1) "SAFEK TUM'AH B'RESHUS HA'YACHID" ACCORDING TO REBBI SHIMON
QUESTION: The Gemara teaches that if a Mikvah that originally contained forty Se'ah of water and was found to be lacking after a person immersed in it, and the Mikvah was in Reshus ha'Yachid, Rebbi Shimon rules that the person is Safek Tamei. Why is he not Tamei for certain, as in the case of Safek Tum'ah of a Sotah, which is a case of Safek Tum'ah b'Reshus ha'Yachid and is considered to be certainly Tamei? Rebbi Shimon explains that we cannot compare the case of a Mikvah to the case of a Safek Sotah, because in the case of a Safek Sotah there is strong reason (Raglayim l'Davar) to consider her Tamei, since her husband had forewarned her not to seclude herself with the other man, and she did not heed his warning.
According to Rebbi Shimon, though, when would there ever be a case of Safek Tum'ah in Reshus ha'Yachid that is Tamei? In every other Safek Tum'ah (besides the case of Sotah) there is no reason to suspect that the object indeed become Tamei!
ANSWERS:
(a) RASHI (DH v'Iba'is) and TOSFOS (DH u'Shneihem) assert that Rebbi Shimon does not agree with the normal rule of "Safek Tum'ah b'Reshus ha'Yachid Tamei," and instead maintains that every Safek in Reshus ha'Yachid is Safek Tamei.
(b) The RAMBAN disagrees. In most cases, Rebbi Shimon agrees that a Safek Tum'ah in Reshus ha'Yachid is certainly Tamei. Accordingly, when there is a doubt whether or not an action that can make an object Tamei actually occurred, we rule that the object in question is Tamei for certain. In contrast, in the case of the Mikvah, the person is considered to be only Safek Tamei, because it is known that the Mikvah was once measured and found to contain forty Se'ah. Rebbi Shimon does not compare a case in which there is a doubt regarding when (or whether) an object's status changed (and we see no specific action taking place that could cause the change in status) to the case of Sotah, because there is a Chazakah d'Me'ikara in such a case which allows us to assume that the status did not change (or that it changed at the latest possible moment).
Why does Rebbi Shimon not say this explicitly? Why does the Gemara say that tjhe case of Mikvah is different from the case of Sotah because of the strong grounds to consider her Tamei (Raglayim l'Davar), instead of saying that the case of Mikvah is different because it has a Chazakah that it was once a valid Mikvah (with forty Se'ah)?
The Ramban explains that even when there is a Chazakah, we still judge the object (in our case, the Sotah) to be Tamei when there is Raglayim l'Davar. Since there is no Raglayim l'Davar to consider the person Tamei in the case of the Mikvah that was found lacking, Rebbi Shimon does not rule that it is definitely Tamei, but rather that it is Safek Tamei.

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2) HILLEL'S QUESTION FROM "KUPAH"
QUESTION: The Mishnah (2a) teaches that Shamai and Hillel argue about the status of a woman who saw Dam Nidah. Shamai says that she is considered Tamei only from the actual moment at which she saw the blood, and we do not assume that the blood was present in the Bayis ha'Chitzon before she noticed it (see RASHI DH Shamai Omer). Consequently, all of the food that she previously prepared is still considered Tahor. Hillel argues and says that the woman is considered to be Tamei retroactively (with regard to Tum'ah and Taharah, but not with regard to being forbidden to her husband) from the last time she had a clean Bedikah.
The Gemara gives several explanations for the argument between Shamai and Hillel. In one of the explanations, the Gemara says that Shamai maintains that if the blood had been present in the Bayis ha'Chitzon for some time before she noticed it, it would have emerged. The Gemara cites support for this explanation from a Beraisa, in which Hillel asks Shamai, "Do you not agree that if one side of a box (Kupah) contains Taharos, and a Sheretz was found later in a different corner of the box, that all of the Taharos in the box are considered to be Tamei retroactively?" Shamai answers that this is true; however, in the case of the box, the Sheretz is "contained," proving that it was present even before it was discovered to be there. A woman, in contrast, is not considered "contained," meaning that if blood had been present in the Bayis ha'Chitzon, it would have dripped out immediately, since there is nothing to contain it and prevent it from emerging.
The RAMBAN has difficulty with Hillel's question. The case of the Kupah seems to pose a challenge to Hillel's own opinion. In the case of the Kupah, the Taharos are deemed to have become Tamei for certain, while even Hillel agrees that any Taharos that were handled by a woman after her last clean Bedikah are only Safek Tamei when she sees Dam Nidah. If Hillel is so certain that these cases are similar, then why are their Halachos not the same?
ANSWERS:
(a) The RAMBAN answers that Hillel is not saying that the two cases are the same. Hillel knew that Shamai would answer that there is a difference between something that is "contained" (Kupah) and something that is not "contained" (a woman). However, he asked Shamai that although there is such a difference, if the Halachah regarding Kupah is that the Taharos are definitely Tamei, then the Taharos of such a woman should at least be deemed Safek Tamei. Shamai rejected this logic, saying that the difference between the two cases is significant enough that the Taharos handled by the woman are not affected at all.
The RASHBA rejects this answer. One of the questions that he asks is that the fact that Hillel phrases his question by saying, "What difference is there between the two cases," shows that Hillel maintains that there is no difference between the two cases.
(b) The Rashba therefore answers that the question is based on a false premise. The Halachah of the Taharos in the Kupah is that the Taharos are only possibly Tamei, and not definitely Tamei. This answer is also given by the RITVA. (Y. MONTROSE)

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