Here we'll talk about a situation where we don't know which of a kosher animal's offspring is a Bechor (the first ones). First, we'll talk about a sheep that gives birth to twins. The Mishnah says: You have a sheep that has never given birth before, so the offspring she's about to have are the first ones. She gives birth to twins, two males. The two heads of the newborn sheep come out of the womb at the same time. They both come out together. The head being born determines whether something is a Bechor or not. So they both come out together, in unison. Rabbi Yose HaGelili says both of them are given to the Kohen (priest). Both animals came out at the same time, so they're both considered to be the Bechor, and they both have that status. So they're both given to the Kohen. As the Pasuk (verse) tells us about the Bechor of kosher animals, the males, in the plural, belong to Hashem. The fact that we have the plural term here, males, tells us that if two males are born at the same time, both are considered to be the Bechor. The Hachamim (the Sages) say no: that's an impossibility. Two animals can't emerge directly and exactly at the same time. It's a davar she'i efshar letzamtzem. It's impossible to have two things that are exactly at the same time. Therefore, the Hachamim say that one of them must have come out first. One of them is a Bechor, but we don't know which one came out first. So the owner is not obligated to give both animals to the Kohen. Rather, the owner keeps one for himself and gives the other one to the Kohen. That's the opinion of the Hachamim.
We'll elaborate more on this idea: the owner keeps one and gives one to the Kohen. We'll have a machloket (dispute) between the Tannaim (Mishnaic sages) about this opinion of the Hachamim. Which animal is given to the Kohen, and which remains with the owner? The Kohen chooses the better of the two animals. The owner gets the inferior one, and the Kohen gets the stronger one. We assume that the stronger one pushed ahead. He probably emerged first and is therefore the Bechor, so he's given over to the Kohen. Rebbe Akiva will disagree, though. He says, “No, you can't assume that the stronger animal came out first.” But he'll say that we have to understand a few legal points before we decide who goes where. Rebbe Akiva says, Meshamnin Beinayhen: the quality between the two of them is first assessed. Which animal is worth more money? We look at the difference between them. One of the two is definitely the Bechor. So if one of them was worth $100 and the second one was worth $120, we know with definite certainty that the Kohen is entitled to at least the value of the inferior animal, which is $100. The question becomes: What about that last $20? The price difference between the inferior and superior animal. The Kohen can't prove that the better animal is the Bechor. We have the idea that when somebody wants to take money from his friend, the burden of proof is on him, the one who's taking the money. The $100 is definite. The Kohen has to prove that $20, which is the difference between the cheaper and more expensive animal. If he doesn't have any direct, concrete proof that the more expensive animal was the Bechor, that money remains in the owner's possession, and the Kohen would not receive it. Rather, he would receive the inferior animal.
But the problem is that the second one, the one that was more expensive, could have been the Bechor. So what do we do about that? According to Rebbe Akiva, the second animal, the one we said remains with the owner, is let out to graze until it develops a blemish. It's possible that it is the Bechor. And if it was the Bechor, it's not allowed to be slaughtered and eaten outside of the Bechor. So the owner has to wait until it develops a blemish. Once that happens, the animal is allowed to be slaughtered and eaten like any blemished Bechor. We have to say this is all according to the opinion of Rebbe Akiva, who says we don't really know; it's a 50-50 chance which animal came out first. But according to Rebbe Tarfon, the other interpretation of the Hachamim, the owner receives the inferior animal because we assume it's not the Bechor. Therefore, it doesn't have to graze or develop a blemish. He could eat it immediately because he assumes that it's not the Bechor.
The Mishnah will now tell us another law that applies to the owner's animal. This is talking about the Matanot (gifts): the leg, the jaws, and the stomach, which are given to the Kohen from non-sanctified animals. The owner is obligated to give the Matanot to the Kohanim from this animal. These parts definitely belong to the Kohen, regardless of the status of the animal. If the animal was the Bechor, the entire animal belongs to the Kohen. If it wasn't the Bechor, then it's a regular Chullin (non-sanctified animals) animal, and the gifts have to be given to the Kohen like with every other non-sanctified animal. That applies to the animal in the owner's possession. Rabbi Yosei is going to disagree. Rabbi Yosei says Poter: the owner is exempt from giving over the gifts to the Kohen. According to Rabbi Yosei, if the animal is the Bechor, we consider it to have been given to the Kohen, who then returns it to the owner in exchange for the animal that he did receive. So that's the way the mechanics work. Let's say the more expensive one here was the Bechor, and it's currently in the owner's possession. So we say that what technically happened is that he gave over the more expensive one. It was given back to him by the Kohen, and he gave the less expensive one in its place. So the Bechor was now in his possession, possibly the real Bechor. And the other one was given over to the Kohen: the one that wasn't really the Bechor, but was the replacement. Now, when he has the possible Bechor in his possession, it's possible it's not the Bechor, but it's on the Kohen to prove that he's entitled to the gifts. And he can't prove it, because it's possible it's the Bechor. If it is the Bechor, the Bechor is exempt from these gifts. Therefore, the Matanot are not handed over according to the opinion of Rabbi Yosei.
Now we'll go back to the laws of the animals, where a person found one of the animals dead. If one of these two animals died, again, two were born, and one of them is definitely the Bechor. They're both from the same mother. What happens now if one of them dies? It says the owner and the Kohen divide the remaining animal. We don't know which of the two animals was born first, so we don't know whether the remaining animal is a Bechor and we would give it over, or is not a Bechor and would be kept by the owner. Maybe the dead one was the Bechor. Therefore, we divide it. We come up with a compromise between the Kohen and the owner. They divide the value. Even though Rebbe Tarfon earlier on said we could assume that the stronger animal was born first, he eventually changed his opinion and said that there's not so much clarity about which animal came out first. So if one of the animals died, the remaining one, even if it's the weaker one, is a Safek Bechor (doubtful firstborn). Therefore, Rebbe Tarfon says that, out of this Safek (doubt) that we have, we should make a compromise and divide the value of the living animal between the owner and the Kohen. That's the opinion of Rebbe Tarfon. According to Rebbe Akiva, the burden of proof is on the one who wants to take money from his friend. The Kohen here wants to take some money from the remaining animal, even if it's only half. He wants to take half of it. He can't do that. He has to prove that the animal is his. He can't do that. He can't prove that the animal was the Bechor. So the entire animal remains with the owner.
Until now, we've been talking about two males that were born to one animal. Now we'll talk about a male and a female who were born at the same time. If you have a sheep that has never given birth before but now gives birth to a male and a female at the same time, we don't know which one was born first. The Kohen is not entitled to anything. In this case, there's a Safek as to whether there's a Bechor here at all. Since it's possible that the male was born second, there would be no Bechor. Only a male could be a Bechor. The Kohen can't prove that there is a Bechor, so it remains with the owner.