Gemara

Kesuvos 13b: An Arus and his Arusah came in front of Rav Yosef. Both said that she is pregnant from him.

(Rav Yosef): There is no reason to be concerned. Firstly, the husband admits. Also, Shmuel taught that the Halachah follows R. Gamliel!

Question (Abaye): Your connote that the second reason suffices by itself. Does R. Gamliel believe her when the husband does not admit?

Shmuel (to Rav Yehudah): Even though the Halachah follows R. Gamliel (regarding a single girl), be lenient only if people in the area do not disqualify her. Here, everyone except for her husband disqualifies her!

Answer (Rav Yosef): If the Halachah follows R. Gamliel, why did Shmuel require a Kosher majority? You must say that the Halachah follows R. Gamliel b'Di'eved. L'Chatchilah, we require a Kosher majority. Also our case is like b'Di'eved!

Kidushin 69a (Mishnah): Ten different lineages came up with Ezra from Bavel: ... Shtuki (one who knows his mother but not his father)...

Aba Sha'ul calls a Shtuki 'Beduki' (one who was checked).

74a: We check by asking his mother. She is believed to say that his father is Kosher.

(Rava): The Halachah follows Aba Sha'ul.

(Beraisa - R. Yehudah): "Yakir" - the father will make known whom is his firstborn;

This teaches that a man is believed to say which son is his firstborn. Similarly, he is believed to say that a certain son was born from a divorcee or Chalutzah (and is a Chalal);

Chachamim say, he is not believed to say that his son is a Chalal.

Rishonim

Rif and Rosh (Kidushin 31a and 4:12): Aba Sha'ul calls a Shtuki 'Beduki'. We ask his mother. If she says that his father is Kosher, she is believed. The Halachah follows Aba Sha'ul.

(Nimukei Yosef, Yevamos 23b DH Gemara): When both are quiet the child is a Safek Mamzer. The Ritva says that when she is Machshir and he says the child is Vadai Pasul for he never had Bi'ah with her, he cannot disqualify her, but the child is a Mamzer, since each of them is Vadai. The Torah believes a father to disqualify even a son born with Chezkas Kashrus, all the more so here. When he is Machshir and she says that the child is Pasul, some say that he is a Safek Mamzer, since each of them is Vadai. Some say that if he says that she never left him, he is believed. The Torah believes a father to disqualify his son, all the more so to be Machshir, for his mother has Chezkas Kashrus. She is not believed to say that she had Bi'ah with another man b'Shogeg, all the more so b'Mezid. If she is Machshir and he is silent, the child is Kosher. If she says that the child is a Mamzer and he is silent, the child is a Safek Mamzer. The Torah (totally) believes her to be Machshir her son, because her Chezkas Kashrus supports this, but not to disqualify him

Ri'az (in Shiltei ha'Giborim Kidushin 32a): The Torah believes a father only to say that his son is a Mamzer, i.e. he realized now that his 'wife' is Ervah to him. He is not believed when he says that the child is not his son.

Rambam (Hilchos Isurei Bi'ah 15:17): If an Arusah became pregnant in her father's house and the Arus denies having Bi'ah with her, the Chazakah is that the child is a Mamzer. Even if the Chazakah would be that it is his son, one who says that his son is a Mamzer is believed.

(Nimukei Yosef, ibid): The Torah believes a father about his children - "Es ha'Bechor...Yakir"

Magid Mishneh: The Gemara in Kesuvos (13b) is Machshir the child of an Arusah who became pregnant even if her Arus does not admit, i.e. he is not here. If he contradicts her, the child is a Mamzer.

Mishneh l'Melech (Hilchos Ishus 15:4): All Poskim agree that if an Arusah is pregnant and the Arus denies having Bi'ah with her, the child is a Mamzer Vadai and may marry a Mamzeres. This shows that we are not concerned lest a woman became pregnant without Bi'ah through semen in a bathhouse.

Rosh (Kesuvos 1:29): We do not Machshir if the Arus does not agree. R. Gamliel is not Machshir when both of them are Vadai (16a). Also, the Halachah follows R. Yehudah, who says that a father is believed to say that one of his children is a Mamzer, all the more so regarding this fetus. Rather, even if we do not know that he agrees, e.g. he died or went overseas, the child is Kosher, for it is b'Di'eved for the child. If we would be stringent, he could not marry a Kesherah nor a Mamzeres.

Teshuvas Rosh (82:1): Normally, a man is believed to say 'Ploni is my son' regarding inheritance (of property that he has now) and to exempt his wife from Yibum. He has a Migo (he could give him a gift or divorce his wife). This does not apply when Ploni was born through Zenus with a single girl, for he is not sure that he is the father. A person is not believed with a Migo when we suspect that he relies on estimation (Yevamos 114b). Also, she probably had Zenus also with others. When we know that something is probably false, a Migo cannot overturn it. It is considered a Migo against witnesses (Bava Metzi'a 81b).

Poskim

Shulchan Aruch (4:27): If an Arusah became pregnant in her father's house and she says that she is pregnant from the Arus and the Arus agrees or is not here, the child is Kosher.

Taz (15): If the Arus died, l'Chatchilah she or the child may not marry a Kohen unless the Arus agreed that the child is from him.

Bach (DH v'Im): A Chacham explains the Tur to say that when the Arus agrees or is not here the child is Kosher, and all the more so she is Kosher to Kehunah (after the Arus dies), for it is b'Di'eved for her. It seems that she is permitted to a Kohen l'Chatchilah only if the Arus agreed. If he was not here l'Chatchilah she is forbidden to Kehunah because most men disqualify her. The Gemara in Kesuvos connotes that the Pesak like R. Gamliel permits (b'Di'eved) as much as the Arus' admission does. This is only regarding the child and her Heter to the Arus, but not to permit her to a Kohen.

Shulchan Aruch (ibid.): If the Arus says that she is not pregnant from him, the child is a Vadai Mamzer. If she is not here or does not remember, the child is a Safek Mamzer.

Bach (DH Arusah): This is when we do not know that she had Bi'ah with the Arus, and she is not suspected of Bi'ah with other men, or when we know that she had Bi'ah with the Arus and she is suspected of others. If we know that she had Bi'ah with the Arus and she is not suspected of others, the child is Kosher even if we did not ask her. In Yevamos it seems that if she is suspected of the Arus but not of others, we need not ask her.

Prishah (38): If the Arus says that it is not his child, surely the child is a Vadai Mamzer. Alternatively, perhaps he is a Safek Mamzer, for perhaps she became pregnant from a Nochri.

Beis Shmuel (44): Perhaps the Shulchan Aruch agrees, and discusses when she is here and does not say that she became pregnant from a Nochri. However, it seems that (even if she is away) we are not concerned lest she became pregnant from a Nochri.

Tur (CM 279): The RaMaH says that a man is believed to say that that his 'wife' is Ervah to him and so his son is a Mamzer, or that his wife's son is not from him, so it follows that her son is a Mamzer. If we know that his wife is a Bas Yisrael, he is not believed to say that her son is a slave.

Gra (279:7): One cannot disqualify his child if the child has children. Likewise, one cannot say 'my son is a slave', for this says that his mother is a Shifchah.

Rejection (Chazon Ish EH 59:27): This is wrong. Since the mother is not disqualified due to the son, just her Pesul is inferred from the son's Pesul, we could believe him only about the son! Also, even if she died he is not believed to say that the child is a slave! Rather, this resembles the Ri'az, but it is slightly different. He is believed only about conception of the child.