Two equal sides

How do you negate the fraction on both equal sides: What do you say to the students who have side z1 and side z2 when you teach that it only has side z1?

As mentioned, if Z2 is relevant then it can certainly be a pirkha. But the pirkha does not need the conjunction with Z1; it is enough that the teachers have Z2. There is always an equal side for the teachers and the teacher (Z1), otherwise there would be no learning even without the pirkha.

Please note, the comments continue below for some reason.

I didn't understand the answer and I'm not sure if the question passed
I'll try to elaborate on the details so that you can point me to the answer

There are four sources A B C D
In A the attributes are x, y~, z1, z2 and it has the law P
In B the attributes are x~, y, z1, z2 and it has the law P
In C the attributes are x~, y~, z1~, z2 and it has the law P~
In D the attributes are x~, y~, z1, z2~ d and they try to learn the law P on it
(The letter d was written only to straighten the writing and has no meaning here)

From the two teachers A B want to learn the law P to the teacher D
All the attributes x,y,z1,z2 are for the purpose of the matter severity and they want to learn the severe law P
[I tried to formulate this “The two teachers of a certain law have two common attributes z1,z2 and in the teacher there is only one attribute z1 and we have another matter in which there is the attribute z2 and it does not have the particular law”].

How will the process be
– Trying to learn from A to D
– Explain what A has x, say in D that he does not have x
– B will prove that he does not have X and has P
– Explain what B has y, say in D that he does not have y
– A will prove
– And the law is repeated, the equal side of which is z1 Therefore we will also learn for D
– Explain what A and B have z2, say in D that he does not have z2
– C will prove that he has z2 and does not have P
– Explain [here is the question] What A and B have both z1 and z2, say in D that he only has z1
– I don't see a negation to this argument, except to say, "The more plausible theory is that there is only one simple reason for the law P, and that is the reason z1, and not the reason z1 AND z2." Is that why you (in the name of the Gemara) negate this argument? Just as in every equal side, one negates the argument, "Perhaps the reason for the law P is x OR y, and therefore the law exists in A, and in B, and not in C, and it will not exist in D."

Exactly. After we have proven that Z2 is irrelevant (because the law P does not exist in C), it is completely removed from the game and is not considered in the conjunction either. Anyone who wants to argue that the conjunction determines the burden of proof on him. This is possible, but the default is that the conjunction does not play a role.
As an aside, there is an interesting point here. This argument is presented as a deduction, but this deduction actually, in the end, repeats and confirms (rescues) the original inference. It is not clear what the status of such a deduction is: is it a deduction or an inference. This could have implications, since in order to refute an inference, it is enough to raise another possibility, but strengthening an inference cannot be done by raising any possibility. The inference must be clear.

Thank you. I took a while to look for an example and I apologize for the gaps. And look, I found this new one
[The summary of the above is as follows. In the usual equal side, two options are presented: one necessary and sufficient attribute (the common one) or two sufficient and non-necessary attributes, and for reasons of simplicity, the one necessary and sufficient attribute is preferred, which is solid enough to establish the inference. In the scenario discussed here, two options are presented: one necessary and sufficient attribute (the common one) or a combination of two attributes that is necessary and sufficient. And the answer that is explained here, for reasons of simplicity, the one attribute is preferred over the combination, which is solid enough to establish the inference. So far. The theoretical considerations for the other side are apparently (mainly according to what I understood from your words; I myself am still pondering the matter) the difference between “one attribute” and “a combination of two attributes” When the conjunction is AND, it is not a sharp distinction at all. Moreover, from the point of view of simplicity, if not the conjunction, then why did the conjunction occur in the teachings].

Pesach 27:2:
Rabbi Yehuda returned and discussed another law.
– What remains forbidden in eating and leaven is forbidden in eating, what remains in burning, even leaven in burning.
– They said to him, a carrion will prove that it is forbidden in eating and does not require burning.
– The Pharisee said to them, It remains forbidden in eating and enjoying, and leaven is forbidden in eating and enjoying. What remains in burning, even leaven in burning.
– They said to him, A stoned ox will prove that it is forbidden in eating and enjoying and does not require burning.
– The Pharisee said to them, It remains forbidden in eating and enjoying and does not require burning. What remains in the fire, even leaven in the fire.
– They told him the milk of the stoned ox, which is forbidden for eating and enjoying and is punishable by a fine and does not require burning.

It seems that here the Midianites accept the idea that the ”combination” of 2 equal sides with AND can be a reason for inference. That is, the assumption that the conjunction determines (otherwise how did the conjunction appear in these appearances of the law) is strong and simple enough to learn from its strength. What is the relationship between this and the above remains to be seen.
(There are two types of inference. Inference that proves to strengthen the inference that brings another teaching that there is no x in it and there is P in it and shows that the property x is not necessary for the law P. And there is inference that proves to strengthen the inference that brings an anti-teaching that there is no x in it and there is P in it and shows that the property x is not sufficient for the law P and for which I have not found an example at the moment and perhaps my eyes are blind. And here in Pesachim inference that proves to refute the inference).
My incredulous feeling tells me that there is something interesting here to dig into (someday when I will expand on it). But maybe everything is simple or known or both.

[It seems that the response I sent was lost in the depths and I am trying to send with updated wording]

Thanks. I took a while to look for an example and I apologize for the gaps. My unreliable feeling says that there is an interesting point here although perhaps it is simple or well-known or both. [The summary of the above is as follows: in the usual equal side, two options are presented: one necessary and sufficient attribute (the common one) or the disjunction of two sufficient and non-necessary attributes, and for reasons of simplicity, one prefers the one necessary and sufficient attribute and this is solid enough to establish the inference. In the scenario discussed here, two options are presented: one necessary and sufficient attribute (the common one) or the conjunction of two attributes which is necessary and sufficient. And the answer that is explained here too for reasons of simplicity, one prefers the one attribute over the conjunction and this is solid enough to establish the inference. The consideration in favor of the one attribute and that the conjunction is not even disjunctive is simplicity. The considerations in favor of the combination are: A. Theoretically (mainly according to what I understood from your words, even if this is not exactly your opinion, as I am still working on it myself) the difference between a "single attribute" and a conjunction is not a sharp difference at all. B. On the contrary, a posteriori simplicity says that if in the teachings there is an equal side of a combination, then it did not happen by chance and therefore it is reasonable to assume that it is the cause of the law and not just one single attribute. C. A combination is certainly less simple than a single attribute, but on this small probability gap one does not build a conclusion].

And see, this is new I found
Pesachim 27:2
Rabbi Yehuda repeated and discussed another law
– What remains forbidden in eating and leaven is forbidden in eating, what remains in burning, even leaven in burning.
– They told him that carrion would prove that it is forbidden to eat and does not require burning.
– The Pharisee said to them. It remains forbidden to eat and enjoy, and leaven is forbidden to eat and enjoy. What remains in the fire, even leaven in the fire.
– They told him that the stoned ox would prove that it is forbidden to eat and enjoy and does not require burning.
– The Pharisee said to them. It remains forbidden to eat and enjoy and is punishable by a fine, and leaven is forbidden to eat and enjoy and is punishable by a fine. What remains in the fire, even leaven in the fire.
– They told him that the milk of the stoned ox would prove that it is forbidden to eat and enjoy and is punishable by a fine, and does not require burning.

It seems that here the Midianites accept the idea that a conjunction of equal sides is solid enough to base the inference on. On the surface, it seems easy and obvious that a conjunction of equal sides can be used to disprove an inference. [And we can see this without any doubt from this, because here we see that it is impossible to base a conclusion on avoiding a conjunction, since we could have learned directly from a negation (what negation is forbidden in eating and does not incur a sin, even leaven that is forbidden in eating does not incur a sin) and we prefer the teaching from a Ṭuṭer (what remains forbidden in enjoyment and eating and incurs a sin, even leaven that is forbidden in enjoyment and incurs a sin), and what rejects the teaching from a Ṭuṭer and is the pirkha for this teaching – oh, he says the teaching is preferable to a pirkha. But perhaps this should be rejected because there is a difference between a law and an absence. And there is a difference between a pirkha and the selection of teachers in which they say, “We see who is similar.”] And indeed, in the scenario that I presented above, there is another additional priority for the conjunction, which is the apo-serial simplicity, since the conjunction appeared in both teachers and this would not be a coincidence.

I think it's a question of relevance. In a place where there are two attributes of severity, then if in both cases there is a combination of severity, this is of course a contradiction, since it is possible that only the combination is sufficient to apply the law and anything less than that is not severe enough. But if it is a casual attribute that is not necessarily severity, this is a different situation.

(It says that the answer was written hours ago, but I only just now managed to see it.)
Allegedly, all the properties we are dealing with are properties that can be the cause of the law. That is, if we are trying to study a severe law, then we are dealing with severe properties. Therefore, they participate in the process of deductive and deductive reasoning and are not trivial. (And especially if “whatever the side is, they are all the same”)
Do you mean to distinguish between properties that accumulate with each other (perhaps: commensurability) in which the combination is absolutely reasonable (and we make it a determinative, and it is possible to build an inference by virtue of the assumption that the combination determines and not a singular property as in the Pesachs) and between a combination of properties that do not themselves have a hidden equal side and in which the combination is not so reasonable (and we do not make it a determinative, and it is impossible to build an inference by virtue of the assumption that the combination determines).

That's one possible difference. But sometimes there are interpretations whose relevance is not clear. That comes up as an option. This is compared to interpretations whose relevance is completely clear (=seriousness).

Your claim is that philosophies whose relevance is unclear can be regular philosophies (even though their relevance to the law taught is unclear) but their combination is more problematic (because their relevance to the law taught and their relevance to each other is unclear)? Can you explain the explanation with this?

[M”M It seems that on the strength of your words, you can explain a certain point in the Sugya there on Pesachim. There, Rabbi Yehuda improves his teaching from one attribute (forbidden to eat) to two attributes (forbidden to eat and to enjoy) to three attributes (forbidden to eat and to enjoy and punishable by a fine) and then moves on to a completely separate attribute (bel tutiro) alone and does not make it a combination of four attributes, and receives a philosophie on this attribute (there is a dependent guilt in bel tutiro and not in the burning).
Tosofats say that it is actually also possible to combine ‘bel tutiro’ As a fourth attribute [and the disambiguation of a dependent guilty person also disambiguates the combination of the four attributes because a dependent guilty person also has the other three attributes of prohibition in eating and enjoyment and the penalty of keret. Even though it was not mentioned in the Gemara here]. And even more than that, in Rashi, you say in the burial, it seems that in practice Rabbi Yehuda's intention is to combine four attributes (disambiguate a teaching from a combination of four attributes is of course more difficult).
According to your words, perhaps we can say more forcefully, that not to let go is not the same matter as the prohibition of eating and enjoyment and the penalty of keret and it does not seem relevant enough to combine this quartet, since the triad of eating, enjoyment, keret seems more coherent in the same matter (although it is true that both Tosafot and Rashi require a known relevance for the combination as you say and here it seems to them that not to let go is also relevant enough).

I don't know how to explain more than I wrote. Complex hypotheses are less good than a simple hypothesis.