Midrash and Principles of Interpretation – Lesson 9

Table of Contents

• Kal va-chomer: “included in two hundred is one hundred” versus ordinary kal va-chomer
• The primitive structure of kal va-chomer
• Talmudic kal va-chomer in Bava Kamma and the move to a two-by-two table
• Deriving a hierarchy rule from the data and two ways of formulating it
• A refutation, “tail,” and the claim that kal va-chomer is not deduction
• The analogy and hidden generalization behind kal va-chomer
• The rows versus the columns: apparent difference and the connection to the dispute between Rabbi Tarfon and the Sages
• The puzzle of refutation: why one refutation collapses the entire kal va-chomer
• Solving the puzzle באמצעות a shared parameter (alpha) and an explanatory theory
• Occam’s razor and the preference for the simpler theory
• A general algorithm: two tables, two hypotheses, and refutation as non-decision
• Binyan av as analogy and its analysis by the same method
• Methodological conclusion: an orderly non-deductive logic for Jewish law, science, and everyday life

Summary

General Overview

The text draws a distinction between a “primitive” kal va-chomer based on one datum and an explicit hierarchy rule, and a Talmudic kal va-chomer built from a two-by-two table with three known data points and a missing fourth square. The claim is that even in a kal va-chomer of “included in two hundred is one hundred,” there can still be a refutation, because the move from the data to a hierarchy rule is a generalization that is not logically necessary but resembles scientific inference, which can be overturned by a counterexample. From this, a thesis is developed that kal va-chomer is not a deductive syllogism, but rather a process of finding a simple explanatory theory, in terms of parameters like alpha/beta, and inferring the missing result according to Occam’s razor, while clarifying that a refutation does not prove the result false but only that the issue cannot be decided.

Kal va-chomer: “included in two hundred is one hundred” versus ordinary kal va-chomer

A kal va-chomer of “included in two hundred is one hundred” is described as a case where the stricter case includes the lighter one plus an additional component, like the relationship between uncovering and digging a pit, where digging also includes uncovering. A kal va-chomer of the type “The children of Israel did not listen to me, so how will Pharaoh listen to me?” is presented as a relationship of severity where you can imagine some counterconsideration, and therefore it looks vulnerable to refutation. The text then argues that even in “included in two hundred is one hundred” there can be a refutation, because once you apply logic to the world, additional assumptions enter and those have to be tested.

The primitive structure of kal va-chomer

Primitive kal va-chomer is defined as an inference that begins with a single factual datum, adds a hierarchy rule, and draws a conclusion, for example: “The Jewish people did not listen to me,” “Pharaoh is less obedient,” and therefore “Pharaoh will not listen.” It is argued that this is the structure of the kal va-chomer arguments that appear in Scripture, following the midrash about “ten kal va-chomer arguments in Scripture,” and that all of them are of the type: one datum, a hierarchy rule, and a conclusion.

Talmudic kal va-chomer in Bava Kamma and the move to a two-by-two table

The Mishnah in Bava Kamma is presented as the setting for the dispute between Rabbi Tarfon and the Sages about the law of horn damage in the injured party’s courtyard: Rabbi Tarfon obligates full damages, while the Sages say half damages. The data are set up as follows: tooth and foot in the public domain are exempt, and in the injured party’s courtyard they are liable for full damages; horn in the public domain is liable for half damages; but horn in the injured party’s courtyard is missing. The claim is that a typical Talmudic kal va-chomer is a two-by-two table in which three squares are given and the fourth is determined, and that in the Talmud most kal va-chomer arguments are of this type, based on three data points rather than one.

Deriving a hierarchy rule from the data and two ways of formulating it

The text argues that when you have three data points, you can choose two of them in order to derive a hierarchy rule, and then use the third as the “single datum,” as in the biblical structure. Two ways are presented: deriving a hierarchy from the tooth-and-foot column in order to determine that the injured party’s courtyard is more severe than the public domain, and then applying that to horn; or deriving a hierarchy from the public-domain row in order to determine that horn is more severe than tooth and foot, and then applying that to the injured party’s courtyard. The claim is that in Talmudic kal va-chomer there is an extra step that does not exist in the biblical kind, because the hierarchy rule is not “obvious to reason” but is derived from two data points.

Refutation, “tail,” and the claim that kal va-chomer is not deduction

An example is brought of adding another “damager” called “tail,” in which in the injured party’s courtyard it is exempt but in the public domain it is liable, in order to show that a refutation can reverse the hierarchy relation between the domains and leave the missing square undecided. It is said that Adolf Schwartz of the rabbinical seminary in Vienna claimed that kal va-chomer is deduction and a necessary syllogism, but the text argues against him that the very possibility of refutations proves that this is not a necessary mathematical inference. The distinction between science on the one hand and logic and mathematics on the other is presented as follows: science is falsifiable, in Popper’s sense, whereas a mathematical proof cannot be falsified, only challenged as having made an error in application. Therefore, the very concept of refutation places kal va-chomer on the side of inductive generalization rather than deduction.

The analogy and hidden generalization behind kal va-chomer

An analogy about tables with four legs is brought in order to show that analogy rests on a hidden generalization of the form “all tables are alike in this parameter,” and that bringing a counterexample breaks the generalization and blocks the inference. The claim is that this is exactly how kal va-chomer works: the move from two data points to a hierarchy rule is a generalization, and the refutation is a counterexample that attacks that generalization.

The rows versus the columns: apparent difference and the connection to the dispute between Rabbi Tarfon and the Sages

At first glance, the text argues, the formulation by columns and the formulation by rows look like two independent inferences, because one assumes a hierarchy between the domains and the other assumes a hierarchy between the damagers. It is shown that when you restore the “half” of horn in the public domain, you get different results: from the rows you get the conclusion of “at least one,” and therefore full damages; from the columns you get the conclusion of “at least half,” and therefore half damages. From this it is suggested, at least initially, that this could explain the dispute between Rabbi Tarfon and the Sages over which inference to apply.

The puzzle of refutation: why one refutation collapses the entire kal va-chomer

A claim is presented that, apparently, a refutation of a column should only undermine the kal va-chomer based on the columns, and a refutation of a row should only undermine the kal va-chomer based on the rows. So one would have expected that in the Talmud two refutations would be required to collapse a kal va-chomer. The text argues that in practice, in the Talmud, one refutation collapses the kal va-chomer, and presents this as a puzzle that needs a principled solution.

Solving the puzzle through a shared parameter (alpha) and an explanatory theory

A solution is proposed according to which the two formulations are not two different inferences but two formulations of one and the same inference, because the hierarchies of the domains and of the damagers sit on the very same parameter. A model is presented in which the injured party’s courtyard needs only “enough alpha” in order to create liability, whereas the public domain requires a stronger force marked as two alpha. From the data it is determined that tooth and foot have one alpha and horn has two alpha, and therefore horn is liable in the injured party’s courtyard as well. The claim is that if the hierarchy among the damagers rested on another parameter, beta, independent of alpha, then it would not explain the three given data points in the table. Therefore, if the hierarchy collapses in one place, the whole theory collapses and an additional parameter must be introduced, which removes the ability to decide.

Occam’s razor and the preference for the simpler theory

The text presents the possibility of an alternative theory in which instead of “alpha and two alpha” you have “alpha” versus “alpha and beta,” and that too still explains the data, but it argues that there is no reason to add beta when a single parameter is enough. It claims that Occam’s razor is not merely a rule of methodological convenience but a principle about the world, according to which the simpler theory has a higher chance of being correct, and illustrates this with examples involving pepper-salt-sugar and the example of a straight line versus complicated curves that fit the same measurements.

A general algorithm: two tables, two hypotheses, and refutation as non-decision

A general algorithm is proposed in which you fill in the missing square in two possible ways, zero or one, build two tables, and look for an explanatory theory for each one; the simpler theory determines the preferred filling. The claim is that when there is a refutation such as “tail,” both possibilities require theories of similar complexity, for example at least two parameters, and therefore there is no preference for zero or one and the law remains unknown. It is emphasized that a refutation does not mean that the result is zero, but only that there is no way to prefer zero over one, and therefore no conclusion can be inferred.

Binyan av as analogy and its analysis by the same method

Binyan av is defined as an analogy according to which, if tooth and foot are similar to horn in one parameter, then they are also similar in an additional parameter, and the claim is that it too can be analyzed through tables and by choosing the simpler theory. It is argued that in the case of binyan av, filling in “one” allows for an especially simple theory, whereas filling in “zero” requires a more complex model, like alpha and two alpha, and therefore preference is given to the filling that yields the simpler model. The text argues that binyan av also collapses through a refutation by the same mechanism, where the refutation cancels the advantage of simplicity and forces more complex models.

Methodological conclusion: an orderly non-deductive logic for Jewish law, science, and everyday life

The claim is made that there exists a “mathematical framework” that formalizes non-deductive inferences, and that the same type of logic governs the forms of inference used in science, law, Jewish law, and everyday life. The text states that all the hermeneutic principles and forms of inference can be translated into tables larger than two-by-two, and that the difference is one of size and data, not of principle. In conclusion, it says that the presentation is not precise in every detail but is enough to get a sense of the principle, and that next time there will be progress toward the topic of “the common denominator,” and perhaps also its formalization.

In the previous lecture I started talking about a fortiori reasoning. And there we saw the distinction between an a fortiori argument of the “if it applies to two hundred, then certainly to one hundred” type, and an ordinary a fortiori argument. An a fortiori argument of the “if it applies to two hundred, then certainly to one hundred” type is when the stricter case contains within it the lighter case plus something extra. Like the relation between uncovering and digging a pit. When I dig a pit, I obviously also uncover it; I’m just doing something more. So there it seems, ostensibly, like an a fortiori argument that has no answer to it, that no refutation could apply to. By contrast, the a fortiori argument of “If even the children of Israel did not listen to me, how then will Pharaoh listen to me?” is a relation of greater and lesser stringency, and it could be that in that particular case there is some consideration showing why Pharaoh specifically would obey or listen to him more than the children of Israel, and therefore there can be a refutation there. I explained afterward that this is not correct. Meaning, even in the “if it applies to two hundred, then certainly to one hundred” type of a fortiori argument there can be a refutation, and we talked a bit about the relation between logic and the world: in logic many things seem absolute and unshakable, but when you apply them to the world, additional assumptions always enter in, and those are what we need to examine. All of this really dealt with a fortiori reasoning in its primitive form. A fortiori reasoning in its primitive form is an argument that starts from one assumption, one factual assumption, one fact—say, the children of Israel did not listen to me. A hierarchy rule: Pharaoh is less obedient than the children of Israel. And a conclusion: apparently Pharaoh too will not listen to my voice. Okay? So I have one assumption, a hierarchy rule, and a conclusion. Let’s look, for example, at an a fortiori argument of a different kind, say the one that appears in tractate Bava Kamma. A fortiori argument, hierarchy, conclusion. One assumption, hierarchy, and from that the conclusion. One assumption, hierarchy, and from that the conclusion. There is the lighter case—that’s the given. There is a hierarchy rule saying that the stricter case is stricter than the lighter one. Conclusion: the same law that applies in the lighter case must also apply in the stricter case. Okay, that is basically the simple structure of a fortiori reasoning. For example, the midrash says that there are ten a fortiori arguments in Scripture. I mentioned this in Humash. One of them is “If even the children of Israel did not listen to me, how then will Pharaoh listen to me?” And all the a fortiori arguments that appear there are of this type: one assumption, a hierarchy rule, conclusion.

Now let’s look at this a fortiori argument in the Mishnah in Bava Kamma. “An ox that causes damage on the property of the injured party—how so? If it gored, pushed, bit, crouched, or kicked in the public domain, it pays half damages. On the property of the injured party, Rabbi Tarfon says full damages, and the Sages say half damages.” We have—the rules are as follows. In the primary categories of damages there are several types. There are tooth, foot, horn, a harmless horn, a forewarned horn, a pit, fire, and so on. Okay, here we’re talking about horn and about tooth and foot. The rule is that horn is liable in the public domain for half damages—an innocuous horn, during its first three gorings. From the fourth goring onward it is a forewarned ox. During the first three gorings it is an innocuous ox. An innocuous ox pays half damages. Okay? Now, an ox that gores, say, for the first or second time in the public domain is liable for half damages. And with tooth and foot—tooth and foot means when the animal eats something for its own enjoyment, or while walking it tramples something. That is called damage by tooth and foot. The rule is that in the public domain, for tooth and foot one is exempt, because I have the right to walk with my ox, and whoever leaves fruit there should watch it. Meaning, this is an area given over to all of us for use. So in the public domain, tooth and foot are exempt; in the courtyard of the injured party, tooth and foot are liable for full damages. Okay, those are the givens.

Now Rabbi Tarfon and the Sages disagree about the law of horn in the courtyard of the injured party. That is unknown. In the Torah it says that horn in the public domain is half damages. It says in the Torah that tooth and foot in the public domain are exempt, and tooth and foot in the courtyard of the injured party are liable for full damages. It does not say what the law is for horn in the courtyard of the injured party. So I ask myself: what is the law? So Rabbi Tarfon and the Sages disagree on this. Rabbi Tarfon says full damages, and the Sages say half damages. What is the idea? So Rabbi Tarfon said to them: “Now if in a place where Scripture was lenient regarding tooth and foot in the public domain, exempting them, it was stringent regarding them on the property of the injured party, requiring full damages—then in a place where it was stringent regarding horn in the public domain, requiring half damages, is it not logical that we should be stringent with it on the property of the injured party and require full damages?” A fortiori. If tooth and foot, which are exempt in the public domain, are liable in the courtyard of the injured party, then horn, which is liable in the public domain—which is stricter—certainly should be liable in the courtyard of the injured party. All right? That is Rabbi Tarfon’s a fortiori argument. The Sages answer him with the rule of “it is sufficient,” but that is not important right now—we’ll leave that aside for now.

What is this a fortiori argument? Is it of the same type as the a fortiori arguments we encountered? Here we are talking about the place itself, no? About the increased stringency. We have two things: place—damage in the public domain, damage in the courtyard of the injured party. And we have two damagers: tooth and foot, and horn. Now I make an a fortiori argument. Is this a fortiori argument constructed in the same way as the ones we have seen until now? No, because there is no relation here of container and contained. No, this is not the “if it applies to two hundred, then certainly to one hundred” type. Never mind that; I’m talking about an a fortiori argument that is not of that type. Like “If even the children of Israel did not listen to me, how then will Pharaoh listen to me?”—there it’s a fact and then a hierarchy, Israel and Pharaoh. Yes. How many facts are the basic facts here? How many facts are we starting from? Three facts. Which ones? That tooth and foot—or horn, or tooth, I don’t remember—which in the public domain is exempt and in the courtyard of the injured party is liable. But first of all, that already gives you two facts, not one. In the public domain exempt, and in the courtyard of the injured party liable—two facts. But really there are three. Horn is liable in the public domain, tooth and foot are exempt in the public domain, tooth and foot are liable in the courtyard of the injured party. Basically we start from three pieces of data, three facts. Okay? From this we want to infer a fourth fact. But a hierarchy rule, for example, you don’t see here. Here there are three facts without a hierarchy rule. There there is one fact and a hierarchy rule.

By the way, what was true of the a fortiori arguments in the Talmud? Most of them are of this type, and not of the biblical type. These are a fortiori arguments based on three data points and not on one. Let’s draw this for a moment so that we can work with it. Good, so we begin like this. We have domain and damager. You can see, right? Domain and damager. So here is the row of damagers. This is tooth and foot, and this is horn. This is the column of domains. This is the public domain and the courtyard of the injured party. Okay? What are the givens? Tooth and foot in the public domain—exempt. Horn in the public domain—liable; really, liable for half. Tooth and foot in the courtyard of the injured party—liable, no, full—liable for full damages. And here: question mark. Okay? That is the given.

Now, this is the structure if we ignore the half for a moment. I’m now going to change it to one just for simplicity; afterward we’ll come back to the question of the half. Liable—let’s ignore for the moment the question of how much one is liable. This is the classic structure of a Talmudic a fortiori argument. In the Gemara, an a fortiori argument always has this structure. It could be, say, canopy versus document in betrothal and marriage. So there are two actions and two outcomes. But that doesn’t matter; here too, the two actions are damage by tooth and foot and damage by horn, and the outcomes are public domain liable, courtyard of the injured party liable or exempt. Okay? But the structure is always a two-by-two table where in three of the cells we have givens. And I am looking for the fourth datum. More than that: the givens are always arranged so that here there is zero, here there is one, here there is one, and here a question mark. Always. And then the a fortiori argument tells us: if that is the structure, the answer here is one. Okay? That is the structure.

Now the question is: what is the connection between this and the biblical a fortiori argument? Why is this called an a fortiori argument and that also called an a fortiori argument? Where is the hierarchy rule here? What is stricter than what? Where does it come from? We have three data points and suddenly we jump to the fourth. There the logic is clear, right? If the children of Israel did not listen to me, and Pharaoh is surely less obedient than the children of Israel, then obviously Pharaoh too will not listen to my voice. The logic is very clear. Here I have three data points, and I suddenly jump to a fourth. Where is the hierarchy rule? Where? How is this constructed?

There? You see that in the public domain, in a certain case, there is a certain outcome, and in the courtyard of the injured party the outcome, on exactly the same case, is more stringent. Meaning there is a hierarchical difference here between the places where the damage occurred. Okay. And then you apply the same mathematical relation also to horn. You say: if the public domain is, say, x, then presumably the courtyard of the injured party will be multiplied by the same—I don’t know what. Okay. So basically what you are saying is this. Let’s look at the right column. In the right column we see the two laws of tooth and foot: in the public domain and in the courtyard of the injured party. From looking at that column, we can actually derive a hierarchy relation: damage in the courtyard of the injured party is more severe than damage in the public domain. There is more reason to impose liability there. Right? Ah, I thought of something. Okay, no. So the courtyard of the injured party is stricter than the public domain. Now I learned that from the right column. Now I go to the left column and say to myself: okay, if horn is liable in the public domain, and I already know that the courtyard of the injured party is stricter than the public domain, then if horn is liable in the public domain, certainly horn is liable in the courtyard of the injured party, because the courtyard of the injured party is stricter. And therefore the result is one. Good.

Second possibility: let’s look at the top row. That is looking at the public domain. There is damage by tooth and foot in the public domain, and horn in the public domain. What do I learn from that row? That horn is stricter than tooth and foot. Let’s now apply that to the bottom row. If tooth and foot are liable in the courtyard of the injured party, and horn is stricter than tooth and foot, then horn certainly will be liable in the courtyard of the injured party. Okay?

So what does that basically tell us about the relation to the biblical a fortiori argument? It tells us that when we have three data points, we can use two of them to derive from them a hierarchy rule, and then what are we left with? The third datum remains as one datum, we already have the hierarchy rule, and we’ve gone back to the biblical a fortiori form. Right? Basically the Talmudic a fortiori argument—let’s call it that now—which is based on three data points: from two of those data points I derive a hierarchy rule, and then I move to the third datum and use the technique I know from the Torah: hierarchy rule plus datum gives me the result. Okay?

Now in fact, as we saw before, I have two ways to make this inference. Right? I have two ways to make this inference. Either choose these two data points of the right column, and from that I get a hierarchy rule about the domains—that the courtyard of the injured party is stricter than the public domain—and then the third datum, horn in the public domain, this datum, yes? It will be my single datum, and the hierarchy rule will be what came from those two columns. Right? That is one possibility. The second possibility is to look at the row—the rows, sorry. From the row I derive a hierarchy rule. The third datum will be the single datum, and that plus the hierarchy rule will give me the result. So I can translate an a fortiori argument of three data points by taking two of them—either the two of the right column or the two of the top row—producing from them a hierarchy rule, and setting them aside. Now I am left with the third datum and a hierarchy rule, and from that I arrive at the conclusion. Okay, so basically it is the same thing. The only difference is that in the Talmudic a fortiori argument, as we’re calling it now, there is another step that is not present in the biblical a fortiori argument. I take two data points and generate a hierarchy rule from them. That we didn’t do in the biblical a fortiori argument. From them I make, create, some kind of generalization. Okay?

Now, when I have a refutation—say I now bring an additional damager, I don’t know, a tail. Okay? And the tail is exempt in the courtyard of the injured party and liable in the public domain. Let’s say those are the data. I found another damager and those are the data. What does that do to the a fortiori argument? It is a refutation of the a fortiori argument. Why? Because you can derive the opposite relation, basically. Right. Because the relation between the public domain and the courtyard of the injured party is no longer clear. From here it comes out that the courtyard of the injured party is stricter, but if we look here then in fact it comes out that the public domain is stricter. Since that is so, I have no way of knowing which is stricter, and therefore in this column I simply do not know what to do. It remains open. If there is a clear hierarchy relation, then yes; if not, then no.

Now, as I already told you, there is a fellow named Adolf Schwartz from the rabbinical seminary in Vienna, who wrote books on the hermeneutic principles. Among other things he wrote a book on a fortiori reasoning. And there in the book he claims that a fortiori reasoning is basically deduction. It is a logical syllogism. It is a logically necessary argument. Now why can’t he be right? First of all, what is a syllogism, what exactly does that word mean? A necessary inference in logic, something like that. In Aristotle, in the Organon, there are forms of syllogisms. There are several specific structures that are logically necessary arguments. He basically classified them. Okay? Now why is it obvious that he is not right that this is a syllogism? Because there are no refutations to a mathematical proof. You can find a mistake in the proof—someone got confused, sure. But if it was a valid proof, nothing will help you: the conclusion is correct, no refutation will topple it. We talked about this in the previous lecture, about the difference between science and logic and mathematics. Science is refutable, because in science I make some generalization on the basis of facts I observed. If you bring me an opposite fact, you show me that my generalization was probably not correct. Science can be refuted. According to Popper, the definition of science is a theory that can be refuted. But in mathematics you cannot refute. What, are you going to bring a counterexample showing that the sum of the angles in a triangle is not 180 degrees? That won’t help. Even if you bring such an example, I’ll say: okay, then in that imagined triangle there was some mistake; you did not apply the theory correctly. I will never give up a result that follows from a mathematical proof. A mathematical proof is not refutable. Facts—can they be refuted? You can show maybe that you didn’t see correctly. But that is a matter of definition. Usually when we talk about refutation, we are talking about a general law, not about something you specifically saw. About a theory, about some generalization.

So therefore, if there can be refutations to an a fortiori argument, that means it does not belong on the logic side, it belongs on the science side. It is built on some generalization, and that generalization I can attack with a refutation, with a counterexample. Where do we see the generalization here? In the move from two data points to a hierarchy rule. Because basically, look: say I take these two data points, and I say that from here I learn that the courtyard of the injured party is stricter than the public domain in the case of tooth and foot, so presumably in general it is stricter than the public domain. Consequently with horn too it should be that way. Right? Here there is a generalization. We saw that it is stricter regarding tooth and foot. Who says that in general, always, the courtyard of the injured party is stricter than the public domain? That is a generalization we make on the basis of the specific case that we know. There is induction here, generalization. And it is that induction that we attack with the refutation. There—you see, the tail. The tail shows us that our generalization was not correct. The courtyard of the injured party is not in general stricter than the public domain. Regarding tooth and foot, yes. In other places, no. Now you tell me whether horn belongs to this side or to that side. And therefore you cannot know the law of horn in the courtyard of the injured party.

Okay? Meaning, the existence of a refutation shows us that behind the a fortiori argument there sits a generalization. Therefore a fortiori reasoning is not a logical rule. It is not a necessary deductive rule. Behind it sits a generalization, and that generalization is never necessary. It can always be attacked. Refutations—counterexamples—will attack the generalization.

I’ll give you an example. We talked about analogy, induction, and deduction, if you remember. Say I make an analogy. This table has four legs. That thing over there is also a table; therefore it too has four legs. That is an analogy from table to table. What have I actually assumed behind the analogy? That all tables have the same number of legs. Therefore if this table has four, then probably that one has four too. Meaning that behind that analogy there is actually some hidden generalization. Sometimes we do not state that generalization, but it is there. We are basically assuming that the example we saw is a representative example, and that what appears in it is an overarching principle, a general principle, and that this is always true. That is exactly what happens in an a fortiori argument. We saw in the first row, in the first column, that the courtyard of the injured party is stricter than the public domain. Since this is a case that is true only for tooth and foot specifically, who says this hierarchy is always so? We are making a generalization. So if we have no refutation, we assume the generalization is correct, as in science. But if we have a refutation, we have refuted our generalization. What does that mean? It is still true that the courtyard of the injured party is stricter than the public domain regarding tooth and foot. But only regarding tooth and foot. Or at least not in all cases. I do not know where yes and where no. But it is no longer a general law. If I find a table that has only two legs, then I can no longer make the analogy from this table to that table. Who knows whether that other table resembles the two-legged table or this four-legged table? I can no longer assume that it is a general law that all tables have the same number of legs. Okay? That is exactly what is happening here.

And therefore, even though a fortiori reasoning has a certain tendency to cast a spell on us, yes? To tell us: this is a necessary argument, you can’t argue with it—not true. Behind the a fortiori argument sits some generalization, and the generalization is never necessary. It can always be attacked.

Now I ask the next question: we saw two formulations of the a fortiori argument. We saw two formulations of the a fortiori argument: one formulation according to the two columns, according to the column, and the second formulation according to the row. Is it the same inference? Two formulations of the same inference, or are these two different inferences? The same, Rabbi? Is it the very same inference, just in different words or under a different description, or are these two different inferences? Between the row and the column? Yes. Different inferences. One speaks about the domain—the inference or generalization is in the domain of place, public domain—and the other speaks about the types of damagers. Right? On the face of it, these are two completely different inferences. Right? Why? Because the inference built on the columns, yes? I derive a hierarchy rule from the right column and then apply it to the left column. That hierarchy rule establishes that the courtyard of the injured party is stricter than the public domain. It has nothing to do with whether horn is stricter than tooth and foot or less strict than tooth and foot. It doesn’t matter. Horn could be less strict than tooth and foot and the inference would still be fine. Right? Because if I say the courtyard of the injured party is stricter than the public domain, then I move to horn. For horn, in the public domain it is one, so in the courtyard of the injured party it is also one. Even if horn is lighter than tooth and foot, that doesn’t matter. I am not assuming anything about the relation between the damagers, only about the relation between the domains. The same thing, just in reverse, in the row inference. In the row inference I assume a hierarchy between the damagers, that horn is stricter than tooth and foot. And I will now apply that to the courtyard of the injured party. And it does not matter whether the courtyard of the injured party is stricter than the public domain or less strict than it. Ostensibly these are two independent inferences.

Indication of this: remember that here originally it said half? I changed it to one only in order to describe it simply. But let’s now do the calculation and you’ll see. Let’s now do the calculation by rows. So from here I derived the hierarchy that horn is stricter than tooth and foot, right? I don’t know by how much, but stricter. Now I go to the bottom row. If tooth and foot are liable for one, and horn is stricter, then what is the result? At least one. Right? One is “it is sufficient for what comes from the law to be like the original case.” So that is one. That is in the row analysis.

Now let’s do the column analysis. In the column analysis we get that the courtyard of the injured party is stricter than the public domain, right? Now we move to the left column. If in the public domain horn is liable for half, then how much will horn be liable for in the courtyard of the injured party? Half. At least half, but half. “At least” is always what we have; I do not know more than that. You see that the results of the a fortiori argument are different. Clearly these are different inferences. If it were the same inference in different words, the result should have been the same result. Wait, why a different result? I didn’t understand. Because I assume that the courtyard of the injured party is stricter than the public domain from the column, right? Now I move here. So if horn in the public domain is liable for half, and the courtyard of the injured party is stricter, then how much will horn be liable there? At least half. Fine—but in the other analysis the result is at least one, not at least half. Because horn is stricter than tooth and foot, tooth and foot are liable for one, and horn is stricter, so certainly one. And the result is one, not half. So we see that these two inferences are different, because in fact they give different results. If they were only two different formulations of the same inference, they should have given the same result, right?

By the way, this is the dispute between Rabbi Tarfon and the Sages. One says that horn in the courtyard of the injured party is liable for half; the other says it is liable for full damages. The dispute is basically over which inference to look at—the column or the rows. So it would seem.

Now let’s go back for a moment to the structure without the half. So we talked there about the tail. Yes. Those are the data. Now we have five data points. And now I ask what to fill in where the question mark is. Sorry, yes. I’m confused. Yes. So what are we supposed to mark here for horn in the courtyard of the injured party? We don’t know, right? We don’t know. Why? Because this thing refutes the a fortiori argument, therefore it is called a refutation. Why does it refute? Because from here we concluded that the courtyard of the injured party is stricter than the public domain, and therefore we wanted to write one here, right? But from here we can see that the public domain is stricter than the courtyard of the injured party, so there are aspects in which it is stricter and aspects in which it is lighter. You do not know what happens with tooth and foot; it remains open. Wait—what about this? There’s a short circuit in the inferences. Exactly. What about this argument? Is it also damaged? I have a refutation. But I said this argument does not depend on whether the courtyard of the injured party is stricter than the public domain or not. Look, I will prove to you from here that horn is stricter than tooth and foot. That remains true; the tail doesn’t disturb that at all. Horn is stricter than tooth and foot. Fine, so now I say here too: horn is stricter than tooth and foot. If tooth and foot are one, horn too is one. But on the other hand the tail interferes, no? Why? Because then you say because with the added stringency of— No, I’m saying that between tooth and foot and tail there is no clear hierarchy. So what do I care? I’m looking for the hierarchy between tooth and foot and horn, and that remains; it is not broken by adding the tail. No, but if you now take horn and tail? Then indeed there I won’t be able to infer conclusions. But here too you can set up the relation between horn and tail as some kind of— Who says? Why? Why? The fact that it doesn’t work on something else means it won’t work here? What is the connection? So indeed from tooth and foot to tail I won’t be able to learn. Correct, there is no simple hierarchy between them. But I don’t care—I don’t want to learn about the tail, I want to learn about horn. And the relation between horn and tooth and foot exists. The tail doesn’t disturb it, doesn’t refute it. What is the problem?

Because the connection between horn and tail creates a counterweight, say—no, maybe it’s different. Why? Why is it different? Why ask why we should take horn and tooth and foot and not horn and tail? Because with horn and tail I have no way to do this. Not only do I not have—no, with horn and tail there’s no problem. It contradicts. No, it doesn’t contradict. It doesn’t contradict. I have no way to do it. Why? Derive a relation. Derive the relation. Let’s look only at this half of the table, just these four cells. Exactly. Just these four cells. Yes. So now what? What do you want to do now? I see that horn and tail are actually at the same level of stringency. Okay. And that foot in the courtyard and public domain are the same thing, right? Therefore in the courtyard of the injured party, just as tail is zero, horn should also be zero. No, who says? They are the same in the public domain, but not the same in the courtyard of the injured party. So what? No, I am making a comparison, like I’m doing, say— But why should you make a comparison? In an a fortiori argument you do not make a comparison; in an a fortiori argument you infer by a fortiori reasoning. Here you are making a comparison. Look. There’s no a fortiori argument here? You claim—wait, I’ll ask. You claim that basically here, looking at these four cells, it cannot be one here but must be zero—is that what you claim? So I will show you that you are not right. In every a fortiori table this is the structure. Here, look. In the public domain, tooth and foot are zero, horn is one. In the courtyard of the injured party, it is one and one. So here, in the courtyard of the injured party, they are alike, and in the public domain they are not alike. There is no a fortiori argument there, but you are making an analogy. No, no, no—I am talking here now about the a fortiori argument. Forget the tail for now. There is no tail. All right? Let us look at these two columns. There is an ordinary a fortiori argument here. Okay? Now here I fill in one, right? In a regular a fortiori argument. According to you, I cannot fill in one? Because you say: if I fill in one here, then what comes out? That in the courtyard of the injured party, tooth and foot are like horn—both are one, right? Say I filled in one. I did the a fortiori argument and filled in one. Now what—at least one. Fine, one for purposes of discussion. Now what comes out? That in the courtyard of the injured party, tooth and foot and horn are the same thing, right? In the courtyard of the injured party, yes. Right? Yes. So why aren’t they the same in the public domain? Exactly, so you are saying that if it is the same in one place, it is the same in all places? No, not that it is the same, but what is stricter. Exactly—so they are at least the same. Exactly! Exactly! And that is what I answer you to your question. Let’s now look at these four cells: you say that here it must be zero, because if it is similar here then it must be similar there as well. It could be zero. No, but it could also be one. Good, and if it is one then that doesn’t contradict anything. So it is one. Because from the a fortiori argument I learned it is one, and from the tail it is not contradicted—it can be one. It does not have to be, but it can be. So what is the problem? Everything is fine, so the result is one. So the tail does not refute the a fortiori argument.

Now this is something strange, because it basically means the following: as we saw with “it is sufficient,” yes, when there was half here—remember there was half here? So I said that the result here would come out differently depending on whether we go by columns or by rows. Now it turns out there is another difference between columns and rows: the refutation. When there is a refutation like this, it will topple the a fortiori argument of the columns, but it will not topple the a fortiori argument of the rows, right? This refutation shows that there is no hierarchy between the domains, and it does not touch the hierarchy between the damagers. That can remain. And therefore I could have filled in the result one not from the a fortiori argument of the columns but from the a fortiori argument of the rows, and I still can write one here. So this refutation does not really throw the a fortiori argument in the trash, only one formulation of it. The second formulation remains.

What happens if I now make a refutation from below? Then it also harms this one. Yes, I now have here another domain—on the moon. Okay? Tooth and foot are liable for one on the moon, and horn is exempt on the moon. Okay? And now I ask—ignore the left column for a moment, yes? Let’s talk only about this. It is the same as if I added a column; I just added a row. Let’s call this now a row-refutation. What happens to the result here? It falls. Why does it fall? Because the row hierarchy here falls, because there is an opposite hierarchy in this row. So the a fortiori argument of the rows falls. But what happens to the a fortiori argument of the columns? It remains. So it turns out that a column-refutation refutes the a fortiori argument of the columns but leaves the rows, and a row-refutation refutes the a fortiori argument of the rows but leaves the a fortiori argument of the columns. So we would have expected that everywhere in the Talmud where a refutation is raised, they would say: what happened? Exactly—for an a fortiori argument to collapse, you should always need two refutations. There is no such thing anywhere. Nowhere. Whenever they raise one refutation, the a fortiori argument has fallen. A column-refutation, a row-refutation—it topples the a fortiori argument. Why? There are two different formulations here. So why does it topple both of them and not just one? An interesting puzzle.

Now let me show you the solution to this puzzle. So this is the a fortiori argument. Now look, I want to argue that from the Talmud we see that the two formulations of the a fortiori argument are not two inferences; they are two formulations of one inference, contrary to what appears at first glance. And let me explain why. When I now make the—let’s start with the a fortiori argument of the columns. I am basically looking at the a fortiori argument of the columns and saying this: the public domain is less strict than the courtyard of the injured party, right? Yes. What does that mean? It means that the courtyard of the injured party has some feature—let’s call it alpha. It has some property; let’s call it alpha. Yes. And the courtyard of the injured party has the property two alpha. You can derive that from tooth and foot? Ah? You derive that from tooth and foot? You infer it from tooth and foot? Yes, from the column of tooth and foot. It’s killing me. Doesn’t work. So this is two alpha. That came out incidentally. So the courtyard of the injured party is two alpha, right? Some feature—without getting into what it is—but there is something stronger in the courtyard of the injured party than in the public domain.

Now I say: in tooth and foot there is something that—it does not operate at alpha; it operates only at two alpha. Meaning, in order to impose liability in the public domain—it works the other way around—in order to impose liability in the public domain, you need a strength of two alpha, and therefore it is harder to impose liability in the public domain, right? In order to impose liability in the courtyard of the injured party, which is stricter, it is enough to have alpha in order to impose liability. Notice that this works opposite to the intuition. In fact the courtyard of the injured party is stricter, therefore it is alpha and not two alpha. Because I am asking: what level of damaging force do you need in order to impose liability? Any damager, because the courtyard of the injured party is very strict, so any damager will be liable there. What level of damaging force does tooth and foot have? Alpha. How do I know? Because it is liable in the courtyard of the injured party? Yes. Since even there it is liable. In the public domain it is not liable because there, to impose liability, you need two alpha, and it doesn’t have that; it has only one alpha. But horn is liable even in the public domain. Therefore if I had to guess what its force is, I would say two alpha. Consequently the result here is one. Because if it has two alpha, then of course it succeeds in imposing liability in the courtyard of the injured party, where alpha is enough. If you have two alpha, then certainly you impose liability, right? And then wouldn’t you want to impose liability doubly? So I said, “it is sufficient for what comes from the law to be like the original case.” Since I have no proof how much to impose on it. One is definitely liable—that is clear. Anything beyond that I do not know; maybe double, maybe one and a half, maybe five times. So the rule in Jewish law is “it is sufficient for what comes from the law to be like the original case”—you go with the minimum. The minimum you can prove. You can prove that one is liable, that is clear. Beyond that, no. That cannot be. Anything beyond that is debatable. You’ll say one and a half—maybe one and a third, maybe one and a quarter, maybe? I don’t know. One is certain. Okay? Therefore you are liable for one.

But notice what we have received. What we have received is that the hierarchy—the hierarchy of the rows and the hierarchy of the columns are in the same parameter. If the hierarchy between the rows were, say, beta and two beta, then it could not explain the table. Because if it has beta, but in order to impose liability in the public domain you need two alpha, and in order to impose liability in the courtyard of the injured party you need alpha, then it does not have alpha, it has beta. It must be that it has the very same thing that is required in order to impose liability in the courtyard of the injured party and in the public domain. Or in other words, the feature found in the domains, which distinguishes and creates the hierarchy between the domains, is also the feature that creates the hierarchy between the damagers. It must be. Because if it were not the same feature, then it would not be reflected in the stringency in the table. You understand what I’m saying? Is that because the relations between alpha and beta could be different? It doesn’t matter to me—yes, right. It could be one point three alpha. Yes. Okay, maybe there is no linear relation between intensities and stringency. Okay. So what did we really get? Notice: we got that the hierarchy between the rows and the hierarchy between the columns are in the same parameter. If the hierarchy between the damagers were in another parameter, then it would not explain the right column as well. Say I want to do the a fortiori argument of the columns. What did I say earlier? That for the a fortiori argument of the columns, it is enough for me to establish a hierarchy between the domains; I don’t need to establish a hierarchy between the damagers. Not true. The hierarchy between the domains dictates that there is the same hierarchy between the damagers. And if it falls—if I have a refutation here from the tail—then that hierarchy also falls. Therefore a column-refutation topples the a fortiori argument, and so does a row-refutation alone. You do not need two refutations to topple an a fortiori argument.

What have I actually done here? What I have done here is to assume, or decode, what stands behind the analogies we are making. I performed abduction—what I called in earlier lectures. I found the theory. I have now—after all, we start with three data points, right? The three data points. I have three data points—this, this, and this. I am looking for a theory that explains the three data points, and then I can derive from it the fourth law that I do not know. Here is the theory: alpha, two alpha, alpha, two alpha. That is the theory. Assuming that this is the theory, I know that here I have to fill in one.

Now let’s suppose that here—well, it’s driving me crazy, it keeps getting stuck, I know… If the hierarchy were with another parameter—beta and two beta—because that is what we are assuming when we say the two inferences are different. We are basically saying that the hierarchy between the public domain and the courtyard of the injured party is in alpha, and the hierarchy between tooth and foot and horn is independent, it is in beta. You understand that if that were the theory, it would not explain the three data points? It does not explain them. Explain to me now why tooth and foot are exempt in the public domain and liable in the courtyard of the injured party. Because they have beta? Beta does not help me both impose liability in the courtyard of the injured party and not in the public domain. I need different levels of alpha, but beta—a property of beta—does not help me. Someone who has more pepper does not become more or less salty. He has pepper, not salt. As long as beta has no relation to alpha— That’s true, that really has nothing to do with it. Exactly. So if I assume, as we assumed earlier, that these two kinds of inference are really two different inferences, what am I actually saying? That the hierarchy between the domains is in terms of alpha—this is alpha and this is two alpha—and the hierarchy between the damagers is in terms of beta. That is another parameter. If that were reality, it could not be correct; it does not explain the columns in the table, the data in the table. In order to explain the data in the table, I must say that both here and here the hierarchy is in terms of alpha, not in terms of beta. I need to say: tooth and foot have the force alpha, therefore for the public domain it is not enough to impose liability—that is zero—and for the courtyard of the injured party it is enough, therefore that is one. Horn has two alpha, so even in the public domain it is liable, and of course also in the courtyard of the injured party. That can be an explanation of the data in the table. But if the hierarchies were here in terms of alpha and here in terms of beta, I have explained nothing in the table—that is not a theory.

In other words, just as in analogy we go to a theory in order to explain the analogy, here too we are doing the same thing. We look for the theory that stands behind the facts we know. We basically create a theory, a generalization, as in science. And then infer a conclusion about what will happen in another case from that theory. For example, we saw that objects with mass fall to the earth—this falls to the earth, that falls to the earth—and we arrived at the theory: all things that have alpha, mass, fall to the earth. Now I ask: will this too fall to the earth? The answer is yes. It has mass. I can infer from the generalization I made that this too will fall to the earth. That is what I am doing here. What I am doing here is exactly a scientific generalization. I am basically saying: there is a parameter—without identifying what it is at the moment—there is some parameter; I call it alpha. That parameter is present at a higher intensity in the public domain than in the courtyard of the injured party. And it is also present at a higher intensity in horn than in tooth and foot. If I assume that, I can explain the three data points I have in the table. So that confirms my theory. And once I have confirmed the theory, I can use it to derive the missing datum, to make predictions, as in science. And therefore the result is one. Okay? Therefore this is really not two inferences but one inference. The hierarchy in the damagers and the domains is the same hierarchy.

Now look. You could say that these are two hierarchies, but that you take the phenomenon under some common denominator. I didn’t understand. Say tooth and foot and horn—basically, when we present a difference in stringency, you say: where does it happen? It happens on a certain substrate, on a basis. A basis, say, of the public domain. And then the place is an active participant. Well then there is no point bringing in beta. There is no reason. I can use alpha alone. There is no point introducing beta by hand. We’ll see in a moment. Theoretically it is possible, but it is less simple. Occam’s razor basically says not to do it. Okay. We’ll see that in another second.

Look. Earlier we had the tail refutation, right? And we asked why that refutation topples the whole a fortiori argument. I understand why it topples the a fortiori argument of the columns. Because the hierarchy between these two no longer exists. But why does it topple this as well? The answer is very simple. Once I inserted this, it means that it is not correct to explain the situation as alpha and two alpha. There is probably also some beta here now, right? Once that is so, everything collapsed. You can no longer assume anything, because even if you assume alpha here and two alpha there, it won’t help you, because here too you need beta factors in order to impose liability. You must introduce another parameter, beta. Once you introduced another parameter beta, you broke the entire a fortiori argument. Now it can be zero and it can be one. It depends how you build the theory. There is a theory that will give you one and a theory that will give you zero. Therefore the column-refutation topples both inferences. The same with a row-refutation. Because once it topples alpha and beta, it is basically saying: you must introduce beta here too, not just alpha and two alpha. You will not succeed in explaining the data with a single parameter. There must also be beta here. And once there is beta here too, now everything is reopened; you can no longer explain anything.

Now let’s try for a moment to suggest another theory that also works. Let’s assume that in the public domain it is not two alpha but alpha and beta. In the courtyard of the injured party it is only alpha. What does that say about the damagers? Here what there is is only—say I have now built the hierarchy between the public domain and the courtyard of the injured party. Let’s erase this. I’m building another theory. I have another theory that can explain the data. Okay? Another theory. The hierarchy is not alpha and two alpha, but alpha and here alpha and beta. Also possible. Okay? Now if tooth and foot impose liability in the courtyard of the injured party, then what does that mean? Clearly they have alpha. On the other hand, clearly they do not have beta, because if they had beta they would also impose liability in the public domain. So they have only alpha. Right? What happens with horn? Horn must be alpha and beta. Right? It has both alpha and beta. Okay? And therefore in the public domain it imposes liability. What will now happen in the courtyard of the injured party? It will also impose liability. It has alpha and beta—specifically alpha—so it will impose liability, right? Meaning that this too is a possible theory. I could have written the theory as alpha and two alpha, alpha and two alpha. I can also write it as alpha and alpha-beta, alpha and alpha-beta. Same outcomes. Why do I choose alpha and two alpha over that? Occam’s razor. If I have two theories, one of which is simpler, I prefer the simpler one. If I have a theory with one parameter and another possible theory with two parameters, of course I will choose the theory with one parameter. Right?

For example, I saw that this fell to earth and that fell to earth. So maybe there is no general law that all bodies with mass fall to earth; maybe only bodies that are media devices and have mass fall to earth, and bodies that are printed and have mass fall to earth. But perhaps something that is not printed and not a communications device—a ball—which has only mass, maybe it won’t fall? It could be; it fits the data, right? Why do I assume it is not so? Because if those two data points can be explained by one single parameter—that they have mass—there is no reason to add more parameters: this is communication and this is printed. It doesn’t matter. I can explain it without that. You do not add parameters for nothing.

It’s a matter of convenience, no? No. Occam’s razor says that the simpler theory is probably also more correct, unless it is proven otherwise. This is an old philosophical dispute. I once wrote an article about this too. People think Occam’s razor is only a methodological principle; it is not really a true claim about the world. If I have a complicated theory and a simple theory, why not choose the simple one? It is easier. That doesn’t mean the simple one is more correct. And if both theories are usable, I’ll use the simple one. And I can prove that this is not correct. Occam’s razor is not merely methodological; it is a principle about the world. A simpler theory has a better chance of being correct, and that is why I use it. We won’t get into the proof right now; this is not the place. But that is the claim.

Occam’s razor basically says that—look what I am actually doing. The beta here will be alpha itself. I simply identify beta with alpha. And then what happens? Here there is alpha, here there is two alpha, here there is alpha and here there is two alpha. The same thing; I am just identifying beta with alpha. There is no reason to make them different if I can assume they are the same parameter. That’s it. That is how you construct the a fortiori argument.

This is the explanation that stands behind the inference of a fortiori reasoning. When Pharaoh and the children of Israel do not listen to my voice, then I understand that Pharaoh’s degree of obedience is lower than that of the children of Israel. I understand that by reasoning. Degree of obedience is the alpha. Pharaoh has one alpha and the children of Israel have two alpha—they are more obedient. Okay? Or more resistant, then it is the opposite. It doesn’t matter. Okay? Here, in the case of tooth and foot and the courtyard of the injured party, I don’t know how to say in what sense it is stricter, but I do know how to say that there is some parameter here by which the courtyard of the injured party is stricter than the public domain. That is a fact—I see it in the laws of tooth and foot. So let’s mark it as alpha. I do not identify what it is. I mark it as alpha and say: let’s use it. That is the assumption. That is enough to infer the conclusion of the a fortiori argument. That is basically what underlies the logic of a fortiori reasoning.

If so, now I’ll propose a general algorithm. I have a table with data. What am I actually supposed to do in order to fill in the missing datum? I am supposed to look for the theory that explains the existing data, choose the simplest among the available theories, and infer from it a conclusion about the missing cell. That is what I am supposed to do. Okay? Or in other words—I will phrase it this way because mathematically it is easier to do—I am basically saying this: in the place of the question-mark cell I put two possibilities. Either… zero or one. Okay? Is it clearer now? We’re sitting down here and can’t really see what—can you see better now? This is blue and this is red. Okay? You know what, I’ll use, if you want, a more… a more… green. Okay. Green… no, I think yellow maybe stands out better here. Yellow you don’t see well. You can see that it’s different but you don’t really see it. Black you see well. Okay, no, black I specifically don’t want. Fine, then let’s use green. Okay, whatever, not important.

So basically we have three data points written in black and two options. I want to choose between the two options: is a zero written here or a one written here? Right? That is what I want to do. What am I going to do? I will build two tables. One table in which there is a one here—I’ll look for a solution or a theory that explains what is written there. A second table where there is a zero here—I’ll look for a theory that explains what is written there. Which theory is simpler? That will determine what the correct filling is. Okay?

Now look. I am basically doing it this way. I take this table and say this: I’ll look for a solution to this table and I’ll look for a solution to this table. There are two tables. Now let’s look for the solution. Look at this table. What is the solution to this table? Understand that for this table you will not find a solution with one parameter. No chance. These are two independent tables; this is linear algebra. You will not find a solution with one parameter. It just won’t be found—say again? A solution with one parameter, just alpha. You won’t succeed. Why? Because this column is the opposite of that column; they are independent. If they are independent, then basically there are two eigenvectors here—whoever knows, I don’t know, linear algebra. You will not find a solution with one parameter. There must be two. Okay? Therefore the solution is what I wrote here. Let us assume that tooth and foot have the parameter alpha, and horn has the parameter beta. To impose liability in the public domain you need beta; to impose liability in the courtyard of the injured party you need alpha. You see that that explains everything, right? If you replace beta with two alpha it won’t help; it won’t be a correct theory. You understand why not? Okay? So therefore that won’t work. So here we found the model.

What happens here? What is the solution here? In the filling with zero. Now I have the same table but with the filling one. What is the solution here? There—you see that this is the solution… what we had earlier. Right? Meaning, now I ask which of the two tables is more correct? Because that is basically the question whether to fill in a zero here or a one here, right? I ask which of the two tables is more correct. The simpler one. Right. Therefore how do I choose after all? Occam’s razor. This table is basically telling me: here there is a theory with two parameters. This table says: I have a theory with one parameter. One parameter is preferable.

By the way, Occam’s razor in its original form—William of Ockham—says that the theory with the fewest entities is the correct one. Today we have already broadened that to “the simpler one.” Simplicity can involve many things, but he talks about as few entities as possible. In that sense, this is really like Ockham’s original razor. Okay? As few parameters in play here as possible. All right? Yes. That makes for a simpler world. Why assume that this dish contains both salt and pepper if I can—look, I… I am measuring spiciness, say, in a dish. I can say that the dish contains salt, pepper, and sugar. The sugar neutralizes the salt, the pepper remains, and therefore it tastes spicy to me. Okay? I can also assume it has only pepper, no salt and no sugar. Why introduce both salt and sugar that cancel each other out? Who says either of them is there at all? Better to assume there is only pepper.

So that is already observational probability, intuition from experience. No, no, no, no, no. No, I said no. An a priori analysis without observation. I checked: there is a spicy taste here—that is the observational datum. Now I ask: what theory explains that? There could be a theory saying: it has pepper. There could be a theory saying: no, it has pepper, salt, and sugar. Yes, when you present it like that it’s obvious, but why would you assume things— That is Occam’s razor. That is Occam’s razor. When you see—what I did here is exactly the same thing. I have three data points here, not just one with the spicy taste—three data points. I offer two explanations for them. One explanation has both pepper and salt; the other has only pepper. Why assume both pepper and salt if pepper alone can explain everything? That is Occam’s razor.

You could say that maybe there really is pepper there, but it’s not like that, not out of nowhere. Also out of nowhere. There is no reason to assume there is something if there is no need to assume it. In science too, maybe this thing fell to earth just because someone kicked it and I didn’t notice. So why did that too fall to earth? Someone kicked that one too and I didn’t notice. Fine. I’m not assuming there were some demons there that I didn’t see. Rather, the simple explanation is probably the correct one—that things with mass fall. Maybe not? Who knows? Maybe all kinds of demons and spirits were moving around here and I didn’t see them. Fine, but I do not assume that. Same thing here—it is exactly the same idea. What I am doing here is simply the way one does abduction in science. How one gets from facts to theory in science. That is what you do. You generalize and check what the simplest generalization is.

Right? If I have a line—say I have a graph like this. A plot of two parameters against each other. Y and X. And this is what came out. When x was one, y came out one. When x was two, y came out two. Three, three. Four, four. That is what came out. Okay? Now I can say what any normal scientist would say: that the line is probably a straight line. Right? But I could also say that the line is this. That too fits all my observations. Why would I choose the straight line? Well, that you could perhaps check, but— No, suppose I can’t check. It is the simple line. A simple line. What is a straight line? A straight line has one parameter—never mind the intercept, with another two—y equals ax plus b. Right? So that is two parameters. That is the simplest. Anything beyond that—for example a parabola, which is the next simplest—has three parameters, ax squared plus bx plus c. And so on. The straight line is the simplest line. Therefore when I make measurements, I have many theories that can explain each such measurement—in fact infinitely many theories—and I choose the simplest one. And it works. So in Jewish law too it is exactly like that, just as in science. I have many theories; I choose the simplest theory. That’s all. Okay?

So if that is so, we now have an interesting technique for how we proceed. We have an a fortiori argument—basically a table of three data points—and I want to fill in the fourth datum. What do I do? I place the two possibilities. I could have done it by looking for an explanation of the three data points and deriving from that explanation the fourth result. A mathematically more convenient way—it is the same thing, but mathematically more convenient—is to say: I will assume both possibilities in the fourth datum, either zero or one, look for explanations for the two tables, and choose the simpler one. Okay.

Now let’s see what happens when we have a refutation. “From all my teachers I have gained wisdom.” Are there any colors left? Yes. Now I want to add our friend the tail, His Honor the Tail. This is not in red; it should be in black. These are fixed data, so no colors—it’s black. Okay? You understand that basically now I again have a data table, and I have to check which filling is correct. So I’ve already forgotten that one thing is called “refutation” and another thing is called “a fortiori” and this one is called “building a prototype from one verse and from two verses”—I don’t care. I have a table, and now this table—whatever you call it, it does not matter—the technique is one. I simply copy these two data points, these two tables, yes. Okay. Once I put here one; once I put here—I mean, I search for a theory for these two fillings. Okay? And once I search for a theory for these two fillings, I choose which theory is simpler.

Now it turns out that the theory you will find for this—can all this be explained by one parameter? In this table. No. No chance, because there are two independent columns here. A beta will appear here too in some form. Okay? I’m not getting into the question of how I find the theory; there is quite a good way to find the theory—I could show you in graph theory how to find the theory. But there is a way to find the theory, and from that I can already tell you, and anyone can see this: there will not be a theory here with fewer than two parameters. When there are two independent columns here—and by the way likewise two independent rows, zero and one and zero and one, right? Between tooth and foot and tail down here there is an opposite relation. Therefore two independent rows, two independent columns—there are two independent parameters here.

What will happen here? The same thing. Here too there are two independent columns and two independent rows, right? Therefore here too it will require at least two parameters. And therefore you can already see there is no preference for the filling one over the filling zero. The theory of this table and the theory of this table are both two-parameter theories. Therefore it is a refutation. Therefore we cannot know what to fill in here; one and zero are both possible. That is how you analyze a refutation.

So basically you notice that we have completely freed ourselves from the need to think in terms of a fortiori, refutation, common denominator, building a prototype, one verse, two verses. Everything, everything, everything can be analyzed in this way. Write the full data table—all the refutations, all the a fortiori arguments—put all the data inside. Please mark for me the cell you want to fill. I will tell you whether that cell should contain one, should contain zero, or should contain “unknown.” “Unknown” means that one and zero have the same standing; you cannot prefer one over the other. If it says one or zero, then it is a conclusive inference, meaning you can infer a conclusion: exempt there or liable there. If between one and zero it cannot be determined which is better, then that is a refutation. And that basically means that you cannot know what happens in that cell. Because a refutation does not mean that it is false. A refutation means that it is unknown. You cannot infer a conclusion, and therefore it remains open.

You need to remember this well: a refutation does not mean that zero should be written. A refutation means: I do not know whether zero or one should be written. Both are equally possible. There is no way to know. Okay? So that is the general technique.

Now let’s do, for example—okay, let’s do, for example, the table for… We have an a fortiori table; I am now going to do a table of binyan av. Binyan av? Yes. A table of binyan av is built like this. Because what is binyan av? Binyan av basically means: if tooth and foot are similar to horn in one parameter, then tooth and foot will be similar to horn in the second parameter too. That is basically… An analogy? Exactly. It is an analogy. And now let’s see how we prove that a one should be written here. Ostensibly very simply. What do we do? As usual, with the regular technique. I now analyze the table in which there is a one here, analyze the table in which there is a zero here, and I see. Alpha, alpha, and here too alpha, right? No need for anything, no change, everything is one. This is as simple as it gets. One parameter, and not even divided into alpha and two alpha—everything is alpha at one level. That explains everything, right? That is as simple as possible. You can already see why in binyan av it makes more sense to put a one there.

What happens with zero? Two? No. Two? No. We said that in order to have two independent parameters, I need two independent rows and two independent columns. There aren’t any here. In fact, you see the table written down here? This is an a fortiori table. Right? Somehow what came out for us here is an a fortiori table, because this is zero, one, and one, and think as if the question mark for the a fortiori argument is here. This is basically an a fortiori table. Therefore I already know the solution to this table. Wait, wait—is this a fortiori? It is not an a fortiori argument, but the table comes out like one. Because basically—just reversed—the question mark is here rather than there. But what was the a fortiori table? Zero here, one here, one here, and a question mark, right? So here the zero is on the bottom; what difference does it make? So call the courtyard of the injured party the public domain and vice versa, switch the places—but it is an a fortiori table. And we already know its solution. Its solution is alpha and two alpha. How do I know? Tooth and foot have two alpha, so they impose liability in the public domain, because in the public domain alpha is enough; if you have two alpha, then certainly you impose liability. And also in the courtyard of the injured party, because there too you need two alpha and you have two alpha. Horn has only one alpha, so in the public domain it manages to impose liability. And in the courtyard of the injured party it does not manage to impose liability, because two alpha are needed and it has only one alpha. So here is an explanation of the table with one parameter, not two. But it is one parameter divided into two levels, alpha and two alpha. That is still less simple than this parameter. Therefore in binyan av too the preferred result is one. Because it is still simpler. This model, in which there is just alpha at one level, whereas here there are alpha and two alpha—that is one parameter but with two levels, it is more complex. Therefore this model is preferable, and therefore in binyan av too the filling is one. That is the explanation of why binyan av works.

So we explained a fortiori reasoning, we explained a fortiori reasoning with a refutation, and we explained binyan av. Binyan av with a refutation will also be the same thing. Look: what will happen with binyan av and a refutation? Let’s make some quiet here and tighten the cloth—doesn’t help, it’s just like that. I try not to move it but it’s always angry with me. Some air spray with that magic stuff. Fine. Let’s go back to our friend the tail, and this time it is a refutation of binyan av. Okay? Same thing: horn, tooth and foot, tooth and foot, public domain, courtyard of the injured party. Okay, so now I analyze the refutation. I already did this on the two ta—now I look for a theory, so let’s see what the theory is. Here, in this case, what happens? How many parameters will we need? You already know on your own. On the face of it, three. Yes, for the whole table. Why? No, one will suffice but it will need—alpha and two alpha. Why? Because there is no independence between the columns here, right? There is no independence. But there are zero and one, so we will probably need two levels. By contrast here, what happens? Look at the bottom one. Same. Right. Here too we will basically need one parameter with alpha and two alpha, right? What does that mean? Exactly. The preference has been erased. The preference of one over zero has been erased. That is how a refutation works. Therefore a refutation topples binyan av too, not only a fortiori reasoning.

Okay. But everything is based on Occam’s razor? Yes. All of science is built on Occam’s razor. For halakhic inferences it sounds less dramatic, but this is how it works. Science works. It is not built on Occam’s razor. Every theory you find in science—you could always invent three more demons involved there, and your explanation is not right at all. Fine, but my explanation is simpler, and therefore I adopt it. Occam’s razor is a very powerful instrument. It is the basis of all non-deductive thought, Occam’s razor.

Now, the logic that describes all these structures is more complicated than what I showed you here. What I showed you here is actually not exact, but I am not going to get into all the details in order to show why it is not exact. I want you to get the principle. The principle is that we have a systematic technique for showing how we infer conclusions in logic that is not deductive. After all, these inferences are not necessary—there are refutations here, right? And still, you see that we have a fully mechanical logic, just like deductive logic, that governs the modes of inference of science, of law, of Jewish law, of everyday life. This is how we infer conclusions. Basically this is an analysis of how we infer conclusions from facts in daily life.

And it is not logic in the sense that it is necessary, and therefore usually logicians do not deal with this type of inference, because they think: this can’t be mathematics; it’s common sense, I don’t know exactly what. I am showing you that no—there is a mathematical framework here, inside which one can formalize all non-deductive inferences. And therefore all scientific inferences, all legal inferences, all halakhic inferences, all of them sit on this logic. Everything. I can translate for you any scientific generalization, any legal generalization, any halakhic generalization, into this. And that is what is there. That is all. What changes are the tables; the tables can even be a hundred by a thousand. Here I made two by two, two by three, but there can also be tables of one hundred by one thousand or one million by one million—it doesn’t matter. On the principled level, this is basically what we do.

Now I’m not going to get into all the details—we wrote a book about this. I’m not going to get into all the details of the matter, but I think this is enough to get an impression of how this whole thing works. Why on the one hand there are non-deductive modes of inference here, and on the other hand it is not arbitrary, the way the child accused the adult. Remember when I talked about that in the lecture? The adult says “common sense,” and the child says, wait, but you have no proof, it’s not certain, it’s not logic. No—there is. Common sense too has patterns for how it works. What is more reasonable, what is less reasonable, what is probable, what is less probable. It’s not that I just decide whatever I want. And basically everything rests on Occam’s razor. Occam’s razor works; that is the optimizing reality we have in this world here. Okay?

Good, I’ll stop here. Next time I’ll move on to something a bit different, the “common denominator,” and we’ll see—if we have time—whether we can formalize that as well.