Gemara

(Rav Ada bar Ahavah): The firstborn of a Leviyah is exempt from Pidyon.

Question: Who is the father?

It cannot be a Kohen or Levi. Their children are exempt, even from a Yisraelis!

Answer #1: He is a Yisrael.

Rejection: "L'Mishpechosam l'Veis Avosam" teaches that lineage follows the father!

Answer #2 (Rav Papa): The father is a Nochri.

This is not only according to the opinion that the child (of a Yisraelis from a Nochri) is not tainted. Rather, it is even like the opinion that the child is tainted. He is a Pasul Levi.

Defense of Answer #1 (Mar brei d'Rav Yosef): Really, the father is a Yisrael;

Pidyon does not follow the usual law of lineage, for it depends on Peter Rechem, i.e. the mother.

(Mishnah): If a woman gave birth for the first time, and the baby became mixed up with a baby of a Kohenes, Leviyah or a woman with previous children, the next child will be a Bechor for Pidyon, but not for inheritance.

Inference: The child of a Kohenes or Leviyah is exempt from Pidyon.

Question: Who is the father?

Answer #1: The father is a Nochri.

Rejection: If so, Pidyon applies (to a Kohenes' son)!

Question (Rava): If a Kohenes' first child was from a Nochri, does Pidyon apply?

Answer #1 (Rav Papa): This is like Rav Ada's law, that a Leviyah's child from a Nochri is exempt from Pidyon!

Rejection (and Answer #2 - Rava): No. A Leviyah's child is exempt, because she retains her Kedushah!

(Beraisa): (Even) if a Leviyah was captured (and presumably Nochrim defiled her) or had Zenus (extramarital relations, with someone Pasul), she may receive and eat Ma'aser (Rishon);

A Kohenes is different. If she had relations with a Nochri or Pasul, she becomes like a Zar!

Answer #2: Mar brei d'Rav Yosef can answer like he did above. The father is a Yisrael.

Answer #3 (for Rav Papa): Really, the father is a Kohen or Levi. The woman is called a Kohenes (or Leviyah) because her son is a Kohen (or Levi).

4a (Rava): Since Kohanim and Leviyim exempted (people, i.e.) Yisraelim in the Midbar (from the need to redeem themselves), all the more so they themselves are exempt.

Question (Rav Safra): Only male Leviyim exempted Yisraelim. Children of a Bas Levi should not be exempt!

(Rav Ada bar Ahavah): The Bechor of a Bas Levi need not be redeemed.

Answer (Mar brei d'Rav Yosef): Kedushas Bechor depends on "Peter Rechem," therefore a Bas Levi exempts her son just like a Levi does.

Chulin 132a (Ravina): The Halachah follows Rav Ada bar Ahavah.

Rishonim

Rambam (Hilchos Bikurim 11:10): A Yisrael born to a Kohenes or Leviyah is exempt from Pidyon. It does not depend on the father, rather, on the mother, for it says Peter Rechem b'Yisrael.

Rambam (11): The son of a Leviyah pregnant from a Nochri is exempt. A Kohenes' son from a Nochri is liable, because she became disqualified from Kehunah through Bi'ah with a Nochri.

Ramban (Hilchos Bechoros 56a): In Chulin, we rule like Rav Ada bar Ahavah. However, this could be like Rav Papa, who exempts a Leviyah's son from a Nochri, or like Mar brei d'Rav Yosef in the name of Rava, who exempts a Leviyah's son even from a Yisrael. However, on Daf 4a, we answer like Mar brei d'Rav Yosef. We did not say 'alternatively, according to Rav Papa...' This shows that the Halachah follows Mar brei d'Rav Yosef. Also, Rav Papa was a Talmid of Abaye and Rava, and the Halachah does not follow a Talmid against his Rebbi. Also Bahag rules like this. In some texts in Chulin, e.g. of the She'altos, it explicitly says that the Halachah follows Rav Ada according to Mar brei d'Rav Yosef.

Rosh (8:2): The proof from 4a is not a proof. There, it could not have explained according to Rav Papa, since this would not answer the question (why the son of a Leviyah, even from a Yisrael, is exempt). Indeed, Rav Papa obligates the son of a Leviyah pregnant from a Yisrael, for it says l'Mishpechosam l'Veis Avosam. The Ramban said that the Halachah does not follow the Talmid against his Rebbi. The Ge'onim say that from Abaye and Rava and onwards, the Halachah follows the Talmid against his Rebbi. However, the proof from Chulin is valid. Even though some texts do not explicitly say that there that the Halachah follows Rav Ada according to Mar brei d'Rav Yosef, surely it means so. If not, why was the ruling given there? Surely it was because Ula ruled that Matanos Kehunah are given to a Kohen, and even a Kohenes. Therefore, it mentions that a Leviyah is like a Levi regarding Pidyon ha'Ben, even if the father is a Yisrael.

Rashi (47a DH v'Lo Teima): The opinion that the child (of a Yisraelis from a Nochri) is tainted, he holds that we attribute the child to his father.

Tosfos (47a DH v'Lo): This opinion attributes the child to his mother, therefore Mamzerus applies.

Question: If a Penuyah (single) Bas Kohen became pregnant through Zenus, and says that Ploni (a Yisrael) is the father, and he contradicts her and says that a Nochri is the father, and he gives proofs for this but they are not absolute, and many believe her, must the son redeem himself?

Answer (Terumas ha'Deshen 267): He is exempt. The Gemara exempts the son of a Leviyah or Kohenes from Pidyon. Even though we conclude that if a Kohenes had relations with a Nochri, she becomes like a Zar and her son must be redeemed, in such a case with a Leviyah, her son is exempt. Even though we believe Ploni that it is not his son regarding inheritance, to exempt Ploni from burying the son, and for Yibum and Chalitzah, like the Rosh says (Kesuvos 1:29), we do not believe him to say that the father is a Nochri, especially here because she was not Ploni's wife. Therefore, we rely on her that the son is from a Kosher Yisrael. R. Yehudah says that a man is believed to say that one of his 'sons' is not really his son, rather, a Mamzer. This refers only to children from his wife. Stam, we would attribute the child to him. When he says that it is not his, it follows that the child is a Mamzer. Here, she is not Plonis wife. Surely he is not believed! We hold like Aba Sha'ul (Kidushin 74a), that a Penuyah who gave birth is believed to say that the child is Kosher even if most of the men in the city are Pesulim. Tosfos (Kesuvos 14b DH k'Man) explains that this is because most women check (lineage of) a man before having Zenus with him. Surely Ploni is not believed to obligate the son to redeem himself, for the son is Muchzak in his money.

Poskim

Shulchan Aruch (YD 305:18): Even if a Kohenes or Leviyah is married to a Yisrael, her son is exempt from Pidyon. It does not depend on the father, rather, on the mother, for it says Peter Rechem. If she became pregnant from a Nochri, a Leviyah's child is exempt, but a Kohenes' son is liable, for she became disqualified from Kehunah through Bi'ah with a Nochri.

Shach (22): The Kohenes need not redeem him, for women are exempt from Pidyon ha'Ben. Her husband (a Yisrael) is exempt for it is not his son. If she was disqualified through Bi'ah with a Nochri, and then became pregnant from a Yisrael, the Yisrael must redeem the son.

Rema: If she says that she is pregnant from a Yisrael Ploni, and Ploni contradicts her and says that she is pregnant from a Nochri, the son is exempt from Pidyon.

Shach (23): The Levush exempts the son because ha'Motzi mi'Chavero Alav ha'Re'ayah. This implies that the same applies if she is married to Ploni. However, this law is from Terumas ha'Deshen, and he connotes unlike this. He says that we do not attribute the son to Ploni in any respect. R. Yehudah's law does not apply, for she is not his wife. He did not say that R. Yehudah's law is different, for there she does not contradict him. This implies that he holds that R. Yehudah holds that a husband is believed even if she contradicts him. The Terumas ha'Deshen concluded 'the son has Chezkas Mamon', but this is merely an extra reason. Even regarding his wife, it does not matter whether or not she contradicts him. Therefore, if he says that it is not his son, we believe that she was defiled through another man, and the son must redeem himself. In practice, this requires further investigation.

Pischei Teshuvah (27): Be'er Heitev of the Maharit says that the Shach said that the Rema's law requires investigation. This is wrong. He said so regarding a husband. Sha'ar ha'Melech (Ishus 15:17, b'Sof) brings many Rishonim who say that a man is believed to say that his 'son' is really a Mamzer even if his wife contradicts him. He questioned why the Shach was unsure, and proved this from Terumas ha'Deshen, but remained unsure. Only the Tosfos Rid says that he is not believed. However, the son is exempt. Firstly, the Tosfos Rid exempts him. Also Bahag and R. Tam (brought in Beis Yosef EH 4) hold that a man is believed to say that his 'son' is really a Mamzer only when this follows from identifying his Bechor (i.e. when it is not his oldest 'son'). Also Ri'az (in Shiltei ha'Giborim) says that one is believed to say that the child is a Mamzer only when he says that it is his son. Therefore, the son can say that he holds like these Poskim who exempt, and ha'Motzi mi'Chavero Alav ha'Re'ayah.

Beis Yosef (DH u'Mah she'Chasav v'Chen): The Tur says that also the son of a Bas Yisrael pregnant from a Nochri must redeem himself. This is obvious.

Bach (18): We learn from a Kal va'Chomer from a Kohenes pregnant from a Nochri.