Gemara

73a (Mishnah): Mid'Oraisa, a Shetuki (one who recognizes his mother but not his father) is Kosher. It says "Lo Yavo Mamzer (bi'Khal Hash-m)." A definite Mamzer is forbidden, a Safek Mamzer is permitted.

73b (Beraisa): A midwife is believed to say 'this baby is the Kohen, this is the Levi, this is the Nasin, this is the Mamzer' (e.g. they were born in one house).

This is only if no one protests. If someone protests, she is not believed.

74a (Mishnah): Aba Sha'ul would call a Shetuki 'Beduki' (checked).

This means that his mother is believed to say that his father is Kosher. R. Gamliel taught so in another Mishnah. Aba Sha'ul teaches that she is believed even when most men in the city are forbidden to her.

Kesuvos 13a (Mishnah - R. Gamliel and R. Eliezer): If a single girl is pregnant and she claims that the father is Ploni, who is Kosher, she is believed;

R. Yehoshua says, we say that she is pregnant from a Nasin or Mamzer, unless she brings proof.

13b (R. Yochanan): R. Gamliel, who permits her (to Kehunah), also permits her daughter. R. Yehoshua, who forbids her, also forbids her daughter.

(R. Elazar): (Even) R. Gamliel forbids her daughter.

Shmuel (to Rav Yehudah): Even though the Halachah follows R. Gamliel, do not rule leniently unless most people in the area do not disqualify her.

(Rav Yosef): The Halachah follows R. Gamliel b'Di'eved. L'Chatchilah, we require a Kosher majority. Regarding a pregnant Arusah, most are forbidden to her, but it is like b'Di'eved.

14b (Mishnah - R. Yosi): A case occurred in which a girl went to the spring and was raped. R. Yochanan ben Nuri ruled that if most men of the city have proper lineage, she may marry a Kohen.

(Ze'iri): We require that the majority of the city and of the passers-by is Kosher. We do not rely on a majority of passers-by, lest we rely on a majority of the city.

Question: A majority of city dwellers is enough, if he separated and went to her!

Answer: Sometimes, she goes to him, and he is Kavu'a. Any such Safek is considered an even Safek. It is a stringency of lineage to require two majorities.

Rishonim

Rif and Rosh (Kesuvos 4b and 1:29,30): The Halachah follows R. Gamliel b'Di'eved. L'Chatchilah, we require a Kosher majority. Regarding an Arus and Arusah, it is like b'Di'eved. We require that the majority of the city and the majority of passers-by is Kosher.

Ran (Kesuvos 4b DH ha'Hu): Why is it b'Di'eved for the child? If we disqualify the child, perhaps the 'father' will separate from the mother, therefore it is called b'Di'eved even for the child. Alternatively, if we disqualify the child, he may not marry anyone.

Rosh (30): We require two majorities when she does not claim anything.

Rambam (Hilchos Isurei Bi'ah 15:11): If a single woman became pregnant through Zenus, and she said that the baby is from a Kosher Yisrael, she is believed. This is even if most of the city are Pesulim.

Question (Ramach): Why did the Rambam omit that she is believed only b'Di'eved, if the child already married, but l'Chatchilah he may not marry unless most of the city were Kesherim!

Ran (30b DH Masnisin): The Rambam holds that she is believed to be Machshir the child, because mid'Oraisa a Shtuki is Kosher. The Torah believed a father about his firstborn, even to disqualify him, but it does not believe the mother. A midwife is believed to identify the Mamzer when women gave birth together and we know that there was a Mamzer.

Ran (cited in Mishneh l'Melech 11): Chachamim enacted to believe a midwife, because no one else knows. The child had no Chezkas Kashrus. One witness is believed like two about something prone to become known to others. This does not apply here, so one witness can contradict her.

Mishneh l'Melech (11): Perhaps mi'Dina (letter of the law), a mother and midwife are not believed. Perhaps the Ran says that it is only an enactment to believe a midwife when she calls the child a Mamzer and permits him to a Mamzeres, but she is believed letter of the law to be Machshir, for the Torah is Machshir a Shtuki. However, from what he says about contradicting her, it seems that even to be Machshir she is believed only due to an enactment. It seems that the same applies to the mother. A mother is believed to be Machshir. A midwife is believed even to disqualify and permit to a Mamzeres, for there was known to be a Mamzeres in the house.

Rambam (18:14): A single woman is believed to say that she had Bi'ah with a Kosher when the Bi'ah was in a place where many pass and most of them are Kesherim. Chachamim made a stringency in lineage and require two majorities.

Magid Mishneh: The Rambam holds that l'Chatchilah, she must claim and there are must be two majorities. B'Di'eved, she need not be divorced if she claims or there are two majorities. The Ramban and Rashba say that when she claims, one majority suffices l'Chatchilah, like Rav Yosef said.

Kesef Mishneh: The Rambam requires two majorities even when she claims. He holds that the girl who was raped claimed that the man was Kosher.

Terumas ha'Deshen (2:37): Surely, we do not believe a woman living with her husband to disqualify her child. R. Yehudah taught that a woman is believed only for seven days, and afterwards only the father is believed. Even if this is only regarding Bechorah, Tosfos (Yevamos 47a DH Kach) says that we know that a father is believed to say that his child is a Mamzer only because he is believed about Bechorah! Regarding a Shtuki we check his mother, because we do not know who is the father.

Poskim

Shulchan Aruch (EH 4:26): If a single girl gave birth and she says that the child is from Ploni and he is Kosher, the child is Kosher.

Beis Shmuel (39): Why is she believed when most men are Pesulim? The child is Kosher mid'Oraisa only when most are Kesherim! Rather, since the Torah is Machshir a Safek Mamzer, even if most are Pesulim, the child may marry Kesherim if she went to the Bo'el (a Kavu'a Isur, which is like an even Safek). Indeed, if we know that the Bo'el came to her, she is not believed to be Machshir the child. The Tur (Siman 6) says that Stam, we assume that the Bo'el went to her, to be lenient. We do not say so to be stringent.

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