Gemara

(Rav Anan): Brothers are exempt from Ma'aser Behemah only if one took kids and the other took (mature) goats;

If each took half of the kids, and each took half of the goats, we say that (Yesh Bereirah, and) each brother received his proper inheritance, so they are obligated.

(Rav Nachman): Even in this case, it is as if each bought the animals he took in exchange for his true inheritance (because Ein Breirah).

(R. Elazar): They are exempt only if one brother took nine animals and the other took 10, but if each took the same number, we say that each received his proper inheritance (Yesh Breirah).

(R. Yochanan): Even if each took the same number, (Ein Breirah, and) we do not say that each received his proper inheritance. (They are exempt).

R. Yochanan similarly taught that if brothers divide an inheritance, they are like buyers. They must return their shares and redivide in Yovel.

Had he taught only about returning land, one might have thought that this is a stringency (for he is unsure whether Yesh Breirah), and we are stringent also to take Ma'aser, or that in Yovel we must return to the original status (before division), but brothers must take Ma'aser (because Yesh Bereirah).

Question (Mishnah): Similarly, if two partners divided, and Reuven took 10 sheep and Shimon took nine sheep and a dog, all Reuven's sheep are Pesulim for Korbanos. (Each one is a Safek Mechir Kelev; perhaps it was in exchange for the dog.)

If Yesh Breirah, we could pick one of Reuven's sheep to be in exchange for the dog, and the rest would be Kesherim!

Answer (Rav Ashi): Indeed, if the animals were worth the same, we could. The case is, the dog is worth more than any one sheep. Each of Reuven's sheep is worth slightly more than each of Shimon's. In all, they compensate for the extra value of the dog. Each is partially in exchange for the dog.

Gitin 24b (Mishnah): If Reuven told a scribe 'write a Get for my wife Leah. I will decide which of my wives Leah I will use it for', it is Pasul.

We do not rely on Breirah (to say that retroactively, it was written for the one Reuven later decides. Perhaps when it was written, he intended to divorce the other wife!

(Rav, Shmuel, Ze'iri and Rav Asi): If he gave the Get, it disqualifies her from Kehunah.

(R. Yochanan): It does not disqualify her.

Question (Abaye): Rav Hoshaya asked about a case of Breirah in which a Get depended on what others will do (which wife will leave first), and Rav Yehudah answered from a case that depends on the man's own will (which wife he will choose)!

Further, Rav Hoshaya challenged the answer from a case that depends on the will of others!

Answer (Rava): Perhaps all equate Breirah that depends on one's own will with that depends on others' will!

Beitzah 38a: R. Oshaya holds Ein Breirah regarding mid'Oraisa laws. For mid'Rabanan laws he holds that Yesh Breirah.

The Halachah follows R. Oshaya.

Rishonim

Rif (Beitzah 21a) and Rosh (Beitzah 5:7): The Halachah follows R. Oshaya, who holds that Ein Breirah regarding mid'Oraisa laws, but Yesh Breirah for mid'Rabanan laws.

Rif and Rosh (Gitin 12b and 3:1): The Get does not disqualify. Only R. Yochanan disagrees. He is an individual. The Halachah follows the Rabim.

Ran (DH v'Hilchesa): Not only does the Get disqualify from Kehunah. Rather, if another man was Mekadesh her afterwards, we are concerned for Kidushin. We hold that Ein Breirah for Torah laws, i.e. we are stringent, but surely we do not say Ein Breirah to be lenient! Therefore, it is Safek divorce. Also the Rambam says so.

Rambam (Hilchos Gerushin 3:4): If Reuven told a scribe 'write, and I will divorce with it whichever I want', and he divorced with it, it is Safek divorce.

Magid Mishnah: The Rambam holds that Yesh Breirah. He calls it Safek Girushin because the Mishnah says that the Get is Pasul.

Kesef Mishneh: The Gemara says that mid'Rabanan Yesh Breirah. I.e. we are in doubt whether Yesh Breirah, and we are lenient for mid'Rabanan laws. For mid'Oraisa laws we are stringent to say that (perhaps) Ein Breirah.

Rebuttal (Yam Shel Shlomo Bava Kama Perek 5, Dinei Breirah. p.47b): The Rambam (below) rules that heirs return their portions and redivide in Yovel. If it were a Safek, we would not take property from them. Perhaps each inherited his proper portion, and they need not redivide!

Rambam (Hilchos Shemitah v'Yovel 11:20): Brothers who divided an inheritance are like buyers. They return their portions to each other in Yovel, but the division is not Batel.

Rambam (Hilchos Isurei Mizbe'ach 4:17): If two partners divided, and Reuven took 10 sheep and Shimon took nine sheep and a dog, if one of Reuven's sheep is worth as much as the dog, we separate that sheep, and the rest are Kosher for Korbanos.

Question (Lechem Mishneh Bechoros 6:19 b'sof): This is like the opinion that Yesh Breirah. The Rambam rules that Ein Breirah!

Answer #1 (Zera Avraham YD 112a, cited in Sha'ar ha'Melech Girushin 3:4 DH v'Da): The Rambam holds that Breirah does not apply to anything destined to be divided, and each stands by itself, e.g. a dough of an Ir ha'Nidachas (Sanhedrin 112a). Since they joined with intent to divide, it is as if it is already divided. However, the Rambam exempts from Ma'aser Behemah when each brother took goats. This is because the Torah equates it to Banecha, which are clear to you.

Rebuttal (Sha'ar ha'Melech): The Rambam (Hilchos Terumos 1:21) obligates taking Terumah from what is owned jointly with a Nochri, because Ein Breirah, even though each stands by itself and it was destined to be divided, just like animals! Also, if coins for a Chatas that cannot be offered were mixed with and coins of Kosher Chata'os, R. Yehudah holds that we cannot separate coins (and discard them, to be Machshir the rest). The Gemara explains that he holds that Ein Breirah, like in the Mishnah of tithing wine (Yoma 55b). According to Zera Avraham, wine is different, for it is not discrete! Also other Sugyos show that we do not distinguish like this. Also, we cannot say that Ma'aser Behemah is different because animals are equated to sons. The Rambam rules that the Get makes Safek divorce, even though a verse requires clarity (Lishmah)!

Answer #2 (Sha'ar ha'Melech DH ul'Inyan): The Kiryat Sefer says that mid'Oraisa, the sheep are permitted, since the lamb in exchange for the dog is mixed and Batel. Mid'Rabanan, living animals are not Batel. For mid'Rabanan laws, Yesh Breirah. If so, what was the question in Bechoros? The Gemara held that Yesh Breirah only for matters such as Eruv and Ma'aser or Yerek, which have no source in Torah. Really, this is not difficult, for we challenged R. Elazar, who holds (Beitzah 37b) that Ein Breirah even for mid'Rabanan laws. Since we hold that Yesh Breirah for mid'Rabanan laws, we needed to say that no sheep is worth as much as the dog.

Rashi (Beitzah 38a DH bed'Oraisa): Regarding Tum'as Mes (that the openings of the house are Temei'im), which is a tradition from Moshe from Sinai, the tradition taught that we do not rely on Breirah.

Poskim

Shulchan Aruch (YD 131:4): If Reuven told a scribe 'write, and I will divorce with it whichever I want', and he divorced with it, it is Safek divorce. The same applies if he said 'write for the one who will walk out the door first. I will divorce her with it.'

Beis Yosef (DH u'Mah): The Gemara asked about a Get for 'whichever wife will leave first' from the case of 'for the one I will decide'. This shows that they are totally the same. I.e. both are Safek divorce.

Beis Shmuel (4): This is like the Rambam. The Ran and Kesef Mishneh say that we are only stringent to say Ein Breirah. However, the Rambam rules that brothers redivide in Yovel, and he exempts from Ma'aser all the lambs opposite the dog, i.e. Ein Breirah. The Yam Shel Shlomo says that for Torah laws we are even lenient to say Ein Breirah, and we are not concerned for Kidushin, but she is forbidden to Kehunah. The Rashba says that it is a Torah law that brothers redivide in Yovel, therefore we are stringent. We must say so for Rav Asi, who says that the Get disqualifies to Kehunah, and that brothers must redivide in Yovel. Why are brothers who divide exempt from Ma'aser Behemah? We can say that it is because Safek Ma'aser is exempt.

Gra (OC 413:1): The Poskim rule that the Get disqualifies. We must explain that Yesh Breirah, for R. Yochanan disagrees since he holds that Ein Breirah. The Yam Shel Shlomo says that it is a Get only to disqualify from Kehunah, but if a man was Mekadesh her, we are not concerned. This is wrong. The Gemara says that they argue about Breirah! Rather, we must say like Rashi, that we are stringent to say Yesh Breirah. However, in many places we say Ein Breirah to be lenient, e.g. R. Yochanan exempts from Ma'aser Behemah, and the Rambam rules like him! Rav Nachman (Kidushin 42b) and R. Yochanan (Bechoros 56b) say that brothers who divided are like buyers. We hold like Tosfos (Gitin 26b DH Divrei and elsewhere), that when one explicitly stipulated, we are stringent to say Yesh Breirah even for Torah laws. It is Pasul for a Get only because "Lah" requires the Get to be Lishmah (at the time it is written). The Yam Shel Shlomo challenged this, for Gitin 25a equates other cases of Breirah to Gitin. We can say that the Sugya is like the conclusion, that we do not distinguish a field from other Breirah. We could not have learned from a field, for R. Yochanan is lenient (to say Ein Breirah) also regarding Ma'aser Behemah.. The rule is, Yesh Breirah for mid'Rabanan laws and Ein Breirah for Torah laws, whether to be lenient or stringent. If one stipulates explicitly, or it is sure to be clarified, Yesh Breirah even for Torah laws, even to be lenient. We are stringent only about (a Get for) two wives. There are difficulties with the Rambam's rulings about Breirah, like the Yam Shel Shlomo said.

Shai l'Mora (in Kovetz Meforshim, Shulchan Aruch Rosh Pinah): Beitzah 38a supports the Yam Shel Shlomo. We say that Ein Breirah for Torah laws. Rashi says that this is a tradition from Moshe from Sinai. The Rambam holds that the Torah is lenient about Sefekos. However, the Ramban disagrees. (If one's barrel broke, and he intended to acquire the shards, he is liable for damage they caused - Bava Kama 29a.) The Nimukei Yosef (13b DH b'Miskaven) says that if he never acquired the shards, we do not say that retroactively he made them Hefker (and he is exempt), for mid'Oraisa Ein Breirah. If it is a Safek, why is he is liable? Ha'Motzi mi'Chavero Alav ha'Re'ayah! Rather, we must say that Vadai Ein Breirah.