Gemara

(Mishnah): If a woman miscarried, but does not know whether or not a child came out, she is Safek Yoledes Zachar or Nekevah, Safek Nidah.

Bava Kama 11a (Ula): If a fetal sac partially left the womb on Sunday, and part came out on Monday, the days of Tum'as Yoledes start from Sunday.

Question (Rava): Are you stringent? This stringency will lead to a leniency, for she will consider herself to be Tehorah two weeks from Sunday!

(Rava): Rather, she is concerned lest she became Teme'ah on Sunday, but she counts days of Tum'ah only from Monday.

Rishonim

Question (Tosfos 11a DH d'Ka): The blood she sees on day 15 (we count starting from Sunday) should be Tahor, due to a Sefek-Sefeka! Perhaps the baby was a male. (If so, whether he was born on Sunday or Monday, day 15 is amidst Yemei Tohar.) And even if the baby was a female, if she was born on Sunday, day 15 is amidst Yemei Tohar!

Answer (Tosfos): We cannot be lenient, for two leniencies will contradict each other. If she will see blood on day 41, a Sefek-Sefeka would similarly permit it. (If the baby was born on Monday, in any case day 41 (is day 40 from Monday, the real day of birth) is amidst Yemei Tohar. And even if the baby was born on Sunday, perhaps it was a female, so day 41 is amidst Yemei Tohar!) Above we were lenient because perhaps the baby is a boy. Now we are lenient because perhaps it was a girl. In any case a (Safek) Isur was done! In Nidah we say that she is stringent for a male, female and Nidah, even though each Isur by itself is due to a Sefek-Sefeka (perhaps it was a child, and perhaps it was a male/female).

Tosfos (29a DH Teshev): She is concerned for Tum'as Yoledes Nekevah, even though there is a Sefek-Sefeka. If we would exempt her (and permit her between days eight and 14) due to the Sefek-Sefeka, we would also be lenient if she sees blood on days 34 and 41, to say that perhaps it was not a male, so a Sefek-Sefeka says that she did not begin Nidah now. These leniencies contradict each other. However, regarding a To'ah (one who does not know when she gave birth), we are stringent even though there are several Sefekos.

Terumas ha'Deshen (2:130): Our cows do not give milk before they give birth. Even so, I do not permit (exempt from Kedushas Bechor) based on this alone, unless there is another reason to permit, e.g. it is not known which calf was born first after he bought the cow. Even if no opinion permitted based on milk alone, this is a Sefek-Sefeka. Even if the cow did not give birth before, perhaps any particular calf was not its Bechor. This is unlike the case of a goat that gave birth to a male and female. Rabanan ruled that the male is a Safek Bechor. Even though perhaps it gave birth beforehand, and perhaps the female was born first, the latter Safek does not permit any more than the first Safek. If a cow gave birth to two males, if it gave birth beforehand, this permits both males. However, such a Sefek-Sefeka does not permit, like Sha'arei Dura says in the name of the Ri about a wolf that was Dores (clawed or bit) an animal in the herd. Perhaps it only bit, and even if it clawed, perhaps the animal we select was not the victim. The first Safek is not Batel even in a majority, and all the more so when one is mixed with one other.

Poskim

Shulchan Aruch (YD 316:3): A milking animal is exempt from Bechorah, for most animals do not milk unless they gave birth.

Rema: Some say that we do not rely on this. The Acharonim rule like this. This is the custom. However, if there is another reason to permit, e.g. a Nochri says l'Fi Tumo (unaware of the consequences) that it already gave birth, we permit. This refers only to cows, where we never heard that they milk before giving birth. Goats often milk without children, so we do not rely on this even regarding a Sefek-Sefeka, e.g. it gave birth to two, and we could say about each that it is not the Bechor, they are forbidden.

Taz (5): The Beis Yosef (57) says that some ask how the Ri (in Terumas ha'Deshen above) can argue with the Gemara, which permits due to Sefek-Sefeka? He answered that the first Safek was known first. I said (110:9) that we forbid when it was possible to know one Safek first, or if both Sefekos are equal, e.g. a mixture. When a goat gave birth to a male and female, why shouldn't we permit due to Sefek-Sefeka? Surely, if both children were males, we should be stringent, since one man owns both. This is like (two who walked on) two paths, and one of the paths was Tamei. If they ask together, we are Metamei both of them. Here he must ask together, for we cannot permit without a Sefek-Sefeka! Perhaps this is why the Rema discusses giving birth to two (males). He changed from Terumas ha'Deshen's case, for then Sefek-Sefeka permits. In practice, this requires investigation.

Maharit (2 YD 2): The Gemara clearly permits a Sefek-Sefeka even for Torah laws, e.g. a Chasan who claims that the Kalah was not a virgin (perhaps she was raped, or perhaps the Zenus was before he was Mekadesh her - Kesuvos 9a). In Zevachim 74a, Shmuel forbids a Sefek-Sefeka, and only R. Yehudah holds like him. Chachamim disagree. The Rashba proves that we permit even when there is Chezkas Isur.

Pri Chodosh (YD 110 Klalei Sefek-Sefeka): If one did Melachah during all of Bein ha'Shemashos at the beginning and at the end of Shabbos, he brings a Chatas in any case, whether Bein ha'Shemashos is day or night (Shabbos 35b). This connotes that the Torah forbids a Safek when it is Tartei d'Sasrei. One is not liable Asham Taluy for Melachah during one Bein ha'Shemashos, for the Isur was not Kavu'a. It is not like two pieces, one of which is Asur. If the Torah permitted Sefekos even when they are Tartei d'Sasrei, we could consider the first Bein ha'Shemashos to be day, and we could not obligate for the latter more than Asham Taluy. Rather, even if the Torah totally permits Sefekos, this is not when Tartei d'Sasrei, for in any case he does an Isur. This is all according to Tosfos in Kerisus, but Rashi in Shabbos (35a) and the Ran in Yoma say that one brings an Asham Taluy for the first Bein ha'Shemashos. The Rambam connotes like this. If so, we cannot resolve (whether the Torah permits a Safek when it is Tartei d'Sasrei, since the Torah forbids the first Safek).

Pri Megadim (110 Dinei Sefek-Sefeka 1): Tosfos says that a Sefek-Sefeka in which one contradicts the other is not a Sefek-Sefeka. In Ginas Veradim (20) I discussed whether this is only when both Sefekos contradict each other, or if it suffices for one to contradict itself. Tosfos holds that it suffices for one to contradict itself. There is no contradiction regarding the Safek that there was no child. However, perhaps there is different, for initially we must judge that perhaps there is no child, and if there is a child, perhaps it is a male. If we would initially say 'perhaps it was a male, and even if you will say that it is a female', there is no second Safek. The primary Heter (during the second week) is because perhaps it is a male, and on day 41 (we permit primarily because), perhaps it was a female. In such a case, it need be (perhaps this should say 'is not' - PF) reversible, for initially we needed to know if there is a child at all. It is different when one Safek does not contradict the other, e.g. a mouse took bread into one of two houses. If it was a small loaf (perhaps less than a k'Zayis remained), or it was a Safek Chametz Safek Matzah, we permit both houses due to a Sefek-Sefeka. We conclude with a Safek - perhaps it is Matzah, or perhaps it ate it. In any case, it seems that we do not distinguish like this. Minchas Yakov (Kuntres ha'Sefekos 11,12) asked from the case of two mixtures. I answer that for (Isurim) mid'Rabanan, we are not concerned if they contradict each other. However, the Terumas ha'Deshen (2:130) and Rema (316:3) connote unlike this. When there are two males, each contradicts the other. See the Taz there, and what I wrote above (11).

Shach (YD 110 Dinei Sefek-Sefeka bi'Ktzarah 19): We do not say Safek mid'Rabanan l'Kula regarding a Safek about an Isur Torah, even if for a side reason the Safek is mid'Rabanan. However, if we are unsure whether there is any Isur here at all, we are lenient.

Pischei Teshuvah (YD 110 19, citing Teshuvah 49 at the end Tevu'os Shor): If so, if one has a Safek whether he used earthenware Kelim for milk or for meat, and the Safek arose when they were not Ben Yomo (any absorbed taste was not from the last 24 hours, so it is detrimental), we can be lenient. He later questioned this, for perhaps this is like Tartei d'Sasrei. Since the Kli is Vadai Asur for one use, i.e. meat or milk, we forbid both of them. He concluded to permit through Kashering three times, even though normally one cannot Kasher earthenware, for there are many reasons to be lenient.

Aruch ha'Shulchan (110:116): Many Gedolim learned from Tosfos that we are not lenient about a Sefek-Sefeka when it is Tartei d'Sasrei. There is not a solid proof from the Gemara, for we find that Chachamim were stringent about a To'ah even though there are several Sefekos. Even so, the reasoning is proper even without a proof.