Commentary for Bekhorot 8:3

וילדה שני זכרים נותן חמש סלעים – that one of them is a first-born male.

If a man's wife had never before given birth and she gave birth to two males, he gives five sela's to the priest.
If one of them dies within thirty days [of birth] the father is exempt.
If the father dies and the sons survive: Rabbi Meir says: if they gave the five sela's before the property was divided up, then what they gave is given; but if not, they are exempt. But Rabbi Judah says: there is a claim on the property.
If she gave birth to a male and a female, the priest receives nothing.
This mishnah deals with a case where twins are born, but it is unclear which child came out first.
Section one: One of these boys is certainly the first born and one is certainly not. Therefore, the father pays five selas to the priest, but does not pay twice.
Section two: A child who dies before he reaches thirty days never has a chance to be redeemed and his father is exempt. In this case, since we don’t know if the first or second born died, the father is exempt, because he can say to the priest, “prove to me that the first born survived and I will give you your five selas.” This accords with the general principle that the burden of proof is upon the claimant.
Section three: If the father dies before he has a chance to redeem one of the sons, then legally the sons must redeem themselves. However, the problem is that each son can claim that he is not the first-born and therefore doesn’t need to be redeemed. According to Rabbi Meir, if they already gave five selas to the priest before the property was divided up, then they cannot get their money back. However, if their father’s property was divided up before the five selas were given to the priest, they are exempt because each son can claim that he is not the first born.
Rabbi Judah disagrees and holds that there is a debt hanging over the father’s property and that debt must be paid off, regardless of the fact that we don’t know which of the sons is a first born.

האב פטור – that he is able to state that the firstborn died. But when the son died prior to [the age of] thirty days [old], he (i.e., the father) is not liable for the redemption [of the first-born son]. But now when the matter is in doubt for perhaps the first-born son died and he (i.e., the father) is exempt, or perhaps, the one who remains is the [actual] first-born son and he (i.e., the father) is liable [for the child’s redemption]. The Kohen removes [the money] from his fellow, for the burden of proof is upon the claimant. And the same rule applies when the remaining son is exempt from redeeming himself when he will be an adult.

מת האב – after thirty days [of fathering a son].

ואם לאו פטורים – for Rabbi Meir holds that the brothers who divided the possessions of their father, they have the law of the bought property (especially – mortgaged property sold), because there is no retroactive designation (though generally, it is accepted with regard to questions of rabbinic law, but not with regard to maters of Torah law) to each one [of the sons] on his inheritance, but they are as if each purchased from the other his portion and these five Selaim were the obligation of their father as a loan by mouth, but a loan by mouth is not collected/paid from the bought property, therefore, if they divided it up until they didn’t give it, they are exempt.

רבי יהודה אומר נתחייבו הנכסים (i.e., the estate is liable) – for he holds that the brothers who divided [the property] are inheritors and there is retroactive designation to state that each one merited in his portion, and a loan by mouth one collects from the inheritors. And the Halakha is according to Rabbi Yehuda.

זכר ונקבה אין כאן לכהן כלום – for one can say that the daughter came out first, and the burden of proof is upon the claimant (see Tractate Bava Kamma, Chapter 3, Mishnah 11).