Gemara

(Mishnah): If a Kohen raped or enticed a Bas Yisrael, she may not eat Terumah. If she became pregnant, she may not eat. If she gave birth, she may eat.

Version #1 (Rav): If a man had Bi'ah with his Arusah (who is Mekudeshes to him) in her father's house, the child is a Mamzer;

(Shmuel): The child is a Shtuki (i.e. of uncertain lineage).

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!

Even though l'Chatchilah Shmuel requires a Kosher majority, b'Di'eved we rely on R. Gamliel. Also here 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: This means that we ask his mother. If she says that his father is Kosher, she is believed.

Rishonim

Rif and Rosh (Yevamos 23b and 7:7): Some say that if (we know that) an Arus had Bi'ah with his Arusah, Rav and Shmuel agree that we attribute the child to him. They argue about an Arusah who is pregnant. The Halachah follows Shmuel, for he holds like Aba Sha'ul, who calls a Shetuki a Beduki. We ask his mother; and she is believed to say that he is from the Arus and is Kosher, even though most men (everyone else) are forbidden to her.

(Nimukei Yosef DH Gemara): When she is believed to be Machshir, this is not only for lineage (to permit the child to marry into Yisrael). It is even for the child to inherit the Arus. Since he is her husband and she claims Vadai, we say that the Arus is Vadai the father, for she is permitted to him. However, if she is single, even though we believe her for lineage like Shmuel and Aba Sha'ul, if he is silent, away or does not remember, we do not believe her for inheritance. Even if he had Bi'ah with her and she was suspected of (more Bi'os with) him, just like she was Mezanah with him, perhaps she was Mezanah with other Kosher men. Her Chezkas Kashrus says only that she had Bi'ah with Kosher men. This applies even if she was a concubine.

Rambam (Hilchos Terumos 8:14): If a woman was raped or enticed by a Kohen and gave birth, she eats Terumah due to her son. Even though it is a Safek (if it is his son) since there was no Kidushin, perhaps it is another man's son, the Chazakah is that it is from the man who (we know) had Bi'ah with her. This is if there are no rumors that she had Bi'ah with anyone else, and everyone talks about her and the Kohen.

Ri Korkus: It seems that the Rambam requires that the Kohen Vadai raped or enticed her, and also there are rumors that he had other Bi'os with her.

Rambam (Hilchos Yibum 3:4): If Reuven was Mezanah with Leah, whether she is single or married, and he claims that she is pregnant from him, even if she agrees, even though the child inherits Reuven, it is a Safek regarding Yibum, and she does Chalitzah. Perhaps she was Mezanah with others. How does he know that it is his child?!

Magid Mishneh: This law is not explained in the Gemara. It is the Rambam's opinion that Reuven is believed to exempt his wife from Yibum or Chalitzah by saying that David is his son only if David is from Reuven's wife. Clearly, when David is from a married woman we attribute him to her husband. However, when she is single we should distinguish whether or not she is suspected of Bi'ah with others! We learned that if a Kohen had Bi'ah with a Bas Yisrael and she gave birth, she eats Terumah. In one version this is only if she is not suspected of Bi'ah with others. In the second version this is even if she is suspected of Bi'ah with others. Regarding Terumah, which is also an Isur mid'Oraisa, the Rambam attributes the child to the (known) Bo'el. Why is Yibum different? The Rashba (Teshuvah 610) says that if we saw Reuven have Bi'ah with her or he admits, we attribute the child to him as long as we do not know that someone else had Bi'ah with her. It seems that he rules like the second version. This is a Safek mid'Oraisa. We should be stringent! In any case when she is not suspected of others I do not know why Yibum should be different than Terumah. Even if she is suspected, perhaps the Halachah follows the latter version! The Gemara in Kesuvos (13b) is like the latter version.

Lechem Mishneh: The Rashba requires that he had Bi'ah with her even though the Gemara says only 'Daima' (she is suspected). He holds like Tosfos (70a DH Aval), who says that regarding the presumed father, 'Daima' means that we know that he had Bi'ah with her.

Question (Ran Kesuvos 4b DH ha'Hu): Why does the Gemara (Kesuvos 14a) call the case like b'Di'eved? Regarding the child it is l'Chatchilah. Regarding her it is truly b'Di'eved!

Answer (Ran): If we will call the child a Safek Mamzer, the Arus will worry (that she is forbidden to him) and will separate from her. Therefore it is called b'Di'eved even regarding the child. Alternatively, it is b'Di'eved regarding the child, for if we disqualify him he cannot marry a Yisrael. It is l'Chatchilah for her, for she could marry a Yisrael.

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.

Rema: He inherits the Arus, even if the Arus says that he does not remember, as long as he does not contradict her.

Chelkas Mechokek (28): Even though the child is a Safek, his brother cannot say 'you must bring a proof that you are my brother'. This is because his mother was Vadai, no one contradicted her, and she had a Chezkas Kashrus. If he would not inherit his father, people would question his Kashrus. This is similar to the Ran, who says that if we will call the child a Safek Mamzer, the Arus will worry and separate from his wife.

Rebuttal (Beis Meir DH Hagah): We hold like the other answer in the Ran, that regarding the child it is considered b'Di'eved for if we disqualify him he cannot marry a Yisrael.

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.

Rebuttal (Chelkas Mechokek 28): The child may marry a Yisraelis and inherits with his brothers. If he is a Kohen the child may do Avodah, and she is permitted to her Arus. Therefore, (in every case) also regarding his mother it is like b'Di'eved and she is permitted to Kehunah after her husband dies.

Beis Shmuel (44): We say that the Arus is the father also to exempt his mother from Chalitzah. If we are Machshir the child only because it is b'Di'eved, why does he inherit his father? It seems that we cannot distinguish. If he is his son (for lineage), he is his son in every respect.

Rebuttal (Beis Meir DH Hagah): I am troubled even by the Nimukei Yosef, who says that the child inherits the father. What is his source? This is unlike the Terumas ha'Deshen (352). Also the Rosh (Yevamos 7:7) connotes that he is a Shetuki, i.e. silenced him from inheriting his father. What is the source to exempt from Chalitzah? Shmuel said not to be lenient l'Chatchilah without two majorities. Exempting without Chalitzah is the ultimate l'Chatchilah!