus. Only in discretization. At least I think so, as far as I know. So in mathematical systems, when a paradox is discovered, that’s a real problem and it needs to be handled, because there there really are no solutions like, okay, use common sense. That is exactly what mathematicians are not allowed to do. Meaning, they have to go all the way with the rigid rules. And therefore mathematicians spend a whole semester proving—mathematicians invest a lot of time in proving this amusing theorem, because they’re not willing to accept something just because common sense says so. You have to show, from the assumptions, how this can be derived logically. That’s the mathematical framework. Meaning, everything has to be precise and certain. There, paradoxes really do matter—like the paradox of set theory that I mentioned earlier, which really did lead to a revolution in set theory. They redefined set theory, axiomatic set theory, in order to build it coherently, in a way free of paradoxes. And meanwhile, it seems to me, as far as is known, there are no paradoxes in it under the new definitions, in the axiomatic theory. In the naive system there are paradoxes, but that’s because the naive system was defined in human language and not in mathematics. And in human language there are paradoxes, whereas in mathematics there aren’t supposed to be. If there are, that’s a proof by contradiction that something here is wrong. What can exist in mathematical systems—and this can exist in mathematical systems—is incompleteness. And not only can it exist; it always exists.

[Speaker C] No, it depends.

[Rabbi Michael Abraham] In systems of a certain kind it holds, those that satisfy the conditions. But I’m saying: paradoxes do not exist in a mathematical system. If they do, then throw it in the trash; it isn’t a mathematical system.

[Speaker C] So incompleteness.

[Rabbi Michael Abraham] There is incompleteness in the sense that there are questions you cannot answer within the system. Singular points. Questions relevant to the system, for which you nevertheless won’t find an answer. Say, by analogy to a legal system, incompleteness in a legal system means that you ask a legal question—whether something is permitted or forbidden—a very relevant question, and you won’t be able to find an answer in the legal system. There is no answer. Okay? You won’t be able to show that answer A is correct or that answer B is correct.

[Speaker B] There’s another incompleteness theorem, even more than that, which is

[Rabbi Michael Abraham] Right, and you can’t prove it. There is another incompleteness theorem of Gödel that speaks about a situation in which there is a correct answer, but you will not be able to prove it from within the system. You won’t be able to prove it within the system, but we know that it is correct.

[Speaker B] And how did you decide that it’s correct?

[Rabbi Michael Abraham] So that’s it—that’s what is so interesting there. You can prove that it’s correct from outside the system. If you assume it’s not correct, you run into a contradiction. And you run into a contradiction not within the system—within the system everything is fine—but from outside you run into a contradiction. That’s how you prove it, in a larger system.

[Speaker B] Exactly.

[Rabbi Michael Abraham] But now, relative to the larger system, you can also define a Gödel sentence. And so on. There will always be one; you cannot complete a Gödelian system by adding a few more axioms to solve the problem, because there is a theorem that says there will be a Gödel sentence in the new system too. Unless you get to a continuum of assumptions, then maybe yes. But if each time you add a certain number of assumptions—a discrete number, a countable number—then there is a Gödel sentence. So that is incompleteness. In short, there is inconsistency and there is incompleteness of a system, and both are problems in a legal system. One is a problem in the positivist sense—the contradiction—which basically means that any conclusion can be derived from the legal system, and that creates a problem. And here, as I said earlier, it seems to me there is no choice but to depart from this logical, positivist way of thinking and understand that one has to use common sense. And within the framework of common sense, it may be that one really can give answers—one answer and reject the other—even though logically both can be proven from within the system. Okay. And regarding a system that has no answers to certain questions, there opinions are a bit divided. Because really, I think that even if we depart from positivism and allow ourselves some kind of expansion, there are those who say that if we allow ourselves some kind of expansion, we will always have an answer. We can always add some line of reasoning, an analogy from somewhere else, something—ultimately there will always be an answer. And there are those who say that even expansions have limits. That is, there are certain things that would not count as an answer of the system. I’ll give you an example maybe.

[Speaker D] If I ask how many asteroids there are in the galaxy.

[Rabbi Michael Abraham] No, I’m talking about a legal question.

[Speaker D] Ah, specifically a legal question?

[Rabbi Michael Abraham] So for example, yes, there was the Foundations of Law Act. The Foundations of Law Act says that every problem, every lacuna in the legal system that cannot be solved by analogy or something else, then one turns to the principles of justice and fairness from the heritage of Israel, or something like that, an ambiguous phrase of that kind. Now, in the Handels case, where they found some lost item—securities papers—on the floor of a bank, I may have mentioned this in one of the recent lectures, there was a dispute there over whether this was a lacuna. Because some judges wanted to say: we can make an analogy from American law or from some other law, it doesn’t matter, and arrive at an answer. So that wouldn’t count as a lacuna. And a lacuna is only when you cannot make an analogy. Now understand that if you take the—no, analogy can also be from other legal systems. Now according to that interpretation of the concept of analogy, there is no such thing as a lacuna. There is no such thing as a lacuna, because I can always make some kind of argument, as long as there isn’t something in the law that contradicts it. Expand it that way—judicial legislation.

[Speaker B] So

[Rabbi Michael Abraham] what—from that legal system over there I’ll make some sort of extension and then come back here? I’m saying maybe, but—but it’s almost imaginary. I almost—meaning, it’s very hard to think of a situation—maybe there is such a thing—where it still couldn’t be judged, because someone can always come and say: I think this is a reasonable analogy, and therefore I think it’s fine. And you can see the dispute among the judges there, between Elon and the majority judges, that this was exactly the issue—whether this counts as a lacuna or not. He argued that it was a lacuna and therefore one should turn to Jewish law, and they argued that it was not a lacuna because it could be completed by analogy. Now this depends very much on what you call analogy. Once you’re talking about logic, logical validity is something clear. When you’re not talking about logic, when you’re talking about common sense, analogies, things like that—okay—then you can almost do whatever you want. Where is the boundary? So in a certain sense this law loses its meaning, because then what does it mean? What was the law talking about? It said that when there is a lacuna, then yes, one should go to principles of whatever, and so on. So what is it talking about if there are no lacunae in the world? Therefore, well, I don’t know, it’s an interesting question. But I’m saying that the difference between—

[Speaker G] A lacuna is simply that each one just explained it the way they wanted to explain it.

[Rabbi Michael Abraham] Obviously, that

[Speaker G] we know.

[Rabbi Michael Abraham] We know what lies behind the statements, but when you look at the law itself, the law speaks about a lacuna, so that means there are lacunae. So tell me: what counts as a lacuna? What’s the criterion? What is a lacuna? What is something that cannot be completed by analogy? I can always find some consideration or other, or some legal system or other, from which I—you know—I can even make an analogy from an entirely fictional legal system. Take a book describing a fictional world in which Asimov rules, with the collection of laws described in the books, and I can make an analogy. Who says that’s forbidden? And if there isn’t such a book, I’ll write it now. Now there will be such a book, and I’ll make an analogy from it. I don’t know.

[Speaker D] Analogy means analogy,

[Rabbi Michael Abraham] it wasn’t defined.

[Speaker F] as not being a lacuna because you can’t make an analogy of that kind, but rather only from within the law itself.

[Rabbi Michael Abraham] No, the claim is that one goes to the principles of justice and fairness only when you cannot make an analogy. The Foundations of Law Act says that when there is a lacuna and you cannot fill it by analogy—

[Speaker F] No, no, also analogy in general.

[Rabbi Michael Abraham] Well, at least look—I don’t know it broadly, but I know the Handels ruling, and there that’s what they argued: they made analogies from legal systems, from foreign legal systems, and they said that if so, then there is no lacuna here. And that is exactly the claim. The claim is that the system of Jewish law does not take precedence over foreign systems. That was really the dispute there. Elon argued that it does take precedence because this is a lacuna. And they said: no, it doesn’t take precedence, because it’s an analogy; we can fill it by analogy, so it isn’t a lacuna. I don’t know, I don’t understand enough in this area, but that’s what was written there; there was a dispute about it. Handels? I don’t know, maybe the 1980s. Handels? Yes, Handels. So I’m saying that in that place—these two problems, the problem of consistency and the problem of completeness—if you depart from positivism, then the consistency problem can usually be solved. Meaning, questions involving contradictions of that sort, we find a solution through common sense and everything is fine. Common sense here—the Rabbi can be…

[Speaker G] Fine, the question is whose mind is

[Rabbi Michael Abraham] defined as straight, okay, who is the reasonable person, okay. But—and by the way, that is necessarily the judge. I don’t even see it as an accusation when the judge sees himself as the reasonable person, because fine, what can you do—what seems reasonable to him, he is the one sitting there wrestling with it. What can you do, who else will do it, me? It’s his authority; he has to do

[Speaker G] it.

[Rabbi Michael Abraham] So he has to decide what is reasonable, and that is his authority; there’s nothing to be done. Yes, when it gets to a judge. So yes, when he has to decide whether the mayor acted reasonably, that’s a different question, that’s… No, but generally I really think he does try to put himself in the mayor’s shoes in order to understand whether it was reasonable from his perspective. In any case, that’s regarding consistency. Regarding completeness—problems that have no answer within the legal system—if I depart from positivism, then as I said earlier, it depends on how far I’m willing to go. Because here I really can see many cases—and we know there are many problems where the legislator or the judges themselves say: there is no answer to this in the law books, and they send the ball back to the legislator. There are such cases. There’s no answer to it; the law books don’t. So they acquit or obligate according to the circumstances, because there’s no problem in that.

[Speaker D] They say it would have been proper to…

[Rabbi Michael Abraham] Or he’ll say that it is permitted, or say that it’s permitted, because there is something unreasonable here.

[Speaker D] Or he’ll say that it should be discussed in the synagogue.

[Rabbi Michael Abraham] No, so he can’t do that—so there is none, it’s lacking, there’s no answer. The law books do not address this issue, including analogies and all sorts of things like that. There are things where judges sometimes—I assume it also depends on which judge—but there are situations where a judge says: yes, it’s not… he still has to decide.

[Speaker H] What? He still has to decide.

[Rabbi Michael Abraham] Yes, but he can decide that if there is no law, then there is no cause of action. Okay? For example, in a criminal case, if the law does not allow me to convict him, then he is not convicted. Fine, that too is certainly a ruling. He’s only saying that here it would have been proper to convict; this is a lacuna, it’s a problem in the law, and therefore he sends it back to the legislator with some comment or something like that. So I’m saying that through expansion, when we expand thought beyond logic, we depart from positivism, and then we can actually also find answers—not only find an answer where there is a contradiction, but also find answers where the system does not address the matter. We make an interpretive expansion of the system, and that is what is called judicial legislation. Something that isn’t possible in the positivist world, but departing from the positivist world does make it possible. And therefore, if I just summarize what we’ve done so far—and afterward I’ll move to an example or two and explain a bit more what it means to depart from the system or depart from positivism—what I want to say is that looking at a legal system, or a philosophical or moral system or whatever it may be, as a system that operates solely by means of logical tools, and anything that does not emerge through pure logical tools—you cannot say that the system requires it or forbids it or, I don’t know, says something about it—whoever goes in that direction, a positivist, in the end cannot make use of any system. It isn’t practical. And therefore I want to argue that in Jewish law too, that’s how it is. And the conception of Jewish law as if it’s basically some collection of rules—we talked about this when we discussed rules and particulars, about intuition; at the end of the series on intuition I spoke about rule-and-particular thinking, and there I touched a bit on this issue—the conception of Jewish law that says that basically there is some set of rules here, and the halakhic decisor only needs to apply his logic in order to derive the conclusion from them, is a problematic conception, a conception that does not stand the test of reality. There are those who want to or think that one can live that way; they are fooling themselves. It isn’t possible, exactly as the positivists fooled themselves in the legal world. Now, even those who say that these are logical rules… What? I’ve never seen anyone say… Fine, the question is how logical it is—that is exactly what hides the question whether this is positivism or not. Because the rules, if they are not logical, then in effect there are no rules. That basically means there are no rules; you depart from them according to what is reasonable, what seems right to you, so that already empties the concept of a rule of content. So really the claim is that I’m saying even more than the claim that a halakhic decisor does not work according to rules, but also against the principled claim that there is some system of rules which, if you gave it to a supercomputer that could process it in an orderly and systematic way all the way through, it would produce an answer to every halakhic question. Even if I, the decisor, am ignorant and don’t know, or all the decisors don’t know, because we are not all-powerful. But the question is whether such a thing exists at all. My claim is that there cannot be such a thing—not only that the decisors do not work in that way. Okay, up to here, that’s what I’m saying.