Topics in Talmudic Logic, Lecture 3

Table of Contents

• Logic versus science and methods of inference
• Rationality without proof and soft rules of thought
• Logical interpretive principles versus textual interpretive principles
• Refutations and their connection to the logical principles
• The difference between deductive and non-deductive logic
• Biblical a fortiori reasoning: one premise and a hierarchical relation
• A fortiori reasoning is not deduction, and the proof is from the refutation
• A fortiori reasoning of “what is included in two hundred includes one hundred,” and the dispute about refuting it
• Talmudic a fortiori reasoning: three data points and a 2×2 matrix
• Refutation in Talmudic a fortiori reasoning and rotating the a fortiori argument
• A fortiori reasoning as theory, independent parameters, and abduction
• Occam’s razor as an outgrowth of a fortiori reasoning and presumptions
• A mechanical method for filling in a missing slot in a table
• Independence between parameters and identifying the parameters

Summary

General Overview

The text presents an attempt to build a rational non-deductive logic that does not rest on certainty, מתוך the distinction between deduction as necessary inference and induction, analogy, and abduction as “soft” forms of inference that add information and therefore are not certain. It argues that a rational, non-skeptical outlook must have criteria for uncertain thinking so that it does not become a “shot in the dark,” and it proposes learning those criteria from three logical interpretive principles: a fortiori reasoning, deriving a principle from one verse, and deriving a principle from two verses, together with the refutations and combinations associated with them. It then spells out biblical and Talmudic a fortiori reasoning, explains how refutation shows that the inference is not deductive, and presents an interpretation of a fortiori reasoning as an abductive process of constructing a simple theory that explains data and predicts a missing case, using the principle of Occam’s razor.

Logic versus science and methods of inference

The text states that traditional logic mainly deals with deduction as necessary inference from the general to the particular, whereas science relies mainly on analogy and induction and therefore does not reach absolute certainty. It defines abduction as moving from facts to theory by means of theoretical concepts that are not directly observable, and distinguishes it from induction, even though the two are often conflated. It presents a “principle of logical uncertainty,” according to which a completely certain argument adds no information, while an argument that adds information cannot be completely certain; so the more information increases, the more certainty decreases, and vice versa.

Rationality without proof and soft rules of thought

The text formulates the challenge of a rational approach that is neither fundamentalist nor skeptical as the need for a system of rules for non-deductive thinking that will allow us to speak in terms of probability, more correct and less correct, without requiring proof. It rejects the automatic identification of proof with rationality, and of the absence of proof with mere guessing, and seeks fixed logical patterns that can justify non-certain conclusions. It presents the planned development as a systematic theory of non-deductive logic based on a fortiori reasoning, deriving a principle from one verse, deriving a principle from two verses, the refutations of these, and combinations among them.

Logical interpretive principles versus textual interpretive principles

The text divides the thirteen interpretive principles into two categories and explains that a fortiori reasoning represents logical interpretive principles, whereas verbal analogy represents textual interpretive principles based on a linguistic “trigger” in the verse. It argues that in a fortiori reasoning and in deriving a principle from a verse, there is no need for a hint in the wording of the verse, because the inference comes from logic and analogy, whereas in verbal analogy, general-and-particular, and the like, the basis is the wording of the verses as a convention or a kind of “code.” It adds that juxtaposition and the proximity of passages are textual, because the analogy comes from closeness in the text rather than from similarity in content.

Refutations and their connection to the logical principles

The text argues that a refutation attacks the logic underlying the inference, and therefore it mainly belongs to the logical interpretive principles; usually it does not belong to textual principles, where there is no hierarchical assumption or similarity of subject matter to put to the test. It notes that it found almost only refutations of a fortiori reasoning and of the two types of deriving a principle from verses, and brings an exceptional example in tractate Chullin of a refutation of general-particular-general, while claiming that there it is not a genuine refutation. It presents the rule that “we do not derive punishments from logical inference” mainly in relation to a fortiori reasoning, and raises a discussion of applying it to monetary law and to other interpretive principles through Tosafot, the Mekhilta, the Babylonian Talmud, the Maharsha, and the Kesef Mishneh.

The difference between deductive and non-deductive logic

The text defines deductive logic as inference in which the conclusion follows necessarily from the premises and cannot be refuted, and it presents the example: “All humans are mortal; Socrates is a human; therefore Socrates is mortal.” It defines non-deductive logic as analogies and inductions that are not necessary, even though they are useful in practice; therefore they are “soft methods of inference” in which one can make mistakes.

Biblical a fortiori reasoning: one premise and a hierarchical relation

The text describes biblical a fortiori reasoning as based on one premise and on a hierarchical relation of greater and lesser stringency, and gives examples such as “Behold, the children of Israel have not listened to me, so how will Pharaoh listen to me?” and “If her father had but spit in her face, would she not be shamed for seven days?” It explains that the conclusion is accepted because the law holds in the lighter case and therefore holds in the more severe case, or the reverse when what is at stake is a leniency. It explains that such an a fortiori argument can be refuted by attacking the hierarchy, through a counter-consideration or an example showing that the relation of “more severe / lighter” is not correct.

A fortiori reasoning is not deduction, and the proof is from the refutation

The text rejects the claim that a fortiori reasoning is deductive inference, because it can be refuted, and presents the refutation as proof that the conclusion does not follow necessarily from the premises. It clarifies that the uncertain element is the rule of hierarchy that connects the premise to the conclusion, unlike deductive rules in which accepting the premises compels acceptance of the conclusion. It distinguishes between what happens in practice and the validity of a forecast in advance, using the example of the Entebbe operation and a letter by Rabbi Shach, and argues that success after the fact does not refute an earlier assessment of the odds.

A fortiori reasoning of “what is included in two hundred includes one hundred,” and the dispute about refuting it

The text presents an a fortiori argument of “what is included in two hundred includes one hundred” as a situation in which B contains A and is not merely more severe than it, and gives the example “If a man opens a pit, or if a man digs a pit,” where digging includes opening, and so it appears to be an even stronger a fortiori argument. It cites Tosafot and the Mekhilta, who derive from here that “we do not derive monetary penalties from logical inference,” and the Maharsha in the later edition, who suggests that the Babylonian Talmud disagrees and that in a case of “what is included in two hundred includes one hundred” one may impose liability because there is no refutation, since it is deduction. It rejects the Maharsha and argues that even in “what is included in two hundred includes one hundred” one can still refute, because one can always challenge the “application assumption” of moving from the mathematical structure to legal reality. It gives examples such as “of his children he passes to Molech, but not all his children to Molech,” and the explanation of the Kesef Mishneh, which allows one to say that the same punishment does not necessarily apply to the more severe case, because perhaps a more severe punishment is required.

Talmudic a fortiori reasoning: three data points and a 2×2 matrix

The text describes most a fortiori arguments in the Talmud as Talmudic a fortiori reasoning built on three data points, from which one fills in a “blank slot” in a basic structure of zero / one / one / question mark. It gives the example from Bava Kamma about tooth and foot in the public domain and in the injured party’s courtyard, and horn in the public domain, and asks about horn in the injured party’s courtyard. It explains that the a fortiori argument proceeds in two stages: in the first stage, a hierarchical relation is extracted from two halakhic data points, and in the second stage that hierarchical relation, together with the third data point, is used to infer the law for the missing slot, unlike biblical a fortiori reasoning where the hierarchy is an a priori rationale.

Refutation in Talmudic a fortiori reasoning and rotating the a fortiori argument

The text presents a refutation in Talmudic a fortiori reasoning as the creation of a counterexample that undermines the hierarchical relation extracted from the data, and emphasizes that a refutation does not prove the opposite; it only negates the proof of the conclusion. It points out that one can formulate two different a fortiori arguments for the same conclusion, “the row argument” and “the column argument,” and asks why the Sages do not “rotate” an a fortiori argument when a refutation knocks out one formulation—that is, why they do not move to the second formulation in order to save the conclusion. It notes exceptions in two places, in Niddah and in Bava Kamma, where the table is asymmetrical, such as a half-payment in one place, and the two arguments lead to different outcomes. It brings in the rule that “it is sufficient for what comes from the inference to be like the source case” in order to explain why, in a certain a fortiori argument, only a minimal result is accepted.

A fortiori reasoning as theory, independent parameters, and abduction

The text proposes that the reason one does not rotate an a fortiori argument is that the argument assumes a unifying theory according to which all the data in the table are laid out along the same single parameter of stringency, marked as alpha, so that the comparisons between the damaging agents and between the domains are in the same “units.” It explains that a refutation forces us to add another independent parameter, beta, and when there is more than one parameter, many possibilities open up for filling in the missing slot in different ways, so there is no unambiguous conclusion. It identifies Talmudic a fortiori reasoning with abduction: three data points are treated as empirical data, from them a theory is built, and from that theory a prediction is derived for a case not written explicitly in the Torah, in the same way that science builds a theory from measurements and predicts situations that were not observed.

Occam’s razor as an outgrowth of a fortiori reasoning and presumptions

The text presents two possible theories that explain the same data and shows that the conclusion depends on the choice of theory, and then states that the deciding principle is Occam’s razor: a theory with fewer entities or fewer parameters is more plausible. It connects this to the three-time presumption in the Talmud, such as an ox that gored three times, and formulates the preference for the explanation of “a goring nature” over three independent outbursts as a preference for a simpler model. It notes that disputes such as Rabbi and Rabban Shimon ben Gamliel about whether a presumption is established after two times or three are connected to the question of how much “soft” persuasion is required in order to act in Jewish law, and distinguishes this from other examples where the threshold is different, such as the ninety repetitions of “Give dew and rain.”

A mechanical method for filling in a missing slot in a table

The text proposes a method of working in which one first fills in the missing slot with the two competing possibilities, and then searches for a theory for each table that explains all four data points, choosing the table whose theory is simpler. It demonstrates that a table in which all the data are explained by a single parameter is preferable to a table that requires two independent parameters, and from that the halakhic answer follows. It argues that the method can also be generalized to larger tables, and that later it will be explained how to do this in a fortiori reasoning, in deriving a principle from one verse, and in deriving a principle from two verses, together with refutations, generalizations, and combinations.

Independence between parameters and identifying the parameters

The text emphasizes that in a model requiring two parameters, there is no dependence between them, and this is reflected in the lack of dependence between rows or columns in the matrix, whereas a model of alpha and two-alpha is simpler because it contains an internal relation. It explains that alpha and beta represent possible parameters of stringency, such as intentional damage in the case of horn as opposed to unintentional damage in the case of tooth and foot, but argues that the method does not actually require identifying the parameters in order to draw conclusions. It states that identifying the parameters will come at a later stage, after the model has been developed.

Okay, so up to this point I basically gave some kind of introduction to what logic is, the concept of logic, what one can try to expect from it and what not to expect from it. I spoke a bit about logic versus science, and I talked about three, three modes of inference: analogy, induction, and deduction. I said that logic usually, logic in the traditional sense, deals with deduction, which is moving from the general to the particular, meaning necessary inferences. And induction, and science, are based on analogy and induction, which are softer kinds of inference. What is abduction, what is that? Abduction is—I’ll say it in one sentence, and you can also argue about it a bit, but that’s at least how I define it. It’s a move from facts to theory. Meaning, say I say that this pen fell to the earth, so from that I learn that all bodies with mass fall to the earth. That’s a generalization, that’s induction. If I say: because bodies fall to the earth, there is apparently some gravitational force behind this, which is a concept I’m introducing even though I don’t see it in the observations. From the observations I understand that there is some theory that includes concepts or theoretical entities, which are actually what generate the phenomena we’re talking about—that’s abduction. In other words, I move from examples to theory. Usually that comes together with induction, and many times people don’t distinguish between them, but they’re two different things.

Okay, so I said that science usually deals with softer modes of inference, induction or analogy, and therefore it is never certain. I talked about the principle of logical uncertainty—that there is always some interplay between the level of certainty and the amount of information I accumulate. And an absolutely certain argument cannot add information for me. An argument that adds information cannot be absolutely certain. The more information it adds, the less certainty there is, and vice versa. That was, broadly speaking, what we’ve done so far.

And now I said that starting today I’m going to try to develop— I also talked, actually, about one more important point. Last time I talked about this model of maturation, and I said that the main challenge facing a non-skeptical worldview—non-skeptical but rational, I mean, not fundamentalist. What does skeptical mean? Skeptical means that I don’t know what is true and what isn’t true. Meaning, anything could be true, and its opposite could also be true, right? So any rational and non-skeptical worldview has to develop a system of soft rules of thought. Just as there is logic for the necessary forms of reasoning we know from logic, right, all the logical schemata—I think one needs to develop mechanisms, or a logic, of soft thinking: analogy and induction. Of drawing uncertain conclusions. Why? Because someone who does this without that, someone who draws uncertain conclusions without such a framework, is just shooting in the dark. It’s guesswork. Who says you’re right or wrong? People always say to you: prove it. Who says you’re right? And I say: I don’t have a proof, but it seems reasonable to me. Why does it seem reasonable to you? On what basis? Do you have some criterion? Just randomly?

So if you propose some criterion, then you’ve actually done something parallel to what Aristotle did with ordinary deductive logic. Therefore you need to show that you are operating rationally even though you are not relying on certainty. You don’t demand certainty. You make do with plausibility. If something is plausible, that’s also good. But show me that you really have some logical basis for assuming what is plausible and what is not plausible, or what is more plausible and what is less plausible. Okay? If you show me that you have some criterion, then I can accept that you’re actually conducting yourself rationally, even if it isn’t certain.

And indeed the feeling that always accompanies these debates is the automatic identification people make between certainty and rationality, between truth, in logic, and rationality. Meaning, if you have a proof then you’re a rational person; if you don’t have a proof then you’re just shooting in the dark. You’re just guessing. And I want to deny that. I want to claim that a person can be rational without having a proof. But okay, that’s just a claim floating in the air. Come and show me whether you have fixed logical patterns of non-deductive thinking. Meaning, not certain, but still you can talk about more correct, less correct, not correct, yes correct, and so on. That’s the point, okay? That’s actually the challenge before you. That was the philosophical introduction.

What I want to do now is simply to show this logic, simply to learn this logic. And I said I would do it on the basis of three of the interpretive principles, three of the principles by which the Torah is interpreted: kal va-chomer, a paradigm built from one verse, a paradigm built from two verses, and the various refutations of them and the different combinations of them. But those are the three building blocks. Okay? I’ll try to show that from these three building blocks one can construct perhaps even the whole non-deductive map—I’m not one hundred percent sure—but a very broad map of non-deductive inferences, and not only in the Talmud. My claim is that these are general forms of inference; they’re not specifically connected to the Talmud. I think everyone understands that, say, analogy like a paradigm construction, or kal va-chomer—leave aside a paradigm from two verses, we’ll get to that—but kal va-chomer or ordinary analogy, meaning a paradigm construction, exists everywhere. You’re always making analogies or kal va-chomer arguments, you do it everywhere, right?

Someone once told me that even animals make kal va-chomer arguments. Say some cat once encountered a jackal, okay? And it’s afraid of it because it understood that it threatens it, fine. Now it’s walking along and suddenly it meets a lion. It hasn’t met one before, it’s not supposed to know what to do with such a thing, whether to fear it or not, but it knows to be afraid. Why does it know to be afraid? Because if the jackal, which is smaller and not so aggressive, is frightening, then all the more so a lion, which is so strong and aggressive, certainly ought to be feared. So it learns, it knows that it should fear the lion even though it has never met a lion. It makes a kal va-chomer argument. I think that’s true, and so on. You can see it. Meaning, there is something here that really is some universal form of thought; it’s not some principle specific to Torah or Talmudic thinking, but something we do in every field, right? That’s clear.

Good. So that’s why I’m choosing these three interpretive methods. I want one more word before I enter kal va-chomer. The thirteen interpretive principles can be divided into two categories. Very often in the midrashim of the Sages, when they want to speak about interpretive principles, they say kal va-chomer and gezerah shavah. For example, with Otniel ben Kenaz in the Talmud in tractate Temurah, the Talmud says that he innovated three thousand kal va-chomer arguments, or one thousand seven hundred, kal va-chomer arguments and gezerah shavah derivations. Does it specifically mean kal va-chomer arguments and gezerah shavah derivations? What about paradigm constructions, what about general and specific? I don’t think the Talmud means specifically kal va-chomer arguments and gezerah shavah derivations. It means to say that he innovated one thousand seven hundred derivations from two categories. One of them is represented by kal va-chomer, and the other by gezerah shavah. What does that mean?

Kal va-chomer is a logical interpretive principle. As I said, it applies in other fields as well; everywhere people use kal va-chomer. Paradigm construction is there too. As we’ll see later, a paradigm from two verses is there too. Those are basically the three logical principles. It may be that “two verses that contradict each other” also belongs there, but that’s a somewhat different topic so let’s leave it aside.

The other principles are what I call textual principles. And gezerah shavah represents them. What does that mean? You have a similar word or an identical word in two accidental contexts, and then you can make a comparison, derive Jewish laws from one context to another. That’s gezerah shavah. That’s not something logical. It’s a convention, it’s a code. The Holy One, blessed be He, tells us: look, I wrote the Torah in such a way that if you find two identical words, know that you can also compare the laws between those two contexts. The Torah could have been written according to a different convention. There’s nothing necessary here, nothing logical. It’s just a convention, a kind of code, okay?

And kal va-chomer isn’t like that. No. That’s why I said: kal va-chomer and paradigm constructions are logical. Gezerah shavah, general and specific, and all the principles of general and specific, “something that was included in a general rule and then went out from the general rule to teach,” and so on—all of those are textual. Textual means that there is something in the text that is the basis of the derivation. In kal va-chomer there is nothing in the text that is the basis of the derivation. The text tells me that there is a law, and from that I make a kal va-chomer argument. There doesn’t need to be any hint in the formulation of the verse that from here one should make a kal va-chomer argument, right? I make a kal va-chomer argument because there is a logic by which I say: if this law is true, then all the more so that law is true. The verse doesn’t have to be phrased in a certain way in order for me to make a kal va-chomer argument on its basis, okay?

In contrast, gezerah shavah and general-and-specific and all that—all of that is based on formulations of verses. The verses are formulated with the same word, or the verses move from a general phrasing to a specific phrasing, so that is general and specific, and so on. And then they tell me: you need to make, or may make, such-and-such a derivation, but it begins with some trigger that is in the text. It’s not logic— not only logic; there is also logic, and maybe we’ll get to that later.

So kal va-chomer and gezerah shavah are really just representative examples of these two types of derivation. Kal va-chomer is a logical derivation; gezerah shavah is a textual derivation. Okay?

Wait—so isn’t a paradigm construction textual? No. No, not at all. When you make an analogy from—right, nothing there has anything to do with the text. I make an analogy because it makes sense, because this is similar to that. Not because Scripture used this terminology or used the plural or used the same word—nothing of the kind. It simply makes sense to me that this is similar to that, so I make an analogy. Okay?

Therefore, for example—just maybe as one indication—you won’t find anywhere in all the literature of the Sages, as far as I’ve checked, a refutation of an interpretive principle that is not one of the three logical ones. There is no refutation. Only of kal va-chomer, a paradigm from one verse, and a paradigm from two verses. I found one example in tractate Hullin of a refutation of general-and-specific-and-general, and even there I showed that it isn’t really a refutation. Why? Because a refutation attacks the logic underlying your inference. It isn’t a kal va-chomer because this isn’t more severe than that, it’s less severe than that. So you can’t make a kal va-chomer argument, right? But with general and specific there is no place for a refutation. I’m not assuming that something is similar to something, or more severe than something, or less severe than something. The verse is phrased in a certain way, and that compels me to infer a conclusion. What does a refutation have to do with that? I haven’t assumed some logical premise here and now put it to the test. Bring me a refutation and let’s see whether I’m right in that logic or not right in that logic. It doesn’t belong here; refutations don’t arise in that context.

So when I speak about refutations in the context of the principles by which the Torah is interpreted, these are only refutations of these three principles: kal va-chomer and the two kinds of paradigm construction. Why? Because these are the logical principles. Therefore I said I’m now going to try to develop a complete systematic theory of non-deductive logic on the basis of kal va-chomer, a paradigm from one verse, a paradigm from two verses, their refutations, and different combinations of them with one another. That is the basis from which I compose all the other modes of inference.

But it seems to me that the Sages regarded all these logical principles with a certain degree of skepticism, at least in terms of punishment for example. “One does not punish based on logical inference.” Kal va-chomer—only with kal va-chomer. Maimonides’ view is that it applies to all interpretive principles, including the textual ones, but Nachmanides already attacks him on that. It’s not—plainly in the Talmud it’s only kal va-chomer. So not all logical principles. Maybe I’ll still comment on that.

A juxtaposition of passages—you’re not talking about that? What? Juxtaposition of passages. Juxtaposition of passages is hekesh. Hekesh is a textual principle. The analogy you make between two things doesn’t arise from their being similar to one another, but from the fact that they’re written next to each other. The text tells you to make the inference, not your logic. Hekesh is a textual principle; it doesn’t appear in Rabbi Ishmael’s list, but that doesn’t matter—it’s a textual principle in every respect.

Can you just explain the difference between non-deductive logic and deductive logic? The question is whether the modes of inference are certain or necessary, or not. Deductive logic is like the examples we had in previous sessions: all human beings are mortal, Socrates is a human being, conclusion: Socrates is mortal. Okay? Non-deductive logic is analogies and inductions. We saw that they are not necessary. I say: this table is brown, this thing is also a table, so it too is brown. It could be that it’s a table and it’s green. That’s not necessary. But on the other hand, although it isn’t necessary, we do make analogies. We don’t avoid making analogies. So that’s what I called soft modes of inference. These are modes of inference that we use, but it’s not certain—you may be mistaken. Okay?

Go on. So I’ll begin with kal va-chomer. The first kal va-chomer, the lighter-and-heavier arguments in the basic or first structure, are the biblical kal va-chomer arguments. There’s the well-known midrash that brings ten kal va-chomer arguments written in the Torah itself. One of them is: “Behold, the children of Israel did not listen to me, so how will Pharaoh listen to me, and I am of uncircumcised lips?” Or: “If her father had but spit in her face, would she not be shamed for seven days?” So what is the meaning of this?

If the children of Israel, Moses says to the Holy One, blessed be He, if the children of Israel, to whom I say something, do not listen to me, then will Pharaoh listen to me? All the more so not. What stands behind this? The assumption is that a priori it is clear to me that Pharaoh is less obedient than the children of Israel. And if the children of Israel, who are relatively more obedient, won’t listen to me, then Pharaoh, who is even less obedient, all the more so won’t listen to me. Okay?

Same thing: “If her father had but spit in her face, would she not be shamed for seven days?” What does that mean? Concerning Miriam: if her father had spat in her face, then she would have been ashamed for seven days. She would have understood that she had done something wrong. Now the Holy One, blessed be He, spat in her face, not her father. That is much more severe. So certainly she should be shamed even more than seven days. At least seven days. There is the rule of dayyo.

In the biblical kal va-chomer, the matter is built on one premise and a hierarchical relation. That is, one premise: “the children of Israel will not listen to me.” Fine? That’s one premise. A hierarchical relation: Pharaoh is less obedient than the children of Israel. More or less, right? Severity and leniency—that is the hierarchical relation between the two things. The conclusion follows from those two. If I have fact A, and a hierarchical relation in which B is more severe than A, the conclusion is that the law that exists in A also exists in B. Because if it exists in the lighter case, then certainly it exists in the heavier one. That is, of course, if the law in question is a stringency. If that law is a leniency, that is called the reverse, from severe to light. Meaning, if the leniency exists in the more severe thing, then all the more so it exists in the lighter thing. Okay? So here I learn the lighter from the more severe, not the more severe from the lighter. With leniencies, the lighter case receives the leniency more easily than the more severe one, right? So the direction of the kal va-chomer changes, but it’s the same idea, the same kind of inference, the same logic.

So we have one premise and a hierarchical rule. Can one refute such a kal va-chomer? You can bring premise B. You can say that the people of Israel also had a much greater desire to hear words of redemption. No—the opposite of his premise. Therefore Pharaoh is less obedient. To refute it, you need to tell me the opposite assumption. Maybe Israel is less obedient than Pharaoh, because Pharaoh is, after all, a king, he’s relaxed, I don’t know exactly what. Israel, with hard labor, have no patience for someone coming to bother them. For example. Okay? What is the relation between one and the other? Which is more this, which is more that—then you have refuted the kal va-chomer. Right. You’re saying this is more severe than that, so place it somewhere else. Exactly. Meaning, you can refute a biblical kal va-chomer. How? If I bring an example in which I show that the hierarchical relation is not correct. Right? I bring an example where in fact A is more severe than B, and B is not more severe than A. Then you can no longer know how to infer from A to B—who knows which is more severe and which is less severe, right? That is the way to refute a kal va-chomer.

Therefore, if I ask whether kal va-chomer is a deductive inference—there are people who thought so, that kal va-chomer is a deductive inference. It’s not true. Kal va-chomer is not a deductive inference. And the proof is that it can be refuted. A deductive inference cannot be refuted. If all human beings are mortal and Socrates is a human being, the conclusion is that Socrates is mortal. You can’t refute that. The conclusion follows necessarily from the premises. If something can be refuted, that means it is not necessary. In other words, the conclusion does not follow necessarily from the premises. There may be a different conclusion even though the premises are correct. How? If the logic that carries me from the premises to the conclusion is not correct. What is that logic? The rule of hierarchy: that B is more severe than A. That is what leads me from the premise to the conclusions, right?

So unlike deductive rules of inference, where the path from the premises to the conclusion is certain and cannot be refuted—there, if you accept the premises, you must accept the conclusion—here, even if you accept the premises, you do not necessarily have to accept the conclusion. I accept the premise that the children of Israel won’t listen to you. But who says Pharaoh won’t listen to you? You assume that Pharaoh is more severe than Israel; perhaps he is lighter. Okay? So there is something here that is not necessary. The logic here is a logic that maybe makes sense, it’s not just nonsense, but it is not certainly true. In other words, it could be wrong.

But according to that assumption, if you say Pharaoh is not obedient, then when Pharaoh let the people of Israel go, that shows he’s more obedient than Israel, because he let them go after the tenth plague. No, that shows nothing. He got beaten up, so he let them go. Is that because he’s obedient? It’s because he’s afraid of getting hit. You can’t prove anything from that. But the blows were also in relation to Israel? No, but it shows that in the end he accepted God’s authority. Fine, and what does that mean? It shows he’s more obedient. No, it means nothing.

You know, it means nothing even without this, but I’ll say: there’s a letter from Rabbi Shach in his Iggrot U’Mikhtavim, the second letter in one of the volumes—there are already several volumes. In the second letter, I once saw, I think it’s in the first volume. He speaks there about the Entebbe operation. So he says there that he opposed carrying out the operation. Meaning, he argued that it is forbidden to endanger soldiers where the chance of success is so small. Okay? You’re sending them thousands of kilometers away, there is no chance you’ll succeed, and to endanger soldiers when your chance of saving the hostages is slim—it is forbidden to do that. That’s what he argued. After they succeeded—the operation did succeed, after all, Yonatan Netanyahu. And you don’t launch an operation for a five-percent chance. Meaning, the fact that something happened in the end does not necessarily mean that the prior forecast was wrong. Rabbi Shach merely said that I think the chance—not only I, everyone understood it that way—that the chance of success was small. True, I also say that it could succeed, but when you ask yourself—you don’t know in advance what will happen—when you ask how to make the decision, most likely you won’t succeed, so don’t go. So in the end it succeeded—so what? You can throw a die and get six three times in a row. So if someone had told you: look, there’s no chance it will come up six three times in a row—was he talking nonsense? Of course not, he was right. True, it can come up, with a probability of one over six cubed. Okay, but that’s a small probability. Fine?

How do you define transitivity, say? Again? How do you define transitivity, say? If I say, for the sake of argument, right—the advice of Haman is more effective than that of Pharaoh, and he is less effective than the children of Israel. Can I do that… No, that would be a refutation. That would be a refutation of the kal va-chomer. The assumption is that this is transitive, certainly. If it isn’t transitive, then it isn’t a hierarchical relation. A hierarchical relation is by definition transitive. Okay? Mathematicians call it an order relation. Meaning, it must be transitive; that’s one of the requirements.

What generally counts as a refutation of a kal va-chomer? When you attack the hierarchical assumption. The hierarchical assumption? Exactly. Yes. You bring an example that B is not more severe than A. Or even attack the very idea of a hierarchical relation, not only with an example. Why on earth do you infer that Pharaoh is less obedient than the children of Israel? There are also aspects suggesting otherwise—even without bringing an example where he was not obedient—but by attacking the very logic at the base of the hierarchical relation. Okay?

Now that’s biblical kal va-chomer. There is a kal va-chomer that is a branch of this kal va-chomer, and it is called—and I already mentioned it in one of the previous classes—“if two hundred includes one hundred.” A kal va-chomer of “if two hundred includes one hundred.” For example, the Torah says, “If a man opens a pit, or if a man digs a pit,” right? About a pit in the public domain for which one must pay damages. So the Talmud asks: if it says “opens,” why does it also need to say “digs”? After all, that is a kal va-chomer. So there is a dispute what is learned from this, between the Mekhilta and the Babylonian Talmud.

What is the assumption underlying the idea that if you wrote “opening,” you don’t need to write “digging”? It’s a kal va-chomer. Let’s say if there is an existing pit and I opened it, then I am liable for the damages it caused. Fine? That is what is written: “If a man opens a pit.” So now if it didn’t also write “or digs,” but someone came and dug a pit. Fine? And now the pit caused damage. Is he liable for the damages, must he pay? Certainly—it’s a kal va-chomer. If there was a pit that you didn’t dig and you merely opened it, then you are liable to pay. So if it is a pit that you dug, all the more so you are liable to pay, right? I would even formulate it more strongly. When you dig a pit, say ten handbreadths deep, among other things you also remove the upper layer, right? You removed everything including the top centimeter, right? So in other words, the act of digging includes within it opening as well. It’s not that digging is more severe than opening. Digging is opening plus something else. Do you understand the difference?

Meaning, Pharaoh and Israel are in a hierarchical relation. Pharaoh is not Israel plus something else. Pharaoh is more severe than Israel. But the relation between opening and digging is like the relation between two hundred and one hundred. Two hundreds isn’t just more than one hundred. It’s one hundred plus something else. Meaning, within two hundred there is one hundred, right? What does this mean? This kal va-chomer is much stronger. “If two hundred includes one hundred.” Why? Say there is a law that I must pay if I opened a pit. Now I ask whether, when I dig a pit, I am also liable to pay? The answer is certainly yes—not because digging is more severe than opening, but because when I dug I also opened. Right? Digging includes removing the upper part, which is opening. And for opening one must pay. I don’t need to get to the question of whether digging is more severe than opening. That is irrelevant. So isn’t that a refutation? What? This case isn’t a refutation? We’ll get to that in a moment. I’ll comment on that. It’s not exact.

If I built nine handbreadths, and the last one came and completed it to ten. No, no, I’m talking about one person, leave me out of a case of two people. One person did it all. Yes. If the second completed it to ten, then ostensibly he opens, he doesn’t dig—he opens. He opens and doesn’t dig. The claim is that a kal va-chomer of “if two hundred includes one hundred” is much stronger.

Now at the beginning of Bava Kamma, on the mishnah of the four primary categories of damages, it says: “There are four primary categories of damages: the ox, the pit, the grazer, and the fire.” And the mishnah says: “This is not like that, and that is not like this,” meaning it creates a need for each category among all the categories of damage. Tosafot asks there: what does this need-for-each-category assume? It assumes, in effect, that in principle it would have been enough to write only pit, and there would have been no need to write fire or ox or the others—I would have learned them from pit by kal va-chomer or by a paradigm construction. Okay? And therefore there was no need to write them, and therefore the mishnah has to explain the need for each one and show that there are differences. If pit had been written, I would not have known fire; if fire had been written, I would not have known pit, and so on. But if I would have known it, if I could have learned it, then it would be superfluous. Right? That’s what the mishnah says.

Tosafot asks: say only pit had been written and there were no such distinctions, no “this is not like that,” meaning I could have learned fire from pit—but even then, isn’t that “one does not punish from logical inference”? I would have learned fire by kal va-chomer from pit, but who says that if one is liable for pit then one is also liable for fire? “One does not punish from logical inference.” Tosafot says—wait, he brings a proof from the idea that we do not impose monetary liability from logical inference. You might have said that in monetary matters we do. True, that’s a good question, because money, plainly speaking, is not a punishment at all, it’s compensation. “One does not punish from logical inference” seems to speak about punishment. So he says he has proof from this derivation of “If a man opens a pit, or if a man digs a pit.” He brings the Mekhilta, and in the Mekhilta it says: “If a man opens a pit, or if a man digs a pit—if one is liable for opening, then for digging all the more so.” A kal va-chomer. “Even so, this teaches you that one does not impose monetary liability from logical inference.” Therefore it had to write digging, since if it had not been written we would not impose monetary liability from logical inference. Fine? Therefore digging had to be written explicitly, even though it is more severe, even though it includes opening. That is what Tosafot says.

The Maharsha, in Mahadura Batra on page 49 I think, or 50, in Bava Kamma, says that the Babylonian Talmud derives something else from “If a man opens a pit, or if a man digs a pit.” And the Maharsha comments there that this Babylonian Talmud is against the Mekhilta that Tosafot brought. Because the Mekhilta says that “If a man opens or if a man digs” comes to teach that one does not impose monetary liability from logical inference, while in the Babylonian Talmud it appears that we do impose monetary liability from logical inference, and therefore it really is superfluous and didn’t need to be written, and then it learns something else from it. But in principle, the law of digging is learned by kal va-chomer from opening. No—yes, we do impose monetary liability from logical inference. Fine?

But the Maharsha says that isn’t fully precise. Because the kal va-chomer from opening to digging is a kal va-chomer of “if two hundred includes one hundred.” About that, the Babylonian Talmud says that we do punish from logical inference. About an ordinary kal va-chomer, even the Babylonian Talmud agrees that we do not punish from logical inference, but here this is a kal va-chomer of “if two hundred includes one hundred.” Why? As I said before: because when you dig, in particular you also opened. Besides that, you also dug. Right? But in particular, you opened. About such a case, says the Maharsha, there can be no refutation. For why do we not punish from logical inference? I don’t punish on the basis of kal va-chomer because perhaps there is a refutation against it. The Maharsha assumes this—there are several explanations—but the Maharsha assumes that perhaps there is a refutation, and therefore I do not punish. But a kal va-chomer of “if two hundred includes one hundred” cannot have a refutation. Why? Because the hierarchical relation is mathematics. It is mathematics, and mathematics admits no refutation. Or in other words: it’s deduction. A kal va-chomer of “if two hundred includes one hundred” is deduction. It is not analogy. Therefore there can be no refutation against it, and if there can be no refutation against it, then why not punish? The two hundred is the digging. What? The opening? The digging? No, digging is the two hundred. The two hundred is digging. Opening is part of the two hundred, because opening is the one hundred; within digging there is opening. But within digging there is also the part that is opening plus something else. So that’s the two hundred.

You discover that the whole digging becomes, in essence… Opening reveals nothing; opening just opens. But I’m saying: when you opened… Right, but the hole already existed before; you only opened it. When I dug, then I also created the hole and also opened it when I removed the upper handbreadth. So I did two hundred: I did the one hundred of opening and also the digging of the lower handbreadths. So I did two hundred. I understood that you opened—you opened the last row. Right, right, you understood correctly. And therefore that is the one hundred and not the two hundred. You did the bottom row—that’s the one hundred—and I also did the nine lower rows, therefore I did two hundred. I did more than you. I did what you did plus. Okay?

So the Maharsha is basically arguing that a kal va-chomer of “if two hundred includes one hundred” is deductive. There is no refutation against it, and therefore one may punish, because there is no fear that you’ll refute it. A certain answer. Yes, a certain answer. There is no refutation against it, and therefore you can punish from logical inference in a kal va-chomer of “if two hundred includes one hundred.”

The truth is that the Maharsha is wrong. A kal va-chomer of “if two hundred includes one hundred” can be refuted. And here I brought this up— I spoke about it in one of the previous sessions, with the Belgian law. I spoke—didn’t I speak about it? I did. So when you apply mathematics to life, or logic to life, there is always some assumption on which one can refute—the application assumption. The logic itself is mathematics, it cannot be refuted. But the assumption that this mathematics applies to a certain real-world context—that is an assumption we could call scientific, not mathematical. I spoke about vectors, if you remember: whether adding forces is mathematics, algebra, arithmetic, or vector calculus. That question is not a question of mathematics, it’s a question of physics. Therefore if I refute the assumption that this is arithmetic, then apparently I’ve shown that this is vector calculus; I haven’t refuted arithmetic. I have refuted the assumption that the addition of forces is described by arithmetic, which is an assumption in physics, not mathematics.

The same thing here. When we try to take some logic of two hundred and one hundred and things like that and apply it in a legal context, one can refute it. How can one refute it? By saying that it is not applicable. Obviously you can’t refute the fact that this is one hundred and this is two hundred, that that contains this—that’s obvious, that’s a fact. But you can refute the kal va-chomer relation I infer from that. For example, there are other views regarding “one does not punish from logical inference.” The Maharsha assumes that we do not punish from logical inference because maybe there is a refutation. And in the Kesef Mishneh… because maybe there is a refutation. Therefore in “if two hundred includes one hundred,” where there is no refutation, then we do punish.

The Kesef Mishneh, for example, in the laws of idol worship, brings a derivation there—maybe I mentioned this too: “one who passes from his children to Molech”—from his children, but not all his children to Molech. Someone who passes some of his children to Molech is liable to death; someone who passes all his children to Molech is not. Now what is that? That’s a kal va-chomer of “if two hundred includes one hundred.” Right? If you passed five of your children to Molech, then in particular you also passed one or two. So you also passed “from your children.” So how can it be that this exempts you from the law of “one who passes from his children to Molech”?

So the Kesef Mishneh there goes on at length about this issue, and one of his claims is that we do not punish from logical inference not because of fear of refutation—and by the way, that really isn’t a plausible explanation, because otherwise I would have had to avoid punishing on the basis of anything that depends on reasoning. Maybe there’s a refutation? Kal va-chomer is no worse than any other reasoning. And in fact they do, yes, punish. He argues that we do not punish from logical inference because it may be that for the more severe act, the punishment received for the lighter one is not enough. If one who passes some of his children to Molech is liable to a certain punishment, then one who passes all of his children to Molech is certainly liable to that punishment—but it may be that he is liable to more; perhaps that punishment is not enough. So leave it—let the Holy One, blessed be He, deal with him. Fine?

Now notice that this explanation itself constitutes a refutation of a kal va-chomer of “if two hundred includes one hundred.” The very fact that there is an alternative explanation—what is he saying? That one can refute the kal va-chomer of “if two hundred includes one hundred.” Why? Because here there is two hundred and one hundred. If I passed some of my children to Molech, I’m liable to death. If I passed all of my children to Molech, then if two hundred includes one hundred, I am certainly liable to death. No, not at all. It may be that death is not sufficient, therefore I am not liable to death. So that refutes the kal va-chomer. It does not refute the logic that one hundred is included in two hundred. It refutes the logic, the application of that logic to the matter of punishing one who passes his children to Molech—which is the application assumption. Okay? The application assumption can always be refuted. And I claim this as a general claim, and those rule-makers who say that “if two hundred includes one hundred” cannot be refuted are mistaken. Not true. “If two hundred includes one hundred” can be refuted just like anything else. There is nothing that is a claim about the world that is completely certain.

And if the entire court acquits… What? So that too… “If the entire court acquits, he is exempt”… So that too is “if two hundred includes one hundred.” Yes. If twenty out of twenty-three convict him, then he is liable to death. When all twenty-three convict him, then all the more so he is liable to death. Right? So here is another example of a refutation of “if two hundred includes one hundred.”

What would we say with a pit? What? Leave me alone about the pit. What would you say there? The same thing I can say there too. Maybe it’s not enough, for example. What? It’s a monetary issue. No—if you assume monetary matters are not related at all, then leave aside “one does not punish from logical inference” in money. Simply say, whatever the case may be: if you assume it is a kind of punishment, then perhaps it is not a severe enough punishment. For example, if he is liable to death for that, then “one who dies does not also pay”—kim lei bid-rabba minei. By the way, that too is said about monetary matters: kim lei bid-rabba minei. Someone who is pursuing and breaks vessels—the mishnah in Sanhedrin says that one who is pursuing and breaks vessels is liable to death as a pursuer, and therefore is exempt regarding the vessels. Why? That’s damages, not punishment. Meaning, in monetary law, as far as kim lei bid-rabba minei is concerned, it is certainly punishment. Tosafot’s novelty is that it is punishment also regarding punishment from logical inference, but that’s…

Anyway, let’s get back to our subject. So, to return—what have we seen until now? We saw that biblical kal va-chomer has ten such examples, ten brought by the Sages—maybe there are more, I don’t know. It’s a kal va-chomer built on one premise and a rule of hierarchy. This kal va-chomer can be refuted if I attack the rule of hierarchy: I show that it isn’t logically compelling, or there is another side to it, or I bring a counterexample—it can be refuted.

Then I said there is another kind of kal va-chomer, called a kal va-chomer of “if two hundred includes one hundred.” A kal va-chomer of “if two hundred includes one hundred” is also like biblical kal va-chomer: there is one premise, a rule of hierarchy, and a conclusion. What changes in “if two hundred includes one hundred” is the rule of hierarchy. The rule of hierarchy is not that B is more severe than A, but rather that B contains A. Okay? That’s not the same thing. B contains A. So whatever exists in A must also exist in B—that is “if two hundred includes one hundred.” Not because B is more severe—just punish on account of the A that is in it. Then when I say that someone dug a pit and I punish him by force of the kal va-chomer from opening, I punish him as one who opened, not as one who dug. Only here he dug—but in particular he also opened; he just did something more. So leave me alone with the something more. For opening one is liable, so punish me as one who opened. There is no need at all to discuss digging. Within digging there exists opening. Okay? That’s the idea of a kal va-chomer of “if two hundred includes one hundred.” And as I said, in the end “if two hundred includes one hundred” is not essentially different from ordinary kal va-chomer. Whoever makes that distinction is not right.

The kind of kal va-chomer that appears in the Talmud is a third type. Most kal va-chomer arguments in the Talmud—usually the kind I’ve described until now appears in language like “all the more so.” When something appears in the language of kal va-chomer or “is it not logical,” it is usually a kal va-chomer built differently. It is a kal va-chomer that starts from three data points, not one, and infers a conclusion from them. Here we’ll move to the board. What board and what shoes. Fine, here we go.

I’ll take the kal va-chomer from Bava Kamma, it’s convenient for me for various reasons, on page 25 there, Rabbi Tarfon and the Sages regarding tooth and foot in the injured party’s domain, horn in the injured party’s domain. So I say: there is horn, tooth and foot, public domain and the injured party’s domain. The Talmud says as follows: tooth and foot in the public domain are exempt. Let’s call that zero—not liable to pay. Fine? What does that refer to? To the ox? An ox that causes damage by tooth and foot, meaning it ate something or stepped on something while walking, in the public domain—the owner is exempt. Fine? But if the ox gored with its horn in the public domain, then he is liable. Actually liable for half, if it is a first-time offender, but let’s leave that aside for the moment. Tooth and foot in the injured party’s domain are also liable. Right? Now I ask: what about horn in the injured party’s domain? That is not written in the Torah. Those three are written in the Torah; these are laws written in the Torah. I ask: what is the law of horn in the injured party’s domain?

This is the universal structure of Talmudic kal va-chomer. In the Talmud, kal va-chomer is always built like this. It differs from biblical kal va-chomer. In biblical kal va-chomer there is one datum and a rule of hierarchy: “Behold, the children of Israel did not listen to me”—datum. Rule of hierarchy: Pharaoh is more severe than the children of Israel. Conclusion: Pharaoh also won’t listen to me. That is a kal va-chomer that starts from one datum: the children of Israel did not listen to me. That’s it. Here this is a kal va-chomer built on three data points. And apparently there is no rule of hierarchy. From these three data points I derive the conclusion. So why is this nevertheless called kal va-chomer? How is this kal va-chomer built?

Basically this kal va-chomer is built like this. Look at this row. In this row we see that tooth and foot are exempt in the public domain and liable in the injured party’s domain. What can we learn from this? The injured party’s domain is more severe than the public domain. It is easier to impose liability in the injured party’s domain than in the public domain, right? Now I look at the bottom row. If horn is more severe—horn imposes liability in the public domain, where it is hard to impose liability—then in the injured party’s domain, where it is easier to impose liability, certainly it should be liable. Therefore this should be one, right?

So notice what happened here. Basically, the wording is the wording of an ordinary kal va-chomer, like biblical kal va-chomer. Why do I need three data points here and not one? I use the first two data points to generate the hierarchical relation. Right? From the first row I extract a rule of hierarchy. After I extracted it, I discovered that the injured party’s domain is more severe than the public domain. Fine? And from here on, now I have one datum: horn is liable in the public domain. So that is the one datum, this is the rule of hierarchy, and now I say: then horn is liable in the injured party’s domain, right? That is the conclusion. This is biblical kal va-chomer, right? There is a rule of hierarchy and one datum. The difference is that in Talmudic kal va-chomer, the rule of hierarchy is not a matter of pure reasoning. The rule of hierarchy emerges from two additional facts. I take two out of the three facts, generate from them a rule of hierarchy, and set them aside. Now that I have a rule of hierarchy, I take the third fact and infer from it the conclusion. At that stage it is like biblical kal va-chomer: I have a rule of hierarchy and one datum. The difference is that in Pharaoh and Israel, that comes from reasoning. It is a priori: I say it’s obvious that Pharaoh will listen to me less than Israel. I don’t need evidence for that, I don’t bring facts to prove it. I simply say it by reasoning. Here I don’t have reasoning; I don’t know what is more severe. I prove it from the legal facts. Therefore I need more legal facts—three. Then the kal va-chomer is done in two stages. First I take these two data points and generate from them a rule of hierarchy. In the second stage I take that rule of hierarchy together with the third datum and fill in the empty square. Okay?

Now you wanted to comment. Probably something I’m about to say. Not necessarily. Okay. But why should the law of horn also carry the same severity, as with tooth and foot, between the domains? There is no necessity. No. Maybe it’s not true. But that’s the analogy we’re making. There is no certainty here. I’ll get to that in a moment. What we have are the data that appear in the Talmud—or in the Torah. Tooth and foot are exempt in the public domain and liable in the injured party’s domain. Horn is liable in the public domain. That’s it. This I don’t know. It isn’t written. Fine?

Now every kal va-chomer in the Talmud is always built like this. It doesn’t matter what the entries of the matrix are, right? What the damaging agents are, or what’s written here and here. The basis is one, zero, one, question mark. Every kal va-chomer is built that way. Right? If we could replace this marker with black, that would be great. But I don’t have one.

Can’t one say that this is stronger even than biblical kal va-chomer? No. In my opinion it’s weaker, but I’ll get to that in a moment. Okay?

Now basically—how do you attack such a kal va-chomer? Right? I need to attack the hierarchical relation, as we did in the biblical case. How do I attack it? For example, I bring a counterexample. Right? If I bring an example like this: suppose there is a damager—I don’t know—horn, tooth and foot, and nose. Fine, I’m just making something up. Okay? And the nose is exempt in the public domain and liable in the injured party’s domain. No, no—the opposite. Exactly. Or together too? No. The opposite. If you do it the opposite—no, then it’s good. The opposite is good. So if the nose is liable in the public domain and exempt in the injured party’s domain, that is a refutation. Why is that a refutation? Because it gives a counterexample in which the public domain is more severe, not the injured party’s domain. So you can no longer infer this conclusion, right? You can’t know. And therefore you can’t infer this conclusion. Notice: the refutation does not prove that here it should say zero. It only refutes the proof that wanted to prove that here it should be one. Exactly. You can’t prove that here it says one. There is no proof here that the law is zero, right? There is only a refutation of the proof that wanted to prove it is one. It’s important to understand that for what follows.

Now look, let me ask you another question. Why would equality also count? If I show that the severity is the same, that could also show a kind of refutation. What? If I show that the severity of the public domain— We’ll get to that later; let’s leave it for now. It’s not a refutation, but in a moment.

Now I want to make another move. I’m still focusing on this, on the basic square, right, without the refutation. Focus on this square. Okay?

Now let’s do another kal va-chomer like this: let’s take this column and learn from it that horn is more severe than tooth and foot. Right? Horn is more severe than tooth and foot. And now these are the two data points—we’ve used these two data points. Now I have one datum: tooth and foot are liable in the injured party’s domain. Right? That is the single factual datum I need, this is the rule of hierarchy, conclusion… Wait, horn… the conclusion? Yes, you reach the same conclusion. So horn… horn is liable in the injured party’s domain. I arrived at the same conclusion, right? How do you refute this? If there is some… I don’t know—show that somewhere else horn is exempt… Say there is a domain on the moon. Fine? On the moon, tooth and foot are liable and horn is exempt. Then I show that horn is not always more severe than tooth and foot. Look, on the moon it isn’t. So maybe only in the public domain it is more severe, and not on the moon. Then I don’t know whether the injured party’s domain is like the public domain or like the moon. It will depend on that. If it resembles the public domain, I can still do it; but since I have shown that there is another axis where that hierarchy does not hold, and I do not know whether to assign the middle column to this or to that, the kal va-chomer is refuted. Fine?

Now I ask another question. Suppose I didn’t find the refutation on the moon. I have the refutation with the nose. Has the kal va-chomer collapsed? Not for that reason—I mean not that kal va-chomer, but the second one has. Right? Apparently not. Look, this argument—wait—the injured party’s domain is more severe than the public domain, horn is liable in the public… I’m just writing what was written here before, and horn is liable in the injured party’s… yes, this… These are the two arguments we went through, right? This argument—let’s call it the row argument—we moved from this row to this row, and this is the column argument. Fine? I took the two items from this column and moved to this column. Is this two formulations of the same argument, or are these two different arguments? Two different arguments whose conclusion is… The conclusion, yes, obviously. But I’m asking whether they’re two different formulations of the same argument, or two different arguments. Two different arguments, right? Completely different. And this argument, in effect, passes through the conclusion that horn is more severe than tooth and foot. How do you refute it? You show that perhaps tooth and foot are more severe than horn. But this argument does not assume anywhere that horn is more severe than tooth and foot. Nowhere here is that assumption written. The only thing it assumes is that the injured party’s domain is more severe than the public domain. If I found this thing, does it in any way refute the hierarchy between these two? Not at all.

Meaning, this refutation of the nose, yes? It refutes this kal va-chomer, but it does not refute this one. But isn’t the datum the result… And the moon refutes this kal va-chomer and not that one. Right? If the refutation refutes the hierarchy of a row—let’s call them a row refutation and a column refutation. A row refutation refutes the row-based kal va-chomer, right? When I found a hierarchy here and it shows me no, the hierarchy is like this. But it does not touch this hierarchy. It doesn’t address the question of the relation between tooth and foot and horn; that is irrelevant. Right? Meaning that the column argument is indifferent to a row refutation. Why? But if the nose is more severe than tooth and foot and still it is zero… Again? How does this go? Tooth and foot here are more than horn, therefore horn should be one in the injured party’s domain. Well, the nose is more severe than tooth and foot and still it is zero, so yes, it’s a refutation. No, that’s an indirect refutation—I’ll get to that in a moment. About the argument itself, you proved nothing, meaning you didn’t refute the premise of the argument. And we said that in order to attack a kal va-chomer, you need to refute the hierarchical assumption, attack the hierarchical assumption. This doesn’t attack it.

Therefore I would have expected that throughout the whole Talmud, when they bring a kal va-chomer—after all, you can formulate the kal va-chomer in two ways, and both lead to the same conclusion. It’s enough that one of them be valid for the conclusion to be correct. If I refuted this argument but this one remains intact, then the conclusion is correct. Why? Because when I refuted that argument I did not show that the conclusion is wrong; I showed that it doesn’t necessarily follow from the premises. But it can still be true—only I didn’t prove whether it’s true. Well, here I did prove it. Since there are two arguments, each of which can lead to this conclusion, it is enough that one of them lead to the conclusion for me to have to accept the conclusion, right? So if I knocked down one of the two arguments, I did not knock down the conclusion. To knock down the conclusion, I would need to knock down both arguments. Isn’t it the opposite? Isn’t it enough to knock down one of them to knock down the conclusion? No, because it’s enough that one of them be valid to prove the conclusion, right? If one of them is valid, then its conclusion is correct. So if you knocked one of them down, the other still remains valid. Unless you bring another assumption that knocks down the… In other words, in order to knock down a kal va-chomer, I would have expected that one would always need both a row refutation and a column refutation; otherwise we haven’t knocked it down.

You don’t find that anywhere in the literature of the Sages except for two places, one in Niddah and one in Bava Kamma on the mishnah I’m talking about, where they rotate the kal va-chomer. But in both of those places the table is not the one I marked here, but rather it looks like this. Also in Niddah, by the way, although the topic is different, the table is like this. Here it is half, not one. Yes, horn in the public domain is liable for half, not one—an ox that is a first-time offender. Horn in the public domain is liable for half, not one. This is an asymmetric table. In both the Niddah sugya and the Bava Kamma sugya, the table is asymmetric.

Now look: in an asymmetric table—I may get into this later, but for now—in an asymmetric table, the two arguments are certainly not identical. And they can lead to different conclusions. Let’s take an example. Let’s now look at the column argument. Then I prove that horn is more severe than tooth and foot, right? Agreed? This is half and this is zero. Horn is more severe than tooth and foot. Now let’s move to this column. If tooth and foot are liable for one, how much should horn be liable for? One, right? Because it is more severe.

Now let’s go to the row argument. Fine? The row argument says that the injured party’s domain is more severe than the public domain. Come down here. If the injured party’s domain is more severe than the public domain, and in the public domain horn is liable for half, then how much is it liable for in the injured party’s domain? Half. One. Why half? Why one? More severe than half. I don’t know by how much. Dayyo: it is enough for what is derived by inference to be like the source. Right—half. No, half. Why one? Why here did you say one and not eight in this kal va-chomer? Here it was one and it’s more severe, so write eight. But why? Dayyo says what is derived by inference should be like the source. No, but yes—we said the rabbinic limit is one. No, my limit is that it is at least one. But the rule of dayyo means: if you have no proof for more, you take the minimum that you proved. The minimum that I proved from here in this kal va-chomer is one. It is at least one. One plus epsilon, where epsilon is as small as you like, so that’s one. Whereas here what I prove is that it is at least half, not at least one. Therefore here you should have written half.

This shows you very clearly that these two arguments, at least in this case, are not only different—they lead to different legal conclusions. The column argument leads to the conclusion of one; the row argument leads to the conclusion of half. They have different results. If you disconnect the rule of dayyo, then… Right, if you disconnect dayyo—but it still wouldn’t come out one without dayyo. It could come out 17. Why? Without dayyo, if what is derived by inference must be more than the source, then 17. Why one? 17. Why one? 3.4. Pi. Pi shekels. Why one? Anyway, I’m speaking at the moment where the rule of dayyo is in place. In any case, this is an even stronger indication of what I said before: that in fact the row argument and the column argument are not two different formulations of the same argument. They are two different arguments. And since they are two different arguments, when one falls, the other does not necessarily fall. In the case of the asymmetric table, they are two different arguments even before the refutations. They are two different arguments because they lead to different legal conclusions. So clearly we are dealing with two different arguments. Okay?

The great wonder, as I said, is that nowhere in the literature of the Sages except in these two examples—in Niddah and in Bava Kamma—do we find them rotating a kal va-chomer. What do I mean by rotating? Suppose someone brought a kal va-chomer, a kal va-chomer on this table, and reached the conclusion that it is one; now they brought him a refutation from the moon. Fine? There is a moon. Okay, so that’s it, apparently the kal va-chomer fell; let’s move on. Or they rescue it somehow, but in principle the kal va-chomer fell. Nobody says: wait a minute, wait a minute—you knocked down this kal va-chomer, I’ll switch to this kal va-chomer and keep the one. That is what I call rotating the kal va-chomer: instead of a kal va-chomer like this, I do a kal va-chomer like this. Or if I started with this and a refutation was made here—what’s the problem? I rotate the kal va-chomer into one like this and everything is fine. They don’t do it. The Sages don’t do it. And the question is: why?

Apparently it follows from this that the Sages understood these two considerations as two formulations of the same argument, and not as two different arguments—so if one falls, both fall. And the big question is why. On the face of it, they look like two different formulations. They assume different assumptions. What connection is there between the assumption that horn is more severe than tooth and foot and the assumption that the injured party’s domain is more severe than the public domain? What does one have to do with the other? If one falls, why should the other fall? There is no connection between them. Two completely different assumptions. Two different arguments.

It concerns all the possible damages of the ox. Almost. Here it’s between domains, and here it’s between damaging agents. What connection is there between this hierarchy and that hierarchy? Two completely different assumptions. So when a refutation knocks down one of them, why does the other fall too? The other should have remained standing. But no—whenever they bring a refutation, the kal va-chomer falls. Is the hierarchy always built according to the table, or was there some prior hierarchy that horn… The table. Only the table. Well then, surely it’s connected to that. To attain that horn is more severe than tooth and foot, you already used the conclusion—how? That’s the question. You’re right about the direction in which one has to look, but one still has to work it out. No, because how did you attain that horn is more severe than tooth and foot? From the fact that it is liable in the public domain and tooth and foot are not. Right. In order to get from that to the conclusion that it is more severe, you also need—basically you already assume that liability in the public domain—no, I assume nothing. Apparently I assume nothing. Well, in this case it is liable and this one is not. And in another case it is not—you assume dependence, not severity. I’m saying: why shouldn’t that severity also remain here? What does it have to do with it? Why not? It doesn’t belong at all to the hierarchy between these two. If this is lighter than that, still this is lighter than that, so horn will be liable here for a quarter and tooth and foot will be zero, because it’s lighter, but the hierarchy between them doesn’t depend on the hierarchy between these. Two completely different things.

So how can it be that when the Sages raise a refutation against a kal va-chomer, the kal va-chomer falls? They don’t rotate the kal va-chomer. So here I begin our move. And I want to make the following claim.

What stands behind this kal va-chomer—when we refute the kal va-chomer, what are we actually doing? We are refuting the hierarchical assumption, right? We are basically saying—suppose I brought such a refutation, so I refuted the hierarchical assumption. Let me go back to the case of one now; let’s leave aside the half for the moment because it will just complicate things. Suppose we have here two laws. Let’s say I made here a column refutation. A column refutation basically says that horn is not necessarily more severe than tooth and foot, right? Because there is… so that means that… But after all, horn is more severe than tooth and foot. Here we see that. So how do you reconcile the two? You say that apparently there is some respect in which horn is more severe than tooth and foot, and that generates this column, but there is another respect in which tooth and foot are more severe than horn. Okay? And that generates that column. Liability in the public domain depends on the first parameter, and liability on the moon depends on the second parameter. Right? That is basically what you are saying. And now the question is: does liability in the injured party’s domain depend on the first parameter? Then you can learn it by kal va-chomer. Or does it depend on the second parameter, in which case you can’t learn it by kal va-chomer. Since you don’t know, you can’t learn it. That is how the refutation works.

So notice what I am actually saying. I am actually claiming that in kal va-chomer we assume that horn is more severe than tooth and foot in some parameter—let’s call it alpha, and this is two alpha. Fine? Let alpha be the unit; it doesn’t matter. So alpha units and two alpha units. Thus horn is more severe than tooth and foot.

Now I say this: tooth and foot are exempt in the public domain and liable in the injured party’s domain. What does that mean? It means: in order to be liable in the public domain, what do you need? What level of severity is needed in order to be liable in the public domain? Horn—two alpha, right? Two alpha. Because then the one that has two alpha really is liable. The one that has only one alpha is not enough; it is not liable. So that is the threshold. You need at least two alpha in order to be liable in the public domain, right? By contrast, in the injured party’s domain even tooth and foot are liable, so how much severity is needed to be liable in the injured party’s domain? One alpha, right? And now you can make the kal va-chomer. Since here one alpha is enough, and this one has two alpha, then certainly liability can be imposed; therefore the result is one. Because one alpha is included within two alpha.

Yes. So what does that mean? Notice: what we suddenly discovered, to our astonishment, is that the hierarchy between horn and tooth and foot is in exactly the same parameter as the hierarchy between the public domain and the injured party’s domain. Both are formulated in terms of alpha. They are not different relations of severity; it is the same relation of severity. Because you are basically telling me—it’s just the mirror image of it—the amount of alpha here tells me what intensity of liability that thing has, right? In tooth and foot or in horn. The amount of alpha here tells me how much intensity is needed in order to impose liability. It’s not how much intensity the public domain has; rather, the opposite: what intensity the damaging agent must have in order to be liable in the public domain. But you understand that it is the same alpha that appears here, right? The relation of severity between the domains is formulated in the same units as the relation of severity between the damaging agents. It has to be. Otherwise the whole thing is irrelevant. If in fact in order to be liable here you need alpha, but in order to be liable here you need beta, fine, that would not explain the table. Because if here you need beta, then how is this liable? It only has alpha, not beta. You must have this too formulated in terms of alpha, right? And this would probably be two alpha. Or in other words, by the very fact that you wrote everything in a two-by-two table, you assumed that all these data are laid out by the same parameters. Therefore the relation of severity between the damaging agents is in the same units as the relation of severity between the domains. Otherwise it won’t work.

And therefore—now look at the refutation. I’m anticipating a bit, but suppose the moon: one and zero. What are you actually saying? You are actually saying there is another parameter in play. Because this you will never in the world succeed in explaining in terms of alpha alone. Tell me how much alpha should be written here. Beta? No. There is no value of alpha—neither zero, nor one alpha, nor two alpha, nor three alpha—you will not find a number of alphas you can write here that will explain the table, right? There has to be some parameter beta here. Right? And this one has beta and this one does not have beta, right?

Can you explain again how the alpha relation arises in each of the columns on the side? Before the refutation? Before the refutation, I say: let this be alpha and this be two alpha because it is more severe. Now I ask, what is the relation between these two? I want to explain the data, right? So I say this: the one that has one alpha here is not enough to impose liability, right? But the one that has two alpha is enough. That means that in order to impose liability in the public domain, the threshold is two alpha. Only damaging agents that have two alpha. By contrast, here even tooth and foot are liable, the one that has only one alpha. That means that here the threshold is only one alpha. And that means exactly that the hierarchical relation between these two is the same relation as between these two—it is expressed on the same scale.

Now look: when I do this, I say this table means one thing. Look now at these data: there is no way in the world to explain them with a theory of a single parameter, just alpha, right? There isn’t. Try to find me how many alphas should be written here so that it continues to explain the table. There is no connection. You can’t. You must introduce a parameter beta. But if you introduced beta, then maybe here too it is not two alpha but beta? Sorry—no, not beta, never mind, sorry, this is two alpha, this is alpha, and here is a question mark. Okay? Now I say this one has—sorry, it works the other way. This one does not have alpha and it has beta, and this one has alpha, right? Agreed? This one has alpha, therefore it can’t impose liability here, because here you need beta. This one has beta, so it can’t impose liability here, because here you need alpha, but only here. Right? And everything is fine. And now I ask: what about this one, the third one?

Now here there are many possibilities. For example, what you know is that it is liable, so you can know that this here is beta. Alpha plus beta. It has both alpha and beta, I don’t know, something like that. And then here too there is alpha plus beta. You can propose many models with two parameters, and each of them will give you a different answer here. Right. It will depend on how you explain the middle table—whether this is alpha or beta or the sum of them or whatever. Therefore you cannot infer an unambiguous conclusion. That is why it is a refutation. But notice: now we already understand that this is a refutation of both sides of the kal va-chomer. Because what this refutation actually does is say to me: you can no longer explain everything in terms of a single parameter. You are forced to introduce another parameter. Once you have introduced another parameter, an entire sea of possibilities opens up, and each one will give you a different answer in this table. Therefore this is a refutation. And you cannot rotate the kal va-chomer. Because once I introduced either a column refutation or a row refutation, I have basically forced there to be at least two parameters.

Look: the two columns—there is independence between the two columns in the matrix, right? This column and this column. So if there is independence, that means there are at least two independent axes here, right? At least two independent parameters. Okay? You cannot explain this with fewer than two parameters. Therefore this refutes the kal va-chomer. And you won’t be able to rotate the kal va-chomer. And that is the reason you said that in the Talmud they never rotate a kal va-chomer. Exactly. “Rotate” means to move from columns to rows or from rows to columns. Meaning: there is a refutation, it knocked down the column reasoning, so let’s use the row reasoning. They don’t do that. Or a refutation of a row—so rotate it, use the column reasoning. They don’t do it. Why not? Now it is clear why not.

Because in the end, what stands behind the kal va-chomer—and this is the conclusion—what stands behind the kal va-chomer is a theory. Here we come to the similarity to science, and we’ll talk about it. What stands behind kal va-chomer is a theory. And the theory basically tells me: there is some parameter of severity that I do not know how to identify, I mark it as alpha, I don’t know exactly what it is. And there is some severity parameter that exists in tooth and foot at a certain intensity, in horn at a certain intensity; the public domain requires a certain minimum, and the injured party’s domain requires another minimum. And this is supposed to explain the three data points known to me. Right? From those three data points I can guess how exactly the theory should be built, what the model should look like. Once I guessed that, I can already fill in this square too, because I understood that here there is alpha and here there is two alpha, so obviously this will be one. Two alpha becomes liable in a place where one alpha is enough to impose liability, right? That is how the kal va-chomer works.

Or in other words, what I am doing here is what I earlier called abduction. Abduction. Because what I am actually doing is taking three data points—look at them as empirical, scientific data. In this case they are laws that I find in the Torah. And I have three laws. I build a theory from them, and from that theory I infer a conclusion about a case I did not find in the Torah. Exactly like in science. I take measurements, I do certain experiments, build a theory on the basis of those experiments, and then from the theory I can explain or try to predict what will happen in situations I have not encountered. Right? That is what we are doing here too. It is exactly the same logic. Talmudic logic and scientific logic work in exactly the same way. And I will show you that these tools of Talmudic logic are also the basis of scientific logic. It’s the same thing, in every field—legal, everything. This is basically the basis of soft logic in general, not just Talmudic logic. Okay?

Basically now, if I want to summarize, I would say this. When I have a table like this—I should have had two colors here—but I have a table like this, and then I say: there is a theory that comes to explain the three data points. I begin from three data points. From those three data points I extract a theory. After I extracted the theory—alpha, two alpha, alpha, two alpha—that is what I call a theory, an explanation in terms of parameters. I infer the conclusion that here it should be one.

Now I have an alternative theory for you. Tell me why it is not good. Another theory. Suppose here you need alpha and here beta, and this is beta and this is alpha. That too is a theory that explains the three data points, right? Look. The one that has alpha cannot impose liability here because here beta is required, right? So that is zero. But here alpha is required, so here it is liable. The one that has beta can impose liability here because this is beta. But here you cannot infer a conclusion. Zero. I can infer a conclusion—it is zero. Why zero? Because here is alpha and here it doesn’t have alpha, it has only beta. So you don’t have an answer, you can’t infer a conclusion. The answer is zero, right?

So now look: the answer in the square with the question mark depends on the question of what your theory is, what your model is. But I have two possible theories, after all: either this one, these two and these two, or this one, these two and these two. Right. And each gives a different result. If I adopt this theory, the result is one. If I adopt this theory, the result is zero. How do I know which theory is correct? You use one more parameter. And therefore—oh, more simply—or less plausible, or less simple. Meaning, behind these things sits a principle called by philosophers Occam’s razor. Alpha? Occam’s razor. The principle of Occam’s razor. William of Ockham was a Christian scholar in the Middle Ages, and he argued that in theory, a theory that requires fewer entities is more plausible. Or more simple. Fewer what? Fewer entities. Say I have a theory that explains a certain set of facts, and I have another theory that explains the same facts: if the first theory contains fewer entities, I need to assume the existence of fewer kinds of entities, then it is simpler. I will adopt it. And the assumption basically is that what is simpler should be preferred—it is more likely correct. Not a simple philosophical assumption, by the way.

Provided that alpha and two alpha is preferable, provided it actually works in practice too. What? Provided it also works in practice. Obviously. If something is very simple but does not fit the facts, that won’t help. There’s a Maimonides about a fool like that. What? Maimonides—yes, exactly. That’s Occam’s razor. Exactly Occam’s razor. The signs of a fool there in tractate Hagigah. So basically that’s the presumption from three occurrences. The presumption from three occurrences is Occam’s razor. When you say that an ox gored three times and you turn it into a forewarned ox—why? Maybe it just went crazy three times. Right, that is one possible explanation. Another possible explanation is: no, this is an aggressive ox and it always gores. Which is simpler? It is simpler to assume that the same isolated event didn’t happen three times, but rather that it has a nature of always goring. So Occam’s razor means I choose the simplest theory. So basically what stands behind the logic I’m going to develop here is Occam’s razor. I choose from among the possible theories the simplest theory.

How can you know it is simpler? How do you decide it’s simpler than… So I’m saying here, in this case, I examine all the possible theories that would explain these three data points. Okay? There are many such theories, infinitely many actually. Okay? The simplest among them is this theory. Why? Because it contains only one parameter. The smaller the number of parameters, the simpler the theory. A space of bases with one element is simpler than a two-dimensional space. Fine?

So I propose—what I am basically saying is this: when we have a table—this is a very important conclusion that will accompany us later—when we have a table of data and we want to fill in… it could also be a table like this, you understand. And of course much bigger.

Zero one zero zero one one one zero one zero zero one zero zero question mark one. Fine, those are my data. I collected data from the Torah. Now I ask: what should be here? We will see how to do it. It can be done, and there is a way to do it in a straightforward way. Okay? But even this is done in the same manner. I look for data—sorry, a theory—that will explain all the data except this, everything I know. Once I have the theory, I have a collection of many theories; I choose the simplest one. The simplest one, for me, is the correct theory. And now I ask what it says about the empty square. That is what I will do, that is basically what I will do. What we’ll now see is how this is done in kal va-chomer, how it is done in a paradigm from one verse, from two verses, refutations, generalizations, and then you’ll see that basically for every size of table one can do this in principle. The interesting question is how it is done in practice.

Let’s take the example of the ox that gored three times and really analyze it. That it is forewarned. Mathematically, what options do you have? One option is that it went crazy on three different independent days. It is basically a harmless ox, not naturally goring. But three times something angered it. Something angered it on Sunday, something angered it on Monday, and something angered it on Thursday. But really it is a normal ox, okay? The other possibility is: not at all, this is an irritable ox. Three times there were simply oxen in front of it, so it gored them. On the other days there simply were no oxen in front of it, but it gores every ox that passes before it. Which is simpler? It is simpler to assume that there is one parameter that explains it, not three, because you would need to assume that on three different days something happened that angered it—each time something else. And there is an alternative: to assume it has a goring nature. So that is a simpler theory, therefore you adopt it.

By the way, with the presumption in the laws of menstruation, all the presumptions based on three occurrences in the Talmud are like this. It’s all built on this. Okay? Maybe the dispute between Maharam of Rothenburg and Rabbeinu Peretz there about three times—that is also connected to the matter. But there is a dispute in the Talmud whether presumption is established after two according to Rabbi; Rabban Shimon ben Gamliel says three. It is really a question, because there are those who want to claim that the difference between two and one is not sharp enough. After all, this is soft logic, so the question is how convinced you need to be in order to go with the conclusion. If I had to bet on what is more likely, then even two is better than one. But to tell you whether it is really true, would I act on that basis, would I impose money on that basis? No. Three is already good enough. Okay?

By the way, in the matter of beginning to say “Give dew and rain,” it is ninety, not three. Okay, so basically what I’m saying is this: I have a table of data. What I need to do in order to fill in the empty square is build a theory that will explain the known data, the three known data points. There are infinitely many such theories. From among all those infinite theories, I need to extract the simplest one. Of course, you don’t need to build all of them; very quickly, in simple cases, you see what the simplest theory is. You choose it. Occam’s razor says the simplest theory is the correct one, and that’s it. Once you have the correct theory, you fill in here whatever is dictated by that theory. Fine?

But usually if we have two medieval authorities of the same stature, the same level—Maimonides and the Rosh—I can’t come and say that Maimonides explains it this way and the Rosh explains it that way, and then decide between them. How is that connected to Maimonides and the Rosh? I didn’t come to decide between them. You’re saying I take two explanations, and once I take them, I need to take the simplest explanation. Right. So how can I determine the simplest explanation if there is— I’m not coming to decide between Maimonides and the Rosh. I’m talking now about having data in the Torah, agreed-upon data. Now I am Maimonides—not Maimonides, I am me. And now I ask myself: what is the law in the fourth case? I do the calculation: I check what the possible theories are, choose the simplest one, and infer from it the conclusion. Okay?

Now look: basically, for future purposes, we’ll do this in a slightly different way. What we’ll do is this: we’ll take the two tables, we’ll take the table twice. Fine? This table and this table. Here we will fill in zero and here we will fill in one. It’s simply more convenient to work that way; in principle it makes no difference. And we’ll find the simplest theory that explains all four data points, not just the three. We’ll find the simplest theory that explains these four data points, okay? And the simpler of the two is the correct one, and then I already have the answer. Whichever of them is simpler, I need to take its table; that also gives me the answer. It’s the same thing, of course, right? It’s the same thing—admittedly, mathematically proving that it is the same thing is not so simple, but in principle, at least intuitively, it is fairly clear that it’s the same thing.

And now basically what I’ll do—look, this is exactly what I did before, only now see how to do it systematically. I begin with this—you know what? I’ll make here the one and here the zero because it’s already prepared for me. Okay? So I fill in one, fill in zero, and I want to know which table is the correct one, this one or that one. It’s just another formulation of the same question, basically.

Now I say: the more correct table is the table explained by a simpler theory. Now this one is explained by this theory, right? Alpha, two alpha, alpha, two alpha explains everything, right? One parameter. This table is explained by alpha, beta, beta, alpha, right? Two parameters. Two parameters. You can do other things too, but with fewer than two parameters you won’t get anywhere here. Okay? Since that is so, this table is the correct table. Then I know the answer is one. Okay? It’s the same thing as what I did before, only the order is different. Before, I put a question mark here and explained only these three data points, and asked in how many ways I can explain these three data points. Then I choose the simplest among them and fill it in. Now I’m doing it this way simply to make it more mechanical. I say: fill in one and fill in zero, explain the four data points as if all four were known to me, explain these four data points as if all four were known to me. I choose which theory is simpler. In this case it is this theory, and therefore this is the correct table. Or in other words, here one should write one and not zero. It is the same thing as what I did before, right? That’s clear to you.

Okay? So that is basically what we’ll do later as well. But what is the dependency between alpha and beta at all? There is no dependency—two different parameters. Once you see there are two independent columns or two independent rows, then there you have it: two independent parameters. This is one unit vector and that is another orthogonal unit vector. But according to the conclusions here there is a dependence between alpha and two alpha. There is some connection. That’s why it’s simpler. That’s why it’s simpler. I don’t understand how you attack this theory. A simpler theory? What? Do you have a simpler theory for this? In this table? Please propose one. No—for one parameter you can explain this too. Please do. Give me a one-parameter theory that explains the left-hand table. Alpha, two alpha, alpha, two alpha, and you need to pass three alpha in order to… That won’t help you at all. With one parameter you won’t succeed. How will you succeed? I say alpha, two alpha, alpha, two alpha—I need at least three alpha to get through. To get through what? You write alpha and two alpha there; that is what has to be passed. To mark one, I mean. No! If you write alpha and two alpha, then you are saying that to get one here, alpha is enough. You can’t now say, no no, but to get one you need three alpha. If you want three alpha to get one, then write three alpha here. What is written here is what is needed in order to impose liability. Okay? So if you tell me alpha is needed here, then to impose liability here you need alpha. You can’t say: no no, but in order to impose liability you need three alpha. If you want to need three alpha, then write three alpha. What is written here is what is needed to impose liability. This isn’t wordplay.

I said the combination of these things is… Then write the combination! So you want the top to say alpha plus two alpha—it would say three alpha. And you’ll see it still won’t work. Not three alpha, that’s not… It won’t work. You won’t be able to write a model defined only by different numbers of alpha, different amounts of alpha here and here, and explain this table. You understand that that can’t be done? Because otherwise the hierarchy would remain preserved. That is exactly the connection between the two directions of the kal va-chomer. And here there is no connection between the two directions of the kal va-chomer because they are independent—the two columns or the two rows. And that is exactly what means that you need two parameters. Right—here it’s two hundred… No, no. Obviously. There cannot be a one-parameter theory here. There just can’t. That’s a mathematical theorem that is very easy to prove. There cannot be such a theory. Okay?

If this is two alpha and this is zero, it cannot be that… This is basically a matrix where you ask how many eigenvectors there are, and it is dimension two. It has to be dimension two. What is not clear in terms of the arguments, though, is what alpha and beta represent in the real world. The level of severity or the level of… A parameter… no. A certain parameter present in tooth and foot and in horn, which determines the level of severity. For example: horn intends to damage. Let’s look here. Horn intends to damage, and tooth and foot are unintentional damage. It could be that the Torah understands that if the animal damages intentionally, that makes the law more severe. Fine? For example. Therefore horn has a severity of two alpha because intention is more severe, while this one damages unintentionally so it has only one alpha, for example. So intention is one specific parameter. You can suggest many things. But the nice thing—and we’ll talk about it—is that this whole method does not assume an interpretation of the parameters. I do not need to identify the parameters, who alpha is and who beta is. I claim that there are two parameters, or one parameter, that explain the matter. That’s enough for me in order to infer all the conclusions; I don’t need to identify them. We’ll get to identifying the parameters later, after we develop the model more fully. Then we’ll get to identifying the parameters too. Okay? Good, let’s continue.