LINEAGE OF A CHILD NOT CONCEIVED THROUGH BI'AH [artificial insemination: lineage]
"Ervas Achoso Gilah" obligates for (each Isur for) a sister who is also his father's sister and his mother's sister.
(Rav Ada bar Ahavah): The case occurs through Isur, e.g. Esav fathered two girls through his mother, and fathered a son through one of the girls.
Yevamos 10a (Rav Ashi): Our Mishnah does not list Imo (one's mother) among Arayos that exempt from Yibum and Chalitzah, for the Tana is R. Yehudah, who forbids marrying a woman that one's father raped (so no brother may marry her).
Question (Ravina): Also according to R. Yehudah, the case occurs if a man raped his son's wife!
Answer (Rav Kahana): The Tana discusses only cases in which the brothers did not result from Isur.
Chagigah 14b - Question: If a virgin became pregnant, is she permitted to a Kohen Gadol?
i. Are we concerned lest someone skilled like Shmuel had Bi'ah with her without breaking the Besulim?
Answer (Ben Zoma): This is not common. We are concerned (i.e. assume) that she became pregnant through (sitting on semen in) a bathhouse.
Tosfos (Yevamos 12b DH d'Afilu): If a girl could do Mi'un after bringing Simanim of adulthood, we could find the case of a mother-in-law doing Mi'un, e.g. she became pregnant through a bathhouse, or they had Bi'ah explicitly not for the sake of Kidushin.
Rashi (Chagigah 14b DH Besulah): The question was whether a Kohen Gadol may marry a girl who says that she became pregnant through a bathhouse, and he will find that she has Besulim. Alternatively, if he married her, found Besulim, and afterwards found that she is pregnant, may he keep her?
Maharil (Minhagim, Likutim 3): Ben Sira was born from Yirmeyah's semen. His daughter bathed after him in a bathhouse, absorbed his semen and gave birth. All testify to her great piety that she was not Mezanah, rather, she became pregnant through a bathhouse. They called him Ben Zera. He was ashamed, and changed his name to Ben Sira, which has the same Gematriya as Yirmeyah.
Teshuvas Rashba (27): The Toras Kohanim says that had the Torah taught only that a Kohen may be Metamei (for burial of) his mother, one might have thought this is because she is Vadai, but he may not be Metamei for his father, for we 'know' that it is his father only due to Chazakah (most Bi'os of a married woman are with her husband). This is difficult. If he is not really his father, the son is a Mamzer, so even if he is a Kohen, he may become Tamei!
Poskim and Acharonim
Bach and Taz (YD 195:5 DH Matzasi and 7, citing Hagahos Semak): R. Peretz says that a Nidah may lie on her husband's sheets, but not on sheets of another man, lest she become pregnant from his semen. We are not concerned lest she become pregnant through her husband's seed, and the son will be a Ben Nidah, for there was no Bi'as Isur. The child is totally Kosher, even if she became pregnant through another man's seed, like Ben Sira. However, we are concerned for Havchanah (to know who is the father), lest a man marry his paternal sister.
Chelkas Mechokek (EH 1:8): It is not clear whether or not a man fulfills Peru u'Rvu with children born from his semen in a bathhouse. (It seems that he is Yotzei, for) Avodas ha'Kodesh calls Ben Sira the son of Yirmeyahu even though he was born in this way.
Chacham Tzvi (Hosafos 9, 17): It is clear to me that he fulfilled Peru u'Rvu. This settles the Beraisa that the Rashba was asked about.
Minchas Yitzchak (4:5): The Chacham Tzvi says that this settles the Beraisa. I.e. he is his father's son, and he is Kosher, so if not for the verse, the son could not be Metamei. Also Sho'el u'Meshiv (3 3:132) says so. The Rashba did not answer like this, for it is very remote for a woman to became pregnant through a bathhouse, and the father is a Kohen, so even without a verse we are not concerned for this.
Birkei Yosef (EH 1:14): Sha'arei Yeshu'ah (EH 1:2) brought a proof that he fulfilled Peru u'Rvu from what the Beis Yosef brought that a Cheresh or lunatic who fathered a child fulfilled the Mitzvah, for there is Sheves (populating the world). The same applies here. I do not understand his proof. The Rema taught that one fulfills Peru u'Rvu through a son who is a Cheresh, lunatic, child or Mamzer, and on this the Chelkas Mechokek said that he is unsure whether one fulfills the Mitzvah through a child born through a bathhouse! Sha'arei Yeshu'ah brought another proof from R. Peretz. (The concern for marrying a sibling shows that the child is totally attributed to his father.) The Beis Shmuel (Mahadura Basra), Minchas Yakov and Mishneh l'Melech already brought this. This is not an absolute proof. Perhaps he means '(even here) we are concerned for Havchanah, which was decreed lest people marry siblings' (even though the concern for siblings applies only to conception through Bi'ah). Also, perhaps he did not fulfill the Mitzvah, because he had no intent at all. He merely emitted semen l'Vatalah! In a Teshuvah, R. Shlomo of London (a Rishon) forbids a Nidah to bathe in the bathhouse in which her husband bathed, lest she become pregnant and the child will be a Ben Nidah. This is unlike R. Peretz. We can distinguish these a little.
Minchas Yitzchak (5:4): The only distinction I can find is that there is more chance of becoming pregnant in a bathhouse than through a sheet. Through bathing, the semen enters her. All the more so we must be stringent when there is concerned for Mamzerus through another man's semen.
Beis Shmuel (10): The SMaK says that a woman should avoid sitting on another man's semen, lest a man marry his paternal sister. This implies that the child is considered his father's son, so he counts towards Peru u'Rvu.
Birkei Yosef (13:10): If a woman became pregnant through a bathhouse, does the Isur to marry Me'uberes u'Menekes Chavero (a woman pregnant with or nursing the baby of another man) apply? Tosfos in Yevamos, Kesuvos (6b DH Rov) and Nidah (64b DH She'ani) hold like Rashi's first Perush in Chagigah, that the question was whether a Kohen Gadol may marry a girl who became pregnant through a bathhouse. Ben Zoma answered that we assume that she became pregnant through a bathhouse, and he may marry her. He marries her now, but not after she gave birth (and finished nursing), for she lost her Besulim through birth. However, perhaps Ben Zoma permitted only before Chachamim decreed about Me'uberes u'Menekes Chavero. This would explain why the Rambam omitted this law. He holds like Rashi's first Perush, so nowadays in any case it is forbidden due to Me'uberes Chavero. However, it seems that Chachamim decreed at once about Me'uberes u'Menekes Chavero and Havchanah (the Isur to marry a woman who had Bi'ah with another man within three months, to avoid Sefekos about who is the father). Also, even though it pertains to danger, perhaps Chachamim did not decree about a matter that is not common. Also, perhaps the question was about Torah law.
Note: Perhaps Chachamim forbade Me'uberes u'Menekes Chavero due to the concern lest she become pregnant from her new husband, wean the previous baby prematurely, and her new husband will not want to pay for milk and eggs for another's baby (Yevamos 42a). Perhaps this concern is only when the father is known, for the new husband feels that the father (or his heirs) should pay. (If a girl was married mid'Rabanan and died, we assume that her father's house would refuse to bury her. They say 'since her husband inherits her, he should bury her!'). Perhaps this does not apply when the father is not known, and especially when her new husband is the richest Kohen in the world (and should be the greatest Chacham), there is no concern lest he let the baby die due to stinginess!
Yabi'a Omer (2 EH 1): In Chagigah, we assume that a woman became pregnant through a bathhouse. Semak brings a Medrash that Yirmeyah fathered Ben Sira through his daughter in this way. Tashbatz (3:263) cites this from Sefer Ben Sira, and was unsure whether we may rely on it. Semak and Maharil (85a) cited it from a Medrash, and proved from it that the child is Kosher. Tzemach David said that it is a mere exaggeration. It is not found in the Gemara or Medrashim.
Piskei Uziel (She'alos ha'Zman 53): The Maharil (CM 1:8) says that Ben Sira was Yirmeyah's son. His semen was in a bathhouse, and entered a woman. The Beis Shmuel supported this from Semak, who says that a woman must be careful not to sit on a sheet that a man sat on, lest she become pregnant, and her child will marry a paternal sibling. Also the Mishneh l'Melech holds that the child is considered the son of the father. There is no proof for this. We hold that we do not learn (Halachos) from Agados (Yerushalmi Pe'ah 2:4). Also, another Agada says that Yirmeyah's daughter became pregnant from her father in this way. Perhaps this is why Ben Sira was called Yirmeyah's son, for grandchildren are like children. The Taz says that perhaps we are only stringent to consider him to be his son, but not to be lenient. The Birkei Yosef (EH 1:13) says similarly.
Note: I did not find the Maharil he cites. The Maharil I cited above says that Yirmeyah's daughter became pregnant from her father.
Mishneh l'Melech (Hilchos Ishus 15:4): The Halachah does not follow Ben Zoma. We hold that a woman can become pregnant only through a full Bi'ah, and surely not through semen in a bathhouse. If a woman gave birth more than 12 months after her husband went away, we say that the child is a Vadai Mamzer, unless we are concerned lest he came through Hash-m's name.
Turei Even (Chagigah 15a DH Chaishinan): If it is possible to become pregnant through semen in a bathhouse, why did we say that the case of Imo does not arise through Heter? We can say that Ploni's mother became pregnant from the semen of Ploni's maternal brother!
Divrei Malkiel (4:107): In Kerisus, we say that the case (of a sister who is also his father's sister and his mother's sister) occurs through Resha'im. It is found even b'Heter, through a bathhouse! Rather, this shows that the child is totally attributed to his father only if he was conceived through Bi'ah. This answers the Turei Even's question. This is difficult for the Tashbatz (2:263), Bach and Taz, who attribute the child to his father in every respect.
Aruch l'Ner (10a DH uv'Hachi): We do not teach cases that arise only through Isur because they are uncommon. Also becoming pregnant through a bathhouse is uncommon! In Chagigah, we said that it is rare that a man can have Bi'ah (that impregnates) without breaking the Besulim. Rather, we assume that she became pregnant through a bathhouse. Only for a pregnant Besulah, conception through a bathhouse is more common than someone like Shmuel. Normally, we are not concerned for conception through a bathhouse, for it is uncommon.