Gemara

(R. Yochanan): "(Al Tischaten Bam... do not marry your daughter to a Nochri.) For he will veer your son away from Me" - your son from a Yisraelis is called your son, but your son from a Nochris is called her son.

(Ravina): We learn that your daughter's son from a Nochri is called your son.

Version #1 - Suggestion: Ravina holds that a slave or Nochri's child from a Yisraelis has proper lineage.

Version #2 - Suggestion: Ravina holds that a slave or Nochri's child from a Yisraelis is a Mamzer. (Since the child is a Yisrael, Mamzerus applies!) (end of Version #2)

Rejection: The child is not a Mamzer, nor is his lineage proper. He is called a disqualified Yisrael.

44b (Rabah bar bar Chanah): All agree that a slave or Nochri's child from a Bas Yisrael is a Mamzer.

Question: Who agrees?

Answer #1: R. Shimon ha'Temani agrees. Even though he says that Mamzerim do not come from Chayavei Lavin, that refers to Chayavei Lavin who can be Mekudeshes to others. A slave or Nochri cannot be Mekudeshes to others, so they are like Chayavei Kerisus.

Objection (Beraisa): A Bas Yisrael's child from a Nochri or slave is a Mamzer;

R. Shimon ben Yehudah says, a Mamzer comes only from Ervah of Chayavei Kerisus.

Answer #2 (Rav Yosef): Rather, Rebbi agrees. Rebbi said 'Only R. Akiva says this (that if Yibum was done after Chalitzah, no Get is needed), since R. Akiva considers a Chalutzah like Ervah', but Rebbi himself argues. Rebbi admits (to R. Akiva) that the child of a Bas Yisrael from a Nochri or slave is a Mamzer.

(R. Ami): R. Yochanan, R. Elazar and R. Chanina all say that the child is a Mamzer.

Objection (Rav Yosef): Does a long list of names prove that this is the Halachah?! Rav and Shmuel in Bavel, and R. Yehoshua ben Levi and Bar Kapara (alternatively - elders of the south) in Eretz Yisrael say that he is Kosher!

(Rav Yosef): The Halachah is, he is a Mamzer, but this is because Rebbi says so.

(Rav Dimi): Rabeinu (Rebbi) says that the child is a Mamzer. R. Yehoshua ben Levi says that the child is Pagum (blemished).

He cannot mean that the child is Pagum to Yisraelim, for R. Yehoshua ben Levi said above that the child is Kosher! Rather, he is Pagum to Kehunah.

All the Amora'im who say that the child is Kosher agree that he is blemished to Kehunah, from a Kal v'Chomer. The Isur of a widow to a Kohen Gadol is not Shavah b'Chol, yet the child is Pagum (a Chalal). The Isur of a Nochri or slave to a Bas Yisrael is Shavah b'Chol, all the more so the child is Pagum!

Question: We cannot learn from there, since the widow herself becomes defiled!

Answer: Here also, R. Yochanan taught that a slave or Nochri defiles a Yisraelis!

Question (Abaye to Rav Yosef): Why do you rely on Rav Dimi, and not on Ravin? Ravin taught that R. Noson and Rebbi permit the child.

Also Rav permits the child (but he refused to marry his daughter to such a man). Rav Masnah and Rav Yehudah also permitted the child.

When such a case came to Rav Yehudah, he would counsel the son to go to a place where he is unknown and marry there. Alternatively, he should marry a girl of similar parentage. (Even though he is permitted, children of Yisraelim were reluctant to marry him.)

45b: People were calling a certain man the son of a Nochri.

R. Yehoshua ben Levi: Didn't his father immerse after seminal emissions?! (This suffices for conversion.)

47a: A man told R. Yehudah that he converted without Beis Din. He did not have witnesses to prove this.

R. Yehudah: You are believed to disqualify yourself, but not to disqualify your children.

Rishonim

Rif and Rosh (15a and 4:30): The Gemara concludes that the child of a slave or Nochri from a Yisraelis, whether she is single or married, is Kosher. Some say that the child is Pagum to Kehunah; earlier, the Gemara said that all Amora'im agree to this. The Kal va'Chomer from a widow to a Kohen Gadol is valid, for a slave or Nochri disqualifies a Bas Yisrael just like a Kohen Gadol disqualifies a widow. Some Rabanan say that the child is Kosher and do not mention Pegam. We are in doubt, for when the Gemara says the final Halachah it does not mention Pagum.

Ra'avad (brought in Sefer ha'Zechus): The Gemara (23a) suggested that Ravina holds that the child is (fully) Kosher. It concluded that he is Pagum. Our Gemara says that all the Amora'im who are Machshir agree that the child is Pasul to Kehunah, so the final Pesak did not need to mention Pegam. Shmuel taught that if we do not know which of 10 Kohanim is David's father, David is a Shtuki, i.e. silenced from Kehunah. Not having lineage from his father disqualifies to Kehunah. The same applies here!

Rebuttal (Ramban, in Sefer ha'Zechus): The Gemara was Docheh and said that perhaps Ravina holds that the child is Pagum. Shmuel taught that a Kohen without lineage from his father is silenced from Avodah, but Isurei Kehunah apply to him and he may eat Terumah. He is not a Chalal! Shmuel's law is a stringency mid'Rabanan. It is not relevant to the question of whether or not the Kal va'Chomer makes a Chalal mid'Oraisa. The Gemara left this question a Safek. Rav Papa (Bechoros 47a) exempts the firstborn of a Bas Levi from a Nochri from Pidyon, according to the opinion that a Nochri's child from a Yisraelis is Mezuham, and even according to the opinion that he is not Mezuham. Mezuham means Pasul to Kehunah.

Note: This is no proof according to Tosfos (Bechoros 47a DH v'Lo), who explains that Mezuham refers to a Mamzer. The Ramban (b'Sof Devarav) cites a Yerushalmi which supports himself.

Ramban (ibid.): Rav did not explain why he refused to marry his daughter to such a man. Surely, he permits even to Kehunah! (The Yerushalmi says otherwise because it holds like Rav Yosef.) Rav Yehudah counseled such people to go to where they are not known. He would not do so if they were Pesulim to Kehunah!

Rashba (45a DH Omar Lei): A Nochri or slave's daughter from a Bas Yisrael is forbidden to Kehunah, but his son and all his son's children are Kesherim.

Chasam Sofer (EH 2:125 DH v'Atah) Rav Yehudah relied on the man to publicize after marrying that his daughters are Pesulim to Kehunah. Such a ruse is permitted in order to fulfill Peru u'Rvu.

Rosh (ibid.): I see no room for doubt. Since there is no challenge to the Kal va'Chomer, all agree to it. The final Pesak did not need to mention Pegam.

Rebuttal #1 (Beis Shmuel EH 4:4): R. Yochanan (Yevamos 77a) says that the daughter of an Amoni from a Yisraelis is Kesherah to Kehunah, even though the Bi'ah is forbidden and disqualifies her. This refutes the Kal va'Chomer. This is why the Rambam is Machshir the child (totally).

Rebuttal #2 (Yam Shel Shlomo 4:38): There is a Pircha against the Kal va'Chomer (that the child of Chayavei Lavin is Pagum, from a widow to a Kohen Gadol). The Gemara rejected this, because the widow herself becomes a Chalalah, and Chalalah is only from Isurei Kehunah. An Isur (e.g. of Kehunah) that is not Shavah b'Chol is more stringent. Also, we seek only to disqualify the daughter to Kehunah, not to make the son a Chalal! The Hava Amina was only to forbid the daughter to marry a Kohen, but not to disqualify her children from a Kohen. The Gemara concludes that the child is Kosher and retracts from this Hava Amina. We conclude that the child is not a Mamzer even if the mother was married. This shows that the Torah totally ignores the father's seed. If his seed does not cause Mamzerus, all the more so it does not cause Pegam! However, I do not say to be lenient l'Chatchilah, since the Rif was unsure. But if she married a Kohen she may remain, and her child is Kosher. The Ramban says that she may remain, but her child is Safek. This is unreasonable. If the child is a Safek Chalal, it is a Safek Isur mid'Oraisa for the parents to remain married!

Korban Nesan'el (300): The episode in which people called a certain man 'a Ben Nochri' suggests that such a child is Pasul. However, perhaps they were saying that he does not inherit his father, or that his mother was disqualified.

Note: The Taz (YD 268:10) says that even if the child is Kosher, people of good lineage would not marry him. The Shach (268 11) remarks that Pidyon ha'Ben does not apply if the father was a Nochri.

Rambam (Hilchos Isurei Bi'ah 15:3): The child of a slave or Nochri from a Yisraelis is Kosher, whether she is single or married, whether the Bi'ah was b'Ones or b'Ratzon. The child of a slave or Nochri from a Mamzeres is a Mamzer.

Beis Yosef (EH 4 DH Goy): His child from a Mamzeres is a Mamzer because whenever one of the parents cannot have Kidushin, the child is like the mother.

Rambam (13:8): If a man says that he converted without Beis Din, he is not believed to disqualify children from his Yisraelis wife.

Magid Mishnah: In any case his child is Kosher! Rather, if he says 'My wife and I converted without Beis Din...' he is not believed about his children.

Rebuttal (Mishneh l'Melech 15:3): The Rambam says that his wife is a (Vadai) Yisraelis! Rather, the Rambam disqualifies a Nochri's child from a Yisraelis to Kehunah.

Poskim

Shulchan Aruch (EH 4:19): A slave or Nochri's child from a Yisraelis, whether she is single or married, is Kosher (not a Mamzer), but he is Pagum to Kehunah.

Beis Shmuel (4): The Bach says that this is more stringent than her child from an Amoni, for an Amoni can have Kidushin. This is wrong. If it was true, the Gemara would have made this Kal va'Chomer!

Be'er ha'Golah (10, on 4:5): When we say that the child is Pagum to Kehunah, if it is a daughter she is forbidden to a Kohen. If she married a Kohen he must divorce her; her son from him is a Chalal.

Gra (7:54): The Rambam holds that the child is totally Kosher. In conclusion, we do not hold like the Kal va'Chomer to make the child Pagum. However, the Sugyos are explained much better according to the Rosh, who says that the child is Pagum.