Q&A: Two Equal Sides
Two Equal Sides
Question
In studying an argument from a common denominator, what happens if the two source cases from which we want to derive a certain law have two shared features, z1 and z2, while the derived case has only one feature, z1, and we also have another case that has feature z2 but does not have that law? Do we derive the law for the derived case or not? [You may have written about this and I read it and forgot, or maybe this is something well known, but in any case I couldn’t find it now and I don’t know. I also don’t have an example, so I’m asking in general terms and using the framework of your explanations and your wording].
The theory that the shared law is caused by feature z2 is ruled out by the other case (which would serve as a “will prove” against deriving from the common denominator z2). That leaves two “simplest” theories for what causes the shared law: either z1, or z1 AND z2. At first glance, on grounds of simplicity, it would seem preferable to say the theory is z1 (and then we would indeed derive the law for the derived case) rather than the more complex theory z1 AND z2. But perhaps such a slight advantage in “simplicity” is not enough to justify deriving a common denominator, and only a more significant advantage would suffice—as opposed to the usual common denominator, where the competing theories are z, the shared feature, versus x OR y, which are the distinctive features in the source cases. (Or perhaps the opposite: z1 AND z2 is actually the simpler theory for explaining the data, because if the reason is only z1, then the appearance of z2 in both cases is merely coincidental.)
Answer
Definitely yes. We derive from the common denominator Z1, and there is a refutation from the law shared by both, and the third case rules that out.
Answer
I didn’t understand the question. That is always the situation in a common-denominator argument. The derived case is never identical to the source cases, otherwise no inference would be needed. Unless the derived case has some feature that seems relevant to us in the discussion (a special leniency that the two source cases do not have). In that case, the situation is entirely equivalent to one where the two source cases have a stringency that the derived case lacks (they have z2 and it does not), and then this is a standard refutation of the common denominator.
Discussion on Answer
As stated, if Z2 is relevant then that certainly can be a refutation. But the refutation does not require the combination with Z1; it is enough that the source cases have Z2. There is always a common side to the source cases and the derived case (Z1), otherwise there would be no derivation even without a refutation.
Please note, for some reason the comments continue down below.
I didn’t understand the answer, and I’m not sure the question got across.
I’ll try to burden it with details so you can point me to the exact answer.
There are four sources: A, B, C, D.
In A the features are x, not-y, z1, z2, and it has law P.
In B the features are not-x, y, z1, z2, and it has law P.
In C the features are not-x, not-y, not-z1, z2, and it has not-P.
In D the features are not-x, not-y, z1, not-z2, d, and they are trying to derive law P for it.
(The letter d was written only to keep the formatting aligned and has no meaning here.)
From the two source cases A and B they want to derive law P to the derived case D.
All the features x, y, z1, z2 are, for the sake of the discussion, stringencies, and they want to derive the stringent law P.
[This is what I tried to formulate as: “the two source cases of a certain law have two shared features z1, z2, while the derived case has only one feature z1, and we have another case with feature z2 that does not have that law.”]
How would the process go?
– Try to derive from A to D.
– Refutation: what about A, which has x—will you say that about D, which does not have x?
– B will prove it, since it does not have x and still has P.
– Refutation: what about B, which has y—will you say that about D, which does not have y?
– A will prove it.
– So the law returns, and the common denominator in them is z1, therefore we should also derive to D.
– Refutation: what about A and B, which have z2—will you say that about D, which does not have z2?
– C will prove otherwise, since it has z2 and does not have P.
– Refutation [and here is the question]: what about A and B, which have both z1 and z2—will you say that about D, which has only z1?
– I do not see how to rule out that refutation, except by saying: “the more plausible theory is that there is only one simple reason for law P, namely z1, and not the reason z1 AND z2.” Is that why you (speaking in the name of the Talmud) rule out that refutation? Just as in every common-denominator argument one rules out the refutation “maybe the reason for law P is x OR y, and therefore the law exists in A and in B, but not in C and also will not exist in D.”
Exactly. Once we have proved that Z2 is not relevant (because law P does not exist in C), it drops out of the game entirely, and we do not take it into account even in combination. Whoever wants to claim that the combination is what determines things bears the burden of proof. That is possible, but the default is that the combination plays no role.
As an aside, there is an interesting point here. This argument is presented as a refutation, but in the final analysis that refutation actually comes back and reaffirms (rescues) the original inference. It is not clear what the status of such a refutation is: is it a refutation or an inference? That could have implications, because to refute an inference it is enough to raise another possibility, but strengthening an inference cannot be done merely by raising some possibility. An inference has to be clear.
Thanks. I got delayed looking for an example, and I apologize for the spacing. And look, here’s something new I found.
[The summary of what was said above is this. In an ordinary common-denominator argument, one sets up two options: one feature that is necessary and sufficient (the shared one), or two features that are sufficient but not necessary; and on grounds of simplicity one prefers the one necessary-and-sufficient feature, and that is solid enough to found the inference. In the scenario under discussion here, the two options are: one feature that is necessary and sufficient (the shared one), or a combination of two features that is necessary and sufficient. And you replied that, as a matter of reasoning, here too grounds of simplicity lead us to prefer the one feature over the combination, and that is solid enough to found an inference. So far so good. The theoretical considerations for the other side are that, apparently (mainly based on what I absorbed from your words; for my own part I’m still mulling it over), the difference between “one feature” and “a combination of two features” when the combination is AND is not a sharp difference at all. Also, from the standpoint of simplicity, if the combination is not what determines things, then why did the combination happen to appear in both source cases?]
Pesachim 27b:
Rabbi Yehuda returned and argued a different law.
– Leftover sacrificial meat is forbidden to eat, and leavened food is forbidden to eat; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: carcass will prove otherwise, for it is forbidden to eat and yet does not require burning.
– He said to them: make a distinction—leftover sacrificial meat is forbidden both for eating and for benefit, and leavened food is forbidden both for eating and for benefit; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: an ox sentenced to stoning will prove otherwise, for it is forbidden both for eating and for benefit, yet does not require burning.
– He said to them: make a distinction—leftover sacrificial meat is forbidden for eating and for benefit and is punishable by karet, and leavened food is forbidden for eating and for benefit and is punishable by karet; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: the fat of an ox sentenced to stoning will prove otherwise, for it is forbidden for eating and for benefit and is punishable by karet, yet does not require burning.
It seems that here the disputants accept the idea that a “combination” of two common sides with AND can be a reason for an inference. That is, the assumption that the combination determines things (otherwise how did the combination appear in these instances of the law?) is strong and simple enough to derive from. What the relation is between that and what was said above still requires analysis.
(There are two types of “will prove” used to strengthen an inference. One is a “will prove” that strengthens an inference by bringing another source case that lacks x but has P, showing that feature x is not necessary for law P. And there is a “will prove” that strengthens an inference by bringing an anti-source case that has x and lacks P, showing that feature x is not sufficient for law P; for that one I have not yet found an example, unless my eyes have grown dim. And here in Pesachim the “will prove” is used to refute the inference.)
My unreliable intuition tells me there is something interesting here to dig into (sometime in the future, when this gets expanded). But maybe it is all obvious, or known, or both.
[It seems a comment I sent sank into the depths, and I’m trying to resend it with updated wording.]
Thanks. I got delayed looking for an example, and I apologize for the spacing. My unreliable intuition tells me there is an interesting point here, though perhaps it is obvious, or known, or both. [The summary above is this: in an ordinary common-denominator argument, one sets up two options: one feature that is necessary and sufficient (the shared one), or a disjunction of two features that are sufficient but not necessary, and on grounds of simplicity one prefers the one necessary-and-sufficient feature, and that is solid enough to found the inference. In the scenario under discussion here, the two options are: one feature that is necessary and sufficient (the shared one), or a conjunctive combination of two features that is necessary and sufficient. And you replied that, as a matter of reasoning, here too grounds of simplicity lead us to prefer the one feature over the combination, and that is solid enough to found an inference. The consideration in favor of the one feature, and for saying the combination is not even a refutation, is simplicity. The considerations in favor of the combination are: (a) that theoretically (mainly from what I understood from your words, even if that is not exactly your view; for my own part I am still hesitating about this), the difference between “one feature” and a conjunction is not a sharp difference at all. (b) On the contrary, a posteriori simplicity says that if in the source cases there is a common side consisting of a combination, that did not happen by chance, so it is reasonable to assume that it is the cause of the law rather than just one isolated feature. (c) True, a combination is less simple than a single feature, but one does not build an inference on such a small gap in plausibility.]
And look what new thing I found:
Pesachim 27b.
Rabbi Yehuda returned and argued a different law.
– Leftover sacrificial meat is forbidden to eat, and leavened food is forbidden to eat; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: carcass will prove otherwise, for it is forbidden to eat and does not require burning.
– He said to them: make a distinction. Leftover sacrificial meat is forbidden for eating and for benefit, and leavened food is forbidden for eating and for benefit; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: an ox sentenced to stoning will prove otherwise, for it is forbidden for eating and for benefit and does not require burning.
– He said to them: make a distinction. Leftover sacrificial meat is forbidden for eating and for benefit and is punishable by karet, and leavened food is forbidden for eating and for benefit and is punishable by karet; just as leftover sacrificial meat must be burned, so too leavened food must be burned.
– They said to him: the fat of an ox sentenced to stoning will prove otherwise, for it is forbidden for eating and for benefit and is punishable by karet, and does not require burning.
It seems that here the disputants accept the idea that a conjunction of common sides is solid enough to serve as the basis for an inference. On its face, it would seem all the more so that a conjunction of common sides can serve as a refutation of an inference. [And even without an a fortiori argument, you can see it from here, because here we see that one cannot base an inference on the absence of a conjunction, since we could have learned directly from carcass (“just as carcass is forbidden to eat and does not require burning, so too leavened food, which is forbidden to eat, should not require burning”), and yet they prefer the derivation from leftover sacrificial meat (“just as leftover sacrificial meat is forbidden for eating and for benefit and requires burning, so too leavened food, which is forbidden for eating and for benefit, requires burning”); and what pushes aside the derivation from carcass and serves as the refutation of that derivation? The preferable derivation from leftover sacrificial meat. But perhaps that can be rejected, since there is a difference between a law and the absence of a law. And there is a difference between a refutation and choosing between source cases, where one asks “to which is it more similar?”] And indeed, in the scenario I presented above there is an additional advantage to the conjunction, namely a posteriori simplicity, since the combination appeared in both source cases, and that would not be accidental.
I think this is a matter of relevance. In a place where we are dealing with two features that are stringencies, then if both source cases contain a combination of stringencies, that is of course a refutation, because it may be that only the combination is sufficient to apply the law, whereas anything less than that is not stringent enough. But if this is just some incidental feature that is not necessarily a stringency, that is a different situation.
(It says the answer was written hours ago, but only now did I manage to see it.)
Seemingly, all the features we are dealing with are features that can be the cause of the law. That is, if we are trying to derive a stringent law, then we are speaking about stringent features. Therefore they participate in the process of refutations and proofs and are not incidental. (Especially if “with a common denominator, we refute from any slight difference.”)
Do you mean to distinguish between features that accumulate together with one another (perhaps: commensurability), where the combination is quite reasonable indeed (so one makes from it a refutation, and one can build an inference on the assumption that the combination determines things, as in Pesachim), and a combination of features that do not themselves share some hidden common side, where the combination is not so plausible (and one does not make from it a refutation, nor can one build an inference on the assumption that the combination determines things)?
That is one possible distinction. But sometimes there are refutations whose relevance is not clear. That can come up as an option. That is unlike refutations whose relevance is entirely clear (= stringencies).
Is your claim that refutations whose relevance is not clear can still be ordinary refutations (even though their relevance to the law being derived is not clear), but their combination is more problematic (because both their relevance to the law being derived and their relevance to one another are unclear)? Could you explain the reasoning behind that?
[In any event, it seems that these remarks of yours can explain a certain point in the Talmudic passage there in Pesachim. There Rabbi Yehuda improves his derivation from one feature (forbidden to eat) to two features (forbidden to eat and for benefit) to three features (forbidden to eat and for benefit and punishable by karet), and then moves to a completely separate feature (“you shall not leave over”) by itself, and does not make it into a combination of four features; and he receives a refutation on that feature (a provisional guilt-offering falls under “you shall not leave over” and yet is not burned).
Tosafot say there that in fact one can also combine “you shall not leave over” as a fourth feature [and the refutation from a provisional guilt-offering refutes even the combination of the four features, because a provisional guilt-offering also has the other three features: it is forbidden to eat and for benefit and is punishable by karet, even though this was not mentioned in the Talmudic text here]. And even more so, in Rashi on “and yet you say by burial,” it appears that Rabbi Yehuda in practice intends a combination of four features (to refute a derivation from a combination of four features is of course harder).
According to what you said, perhaps one could say more forcefully that “you shall not leave over” is not of the same kind as the prohibition of eating and benefit and the punishment of karet, and it does not seem relevant enough to combine that foursome, whereas the trio of eating, benefit, and karet seems more cohesive as one topic. (Though in truth it may be that Tosafot and Rashi also require known relevance for a combination, as you suggested, and here it seemed to them that “you shall not leave over” is relevant enough.)]
I don’t know how to explain it any more than what I wrote. Complex hypotheses are less good than a simple hypothesis.
How do you rule out the refutation from the two common sides: what about the source cases, which have both side z1 and side z2—will you say the same in the derived case, which has only side z1?