Topics in Tractate Makkot, Chapter 3 – Lesson 10 – Rabbi Michael Avraham

[Rabbi Michael Abraham] Okay. So last time we talked about — I mentioned punishment derived by logical inference, and I said I’d come back to it, and I want to touch on that a bit today. We have two sessions, so we’ll see whether I manage to get to something else next time, or whether I’ll have to continue this. We’ll see how far we get. If there’s time, then maybe I’ll touch on a prohibition linked to a positive commandment. Anyway, regarding punishment, the rule is that punishments are not derived by logical inference, and warnings are not derived by logical inference. Okay? Meaning, punishment derived by inference means that I have a lighter offense and a more severe offense. For the lighter offense, a punishment is written explicitly. If I make an a fortiori argument, then I could punish for the more severe offense too. If the lighter offense carries a punishment, then the more severe one certainly does. That’s punishment derived by inference. Warning derived by inference means that I learn the warning itself through an a fortiori argument. And there, there is a warning but no punishment. So I make an a fortiori argument regarding the punishment. There’s also a case of warning derived by inference, where the warning itself is learned by an a fortiori argument. And then there’s also punishment, but warnings are not derived by inference — meaning, that does not count as a warning for purposes of punishment, and without a warning you can’t punish either. So I want to touch on this a bit. Let’s say for the sake of discussion, punishment by inference — it’s very similar to the rule that warnings are not derived by inference, but it’s the same basic idea. I’ll start, maybe, with what “from logical inference” means. In the simple understanding, what does it mean? An a fortiori argument, right? When the Talmud says, “But is this not a logical inference?” what does that mean? It means it’s an a fortiori argument, right? So “logical inference” in Talmudic language basically means an a fortiori argument. So “punishments are not derived by logical inference” means punishments are not derived through an a fortiori argument. Clear? Okay. So basically we have a rule that punishments are not derived by logical inference — meaning we do not learn a punishment through an a fortiori argument. Later we’ll see that Maimonides has a somewhat different approach here, and for him “logical inference” means any interpretive principle, not just an a fortiori argument. None of the interpretive principles can be used to derive punishment. But right now I’m talking about an a fortiori argument.

So what’s the meaning of this? To understand why punishments are not derived by inference, I want to present a few kinds of a fortiori arguments. One kind — you can divide a fortiori arguments into two large categories. One kind is the biblical a fortiori argument. The midrash says there are ten a fortiori arguments in the Bible itself. “If her father had but spit in her face, would she not be shamed for seven days?” — then all the more so if it’s the Holy One, blessed be He. Or, “Behold, the Israelites did not listen to me; how then will Pharaoh listen to me?” — that too is an a fortiori argument. And so on; there are all kinds of a fortiori arguments that Scripture itself makes. Okay? By contrast, there are a fortiori arguments in the Talmud. For example: if tooth and foot, which are exempt in the public domain, are liable in the injured party’s courtyard, then horn, which is liable in the public domain, is it not obvious that it should be liable in the injured party’s courtyard? That’s an a fortiori argument. What’s the difference between that and the earlier a fortiori arguments? This one is based on three data points. The earlier ones are based on one. Right? “The Israelites did not listen to me” — one datum, and the conclusion: Pharaoh won’t listen to me either. Right?

[Speaker B] That’s a story, right? I mean, you can learn from it, but it’s a kind of story, not in the context of commandments. “Behold, the Israelites did not listen to me.”

[Rabbi Michael Abraham] Doesn’t matter. It’s still an a fortiori argument. On the level of logic it’s an a fortiori argument. What —

[Speaker B] Does it matter whether it’s a commandment?

[Rabbi Michael Abraham] Am I learning a law from it? No, I’m not learning any law. It’s an a fortiori argument, but not in a halakhic context. Okay. An a fortiori argument appears in many places, not only in a halakhic context. Even in our everyday lives we make a fortiori arguments all the time. If a spacecraft has a thousand-horsepower engine and it manages to escape the earth’s gravitational field, what about a spacecraft with two thousand horsepower? Will it escape or not? A fortiori, right? If with a thousand you succeed, then with two thousand you certainly succeed. And so on.

So the a fortiori arguments that appear in Scripture are arguments that have one datum, a basis, yes? “The people of Israel did not listen to me” — that’s the datum. The conclusion: Pharaoh won’t listen to me. Or, “If her father had but spit in her face, she would be shamed for seven days,” then certainly the Holy One, blessed be He, at least seven days too, right? That’s one datum from which I infer a conclusion because the conclusion is more severe than the datum. In a Talmudic a fortiori argument, in most cases there are three data points, not one. Tooth and foot are exempt in the public domain and liable in the injured party’s courtyard — two data points: exempt in the public domain and liable in the injured party’s courtyard. Third datum: horn is liable in the public domain. Then I make an a fortiori argument and learn that horn is liable in the injured party’s courtyard too. Okay? So that’s an a fortiori argument based on three data points, from which I derive the fourth.

Now, within the biblical a fortiori arguments, the ones based on one datum, you can identify two types.

[Speaker C] One more time? You said Pharaoh not listening is an a fortiori argument, and Israel listened more readily, so why wouldn’t Pharaoh listen? Okay, and that’s a kind of datum in itself.

[Rabbi Michael Abraham] It’s not a datum, it’s a hierarchical rule. Obviously the datum by itself doesn’t give you the conclusion. You need a hierarchical rule that says the conclusion is more severe than the datum. So if the datum is “did not listen to me,” then the more severe case certainly won’t listen to me either. So there’s a datum and a hierarchical rule. In the Talmudic case there are three data points, and no explicit hierarchical rule. Just three data points. Okay? What’s really happening in that case is that from the three data points you take two, extract a hierarchical rule from them, and then you’re back to the standard case. You have a datum plus a hierarchical rule that came from the earlier two.

Take tooth and foot: tooth and foot in the public domain are exempt, and in the injured party’s courtyard they’re liable. What does that mean? That the injured party’s courtyard is more obligating than the public domain, right? Erase those two data points, and now we have a hierarchical rule: the injured party’s courtyard is more severe than the public domain. Now we have a datum: in the public domain, horn is liable. So in the injured party’s courtyard, which is more severe, horn will certainly be liable. So once again we’re back to a datum and a hierarchical rule. The difference is that in the Talmudic a fortiori argument, we extract the hierarchy rule from two data points. In the biblical a fortiori argument, the hierarchy rule is a matter of reasoning. Okay? That’s why these are two types of a fortiori argument.

Now look at the a fortiori argument we saw, okay?

[Speaker C] Is this thing working? I don’t know.

[Speaker D] Yeah, it’s working. I see a light here. A blue light — usually it’s green, if I’m not mistaken.

[Speaker C] Working? I don’t know, it’s working. Not working. Now it lit up. Wait, there’s the light. But blue —

[Speaker D] Again. I can’t get it to work. There, now it’s green.

[Speaker C] Yeah, okay.

[Rabbi Michael Abraham] “The Israelites had light in their dwellings.” So the first a fortiori argument: “And Moses spoke before the Lord, saying: Behold, the Israelites did not listen to me, so how will Pharaoh listen to me? And I am of uncircumcised lips.” That’s the a fortiori argument we saw. A second a fortiori argument: “If a man opens a pit, or if a man digs a pit, and does not cover it, and an ox or a donkey falls into it, the owner of the pit shall pay; he shall return money to its owner, and the dead animal shall be his.” Okay? Now on that the Talmud says: if one is liable for opening, then for digging all the more so, right? “If a man opens a pit” or “if a man digs a pit.” So you’re liable both if you opened the pit — an already existing pit where you just removed the cover, that’s called opening a pit — or if you dug the pit, meaning you created the pit.

So the Talmud says: if for opening one is liable, then for digging all the more so. This too is a biblical-style a fortiori argument based on one datum, right? Not that Scripture itself makes the a fortiori argument here — in the first case Scripture itself makes it: “The Israelites did not listen to me, so how will Pharaoh listen to me?” Here Scripture doesn’t make the a fortiori argument; the Sages do. But they too are making an a fortiori argument based on one datum, okay? What’s the difference between them? There is a difference. What’s the reasoning behind the first a fortiori argument? “The Israelites did not listen to me, so how will Pharaoh listen to me?” If —

[Speaker C] If the Israelites didn’t listen to my voice, though they wanted to listen —

[Rabbi Michael Abraham] It’s less likely that Pharaoh will listen to my voice than that Israel will listen to my voice, right? Why? Because I know the actors involved, right? Yes, I know the actors involved, and the chance that Pharaoh will listen to me is lower than the chance that Israel will listen to me, right? What happens with opening and digging?

[Speaker C] What’s the hierarchical relation there?

[Rabbi Michael Abraham] Why is opening less severe than digging? Let me try to phrase it differently. What you’re saying is right, but there’s something even stronger that can be said. When you dig a pit, in particular you also open it, right? Think about it: when I dig the pit, what does it mean to open the pit? The pit already exists and I remove just the top centimeter that was covered, right? Now if I dig the pit, then I dig all ten handbreadths, including that top centimeter, right? Meaning that digging the pit is an action that contains opening within it. So the hierarchical relation between opening and digging is not the result of some reasoning; it’s a relation of inclusion. Pharaoh listening and Israel listening are not contained one within the other — they are two separate things. We just have a line of reasoning that Pharaoh is less likely to listen than Israel. Okay — here it’s a mathematical relation. When you dig the pit, you dig 9.9 handbreadths, and then you also dig the last 0.1 handbreadth, okay? Which is really the opening. Meaning that the act of digging contains opening plus something more. Fine? It’s not just that it’s more severe; it literally contains the act of opening itself plus something else.

Basically I’d put it like this: on the first a fortiori argument, there can be a refutation. Maybe Pharaoh got hysterical one day and suddenly he’ll listen to me more than Israel does. They feel close enough to me that they allow themselves to refuse me; Pharaoh is afraid, I’m striking him with plagues and so forth. There are possible refutations of the a fortiori move from Israel to Pharaoh. Can there be a refutation of the relation between opening and digging? It seems not. Because digging is not merely more severe than opening; it simply — say I punish for opening. Say the Torah writes a punishment for opening. Now I want to know whether to punish also for digging. The answer is obviously yes. Why? Because you can punish for the opening contained within the digging. Don’t punish for the digging — but inside the digging there is also opening, so punish for the opening element within the digging. You don’t even need to assume that digging is more severe than opening; you only need to note that inside digging there is opening too, that’s all. This is an a fortiori argument of “two hundred automatically includes one hundred.” Obviously. Because he didn’t open a pit — he opened, but not a pit. Fine, but practically speaking he also performed an act of opening. So what would that mean? Say I somehow managed to dig the pit from below and only at the end removed the last top centimeter — it wouldn’t matter, right? The point is that the act of digging includes opening. So this is basically a mathematical a fortiori argument. It’s not an a fortiori argument that can — can it be refuted? Okay?

Fine. Now we need to understand this well. Let me step back for a moment. Is an a fortiori argument a deductive argument? Meaning, something necessary that you can’t argue with? Well, you can argue with it, right? After all, sometimes we do find a refutation. But against a mathematical or logical argument there is no refutation. When I say: all human beings are mortal, Socrates is a human being, therefore Socrates is mortal — can you refute that? No. Because if all human beings are mortal, then Socrates in particular is mortal, right? It’s already included in the premises. You can’t refute that. That’s logical or mathematical certainty. An a fortiori argument is a different type of argument. It’s something that can be refuted. I have a line of reasoning that one thing is more severe than another, but there may be countervailing reasons saying: no, the relation is not that simple. Okay? But regarding an a fortiori argument of the type opening and digging, at first glance it seems there’s no way to refute it. Because it’s a logical a fortiori argument, a deductive a fortiori argument. It’s necessary. Right?

Now, in the Mekhilta on this verse it says the following: “If for opening one is liable, then for digging all the more so — rather, from here we learn that punishments are not derived by logical inference.” Yes, the Mekhilta on this verse asks why the Torah writes both “if a man opens” and “if a man digs.” Let it write only opening, and we’ll learn digging by an a fortiori argument. And we saw that a fortiori argument earlier, right? Rather, it teaches you that punishments are not derived by logical inference. What does that mean? If it had written only opening, yes, then I would have learned digging by an a fortiori argument. But the rule is that punishments are not derived by logical inference, so I would not have imposed liability for payment on someone who dug a pit, even though digging is more severe than opening and there is an a fortiori argument. But punishments are not derived by logical inference. Right? That is why the Torah wrote digging too. And then what happens? It teaches us why digging had to be written. Because if digging had not been written, I really wouldn’t have known to punish for digging. From here we learn the rule that punishments are not derived by logical inference. What? Because punishments are not derived by logical inference. I would not have learned to punish. You’re asking what the reasoning behind this is? Why punishments are not derived by logical inference? No — that’s what I said.

[Speaker F] No —

[Rabbi Michael Abraham] I would punish only for opening. Why would you punish for digging? Because there’s an a fortiori argument from opening? So what if there is? Who says you punish when there is a pit? You punish for opening. How do you know about digging? Because there’s an a fortiori argument. But punishments are not derived by logical inference. An a fortiori argument is not enough in order to punish. Okay? That’s what the Mekhilta says.

Now the Talmud — maybe I’ll actually start there first — in the Talmud they learn something else from opening and digging. In Bava Kamma, page 49 I think, they derive something else from digging and opening. And the Maharsha there basically says: in the Mekhilta it says, “From here we learn that punishments are not derived by logical inference.” “In the Mekhilta” means in the Mekhilta, not “in our tractate.” That’s a mistake; somebody must have opened the — there was probably a comma there. That happens a lot. “In our tractate” usually means in our Mishnah, but here it means in the Mekhilta. So: there it says, “From here we learn that punishments are not derived by logical inference,” but in the Babylonian Talmud it seems that one does derive punishment by inference. After all, the Babylonian Talmud says: “If for opening one is liable, then for digging all the more so.” So you see that punishments are derived by logical inference.

So he says: and if one can answer the reasoning, here it doesn’t really belong to say this is an a fortiori argument, because with digging he is also performing opening; he is just doing something else in addition to the opening — namely, he digs and opens. So even if opening had not been written, why would you think he should be exempt? In any case he performed an opening. He says: an a fortiori argument from digging to opening is an a fortiori argument of “two hundred automatically includes one hundred,” right? An a fortiori argument of “two hundred automatically includes one hundred” cannot be refuted. So if now you dug — say the Torah had written only opening, and I dug a pit — would I know that one is liable for that? The answer is obviously yes. Why? Because when I dug, in particular I also opened, and the Torah explicitly says that one is liable for opening. I don’t need the a fortiori argument. They would punish me for the opening involved in the act. Okay?

Therefore he says: where does it make sense to apply the rule that punishments are not derived by logical inference? To an a fortiori argument where each case is a separate matter, and it just seems more reasonable to impose liability — like Pharaoh. “The Israelites did not listen to me, so how will Pharaoh listen to me?” There the hierarchy is based on likelihood. But it’s not a mathematical rule like opening and digging, where one contains the other. Okay? So there it makes sense to say punishments are not derived by logical inference, but here it doesn’t. Whereas here — just because he did another thing as well, namely he also dug, should that exempt him from the opening that he performed? He opened, and the Torah imposes liability for opening. So the fact that besides opening he also did something else — should that exempt him? Why would it?

Good. So that’s interesting.

[Speaker D] So why then — how —

[Rabbi Michael Abraham] does the Mekhilta still say here that punishments are not derived by logical inference? The Mekhilta learns from here, from the fact that the Torah wrote both opening and digging, that punishments are not derived by logical inference. Which means that the Mekhilta understands that if only opening had been written, we would not have learned liability for digging. According to the Maharsha — meaning, we wouldn’t have learned liability? But opening is included inside digging! This isn’t an a fortiori argument. You’re liable for the opening contained in the act. So apparently the Mekhilta doesn’t agree with the Babylonian Talmud, at least as the Maharsha understands it. It’s a dispute between the Mekhilta and the Babylonian Talmud. The question is whether in an a fortiori argument of opening and digging we apply the rule that punishments are not derived by logical inference. That is the dispute.

What is the dispute? The Babylonian Talmud’s explanation we understand from the Maharsha. What does he say? He says: true, punishments are not derived by logical inference — but here this is not “logical inference.” You are punishing for the opening included in the act. So what does the Mekhilta hold? The Mekhilta says that even in this a fortiori argument one does not derive punishment by logical inference. Why? What lies behind this is basically the question of why punishments are not derived by logical inference in the first place.

Yes, Ginat Veradim of the Pri Megadim — it’s a book of rules by the Pri Megadim — he really says that from here one cannot bring proof that punishments are not derived by logical inference, same as the Maharsha. Because this is an a fortiori argument of “two hundred automatically includes one hundred.” Yes, an a fortiori argument of “two hundred automatically includes one hundred” is the mathematical kind. Just as you learn from one hundred to two hundred — with two hundred you impose liability because of the hundred that is inside it, not because two hundred is more than one hundred, but because inside two hundred there is also one hundred, and you impose liability for the hundred. Okay? That is the mathematical a fortiori argument. So in the language of the rule-books, this is called an a fortiori argument of “two hundred automatically includes one hundred.” Okay? Because the two hundred includes the one hundred. Inside the two hundred is the one hundred. Not that the two hundred is more severe than the one hundred. Okay?

So we really do see in the Mekhilta a statement that even in an a fortiori argument of “two hundred automatically includes one hundred” one does not punish by logical inference — which is very strange. So look: the question is why, really, punishments are not derived by logical inference. One possibility is that punishments are not derived by logical inference because — as later authorities write — of the concern that there may be a refutation of the a fortiori argument. Who knows? There are a fortiori arguments that can be refuted. Now if you punish, it could turn out that you flogged or killed a person on the basis of an a fortiori argument for which a refutation may later be found. You don’t want to take that risk, especially since we’re talking about execution more than flogging. Because with flogging, the punishment is not written explicitly at all. There’s nothing to learn by an a fortiori argument, there’s no punishment to derive by inference in the case of flogging. Regarding flogging, the issue is only warning derived by inference. As for punishment by inference, the punishment is written neither in the lighter case nor in the more severe one. The punishment of flogging is not written anywhere. Every prohibition gets flogging, not because the punishment of flogging is written there. A death penalty has to be written in the Torah. So if it’s written in the lighter case, I learn by an a fortiori argument to punish in the more severe case as well. And punishment by inference is speaking about death. So where there is concern about a possible refutation — and any a fortiori argument might be refutable — you don’t kill a person. Who knows? Maybe afterward a refutation will be found, and you’ve already killed him. So that’s the accepted explanation.

And then of course it’s clear why, in the a fortiori argument of opening and digging, one would indeed punish by logical inference. Why? Because there there can be no refutation. It’s a mathematical a fortiori argument. The whole issue with the rule that punishments are not derived by logical inference is the concern that there may be a refutation. But in the a fortiori argument of digging and opening there is no concern that there may be a refutation. You can’t find a refutation to “two hundred automatically includes one hundred.” What are you going to say — that digging is less severe than opening? Digging contains opening itself, plus something more. So there can’t be a refutation here. It’s an a fortiori argument of “two hundred automatically includes one hundred.”

But according to the Mekhilta, the rule that punishments are not derived by logical inference also applies to an a fortiori argument of “two hundred automatically includes one hundred.” How can that be understood? So the Maharsha himself elsewhere, in Sanhedrin 64, explains that punishments are not derived by logical inference for another reason — not because of concern that there may be a refutation. Rather, it may be that the more severe act requires a more severe punishment, and the lighter punishment is not enough. Therefore punishments are not derived by logical inference. That’s a different explanation. Again: the earlier explanation was that punishments are not derived by logical inference because maybe there’s a refutation. This explanation says punishments are not derived by logical inference because maybe the punishment given for the lighter case is not enough to punish the more severe case. It may be that the more severe case requires a greater punishment. Therefore punishments are not derived by logical inference.

[Speaker C] If it’s more severe, then why not punish at all?

[Rabbi Michael Abraham] Right — leave it to the Holy One, blessed be He. There will be heavenly punishment here. Why? Because the assumption is, once again, that someone liable to death cannot simply be punished with flogging. Flogging is not half of death; flogging is a different kind of thing. And if you are liable to death, then flogging doesn’t apply, even though apparently flogging is lighter than death. It’s lighter, yes — but it is also a different kind of punishment, it addresses something else, works in a different way, however you want to define it. And therefore someone liable to death cannot simply be flogged. Now if that’s so, then one can understand the Mekhilta’s position.

[Speaker B] And where did we learn, for example, that those liable to karet do get flogged? I didn’t understand. Those liable to karet do get flogged?

[Rabbi Michael Abraham] That was in the first lectures. So what?

[Speaker B] But it’s also not written explicitly.

[Rabbi Michael Abraham] So? What of it? What do you mean —

[Speaker B] What is that?

[Rabbi Michael Abraham] There’s a source for it, doesn’t matter, we covered that there. But why does it matter? How is that connected to punishments not being derived by logical inference?

[Speaker B] About that — the question is whether it’s built on an a fortiori argument. No.

[Rabbi Michael Abraham] So the Mekhilta can perhaps be understood according to the Maharsha’s explanation in Sanhedrin. What does he say? The Maharsha here in Bava Kamma explains the Babylonian Talmud’s position. The Babylonian Talmud’s position is that punishments are not derived by logical inference because of the concern that there may be a refutation, and therefore in an a fortiori argument like digging and opening, where there is no concern of refutation, there one does derive punishment by inference. But the Mekhilta, which says that even in the a fortiori argument of digging and opening — even in an a fortiori argument of “two hundred automatically includes one hundred” — one does not derive punishment by inference, that cannot be because of concern over refutation. Rather, it is because the lighter punishment may not be enough to punish the more severe case.

Ugh, electrical trouble again — I forgot to charge this. Okay. So in a case where, say, a punishment had been written for opening, but not for digging, and now I want to know whether if somebody dug a pit I punish him — then the Mekhilta would say no, I can’t know that. Punishments are not derived by logical inference. Why? Because maybe digging is a more severe act than opening, and therefore maybe it requires a more severe punishment. If so, the punishment given for opening may be insufficient, and then you don’t punish. This argument can also be made about an a fortiori argument of “two hundred automatically includes one hundred,” because the two hundred is still more severe than the one hundred, not only inclusive of it. So therefore one can say this there too.

There are additional examples of this as well, both in the case of conspiring witnesses and in passing one’s children through to Molekh. How good it is that there are idol worshippers, so we can sharpen all sorts of questions like these. What would we do if there had never been worshippers of Molekh? So Maimonides writes in Laws of the Sanhedrin, chapter 20: “If the person against whom they testified was executed, and afterward the witnesses were shown to be conspiring witnesses, they are not executed by logical inference. As it is said: ‘as he intended to do’ — and he has not yet done it. And this matter is part of the received tradition. But if the one against whom they testified was flogged, they are flogged. Likewise, if money passed from one person to another on the basis of their testimony, it is returned to its owner, and they must pay him.” And Abraham says: “This is a mistake.”

Maimonides — the rule with conspiring witnesses is that two witnesses come and say so-and-so is liable to flogging, or death, or whatever, monetary payment. Then other witnesses come and prove the first ones to be conspiring witnesses, and the conspiring witnesses receive the same punishment they sought to impose on the defendant: flogging, money, death, and so on. But the rule is: “as he intended,” and not “as he did.” What does that mean? If the court ruled on the basis of the first witnesses and then they were proven to be conspiring witnesses, then the penalty is imposed on them. But if the court has already carried out the punishment, then not. Then we do not apply to them the rule of “as he intended.” We do not give them the punishment of “as he intended.” “As he intended,” not “as he actually did.” Meaning, if they already did it, not merely intended it, then no.

Incidentally, there isn’t actually a midrashic formula saying “as he intended and not as he did.” That’s Rashi in Makkot; there’s no Sages’ formulation like that. The language in the Sages is: “if they caused him to be killed, they are not killed.” But never mind, that’s the familiar expression.

Now notice that this a fortiori argument is an a fortiori argument of “two hundred automatically includes one hundred.” Why? Because the witnesses intended something against someone, then the court sentenced him to death, and now he was actually killed, right? Now they are not punished. Because only if they intended it but it was not carried out do we apply “as he intended,” but if it was also carried out, then no. But this is an a fortiori argument of “two hundred automatically includes one hundred.” If it was carried out, then there certainly had been intent beforehand, only after the intent it was also carried out. Exactly like opening and digging, right? It’s an a fortiori argument of “two hundred automatically includes one hundred.” So we see here in Maimonides that even in an a fortiori argument of “two hundred automatically includes one hundred,” one does not derive punishment by logical inference. That’s what Maimonides says: “they are not executed by logical inference.” Maimonides only claims that in the case of flogging and money, then we do apply it even if it was done, not only if it was intended. And the Raavad says no. But regarding death, Maimonides says punishments are not derived by logical inference.

[Speaker C] Like the Mekhilta.

[Rabbi Michael Abraham] Yes, exactly like the Mekhilta. So what does that mean, basically? That even in an a fortiori argument of “two hundred automatically includes one hundred,” punishments are not derived by logical inference according to Maimonides, right? Like the Mekhilta.

Now the Kesef Mishneh there asks exactly this question, and he says: “And it is possible to give a reason for our master’s words: that we say ‘as he intended and not as he did’ only where they caused him to be killed on the basis of their testimony, because their sin is too great to bear; it is not fitting to give them a court-imposed death which would atone for them, but rather it is fitting to leave them to be judged after death with terrible punishments.” What is he saying? That if the punishment was actually carried out, then the conspiring witnesses deserve such a severe punishment that the ordinary “as he intended” may not be enough. And then their judgment is left to Heaven; the Holy One, blessed be He, will take care of them. In other words, exactly the Maharsha’s reasoning that we saw in Sanhedrin 64 — that is exactly how the Kesef Mishneh explains Maimonides. Maimonides is following the Mekhilta’s approach.

What does that mean? That even in an a fortiori argument of “two hundred automatically includes one hundred,” punishments are not derived by logical inference. Why not? Because the two hundred is more severe than the one hundred, and therefore the punishment given for the one hundred may not be enough for the two hundred. Not because of concern over refutation, as the Maharsha explains the Babylonian Talmud. And he brings an example for this. The example is: one who gives all his offspring to Molekh is exempt. Which is not something that can be said in the case where the accused was executed on the basis of their testimony. What did he mean with this Molekh case?

This is Maimonides, Laws of Idolatry, chapter 6: “A person is not liable to karet or stoning unless he gives his son to Molekh and passes him through the fire on foot in the prescribed manner. If he gave but did not pass him through, or passed him through but did not give him, or gave and passed him through but not in the prescribed manner, he is exempt. And he is not liable unless he gives part of his offspring and leaves part, as it says: ‘from his offspring he gave to Molekh’ — part of it and not all of it.” When he gave part of his offspring to Molekh, then he is liable; but if he gave all of his offspring to Molekh, then he is exempt.

Now you understand that this too is an a fortiori argument of “two hundred automatically includes one hundred,” right? If, say, you have five children and you gave two of them to Molekh, then you’re liable. But if you gave all five, then not. Now why not? If you gave all five, then certainly you also gave two. So why shouldn’t you be liable for giving the two? The fact that you gave another three exempts you? That is the example the Kesef Mishneh brings for what he said above. Why? Because maybe giving all of one’s offspring to Molekh is something that demands such a severe punishment that the ordinary punishment for passing offspring to Molekh is not sufficient for it. Therefore one cannot learn by a fortiori reasoning in such a case.

And with flogging, do we know that the punishment is fitting for them? What? And with flogging, do we know that the punishment —

[Speaker G] is indeed appropriate for them?

[Rabbi Michael Abraham] Yes, that depends on the dispute between the Raavad and Maimonides. It’s not important right now. But I’m talking about death now. Okay?

There are also the Hagahot Maimoniyot there, quoting the Sefer Mitzvot Gadol. The wording of the Sefer Mitzvot Gadol on this is: “And there is a reason for this in answer to the heretics: because those executed by the court are atoned for through their execution, and this person committed so great a sin that the Holy One, blessed be He, does not want him to have atonement.” Meaning, an offense so severe that the lighter punishment is not enough for it; we need a harsher punishment.

Now the Kesef Mishneh says: and one may also give another reason — since God stands in the congregation of God, had this person not truly been liable to death, the Holy One, blessed be He, would not have allowed one soul from Israel to be destroyed. Since the Holy One, blessed be He, allowed the court to agree to kill this person and he was killed, he must truly have been liable to death. Therefore the witnesses are not subject to the death penalty — which cannot be said in the case of flogging.

[Speaker C] Yes, that means —

[Rabbi Michael Abraham] meaning, that is an explanation already brought by Nachmanides on the Torah. And the claim is that if he was killed, the Holy One, blessed be He, would not have allowed him to be killed unless he truly deserved it. So apparently he really did deserve it, and therefore the conspiring witnesses are not executed. Because maybe they didn’t lie after all. In the law of conspiring witnesses there is a novelty: we believe the second pair and not the first, even though on the face of it it’s two against two. Okay? So he says: here we have an indication that דווקא the first witnesses were right and not the second ones, and therefore we don’t execute them. Fine, that’s another explanation.

What’s bothering him? Why does he need another explanation? What bothers him about the previous explanation? Exactly the dispute between the Babylonian Talmud and the Mekhilta, right? The question is whether the rule that punishments are not derived by logical inference is because the lighter punishment is insufficient for the more severe offense — that was the explanation he proposed — or the simpler possibility, which says that punishments are not derived by logical inference because of concern that there may be a refutation. And if that’s the point, then in these a fortiori arguments of Molekh or conspiring witnesses, that doesn’t apply, because these are a fortiori arguments of “two hundred automatically includes one hundred,” and there certainly is no refutation there. Therefore he brings another explanation. Okay.

Now I just want to note something — just an interesting methodological remark. Usually when somebody discusses the rule that punishments are not derived by logical inference, they bring the words of Rabbi Yosef Engel in Lekach Tov, and the Talmudic Encyclopedia also brings this, and others. He claims there are three reasons, three reasons why punishments are not derived by logical inference, three positions. One position is the concern that maybe there is a refutation, and therefore we saw that in an a fortiori argument of “two hundred automatically includes one hundred,” one would indeed derive punishment by inference. A second position is that maybe the lighter punishment is not enough for the more severe offense — what we saw as the position of the Mekhilta and Maimonides. A third position is that this is learned from the verse “his sister, whether his father’s daughter or his mother’s daughter.” The Talmud asks: why do I need “his father’s daughter and his mother’s daughter” — if it is already his father’s daughter, why also say “his father’s daughter and his mother’s daughter”? The Talmud says: to teach you that punishments are not derived by logical inference. So this is learned from a verse. Incidentally, there too it is an a fortiori argument of “one hundred in two hundred,” because one’s sister from both father and mother is, in particular, also one’s sister from one’s mother. Right? So that too is — but for our purposes, there is a verse from which we learn that punishments are not derived by logical inference. One could also bring digging and opening, or the Mekhilta; that too is a source for this rule. So that’s a third position.

This is just a remark I can’t help making. This is nonsense. There are not three positions here. There are two positions and a source. After all, the common yeshiva assumption is that if a source is brought, that means there is no explanation. So we have two explanations — that makes two positions — and the third “position” is simply: here is a source. And if there’s a source, apparently there is no explanation, because if there were an explanation, why would you need a source? That’s called reasoning, right? That’s the common yeshiva assumption: when you bring a source, that means it disagrees with the approaches that hang it on an explanation. And that’s nonsense, obviously. Right? Just because you bring a source, does that mean there can’t be an explanation? There are two possible explanations. The source is the verse about the sister, or from digging and opening, it doesn’t matter. When you ask yourself what the explanation is, the explanation can be either that the lighter punishment is insufficient, or concern that there may be a refutation. There are two explanations and a source. That is not three positions, not three explanations.

Especially since one might say that we do not expound the reason of the verse. If there is a source, then we don’t derive the reason of the verse and extract explanations. But here that’s not true, because this thing is not written explicitly in the Torah. We learn it from exposition. The Torah writes opening and digging, and “to teach you that punishments are not derived by logical inference” — because otherwise, why write digging? That’s exposition. Okay? Regarding exposition there is no such thing as “we do not expound the reason of the verse.” In exposition, of course we do expound the reason of the verse. The expositor himself expounded the reason of the verse, so why shouldn’t I interpret his exposition according to the reason of the verse? Right? The rule that we do not expound the reason of the verse is said only about laws explicitly written in the Torah. There I do not expound the reason of the verse. But if I derive a law from exposition, there is no such rule. Therefore, if you bring me a source and two explanations, that is not three positions, it is two positions. Okay? If there were a source written explicitly in the Torah and you brought two explanations, then there would be room to say maybe no — it’s a scriptural decree, plus two other positions that offer an explanation. Okay, so that’s just a methodological note.

For our purposes: until now I assumed that an a fortiori argument of “two hundred automatically includes one hundred” cannot be refuted. But that’s not true. Even such an a fortiori argument can be refuted. One of the refutations you’ve already encountered in what I said until now. Look at the reasoning of the Maharsha and the Kesef Mishneh. What do they say? Punishments are not derived by logical inference because maybe the lighter punishment is not sufficient to punish the more severe case, right? Now you say that even regarding an a fortiori argument of “two hundred automatically includes one hundred.” So what you’re actually telling me is this: if a man has five children, and he passes two of his sons to Molekh, then he is liable to death, right? That is what is written: “from his children” to Molekh. Fine. Now I ask: what happens if he passed all five? Then he is exempt. Why is he exempt? In an a fortiori argument of “two hundred automatically includes one hundred”? Because maybe the punishment for passing all one’s children is so severe that the punishment given for passing some of one’s children is not enough for it. Right?

Do you understand that this itself is a refutation of the a fortiori argument? This second explanation is itself a refutation of the a fortiori argument, right? What am I saying? I learn by an a fortiori argument: if passing some of your children incurs the death penalty, then all the more so passing all of your children incurs the death penalty. Not true — I have a refutation. If I pass all my children, maybe that is so severe that the punishment for the lighter case is not enough for the more severe case, and therefore one cannot learn from the lighter case to punish the more severe one. But do you understand that all I’ve done here is refute the a fortiori argument? How can that be, if it’s an a fortiori argument of “two hundred automatically includes one hundred”? It sounds like an oxymoron. The whole point of an a fortiori argument of “two hundred automatically includes one hundred” is that there should be no refutation — it’s a necessary argument. If he passed five children, then in particular he passed two, no? No, that’s not the point. The point is that the punishment is not severe enough.

[Speaker B] That reminds me of the Sanhedrin case where if all the judges convict a person of death, then he is exempt from death.

[Rabbi Michael Abraham] Right, right. Same thing. There too one could ask about an a fortiori argument of “two hundred automatically includes one hundred.” And there too there’s some logical explanation: if all of them convicted him of death, then apparently the deliberation was biased. Twenty-three judges never all agree that someone deserves death. That almost never happens. So if it happened, we suspect the deliberation was biased, maybe one person was too charismatic and everyone followed him. Okay? Then we don’t allow it. That’s a reasonable explanation. But that reasonable explanation itself is a refutation of the a fortiori argument of “two hundred automatically includes one hundred.” Because if twenty judges are enough to put him to death, then how can it be that twenty-three are not? And inside twenty-three there are also twenty.

First of all there’s this paradox that exists in capital law. Suppose I’m one of the twenty-three judges in a capital case. Fine? Now twenty-two convicted him and sentenced him to death. Now look — if I think he deserves death, then I’ll say he deserves death, and what will happen? He’ll go free, because everyone convicted him. But if I think he is exempt and I say he is exempt, then they’ll actually kill him, because now there are twenty-two and not twenty-three. So they kill him. So it comes out that in order to realize what I actually think, I have to lie. If I think he is exempt, I need to say he is liable, and if I think he is liable I need to say he is exempt.

[Speaker C] And who says I don’t know that?

[Rabbi Michael Abraham] Then they add two more. That was the whole —

[Speaker B] up to here.

[Rabbi Michael Abraham] No, I think even in such a case you still have to say the truth. Want to say the truth? Say the truth. Don’t try to be smarter than the Torah. If you think he is liable, say what you think. And then the Torah says: wait, if twenty-three think he is liable, then something in the deliberation is biased. If you lie, you are going against the law. Because the whole idea is that if you truly think he deserves death, then there really is a flaw in the judgment. Now if you lie and conceal that, then it comes out that we’ll kill someone even though there was a defect in the judgment. So obviously the judge has to say what he really thinks, not lie. Okay? Good, but that was just a side remark in parentheses.

For our purposes, what am I really trying to say? That from all these examples we see that somehow, surprisingly, there are refutations even against an a fortiori argument of “two hundred automatically includes one hundred.” And let me give you an example of where I first saw this. There is a legal philosopher named Chaim Perelman, a Belgian Jew, from Antwerp I think, or Brussels, I don’t remember, from a Belgian university, who wrote all sorts of books on rhetoric and legal logic. In one of his books he brings a case of a law in some Belgian town called Vandervelde — I don’t know if I’m pronouncing it correctly, but that’s how it’s written. Okay? The law there was that you are forbidden to sell two liters of wine to a laborer. To me? To a laborer! A laborer goes to a pub and asks for two liters of wine — you’re not allowed to sell it to him. What’s the rationale? The rationale is that he shouldn’t spend his whole weekly salary on wine; he’s supposed to bring it home, make a living. Okay? That was the rationale.

Now a laborer came to a tavern and asked to buy ten liters of wine. So the seller says to him: listen, I can’t, the law forbids it. No — the law forbids selling two liters. I’m asking for ten. So he says: what do you mean? “Two hundred automatically includes one hundred.” If I sell you ten, then in particular I’ve sold you two, I’ve committed five violations, not one. Five violations takes me out of the game, right? Every two liters is one violation, so I’ve really committed five violations. So obviously if the law forbids selling two liters, it also forbids selling ten liters. So they went to court, and the court held that the consumer was right, that the buyer was right — he should be sold the ten liters.

[Speaker H] Maybe the law says only two, and not from two and upward?

[Rabbi Michael Abraham] What, I didn’t understand?

[Speaker H] Maybe it doesn’t say in the law “starting from two,” so he can’t extend it to ten?

[Rabbi Michael Abraham] It doesn’t say it in the law, but I’m learning it by an a fortiori argument. If it were written in the law “from this amount upward,” I wouldn’t need an a fortiori argument. The law itself would say it’s forbidden to sell ten. But if the law says it’s forbidden to sell two — not “from” — then I make an a fortiori argument and forbid ten too.

[Speaker H] That I understood. But the court rejected it.

[Rabbi Michael Abraham] Why? One second. The court rejected it because maybe —

[Speaker C] Because someone buying ten has enough money, and maybe he’s using the rest for other things?

[Rabbi Michael Abraham] Maybe he spent all his money on the ten — I’m not sure that’s a great argument. I think the point is — I don’t remember exactly what he said, but I think this was more or less the spirit of it — he basically says this: if a person wants to invest his money in wine, not to drink it all — ten liters you don’t drink immediately. You want to invest in wine. That’s allowed, isn’t it? There’s freedom of occupation. I want to be a wine merchant, okay? I want to invest in wine. Do I have to remain a laborer all my life? Or because I’m a laborer am I forbidden also to invest in wine and start selling wine? I’m allowed, right? So the law does not forbid me to be a wine merchant. And if I buy wine in that quantity, it may be that I’m buying it for commerce, not for drinking. That the law does not forbid.

Now do you understand what this ruling actually says? That there was a refutation here. Against an a fortiori argument of “two hundred automatically includes one hundred.” Because it is an a fortiori argument of “two hundred automatically includes one hundred” — if you buy ten liters, then in particular you bought two, right? So apparently there should be no refutation here; this is logic. You can’t refute such a thing. And yet you see that you can. We also saw earlier examples that you can: “from his children” and not all his children; the severity of the punishment and all sorts of things like that; or conspiring witnesses, where if it was carried out — if someone was killed on the basis of their testimony — then maybe somehow it becomes clear that they didn’t lie, because otherwise the Holy One, blessed be He, would not have let it happen. That itself is a refutation of the a fortiori argument. Any further argument anyone gives you for why you can’t make the a fortiori move here is in fact a refutation of the a fortiori argument of “two hundred automatically includes one hundred.”

There is really a general phenomenon here that tells us that even an a fortiori argument of “two hundred automatically includes one hundred” can be refuted. But logic —

[Speaker C] You can’t say just anything about logic.

[Rabbi Michael Abraham] On the contrary, supposedly you can’t. Logic is exactly the sort of thing you can’t say just anything about. Supposedly it’s ordered, it’s the clearest and most certain thing there is. The question is whether you’re doing the logic correctly. But here, seemingly, you are doing the logic correctly. Two is included in ten. You don’t need years of study to know that. Okay, we’ll get to that in a second.

There’s another example here. Someone once brought me the fact that Justice Neal Hendel, who sat on the Supreme Court, cited an article of mine in a ruling. And my article dealt with exactly this issue of the a fortiori argument of “two hundred automatically includes one hundred.” That’s the article on which I’m basing this lecture. So he writes there as follows: there was some judge named Poznanski who corresponded with some lawyer from the Israel Land Authority, or something like that — I don’t remember exactly what — who was appearing before her in court. Meaning, he was a party in a case being heard before her, and she corresponded with him by text messages or something like that, which is against the rules of ethics. The commissioner for judicial complaints dealt with it, and they brought her before a disciplinary court for judges. Fine? Now the disciplinary court suspended her for a fixed period. Fine? And the question was whether the disciplinary court has the authority to suspend her for a fixed period. Now we know it has the authority to remove her from office. It has the authority to remove her, meaning to suspend her —

[Speaker E] completely, forever.

[Rabbi Michael Abraham] Now they suspended her only for two years, say — or whatever, two years, three years, something like that. Are they allowed to do that?

Now at first glance this is an a fortiori argument of “two hundred automatically includes one hundred.” If they can suspend her for ten years, they can also suspend her for three years, right? What difference does it make? It’s just “two hundred automatically includes one hundred.” Look what he writes, Neal Hendel: “The main argument on which the parties based their position, presented in the form of the rule ‘two hundred automatically includes one hundred,’ namely that the whole contains its part, is based on the possibility of permanently removing a judge from office. According to their view, this possibility also includes the possibility of removing a judge from office for a fixed period — suspension. The flaw underlying this position is the assumption that these are two punishments identical in nature, differing only quantitatively — the length of the period of removal from office.” Meaning, the claim is that these are both suspensions: one longer and one shorter, but of the same type, and the difference is only quantitative. He says that’s not true.

“The rule ‘two hundred automatically includes one hundred’ refers to coins from the Talmudic period, where one maneh was worth one hundred zuz. The relevant discussions in the Babylonian Talmud and in Maimonides concern a claim for repayment of a debt, where one witness testifies to a loan of one maneh, that is one hundred zuz, and the second witness testifies to a loan of two hundred. In such a case it was ruled as follows: the borrower must repay the lender one hundred zuz. Why? Because both witnesses testify to a loan of one hundred zuz, since within two hundred zuz there is also one hundred zuz. Talmud, Sanhedrin. This is an evidentiary rule, not a punitive rule. And in any case it cannot teach what counsel for the parties sought to derive from it.” And here he cites that article of mine.

“A related argument by counsel seeks to transfer the evidentiary principle into the punitive sphere. According to this line, the more severe punishment contains within it the lighter punishment. But as will be shown, that is not the case where two different punishments are involved.” And then he starts discussing different punishments, maximum punishment, and he brings various proofs from the wording of the law and so on.

Now I don’t entirely understand his claim. He uses my article, but I don’t think that’s what it says there. You know, like Agnon said when people asked him all sorts of questions, he said: ask Kurzweil. Meaning, the person authorized to interpret my stories is Kurzweil, not me. Once I wrote it, that’s that. So here too — but I don’t think, at least I don’t interpret what I wrote the way Neal Hendel did. Because I don’t think it matters whether it’s evidentiary or punitive. The logic is that inside two hundred there is one hundred. What difference does it make whether it’s evidentiary or punitive? If the logic is right, it’s right everywhere.

The point is something else. The point is that suspension for a fixed term is a punishment. Permanent removal from office is not a punishment. It’s simply saying she cannot be a judge. That’s a determination of fitness. Meaning, someone who behaves that way cannot serve as a judge. Consequences? So what? I remove her from office not as a punishment. I remove her because it’s like the judicial appointments committee: I’ve reached the conclusion that she’s simply unfit for the role, so I remove her. I am not punishing her; I don’t have authority to punish her. I have authority to determine whether she is fit to serve in the role. Therefore the disciplinary court can determine that she is unfit to serve. Now when they come to suspend her for two years — suspension — if you’re saying she’s unfit for the role, then what happens after two years? She suddenly becomes fit? What happens after two years? Obviously those two years are not a determination of unfitness; they are punitive suspension. But that court has no authority to impose punishments. It only has authority to determine whether the judge is fit for her office or not. Therefore you can’t make an a fortiori argument of “two hundred automatically includes one hundred” here, because this is simply not two hundred and one hundred; it’s one hundred oranges versus two shekels. Two shekels are not contained in one hundred oranges. Two oranges are part of one hundred oranges, but two shekels are not less than one hundred oranges. There is nothing to compare; these are different things. Therefore a punishment of two years cannot be compared to a determination that the judge is unfit for office, even though that determination in practice brings about her suspension for ten years, or however many years she had left until retirement.

[Speaker C] If she’s unfit, they should cancel her pension and so on.

[Rabbi Michael Abraham] There are consequences, obviously. But the determination is not a punitive determination. It’s a determination of whether she is fit or unfit. So if you, as a court, have no authority to punish her, then where did you get the authority to suspend her? That is the problem. Not because this moves from the evidentiary domain to the punitive domain. That’s not the point. The point is that it isn’t even in the punitive domain at all. Suspension is punitive, but removal from office is not punitive. It is simply a determination that she is unfit for the role.

So again, for our purposes this is another example of an a fortiori argument of “two hundred automatically includes one hundred” which nevertheless can be refuted. “Two hundred automatically includes one hundred.” What does that actually mean? It means that one has to be very careful with logical arguments — apropos what you said earlier. But why? Not because there is a problem with logic. If you do the logic properly, then there are no games; it’s necessary. The problem is always in the process that logicians call formalization. What is formalization? In English, formalization. Form is shape — formalization and making-formal are translations. What is formalization? I take an argument in everyday language, whatever it may be, and transfer it into a logical pattern. I translate it into a logical pattern. That is called formalization. I turn it into a formal argument. And now I analyze it on the logical plane. And on the logical plane, it is a necessary argument. Okay?

Now since the logical plane merely represents the argument from everyday language, the logical proof is naturally taken by us as proof also on the everyday, non-logical plane. But that’s a mistake. Why is it a mistake? Because the very process of formalization may be where mistakes crept in. Suppose I say “inside two hundred there is one hundred,” yes? I say the a fortiori thing: it is forbidden to sell two liters of wine. Now the question is whether it is also forbidden to sell ten. What is the formalization? Whether two is smaller than ten or not. Or whether two is included in ten or not. Obviously yes. Which was to be proved, right? I formalized the problem as an arithmetic problem: which is larger, or which is contained in which. And in arithmetic, obviously, ten is larger than two, or includes two, which was to be proved. The assumption, of course, is that there is a match between the legal everyday problem, or whatever it is, and its formal logical formulation. Then everything I do on the logical plane is supposed to be true also on the everyday plane, because the formalization assumes that it is the same thing, just translated into another language. But the process of translation is a very problematic process. It is not at all clear that in translation you didn’t miss something.

Yes, this reminds me of the dispute between Maimonides and the Raavad. In the laws of reciting the Shema, Maimonides says that one may recite the Shema in any language. The Mishnah at the beginning of chapter 7 of Sotah says: these are things said in any language. So the Shema may be recited in any language. You can say, “Listen Israel,” yes, “Listen Israel,” something like that, and you’ve fulfilled your obligation. Okay? Now Maimonides says: provided that he articulates its words and letters carefully. One may read in any language, provided he is careful with its words and letters. So the Raavad says: but all languages are interpretations — and who is there that can be exacting about an interpretation? What does it mean to be precise with the letters when the word is your own word, in English? You’re translating the verse into English, and that translation is a kind of interpretation. What does it mean to be exacting about the letters of the interpretation? What sanctity is there in this wording rather than another? One could have translated it differently and that would also have been fine. So what sanctity is there in these letters specifically and not others? As long as what you say is substantively equivalent to the original text, all is fine. What does it mean to be exacting about the letters of an interpretation?

[Speaker C] It’s not relevant.

[Rabbi Michael Abraham] Mathematical language — people think that solving a problem in physics is just a mathematical problem. And that’s a mistake. The real physics enters in the translation between the physical situation and the mathematical language. That’s the physicist’s work. From that point on it’s the mathematician’s work. The physicist does it, but really it’s mathematical work. Okay? The substantive problem is how you translate, how you model, how you formalize a problem in physics and turn it into a mathematical problem. That’s the physicist’s work, not solving the mathematical problem. Okay?

I’ll maybe give an example of that in a moment. But the point is that the process of formalization always contains the real difficulty. Let me maybe indeed give an example from physics. I once taught mechanics in physics while I was doing my master’s degree. So I taught mechanics. I opened the tutorial and gave them an introduction: what is a scientific theory? According to Popper, a scientific theory is a theory that can be subjected to a falsification test. You can propose an experiment by which the theory might be refuted. Fine? If the theory can be subjected to a refuting experiment, then it is a scientific theory. That’s Popper’s definition. Okay?

Now I ask: is the proposition two plus two equals four a scientific proposition?

[Speaker C] Why not?

[Rabbi Michael Abraham] Is there no experiment that could refute it? Look, I’ll propose an experiment. Take a basket and put two oranges into it. Now take two more oranges and put them into the basket. Count how many there are altogether. If you get three, then you’ve refuted the proposition that two plus two equals four. Now notice: we know this won’t happen. Fine. But in principle it is falsifiable. What’s the problem? With gravity too I know it won’t happen. If I let go of this, it will fall down, right? I’m sure. But still the theory of gravity is a scientific theory. Why? Because in principle there is an experiment such that if it fails, the theory is refuted. That’s what makes it a scientific theory.

Now here too, apparently, the theory that two plus two equals four is a scientific theory. And if it comes out three and a half kilos or five kilos, then you’ve refuted the proposition that two plus two equals four. So it is a scientific theory. It can be subjected to a falsification test.

I’m telling you: not true. It is not a scientific theory. I’ll tell you why. Suppose you put the oranges in the basket, two, and then added two more and it came out three. What would you conclude? That two plus two really does not equal four? Never in your life. You would conclude that there was a mistake in the experiment. We didn’t count right, one orange disappeared, I don’t know. There was some mistake in the experiment. And even if we found no mistake at all, we would never give up the proposition that two plus two equals four. Meaning, de facto, it is not really falsifiable. We are not really going to give up the proposition that two plus two equals four because of any experiment in the world.

Why not? No, that would be psychology. I want to offer a philosophical justification, not a psychological one. I want to argue that in fact we would also be right not to give it up. Why not? Because the theory that two plus two equals four — and here I come to what you said earlier: how do I know that two plus two equals four? People usually think it’s from experience, that I simply see it. That’s not true. I know it because I understand the concepts of two and plus and four, and I understand that this is the relation among them. Sometimes there are didactic aids that can illustrate it for me through various examples, but those are only didactic aids. At the fundamental level it arouses an understanding that is already in me a priori from birth. It is not the result of experiment. I know it even without the experiment. It may be that examples help me conceptualize it and formulate it, and often they do help, but they only help. It’s not that I really — unlike the law of gravity, for example, where I learn it from experience. It’s not that experience only helps illustrate what I already knew; I didn’t know it. I would not know it without observation. Right? That I learn from experience. For Kant, that’s called an a posteriori proposition. But mathematical propositions are a priori propositions.

Now why am I saying this? Because if that’s the case, then no experiment is supposed to refute it. I know it independently of the experiment; it’s not a result of the experiment. It’s an a priori truth. It has nothing to do with measurements and observations. So if an experiment happens in which I did two plus two equals four and got three — say I found no error at all, nothing — my conclusion would not be that two plus two does not equal four. My conclusion would be that the move from adding oranges to a basket to formalizing that as the arithmetic exercise two plus two equals four — that formalization is incorrect. And arithmetic is not the correct mathematical theory for describing the addition of oranges into a basket. That would be the conclusion.

So notice: what did I refute in the experiment? Not the mathematical proposition that two plus two equals four, but the mathematical formalization of the physical claim — that adding oranges to a basket is adequately described by arithmetic addition. And that is a claim in physics, not mathematics.

Think, for example, about when Einstein found that space is not flat, that the sum of the angles in a triangle is not really 180 degrees in our world. It seems that way to us because it’s close, but it’s not really 180 degrees. Did he refute Euclidean geometry? Of course not. Euclidean geometry remains intact. What he said was that Euclidean geometry is not the correct theory — our world is not, in the language of logic, a model of Euclidean geometry. That’s all. Meaning, he refuted the physical claim, the claim about the world, not the mathematical claim. He refuted the claim that our world is described by Euclidean geometry. Not true — our world is described by a different geometry. But you can never refute Euclidean geometry, because Euclidean geometry is not the result of observation, like every mathematical theory. Okay? Therefore mathematics is not science, because it is not subject to a falsification test. Science is something subject to a falsification test. Mathematics is not subject to such a test. If I carry out a falsification test and refute some mathematical law, that only means that what I did in the experiment is apparently not described by that mathematical law. It does not mean that the mathematical law is false.

Why did I say this at the beginning of a mechanics course? Because I told them: look, I’ll refute for you the proposition that five plus five equals ten. Okay? Take this Bible. I apply to it a force of ten newtons northward and another force of ten newtons eastward. Now I ask: what is the total force acting on the Bible? Apparently five plus five equals ten. But you know, right? It’s not ten, it’s five root two. Because one force points here and one points there, so the resultant is a diagonal force whose magnitude is seven and something. Okay? So here we have refuted the mathematical law that five plus five equals ten. No. We refuted the physical law that says the addition of forces is described by arithmetic addition. Not true. Arithmetic is not the right mathematical theory for describing the addition of forces. What is the right theory? Vector addition, right? Vector theory. Okay?

So that is a good example of the fact that whenever we refute something — when you are talking about the world, you are never really a mathematician, you are always a physicist. Mathematics does not deal with the world. When you deal with mathematics, you deal with abstract, Platonic ideas, however you want to put it; you are not dealing with the world. When you talk about the world, you have always made one additional move beyond mathematics, and that move is formalization. You are really saying: the piece of the world I am dealing with is described by this mathematical model. And now you can talk about it in mathematical language. But that connection between the world and the mathematics is what is called formalization. That is where the problems may sit.

Let me give you another example. This is, I think, a very important lesson. There is a theorem in topology, which is a branch of mathematics, that distinguishes between convex and concave shapes. Fine. In a loose, everyday definition — not the mathematicians’ definition — convex shapes are shapes that bulge outward, and concave shapes are shapes that curve inward, like a bowl. Okay? A ball is a convex shape; a bowl is a concave shape. Or a crescent moon, like a banana — yes, that’s a concave shape. Now of course if you think about a banana, this side of it is convex; only this side is concave. And a convex shape is something convex from every direction. A concave shape is a shape that is not convex, one that has at least some part that is not convex. It need not all be concave; “entirely concave” is hard even to picture.

In any case, there is a theorem in topology which says that the intersection of two convex shapes — or really of any set of convex shapes — is a convex shape. Say a triangle and a circle; their intersection is a convex shape. Both the triangle and the circle are convex shapes, and for our purposes a straight line also counts as convex. Okay? So the intersection of two convex shapes is also convex; and likewise three convex shapes, ten convex shapes, one hundred convex shapes — it doesn’t matter. Take their intersection, the result will be a convex shape.

Now if I asked you to prove this, say for two convex shapes, let’s try to prove it — would you have an idea how to prove it? You would try examples, right? You’d experiment and check. But you can never know that you thought of all the possibilities, right? Examples only show the cases you happened to think of. That’s not enough for a mathematician. Mathematicians want a proof, something certain, not like physicists who try a few examples and generalize. So look — it’s very hard to prove this unless you start from the definition, like mathematicians do.

Let’s define a convex shape. A convex shape is — now this is a mathematical definition, not “a shape with a belly.” The mathematical definition is: a shape such that for every two points inside it, if you connect them with a straight line, the whole line lies inside the shape. Right? Think of a bowl: pick two points, and the line connecting them lies partly outside the shape, right? So the shape is concave. A convex shape like a ball: every two points you choose, connect them with a line, and the whole line lies inside the shape. That is the definition. Fine?

Once that is the definition, the proof is one line, one line. Take two convex shapes. Given that they are convex, and an intersection is formed between them. I want to prove that the intersection is convex. Very simple: take two points in the intersection and draw a line between them. What do I need to prove? That the whole line lies inside the intersection, right? That’s the goal. Now if those two points lie in shape A, and A is convex — that’s given — then the line connecting them lies in A. And those same two points also lie in B, because they lie in the intersection, and B is convex, so the line lies in B too. But if the line lies in A and in B, then it lies in the intersection. Which was to be proved. Abracadabra — so easy. I struggled with that for two days until I managed to read it in a book; I didn’t come up with it on my own. And in a high-school-level book too — I once found it in a used bookstore at Harvard in the United States. There was some cute little booklet there. So I tried to prove it, couldn’t. And then he gives the definition and boom — two lines and you’re done.

Now I’ll ask you a more interesting question, one not asked in that book because it’s a math book. I’m asking a question in physics or philosophy: did we really prove what we wanted to prove? The big question is whether we formalized correctly the concept of a convex shape. We have a mathematical definition and we have an everyday definition involving the “belly,” right? Who says the mathematical definition fully captures the everyday definition? It sounds very plausible to us, but there’s no proof of it, right? Mathematics doesn’t even deal with that; it starts from the definition and doesn’t ask whether the definition is accurate. That is not a legitimate question in mathematics. Okay? That’s the definition, and from there on we work.

Now do you understand that if I now ask you about real life, not about mathematics — whether the intersection of two convex shapes is convex in real life — you have no way to answer me. You don’t know. Only if you assume that the formalization is correct — meaning, that the mathematical definition really is equivalent to the everyday definition — then you can prove it at the mathematical level. Okay? But you are assuming here that the mathematical definition fully captured our everyday definition, right? Who says that assumption is true? You didn’t prove that assumption.

In other words, what formalization did for us is basically remove all the problems we had in the issue and leave us with the pure problem, which is easy to handle. That’s mathematics, it’s simple. But it didn’t solve the difficulties; it only hid them under the rug. It hid them under the definition. Inside the definition, all sorts of difficulties are hidden. But from the definition onward there is no problem at all; everything is perfectly tidy.

Do you see that this is exactly the problem we are discussing here? This is the problem of an a fortiori argument of “two hundred automatically includes one hundred.” Why does it feel as though there can be no refutation there? Because at the formalized level, two is part of ten. You can’t argue with that. Or opening is part of digging. Obviously. It’s a simple definition. Clear, right? But when you apply this to the world — or here, to the world of law — you are assuming something more. You are assuming that the formalization correctly describes the meaning of the things in the world, and that you didn’t miss anything in the move from the world to the logical form of the problem. But in formalization there can always be problems.

And all the refutations — notice — all the refutations in every example I brought until now, what are they really showing? The mathematics itself cannot be refuted. What they show is that the formalization was incorrect. Those are refutations saying: you formalized it incorrectly. Once you formalized it, of course two is included in five, and one in two, and everything is simple and undeniable. But your formalization was wrong. The translation of the problem into mathematical language was the thing that was wrong.

Take punishment. The suspension for two years versus removal from office. When we formalized it, we said: removal from office is twenty years, say she has twenty years left to retirement. Okay? Twenty years, and suspension is two years. So — two is included in twenty years, “two hundred automatically includes one hundred.” The answer is: not at all. You formalized it as though this is two in some units and that is twenty in the same units, and you assumed they were the same units. But they’re not. These are units of punishment — the punishment of two years’ suspension. And those are not units of punishment at all; they’re a determination. The twenty years are not relevant here. They are determining that she is unfit for office, that’s all. The twenty years are only an incidental consequence. So what is there to compare here? Where was the problem? No one disputes that two is included in twenty. The problem was in the formalization — how you moved from the legal problem to its logical-mathematical description. Once you made the move, everything is clear: two is inside twenty, all fine. The move itself is what is problematic.

Let’s go back to the rule that punishments are not derived by logical inference. I say: if for opening one is liable — or “from his children” to Molekh and not all his children to Molekh — what model did I make? Fine, this is two and this is five, and two is part of five. Obvious, right? “Two hundred automatically includes one hundred.” But no — why should that be? Because passing some of his children to Molekh is a lighter offense. The five tells you it is more severe, but that does not mean the punishment it receives is simply more severe. With respect to punishment, it is not correct to describe this as five versus two. Because regarding punishment, this may not be a quantitative difference at all. The punishment for the five may be a different punishment entirely. If you give the punishment of the two to the five, you may be completely missing the point; it may be a difference in kind. The punishment given for the two is of one type, and the punishment given there is of another type altogether. Flogging and death: flogging is not light death. It is another punishment, of another type. You cannot say that flogging is included in death. Your formalization is incorrect.

Meaning: all the refutations we saw in all the examples against an a fortiori argument of “two hundred automatically includes one hundred” are refutations that attack the formalization. They are refutations that say: you formalized it incorrectly. Once you formalized it, then obviously two is included in five, and one is included in two, and everything is simple and undeniable. Your formalization was incorrect. The art of the physicist is the art of models. Meaning, you take a problem from life and create a mathematical model for it that will describe it, and within that model you can try to solve it.

Yes, one more example, though I’m already out of time to continue. There was once someone who came to me doing a doctorate in mathematics at the Technion under Hershkowitz — the one who later became a minister — he was a professor and supervised him. And he was looking for mathematical problems in Jewish law, in the Talmud, and he wanted to do his dissertation on that. So he asked me whether I had any such problems. I gave him a few. One of them was a Mishnah in tractate Mikvaot. The context is interesting, so not by accident I’m telling you the context — you’ll soon see why.

The Mishnah in Mikvaot talks about a water channel — yes, waters valid for a ritual bath, not drawn water — passing through a pit. In the pit there is drawn water, forty se’ah of drawn water. Drawn water invalidates a ritual bath. Fine? Now the waters enter through the pit and exit; the water channel passes through the pit and comes out, at a given rate. Now the incoming water is constantly mixing in the pit, and what exits contains both drawn and non-drawn water. Slowly, in the end, there will remain in the pit only a small amount of drawn water, less than three logs, and three logs is the threshold that invalidates. Less than three logs of drawn water does not invalidate. The question is: when may I assume there are already fewer than three logs of drawn water in the mikveh? After how much time? I’m given the flow rate of the channel and the size of the pit, and that’s it. Now they ask: after how much time can one immerse in that pit, such that one can assume there are fewer than three logs of drawn water there?

Now the medieval authorities make all sorts of calculations, this way and that way, and each builds his own model. And again, these are wonderful, fascinating processes of formalization. Because each one builds a different model and they reach very different answers from each other. Very, very different. The Mishnah only says that when there are fewer than three logs you may immerse; it doesn’t say when that happens or how. That’s the commentators there. Okay?

Now I proposed to him a model which of course the medieval authorities could not use, because it’s a model from calculus. What is it? A differential equation. I set up a differential equation. There is a water flow rate. The assumption is that the water mixes uniformly in the pit — an assumption, not true, but an approximation. Say if the stream flows slowly, the slower it flows, the more plausible it is. Okay? It mixes uniformly, and all the time the exiting water comes out in the same proportion as the drawn water is distributed in the pit. That’s the proportion of drawn water in what exits. Fine? That’s the model. One can write a differential equation, solve it, and the result comes out differing by thousands of percent from the medieval authorities in many, many examples. Of course it depends on the flow rate and lots of other things, but it’s simply not close. It bears no resemblance at all to the answers of the medieval authorities.

Now what I did is also not an exact solution; it’s a model. Because that model assumes uniform mixing of the water throughout the pit. Right? That as soon as water entered here, at the other side of the pit it is already felt that more water entered, and therefore a lower percentage of drawn water will come out. Okay? But that’s not realistic. It takes time for the water to mix. Fine? But it’s a model, a simple model, what’s called an adiabatic model, a model of slow processes. Okay? That too is a model.

Now none of these answers is truly the correct answer. But in order to solve the problem I have to model it mathematically, and from there on I have an equation and I solve the equation. The medieval authorities also did mathematical modeling, just not with a differential equation but with arithmetic of addition, subtraction, and multiplication. Okay? That doesn’t matter, but it is still a mathematical model.

Now the model does not really tell me what happens in reality. It is a model I choose in order to describe what happens in reality. But a model has assumptions — for example, my assumption of uniform mixing. That assumption I could test and see whether it is true. Say I dyed the drawn water. You could run an experiment: dye the drawn water, and let’s see after how much time only three logs remain. One can measure it. Okay? And then what? Suppose I would discover that I was wrong. What would that mean? That I didn’t solve the differential equation correctly? No. It would mean that the differential equation does not correctly describe the problem, because I built an assumption into the model. I assumed uniform mixing, which is really an assumption in the formalization of the problem — in the move that turns the problem into a mathematical one.

Okay? Therefore refutations always attack the thing that led me to the a fortiori argument. If B is more severe than A, then obviously everything in A is also in B. The question is whether B really is more severe than A. That is the question. It is not obvious that you are right that B is more severe than A. The refutations, for example, show that no — you assumed that B is more severe than A, but I have a refutation showing that not necessarily. There are aspects in which A is more severe than B. And it is the assumption behind the a fortiori argument, the assumption that built this model according to which B is more severe than A — that is what I am attacking. The assumption of the formalization. Okay?

Good. Next time I don’t know whether I’ll get to a prohibition linked to a positive commandment. Maybe I’ll still get to two more aspects of this, and then we’ll finish. What?

[Speaker E] A class at this hour? Very nice. Someone signs in at the end. You left early? Yes, but you’re registered, marked present. Great, excellent. Okay —

[Speaker C] We’ll begin. Good evening, everyone. We can see this great process taking place in the world today. The revelation of heavenly glory, the sanctification of God’s name spreading everywhere. A person sometimes asks himself: where am I in all this? What is my role? And the answer is that each and every person —

[Rabbi Michael Abraham] is a part —

[Speaker C] of this great mosaic. Every good deed, every Torah study session, every strengthening of faith — that is one more —

[Speaker D] layer in the building of Israel’s redemption.

[Speaker C] We see that people are thirsty for the word of God.

[Speaker D] Even at hours like these, even in situations like these.

[Speaker C] And that itself is the greatest sanctification of God’s name. That Jews do not give up their bond with the Holy One, blessed be He. That is our strength, and that is what will bring us salvation. We can see it tangibly, how hearts are opening. How suddenly things that once seemed distant become close and relevant. This is not just a historical process; it is a divine process. And everyone who takes part in a class like this, in this kind of study, is really taking part in repairing the world. The Holy One, blessed be He, looks upon us from above and rejoices in His children who engage in His Torah.

[Speaker D] Let us try today to go deeper into this point of individual providence within concealment.

[Rabbi Michael Abraham] May we always merit —

[Speaker C] to be —

[Speaker D] among those who sanctify His name in the world with joy and a glad heart.