1) THE "BECHOR" OF AN ANIMAL OWNED IN PARTNERSHIP WITH A NOCHRI
QUESTION: The Gemara (end of 2b) cites a Beraisa that discusses the case of a Jew who arranges with a Nochri that he will take care of the Nochri's animal and keep a percentage of the offspring, while the animal itself remains the property of the Nochri. Rebbi Yehudah rules that when the animal gives birth to its firstborn, the Jew must appraise the value of the baby animal and give half of that sum to a Kohen. The Chachamim disagree and rule that since the Nochri has a share in the animal ("Yad Nochri ba'Emtza"), it is exempt from the laws of Bechor.
Rebbi Yochanan (according to the Girsa of the BACH) explains that both Rebbi Yehudah and the Chachamim derive their opinions from the same verse, "Kadesh Li Chol Bechor" -- "Sanctify to Me every firstborn" (Shemos 13:2). The Chachamim derive from the extra word "Kol" ("every") that one must own the entire Bechor in order to be obligated to give it to a Kohen. Had the verse said only, "Kadesh Li Bechor," it would have implied that even when one owns only part of the animal, it must be given to a Kohen.
Rebbi Yehudah understands the verse differently. Had the verse said only, "Kadesh Li Bechor," it would have implied that one must give a Bechor to a Kohen only when he owns the entire Bechor. The word "Kol" teaches that he must give the Bechor to a Kohen even when he owns any part of it.
The Gemara later quotes Rav Papa who says that in the same way that the Chachamim derive from "Kadesh Li Chol Bechor" that one is obligated only when he owns the entire Bechor, they derive from "v'Chol Miknecha Tizachar" (Shemos 34:19) that one is obligated only when he owns the entire mother as well.
The Gemara here seems to contradict the Gemara in Chulin (135b). The Gemara there explains that the verse, "Kol ha'Bechor... bi'Vekarcha" -- "every firstborn that will be born in your herd" (Devarim 15:19), uses the singular form of "your herd" to imply that the obligation of Bechor applies only to an animal that has one owner. Nevertheless, the Torah also states, "u'Vechoros Bekarchem" -- "and the firstborn of your herd" (Devarim 12:6), using the plural form of "your herd," to teach that the Bechor of a jointly-owned animal is also obligated. Why, then, does the Torah use the word "bi'Vekarcha," in the singular form? The Gemara answers that this verse teaches that an animal that is owned in partnership with a Nochri is exempt. An animal that is owned in partnership with another Jew is obligated.
Why does the Gemara here need to derive from the verse, "Kadesh Li Chol Bechor," that an animal owned in partnership with a Nochri is exempt from the laws of Bechor, when the Gemara in Chulin derives this from the verse, "bi'Vekarcha"?
Moreover, the Mishnah (2a) derives from a different verse -- "Hikdashti Li Chol Bechor b'Yisrael" (Bamidbar 3:13) -- that the laws of Bechor apply only to an animal owned in partnership with a Jew, but not to an animal owned in partnership with a Nochri. Why does Rebbi Yochanan suggest that this law is derived from the verse of "Kadesh Li Chol Bechor"?
ANSWERS:
(a) TOSFOS (2a, DH veha'Mishtatef) answers that from the verse, "b'Yisrael" (Bamidbar 3:13), the Gemara derives that the laws of Bechor do not apply when the Nochri owns a share in the fetus. From the verse, "bi'Vekarcha" (Devarim 15:19), the Gemara derives that the laws of Bechor do not apply when the Nochri owns a share in the mother. Tosfos points out that this exemption could have been derived from "v'Chol Miknecha Tizachar" (Shemos 34:19), as Rav Papa (3a) says, but the Gemara in Chulin chose to derive it from "bi'Vekarcha" since it already cited the similar verse of "Bekarchem" (Devarim 12:6).
Why, though, are two verses -- "bi'Vekarcha" and "v'Chol Miknecha Tizachar" -- needed to teach that the laws of Bechor do not apply when a Nochri owns a share in the mother?
(b) The MAHARIT ALGAZI (page 2, DH v'Hineh Ra'isi) cites the Tosefta in Bechoros (2:2) that says that the laws of Bechor do not apply to an undomesticated animal, an animal of Hekdesh, or an animal of a Ger who died without heirs. The Maharit Algazi explains that the reason why these animals are exempt is as follows. Even though a Bechor becomes Kadosh automatically upon birth even before the owner declares it to be Kadosh, there is a requirement that it have an owner who is at least capable of being Makdish it (even though he does not actually have to be Makdish it). Furthermore, when the owner gives it to a Kohen, this act is equivalent to verbally declaring it as Kadosh.
The Maharit Algazi explains that this Halachah is derived from the verse of "bi'Vekarcha." Since the verse of "v'Chol Miknecha Tizachar" is used to teach that the firstborn of an animal owned in part by a Nochri is exempt from the laws of Bechor, the verse of "bi'Vekarcha" can be used to teach that the animal must have an owner who can be Makdish it, and if it has no owner who can be Makdish it, then it is exempt from the laws of Bechor. (See also CHAZON ISH 2a.) (D. BLOOM)

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2) HALACHAH: SELLING A "BECHOR" TO A NOCHRI
QUESTION: Rav Mari used to transfer ownership of the ears of the animals of his flock to a Nochri in order to exempt himself from the laws of Bechor, but he nevertheless would prohibit working with the firstborn animals and he would give them to Kohanim. The Gemara relates that, as a punishment, his flock died out.
The Gemara asks why he transferred partial ownership of the animals to a Nochri if he prohibited working with the Bechoros and gave them to Kohanim. The Gemara answers that he did not want someone to accidentally slaughter a Bechor in his flock and use it for personal use.
The Gemara gives two reasons for why he was punished. One reason is that he uprooted the Kedushah of Bechor from the animals by giving part of the animals to a Nochri. Another reason is that he caused onlookers to think that they, too, could give partial ownership of their animals to a Nochri and exempt themselves from giving the Bechoros to Kohanim when, in truth, they did not know how to properly transfer ownership to a Nochri.
Is one permitted to exempt his animals from Bechorah in the manner that Rav Mari did?
ANSWER: TOSFOS (DH d'Ka) and the ROSH (1:2) quote RABEINU TAM who explains that the Gemara's first reason for why Rav Mari was punished is not applicable when one sells the ear of the mother animal to a Nochri (in contrast to the Bechor's own ear), since no act was done to the Bechor itself to remove its Kedushah. The Gemara's second reason -- that onlookers might attempt to transfer ownership to a Nochri without making a proper Kinyan -- applies only to selling the ear of the unborn fetus, and not to the sale of the ear of the mother. Selling the ear of an unborn fetus is complicated, since it is a Davar she'Lo Ba l'Olam, and one must know the proper manner in which to execute such a transaction (see Insights to Bechoros 2:1), such as by stipulating that one is selling "the mother, for the use of its fetus' ears." An unlearned person will unknowingly invalidate the sale by selling the ear in an incorrect way. In contrast, selling the ear of the mother of the fetus is a relatively simple matter.
Rabeinu Tam suggests further that today it is preferable to prevent the Kedushah of Bechor from taking effect, since the animal cannot be brought as a Korban. Moreover, while one waits for the firstborn animal to become blemished so that it will be permitted to be eaten, it is common to accidentally transgress the prohibitions against making the animal work and sheering its wool, as the Gemara discusses later (53b). Rav Mari was punished for not exempting his animals from the Kedushah of Bechor by cutting their ears as they exited the womb instead of selling them to a Nochri, since that method at least would have left "Kedushas Kehunah" on the animal (3b). Today, since we are not experts at cutting an animal's ear at the moment it exits the womb, we have no means to exempt a Bechor from Kedushah other than by selling it to a Nochri. Therefore, selling it is permitted.
HALACHAH: The SHULCHAN ARUCH (YD 320:6) writes that today it is a Mitzvah to prevent the Kedushah of Bechor from taking effect on an animal by selling a portion of its mother to a Nochri while the Bechor is still a fetus, as Rabeinu Tam suggests. A part of the mother is sold to a Nochri by taking a small amount of money from the Nochri in exchange for the portion of land upon which the mother of the unborn Bechor is standing. The Nochri then acquires the part of the mother that is sold to him through Kinyan Chatzer (as the ROSH (1:2) explains). This is the commonly accepted practice today.

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