A Further Look at ‘the Common Denominator’ (Column 346)
The standard 'common denominator' schema and its basic difficulty
The column begins with the basic formula of a binyan av from two verses: two source-cases, each of which shares the law in question but also has a unique feature that prevents learning from it alone. In the example of vineyard and standing grain in Berakhot, each requires a blessing before eating, but vineyard is special because of gleanings and grain is special because of challah, so each can be rejected on its own. Seemingly only their combination lets us derive the rule for other foods. But this immediately raises the central difficulty: if each source-case can always be pushed aside by saying that its law may stem from its own unique feature, it is unclear how the mere combination of two sources succeeds where each source alone fails. And conversely, if the two sources share some additional feature that the derivative case lacks, the inference is obviously refuted. The question is why such a shared feature refutes the derivation, while two separate unique features do not refute it in the same way.
Occam's razor explains why a common denominator works, and why a shared extra feature defeats it
The explanation the column offers for the ordinary logic of a 'common denominator' is one of elimination together with a preference for the simpler theory. Each source-case rules out the possibility that the law depends on the other source-case's unique feature, so what remains is the possibility that the law depends דווקא on what the two share. In terms of competing theories, one can explain the law either by 'one of the two unique features' or by 'one shared feature.' The column argues that the second theory is simpler and therefore preferable. This also explains why a refutation works precisely when the two source-cases share an additional common feature: then an alternative theory arises that is no less simple, according to which the law depends on that other common feature, and there is no way to decide in favor of the original 'common denominator.'
The 'more severe side' refutation does not destroy the entire rule; it exposes a difference between legal and factual features
From here the column moves to sugyot such as Ketubot 32, where the Gemara rejects a 'common denominator' derivation with the claim that 'they have a more severe side.' Tosafot already note that if one accepts such a refutation at face value, it becomes almost impossible to derive anything by 'common denominator' anywhere. R. Michael Abraham proposes to solve this by distinguishing between two kinds of features: factual features and legal features. When the unique features of the two source-cases are factual, each source really does rule out the possibility that the law depends on the other's feature, and the ordinary logic of the 'common denominator' remains intact. But when the features are legal, both may simply be different legal results of one hidden factual feature shared by the two source-cases. In that situation, even though the visible features differ, logically it resembles a case in which the two sources share a common feature that defeats the inference. In his view, as far as he checked, this is how the places in the Talmud that use the 'more severe side' refutation should be read.
The sugya of 'the owner handles the carcass' presents three source-cases, not just two
From here the column reaches the Bava Kamma sugya about the rule that the injured party handles the carcass, and therefore depreciation that occurs in it after the damage is borne by the injured party. The Gemara brings three sources: a person who damaged an ox, a paid guardian who was negligent, and damages caused by a pit. The tzrichuta that accompanies these sources shows that, seemingly, none is sufficient on its own: a person damaging an ox is an uncommon case, a guardian deals with damage that occurs indirectly rather than by his own hands, and a pit is a case of one's property causing damage. At first glance this looks like a regular expansion of the 'common denominator,' except that here there are three source-cases rather than two.
Two problems in the tzrichuta: the 'more severe side' refutation and the missing source-pairs
The column points to two difficulties in this sugya. First, when the Gemara considers learning from the first two sources together and rejects the move, it effectively uses the 'more severe side' refutation: each source has its own unique feature. But here those features are factual, not legal, and therefore, according to the earlier analysis, it should דווקא be possible to learn from them via a 'common denominator.' Second, the tzrichuta itself looks incomplete: the Gemara checks learning from the first source alone, the second alone, the two together, and the third alone, but it does not check the two remaining possibilities of learning from the first and third or from the second and third. If this were an ordinary 'common denominator' structure, that gap would be very hard to justify.
The proposal: here there is no 'common denominator' at all, only a union of several simple analogies
To resolve both difficulties, R. Michael Abraham proposes a completely different reading: it may be that these three sources share no relevant common denominator at all. If so, this is not an inference of the ordinary sort, in which 'one shared feature explains the law,' but a combination of several separate binyan-av derivations: the law applies wherever one of the source-features is present, not because there is one side shared by all of them. On this reading the first difficulty disappears, because there is no competing simple theory based on one shared feature; and the second difficulty disappears as well, because once no pair of sources has a common denominator, there is no reason to test every pair separately. The halakhic upshot of this reading is sharp: the rule that the owner handles the carcass should not apply to every case of damage that leaves a carcass, but only to cases that resemble at least one of the three sources in the relevant respects.
Thief and robber support this reading but also raise a new difficulty
The column suggests that this may also explain why, later in the Gemara, a thief and a robber are excluded from this rule. In their case the damage is done directly, it is common, and it is not a case of one's property causing damage, so none of the three source-features is present there. But this immediately raises an objection to the explanation: seemingly the three sources do share a very simple common denominator — in all three there is damage and there is a carcass. If so, perhaps that itself should have been the 'common denominator' from which the rule is derived.
Why 'there is damage and there is a carcass' is not enough as a common denominator
The column considers trying to say that this is not a real common denominator but merely the subject matter itself, and then rejects that move: in other sugyot too, the common denominator is very close to the very topic under discussion. It therefore offers a different, more cautious answer. Since later in the sugya there are verses teaching that with a thief and a robber the injured party does not handle the carcass, even though there too there is damage and there is a carcass, the Gemara clearly knows that 'there is damage and there is a carcass' cannot be the common feature that determines the rule. Had that been the common denominator, thief and robber would have been included as well. Therefore, if there is a common denominator here, it must be something more specific and ancillary; and since no such feature is found, it is better to read the sugya as a union of several simple analogies rather than as a true 'common denominator.'
The conclusion: three structures that look alike but run on different logic
In sum, the column distinguishes among three kinds of inference. The first is the regular 'common denominator': there is a feature shared by the source-cases and by the derivative case, and one can therefore extend the law to a broad class of derivatives. The second is a case in which there is a feature shared by the source-cases but not by the derivative, and then the inference is refuted. The third is a case in which there is no shared feature even among the source-cases themselves, in which case this is not a 'common denominator' at all but a combination of several separate binyan-av derivations. This distinction also prepares the ground for what follows: not every derivation from several source-cases works by the same logic, which matters for the broader questions of torts, Shabbat, and the fourth type that he wants to discuss next — 'conceptual construction.'
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Contents of the Article
With God’s help
Another Look at the Common Factor
For my daughter Rivka, may she live and be well
In the previous column I dealt with the relationship between primary categories and derivatives on the Sabbath, and I mentioned the difference between tort law and the Sabbath (see around notes 6-7 there): in tort law there are derivative cases that cannot be learned from a single primary category, and they are learned from two primary categories by means of an inference of ‘the common factor’; by contrast, on the Sabbath we have not found such derivatives. I noted there (in note 7) that this depends on the question of what the logic of the inference of ‘the common factor’ is. From its very name, such an inference appears to be based on a point of similarity shared by the two source cases (the primary categories), but in my article in __Meisharim__ 2 (and in greater detail in the eighth book of the Talmudic Logic series) I showed that there is another kind of similar inference, which I called ‘conceptual construction,’ and it is not based on a common factor shared by the two source cases. In recent days, when I was studying the sugya of ‘the owner takes care of the carcass’ (Bava Kamma 10b-11a) with my daughter Rivka, may she live and be well, the answer to a question that has troubled me for more than twenty years regarding this sugya became sharper for me.
Let me preface this by saying that ‘the common factor’ is a very foundational inference in Talmudic logic, far more than people generally realize, and therefore I have already written quite a bit about it.[1] It contains quite a few challenging puzzles, and various shades of inferences that are similar yet different; when one solves those puzzles and uncovers the shades and sub-shades of the inference, one reaches very deep foundations in talmudic analysis and in thought generally.
Here I would like to touch on the logic of ‘common factor’ inferences, and afterward point to the sugya of the owner taking care of the carcass, which challenges the usual conception of ‘the common factor.’ Finally, I will return once again to the question of the absence of ‘common factor’ inferences in the labors of the Sabbath. Following the discussion requires a certain amount of concentration in order to keep the differences among the various cases and situations straight.
The common factor: the basic schema
At the beginning of the __Sifra__ there appears a baraita containing a list of Rabbi Yishmael’s thirteen hermeneutical rules. Two of the rules in that list (some count them as one rule) are: ‘constructing a paradigm from one text’ and ‘constructing a paradigm from two texts.’ According to the accepted interpretation, the first rule is nothing but analogy. We compare one law (the target case, the derivative) to another (the source case, the primary category), and thus learn some rule from one to the other. The second rule is based on comparison to two different source cases (primary categories), each of which by itself is insufficient to teach the rule. This is the logic of ‘the common factor.’ I should note that some commentators explained that even ‘constructing a paradigm from one text’ is based on the logic of ‘the common factor,’ and the difference between the two rules is only whether the two source cases appear in the same biblical passage or in two different passages (see __Talmudic Encyclopedia__, entry ‘Binyan Av’).[2]
For our purposes כאן, I will bring an example of this logic from the sugya in Berakhot 35a, which deals with the blessing recited before eating:
We have found the case of a vineyard; from where do we know the other kinds? Let it be derived from a vineyard: just as a vineyard is something from which one derives benefit and it requires a blessing, so too anything from which one derives benefit requires a blessing. This can be refuted: what is unique about a vineyard? It is subject to the law of gleaned clusters. Standing grain will prove otherwise. What is unique about standing grain? It is subject to the law of the dough-offering. The vineyard will prove otherwise. And thus the argument returns: the aspect of this is not like the aspect of that, and the aspect of that is not like the aspect of this; their common factor is that one derives benefit from them and they require a blessing, so too anything from which one derives benefit requires a blessing.
There are two source cases here: a vineyard (grapes) and standing grain, and we want to learn from them to all other things (for example, other fruits and vegetables). Just as one recites a blessing over grapes and grain, so too one must recite a blessing before eating anything else. Why are two source cases (primary categories) needed? Because each of them has a unique feature that does not allow us to learn from it alone: a vineyard is unique because it is subject to the law of gleaned clusters (leaving the small clusters for the poor), and standing grain is unique בכך that it is subject to the law of the dough-offering. The combination of the two together can teach us, regarding all things from which we derive benefit, that there is an obligation to recite a blessing before the enjoyment.
The general structure of such an inference is always as follows: we have two source cases (primary categories), A (vineyard) and B (standing grain), in both of which the rule P under discussion applies (the obligation to recite a blessing). Each of them has a unique feature that could generate rule P, and it exists only in one and not in the other. A has feature x (it is subject to gleaned clusters) and not y (it is not subject to the dough-offering), while B has feature y (it is subject to the dough-offering) and not x (it is not subject to gleaned clusters). The assumption is that we are dealing with a target case (derivative) C, which has neither x nor y.
The structure of the inference is as follows:
- We begin by trying to derive rule P from source case A to derivative case C. The derivation can be either by analogy or by an a fortiori inference.
- We reject it on the basis of the stringency x that exists in A but not in C (perhaps rule P applies in A specifically because of that stringency).
- We try to derive rule P from source case B to derivative case C (by analogy or a fortiori).
- We reject it on the basis of the stringency y that exists in B (as above, סעיף b).
- And thus the argument returns: in the end we derive rule P from the two source cases A and B together to derivative C. From the Gemara’s formulation it is clear that the rule is learned from the common factor of the two source cases, z, namely that they involve benefit. This factor exists in the derivative as well.
Schematically, such an inference can be described as follows:
Figure 1: Basic schema of the common factor. A and B are the primary categories, and C is the derivative. The features of each of the primary categories and of the derivative appear in parentheses, with an overbar indicating that the feature does not exist in the primary category or derivative. The arrows express the attempt to derive rule P from the relevant primary category to the derivative. The derivation from both primary categories together is not marked separately, because it is done without any addition beyond the two individual derivations: this general structure is enough to apply rule P to derivative C.
The common factor: the difficulty
At first glance there is an incomprehensible logical miracle here. Each of the source cases (the primary categories) by itself cannot teach rule P regarding the derivative, but the combination of the two, without any additional element, does succeed in doing so. How exactly does the combination succeed where each of its components by itself fails? Put differently, I would formulate the difficulty as follows: seemingly this logic is invalid, because one can refute the inference with the claim (the refutation) that perhaps rule P exists in primary category A because of the unique feature x, and in primary category B because of y. One must remember that derivative C has neither of these features, and therefore it is not clear how the two primary categories teach rule P in derivative C.
The common factor: refutation
I should note that the Gemara there in Berakhot immediately afterward refutes this inference with the claim:
What about their common factor? Both have an altar connection!
The two source cases (vineyard and standing grain) have a shared unique feature, w, that does not exist in the derivative: some sort of connection to the altar. In such a case, one must add to the schema above the feature w, which exists in both source cases but not in the derivative, and therefore there is no possibility of learning from them. Indeed, the rule is that if the two source cases have a shared feature, that refutes the inference.
The resulting structure is now as follows:
Figure 2: A refutation of the common factor. Unlike Figure 1, here a shared feature w has been added to both primary categories, and it does not exist in the derivative. In such a case, of course, one cannot derive rule P from the two primary categories to the derivative.
This of course sharpens the difficulty from the previous section: why, in the ordinary case (when there is no feature w, as in Figure 1), where each of the primary categories has its own unique feature that does not exist in the derivative, is the inference not refuted in a similar way?
The common factor: the logic
It is customary to think that the justification for the logic of the inference shown in Figure 1 is a process of elimination: primary category B proves that rule P does not depend on feature x (for B does not have that feature and yet P still applies there), and primary category A proves that rule P does not depend on feature y (for A does not have that feature and yet P still applies there). We are forced to conclude that what determines rule P is feature z, the common factor of the two primary categories. What we have done here is an elimination that shows what the relevant feature is that determines rule P. It is now clear that rule P will also apply to derivative C, for that factor exists there as well.[3]
But that does not really solve the problem, because it is not clear how the possibility is ruled out that rule P is determined by x or y (a situation in which the presence of either one suffices to create liability for P).
It seems that the logic of ‘the common factor’ is based on a comparison between two possibilities:
- The refutation above actually proposes a theory that explains the rules as follows: rule P can apply when at least one of the two features exists, either x or y.
According to this theory, rule P would not apply to derivative C, because it has neither of those two features.
- There is an alternative theory, according to which rule P does not depend on x and y at all. It applies when feature z is present.
According to this theory, rule P would apply to derivative C, because it too has feature z, which is the common factor of the two primary categories.
The claim at the basis of the inference of ‘the common factor’ is that the second theory is simply more parsimonious, because it grounds rule P in a single parameter rather than in one of two, and is therefore preferable (the principle of Ockham’s razor favors the simpler theory). From this it follows that rule P also applies to derivative C.[4]
It is now clear why, in a situation where there is a shared unique feature of the two primary categories that does not exist in the derivative (as in Figure 2), the inference is refuted. In such a situation there are three competing theories: the two previous ones, plus theory C: rule P depends on feature w. According to this theory, rule P depends on feature w, and that does not exist in the derivative, so it follows that rule P does not exist in derivative C. Note that this theory too depends on only one parameter, and therefore it is no less parsimonious than theory B, so it cannot be rejected. This is the meaning of the refutation, and therefore it exists only in situations described in Figure 2 and not in situations described in Figure 1.
The objection from a stringent feature
In several sugyot we encounter a phenomenon that seemingly challenges this entire schema. For example, the Gemara in Ketubot 32a attempts to derive that one can be flogged and also required to pay for the same offense (according to Ulla). Once again, the same structure I described above appears:
It follows that Ulla holds that wherever monetary payment and lashes are both present, he pays the money and is not lashed. From where does Ulla derive this? He learns it from one who injures his fellow: just as one who injures his fellow, where monetary payment and lashes are both present, pays the money and is not lashed, so too wherever monetary payment and lashes are both present, he pays the money and is not lashed.
What is unique about one who injures his fellow? He is liable for the five indemnities…
Rather, he learns it from conspiring witnesses: just as conspiring witnesses, where monetary payment and lashes are both present, pay the money and are not lashed, so too wherever monetary payment and lashes are both present, he pays the money and is not lashed.
What is unique about conspiring witnesses? They do not require prior warning…
Rather, he learns it from both of them: their common factor is that where monetary payment and lashes are both present, he pays the money and is not lashed; so too wherever monetary payment and lashes are both present, he pays the money and is not lashed.
The Gemara derives this from two primary categories: one who injures another and conspiring witnesses. The schema is exactly as I described above, and I will not repeat it here.
But here a most surprising refutation suddenly appears, rejecting this inference:
What about their common factor? They each contain a stringent feature.
Seemingly, the Gemara here raises theory A from above and undermines the logic of ‘the common factor.’ Where has Ockham’s razor, which prefers the simpler theory B, disappeared to?
Tosafot, s.v. ‘for they,’ there, raise this difficulty:
‘For they each contain a stringent feature’ — this is difficult, for if so we will never again derive anything from the common factor anywhere, since with regard to all of them one could raise a refutation, either from a stringent feature or from a lenient feature.
Tosafot notes that if one accepts this kind of refutation, no room remains for inferences of ‘the common factor.’ All of them collapse by definition under such a refutation, because in all of them there is a stringent feature in each of the primary categories (see Figure 1). In effect, the Gemara here raises as an alternative theory A, according to which rule P depends on the presence of features x or y, and not on feature z (the common factor), as theory B suggests. The Gemara here does not prefer theory B despite its simplicity, and therefore it is not clear how one can derive anything at all by constructing a paradigm from two texts.[5] Tosafot here and the Ritva in Makkot 4a (where the Gemara gives a similar example, also in the name of Ulla, and he elaborates and brings several opinions of the early authorities on the matter) offer several answers to this difficulty, and they all seem forced.
The objection from a stringent feature: the logic
In my view, this can be resolved easily if we return to the logic of the common factor. To explain this, we must distinguish between two kinds of unique features that appear in hermeneutical inferences (analogy and a fortiori): legal features and factual features. In the cases we have brought here, the features {x,y,z} were legal features; that is, the primary categories and the derivative had special legal characteristics (such as liability for gleaned clusters or for the dough-offering, and by contrast a connection to the altar). But in other cases these features are factual. For example, at the beginning of Bava Kamma, the characteristics of the primary categories of damages are all factual: a pit — its making is for damage; fire — another force is involved in it; horn — its intent is to cause damage; and so forth. These are not laws unique to those categories of damages, but factual characteristics of them. We must remember that every law is based on some factual feature. The reason that conspiring witnesses do not require prior warning, or that a vineyard is subject to the law of gleaned clusters, stems from some factual features of the legal situation under discussion. For example, there is something factual about a vineyard that causes it to be liable, legally, for gleaned clusters. That is, underlying legal characteristics there are always hidden factual characteristics that generate them. In light of these two assumptions, we can understand the objection from a stringent feature and why it does not contradict all inferences of ‘the common factor.’
When I have two primary categories, each of which has a different factual feature, then one can perform the elimination I described earlier and prove that neither of those features is the relevant feature for rule P (theory B is simpler than theory A, and is therefore preferable). But if the different features of the two primary categories are legal ones, then it is possible that both primary categories share the same factual feature (w), which in each of them causes a different legal feature. For example, it is possible that in one who injures another and in conspiring witnesses there exists the same factual feature (say, both intentionally harm a person), but in conspiring witnesses its legal consequence is that they do not require prior warning (y), whereas in one who injures another its legal consequence is that he pays five indemnities (x).
If such a possibility exists, the inference of ‘the common factor’ collapses. Look at Figure 1 of ‘the common factor,’ and you will see that it is possible for both primary categories to have the same factual feature w, while their different legal features, x and y, are merely the legal result of the existence of w. Therefore, it may be that the relevant figure is Figure 2 rather than Figure 1. As we saw, in the situation described in Figure 2 one cannot derive from the primary categories to the derivative (as we saw in the sugya in Berakhot, that if the two primary categories have a shared feature one cannot derive from them by the common factor). As far as I have checked, throughout the Talmud, whenever the objection from a stringent feature appears, it concerns legal rather than factual characteristics.[6]
We are now ready to arrive at the sugya of the owner taking care of the carcass.
The sugya of the owner taking care of the carcass
The Gemara in Bava Kamma 10b rules that if Reuven’s ox damaged Shimon’s ox, then the carcass belongs to Shimon, the injured party; from this it follows that if it suffered some depreciation afterward (for example, it rotted or crumbled), that loss is borne by the injured party. He is the one who was supposed to take care of the carcass, for example by selling it at its higher value immediately after the damage, thereby preventing the depreciation that occurred afterward.
Three sources are brought there from which this law can be learned:
As the Sages taught: ‘Compensation for damage’ teaches that the owner takes care of the carcass. From where are these words derived?
- Rabbi Ami said: For Scripture says, ‘One who strikes the life of an animal shall pay for it’ — do not read it as ‘shall pay for it’ but as ‘shall make it whole.’
- Rav Kahana said: From here: ‘If it is indeed torn, he shall bring it; for the torn one he shall not pay’ — up to the point of its becoming torn he pays; for the torn animal itself he does not pay.
- Hezekiah said: From here: ‘and the dead animal shall be his’ — that is, the injured party’s…
Rabbi Ami brings a source from the verses dealing with a person who damaged an ox. Rav Kahana brings a source from the verses dealing with a paid bailee who was negligent. And Hezekiah brings a source from the verses dealing with pit damage in the public domain. The Gemara assumes that these are not conflicting sources, but that all three are needed, and it constructs a necessity analysis to explain why. I will say in advance that such a necessity analysis really means that each of these sources by itself could not have taught us this law, because a refutation would arise due to its unique characteristic, and therefore all three are required. This means that we have three primary categories, and from their common factor we want to learn to all the other cases. The logic of ‘the common factor’ (all the examples and cases I brought above) can, of course, also exist when there are three or four primary categories, mutatis mutandis. The case of two is merely the simplest case.
Let us examine the necessity analysis brought by the Gemara:
And this is necessary. For had the Merciful One written only ‘one who strikes an animal shall make it whole,’ that is because it is uncommon; but torn prey, which is common, I might have said not. It is necessary. And had He informed us only about torn prey, that is because it happens by itself; but ‘one who strikes an animal,’ where it is done directly by hand, I might have said not. And had He informed us only these two, this one because it is uncommon and that one because it happens by itself; but ‘and the dead animal shall be his,’ which is common and done by one’s act, I might have said not. And had He informed us ‘and the dead animal shall be his,’ that is because it is one’s property that causes the damage; but here, where he harms with his own body, I might have said not. It is necessary.
One must remember that the rule that the owner takes care of the carcass is a leniency for the damager and a stringency for the injured party. The source from a person who damaged an ox deals with an uncommon case (ordinarily a person does not simply damage an ox), and therefore perhaps there the Torah was not exacting about placing the obligation to take care of the carcass on the damager, although it would have been appropriate to do so. The source from a paid bailee deals with a case where the damage happens by itself (the bailee was merely negligent and did not close the gate, and the damage occurred on its own and not through the bailee himself), unlike the previous case, where it occurred by the damager’s own hands; therefore there is more room there to be lenient with the damager. And the third source, from pit damages, deals with a case of one’s property causing damage (as opposed to a person who causes damage and a bailee who was negligent, where the damage occurred through a person and not through property).
The difficulties
The structure of the necessity analysis raises two conspicuous difficulties:
- When one examines the possibility of learning from the first two sources (and dispensing with pit), this is rejected by the objection from a stringent feature (each of the sources has its own unique feature). As we have seen, this seemingly contradicts the logic of the common factor (because this is a situation described in Figure 1, and not the situation in Figure 2).[7] Note that here the features x and y are factual, and therefore here one cannot explain the objection from a stringent feature in the way I explained above. Here the objection from a stringent feature ought not to work, and the inference from the two primary categories should be valid (that is, there should be no need for the third source, from pit).
- The necessity analysis is not fully complete. The Gemara examines only the possibility of learning from the first source alone, from the second source alone, from both of them, and from the third source alone. But there are two more possibilities that the Gemara ignores: learning from the first and third sources, or from the second and third sources. Why does the Gemara not complete the necessity analysis?
Explanation: an inference from several source cases to a derivative without a common factor
In light of what I explained above, it is possible to suggest a way of resolving both of these difficulties. The Gemara I cited shows that each of these three sources has a unique feature. The question is whether they have any common factor. Let us assume for the moment, for the sake of the discussion, that they have no common factor at all, and we will see that this resolves both difficulties.
The schema of the inference now looks like this:
Figure 3: A threefold inference without a common factor. Each of the primary categories has a unique feature, and no pair among them has a common factor (note that the feature z here is not shared. It has the same status as x and y; it is simply added because this is an inference from three primary categories). The derivative has none of the unique features of the primary categories.
By looking at this schema, one can immediately see that the two difficulties are resolved of themselves. First, there is no option here of theory B, which would ground rule P in a single unique feature shared by the three primary categories (because there is no such feature). Here there is no choice but to adopt theory A, which says that rule P applies whenever one of the three unique features exists. Here we have no simpler theory. That resolves difficulty 1. But that in turn also resolves difficulty 2, because it is now clear that there is no need at all to continue the necessity analysis. Once we have reached the point where no pair among the three source cases (the primary categories) has a common factor, it is self-evident that one cannot learn from any pair composed of them to other cases.
Note that this proposal has a legal implication. In the ordinary case of an inference of ‘the common factor’ (Figure 1), the result applies to all derivatives regardless of their particular features. More precisely, it applies to any case that has feature z, regardless of the features of the primary categories (x or y). By contrast, here I am claiming that rule P (that the owner takes care of the carcass) exists only in situations where one of the three features of the primary categories is present: x, y, or z (uncommon damage, damage that happens by itself, or damage by one’s property). If there were a case in which none of those three features were present, the rule that the owner takes care of the carcass would not apply there. In such a case, the damager would take care of it, and the depreciation would be at his expense.
Perhaps this is the explanation for why, later on, the Gemara determines that with regard to a thief or a robber, the rule that the owner takes care of the carcass does not apply. If you look carefully, you will see that in the case of a thief and a robber none of these unique features exists: this is damage done by a person (not as in pit damage), it does not happen by itself but by direct action (not as with a bailee), and it is also common (not as with a person who damages an ox). Note that only the case of a person who damages an ox is uncommon, because a person does not ordinarily inflict damage for no gain. But a thief and a robber, of course, do this in order to profit, and therefore there it is called common damage.[8]
The meaning of this is that such an inference is a simple combination of three inferences by analogy from one text. Each of the three primary categories teaches regarding the derivatives that resemble it, and the resulting derivatives are the union of three groups of simple derivatives. There is no connection here among the source cases, and therefore no connection among the groups of derived cases either. If so, the situation in Figure 3 describes a kind of simple analogical inference which, upon examination, turns out to have no connection at all to inferences of ‘the common factor.’
A difficulty with this explanation
I assumed that the three primary categories have no common factor, and this indeed resolves all the difficulties. But it is not clear why the very fact that this is a case of damage and that a carcass lies before us is not itself a common factor. One could say that after eliminating the features of the three sources, we learn that none of them is relevant to the rule that the owner takes care of the carcass. We would then conclude as follows: in all cases where there is damage and a carcass is produced, the injured party takes care of it.
Yet at first glance it seems that this cannot be considered a common factor, for that is the very subject under discussion. The features {x,y,z} under discussion are supposed to be secondary ones (such as the fact that this is damage that happens by itself, damage by one’s property, or uncommon damage), and they are what generate the rule with which we are dealing. But the fact that this is a case of damage and that there is a carcass is not a unique feature; it is the very situation we are discussing. We are looking for special characteristics of that situation, or of partial groups of situations within it.
But that is not precise. Notice that even in the cases I described above, the common factor is not a secondary feature. In the sugya in Ketubot, the common factor is that in both primary categories (one who injures another and conspiring witnesses) the person receives lashes, and from this they infer that the rule of receiving lashes and paying applies there. The fact that in both primary categories (and in the derivatives) the person receives lashes is the common factor, even though that is the very subject of our discussion. It is not a secondary feature. The same is true of the vineyard and standing grain in the sugya in Berakhot. There the common factor is that in both there is benefit. And again, that is the very subject under discussion (a blessing over enjoyments). If so, the difficulty returns: why can the fact that in all three sources in our sugya in Bava Kamma there is damage and the owner takes care of the carcass not be considered a common factor?
Perhaps the explanation is the following. Later in the Gemara, one sees that the rule that the owner takes care of the carcass does not apply to a thief or a robber. But a thief and a robber also have the common-feature characteristic I described here: damage occurred and there is a carcass. Since there are verses there that teach that the injured party does not take care of the carcass, it was clear to the Gemara that this itself cannot be regarded as the common factor (for otherwise, even in the case of a thief or a robber, the injured party would take care of the carcass). In other words, from the law of a thief and a robber it follows that it is not true that in every situation in which damage occurs and a carcass is produced, the injured party is the one who takes care of the carcass. Therefore, if there is some common factor to the three sources in the case of one who causes damage, it would have to be a specific and secondary one, something beyond the mere fact that all three are situations in which there is damage and a carcass. Since the Gemara sees that there is no such common factor, we return to the logic I explained above, and the two difficulties we raised remain resolved.
Summary
We have seen three kinds of inference that look similar to one another, but upon closer examination turn out to be fundamentally different:
- The ordinary case (Figure 1): when there is a common factor shared by the primary categories and the derivative. In such a case, one can derive from the two primary categories all the derivatives that have the feature that is the common factor of the primary categories, including those that are not fully similar to either of the two primary categories.
- There is a common factor only for the primary categories, and it is not found in the derivative (Figure 2). In such a situation the inference is refuted. One cannot learn from the primary categories any derivative that does not contain the common factor shared by the two primary categories.
- There is no common factor even among the primary categories (Figure 3). As we saw, this is an inference that consists of a combination of three analogies from one text, and it has no connection at all to ‘the common factor.’
In the next column I will try to identify a fourth type of inference from two source cases, which looks very similar to type 3 — that is, here too there is no common factor shared by the two primary categories — and nevertheless one can derive from it conclusions for derivatives that are broader than the primary categories themselves. I call it ‘conceptual construction.’
1.
Footnotes
- The Zoom course on talmudic analysis (Part I), which concluded not long ago, was based largely on a discussion of the common factor and various extensions of this inference.
- Immediately after the baraita of Rabbi Yishmael there appears the ‘baraita of examples’ (sometimes called the ‘scholion’), which illustrates all of these rules. The example given there for constructing a paradigm from one text is also based on ‘the common factor.’
- This is essentially a scientific generalization of the kind we also make in other areas of life. Think of a person who sees a stone falling toward the earth. He wonders whether a pencil will also fall downward, and says that one cannot infer this from the stone because the pencil is made of a different material (perhaps only things made of stone fall toward the earth). Now he takes a tree branch and sees that it too falls toward the earth, and it is made of the same material as the pencil, so one can infer that the pencil too will fall. But even that inference can be rejected on the grounds that the branch is a natural object and not an artificial one (perhaps only such objects fall toward the earth). In the end we rule out the possibility that the material or the naturalness are the parameters relevant to falling toward the earth, and conclude that what is responsible is the common factor shared by the stone and the branch, namely that both have mass. That factor exists in the pencil as well, and therefore the conclusion is that it too will fall downward.
- This, of course, is also what justifies the scientific generalization described in the previous note.
- The Gemara says that the possibility of making the objection from a stringent feature exists only according to the Rabbis against Rabbi Yehuda, but Rabbi Yehuda disagrees with them and does not make such a refutation. The difficulty here is only according to the view of the Rabbis.
- I emphasize that my claim is that everywhere an objection from a stringent feature arises, the features of the two primary categories are legal and not factual. But I am not claiming that in every such case the Gemara raises an objection from a stringent feature. Sometimes even in such cases no objection from a stringent feature arises, and apparently that is because the law follows Rabbi Yehuda and not the Rabbis (and therefore the anonymous voice of the Gemara follows his approach). That is, as a matter of law we do not raise an objection from a stringent feature even when the two primary categories have unique legal characteristics. The rationale is that if the legal features of the two primary categories are different, the burden of proof that they both share a common factual feature rests on the one raising the objection.
- This question was raised by my student Nadav Ettinger when we studied tractate Bava Kamma about twenty-five years ago in the yeshiva in Yeruham.
- The Gemara there also asks about a borrower, and in the conclusion, even in the case of a borrower the owner takes care of the carcass, as in the case of one who causes damage. A borrower is one of the types of bailees, and seemingly the three conditions are present there as well (it is uncommon, it happens by itself, and it is not damage by one’s property). Even so, it still requires further consideration why a different initial assumption arose here.
Discussion
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.
This came out as a file. Was that intentional? Because it’s a bit less convenient to read.