2019-04-22 – Between Midrash and Logic – Lesson 9

Table of Contents

• The division between logical hermeneutic principles and textual hermeneutic principles
• Gezerah shavah, inclusions derived from “et,” and the relationship between text and reasoning
• Choosing word-pairs for gezerah shavah: tradition and a textual criterion
• Refutation as an indication: why there are refutations in logical hermeneutic principles but not in textual ones
• General and specific as broader generalizations beyond the barrier of refutations, and their relation to binyan av
• Beginning with the logical hermeneutic principles, and a reservation regarding two verses that contradict one another
• Three types of a fortiori reasoning: “included within two hundred is one hundred,” conceptual a fortiori reasoning, and formal hermeneutic a fortiori reasoning
• A fortiori reasoning in Scripture and what makes it unique compared to other hermeneutic principles
• Formal hermeneutic a fortiori reasoning in Bava Kamma: constructing it from a matrix of three data points
• A fortiori reasoning as an argument that adds information: not deduction, close to analogy, and the role of refutation
• Two stages in a fortiori reasoning and the critique of identifying it as a syllogism

Summary

General Overview

The text divides the hermeneutic principles into two types based on what activates the interpretive move: logical principles, which begin from a substantive connection and reasoning between the law being derived and the source law teaching it, and textual principles, which begin from a linguistic trigger within the verse. It argues that in logical principles there is a concept of refutation, because the inference rests on an assessment of leniency and stringency that can be challenged, whereas in textual principles there are no refutations, because the comparison is imposed by the text itself, and only at a second stage does reasoning guide us as to what issue to apply it to. It proposes that general-and-specific comes to create broader generalizations than binyan av, beyond the barrier posed by refutations, and then moves on to an orderly opening discussion of a fortiori reasoning and a logical analysis of it as an argument that adds information and therefore is not pure deduction.

The Division Between Logical Hermeneutic Principles and Textual Hermeneutic Principles

The text defines logical hermeneutic principles as principles in which the trigger for interpretation is logic and a substantive connection between a teaching law and a derived law, such as a fortiori reasoning, binyan av from one verse, binyan av from two verses, and perhaps also two verses that contradict one another. The text defines textual hermeneutic principles as principles grounded in a trigger found in the text, such as gezerah shavah and general-and-specific in its various forms, where changes of wording and the appearance of a word or pattern compel interpretation even without prior reasoning. The text illustrates this with the a fortiori argument, “If her father had but spat in her face, would she not be humiliated for seven days?”, where the connection between the source and the derived case is substantive, whereas in the gezerah shavah of “to her” and “to her,” the comparison begins from the word itself and not from an analogy between a woman and a slave. The text argues that even when reasoning exists in textual principles, it enters only at stage two, in order to determine with respect to what issue the comparison or inclusion should be applied, and not in order to generate the comparison itself.

Gezerah Shavah, Inclusions Derived from “Et,” and the Relationship Between Text and Reasoning

The text argues that the basis for comparison in gezerah shavah is verbal similarity, and reasoning only filters which areas the comparison is relevant to, because there is no comparison “for every matter and purpose.” The text brings the interpretation of “You shall fear the Lord your God” as including Torah scholars, and attributes the general rule of inclusion to Shimon HaAmsuni, while arguing that the word “et” requires some inclusion but does not define its content. The text uses the fact that Shimon HaAmsuni used to interpret every “et” in the Torah and stopped at “You shall fear the Lord your God” in order to show that the trigger is textual and not conceptual, and afterward the interpreter uses reasoning to direct what thing is “most similar or least far removed” from the Holy One, blessed be He. The text formulates the point that the substantive connection is not the trigger in textual interpretations, but only a supporting condition at the stage of direction and application.

Choosing Word-Pairs for Gezerah Shavah: Tradition and a Textual Criterion

The text presents the claim of the Sages that gezerah shavah may be made only if one received it from his rabbi, and his rabbi from his rabbi, back to Moses our teacher, so that ostensibly there is no problem of selection. The text notes that Nachmanides comments that this claim does not entirely stand up to scrutiny, because there are interpretations about which there are disputes, and not all of them were received by tradition; therefore he suggests that only some of the components are transmitted by tradition, such as transmitting the direction “from here to there” without specifying what exactly is to be derived, or transmitting the law without the source and the interpretation comes only as support. The text also cites a scholarly proposal by Michael Chernick, according to which early gezerah shavah interpretations were made between unique word-pairs in Scripture that appear only in the two required places, as a textual criterion that guides the choice.

Refutation as an Indication: Why There Are Refutations in Logical Hermeneutic Principles but Not in Textual Ones

The text argues that in logical hermeneutic principles there are refutations because the conclusion is built on reasoning about stringency or similarity, and a refutation tests whether the relationship of leniency and stringency really holds up in light of additional data. The text argues that in textual hermeneutic principles there are no refutations because the comparison arises from the command of the text itself and not from the interpreter’s analytical decision, and therefore claims of the form “What about a woman, who is different because…” do not undermine the comparison itself. The text qualifies this by saying that in a gezerah shavah that is not mufneh, one can refute it, and explains that this is because then it reverts to the pattern of binyan av. The text adds that deciding whether a word is mufneh also involves a reasoned consideration, because the claim that it is superfluous depends on the understanding that “even without this word I could have known it.”

General and Specific as Broader Generalizations Beyond the Barrier of Refutations, and Their Relation to Binyan Av

The text presents the principles of general and specific, specific and general, and general and specific and general, as generalizations of different radii, and says that it will discuss them later. The text explains that binyan av is a limited generalization because a refutation sets its boundaries, and wherever there is a special feature that distinguishes the derived case from the source case, the generalization stops. The text argues that the principles of general and specific are meant to generate broader generalizations that continue even beyond the points where binyan av would stop because of a refutation. The text states that in general and specific “there will always be a refutation of the generalization,” because that is precisely its point of uniqueness, and that if there is no refutation then it is really a case of binyan av and not general and specific.

Beginning with the Logical Hermeneutic Principles, and a Reservation Regarding Two Verses That Contradict One Another

The text states that it will begin with the logical hermeneutic principles: a fortiori reasoning and the two forms of binyan av. It leaves aside two verses that contradict one another, because at least from its perspective it “doesn’t enter that playing field” and it does not see how it connects to the other three principles, even though it may be logical. The text says that the other three principles “play on the same field,” and moves from introductory remarks to examples within the principles themselves.

Three Types of A Fortiori Reasoning: “Included Within Two Hundred Is One Hundred,” Conceptual A Fortiori Reasoning, and Formal Hermeneutic A Fortiori Reasoning

The text divides a fortiori reasoning into three types and develops especially the first and the third. It presents the type of “included within two hundred is one hundred” through the example, “If one is liable for opening, then for digging all the more so,” and attributes to the Maharsha the claim that opening is included within digging, so this is a mathematical relation in which “one hundred is included within two hundred.” The text also attributes to the Maharsha the claim that such an a fortiori argument has no refutation, and therefore in his view one might even “impose punishment on the basis of legal reasoning,” and it brings his disagreement with Tosafot and the Mekhilta around the duplication “if a man opens… or if a man digs” and the rule “we do not impose punishment on the basis of legal reasoning,” while also mentioning the discussion whether monetary penalties may be imposed on the basis of legal reasoning and the dispute among the medieval authorities (Rishonim) and Tosafot about conspiring witnesses. The text in practice disputes this distinction and brings an example from the Hundeverbode law in Belgium, where a court permitted three liters of wine even though two liters had been prohibited, to show that even in “included within two hundred is one hundred” a “refutation” can appear by way of the purpose of the law. It adds that the alternative approach of the Kesef Mishneh regarding punishment based on legal reasoning also creates another kind of refutation concerning the transfer of the lighter punishment to the more severe case.

A Fortiori Reasoning in Scripture and What Makes It Unique Compared to Other Hermeneutic Principles

The text argues that a fortiori reasoning is an exceptional principle because Scripture itself explicitly makes use of it, unlike gezerah shavah, which does not appear as an interpretive operation within Scripture itself. The text brings examples of a fortiori arguments from Scripture 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 spat in her face, would she not be humiliated for seven days,” and “Behold, the money that we found in the mouths of our sacks,” and defines them as a fortiori arguments based on reasoning. The text mentions the Raavad’s claim that even two verses that contradict one another is a principle that the Torah itself uses, and says that this is a separate issue.

Formal Hermeneutic A Fortiori Reasoning in Bava Kamma: Constructing It from a Matrix of Three Data Points

The text defines “formal hermeneutic a fortiori reasoning” as an a fortiori argument of the Oral Torah that is based on laws and not on direct reasoning, and illustrates this with the Bava Kamma table of the public domain and the injured party’s courtyard, as against tooth and foot and horn. The text sets out the data: tooth and foot in the public domain are exempt, horn in the public domain is liable, and tooth and foot in the injured party’s courtyard are liable, and from there it infers the fourth datum, that horn in the injured party’s courtyard is liable, with a dispute between Rabbi Tarfon and the Sages whether the liability is half-damages or full damages. The text shows two ways of constructing it: deriving a relation of stringency from one row and then applying it in the second row, or deriving a relation of stringency from one column and then applying it in the second column. The text argues that in this type of a fortiori reasoning, instead of external reasoning there enter “two additional halakhic data points” that generate the reasoning from within the laws themselves.

A Fortiori Reasoning as an Argument That Adds Information: Not Deduction, Close to Analogy, and the Role of Refutation

The text argues that hermeneutic principles, and especially formal hermeneutic a fortiori reasoning, add new information, and therefore are not ordinary deductive logic, which does not add information but only exposes what is already implicit in the premises. The text notes that arguments that add information are not certain, and therefore are vulnerable to refutation, and it parallels this to the distinction between mathematics, which has no refutations, and science, which is decided by experiment and falsification, illustrating this with examples of adding oranges as opposed to adding vector forces. The text explains that this kind of a fortiori argument is “a kind of analogy,” in the sense of an analogy in hierarchical relations of leniency and stringency between rows or columns, and not an analogy of identical laws. The text defines refutation as the discovery that the relation of leniency and stringency that had been assumed is not correct in light of other data, such as finding another category of damager or another domain that would reverse the relation of stringency.

Two Stages in A Fortiori Reasoning and the Critique of Identifying It as a Syllogism

The text cites Adolf Schwarz and his claim that a fortiori reasoning is the Greek syllogism, and defines this as a mistake, because “it is written nowhere that horn is more stringent than tooth and foot”; rather, that is a generalization learned from the data. The text explains that a fortiori reasoning is built in two stages: a first stage of generalization from two data points in order to create a general principle, and a second stage of clean deduction once the principle has already been formulated. The text argues that the refutation does not attack the deductive inference in the second stage, but attacks the generalization in the first stage from which the general premise was created. The text compares this to John Stuart Mill’s critique of deduction, according to which even “all human beings are mortal” is a generalization from observation and therefore can be breached if an exception is found, so the certainty applies only to the move from premise to conclusion, and not to the premise itself.

Last time we spoke about the hermeneutic principles in general, and I divided them into two types of interpretive principles: logical principles and textual principles. The logical principles are a fortiori reasoning, deriving a rule from one text, deriving a rule from two texts, and maybe also two verses that contradict one another. And the textual principles are principles whose basis is a trigger in the text. For example, if I make an a fortiori argument, say, “If her father had spit in her face, would she not be humiliated for seven days?” So if she is humiliated before her father, then she needs to be humiliated for seven days. Then if she is humiliated before the Holy One, blessed be He, that should also be at least seven days, maybe even fourteen, as the Talmud says there. What is that kind of consideration based on? When I derive the new idea that if she is humiliated before the Holy One, blessed be He, then she should be humiliated for fourteen days, how is that connected to the law I started from? To the law that says that if she is humiliated before her father, then it is seven days? So clearly the connection between the law being derived and the law from which it is derived is a substantive connection. Since her father has some kind of binding status—yes?—such that one may not humiliate him, and that obligates seven days, then the Holy One, blessed be He, whose status is even higher, certainly obligates some kind of humiliation of at least seven days. The connection between the law being derived and the law from which it is derived is a substantive connection. Okay?

By contrast, in a verbal analogy we say there is a verbal analogy of “to her” and “to her” from woman, so I compare a slave to a woman. In both places the word “to her” is written, and the principle of verbal analogy says that when the same word appears in two passages, we compare them. What is that based on? We don’t start from some reasoning that says a woman resembles a slave in some sense. There is no such reasoning. We start from the text. In the text it says “to her” here and “to her” there. The text instructs us to make a comparison. We don’t make the comparison because of some reasoning we have—because this is similar, because this is more stringent, because this is more lenient, all kinds of reasoning connected to how we look at things. We begin with what the text says.

Actually this example raises all the difficulties, because slaves and women always appear together in the words of the Sages even before they come and ask about the verbal analogy of how we know that acquisition works this way or that way. Yes, they appear there also many times with minors and various things. That has nothing to do with the comparison of those who are not legally obligated. Usually it’s in the context of those who are not legally obligated, but it doesn’t seem to me that that’s relevant here. What, because these are modes of acquisition? Yes, those are different laws from the degree of obligation, also different from the degree of obligation in commandments. I don’t think that’s the similarity here. But never mind—so take another verbal analogy. It doesn’t matter.

But the question is: to what extent—especially with verbal analogy, which looks like something completely unrestrained—did they really not find some basis of analogy before they even decided on the verbal analogy? I think not. Because otherwise, why do you need the verbal analogy? Analogy is deriving a rule. Without that… So it’s like Maimonides’ hierarchy of categories. You can’t make a textual principle if there’s no aspect of content. No—so I’ll get to that in just a moment. But that’s why I’m saying… The wording “certainly there is no difference at all and it definitely can’t…” There is some… there is no connection. There is a connection, but it’s not strong enough to stand on its own. But that connection is not the trigger for the exposition. I’ll get to that in a moment. It’s not the trigger…

The trigger for the exposition is the similarity between the words. That is the trigger for the exposition. After I have an exposition that compares slave and woman, I ask myself: in what respect am I comparing them? After all, I’m not comparing slave and woman in every respect, right? There are things… I’m not allowed to enslave a woman, even though I can enslave a slave. It’s not a similarity in every respect. I also don’t write a woman as compensation for an eye just because a slave is written with regard to an eye as well, right? Meaning, there are things that I exclude on the basis of reasoning, because they tell me in what respect to make the comparison. But that is not the basis of the comparison. The basis of the comparison is textual. From the fact that it says “to her” here and “to her” there. Reasoning enters at stage two. After you ask yourself: “Okay, the Torah tells me to compare slave to woman. In what respect?”

Or: “You shall fear the Lord your God”—to include Torah scholars. So the word “et” comes to include, as Shimon HaAmsuni says. That’s an example we’ve already returned to several times. Fine? Maimonides brings it in the second root. So the word “et” comes to include. But the word “et” doesn’t tell you what to include. What? Maybe Roman standards? As something like the Holy One, blessed be He? What am I supposed to include? Clearly this is where the interpreter’s reasoning comes in. The interpreter says: what is most similar, or least distant, let’s put it that way, from the Holy One, blessed be He? So he says… Torah scholars. Therefore the verse probably instructs me to include fear of Torah scholars.

So clearly reasoning is involved, but the reasoning is not the trigger for the exposition; rather it directs the exposition. The word “et” causes you to include something regardless of any reasoning. By the way, with inclusions based on “et” we see this most clearly, and I already mentioned this too. Shimon HaAmsuni used to expound every “et” in the Torah. So if he expounded every “et” in the Torah, then clearly he began with the rule, not with the idea, because in every place where “et” was written he expounded it. If he had begun with the idea and only afterward found the “et,” then there would be some occurrences of “et” that he expounded and some he didn’t. More than that: he came to “You shall fear the Lord your God” and stopped. There he got stuck, because what can be likened to the Holy One, blessed be He? Nothing. He couldn’t figure out what could be included as resembling the Holy One, blessed be He. So why did he begin at all? Why did he look for something in the first place? Rather, clearly, if the word “et” is written, the textual trigger alone is enough to make the exposition. Afterward, of course, the reasoning enters and tells me: okay, what do I now do with this textual trigger—what do I compare, include, or whatever. But I think that comes at stage two.

The question of how one chooses pairs of words on which to make a verbal analogy is a very interesting question. The Sages claim that verbal analogy is used only if one received it from one’s teacher, and the teacher from his teacher, all the way back—as Rashi at least always adds—and his teacher from his teacher all the way back to Moses our teacher. Fine? Then the question doesn’t arise. If there is already a tradition, then there is no problem. But Nachmanides already comments that this does not exactly stand up to scrutiny. It is clear that there are expositions over which there are disputes, and they were not exactly received by tradition. So he wants to claim that only some part was received by tradition. Maybe they told you: make an exposition from here to there, but they didn’t tell you what to expound. Or they gave you the law but didn’t tell you where it comes from, and you construct the exposition that supports the law. Or all kinds of things of that sort. Nachmanides among the medieval authorities (Rishonim) already comments on this.

But in any case, it starts from the text. There’s one suggestion I once saw—I’ve forgotten his name—someone who wrote a book on verbal analogy, Michael Chernik. Michael is a scholar, he’s at an institute in Lod, some institute—what institute is it? I don’t remember anymore. In Lod, a publisher of scholarly Torah books and so on. So a book on verbal analogy came out there, and there he has a very interesting claim. Of course he deals in archaeology, as studies often do. And in that archaeology he discovers that the earliest verbal analogies were all made between pairs of words that are unique in the entire Bible. So one makes a verbal analogy only where the pair of words being compared appears nowhere else in all of Scripture, only in those two places. I don’t remember exactly anymore. He brings there the list of verbal analogies. And that’s a very interesting point, because it really says that if so, then at least at that stage there was a textual criterion for how I choose which pairs of words to expound and which not. Because that is really what you assumed earlier—that something has to guide me; I don’t make a verbal analogy out of every two words. That’s why you said that somehow reasoning has to be involved here. So here is another solution, one solution. Another solution is what I said earlier—just a second—that it comes by tradition, and then too I don’t need to make decisions of that kind. I don’t know, early Tannaim. There are verbal analogies among the Tannaim. There are verbal analogies in Tannaitic sources, obviously there are. He’s speaking, as far as I remember, about early Tannaim, not in the Talmud, that’s how it seems to me if I remember correctly—but that would have to be checked. I looked at this many years ago; I don’t remember anymore.

So the difference between logical principles and textual principles lies, is rooted, in the question: what is the trigger for the exposition? Is the trigger for the exposition logic, the substantive connection between what is learned and what teaches it? Fine? Is there some similarity or relation of leniency and stringency between them, something I derive from the contents involved in the exposition—logic, my understanding of the contents. The same, by the way, with two verses that contradict one another: there too I understand that there is some contradiction because I understand the contents involved. Therefore I think that perhaps that too can be included among the logical principles.

By contrast, expositions like verbal analogy or general and particular and things like that—these are textual expositions. In verbal analogy, as I said earlier, we don’t begin from the similarity, from the contents, but from their textual expression. If a word appears here and that same word appears there, we make a verbal analogy. The same is true of all the principles of general and particular: general and particular, particular and general, general and particular and general—all these principles too, simply because the Torah changes the wording from a singular formulation to a plural formulation, that itself instructs me to expound. Not because in my own logic I understand that something should be included. Therefore all those are textual principles.

I said that one of the strongest indications of the distinction between these two types of principles is the existence of refutations. In logical interpretive principles there are refutations. If you think A is more stringent than B, let’s check whether that withstands scrutiny. That’s what you think. But if, for example, specific laws show me that A is not more stringent than B, then there is a refutation; apparently you were mistaken. Everything begins from your reasoning, so you need to check whether there is a refutation—your reasoning is not correct. But in textual principles there are no refutations. You will not find refutations against textual principles. There is no such thing. Why? Because when the Torah tells us to make a verbal analogy, what will a refutation do? “What about a woman, who has such-and-such…” I don’t know, that she can be subjugated? It’s irrelevant. The Torah tells me to compare them, not that I compare them because I think they are similar. So you can bring me a refutation and show me that I’m mistaken, that they are not similar—but from the outset I am not proceeding on the basis that they are similar. I am not comparing them because they are similar. I compare them because two identical words are written, and the Torah instructs me to compare them. So it is not my decision; therefore a refutation cannot overturn it. Only decisions that are mine, from my own reasoning, can be overturned by a refutation. Fine? But things that are textual begin from the text, and a refutation will not overturn that.

Yes. What’s different is, there seems to be a case of verbal analogy where if it is not free, then it can indeed be refuted. Right, because if it’s not free, then as I said, it is deriving a rule. It is deriving a rule, but then it returns to the same point that verbal analogy has to involve a free word, so certainly there is here something very, very prominent—it is more the category of textual pointing, very sharp, textual. Right. Fine. But of course, deciding whether it is free or not—how do you decide whether it is free or not? By your reasoning, right? Whether it is free involves reasoning that can be debated. Fine. There is no free… although I don’t know—no, there are, there are, there are debates about this. Free on one side, free on both sides—this one says it is free on one side, that one says it is free on both sides—there are debates over free words. They say they are required for the body of the law, required for this, required for that. That’s what I mean. In other words, the freeing too is itself something somewhat related to reasoning. Because when you say that a certain word is superfluous, you are really saying that even without that word I could have known what it says. How could I have known it? From your reasoning. So the consideration of whether something is free or not does not completely solve the problem, because there is still some entry point here for reasoning. But fine, that is the basic division. The lines are less sharp than I describe them here, but it seems to me that they do exist.

So the concept of refutation is really the indication for the distinction between textual interpretive principles and logical interpretive principles. And as I think I mentioned last time—and if not, then I’ll add it only now, though I think I did mention it—general and particular, when we use the principles of general and particular, are principles of generalization. Generalizations at different levels. General and particular, particular and general, general and particular and general—each of these is a generalization in circles or radii of different breadth. That is really the difference between these principles. We’ll discuss them more extensively later.

Why do we need them? After all, we already have generalization through deriving a rule. Deriving a rule from two texts, from one place—those are also kinds of generalization. The answer is that we need… yes, now I remember that I said this because Naftali asked something here. The reason we need it is because deriving a rule is a logical principle, and a logical principle gets stuck where there is a refutation. That is, if you have a certain source and you derive from it, by deriving a rule, that the law in question exists there too, in the second context as well—now a refutation comes and says: what about the second context, which has some special characteristic? You cannot derive from it with respect to that. Then we do not make the derivation. Meaning: the generalization whose basis is deriving a rule is a very narrow generalization. It is a generalization whose boundaries are determined by refutation. It will not extend into places that are special relative to the teaching source, places where there is a refutation characterizing them as opposed to the teaching source. So the generalization of deriving a rule is a very narrow generalization.

General and particular are principles whose purpose is to generate broader generalizations than deriving a rule. Generalizations that break through the boundary of refutations. Meaning, where the refutation would stop the simple generalization—if it had been based on deriving a rule—the refutation says to me: wait, you can’t get there, because it’s not similar. You are making the generalization on the basis of thinking it is similar. But the refutation tells you: no, there it is not similar; that is beyond the radius of generalization. That is what general and particular are for. General and particular always, always, make generalizations beyond the barrier of refutations. That means there will always be a refutation against the generalization of general and particular, because if there were no refutation and yet you still make it, that means—it would not refute the general and particular. On the contrary: if there is no refutation there, then it is not general and particular. It is deriving a rule.

In other words, generalizations that are not stopped by a refutation I can make even without general and particular. What do you mean? I make them from my own reasoning. If it is true here, then it is true in all similar places—unless there is a refutation and those places are not similar. Then I will not do it with deriving a rule. For that there is general and particular. General and particular tell you: go there even though it is not similar. Now the question is how far into the depth… of the refutations to go. How many refutations are there? One, two, three? Depending on how many refutations there are, those are the different principles of general and particular. There is general and particular, there is particular and general, and there is general and particular and general. These principles differ from one another in the question of what the radius of generalization is beyond the zone of refutation. Fine, that is basically the relation. Here you can see very clearly the relation between a logical principle and a textual principle, because they simply complement one another. The radii of generalization are different, because in deriving a rule the radius of generalization is very narrow. In all places where there is a refutation, you can no longer do it, because it all begins from my reasoning. And general and particular also do something rather narrow, particular and general do something very broad, and general and particular and general do something in between. How do you define what “in between,” “narrow,” and “broad” mean? We’ll get there when we discuss the principles of general and particular, because that is defined very precisely by the Sages. It appears explicitly in the Talmud. So we’ll talk about that when we get there.

But I want to begin with the logical principles. In the logical principles, as I said earlier: a fortiori reasoning and the two forms of deriving a rule. I am leaving aside for the moment two verses that contradict one another, because at least for me it does not enter the same field. Even if it is a logical principle, I don’t see how it connects to those three principles. But those three principles do play on the same field.

Okay, so I’ll bring a few examples so we can see what we’re talking about. Here we’re really beginning to enter into the principles themselves; we’ve more or less finished the general introductions. Now I’m beginning to go into the principles themselves. So let’s perhaps start with a fortiori reasoning. Fine?

There are, one could say, three types of a fortiori reasoning. One type is the a fortiori of “included in two hundred is one hundred,” as the authors of rule-books put it—the Ginat Veradim and the Maharsha and various authors of principles, that’s what they call this type of a fortiori reasoning: included in two hundred is one hundred. What does “included in two hundred is one hundred” mean? For example, if I say—as the Talmud says—if one is liable for opening a pit in the public domain, then for digging one all the more so. Right? If when you open a pit in the public domain—an existing pit—you open it and you’re liable to pay if someone is injured by that pit, then what happens when you actually dig a brand new pit? Not opening an existing pit, but digging a new pit. Then certainly you should be liable. If one is liable for opening, then for digging all the more so.

Now what is the relation between opening and digging? The Maharsha, in the second edition on Bava Kamma 47, I think, says there that the relation is “included in two hundred is one hundred.” Opening is actually included in the act of digging. The act of opening is simply part of digging. When I dig a new pit, I dig the pit from above—I don’t start from below, of course; this isn’t the capitaria in Tiberias—but I start from above, and I dig the whole depth of the pit, including in particular the upper layer. So the act of digging actually contains within it also the act of opening. Right? It’s simply included within it. Therefore it’s obvious that everything I know about opening a pit will also be true about digging a pit, and more. Right? But at least this is an a fortiori relation that is simple.

The Maharsha claims that on this there is no refutation. It is an a fortiori argument to which there is no refutation. It cannot be refuted. How could it be refuted? One hundred is included in two hundred. What exists in one hundred also exists in two hundred. A refutation can at most show—we’ll see this in more detail later—that the relation of leniency and stringency is not a correct relation. That there is no relation of leniency and stringency between the law being learned and the teaching law. That is the purpose of a refutation. But in an a fortiori of “included in two hundred is one hundred,” there can never be a refutation. The relation of leniency and stringency is a mathematical relation. This is one hundred and that is two hundred. This is opening and that is opening-plus, so it is contained within it. Therefore, says the Maharsha, there can be no refutation here.

So for example, the Maharsha claims that “if one is liable for opening, then for digging all the more so”—this is in monetary law. There is a Tosafot and a Mekhilta there; there is a dispute between Tosafot and the Mekhilta in Bava Kamma 2. It seems that he disagrees with the Mekhilta, because Tosafot brings some Mekhilta that says: if one is liable for opening, then for digging all the more so—so why does it say, “If a man opens a pit” or “if a man digs a pit”? Why do we need both? Tell me only opening and I will know digging too by a fortiori reasoning. So Tosafot says: to teach us that we do not derive punishments from logical inference. The Torah wrote both in order to teach that we do not derive punishments from logical inference. But the Maharsha claims that the Talmud there on 47, I think, or 51 or somewhere there later in Bava Kamma, derives something else from that duplication of digging and opening. So the Maharsha says: why? Because the Talmud says that in monetary matters we do derive punishment from logical inference. Fine? So…

The Maharsha’s claim is essentially that from this a fortiori we can impose punishment by logical inference. Why? Because there is no refutation against it—not only in monetary law, really, but in general. Because there is no refutation against it. The whole problem with why we do not derive punishment from logical inference—it’s not that we do not punish by force of an a fortiori argument, but that perhaps there is a refutation against the a fortiori. But an a fortiori of “included in two hundred is one hundred” will never have a refutation against it. Therefore you can punish on its basis. So that is the second explanation. The Maharsha holds that the reason is perhaps there is a refutation, not the explanation of the Kesef Mishneh. But why do we not derive punishment from logical inference in monetary law? He isn’t talking about monetary law; Tosafot is talking about monetary law. The Minchat Chinukh—someone will pay this or he’ll pay that. The simple assumption is that tort compensation is a kind of punishment. A very problematic assumption. One could give a separate lecture on that, but that is the assumption there. But if there is no such thing, then does the principle “we do not derive punishments from logical inference” apply to all monetary law? Not necessarily. It is a dispute among the medieval authorities (Rishonim) whether monetary penalties are derived from logical inference or not. Tosafot there on the spot talks about this, and with conspiring witnesses too; there is a Tosafot on this at the beginning of Bava Kamma. Tosafot in Bava Kamma. The question is whether it is money or a fine, and Tosafot ties to that the question whether one derives punishment or does not derive punishment.

The a fortiori regarding conspiring witnesses—that if one merely plotted, then he is liable, so certainly if he actually did it, he should be liable. A fortiori. So the question is whether monetary penalties are derived from logical inference or not. In short, it is a dispute among the medieval authorities (Rishonim). But it doesn’t matter, we won’t go into it now. For the Maharsha it isn’t important. In an a fortiori of this kind, one may derive punishment from logical inference wherever you want, because there is no refutation against it. Maybe that goes against the idea, because the Mishnah says explicitly: “as he plotted,” and not “as he did.” The Mishnah does not say “as he plotted and not as he did.” That exposition appears nowhere in rabbinic literature. It is only Rashi, not even an Amora. Only Rashi at the beginning of Makkot. And only for conspiring witnesses is it like that? There are, yes, similar principles in other formulations, but “as he plotted and not as he did” is only Rashi. He can say something else at least with conspiring witnesses. He can say that the logic is not that there is no refutation; there really may be no refutation, but the logic is not that. You’re suggesting another possibility. I’m not entering the sugya of conspiring witnesses. One could discuss those suggestions; there is much to elaborate. The Maharsha claims, interprets it this way. For me it’s only an example. He wants to claim that an a fortiori of “included in two hundred is one hundred” has no refutation against it, and since there is no refutation against it, the implication is for example in his view that one may also derive punishments from logical inference. I don’t care—I’m not now entering the sugya of deriving punishments from logical inference. I’m only bringing an example to show what one does with such a claim. By contrast, another a fortiori argument on which there can be a refutation, says the Maharsha there, indeed punishments would not be derived from logical inference.

So that is a fortiori reasoning of the first type, the a fortiori of “included in two hundred is one hundred.” Perhaps one more comment on this matter—I mentioned this also at the Nitzotzot conference, for those who were there. There are refutations even against an a fortiori of “included in two hundred is one hundred.” Unbelievable, but there are. I gave an example there. There is a Belgian law called the Vandervelde law. Some place in Belgium where there was a law forbidding the sale of two liters of wine in taverns. And a question was raised before the court there: are three liters permitted? Seemingly that is “included in two hundred is one hundred.” If two are forbidden, then surely three are forbidden, right? An a fortiori with no refutation, says the Maharsha. So now listen to the refutation. The court there said it was permitted. Not two. Because what was forbidden was selling two liters, since that almost consumed the entire weekly wage a worker earned. He wouldn’t bring his wages home; they would have nothing to eat afterward. So they didn’t want him buying wine with it, and it was forbidden to sell it to him. But three liters cost much more than a worker’s weekly wage, so he wouldn’t buy that. Therefore that was permitted to sell. Maybe if he came once every two weeks to buy, I don’t know. The Belgian legislator probably didn’t think of that. But in practice we see here an example of a case where an a fortiori of “included in two hundred is one hundred,” which seems to be an a fortiori with no refutation against it—that’s not true. There is no a fortiori with no refutation against it. None.

And one of the refutations, by the way, is exactly the alternatives you kept mentioning earlier to the Maharsha. When we say that punishments are not derived from logical inference, the Maharsha says why? Because perhaps there is a refutation. Let me give you a refutation against what he says. It could be that punishments are not derived from logical inference because the punishment is insufficient, as the Kesef Mishneh says. Really, you deserve the first punishment, the basic punishment, but it will not be enough. This is after all a more severe transgression, a transgression learned by a fortiori reasoning. So perhaps that one needs a more severe punishment. Therefore don’t punish him with the lesser punishment. So here we have a refutation of the a fortiori argument. Notice, in practice this comes out as a refutation. Because if you want to learn the punishment by a fortiori reasoning—if for A one receives lashes, then B is more severe than it, so certainly one should receive lashes for it. Now suppose B is “included in two hundred is one hundred” compared to A—it is literally opening and digging. Fine? Suppose that for opening one was liable to lashes, okay? And for digging too one wants to claim liability by force of the law. Then he says: not at all. It could be that because digging is more severe, lashes are not enough, so you need more than that. So notice that what we have here is a kind of refutation. That is a refutation. The conclusion you drew is not correct to draw. That too is a kind of refutation.

So here too, even in a place where there is an a fortiori of “included in two hundred is one hundred,” the very alternative suggested to the Maharsha produces a refutation against this a fortiori. It doesn’t contradict the fact that it is more severe; that’s why I say it’s a special kind of refutation. But in Vandervelde, for example, it does contradict that. It really contradicts that, and that is why it is even more interesting. In various ways one can attack even an a fortiori of “included in two hundred is one hundred,” because in rabbinic literature one sees rather clearly that they did not distinguish. An a fortiori—just a second—that an a fortiori of “included in two hundred is one hundred” among the Sages is considered an ordinary a fortiori in every respect. These innovations, that there is no refutation against it and so on, are inventions of later authorities (Acharonim). In plain terms, in the Sages there is no such distinction.

It says, “his sister, the daughter of his father or the daughter of his mother.” They say: if it says the daughter of his father, then clearly that also includes one who is the daughter of both his father and his mother, right? This teaches you that punishments are not derived from logical inference. What do you mean? This is a law of a fortiori “included in two hundred is one hundred.” What does “we do not derive punishments from logical inference” have to do with it? Rather, you see that even against an a fortiori of “included in two hundred is one hundred” there can be a refutation. It is established that they say with one who passes his seed to Molech: “from all his seed,” and not all his seed—only the Kesef Mishneh from earlier here… In any case, that is the first type of a fortiori.

The second type of a fortiori is a fortiori from reasoning. The midrash brings ten a fortiori arguments that appear in Scripture. Right? “Behold, the children of Israel have not listened to me, so how will Pharaoh listen to me, and I am of uncircumcised lips,” for example. Or “If her father had spit in her face, would she not be humiliated for seven days?” Or “Behold, the money that we found in the mouths of our sacks”—that’s exactly from the Torah portion of a week or two ago. So all these are a fortiori arguments that appear in Scripture.

Now what does it mean that they appear in Scripture? Notice: all the hermeneutic principles, almost all of them, are principles given to us as a law to Moses at Sinai, and Scripture itself serves as the basis on which we apply those principles. But not that Scripture itself uses those principles. Fine? But there are certain principles in which Scripture itself makes use of them. In Scripture itself one sees a fortiori reasoning. In Scripture itself one does not see verbal analogy. In Scripture itself the word “to her” appears here and “to her” appears there, and I make a verbal analogy because I have the law to Moses at Sinai of the thirteen interpretive principles. But a fortiori is a principle that appears explicitly in the Torah; the Torah itself makes an a fortiori argument. Therefore it is a very exceptional principle. Fine? Essentially, the Torah itself makes an a fortiori argument. The Raavad claims that also two verses that contradict one another is a principle that the Torah itself uses. But that is a topic unto itself. He claims that a fortiori and two verses are the only principles in which the Torah itself makes the exposition.

Now all the a fortiori arguments that appear in these midrashim, that appear in Scripture—the midrashim merely collect them—those are a fortiori arguments based on reasoning. Meaning, the greater stringency of the Holy One, blessed be He, in relation to her father is an assumption of the interpreter, an assumption based on reasoning. He did not prove it from some laws in the verses. He assumes it—a simple reasoning, yes—but he assumes that the Holy One, blessed be He, is more stringent than her father. So the basis for the a fortiori is a basis of reasoning. That is the second type of a fortiori. The a fortiori of “included in two hundred is one hundred” is of course also an a fortiori of reasoning, but it is a stronger a fortiori in terms of reasoning. You could say that those two are really one type split into two—not important.

The third type, with which we will deal from here on, let’s call it for the purposes of discussion here a formal a fortiori. An a fortiori in which the principle itself is expressed; that the verse itself does not do. This is an a fortiori of the Oral Torah. And this a fortiori is based not on reasoning but on laws. Fine? For example, the famous a fortiori of the Sages, the Mishnah in Bava Kamma—the tables here will accompany us a bit from here on, so it’s worth getting used to them. Public domain and the damaged party’s courtyard, tooth and foot and horn. Yes, there, take a look. What? Ah, horn. There’s an eraser here on the table. Great. Ah yes, yes. Maybe I said that just to get some backup. Moral support.

Anyway, the Mishnah in Bava Kamma 24 says: tooth and foot in the public domain are exempt. Let’s mark that as zero—not obligated to pay, fine? Horn in the public domain is liable. Let’s mark that as half, but call it one for the moment. Liable. Fine? Tooth and foot in the damaged party’s courtyard are also liable, okay? And the question is: what is the law of horn in the damaged party’s courtyard? That is essentially the question the Talmud discusses—Rabbi Tarfon and the Rabbis. But clearly it is liable here. The question is whether it is liable for half or liable for one; that is the dispute of Rabbi Tarfon and the Rabbis. Ox or son… What? No, I’m reading it. Ah, “ox” is tooth and foot. Tooth and foot, yes. So with horn in the damaged party’s courtyard, it is liable. Fine? That is the a fortiori.

What is this a fortiori based on? This a fortiori is not based on reasoning. No reasoning entered here anywhere. I have three laws, and from these laws I derive a fourth law. Why? Tooth and foot are common damages, horn is uncommon damage, the public domain is a place… No, no, no, no. If you need that, then you don’t need the top row. Then just do it straight to here. Take tooth and foot in the damaged party’s courtyard—it is liable, even though it is common; then horn, which is less common, surely should be liable. Why do you need these two? Why do I need what? These two? No, that’s the opposite a fortiori, to strengthen it, even though… even though… What do you mean? It isn’t “even though”; it’s the same direction. Horn is more stringent than tooth and foot. Clearly that doesn’t come from reasoning. I’ll explain in more detail in a moment where reasoning does nevertheless enter here. This is an a fortiori that does not come from reasoning.

I have three halakhic data points, and from them I derive a fourth halakhic data point. Notice that all the a fortiori arguments I mentioned earlier, a fortiori arguments based on reasoning—“if her father had spit in her face,” or “behold the money that we found in the mouths of our sacks”—all those a fortiori arguments are based on one datum, not on three. This a fortiori is based on three data points and yields from them a fourth. Those a fortiori arguments are based on one datum. “If her father had spit in her face,” she must be humiliated for seven days. That’s it. There are not two data points there. So if the Holy One, blessed be He, spits in her face, who is more stringent than her father, then certainly she must be humiliated at least seven days. Right? Meaning, there is one datum and a reasoning. Those are the first two types of a fortiori.

Rabbi, that’s not true. There too there are three data points. What are the data points the rabbi isn’t counting? That God is more important than a person. That is the reasoning. What do you mean by data points? Those aren’t data points. The reasoning that the stringency of God is greater, more important than a person. Okay, then the other data points would be that her father spits in her face and that she must be humiliated. There is… no. “Her father spits in her face” is the entry in the matrix. “Her father spits in her face”—the law is humiliation, one. Now I ask what happens when the Holy One, blessed be He, spits in her face. Then I say also one, because He is more stringent. There is no other… there is no other column. You have to remember, there are two axes here: this axis, the axis of domains, and the axis of damagers, but there are also the laws. Those are of course also part of the data. Okay.

So all the a fortiori arguments of the first two types are a fortiori arguments based on one datum and a reasoning of a leniency/stringency relation. The formal a fortiori, with which we will deal from here on, is an a fortiori based on three data points without reasoning. At least for the moment. Later I’ll refine that a bit. But for now: three data points. If I have a structure like this of three data points, I can derive from them a fourth halakhic data point, learn from them a fourth halakhic data point.

Now, what is common to all a fortiori arguments, and really to all interpretive principles? That I derive new laws from them. In contrast to logic—and we already talked about this in previous sessions—logic can never add information that I did not already have. Right? Therefore it is clear that the logic of these principles is a logic that adds to me… a logic that describes the addition of information or the accumulation of information. In contrast to ordinary deductive logic, which is a logic that never adds information to me; it only exposes to me information I already had. We already talked about that. Fine?

So here we really do see: we have information. These three laws are given; that is the information I have when I set out. And when I finish the process, I have another fact. I have accumulated additional information. Meaning, a fortiori reasoning is an inferential tool that adds information I did not have. What does that basically mean? That clearly it is not deduction. Right? Because deduction never adds information. Deduction is empty of information. This is either analogy, induction, or from that family. It is a family of arguments that add information.

What characterizes those arguments? We talked about this too. What characterizes those arguments is that they are not certain. Deduction is a certain argument. All human beings are mortal; Socrates is a human being; Socrates is mortal. You can’t argue with that; it is certain. That is basically parallel to the a fortiori of “included in two hundred is one hundred.” Fine? It is certain, so it doesn’t add information. That is why it is certain, because it doesn’t add information. It’s like with the hot-air balloon. Fine? These a fortiori arguments and these kinds of arguments are arguments that add information, and precisely because of that it is clear that… clear that they cannot be certain. Therefore all those who identified a fortiori with deduction, with what is called a syllogism in Aristotelian logic, were mistaken. It’s not true. Because if it were a syllogism, there could be no refutation against it; it would be certain. If there is a refutation against an a fortiori argument, that means the inference is not certain. So that means it is not deduction. Because deduction has no refutations. The inference with Socrates has no refutation, none. Right.

And once there is a refutation—no, therefore it won’t be deduction. It assumes all kinds of additional assumptions that it does not state explicitly. You decided that this is “included in two hundred is one hundred” because you thought about the relevance of the data to the outcomes, so you already introduced other things. Exactly because of that. But there is no… mathematics, look, this is the difference between mathematics and science; we talked about that. Mathematics has no refutation. You can’t measure something and discover that a mathematical law is not true. There is no such thing. You can discover that a physical law is not true. You cannot discover that a mathematical law is not true. There are no refutations against mathematics. There are refutations against science, and because science is not certain, you have to keep checking yourself against facts, against data, to run experiments, and the experiment is the refutation or lack of refutation.

People tried to prove whether Fermat’s Last Theorem was true or not true. That was a mathematical theorem. Not refute—it was proved, of course there are proofs in mathematics. Refutations, though—there are no refutations in mathematics. But what do you mean you can’t refute something in mathematics? Can you run an experiment and discover, run an experiment and discover, that some mathematical law is not true? We talked about this, didn’t we? About the oranges in the basket? How did the Pythagoreans discover that there are irrational numbers? Well? Well, they thought there weren’t, and suddenly discovered there were. They discovered they were mistaken—so what? That isn’t a refutation. That isn’t a refutation. What does it mean, they thought there were? “Thought there were” and “thought there weren’t” is a claim of existence, not a mathematical claim. No, they made a claim. If you derive it from assumptions, that’s mathematics. You can say, I claim there are triangles of such-and-such kind. That’s not mathematics. Mathematics is deriving certain conclusions from certain assumptions. That is mathematics. That exactly one straight line passes through two points—that’s not mathematics. That’s physics. That’s physics. No, but what, all geometry you remove from the… Certainly. You can’t measure geometry. You can measure the geometry of the world. You cannot measure whether geometry as such is a correct theory. That has no meaning.

We talked about this: how do we measure or refute the law that two plus three equals five? Right, so someone once suggested—not here; sometime in the past—someone suggested: take a basket, put two oranges into it, fine? Now take another three oranges and put them into the basket as well. Now count how many you have altogether. If it comes out five, then fine. If it comes out six, then you have refuted the law that two plus three equals five. But that’s complete nonsense, because even if you find six inside, you have not refuted the law that two plus three equals five. What you refuted was the physical assumption, not the mathematical one, that adding oranges into a basket is described by addition. Apparently algebraic addition does not describe adding oranges into a basket.

I gave you this example—I think we talked about it, didn’t we? Didn’t we talk about vectors? For example, the best example for this is forces, adding forces. Take some object, apply to it a force northward, fine? Ten newtons, the magnitude of the force northward. Okay? Now I apply another force westward, also ten newtons. Fine? What is the resultant force on that body? Ten root two, right? A vector like that, the Pythagorean theorem, ten root two. Fine? It’s clear to everyone that that is not twenty. Leave the physical details aside for now—it’s not twenty. It would be twenty if I added another force of ten in the same direction; that would be twenty. But a force like this is not twenty. Why not twenty? Have we refuted the law that ten plus ten equals twenty? Here’s an experiment—we have refuted the law that ten plus ten equals twenty. Not so. What have we refuted? We have refuted the physical hypothesis, not the mathematical law. The physical hypothesis that adding forces is described by algebraic addition. That is not true; it is described by vector addition.

But this isn’t only a game among the Sages. They don’t say, look, this is an ox that gored here and an ox that gored there. Everything is under the same plane, on the plane of halakhic reasoning. Fine? Legal reasoning, so what? I didn’t say that this is science, but I did say that this is an inference that adds information. Yes, but it isn’t… but it’s not that you’re trying to do… as if you’re saying you can’t refute it because you can’t do something in reality that proves the formula. No, the opposite. There you can refute it, I’m claiming. No, I mean in mathematics. Ah, in mathematics reality can’t speak about abstract formulas. Right.

Now when everything speaks in abstract formulas, then what? Take it—if you have a verse in the Torah that refutes the a fortiori, what’s the problem? Or a Mishnah or whatever. What difference does it make? Exactly. Facts—there too there are facts, halakhic facts. You infer that tooth and foot are more stringent than horn? Come, I’ll show you that in shared property tooth and foot are more lenient than horn. So here, I’ve refuted you. I brought you a fact, I ran an experiment, you failed the experiment, I refuted your theory. That is completely parallel.

Good, so if that’s the case, once we see that we have an inference that adds information, an inference that describes the accumulation of information, then first, it is not necessarily certain—indeed it is certain that this inference is not certain, yes? And second, there probably can also be refutations against it. The refutations will show us whether it is right or not right, but it is never certain. That is one of the reasons, as we said earlier, that punishments are not derived from logical inference. Punishments are not derived from logical inference because perhaps the a fortiori is not correct. Why? Because a fortiori is not a necessary inference. It may be incorrect—you cannot know. Even if you did not find a refutation, that still does not mean there is no refutation. You didn’t find one; you didn’t think of one; but perhaps there is. You can never know. Right, and so too in science—you can never know whether your conclusion is correct, since there may always be something else you didn’t measure or didn’t think about or whatever. It isn’t certain. In mathematics, after you’ve done a proper proof, you know the result is correct. That’s it. It doesn’t depend on any future measurement. It isn’t something that can be refuted. At most one can find a mistake in the proof—that you simply missed something. But there will never be an external refutation, something from outside showing you that the mathematical law is not correct. At most they will review the proof with you and show you where you missed something. Fine?

Okay, so this a fortiori basically rests, as I said earlier, on three data points. How exactly is it constructed? This a fortiori is built as follows. I look—one possibility—I look at this row and say that from this row it emerges that horn is more stringent than tooth and foot, right? It is easier to impose liability for horn than for tooth and foot—that is called more stringent, yes? Easier to impose liability. Okay. So now I go down to this row and say: if tooth and foot are liable in the damaged party’s courtyard, and the assumption is that horn is easier to impose liability for than tooth and foot, then if even tooth and foot are liable, horn, which is even more stringent, certainly should be liable. Therefore the filling in is one. That is one way to make the a fortiori.

A second way to make the a fortiori is to go like this, taking this column. What emerges from this column? That the damaged party’s courtyard is a place where it is easier to impose liability than in the public domain. In other words, it is more stringent in the sense that it is easier to impose liability for damage that occurs there. Now let’s move to here. So also with horn—even horn, which is harder to impose liability for, is liable in the public domain, so in the damaged party’s courtyard it certainly will be liable, because in the damaged party’s courtyard it is easier to impose liability. Right? So that is an a fortiori of the second pattern.

How is such an a fortiori built? Such an a fortiori, basically as I described just now, begins either from a column or from a row and infers a conclusion regarding the second column or row, right? Basically, there is here some sort of analogy. I am really making an analogy between tooth and foot and horn. I’m saying that if in tooth and foot there is some relation between the domains that is such-and-such, then in horn too there should apparently be that same relation between the two domains. You see? Basically an a fortiori is a somewhat different kind of analogy, that’s all. Only I’m making an analogy not in the specific laws but an analogy of hierarchical relations, relations of leniency and stringency. If here within tooth and foot there are relations of leniency and stringency such that the damaged party’s courtyard is more stringent than the public domain, then within horn too, in my view, it is similar to tooth and foot, and so within it too the damaged party’s courtyard is more stringent than the public domain. Fine? So I am really making some kind of analogy. Therefore a fortiori is not deduction, as some wanted to say, but belongs rather to the family of analogy in terms of its logical family. It’s not exactly analogy, yes? It’s a kind of analogy.

What does a refutation do? Suppose I found another domain, another—sorry—another damager, say a pit, it doesn’t matter at the moment, fine? Suppose a pit is liable in the public domain and exempt in the damaged party’s courtyard, for the sake of discussion, just as an example. Okay, so that refutes the a fortiori, right? Now we’re really saying that you can’t fill in a one here; that refutes the a fortiori. Of course I could also make a refutation like this if I found another domain—on the moon, tooth and foot are liable and horn is exempt. Fine? For example—never mind—some additional place, another domain. Okay, that too is a certain kind of refutation.

What does the refutation do? As I said earlier, a refutation reveals to me that the relation of leniency and stringency that I assumed is not correct. Fine? Now, in a fortiori arguments of the first two types, these are a fortiori arguments based on one datum and a reasoning of a leniency/stringency relation. “If her father had spit in her face, would she not be humiliated for seven days?” The datum: if her father spits in her face, she is humiliated for seven days. The reasoning: that the spit of the Holy One, blessed be He, is more severe than the spit of her father. So from here, that the spit of the Holy One, blessed be He, also requires at least seven days. Right?

What does this a fortiori do? This a fortiori, instead of reasoning, adds two data points. That is what it does. There I had one datum, as in that a fortiori. In that a fortiori there was only one datum and a reasoning. In this a fortiori there is one datum, and instead of the reasoning there are simply two more halakhic data points. What do these two halakhic data points do? They show me the reasoning. I take these two halakhic data points and they show me that horn is more stringent than tooth and foot. So if that reasoning comes from my own thinking, I don’t need two halakhic data points; one datum is enough for me to make an a fortiori. But where I would not know that from my own reasoning, then I can learn it from two additional halakhic data points. Therefore three data points are needed here.

And indeed, an a fortiori of this type is always built by taking either a column or a row, extracting the reasoning from that row or column, and then making an a fortiori of reasoning with the third datum that remains. For example, I took this row here and extracted from it a reasoning: that horn is more stringent than tooth and foot. Now I’ve returned to one of the first two types of a fortiori. I have an a fortiori of one datum, which I haven’t yet used, and the reasoning I extracted from the first two data points. So “horn is more stringent than tooth and foot” is the reasoning; “tooth and foot are liable” is the datum; if tooth and foot are liable, then horn, which is more stringent, certainly will be liable. And the same with columns. Right?

So basically this a fortiori is one that replaces the reasoning with two factual or halakhic data points. It simply derives the reasoning from two halakhic data points. And here exactly is the sting of the refutation. Because when I produce a refutation, what does the refutation do? The refutation basically says to me: the reasoning you extracted from here is not correct. Look, here it is the opposite. The reasoning you extracted from here—what does it basically say? That the damaged party’s domain is more stringent than the public domain. Now you want to take this datum with the reasoning that emerged from here and produce the result. Right? But notice: that reasoning is not correct. Here, for example, the public domain is more stringent than the damaged party’s domain. So the reasoning you extracted from those two data points is not correct. Okay?

Therefore many times a formal a fortiori has many refutations. An a fortiori from reasoning—I don’t even remember whether there are refutations against it. I don’t remember if there is an example of a refutation against an a fortiori from reasoning. Because with an a fortiori from reasoning, the reasoning says this is more stringent—it’s not speculative. It’s obvious that the Holy One, blessed be He, is more stringent than her father; there won’t be a refutation against that. Okay? There is a book, Halikhot Olam, by Rabbi Yeshua Halevi, who wrote refutations on all thirteen a fortiori arguments in the Torah. What, did he write them himself? Fine, no, I think he brought them from the Sages, refutations of the Sages. Well, I don’t know, I don’t remember at the moment. Yes?

But if we begin now to think that there could be a refutation against an a fortiori from reasoning, just as in this a fortiori we derive a reasoning from three factors instead of one—two. The reasoning comes from two, right. So if something is based on reasoning, can’t we derive it from two other factors? For example, if I find a place where God honored a person more than Himself? That would be a refutation. That would be that kind of a fortiori. No, I’m saying: “If her father had spit in her face,” we also assume by reasoning that God is more important than a person, therefore it would be more severe. But if we were to find another place where it says, for example, that God waived His honor before a person… that would… No, I’m saying there could be a refutation. I’m not saying there couldn’t. That’s why I divided earlier: the a fortiori of “included in two hundred is one hundred”—that is seemingly deduction. There there is no refutation, although we already saw that perhaps there is even there. Fine? But in the idea at least there is no refutation there. Fine? The reason-based a fortiori, the second type—even though it is based on reasoning, there can be a refutation against it. The refutation will reveal that my reasoning was not correct. Only usually, if this is a reasoning that seems obvious to me, it is less likely that it will really turn out to be incorrect. In this case I have no reasoning at all. How do I know whether horn is more stringent than tooth and foot or not? I simply derive it from two data points of the Torah.

Good. So that is speculation. Maybe I will suddenly find opposite data points that show me I’m wrong? Here there is no second reasoning. Therefore it is obvious that in a formal a fortiori I expect there to be more refutations than in the other types of a fortiori, although there can be in the others too, as I said earlier. What does the refutation do? It is very important to understand this.

There are books by Schwartz—Adolf Schwartz, who was at the rabbinical seminary in Vienna. So he wrote books already a hundred years ago or something like that, books about the interpretive principles. He wrote them in German; some were translated into Hebrew, not all. He claims that a fortiori reasoning is the Greek syllogism, that it is deduction. Fine? And… why? Because he says: what do you mean? If horn is more stringent than tooth and foot, and tooth and foot are liable in the damaged party’s courtyard, then obviously horn too will be liable in the damaged party’s courtyard. How is that different from Socrates? All human beings are mortal, Socrates is a human being, therefore Socrates is mortal.

His mistake, of course, is that nowhere is it written that horn is more stringent than tooth and foot. Where is that written? You infer that from these two data points. Notice: that is a generalization. After you have already formulated the rule that horn is more stringent than tooth and foot, then you’re right, that is deduction. If horn is more stringent than tooth and foot, and tooth and foot are liable, horn is certainly liable. But where did you get that assumption itself from—that horn is more stringent than tooth and foot? That is a generalization. You can say because it pays half-damages whereas tooth and foot pay full damages. So what? There are many examples, though there are refutations against that. What? That you say horn… why do you think horn is more lenient? No, I understood—there’s a refutation against that, that all… no, who says? There is a reasoning: the Holy One, blessed be He, and his father—not because I observed something, but because I think that’s how it is; that is reasoning. But that is intuition. Why? Because God created, because I saw that He is this, because I saw that He is… No, not because I saw anything at all—because I understand what the Holy One, blessed be He, is, and therefore I think He is more stringent than his father. The concept… yes. In an analytic sense almost, one could say. That too can be refuted. That too can be refuted. The Holy One, blessed be He, does not stand with him because… the Holy One, blessed be He, does not stand with him because… So what? Is it more frightening? The question isn’t whether it’s more frightening; the question is what is more stringent.

Anyway, the point is: where exactly did Schwartz miss it when he identifies this with a syllogism? With deductive logic. He misses it because this a fortiori comes… this a fortiori is built in two stages. He missed the first stage. First stage: I take two data points and make from them a general principle—that horn is more stringent than tooth and foot. Those two data points are specific data points: that horn is liable in the public domain and tooth and foot are exempt. Those are the data I have. From that I derive a general rule: that horn is more stringent than tooth and foot in every respect. And now I apply it here. Right? That is comparison or induction here. What? Analogy or induction? You can present it either way. And we talked about the fact that analogy and induction are built one on the other, one unit. No, because if we make an analogy, then it is equal, not more stringent in that case. It is equal in terms of hierarchical relation, not in terms of the laws. When you talk about a hierarchical relation, that is analogy. When you talk about the laws, that is a fortiori. Fine?

So basically every a fortiori is built—let me repeat—we spoke about John Stuart Mill’s challenge to deduction, remember? Deduction says: all human beings are mortal, Socrates is a human being, therefore Socrates is mortal. Which is seemingly necessary. Stuart Mill asks: what do you mean it’s necessary? How do you know that all human beings are mortal? How do you know? Have you seen all human beings? For example, everyone sitting here—I don’t know if they are mortal yet; thank God, may they live to one hundred and twenty. I haven’t yet seen them be mortal. You already said that, you already said that. What did I say? One hundred and twenty until one hundred and twenty. That’s probably connected to… How do I know? Generalization. Right? That means that every deduction is built on a generalization. So deduction too is not necessary.

This isn’t a fact that… once you act from assumptions that everyone will disappear and only then… seemingly the fact is that we don’t do that. So how do we know? Stuart Mill says: how do we know? From observation. From generalization. We already know people who existed in the past and died. We assume that people today do not differ essentially from people in the past, and therefore the assumption is that all human beings will die eventually. Meaning that all are mortal. That means there is a generalization sitting here. So when I make a deduction that is seemingly a necessary argument—“all human beings are mortal, Socrates is a human being, therefore Socrates is mortal.” Fine? Just a second, just a second. That is seemingly completely necessary. Not so. The assumption “all human beings are mortal” is itself the result of a generalization. And generalization is not necessary. So if I were to find, for example, a human being who is not mortal, that would be a refutation, right? Suppose Elijah the Prophet were revealed to me, fine? So what? That would be a refutation of the conclusion that, say, I could present Elijah the Prophet as mortal by the same reasoning. A refutation of what? What does it refute? It refutes the assumption that all human beings are mortal. Right—not the inference itself. The inference itself is necessary. That may be what you wanted to say. The inference itself is necessary, but the assumptions on which the inference is built are not necessary. Where did they come from? They came from generalization. Generalization is not something certain. If I find a refutation, then I will refute the assumptions, even though if the assumptions are correct, then the conclusion is certainly correct. It’s necessary. Necessary, yes—but the assumptions themselves are not certain.

The exact same thing is happening here. Exactly the same thing. So here too, after you have already arrived at the conclusion that horn is more stringent than tooth and foot, then this is simple deduction. If horn is more stringent than tooth and foot, and tooth and foot are liable, then horn is certainly liable. But how do you know that horn is more stringent than tooth and foot? Where did your assumption come from? From generalization. Why? Because all you know is only from the laws of the public domain. You know that horn is liable and tooth and foot are exempt. How do you know that this is true in every place, in every matter? That is a generalization. That generalization is exposed to attack by refutation. If I now find a place where tooth and foot are liable and horn is exempt, I’ve discovered that my generalization is incorrect. Rather, horn is more stringent than tooth and foot in the public domain, but it also has aspects in which it is more lenient than tooth and foot. And now the question is which aspects matter in the damaged party’s courtyard. I don’t know. Fine? And therefore I have refuted the a fortiori.

So the point that is very important to understand here is that a fortiori reasoning is an argument built in two stages. The first stage is a stage of generalization. I take two data points and derive from them a general principle—either the column or the row. I derive from them a general principle. After I make that generalization—pure deduction. That’s true. But the first stage is a stage of generalization, and therefore there are refutations. What does the refutation do? The refutation does not attack the inference. The inference cannot be attacked; it is necessary. The refutation attacks the assumption, the generalization that generated the assumption. That assumption—who says it is correct? I’ll show you a place where it isn’t correct. A refutation always attacks the generalization that underlies the a fortiori.

Okay, now let’s move on with the time—we need to wrap up.