On Positivism in Halakha and in General

Table of Contents

• [0:03] Placing the question in a broader context
• [1:23] The existence of a deductive system
• [3:17] The judge’s role in mechanical derivation
• [4:50] The halakhic decisor reduced to logical inference
• [9:30] The inscription on the American Supreme Court
• [11:03] Rav Nachman and the definition of the judge’s role
• [25:41] Conclusion: the question of mechanical authority
• [27:54] The question of bribery and its influence on legal rulings
• [29:55] The puzzle method and determining halakhic rulings
• [31:47] The Torah’s view of ideal and non-ideal situations
• [33:19] Introduction to logic and its connection to positivism
• [35:24] Differences between validity and falsity in arguments
• [??:??] Soft tools for legal derivation (NONE)
• [??:??] The story of the stolen ox in the Talmudic text (NONE)

Summary

General Overview

The text places the question of the work of a halakhic decisor within a broad framework concerning the nature of human thought, and asks whether drawing conclusions can be reduced to a logical deductive operation like a computer, in the style of the thesis attributed to Alan Turing. In the legal and halakhic context, it presents a positivist view according to which a judge or halakhic decisor must confine himself to mechanical derivation from the law or the sources of Jewish law so as not to become a legislator. Opposed to that are various justifications for departing from deduction, including practical necessity and a substantive justification for using “soft” tools like analogy and intuition in order to uncover the will of God. Later, the text offers an introduction to deductive logic and a distinction between the validity of an argument and the truth of its claims, using John Stuart Mill’s critique of deduction and a discussion of induction and analogy, in order to argue that the position of “purely logical halakhic ruling” is an illusion or an empty act of quotation. Finally, it mentions Nachmanides’ critique, in the introduction to Milhamot Hashem, of comparing Torah to mathematics and calculation.

A Broader Framework: Human Being, Computer, and Deduction

The text argues that one can ask of any reasoner whether his activity can be described as a deductive logical argument or as something broader. It formulates the question as a thesis in the style of Alan Turing: if all human inferences can be translated into a deductive process, then in principle it would be possible to program a computer with a person’s set of premises and simulate his thinking, even if technically this is “complicated” and we do not have all the foundational assumptions. The text presents this as a major debate within the world of artificial intelligence and as a philosophical question about what a human being is, and whether a human differs essentially from a computer or is simply “a very, very sophisticated and complex computer.”

What a Halakhic Decisor Does and What a Halakhic Decisor Should Do

The text distinguishes between the descriptive question of how a halakhic decisor works and the normative question of what a halakhic decisor ought to do. It says that if a person can be reduced logically to an axiomatic system, then a halakhic decisor should also be such a system. But even if a person is not such a system, one can still argue ideologically that a halakhic decisor should restrict himself to deductive derivation alone. The text compares the discussion to judges and asks whether a judge’s role is to derive conclusions mechanically from the laws or whether “judicial legislation” is permitted, arguing that when a judge adds something that is not derived from the law, he is functioning as a legislator, whereas the legislator is the elected body, not the judge.

Positivism, Authority, and Judicial Legislation

The text describes positivist conceptions that were more widely accepted in the past, according to which even if a person is capable of non-deductive reasoning, the judge should limit himself to logical inferences so as not to exceed his authority. It formulates this as a question of “roles and powers” and as an ideological issue, and parallels this to Jewish law by arguing that the legislator is the Holy One, blessed be He, or the Sages, and therefore the role of the halakhic decisor is to derive logically from the halakhic corpus “as it has been received until now” the norm that applies to the case. The text sharpens the question of how a non-positivist conception can be justified—that is, how to legitimize the activity of a judge or halakhic decisor beyond deductive derivation, when on the face of it this seems like taking legislative authority.

“By Laws and Not by Men” and the Gap Between Declaration and Reality

The text cites the inscription at the Supreme Court in the United States about being governed “by laws and not by men” as a positivist conception according to which the conclusion follows from the law and not from the identity of the judge. It points out that in practice there is an intense political struggle over the appointment of judges in the United States because people know that political identification will affect rulings and the character of the country, and presents this as an example of the fact that even though a non-activist method is declared, “the person has an impact.” The text presents one possible response: that if the person were a machine, it would work, but in practice the human being “fails” and rules in accordance with his worldview.

Halakhic Examples: Rav Nachman, Fines, and the King’s Justice

The text brings Rav Nachman, who says that he is “like Shvor Malka,” and describes actions in which he imposed fines and laws even “after the person had already committed the offense,” in a way that seems like retroactive legislation. It mentions Chanina ben Menachem, who dedicated his life to the question of the Amoraim and halakhic decisors in the time of the Talmudic text “as people who created laws on the move.” The text raises the Ran, in his derashot, in relation to the idea that “the king’s justice” is a mechanism for generating answers to questions that have no solution within ordinary Jewish law, and presents an interpretation according to which the Ran assumes that there are, in a substantive sense, lacunae in Jewish law, and therefore there is a parallel system whose role is “to plug the gaps.”

Justifications for Departing from Positivism: Necessity Versus Substantive Justification

The text presents the “justification of necessity” as the claim that there is no choice, because it is not always possible to derive an answer deductively from the cases, and one cannot go back to the legislator for every case. It distinguishes between technical necessity and a “lacuna” as a substantive claim that there is no answer in the law. The text argues that even the claim that “it can’t be done” already assumes in advance that the law is supposed to address the situation, and it suggests a principled possibility according to which, when no conclusion can be derived, there simply is no ruling, and one may say “passive omission” or that the law does not apply. It states that the central issue is a substantive justification according to which not only is it impossible, but one should not, because sometimes “the results emerge from the laws through non-deductive tools,” and this still reveals the will of God by means of analogy, intuition, and “soft” tools, as part of the way the Holy One, blessed be He, “encoded” the conclusions within the laws so that they cannot be extracted only through rigid logical tools.

Torah-Level Interpretation Versus an Alternative Mechanism

The text presents a sharp distinction between adding a rabbinic fine or an alternative system like “a religious court may administer lashes and punishments not according to the formal law,” and a “more radical” situation in which the Sages interpret Torah law itself in a non-deductive way and even “apparently against the rules,” because it is clear to them that this is the Torah’s intention. The text brings a story from Ketubot about a robber who stole an ox, plowed with it, and returned it, and Rav Nachman obligated him to pay the value of the improvement, or part of it, in order to fine him because he was a “habitual robber” taking advantage of the rule that “all robbers pay according to the value at the time of the robbery.” Rava objects that the clear Jewish law exempts him from paying for the profits produced by the robbery. The text presents an interpretive dispute as to whether Rav Nachman acted by way of a rabbinic fine or as an interpretation of “and he shall return the stolen item that he stole,” and notes that Maimonides writes that if there is a habitual robber, “this is the law… including the improvement,” and marks this as “Torah-level.”

Degrees, the Illusion of Logical Halakhic Ruling, and Rabbi Soloveitchik

The text argues that in practice there is no such thing as a “pure logician” in halakhic ruling, and that this is a matter of degrees, even among those who claim to limit themselves, and sometimes it enters “without his even noticing.” It cites Rabbi Soloveitchik as saying that there was never a halakhic decisor who “didn’t decide before he checked,” and describes a situation in which the halakhic decisor understands what is right and then backs it up with sources and arguments. The text distinguishes between halakhic ruling and “quotation,” and argues that stating a law without its rationale is an act of quotation that one could say “even from the bathroom,” whereas “halakhic ruling means you innovated something… you created something.” It connects this also to the statement “both these and those are the words of the living God” as a sign that this is not a purely logical process.

Bribery, “Follow the Majority,” and an Ideal Society

The text uses the verse “and you shall take no bribe, for bribery blinds the eyes of the wise” to ask how bribery could have an effect if halakhic ruling were purely deductive, and suggests that the influence of bribery is evidence that this is not pure logic. It adds that the reality of majority and minority in “follow the majority” hints that there is not one logically derived answer that would lead every reasonable judge to the same conclusion like a computer. The text suggests one interpretive possibility: that the Torah recognizes that human beings are not perfect and therefore sets up safeguards. It cites the Tur in section 2 in the name of Rabbeinu Yonah, that the Torah of “and he shall return the stolen item that he stole” fits an ideal society in which theft is rare and the thief regrets what he did, whereas in a world that requires deterrence this is not deterrent enough.

Introduction to Logic: Aristotle, Validity Versus Truth, and Three Types of Inference

The text attributes the first conceptualization of logic to Aristotle’s Organon and to the presentation of patterns of syllogisms, and defines a valid deductive argument as one in which “it is impossible to accept the premises and not accept the conclusion.” The text presents a discussion of the possibility that coercion is “psychological coercion” in the context of evolutionary psychology, alongside an opposing position. It demonstrates that an argument can be valid even if its premises and conclusion are false, and distinguishes that truth and falsehood are properties of propositions, tested by comparison to the state of affairs in the world, whereas validity and invalidity are properties of relations between propositions. The text defines three mechanisms of inference: deduction as moving from the general to the particular, induction as generalizing from the particular to the general, and analogy as comparison at the same level of integration.

John Stuart Mill, Popper, and the Claim That Deduction Hides Non-Deduction

The text brings “Mill’s challenge to deduction” and argues that a conclusion like “Socrates is mortal” rests on a general premise, “all human beings are mortal,” whose source is an inductive generalization, so that a non-deductive step is hidden inside the process. The text presents a broader formulation according to which in practice “all we do is analogies,” and induction and deduction are just the breaking down of analogy into two steps. From this it concludes that one cannot arrive at absolute necessary certainty, because one can always ask where the premises came from. The text brings Karl Popper in the context of refutation and the white raven that undermines a generalization about black ravens, and describes an idea of a continuum of “degree of confidence” instead of absolute truth, while noting that Popper himself may not have gone that far. The text argues that even in a halakhic system, rules like “women are exempt from all positive commandments dependent on time” are not written in the Torah but are created out of examples and generalizations. Therefore, anyone who thinks he is acting positivistically “is living in an illusion,” because he is hiding non-deductive components in the premises.

Positivist Halakhic Ruling as Quotation and Emptying Out the Content

The text portrays a “true positivist” as someone who assumes a premise such as “everything Maimonides wrote is true” and then derives a halakhic ruling from it, for example in matters of twilight as opposed to Rabbeinu Tam. The text argues that such a move is “impeccable” from a formal standpoint if the premise is accepted, but it does not create halakhic ruling; it only creates quotation, and therefore “this act is empty of content.” The text says there is a subtle tension between the claim that this is empty of content and the claim that it is also incorrect, and in any case establishes that positivist halakhic ruling tends to shrink into mere quotation.

Certainty Versus Quantity of Information: Logic, Mathematics, and Science

The text presents the “philosophical certainty principle,” according to which the level of certainty multiplied by the amount of information in an argument is constant, and argues that deduction is certain because the conclusion is hidden in the premises and therefore adds no information. The text uses a joke about a hot-air balloon to describe a statement that is certainly precise but useless, and explains that analogy and induction are not necessary in the same sense, because the conclusion adds information beyond the premises and is therefore exposed to failure. The text notes that the relation between analogy and induction is not simple, because one can view induction as a collection of analogies, and analogy as hidden induction through the choice of a relevant parameter of similarity. It distinguishes between mathematics and logic, which deal with relations among propositions and with the “if-then” of validity, and science, which deals with truth and falsehood and with accumulating information about the world.

Halakhic Devices, Nachmanides, and Baal HaMaor

The text asks how people involved in Jewish law “get around” the problem of certainty without adding information, and raises a positivist possibility according to which “the system of premises” is sophisticated and contains everything, so the halakhic decisor merely uncovers it. It raises another possibility according to which the halakhic decisor does not arrive at certainty, but rather rules on the basis of probability while being aware that “doubt is wrapped around your heels.” The text notes that the one who first attacked positivist halakhic ruling was Nachmanides in the introduction to Milhamot Hashem, with the statement that “the wisdom of our Torah is not like the wisdom of geometry and calculation,” and presents a direction according to which Nachmanides seeks to rely on the authority of the halakhic decisor and on tradition. The text describes how Nachmanides argues that Baal HaMaor objected to the Rif with good logical arguments, but failed to grasp that the Rif “is an elder and has a tradition,” and it concludes by promising to continue with this next time.

So this question can also be framed in a broader context, not only regarding a halakhic decisor, but regarding any person who draws conclusions from assumptions or from some kind of underlying framework. We can always ask ourselves whether it can really be translated into a deductive action—that is, to describe what he is doing in the form of a logical argument—or whether we are dealing here with some broader kind of action. In the broadest sense, this can be translated into questions about what a human being is at all. Yes—if I put this in a really broad context, there is Alan Turing’s thesis: are all the inferences a person makes in fact inferences that can be translated into a deductive process, into a logical argument, and then, if I could program a computer with the collection of all the assumptions that exist within us or within some person, then I could make a full simulation of that person’s thinking by means of a computer? It may be that I can’t do that technically because it’s complicated and there are lots of basic assumptions and we don’t know them all and so on, but on the principled level there is such a system. A non-constructive existence claim. There is such a system—or perhaps there is no such system. It’s a big debate. Even within the world of artificial intelligence there is such a debate, though there it’s phrased a little differently, and so this really touches on much broader philosophical questions than how one issues halakhic rulings; rather, how do people think at all? What is a human being? Meaning: is a human being really essentially different from a computer, or is a human being just a very, very sophisticated and complex computer? You mentioned heuristics? What do you mean? The heuristics of a halakhic decisor, beyond rational and logical considerations? No, I’ll still get to that; it’ll take me a little more time until I get there. At this stage I’ve only said that the general goal is to describe how a decisor works, and I’m placing this within some broader context, and it really is not detached from the question of what a human being is or how a human being thinks in general. After that you can narrow it and ask what a decisor does. It’s obvious that if a person—if all thought, let’s already place these questions side by side—if every person can basically be translated into or reduced to logic, meaning it’s possible to describe him in the form of an axiomatic system, then a decisor is also like that; a decisor is supposed to be a human being, that’s the hope at least. But if a human being is not like that, you can still leave open the question of what a decisor is supposed to do. It may be that a person can also make broader or less rigid inferences than logical inferences. Maybe. But the question of what a decisor does, or what a decisor ought to do, is another question. Someone might say on the ideological level, on the theological level, whatever, that a decisor ought to restrict himself only to deductive derivation, even though a person can think in other ways too. And why? You can see a lot of logic in that. The same debate also exists regarding judges—judicial legislation and things like that. The question is whether the judge’s role is really only to derive mechanically from the laws the conclusions or results regarding the case before him. Do you have arguments about punishment… No, no, no, you’re already dragging me into details now that are less important. In all these areas, in all of them, I’m asking this question now without distinguishing between whether I’m discussing punishment or guilt; that’s not important. In any legal discussion whatsoever. There are conceptions—there were conceptions, today it’s less popular—there were conceptions that said, what is called positivism in the legal context, that a judge, even if a person can make inferences that are not only logical inferences in the narrow sense, a judge ought to restrict himself only to logical inferences. And why? Because every time a judge doesn’t do that, he is really functioning as a legislator, because he is adding something that cannot be derived from the laws, and that is neither his role nor his authority. The one elected to be a legislator is the legislative body, not the judge. And therefore, even though perhaps one can think differently, or even recognize the legitimacy of different kinds of thinking, that still doesn’t mean that that is what a judge ought to do, or what a decisor ought to do. So if we assume that a person is only that and cannot do anything beyond the work of a computer, then the question does not exist. Meaning, if you think that a person is only a complex machine, then there’s no point asking the question. If I think that a person is something beyond that, now the question can be asked. The answer is still not clear. One can ask whether the decisor—or the judge in the legal context—should restrict himself only to logical inferences or not, when behind this question there actually stands, at least in what I’ve said so far, an ideological question. An ideological question, or a question of roles, of authorities. The question is whether the judge is allowed also to perform quasi-legislative actions. If he is not merely deriving from the laws but adding something of his own, then in fact he is acting as a legislator and not only as a judge. And for that he was not authorized—meaning, he was not elected to be a legislator, he was elected to be a judge. A judge is supposed to implement what the legislative body determines. And therefore even if he can, I don’t permit him to do that. And the same in Jewish law. One can say: that is not your mandate as a decisor. The Holy One, blessed be He, is the legislator, or the Sages, or whatever, if this is rabbinic law, the authorized body. That is the legislative body. You as a judge or as a decisor—what you are supposed to do is derive logically from the Jewish law as it has been received up to this point—it doesn’t matter, one can argue about what exactly counts as binding Jewish law and what was accepted and what wasn’t. Assuming there is such a corpus, your role is to derive deductively from that corpus the answer to the question before you. What norm applies to the situation before you. Judge, or judge in secular law, or decisor, or whatever it may be. Why? Because you are not the Holy One, blessed be He. All you can do is tell me what the Holy One, blessed be He, wants. Derive from the laws established by the Holy One, blessed be He, what that means—what I am supposed to do. That’s the restrictive side. What could someone who says otherwise say? What—am I bound to a rabbinic judge, or to a judge in the legal context for example? How can it be at all that a person—or a conception—would grant legitimacy to an action of a judge or a rabbinic judge that goes beyond deductive derivation? Why? What permission does he have to be a legislator? There is an obvious legislative body—the Holy One, blessed be He, say in the halakhic context; let’s talk now about Torah-level law, and leave rabbinic law aside—and the legislator in the legal context. How dare you, as an interpreter or as a judge or as a rabbinic judge—the decisor—take for yourself the authority to legislate? The Sanhedrin or a legislative body is something else. I said rabbinic law, leave that aside now. So seemingly the obvious answer is, and we can close up shop here, that if a person is a machine, then the question doesn’t arise. We can’t do anything other than derive logically. If a person is not a machine, like a judge… yes, a judge is also a human being, yes, at least… if a person is not a machine, then we can ask what his role is, but here too, as I say, seemingly the answer called for is that even if the judge or decisor or interpreter is not a machine, he still must restrict himself to a mechanical action, because otherwise he exceeds his role—he legislates. So what argument could nevertheless say that it is not right to be—let’s already use the word—positivist, that it is not right to restrict oneself to a positivist conception, to see only logical work as the role of the decisor or the interpreter or the judge? I think taking it to the case of a legislating judge, I think that deviates a bit from the point… Meaning, from what I understand. The judge knows that he comes to interpret the matter. One person will interpret it this way, another will interpret it that way, and both of them, both of them study, and their authority comes from the law, also the authority that the person accepts. It’s not written in the law exactly what you need to do. There is some kind of residual authority: you interpret it, and through that you determine your authority. Why? It follows from the interpretation you give that clause. Now it could be that in the Knesset, when they passed a certain law, the law—and the Knesset members who passed it and discussed it in the parliamentary record—said such-and-such, but when it reaches the court they go in a completely different direction, and therefore it is important to know the background of the law. You’re describing facts to me, but I’m asking about norms. I’m asking what the court should do, not what it does. Because if in practice it gives an expansive interpretation… By virtue of its authority to interpret the law… That’s exactly what I’m asking: does it have authority to do that? That’s the question. What you’re saying is that in our courts, in one measure or another—Aharon Barak was the extreme representative of the approach of judicial legislation, but there are different levels—you’re telling me de facto, in practice, that the courts adopt a non-positivist or not fully positivist approach, okay? But I’m asking what the justification is. Why? I want to point out that there is a well-known fact that on the U.S. Supreme Court there is an inscription, and the inscription is in Latin; I don’t remember how it is in Latin—not that I could say it and I just don’t want to tell you, I simply don’t remember—but the inscription says: by laws and not by men. Meaning, a completely positivist slogan. The idea is that it shouldn’t matter who the judge is; the conclusion follows from the law. I don’t want to entrust my fate to people; I entrust my fate to the laws. And this conception is seemingly—yes, that is the heading of the Supreme Court—even though anyone who knows the history of the U.S. Supreme Court knows there is a very, very strong political struggle, no less than in Israel, over appointing judges to the Supreme Court, because people know that a judge’s political identification will affect his rulings, and afterward will also affect the character of the state. So the question Michael asked, yes: by laws and not by men—and nevertheless how does this work on the ground? And this is a very striking example that even though they declare that it is by laws and not by men, and the method is a method not of judicial activism, a method that says the laws are important—that is the American method—still, the human being has an effect. And the question is how can that be? So Rabbi Miki will say: he simply didn’t notice, he slipped. If the human being were a machine that would be fine, but he failed. And he failed and ruled according to his political outlook. Do you want to talk about how worldviews are factored into this topic? We’ll talk about that; that’s going much farther ahead. At the moment I’m really only making an additional remark—what about Jewish law on this topic? So for example Rav Nachman says that he is like Shapur Malka. Meaning, you asked whether the judge is a legislator or a judge? Rav Nachman says: I’m a legislator. Yes, I’m like the king; I can do whatever I want when I sit as judge. Meaning Rav Nachman did things that from a legal perspective are unimaginable. He imposed fines and established laws after the person had already committed the offense. A person did something, and then Rav Nachman said: I decide that this will be the Jewish law—even after, after he had already done it. What? Even though he had not been warned beforehand. Yes, it wasn’t a law before then. He created a retroactive law that obligated the person in some halakhic requirement. Anyone interested in this topic—there is a Jew who dedicated his life to this issue, his name is Hanina Ben Menachem. Admittedly he isn’t from here, he’s from Jerusalem, but one may read him too. An observant Jew, a God-fearing Jew, and he dedicated his whole life to this question: the Amoraim, the decisors in the time of the Talmud, as people who created laws on the fly. Yes. So the question we will stand on right now is: what could be the justification for a non-positivist conception? Assuming there is such an option at all, still, seemingly there is no justification for it. Even if there is such a possibility, there is no justification for it. Good. So what you suggested earlier perhaps—or at least what I understood from it—that is a justification of necessity, what jurists call a necessity justification. Meaning, there is no choice. I don’t always succeed in deriving by deductive means the answer to the question before me, and I must make expansions. You can’t, every time a case comes before me, run back to the Knesset and tell them, “Friends, legislate the law accordingly so you can tell me what I need to do here,” because that doesn’t work. Here too the question is how broadly you expand your considerations. The fact is that in court one judge decides this way and another goes the opposite way. The question is what system of considerations each one brought into the data when he comes to his ruling; that’s the question of the one who asks. Right, right. Going to the legislator won’t help either because again that leaves us dependent on people if the law is legislated retroactively. No, no, but it won’t solve the present problem. Right, we always fight yesterday’s wars, as they say, or prepare for yesterday’s wars. So the first argument is that it’s impossible. Meaning, it’s not practical. Even if in principle we ought to do it, it isn’t practical. Is that basically the lacuna argument? That judging here… No, a lacuna is a substantive claim. A lacuna means there is no answer in the law. Here I’m saying there may be an answer; I don’t know it, I can’t find it, I don’t always succeed. Yes? Sometimes there really is a substantive claim, there is a lacuna in the law, and again, technically I can’t keep going back to the legislator all the time, and it’s retroactive even if I do, as you rightly noted, and therefore this may perhaps still join the issue of lack of ability. But if we move to Jewish law rather than secular law, then the claim is that there is no such thing as something outside Jewish law; there is no such creature. Since the Holy One, blessed be He, legislated, we assume all the ways… What? “It is not in heaven,” “you shall not add.” Yes, so everything—everything is there, so that claim simply will not stand. But it’s not defined—not defined. It may be there but not defined in the language in Orach Chayim. Not defined because you didn’t work hard enough. No, there’s another solution here, but I’m just about to say it also in the legal context. When people say “it’s impossible,” that’s not quite precise. What does “impossible” mean? The moment you didn’t reach a deductive conclusion from the laws, there is no ruling. Meaning: sit and do nothing, or the court disperses—I have nothing to say. The law says nothing about this. In principle one could also act that way here. The “impossible” itself also assumes something. When I say “it’s impossible,” I’m really saying there had to be some statement here that is derived from the law, but I don’t find such a statement. I don’t succeed in deriving such a statement from the law, and therefore it’s impossible. So I need to do some kind of quasi-legislative act. But even here, behind that too there is some non-trivial statement. Because it really means I’m already assuming that the law is supposed to speak also about this situation. Also in the halakhic context, there was room to say: true, the Holy One, blessed be He, cannot add laws—what does “cannot” mean? He established that He won’t add laws, “it is not in heaven.” Very well. So whatever I cannot derive deductively from the laws is not forbidden, or not obligatory. Meaning there is no problem. So the halakhic legal system doesn’t speak about it. That too is a way out. Meaning it’s not that if the Holy One, blessed be He, doesn’t speak, then there’s no choice and I have to do the job. No. You can restrict yourself to what was said. And whatever wasn’t said—there is no issue, there is no halakhic norm that applies to it. That’s all. It also depends on the question whether, for example, “might makes right” is a withdrawal of the court or a decision. Right, but as a result—I don’t know if it will depend. No, this is a particular thing within the system. If that is indeed how the matter is understood, then “might makes right” can be an example of it. But the Ran in his homilies, who speaks about the possibility that the Holy One, blessed be He, perhaps did leave… the assumption was that there is an answer to everything. No, the point is: then the king’s justice is a mechanism to produce answers to questions whose solutions you do not find within Jewish law. Yes, but non-halakhic answers, exactly. It’s a somewhat parallel system. But it exists as a halakhic system. Its role is to seal the breaches that essentially exist in Jewish law. On the contrary, there you can see that the Ran really understands that there are essentially breaches in Jewish law. On the contrary, from the Ran the opposite emerges. From the Ran it emerges that what Jewish law did not say—there is no Jewish law. And precisely because of that—but after all you can’t live like that. You said fine, so for that there is the king’s justice to close the gaps, which is also Jewish law. Yes, no—but it’s a parallel system. Meaning, if one could close those lacunae with halakhic tools, there would be no need to create the king’s justice. Then the court would do it. The court would do it as part of its ordinary interpretation. When the Ran creates a parallel system here, he is basically assuming that the halakhic system that you see by itself is by definition full of lacunae. It does not speak about some things. There are things it does not speak about. That’s not exactly what the Ran says there; we’ll talk about that later. Okay. So those are the accepted justifications. Justifications of necessity—necessity with a kaf, of course. Sometimes there are also justifications of power—with a kuf—the kind used to accuse legal imperialists, that these are justifications of power with a kuf. The point I want to focus on here, because I think it’s the important one for what follows, is not that, but rather a substantive justification. The claim that not only is it impossible to derive deductively from the laws the outcomes, but one need not. Very often the outcomes emerge from the laws by non-deductive tools, and still in that way I am uncovering the will of God. Only the tool by means of which I do it is not a deductive tool. It is not a logical tool, but analogy or intuition, or you can call it by many names, it doesn’t matter, but softer tools, you might call them. Not rigid logical tools. And the claim is that when the Holy One, blessed be He, encoded, in a certain sense, all His conclusions—all the conclusions He wanted us to reach—within the laws He gave us, this is encoded in such a way that extracting the things from the laws cannot be done only with logical tools, but also with softer tools, more creative tools, broader tools. I’ll define them a bit more later. Now this justification is a different justification from the previous one. It’s important to understand that. And this is our topic. Our topic is this last thing. Everything I said until now was only the framework. Because the claim here is really that in essence it is impossible to derive the conclusions deductively from the laws even though they are there. Meaning, although when I derive those conclusions I claim that this reveals the will of God. Not that I’m inventing new laws now—that I too am the Holy One, blessed be He, that I too can legislate. I’ll sharpen that a bit. The intention is: even if I can. Meaning, let’s assume I can reach a conclusion that is very precise logically. I know it. I did the work and I can reach a conclusion. I am not obligated to cling to it. I can say something else. Right? That’s the idea. There are many examples of this, even conceptual examples before we enter cases. The “fifth section” of the Shulchan Arukh, the fifth part of the Shulchan Arukh, of the Chazon Ish and things like that. Seemingly, sometimes it’s a lacuna, and the four sections say nothing, so you need to complete it from your own authority. Sometimes in extreme cases it’s not a lacuna, but rather what Rabbi Shabtai said before: that from the four sections of the Shulchan Arukh one answer emerges, but the decisor in his feeling understands that what is required is something else. I’ll tell you a story from the Talmud. A story in Bava Kamma… sorry, Ketubot. There was a person who stole an ox. And returned it, but in the meantime he had plowed with that ox. So now when he returned it—he returned the ox—then of course he recited the blessing, “Who sanctified us with His commandments and commanded us to restore stolen property,” and he restored the stolen property—he’s a great righteous man, because it is a Torah-level positive commandment: “and he shall restore the theft that he stole.” Rav Nachman said to him: wait a second, but you plowed the field—pay the value of the improvement, however much the field improved. So the man said to him: wait, but what do you mean? Did the ox improve the field? I worked with the ox. So no, fine, half. Pay half the value of the improvement in the field. Rava said to him: but it says all robbers pay according to the time of the robbery—that is clear Jewish law. There is no lacuna here and no nothing and no nonsense. It is completely clear: if a robber profited from the robbery, what is the law? He does not pay. The Torah gave the robber the possibility of profiting from the robbery. Why? That needs discussion, but it is completely clear. Rav Nachman said to Rava: first of all, I told you not to ask me questions when I’m sitting in judgment. That’s the first thing. The second thing, he said to him: this man is a veteran robber, a known robber, and this is what he does all the time. He doesn’t want to buy an ox, so what does he do? He steals an ox, works with it, returns it, and also fulfills a commandment. So I fine him so that he won’t do it. There, you see—something like that, there is no such fine, there is no such thing in Jewish law. Now the point is: what did Rav Nachman say? The Jewish law written there, clearly written there—it doesn’t require much logic—it’s very simple: all robbers pay according to the time of the robbery. In this case it is not right; it will not fulfill its role. Let’s say that’s like the Ran regarding the king’s justice. I’m not sure whether this is with or without a lacuna, but I’m now going to note what I noted about the Ran—the same thing. Because you have to notice here: Rav Nachman, Rav Nachman, Rav Nachman—two entirely different people. Rav Nachman did not say here that this is his interpretation of the Torah law of “and he shall restore the theft.” I’m not sure. He established here some kind of rabbinic fine, he didn’t… Yes, but I think he said that “and he shall restore the theft that he stole” will not realize itself in the form of the value at the time of the robbery, and therefore one must add the improvement so that he restores the… So that’s like what I noted about the Ran with the king’s justice. It’s a parallel system. I’m speaking about something more radical. I’m talking about a situation where Rav Nachman sits in judgment. No, I understood what you’re saying, but I think then it would be a great injustice, because the law is determined after the fact. Fine, sometimes there will be a problem of retroactive legislation. But not even retroactively—rather going forward. What does “and he shall restore the theft that he stole” mean? “And he shall restore the theft that he stole” is flexible in the way you described; it is something open to more flexible interpretation. “And he shall restore the theft that he stole”—after all, you derived benefit from the ox; pay also for the benefit. It may be that that is what Rav Nachman is saying, and therefore this is not retroactive legislation, but rather that sometimes that is how one should interpret what is written in the Torah. Okay, so that will depend on how one interprets what Rav Nachman did. Either Rav Nachman interpreted “and he shall restore the theft,” and then in fact one could perhaps say that this became binding Jewish law. And this could now enter the Shulchan Arukh. And this is what Maimonides writes. If there is a habitual robber, then this is the law of “and he shall restore the theft,” including the improvement. That is what Maimonides writes. Okay, Torah-level law. Torah-level law. And one could say that Rav Nachman—this seems to me the simpler interpretation there, I don’t know, but at least I think so—Rav Nachman simply did what it is the role of the Sages to do, or what it was the role of the king to do when there was a king. Because a court may flog and punish not according to the formal law. Essentially this is some sort of substitute for the king—they function in place of the king. When there was no king, then essentially the court took that authority. But then you are acting within an alternative framework; you are not acting by interpretation of Torah law. Now I am speaking precisely in the substantive sense. I’m talking about someone who comes and says: this is “and he shall restore the theft,” not that he is adding something else. Add something else, fine—the Torah gave authority to the Sages to solve problems, or to see when the Torah-level law doesn’t work. I’m not speaking about that. I’m speaking about a situation in which the Sages interpret Torah law in a way that is seemingly against the rules, but it is clear to them that in such situations this is what ought to be. This is the Torah law. It says “an eye for an eye”—money. It depends how you understand the meaning of that exposition, but at least one could say that here the Sages understood that despite what is written, “an eye for an eye”—no, that cannot be right; we say it means money. Fine, there is also an a fortiori comparison there, it doesn’t matter. But on the principled level I mean this sort of thing, where the Sages make a non-deductive interpretation of things because it is clear to them that this is really the intention. Not that they are closing a gap by means of the alternative mechanism of rabbinic enactments or the king’s justice or things like that, but that the Torah-level law itself is interpreted seemingly outside the laws. Meaning, what is the difference between the two things? A huge difference. If I say this is a rabbinic fine, an alternative system, any appointed judge can do this. There is no problem here; it does not derive from some prophetic-intuitive greatness. Who is permitted to make such a conclusion? Any appointed judge can impose such fines right and left. If I say this is interpretation—and a new interpretation in a Torah-level matter—then here the judge has to decide about himself that I am capable, and permitted, and required, and obligated to go and give a different interpretation to what is written in the Torah than what is accepted, than what is written in the Mishnah, than what is written in the Talmud. As Rabbi Moshe said earlier, if it is retroactive, it reveals that this was the law. No, no, again—if it is retroactive revelation then it is not retroactive legislation. But the point is, the point is, that there must be someone who allows himself to do that. So it seems to me that this is the end of the focusing—that is where I wanted to arrive in these introductory remarks. Meaning, basically, the question in brief is this: if I gather everything we said, then the question is whether, assuming that a person can think not only like a machine, there is still room to ask whether as a judge or as an interpreter he should nevertheless restrict himself to mechanical thinking alone. Seemingly the answer is yes, because that is his authority—he cannot legislate—but in practice, as you rightly remarked, that doesn’t happen anywhere. It’s an illusion. Even those who think it happens in their case—it doesn’t happen there either. It’s an illusion. Later we’ll see that it’s more a question of degree than a question of essence, but we’ll discuss that later. But it does come to expression in the authority—the question of authority, yes. He says: I cannot do such a thing; I’m not the kind of person who can do that. Yes, but that too is still a matter of degrees, because there are things that even they will do, either because they didn’t notice or because they won’t see it as enough of an innovation, so they’ll say, fine, this is interpretation. But it is still not deductive interpretation. I think that in the end it’s a question of degrees. There is no situation in which you are only a pure logician—it simply does not happen. Work that is purely logical, nothing but pure logic—I don’t think there is such a halakhic claim at all. Meaning, even in a straightforward monetary case it isn’t like that. We may discuss that later. Rabbi, there are judges suspected of deciding before they examine. I don’t know if “suspected” is the right word. They decide first. Rabbi Soloveitchik says there has never been a decisor who didn’t do that. What? Who didn’t decide before he examined. Didn’t decide before he examined the Jewish law, you mean? Yes, meaning before he checked the sources and saw the proofs. Rather, he understood what was right, and then he checks the proofs and backs it up with arguments or sources or things like that. That is not something disqualified—it’s a question of… But that isn’t connected to the positivist position; even a person who says that would want exactly—right, the question is how he justifies it. Even retroactively he can justify it logically. But especially a person you don’t like—his behavior, his wickedness—and then you say: he’s wicked, you can’t trust him. Those are dangers. Dangers—that’s a second-order question. After you say that we do not work positivistically, you can say, okay, now we have to be careful, but what considerations do we take into account beyond logic? Obviously I must not factor in my sympathy for one litigant, and one has to be careful because sometimes that creeps in without my noticing—you’re right. But that’s a second-order question. I’m speaking now about what is right to do; afterward we’ll see what one has to be careful about, okay? I want to raise a point. For example, the Torah says, “And you shall not take a bribe, for the bribe blinds the eyes of the wise.” So what is the plain meaning of the Torah? That if a person takes a bribe it will influence his judgment. I ask: if a person really were ruling in almost pure logic, could bribery influence him? Let’s assume—you know better than I do, if I bring another practical difference—if the ruling is pure deduction, then it can be subjected to review, so clearly his fellow judges will say to him, look, there’s a mistake here: you said two plus two is five. That’s not right; two plus two is four. Now the question is this: in such a state, where judges are machines—if we say there is an ideal state where judges are machines—would bribery be dangerous? The question is, what, are you going to bribe a computer? What would that help? He might add perhaps… What is the meaning of following the majority? The Torah says—and we need to look at this—that it works. That’s a sign, a sign that thought is not… And from here there is proof that we are not dealing with logic. Yes, I would add to that: what is the meaning of “follow the majority,” “incline after the many”? If every situation has an answer that can be derived logically from the assumptions, then there would be no majority and minority. Every reasonable judge would reach the same conclusion, just as a computer reaches a conclusion. Not necessarily; you could say: this is the tradition I received, and he says this is the tradition I received, and he says this is the tradition I received. But then, right, with bribery too you could say, look, the assumptions are different because… But basically you formed different assumptions because of the bribe, and we’ll discuss that later, because there really is no such thing as a logical ruling. It only appears that way. There is no such concept. It’s not only that it is not right to do—it doesn’t exist. It’s an illusion. Someone who thinks he’s doing that is mistaken. He is not doing that—not mistaken because it is forbidden, but how does he imagine… Yes, that’s why I’m saying—what do you mean? But in the puzzle method, how does he work in the puzzle method? In the puzzle method he wants to decide in advance. He searches for a certain responsum, exactly, he finds a similar case, and says: here, this is it. No—if you take Rabbi Soloveitchik, then if you interpret him in a positivist way, then that’s what is written. But understand that in that case this is not ruling. It is written in Jewish law that a point of law without its rationale may even be stated in the restroom. They ask a decisor who is sitting in the bath on Sabbath eve a question of Jewish law, and if he doesn’t need to explain the reason, then he can answer even in the bath. Now, that is a sign that if I say “this is the Jewish law because it is written in such-and-such responsum,” that is not Torah. It is not Torah, it is not a legal ruling, it is nothing. What is it? Merely a quotation. Exactly. A ruling is when you have innovated something. You have created something. The Shulchan Arukh writes this quite a few times, that a ruling is only something new. There are also sayings like “these and those are the words of the living God,” which also shows that this is not a process of pure logic. Otherwise there would be no room for both. Right, at least on the practical level—the fact is that it happens, that there are different opinions. That does not mean that in principle it is correct. Practically it happens, that there are differing views. The prohibition of taking a bribe can also be explained by saying that in principle we expect deductive rulings, but we are aware that human beings are not perfect and therefore we establish safeguards, and assume that one who rules imperfectly will at least not be biased. So you are assuming that the Torah speaks about non-ideal situations. That is an interpretive assumption. It could be that the Torah does not speak… No, “these and those” is not written in the Torah. “These and those” is the words of the Sages. Here this is the Torah—the issue of bribery. But the question is how the Torah relates also to non-ideal situations. A person is not supposed to steal, and if he stole then he must restore. And if one needs to rule perfectly but he is human and he cannot manage it… The Tur in section 2 explains—section 1 and 2—that for example the Jewish law that a robber returns the stolen object and no more—after all, that doesn’t deter anyone. Because a bank robber—suppose they catch him half the time, which is a lot—then he will make a huge profit because he returns half and the other half stays with him. So the Tur says—really this is Rabbeinu Yonah—that the Torah that said one must return it is speaking about an ideal society, where robbery is rare and a person who robbed stumbled and already regrets it and all that, so the punishment of “and he shall restore the theft that he stole” is a reasonable punishment. So regarding interpretation of the Torah—whether the Torah speaks about the practical world or whether the Torah speaks about some ideal—that is a question. Meaning, failures happen even in an ideal world, but the way one addresses how to correct them is an enormous difference if one has to weigh deterrence or not weigh deterrence. So therefore the question of bribery is indeed a question of appointing excellent judges. To the excellent judges the Torah says, “And you shall not take a bribe.” Or as the Chiddushei HaRim says, it says “You shall not steal.” So he says in Yiddish, when speaking in the plural that’s for a very important person—“you,” honorifically. So he says: important person, don’t steal. You are righteous, the Gerer Rebbe, the known upright man—you too need “do not steal.” Okay, now I want to sharpen the point even more. I’ll give a short logical introduction in order to put things into a somewhat tighter, sharper framework. So in order to define the matter, I want to speak a bit about what a logical argument is. What exactly is a logical argument, because this touches the very core of positivism. What is positivism and what is a deviation from positivism. Well, the main—or first main—conceptualization of logic was done in Aristotle’s Organon. And there he presents several patterns of inferences, syllogisms. Meaning, several general forms of inference. If every X is Y and A is X, then A is Y. If every X is Y and A is not Y, then A is not X. Or if X then Y, and X, conclusion Y. Okay? Meaning, there are various forms of inference, and he classified them there; he presented different forms of inference and more or less showed the main lines of deductive logic. Obviously Aristotle did not invent logic; Aristotle conceptualized it. Meaning, even before Aristotle everyone understood that if all frogs are green and this thing is a frog, then it is green. Everyone understood that. What Aristotle contributed here is that behind such arguments that everyone uses there sits some formal pattern of logic. So what does that actually mean? It basically means that if I now construct a logical argument, make a logical inference, then I posit premises and derive from them some conclusion. A valid deductive logical argument means that the conclusion necessarily follows from the premises. Or in another formulation: it is impossible to accept the premises and not accept the conclusion. Okay? If you accepted the premises, you are forced—logically forced—to accept the conclusion. I’m sorry, yes, I wanted to comment from another direction. Where does this thing come from? I am forced. I accepted the premises, I am forced to accept the conclusion. So there is a fascinating field—I don’t know whether you know of it—evolutionary psychology. Meaning claims that certain things are imprinted in our brains, things about which you can’t really say true or false; rather, they simply came in order that we survive—they helped the human species survive. And one can definitely argue that there is no substantive truth in these rules at all. Rather, what is there? What stands behind them? Life experience. You say, look, the thing that preys on me runs fast and runs faster than I do and is stronger than I am, so that means something regarding the need to defend myself. Then when I say that something devoured a person, it was bigger and stronger than I am, therefore what do I need to do? Band together—people join together, go out with weapons. It’s simply that whoever thought that way survived. But that says nothing about truth. More than that: there are things for which the premises that I know because I’m talking about large things—animals, wolves, bears and so on—are not true when I speak about particles. Because we are not experienced in that perception and not accustomed to it. And therefore it may be that logical perception has no inner, substantive truth. When I say “I am compelled”—I accepted premises, I am compelled to accept conclusions—the question is whether that is really logical compulsion or psychological compulsion. Right, psychological compulsion. On this matter I completely disagree with that view, but that’s a debate that would have to be expanded. Yes. Okay. In any event, let’s take for example a logical argument. And it’s interesting in certain respects because it will sharpen the difference between truth and validity. Say: all frogs have wings. First premise. Second premise: this thing is a frog. Therefore this thing has wings. This argument is a valid argument, right? You cannot accept the premises and not accept the conclusion. Okay? But on the other hand, both premises and the conclusion are false claims. Frogs do not— not all frogs have wings. This thing is not a frog, and this thing also has no wings. Meaning this is a valid logical argument, both its premises are false, and its conclusion is also false, even though the argument is valid. What does that mean? It means that the concept of validity and the concept of truth barely speak to one another. Validity is a relation between claims. If the conclusion follows from the premises, that’s valid. If the conclusion doesn’t follow from the premises, that’s invalid. Invalid doesn’t mean untrue; it only means it doesn’t necessarily follow from the premises. Okay? But these are relations between claims. The conclusion is a claim and the premises are claims. Relations between claims are measured in terms of valid or invalid. True or false are properties of claims, not of relations between claims. When I want to know whether a certain claim is true, I need to compare the content of the claim with the state of affairs in the world. If I want to ask whether this thing is a frog, or whether this thing has wings—I say: this thing has wings. Claim. In order to know whether it is true or not, what do I need to do? Look and see whether there are wings or not. If there are no wings, that means the content of the claim does not match the state of affairs in the world, and that is a false claim. If it does match the state of affairs in the world, that is a true claim. Meaning, truth or falsity characterizes claims, not arguments. And in order to know whether a claim is true or false, usually that’s a matter of observation. You need to look and see whether it is a true claim or a false claim. Factual claims. Obviously one could argue about what counts as wings. Fine, okay. But then it’s really a question of definition and not of truth. The whole barrier—I mean to say—is a barrier that also exists in the premise. Yes. Fine. Right. So the question of truth or falsity, first, characterizes claims and not arguments; second, it is measured by comparison, observation and comparison. I look at the state of affairs, compare it to the content of the claim, and see whether it fits. If it fits, the claim is true; if it doesn’t fit, the claim is false. Validity and invalidity are properties of arguments, not of claims. Of a conclusion’s following from premises. If it follows necessarily from the premises, the argument is valid; if it does not follow, the argument is invalid. Yes? Another thing: the statement that an argument is valid or invalid is not the result of comparison—at least not in the simple sense. I’m not comparing it to something else. I’m not asserting something; an argument asserts nothing. An argument says: if this is true, then that is true. I did not say this is true and I did not say that is true. A valid argument can also have false premises, as we saw before, and also a false conclusion. I’m not asserting anything—not the premises and not the conclusion. I’m only saying that if you accept the premises, you must also accept the conclusion. It’s a kind of if-then; it’s something conditional. So how do I know that an argument is valid? I simply know because this logic—now this is the earlier debate—whether from evolution or because that’s really how it is, one can argue. But I know it not because of observation and not because of comparison. I simply know because that’s how it is. Logic is imprinted in us in some sense, if you want to put it that way. Okay, so that’s the difference between truth and falsity on the one hand, and invalidity and validity on the other. What does this actually mean? It effectively means that if I know the truth and falsity of the arguments—of the claims, sorry—that are involved in an argument, that still doesn’t tell me whether the argument is valid or invalid. And vice versa: if I know that an argument is valid or invalid, that still doesn’t necessarily mean that I know something about the truth or falsity of the claims involved in it. Okay. There could be, for example, a reverse case from what I mentioned before: an argument all of whose claims—the premises and the conclusion—are true, but the argument is invalid. This thing is black, and this thing is white, therefore it is a cellphone—no, sorry, therefore it is crumpled. Because that is a true claim, that it is crumpled, and the premise is true—it is white—and the second premise is also true—it is black—and the conclusion that it is crumpled is also true. But the conclusion doesn’t follow from the premises. So that’s independence in the opposite direction. Meaning, the truth of claims does not mean the argument is valid, just as the falsity of claims does not mean the argument is invalid. Okay. Seemingly these are two independent planes. The truth and falsity of claims, and the invalidity and validity of arguments, are independent planes; they don’t speak to one another. So why do we deal with validity and invalidity of arguments at all? Why is that interesting? It’s interesting because nonetheless there is a connection. You could say it’s intellectually interesting but not important for understanding the world or knowing the truth and falsity of claims—but no, it is important. It is important because there is one connection between truth and falsity and validity and invalidity, only one: if the two premises are true and the argument is valid, then the conclusion must be true. There cannot be a valid argument whose premises are true and whose conclusion is false. This “cannot be” is internal to the rules of argument. Yes, again, whether evolution or not, it is not observation and not… It is inherent, it’s from within the argument. Yes, because truth is like “it has wings,” meaning the property of being true is like the property of having wings, of an argument. Yes, if you assume that looking at an argument with the eyes of the intellect is like looking at something and seeing whether it has glasses, whether it has wings. Yes—well, it’s a question of how one defines observation, yes. That’s Platonism in mathematics—whether I can observe patterns of argument. So that connection is really the focal point, because in practice the whole importance of validity and invalidity of arguments comes from the fact that by means of them I can prove something to you about claims. How? I take two premises or several premises that you accept as true, I build a valid argument from which I derive the conclusion, and there—I have forced you to accept the conclusion. That is no longer evolution. Meaning, if I make claims about gravity between stars, then the sun listens to that even though it did not undergo evolution. No, evolution says that you will see the sun as listening to it because you are built by evolution. What the sun itself does—I don’t know; I know what I see in the sun. That’s bias, yes, the Kantian conception seemingly. There’s a place here that is not in the sun. Why do I want to define these points? Because this is, notice what we’re really doing here. The role of arguments is to derive conclusions from premises, when my ultimate goal is to know whether the conclusion is true or false—that’s my goal, that’s what I want. Playing with patterns of arguments, whether they’re valid or invalid, is good for logicians, researchers in logic—that’s their amusement. But for people who work in other fields, what matters about arguments is that… because it is a means to get to the truth and falsity of claims. How? I take premises that are true, construct a valid argument, derive by means of it a conclusion. If you accept the premises, you are forced by reason to accept the conclusion. Okay, I’ve convinced you of the conclusion. That’s what is called a proof. A proof is to present a valid logical argument based on true premises—or at least premises that my interlocutor assumes are true—and by that means arrive at the conclusion I want to persuade him of. That is called a proof. Now of course there is another possibility that appears here, and this happens a lot to anyone with a bit of experience in argument. Say I want to convince someone of the conclusion that this phone has wings. Fine? So I say to him: look, after all, all black objects have wings. You agree, right? And this thing is black. Therefore this phone has wings. So he says, wait, that can’t be, after all this phone has no wings. That’s what he says to himself, my interlocutor. And then he rechecks his premises. He says, wait, no, no. Not all black things have wings. I retract; I don’t agree to that premise. Okay? So the logical argument is valid. He isn’t challenging the validity of the argument. He says: I don’t accept the premises. Sometimes I thought I accepted them, but since the result was so implausible in my eyes, I reexamine the premises I thought I accepted, and I change my position. Those premises are not true in my opinion, and then I am not forced to accept the conclusion. Okay? This is actually an expression of what is called Mill’s challenge to deduction. The challenge of John Stuart Mill, yes—a philosopher from the eighteenth century I think, or something like that—who challenged the concept of deduction. And people think deduction is necessarily true. A deductive argument, yes? A valid logical argument is necessarily correct. And what does “necessarily correct” mean? It is necessarily correct—the conclusion is necessarily correct if the premises are correct. For example: all human beings are mortal, Socrates is a human being—those are two premises; conclusion: Socrates is mortal. That is the generic argument always used to illustrate a logical argument. Conclusion: Socrates is mortal. So John Stuart Mill says: you can’t convince anyone with that that Socrates is mortal. Why? Because you have to assume the premises. Now convince him of the premises. How did you arrive at the premise that all human beings are mortal? How did you arrive at that? Apparently by generalization. You saw human beings. All those you saw so far were mortal. You inferred the general conclusion that all human beings are mortal. Right? So it turns out that in order to arrive at the conclusion that Socrates is mortal—you begged the question. Exactly. Or you covertly used a non-deductive logical mechanism, by making a generalization. Generalization is not deduction. Or perhaps, if I formulate this now more generally, in logic people distinguish among three patterns or mechanisms of argument. Deduction is moving from the general to the particular. All human beings are mortal, so in particular Socrates, who is one of them, is mortal. That is moving from the general to the particular. That is necessarily true. The “if-then” is necessarily true, or the conclusion. Yes? There is movement from the particular to the general. Socrates is mortal, therefore all human beings are mortal. That is called induction. Yes? Or generalization. This is not mathematical induction, of course, but non-mathematical induction—scientific induction or non-mathematical generalization. Mathematical induction is something else. And the third pattern is a move that remains at the same level of integration. Meaning between particular and particular, or between general and general. If all donkeys have four legs, then all horses also have four legs. That is analogy. It is not… What? That’s analogy. Okay? Analogy means this is similar to that, so if that has four legs then apparently this also has four legs. I deliberately took an analogy between two generalities and not between two particulars in order to show that analogy is not from particular to particular. Analogy is simply staying at the same level of integration. If it is from a particular, then to a particular; if from a general, then to a general. Generalization is moving from the particular to the general, deduction is from the general to the particular, and analogy is remaining at the same level. Perhaps one could make a parallel between comparables? Yes? So that’s comparison, generalization, and specification. Deduction is specification; it’s moving from the general to the particular. Now deduction is an argument that is necessarily correct. As we said earlier, this is basically a valid logical argument. The conclusion follows necessarily from the premises. Analogy and induction do not. Right? There is in them a measure of speculation. The fact that donkeys have four legs—who says that means horses have four legs too? They are similar or they are not similar in that respect. It is not a necessary argument. It has a certain probability; we use it and we are not always wrong when we use it. Science is built on arguments of this kind. But it is not necessary. Science is not mathematics. Induction too belongs to the branch of science, not to the branch of mathematics. Yes? I took some examples and inferred a conclusion about the general category containing those examples. Here there is Popper’s classic idea that you refute it with one example, and never can prove—prove it. Yes, right. Induction cannot be proven. If it could be proven, it would be deduction, not induction. But on the other hand, analogy and induction are mechanisms that are not necessary, not certain. The conclusion does not necessarily follow from the premises. Deduction is an argument in which the conclusion necessarily follows from the premises. But on the other hand, analogy and induction are the cases in which one actually needs logic for them, because to say that if all human beings are mortal and Socrates is mortal—one doesn’t need the “then”; we already said it within the “if.” To that I’ll come in just a moment. I don’t agree with your first formulation, but with what you meant I do agree. So John Stuart Mill, returning to his challenge, basically says this: let’s take deduction. If all human beings are mortal and Socrates is a human being, then Socrates is mortal. Fine. You think you proved to me that Socrates is mortal? How do you know that all human beings are mortal? Explain to me how you know your premise. Apparently this came from induction, right? You saw several human beings, saw that they were mortal, made a generalization—apparently all human beings are mortal. So what comes out? That the conclusion “Socrates is mortal” is composed of two logical steps, not one. The first is a generalization that leads me to the premise that all human beings are mortal. Now this serves as the conclusion: all human beings are mortal; now this conclusion serves as a premise for the second step. The second step is a deductive step: from the general I go to the particular and say all human beings are mortal, so in particular Socrates is also mortal. But if so, then in order to arrive at the conclusion that Socrates is mortal I didn’t proceed only by means that are necessary or certain. Hidden here was a step that was not deductive; it was speculative. There is a measure of speculation in it. Okay, and basically if I formulate this more broadly now, I would say this: in fact we make only analogies. We don’t make deductions and we don’t make inductions—we make only analogies. It’s just that the way to make an analogy can be broken down into two steps. I saw several human beings, saw that they were mortal; from this I infer that Socrates too, who is a human being, is mortal. That’s an analogy, right? From the human beings I saw to Socrates. How do I make that analogy? I take the several human beings that I saw who were mortal, make a generalization, and say: if so, then all human beings are mortal—that’s a generalization—and then I say, good, if all human beings are mortal then in particular Socrates is also mortal. Meaning, the analogy is done in two steps: the first step is induction, which creates what is called the major premise of the argument—“all human beings are mortal”—and then I make a step of specification and say: if all human beings are mortal, then in particular Socrates. So induction followed by deduction is really a breakdown of a process of analogy. And therefore in fact what I’m saying—this is almost a jump to the end of the series—is that no argument of ours is really actually necessary, or no conclusion of ours can really be absolutely necessary and certain. Since it always stems from some premises, and you can always ask where those premises came from. Premises from the Holy One, blessed be He, usually do not arrive in the form in which you use them. When you conceptualize and define them—every positive commandment bound by time, women are exempt. How do you know? It is not written in the Torah. In the Torah there are examples, and the Talmud now infers conclusions from the examples and creates a rule, but the rule is not written in the Torah. You made a generalization and then a specification, and indeed there are exceptions to this rule. The Talmud lists them in Kiddushin. Okay? So therefore always, even though you think you are a positivist—and here I’m returning to everything I said at the beginning—you think you are doing pure logical work, but generally you are living in an illusion. You conceal the fundamental parts in which there are in fact non-deductive elements. So let me just close the circle and say: what would a true positivist do? He would apparently say: whatever a certain decisor wrote is true—that I assume. Why is it true? That’s the assumption, that whatever Maimonides wrote, say, is true. And Maimonides wrote such-and-such, so that thing is true. So then one must rule such-and-such. One must rule all kinds of things—for example, that twilight ends three-quarters of a mil after sunset and not, like Rabbeinu Tam, four mil after sunset. Because everything Maimonides wrote is true, and Maimonides wrote about twilight, so I accept his opinion against Rabbeinu Tam. This is something spotless. It really is something spotless, because there the matter is written explicitly: the premise is that everything Maimonides wrote is true, and that is what I assume—not because I generalized it and not from anything else. It’s just such a premise. Let’s assume even that the premise is correct, that everything Maimonides writes is true. What is the problem with what I did here? Again, I did not produce a halakhic ruling. Rather, what did I do? I did something that can also be said in the bath. I repeated something; I quoted what Maimonides writes. I did nothing here. So this act is empty of content. Meaning, a positivist ruling in the end will be empty of content. No, that is one side of the coin. But at least I think we agree here that in the end we will want to argue that it is also not correct, not only that it is empty of content. And if it is correct, it is empty of content. But that is a very delicate tension. Right. We’ll discuss that later. Maybe I’ll just summarize this point. There is basically—I can put it like this. If I look at the three forms of inference: deduction, induction, and analogy—then it seems to me, what is stronger? What is the strongest of the three, say? Deduction. Deduction is the strongest, right? And between analogy and induction? Induction. Yes? Why? I would say induction is weaker, because it is basically many analogies. Right. Meaning, it basically takes a bigger risk. It is more easily refutable. You are making a stronger claim. You saw several horses that have four legs. Analogy means to assume that the next horse also has four legs. Either you are right or you are wrong. Okay? But if you say that all horses have four legs, then you have reached a much stronger conclusion. It contains much more information than was in the premises from which you started. A more speculative conclusion, taking a greater risk than the weaker conclusion. Yes? So it is more dubious, meaning more speculative. In fact what I now want to argue—well, there are issues here; perhaps I’ll mention this too because we’ll need it later—the picture is not so simple. Because how do I make an induction? A collection of analogies, apparently, right? How do I arrive at the conclusion that if one certain horse has four legs, then all horses have four legs? I make an analogy to this horse and this horse and this horse, and I say it’s clear I would make the same analogy for all horses; therefore all horses have four legs. One could say that induction is really nothing but a collection of analogies. On the other hand, one can also look at it the other way around. I make an analogy between this horse, which has four legs, and I say the other horse also has four legs. Why? Why didn’t you assume that this sheet of paper has four legs? Because you think that the fact that they are both horses is a relevant parameter of similarity. Right? And therefore, if this one has four legs, that one also has four legs. By contrast, the similarity that this horse is white and this paper is also white—that is an irrelevant similarity with respect to the number of legs. So what if they’re both white? That has nothing to do with the number of legs they have. So basically you have covertly assumed that horse-ness is characterized by having four legs. Certainly, if you assumed that there is a property of horse-ness. Yes, right. You hear that one rabbi is a thief and another rabbi is a thief. So the question is whether the fact that they are rabbis is something they have in common. Or that they’re not rabbis… No, let’s assume yes—even suppose yes. No, the point is that you are assuming some property here, and the question is whether it exists. Yes, I’m saying: the similarity—but that’s the question. Their being rabbis—the question is why they really are both rabbis; what defines rabbis? Meaning, perhaps they are similar in irrelevant parameters. Then the claim is essentially that the relation between analogy and induction is not as simple as I said before, because one can see induction as a collection of analogies and one can see analogy as hidden induction. In fact, when you compare this horse to that horse, you are really assuming something about all horses; after all there is nothing special about this horse specifically. You are saying: since they are both horses, if this one has four legs, that one also has four legs. So in effect you have covertly said something about all horses. The relation between analogy and induction is a two-way relation. So say that they’re really the same thing. What? Say that they’re really the same thing. Yes, perhaps a collection of analogies. Yes, right. And then the question of which is stronger is no longer so simple. But what I really want to say is this, and this is the philosophical principle of certainty. The philosophical principle of certainty basically says, it seems to me, that the level of certainty multiplied by the amount of information in an argument is constant. Meaning, if the argument is completely certain, it contains no—adds no information at all. There is no innovation there. Right, it cannot add information. Therefore deduction—why is deduction certain? Why in deduction, if I accept the premises, must I necessarily also accept the conclusion? If Socrates—if all human beings are mortal and Socrates is a human being, the conclusion is necessarily that Socrates is mortal. One cannot disagree with that. Why can’t one disagree with that? Because the fact—the conclusion that Socrates is mortal—is hidden within the premises. After all, if you said that all human beings are mortal, then in particular you already said that Socrates is mortal. So in fact you begged the question. You did not add any information in the conclusion that wasn’t already in the premises. Precisely because of that it is necessary. And that is why it is necessary that if you accepted the conclusion—implicitly you already accepted the conclusion, sorry, implicitly you already accepted the conclusion. You are only bringing it from potential to actuality through the logical argument. But basically it was already in the premises. This argument added no information. It’s the joke about the hot-air balloon, which you probably already know: two people lost their way in a hot-air balloon. One of them asks someone they see below plowing the field, “Tell us, where are we?” “Above my field.” So one says to the other there in the hot-air balloon: the guy down there must be a mathematician. Why a mathematician? Because what he says is completely precise and certain, and it doesn’t help us at all. Behind this joke there lies a bitter truth—or not bitter, but a truth. Truth, truth. A logical argument never actually helps us at all. That’s why it is certain. It is certain because it adds no information. So by means of a logical argument you will never add information. Because otherwise it wouldn’t be a logical argument. Do you understand what lies behind this? And this is very important in Jewish law. What is called added information? Something about which I am doubtful. Exactly the point. Something that is not certain, something you cannot be sure is correct. And therefore, now analogy and induction—why are these non-necessary arguments? Because the conclusion contains more information than was in the premises. I saw two horses with four legs; I say all horses have four legs. Obviously the statement—the claim that here serves as a conclusion—contains much more information than was in the premises. This is an argument that added me a lot of information, took a risk, was speculative. And because of that it cannot be necessarily correct. Okay? If it were necessarily correct, then there would be no doubt about it and it wouldn’t be speculative. So therefore the amount of information added by the argument—the greater it is, the more exposed the argument is to possibilities of failure. Meaning it is less certain. There is a tradeoff between the degree of certainty of the argument and the amount of information the argument adds for me. If its certainty is full, the information it adds is zero. If it adds a lot of information, then this argument is very dubious, very speculative, not cautious. It adds me a lot of information; I made a very broad generalization. Okay? The game is always how I pay in the coin of certainty in order to gain information. That is basically the game—the scientific game, one might call it. In logic or mathematics one does not play with that. Logic and mathematics are fields in which one does not give up the level of certainty. And because of that, no information is ever added to us in logic and mathematics. Whatever was in the premises—the mathematical or logical field merely exposes what was within the premises. It will never add me—sometimes it is very complicated to expose it; that doesn’t mean one need not be intelligent to be a good mathematician. I assume we agree—not that one need not, but that one does need to be intelligent. But it means that in the end, after you did all the steps, after you went through the entire proof of Andrew Wiles for his proof of Fermat’s theorem, after you went through and saw that indeed each stage follows from the previous one, you basically understand that you knew this all along. It was already in the premises. Nothing new was added for you. The role of mathematicians is to expose or quarry more and more information out of the premises. But they can never tell me whether that information is true or not. They can only tell me whether it follows from the premises. If you ask a mathematician, for instance, what is the sum of the angles in a triangle, he will have to tell you: I don’t know. It depends on your premises. If your premises are Euclidean, the sum of the angles is 180 degrees. If your premises are different, then give them to me and I’ll calculate for you the sum of the angles as you wish. One can even construct a space as you wish. Give me the sum of the angles and I’ll construct for you a space that yields it. Okay? So a responsible mathematician can tell me only the if-then. He can never tell me that a certain proposition is true. He can tell me: if your premises are like this, then the conclusion is like that. Who tells me whether this proposition is true? For that, scientists are responsible. Scientists are responsible for determining whether certain claims are true or not, for accumulating information about the world. Science in Hebrew comes from “knowledge”—yes, information. Science deals with information. Mathematics does not deal with information; mathematics deals with relations between claims. Truth and falsity? What? No. Truth and falsity are actually not the issue there—rather validity and invalidity, relations between claims. Science deals with truth and falsity; mathematics deals with validity and invalidity of arguments. If you have such premises, then this conclusion does or does not follow from them. That is the mathematician’s business. How do people of Jewish law think they escape this? I think it’s worth leaving that for next time. Yes. Because people of Jewish law understand this—you can see it in responsa—and try to escape it. Do you have any thought about a tactic for escaping the problem? Meaning, that something should not be zero-information, and yet should still be sure and clear? One could say that in fact—after all, they don’t have scientific refutation, to come and test and see things. They don’t have that option. One could say that the system of premises is so sophisticated that everything is really in it, and all we do is expose it. We are basically a mathematical system, because we have the positivist conception. Yes. And if we are not positivists—Rabbi meant to ask, assuming the decisor does not work in a positivist way—then what does he predict? But one cannot say that the decisor is some kind of prophet. Really, who says he can arrive at certainty? But who says he can arrive at certainty? Is he sure he can arrive at certainty? It could be that he does not arrive at certainty, but fine—that is halakhic ruling. What seems to you reasonable and right, but not certain. But you always need to know that doubt is on your heels. Because if you are not a positivist—meaning, if you understand that you are not a positivist—then right, then halakhic ruling may not be secure and certain. Because the first person to attack positivist ruling was Nachmanides in his introduction to Milchamot Hashem. I don’t know whether anyone preceded him in this. “For the wisdom of our Torah is not like the wisdom of geometry and arithmetic.” And the interesting question is why. He attacked it, and then what is he really saying? Go more by feeling? The definitions are softer? No. What he wants to say is: go by the authority of the decisor. And that too is another escape from the whole story. But here too I actually have a few things to say about the matter. It’s not only authority—there is, I think—we’ll talk about Nachmanides himself and what he writes there. Nachmanides writes that the Baal HaMaor, who criticized the Rif, had good arguments for criticizing the Rif, logical arguments for criticizing the Rif, and he did not grasp that the Rif was an elder and had tradition and so on. “Ancient matters from the mouth of a fresh elder.” “Ancient matters from the mouth of a fresh elder,” yes. Was the Baal HaMaor young? The younger Baal HaMaor. Good. We’ll continue with this next time. Shalom.