1) "TUM'AS MES" OF DOORWAYS
QUESTION: The Gemara says that according to Rebbi Oshaya, Bereirah does not work for any matter which is mid'Oraisa. Therefore, in the case of a corpse inside a house in which there is a doubt through which doorway the corpse will be removed, all of the doorways become Tamei. The actual path of the corpse's exit cannot be determined retroactively because this is a question of Tum'ah d'Oraisa, and Bereirah does not work for a matter which is mid'Oraisa. RASHI writes that this Tum'ah is d'Oraisa because it is a Halachah l'Moshe mi'Sinai.
Rashi earlier (10a, DH Kulan Teme'in) writes that the Tum'ah in this case is only a Gezeirah d'Rabanan. How does Rashi there understand the Gemara here which says explicitly that the Tum'ah is mid'Oraisa? (SHITAH MEKUBETZES 10a)
ANSWER: RAV SHLOMO KLUGER (in BIGDEI YOM TOV) answers that in the Gemara earlier (10a) Rashi is explaining the opinion of Rava who says that the argument between Beis Shamai and Beis Hillel in the case of Tum'ah of entranceways involves Tum'ah l'Mafrei'a. Beis Hillel maintains that it is possible to show, retroactively, which doorway was Tamei in the first place by opening it later, thereby being Metaher all of the other doorways retroactively (the Gemara here understands the argument differently). Rava certainly does not argue with the conclusion of the Gemara here that Bereirah does not work for a Halachah d'Oraisa, so he must understand that this Tum'ah is only mid'Rabanan.
The Gemara here (Rebbi Oshaya), however, maintains that the Tum'ah is mid'Oraisa. (The SHA'AR HA'MELECH gives a similar approach and explains that Rava maintains that the Tum'ah is mid'Rabanan.)
2) THE "TECHUM" OF AN ANIMAL FOR SALE
OPINIONS: Shmuel rules that the ox of a butcher ("Shor Shel Patam") has the Techum of "every person." This means that the animal acquires the Techum of the one who buys it on Yom Tov. The ox of a shepherd ("Shor Shel Ro'eh"), on the other hand, acquires the Techum of "the residents of the town." Since a shepherd raises his animals for himself and occasionally sells them to the residents of his town, the animal acquires the Techum of the townspeople. If the resident of another town buys the animal from the shepherd, he may not take it back to his city because it is limited to the Techum of its town.
Why do the animal of a butcher and the animal of a shepherd have different Techumin?
(a) RASHI and the RAMBAM explain that a butcher expects his animal to be sold on Yom Tov, and, therefore, before Yom Tov he has in mind that the animal should acquire the Techum of the buyer. (Apparently, his intention is effective because of Bereirah.) In contrast, a shepherd does not expect to sell his animal to a buyer from a different town, but only to his neighbors. Therefore, he has in mind that his animal should acquire the Techum of the people of his town and not his own Techum (if it differs from the Techum of the town).
The RAMBAN and RAN ask why Shmuel says that Bereirah works in the case of a Shor Shel Patam when Shmuel himself maintains that Bereirah does not work even for a Halachah d'Rabanan such as Techumin (37b).
1. The BA'AL HA'ME'OR answers that the reason why Shmuel maintains that Bereirah does not work in the case of a barrel of wine owned by partners (37b) is because it is not known for certain that they will divide the barrel on Yom Tov. In the case of the butcher's ox, the butcher knows for certain that he will sell the animal on Yom Tov; the only question is to whom he will sell it. In such a case, since it is known that the animal will not have the owner's Techum on Yom Tov and the only question is whose Techum it will have, even Shmuel agrees that Bereirah works.
2. The RA'AVAD (Hilchos Yom Tov 5:15) answers that Shmuel maintains that Bereirah does not work only in a case of partners who own something together. Since they are particular about each other's portion, each partner wants his own portion to go with him to where his Techum reaches. Therefore, the partners cannot determine later which portion each one owned at the onset of Yom Tov. In contrast, in the case of a butcher's ox the salesman does not want the ox to have his Techum. On the contrary, he wants Bereirah to work so that the buyer can take the animal home.
(b) The RAMBAN and RAN explain that a butcher's ox is like an animal of Hefker; the butcher removes his ownership from it as far as the Techum is concerned. In contrast, the shepherd gives his ox to the townspeople so that they all have joint ownership of it. As such, it is limited to their common Techum (and if two people in the town have Techumin in opposite directions, the ox may not be moved at all on Yom Tov). (The RASHBA suggests that the rule that the butcher's ox is considered like Hefker might be a Takanah d'Rabanan for the benefit of the wholesale butcher, so that he will have buyers and will be able to sell his animals.)
Since the Halachah is that Bereirah works for matters which are mid'Rabanan, in both cases (Shor Shel Patam and Shor Shel Ro'eh) the animal acquires the Techum of the buyer even if he is from a different town.
(c) The BA'AL HA'ME'OR in the name of "Yesh Mefarshim" writes that when the Gemara says that the ox of a butcher acquires the Techum of "every person" it does not mean that it acquires the Techum of the person who buys it. Rather, it means that it is considered to have the Techum of every person who could possibly buy it, and therefore the animal cannot be moved at all outside of the city.
Similarly, when the Gemara says that the ox of a shepherd has the Techum of "the residents of the town," it means that since Bereirah does not work according to Shmuel, the animal has the Techum of all of the people of the city. Everyone is a possible owner since it is not known who will buy the animal. Since the animal has the Techum of every resident of the city, it may not be moved at all if some of the residents have Techumin in opposite directions.
(The question of whether the ox follows everyone's Techum if Bereirah does not work depends on a basic understanding of what "Ein Bereirah" means. Does it mean that Bereirah does not work at all, or does it mean that the true owner cannot be determined retroactively, but one of the possible purchasers was indeed the owner. See Insights to Eruvin 37:1.)

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