A Further Look at 'the Common Denominator' (Column 346)

This came out as a file. Was that intentional? Because it’s a bit less convenient to read.

Because there are diagrams.

Regarding the sugya in Ketubot, it may be that the difference lies between a “unique aspect” and a “stringent aspect.” A unique aspect is not a refutation, because it is not a genuine shared property.
The obligation of gleanings or challah are unique characteristics (but they are not a stringent aspect); by contrast, unusual punishment (in conspiring witnesses or one who injures, even if the unusual nature of the punishment is manifested in different ways) is a genuine shared characteristic indicating that this is a higher-grade offense, and one cannot derive from it to lighter offenses. (And Rabbi Yoḥanan ben Zakkai in Sotah and Rabbi Yehuda in Makkot, who hold that one does not refute on the basis of a stringent aspect, may maintain that the severity of the punishment does not reflect the severity of the act, but is like any other unique characteristic.)

1. I liked your answer that a “stringent aspect” is invoked only with halakhic stringencies. I think this can also be grounded in the fact that something that is not a halakhic stringency but rather a distinction in severity based on reasoning does not really get treated as something more severe in the same way a halakhic stringency does.
You can see this in the matter of a fortiori reasoning: in a situation where there is a stringency based on reasoning that is raised as a refutation to a kal va-ḥomer, one can say, “I will incorporate all this into the kal va-ḥomer.” Moreover, sometimes the rejections of a derivation stem from considerations that are better defined as a factual difference rather than as something more severe in the reality of the source that teaches about the target. (Any minimal refutation is enough in a binyan av.)
2. But I did not understand what fault you found with the Ramban’s explanation such that you called it forced. I understand why Tosafot is forced, since they needed a new explanation for every sugya. According to the Ramban, if this is a common denominator that is really a response to a refutation of a kal va-ḥomer (namely, that there is a stringency in the target that is not in the sources, as he defines it), it is very logical that one cannot say “stringent aspect,” because, as I said before, you can incorporate the stringent aspect into the kal va-ḥomer. But if we are dealing with a binyan av, any refutation whatsoever is sufficient—even a mere distinction that is not clearly a stringency—and therefore even a theory that there is some other stringent aspect in each case that causes the law is enough to reject the derivation. After all, the burden of proof is always on the one “inventing” the derivation, and even if he advances a preferable theory, his words are still not proven, since the simpler explanation is not always the correct one.
And in the sugya in Pesaḥim 77a they make a “stringent aspect” out of the fact that it is always frequent and entirely consumed. True, one could say that this is a halakhic definition, but even if so, it stands to reason that the Gemara is not arguing that because it is frequent, therefore it has some superiority that may be indirectly related to the superiority of the Paschal offering, for which one incurs karet; rather, this too is the factual stringency in it that causes us to say that it overrides Shabbat.
In other words: even if you say that “entirely consumed” and “frequent” are halakhot, it is not likely that they are a sign of stringency; rather, they represent the stringency itself. Therefore, in my humble opinion, the Ramban’s explanation fits the sugyot of the Talmud better (although the idea of your answer is really beautiful, and thank you very much for this article!).

And I will add, in explaining the Ramban’s view, that when we are dealing with a common denominator based on a kal va-ḥomer, the common denominator is nothing more than a conceptual construction, as you defined it, whose purpose is to prove the irrelevance of the stringencies in the source cases. The source of the stringency in the target case does not stem from its being equal to the source cases, but from its own original severity, and this also explains why the refutation of “stringent aspect” is irrelevant here (aside from the fact that one can incorporate that same shared stringent aspect into the kal va-ḥomer).

The problem is that according to the Ramban we are forced into ad hoc answers. There is no criterion for when the stringency in the target differs from the two stringencies in the source cases. Moreover, when there is a stringency in the target, that does not change the logic, since stringencies in both directions are enough to refute the inference. You do not have to prove that it is incorrect; it is enough to show that there is no necessity that it be correct.
I do not see why a common denominator based on a kal va-ḥomer is a conceptual construction. I really do not think so.

An explanation of the difference between an ordinary common denominator—one really based on deriving from the sources—and a common denominator that is essentially a kal va-ḥomer.
I am trying to argue as follows:
Suppose that
A has the stringency X and does not have Y or Z, and the law is unknown.
B has the stringency Y and not the others, and the law is that it is forbidden.
C has the stringency Z and not the others, and the law is that it is forbidden.
D has the stringency Y and not the others, and the law is that it is permitted.
We want to derive by kal va-ḥomer for A that it is forbidden because of the stringency X.
Of course, this will be rejected by saying that B has Y, and perhaps that is the reason it is forbidden.
Now there are two possibilities: if we bring D, we actually prove that the stringency Y is not what causes the prohibition—for in D it is permitted despite the stringency Y—and then it is clear that the kal va-ḥomer stands even though there is a stringency on both sides.
But even without D, one can support the kal va-ḥomer by means of C: the claim of the refutation is that the stringency Y causes the prohibition. The converse statement is that when there is no stringency Y, it is permitted. Through C we prove that this is not true, because there is a case where Y is absent and it is still forbidden. From this it is proven that Y is not relevant to the prohibition, and one can return to making the kal va-ḥomer. (For we have no proof that X is irrelevant—but if the stringency X is found in one of the source cases, then a kal va-ḥomer can no longer be made.)
If so, the proof from C is only that the characteristic Y is not the cause of the prohibition—seemingly a conceptual construction, not a derivation based on what is shared, which perhaps does not exist at all.
And then the source of the law in A is not B, but the stringency X found in it, and therefore one can even derive that A has a greater stringency than B, of course within the limits of the law of dayyo. (But as is known, sometimes dayyo is said only partially, on one issue and not on another, and it is also not said where the kal va-ḥomer has been refuted.)

You have constructed here something that is not an ordinary common denominator based on a kal va-ḥomer. You added unnecessary elements. An ordinary common denominator based on a kal va-ḥomer is much simpler and is not a conceptual construction.

I came late to the party, and I’m writing on the assumption that in your role as the sole master of this holy mountain you still haven’t packed up the microphone. I was seized by a reading frenzy, and as people say, writing follows reading. The discussion is mighty and delightful, but the conclusions don’t sit right with me without further explanation, and so I have come to ask for it. [If I’m consuming too much attention for one week and have overstepped my bounds, then please strike me too with a marker, and I will accept the judgment with love.]

First, a few typo corrections, so that at least I can provide some benefit.
1. Page 4. It says, “The Gemara in Ketubot tries to derive that one can receive lashes and pay for the same offense.” It should say that one cannot. It is known that one does not receive lashes and pay; the only question is whether he pays and does not receive lashes (Ulla), or receives lashes and does not pay (Rabbi Yoḥanan). This typo recurs at the end of the column in the chapter ‘A difficulty with this explanation’ (the common denominator is that in principle there is liability for money and lashes, and they infer that according to Ulla the liability chosen for implementation is monetary). It may be that I did not understand correctly.
2. There, note 5: “The Gemara says.” Better to say: “The Gemara in Makkot 4 says” (it is mentioned only after the note).
3. Page 6. “And the third source is from damage by a pit.” It should be “an ox.” And so too below and on page 7 (3 times total). Thus Rashi explained in s.v. “and the dead animal shall be his,” and with this he explained that it is not by itself; see also the emendation of the Maharshal. For the Gemara says, “Let the Merciful One write ‘ox under ox’ and be silent.” As to the substance, perhaps this still requires investigation.
4. Page 6. In the second difficulty in the structure of the tzrikhuta it says “to derive from the first and second source”; it should be “to derive from the second and third source.”

The explanation of “stringent aspect.”
By my life, it looks sparkling like crystal, and it suits you well. But when one rubs the pearl, I’m not sure more than mere gravel remains, with all due respect. And on three fronts I shall strike it:

A. There is no logic at all in such a refutation of “stringent aspect” (according to the Sages against Rabbi Yehuda). For clearly “the burden of proof that they both have a shared factual characteristic” lies on the refuter, as you wrote in note 6. If he has an idea for such a factual characteristic, let him kindly present it (with explanatory relevance, of course, and not just any triviality).

B. If we are allowing such speculation, then in every common denominator in the world there is a stringent aspect. Perhaps even in the primary categories of damages there is some hidden shared factual characteristic unique to all four of them, and one cannot derive from there to anything else? Surely you will tell me that only if there is a halakhic stringent aspect is there room to speculate that there is a shared factual characteristic. But if so, please explain why the very law being derived cannot function as the “stringent aspect” that would ground the conjecture that they have a shared characteristic. What about the primary categories of damages, where we know of them that one is liable to pay damages, so we can conjecture that they have a shared factual characteristic, and then who can guarantee to us that it also exists in a rooster that struts and breaks vessels? The same applies to every common denominator in the world. (Please note this carefully. And do not answer me from your discussion at the end of the column about the difficulty in this explanation.)

C. To explain why the Gemara does not always refute with “stringent aspect” (which is no difficulty according to the forced explanation of Tosafot, that “stringent aspect” is unusual), you said in note 6 that the halakha follows Rabbi Yehuda, who does not refute with “stringent aspect.” So sometimes the Gemara feels like arranging things also according to the Sages, and sometimes not? And how does it know that Ulla also works according to the Sages and not only according to the halakha following Rabbi Yehuda, such that it invents a verbal analogy for him? And if so, if the verbal analogy is free on both sides (for otherwise one certainly can respond and it would accomplish nothing), then the Gemara should have continued: “And Ulla, according to Rabbi Yehuda in accordance with the halakha, what does he do with ‘under, under’?”—for he does not need the verbal analogy. (And then perhaps they would answer: the verse took the trouble and made the verbal analogy.) It is hard for me to check right now, since I am without books except for Google, so I will ask your leave, as the master of this discussion: is it clear that the dispute between Rabbi Yehuda and the Rabbis is about every common denominator in the world, or perhaps they disagreed only in the specific case discussed there (whereas in other cases, when relevant, everyone would refute with “stringent aspect”)?

The sugya of “the owners deal with the carcass.”
D. The tzrikhuta in “the owners deal with the carcass” seems to me easy to explain. Factual and logical reasons to be stringent can accumulate. Very simple (and this differs from your explanation). In “and the dead animal shall be his” there is an accumulation of reasons to be stringent with him (both common and not by itself), and therefore perhaps the Torah did not grant him leniency there. Only from all 3 source cases together do we see that every pair of reasons for stringency does not combine to create stringency. (And an accumulation of 3 reasons does not seem to the Gemara sufficiently different from 2 reasons. If that is the problem, I have explanations.) In my opinion this is simple and clear. And therefore it is also understandable that one need not complete the tzrikhuta. On Shabbat I will get to a place of books, fig, and pomegranate, and try to check.

E. You wrote that because these are 3 binyan av derivations, at least one of the characteristics is always needed in order to be lenient. Perhaps I simply do not know famous laws—but what is the law in a regular case of a person who damages, where he broke a table and the pieces of wood are worth money? This damage is common, direct, and by his own body. According to your approach, does depreciation of the wood go to the damager? (I understand that only one who kills his fellow’s ox is uncommon, but one who damages his fellow’s property is common indeed.)

F. Your explanation simply seems strange to me. It is as clear as day that the common denominator among them is that they are liable for damages and the owners deal with the carcass—so too anything liable for damages, the owners deal with the carcass. This is the common denominator of the three primary categories, and the assumption “for the sake of discussion” that they have no common denominator is simply just not true. The question why we derive that “the owners deal with the carcass” from the above trio and do not derive the opposite from a thief and a robber is a separate question that I am convinced the commentators addressed. And apparently the simple answer is that we derive damages from damages and not from something involving removal from its owner’s domain. Is that so, Naomi? I am beginning to suspect that I simply did not understand what you are claiming, and may it be God’s will that that is indeed the case.

It is hard for me, alas, to get back into the details of the party again (especially in the sugya of “the owners deal with the carcass”). I will try briefly and from above.
A. Not at all. Note that the refuter here is in the position of defense, not attack. After all, the common denominator is trying to prove something, and the refuter claims against it that it has no proof.
B. Not correct. When the two source cases have a halakhic stringent aspect, it is clear that at its base there is a factual stringent aspect. The question is whether it is the same factual stringent aspect or not. In an ordinary common denominator there is no basis to suspect that there is a shared factual stringent aspect absent from the target, for we have no halakhic indication of its existence. That is exactly my claim here.
C. This is very characteristic of Talmudic sugyot. Sometimes they make a tzrikhuta and sometimes they do not. Sometimes they bring the dissenting view in the same sugya and sometimes they do not. (That is what happens in every situation where some Rishon discusses contradictions between sugyot.)

“The owners deal with the carcass”:
E. This is the law of best-quality payment, namely that movable property equal in value to money may be paid.
F. That common denominator is not a “side” at all, but the very law being derived. True, one sometimes finds in the Talmud that they treat this as a common denominator (“so too every commandment applies immediately and for future generations”), but on its face that is a different matter. But this is a halakhic common denominator, not a factual one, and therefore it is exposed to the refutation of “stringent aspect.”

A and B I still need to think about, and my silence is not yet admission. As for F, I do not understand, but that is what there is.

E. I did not understand, and this is probably a famous law that I apparently do not know. I’ll write at length so that you can answer briefly. The Gemara asks: after all, the injured party can always be paid with carcasses, and it answers that we are talking about depreciation of the carcass (from the time of the damage until the time of payment). Therefore I am asking about depreciation of the wood. You write that if none of the three leniencies are present (uncommon, by itself, his property), then one cannot derive, because there is no common denominator here at all, but three separate derivations. So presumably, in the case of a person who damaged a table, there is no law of “the owners deal with the carcass”; rather, the owners deal with the wood, and the wood belongs to the damager from the time of the damage, and if it depreciated, it depreciated for the damager.

E. Correct. And this fits very well with the law that a damager pays from the best-quality property (= money or movable goods worth money). Therefore he can take his own wood and pay the injured party with it. It is therefore obvious that if the wood depreciated, it depreciated for the damager (because the determining moment is the time of payment). As for who handles the wood, I don’t know.
By the way, all this is under the assumption that a person who damages a table is indeed common. It may be that all cases of a person who damages are uncommon.

I assume that a person who causes damage is the most common thing there is. If there is no common denominator but rather 3 separate derivations, then in the case of a person who damages wood, depreciation of the wood goes to the damager (whereas in the case of a person who kills an ox, depreciation of the carcass goes to the injured party). If there is a common denominator, then in every case the depreciation goes to the injured party. This is a major practical difference for your novel interpretation, and there are very likely explicit proofs about it. In the future I need to learn to arrive on time and not start dancing on the floor after the lights have already gone out. [And since I’m already chattering, I’ll say that I especially enjoy and am enlightened by your discussions of logical structures in sugyot. For in my eyes, in general all your teaching is refined, etc., except that within it ride four horsemen of the apocalypse, and whenever I hear the pounding of one of their hoofs from afar I put my sensitive belly on alert. And these are they: first, your denial of consequentialism; second, your belief in free will; third, your devotion to the categorical imperative; and the fourth, the little colt, autonomy in halakha. When you deal with matters of logical structures in sugyot, there is none of the dust raised by deontology, and then my stomach and I are very pleased, and I pat it with enjoyment and purr in satisfaction.]

I came back to this, and I still cannot get the new explanation of “stringent aspect” straight in my mind. Please explain it to me once more, like the waters of Shiloah that go softly.

A. You explained that the refuter is in the position of defense, but also that he too needs a basis for suspicion. He needs to suspect two things: first, that the theoretical factual characteristic is shared by the two source cases; second, that this factual characteristic is not present in the target case. For the first suspicion (the sharing), there is a basis by virtue of Occam’s razor. Is there a basis for the second suspicion—that this characteristic is not present in the target?
If the target has no halakhic stringent aspect of its own, then there is a basis also for the second suspicion, since if that characteristic were present in the target, we would expect it to produce there some halakhic stringent aspect. But if the target does have a halakhic stringent aspect of its own, then there is no longer a basis for the second suspicion (on the contrary, the razor would tell us not to suspect it). Yet the refutation of “stringent aspect” exists even if the target has a halakhic stringent aspect of its own (Sotah 29b, where Dor Aḥer refutes with “stringent aspect” even though second-degree impurity in a loaf also has a stringent aspect, namely that it invalidates non-sacred food). The conclusion is that for the second suspicion one does not need a basis; mere possibility suffices, since the refuter is in a defensive position.
But if so, then even in an ordinary binyan av from one verse there is the mere possibility for the second suspicion—namely, the possibility that the factual reason for the law is not present in the target—and it follows that the refutation of “stringent aspect” would refute every single derivation in the world.

B. Clarifications following note 6, which is significant.
1. You suggested that the halakha follows Rabbi Yehuda, who does not refute with “stringent aspect,” and therefore the anonymous Gemara follows his view. That is only one possibility, and there is no reason to think the halakha follows Rabbi Yehuda here against the Sages, right? As a matter of halakha, one is not lashed for a prohibition with no action, unlike Rabbi Yehuda; and according to the sugya in Makkot 4b it emerges that in halakha one does refute with “stringent aspect,” or that in halakha they hold that conspiring witnesses are a fine, or both.
2. According to the old understanding, the refutation of “stringent aspect” is a local matter that depends on judgment in each derivation individually. And you are innovating that it is a principled refutation that Rabbi Yehuda does not use and the Sages do. So every place where the Talmud does use “stringent aspect” is not according to Rabbi Yehuda; and when they rely on “stringent aspect,” for example to make a tzrikhuta explaining why a verse was indeed written, as in Pesaḥim 77b, then the matter is not reconciled according to Rabbi Yehuda. And every place that contains a common denominator and the refutations are halakhic (in almost every place I looked, that is what I saw, at least 10 places), the sugya is not according to the Sages against Rabbi Yehuda, because according to them one can refute with “stringent aspect.” Yet in no place in the Talmud that I have seen does it say, “And according to the Sages against Rabbi Yehuda, what can be said?” (or “And according to Rabbi Yehuda, what can be said?”). Seemingly, the likelihood of such systematic ignoring of the dispute is low if your hypothesis is correct.

You yourself write that the target has no stringency of its own, and therefore there is no reason to suspect that it contains the factual stringent aspect of the source cases. This is a simple rationale, and this is the situation in most places where they refute with “stringent aspect.” In cases where the target also has a stringent aspect, there really is room for such suspicion, and therefore there is no room for the refutation of “stringent aspect.” As for the sugya in Sotah that you mentioned, I do not have time right now to get into it, but in my opinion there must be some specific explanation there. (One has to see what the stringent aspect in the target is, and perhaps it is clear for some reason that at its base it does not contain the factual stringent aspect of the source cases.)

B. 1. The fact is that in most sugyot they do not refute with “stringent aspect.” From this it appears that probably here the halakha follows Rabbi Yehuda. As for our case? You yourself said that perhaps they hold conspiring witnesses are a fine—and that indeed is the halakha; see Rambam, Hilkhot Edut 18:8.
2. All true. I just did not understand why, when they make a tzrikhuta (with halakhic characteristics), this contradicts Rabbi Yehuda’s view. Quite the contrary: when there is such a tzrikhuta, one can derive from the two source cases to every other case, because as a matter of halakha one does not refute with “stringent aspect.” Moreover, even according to the Rabbis this is reconciled, for in such a tzrikhuta there is a stringency in each of the two sources relative to the other, but not necessarily relative to other contexts. In those other contexts, the stringency found in one or the other source may exist. In a context where the two stringencies really would both be absent, then indeed we could not derive, according to the view that does refute with “stringent aspect.”

I have no claims of puzzle-solving brilliance, but I will continue to present the side. At most you will answer, “Indeed so.”

A. It follows that, on your view, the refutation of “stringent aspect” is a uniquely peculiar kind of refutation. In ordinary refutations, nobody cares whether the target also has some stringency or a hundred stringencies (which is why one also refutes a kal va-ḥomer). But according to you, in the refutation of “stringent aspect” that suddenly does become relevant. And who says that “this is the situation in most places where they refute with ‘stringent aspect’”? Nobody checked whether one can find some special law in the target (because, as noted, usually this is not interesting). You are basically saying that if one starts from a kal va-ḥomer and then makes a common denominator (as in Sotah 29b), then one cannot refute with “stringent aspect.”

B.
1. It is indeed a fact that usually they do not refute with “stringent aspect,” and the simple explanation is that there it simply is not applicable. For the straightforward assumption is that the halakha follows the Sages against Rabbi Yehuda. Therefore the straightforward assumption is that there is no such general dispute whether one does or does not refute with “stringent aspect”; rather, each derivation stands on its own. It is possible that Rabbi Yehuda elsewhere would refute with “stringent aspect,” and in that very same place the Sages against Rabbi Yehuda would not refute. Therefore we do not see in the Gemara (as far as I managed to find by searching for “and the law reverted” or “stringent aspect”) questions of the type “And according to the one who refutes with ‘stringent aspect,’ what can be said?” or “And according to the one who does not refute with ‘stringent aspect,’ what can be said?” One sees in the Gemara a lack of formal systematicity, and therefore if one assumes there is systematicity, one has to explain each of the exceptions (in both directions, as I presented them above). Seemingly, a systematic explanation here is not an advantage but a disadvantage.
2. I meant the cases where the Gemara explains that a certain verse is a “tzrikha” because without it we could not have derived it from two other source cases, because they are subject to the refutation of “stringent aspect,” and therefore the verse was indeed written. It follows that according to your view, according to Rabbi Yehuda we really could have derived it, and the verse is still unnecessary (and one would have to say “a matter that can be derived,” etc.), and then the entire effort of the Gemara to make those tzrikhutas there was for nothing. That is what happens in Pesaḥim 77a (above I mistakenly wrote amud b).

A. Indeed. But in any case this is a special refutation, and therefore there is a dispute about it.
B. 1. I have said what I had to say.
2. I do not have time right now to examine Pesaḥim.

There was a difficulty with the Gemara about “the owners deal with the carcass,” where they did not address the common denominator of two source cases (a person who damaged an ox and a person who was negligent in guarding his own ox), and in order to answer it you answered, “because they do not have a common denominator,” and even explained that the Gemara too is not aware of a common denominator there, and therefore it raises the refutation of “stringent aspect.”
What is difficult for me is that precisely within this very logic of induction and syntheticity and the common denominator, although we do not identify the microscopic parameters—in other words, although we do not identify what the common denominator is—we nevertheless assume that there is one (because of Occam’s razor), and then infer by the force of induction.
That is, we need a positive reason to say that there is no common denominator, and not merely “we do not know of a common denominator,” because even though it is not known to us, we still assume there is one.
And therefore your answer falls. True, from a thief and a robber we see that the common denominator “damage occurred and there is a carcass” is not correct, but still we should assume that there is some other common denominator!
You tried to explain that the aspect “damage occurred and there is a carcass” does not count as a common denominator, because it is the very subject under discussion, so of course it will be present in all the discussions and objects, since that is what we are discussing. It is not a special common denominator.
This seems to me completely true and understandable. But then the other sugyot that present as the common denominator the very subject under discussion itself are not understandable—such as standing grain and a vineyard, both of which are “for enjoyment.” Of course both are for enjoyment, because that is the discussion. Seemingly, one should look for something else.

The marking disappeared:

A. There was a difficulty, etc.

B. You tried to explain, etc.

Correction:
Second line: that the Gemara* too does not know* of a common denominator there, etc.

Fourth line from the end: does not count as a common denominator* because it is the very subject under discussion* so of course*

Indeed, I have always wondered about those passages in the Gemara that present the subject under discussion as the common denominator. Thus, for example, in the example brought in the baraita of examples: every commandment applies immediately and for future generations. There must apparently be some common denominator behind this.
As for your question, perhaps the Sages had an intuition that there was a common denominator, or they even knew what it was, while I, when analyzing their inferences, do not know what it is and merely call it alpha and beta. And from this it follows that if there is a case where they reach the conclusion that there is no common denominator, then such an inference cannot be made.
Think about the example brought by some of the authors of the principles: to derive that a doorpost should be obligated in tzitzit—“If a four-cornered garment, which is exempt from mezuzah, is obligated in tzitzit, then is it not all the more so that a doorpost, which is obligated in mezuzah, should be obligated in tzitzit?!” Why is this not correct? Because it is clear to us that there is nothing at all in common between a four-cornered garment and a doorpost (in fact, between the parameter that obligates mezuzah and the one that obligates tzitzit), and therefore one cannot derive from one to the other. Thus you see that the Sages can indeed reach the conclusion that there is no common denominator.