Gemara

Rav: A chick born on Yom Tov is forbidden that day;

Shmuel or R. Yochanan: It is permitted.

Rav forbids because it is Muktzah. Shmuel or R. Yochanan permits, for just like its birth permits (enables) its own Shechitah, its birth permits regarding Muktzah.

Question (Rav Kahana and Rav Asi, against Rav): Why is this different than a calf born on Yom Tov (which is permitted)?

Answer (Rav): A calf is Muchan along with its mother through Shechitah (i.e. even had it not been born, it could have been permitted through Shechitah of the mother).

Question (Rav Kahana and Rav Asi): Why is a chick different than a calf born to a Tereifah? (It is permitted even though Shechitah of the mother would not have permitted it!)

Rav did not answer.

Question (Rabah or Rav Yosef): Why didn't Rav answer that a calf was already Muchan along with its mother for dogs to eat? (Tereifos are normally fed to dogs.)

Answer (Abaye): (The opinion that forbids Muktzah, i.e. R. Yehudah, holds that) something Muchan for people is not considered Muchan for animals (in the following Mishnah). All the more so, something Muchan for animals is not considered Muchan for people!

Mishnah: We may cut gourds in front of animals, and a Neveilah in front of dogs (on Shabbos);

R. Yehudah says, if it was not a Neveilah when Shabbos began we may not cut it, since it was not Muchan.

Answer (Rabah or Rav Yosef): Indeed, something Muchan for people is not considered Muchan for animals, for a person does not give to dogs something that he can use himself;

Something Muchan for animals is Muchan for people, for a person plans to use anything he will need.

Support (for Rav - Beraisa #1): A calf born on Yom Tov is permitted, whereas a chick born on Yom Tov is forbidden. This is because a calf is Muchan along with its mother through Shechitah, but this does not apply to a chick.

Support (for Shmuel or R. Yochanan - Beraisa #2): A calf born on Yom Tov is permitted, for it is Muchan along with its mother through Shechitah. A chick born on Yom Tov is permitted. Just like its birth permits Its own Shechitah, its birth permits regarding Muktzah.

(Beraisa): A chick born on Yom Tov is forbidden;

R. Eliezer ben Yakov says, it is forbidden even on a weekday. "Every Sheretz that swarms on the ground (is forbidden)" includes chicks that did not yet open their eyes.

Rishonim

The Rif brings the opinions of Rav and Shmuel. He permits a calf born on Yom Tov if we know that it was a full term pregnancy.

Yam Shel Shlomo (Beitzah 1:18): The Rif does not explicitly limit the Heter to when the mother was ready to be eaten, but since he forbids Muktzah on Yom Tov, this is his opinion. He relies on what he cites from Rav 'it is Muchan along with its mother'.

Rambam (Hilchos Yom Tov 2:1): A chick born on Yom Tov is forbidden because it is Muktzah. If a calf was born on Yom Tov, if the mother was ready to be eaten, the calf is permitted, for it is Muchan along with its mother. Even had it not been born, it could have been permitted through Shechitah.

Magid Mishnah: The Rambam does not exclude if the mother is Tereifah, for also a Tereifah is ready to be eaten by dogs. Rather, he excludes a cow that the owner wants to keep for plowing or milk; it is Muktzah for people or animals.

Question: Shmuel and R. Yochanan permit Muktzah. Why do they need to say 'just like the birth permits itself...?'

Answer #1 (Rosh 1:6): They say that even Rav, who forbids Muktzah, should permit here.

Answer #2 (Rosh ibid.): This is similar to figs and raisins (even R. Shimon forbids such Muktzah).

Rosh (ibid.): Some say that the Halachah follows Rav because two Beraisos are like him. Also, the Halachah always follows R. Eliezer ben Yakov. Also, Rabah and Rav Yosef suggested an answer for Rav; this shows that they hold like him. Others rule like Shmuel because Rav Kahana and Rav Asi challenged Rav and Rav could not answer. The Ge'onim ruled like Rav. Also the Rif is stringent like Rav (out of Safek), therefore he did not need to explicitly say whom the Halachah follows. Even though Rav did not answer, Rabah and Rav Yosef answered nicely for him.

Question (Korban Nesan'el 50): Also Rav is unlike R. Eliezer ben Yakov, for Rav forbids only on Yom Tov! We must say like Tosfos, that they discuss when the chick opened its eyes on the first day. If so, R. Eliezer ben Yakov's law has no relevance to Rav's law! Yam Shel Shlomo (1:17) asks that R. Eliezer ben Yakov connotes that if it opened its eyes it would be permitted on Yom Tov!

Answer (Korban Nesan'el ibid.): R. Eliezer ben Yakov forbids ''even on a weekday'. The Rosh infers that it is obviously forbidden on Yom Tov, like Rav.

Ran (DH Itmar): Since Rav and Shmuel are both unlike R. Eliezer ben Yakov, this shows that the Halachah does not follow R. Eliezer ben Yakov (this is an exception to the rule).

Hagahos Ashri (DH u'Chegon): R. Yo'el holds that a chick without big feathers is forbidden like "Sheketz".

Ba'al ha'Ma'or (3b): The Rif did not decide whether or not the Halachah follows Rav because the Gemara was indecisive whether he argues with Shmuel or R. Yochanan. We follow Rav when he argues with Shmuel, but we rule like R. Yochanan against Rav. It appears that the Halachah is unlike Rav, for also Rav Kahana and Rav Asi are lenient. Rabah and Rav Yosef, who are Basra, say that Rav admits when there are two reasons to be lenient, i.e. it is Muchan along with its mother and its birth permits itself. There are two reasons to be lenient also regarding a calf born to a Tereifah, for it is Muchan for dogs along with its mother.

Rebuttal (Milchamos Hash-m): The She'altos and BaHaG rule like Rav, for Rabah and Rav Yosef hold like Rav, and they are Basra. Rav does not agree that 'its birth permits itself'. Rav forbids an egg because of Hachanah. Here the issue is not Hachanah, rather, Muktzah. When it is Muchan through its mother it is not Muktzah. Shmuel and R. Yochanan permit Muktzah. They say that the Mishnah decrees to forbid an egg lest people permit liquids that exude. When a living being (is born and) permits Its own Shechitah, they do not forbid. Alternatively, they say that the decree does not apply to living beings. Rather, the egg is forbidden because of Muktzah, like Rav Nachman who permits Muktzah on Shabbos and forbids Muktzah on Yom Tov.

Rosh (1:7): A calf born on Yom Tov is permitted, for it is Muchan along with its mother. This is if the mother was ready to be eaten; if the owner wants the mother for offspring, the calf is forbidden according to the opinion that forbids Muktzah. Hachanah does not apply here. It applies regarding a laid egg, for that enables it to develop a chick. However, a calf found inside a cow after Shechitah is as good to eat as a born calf. To slaughter on the day of birth we must know that it was a full term pregnancy, like it says in Shabbos (136a), and that the calf stepped on the ground to preclude a Safek that its limbs were crushed (during birth), like it says in Chulin (51b).

Rebuttal (Beis Yosef OC 498 DH Egel): The Halachah does not require that it stepped on the ground. Perhaps the Rosh discusses when it was a difficult labor, or he is stringent to avoid all doubts.

Yam Shel Shlomo (ibid.): The Rosh requires that it stepped on the ground when we do not know that it was a full term pregnancy.

Rebuttal (Taz YD 15:2): The Rosh writes 'like it says in Chulin', and there it does not discuss the term of pregnancy. Also, a full term pregnancy precludes Safek Nefel. This is unrelated to the concern that limbs were crushed, i.e. during birth.

Poskim

Shulchan Aruch (OC 498:5): One may slaughter a calf born on Yom Tov if the mother was ready to be eaten and we know that it was a full term pregnancy.

Yam Shel Shlomo (ibid.): Tosfos permits Muktzah on Yom Tov but forbids Nolad. However, Tosfos agrees that Nolad does not apply to living animals. This is because even after it is born it may not be eaten, and before birth it was just as fitting to eat as after birth.

Magen Avraham (8,9): Tosfos permits even if the mother was Tereifah, for it is Muchan for dogs. It seems that in this case the calf would be forbidden nowadays that Tereifos are not for dogs (rather, we sell them to Nochrim). Also, today we are not proficient to know when it was a full term pregnancy.

Rebuttal (Kaf ha'Chayim 41): Eliyahu Rabah permits even nowadays. Since Treifos are cheap, people throw pieces to dogs. Also others permit even nowadays. However, if the mother was Safek Tereifah it is forbidden. One would not give it to dogs, for if it gives birth we will know that it is Kosher.

Rema: We also require that the calf stepped on the ground, lest we find internal problems (Safek Tereifah), and the Shechitah was an unneeded Melachah on Yom Tov.

Magen Avraham (10): We are concerned only because of Melachah on Yom Tov. On a weekday it is permitted even if it did not step on the ground.

Shulchan Aruch (513:8): If a chick was born on Yom Tov it is forbidden.

Mishnah Berurah (36): Even those who permit Muktzah on Yom Tov forbid it, for it was totally useless before it was born. Even dogs would not eat a chick in its shell.

Shulchan Aruch (ibid.): If it was born on Shabbos that is Erev Yom Tov, it is forbidden on Yom Tov.

Taz (11): It is forbidden mid'Oraisa because of Hachanah. However, it is unlike Hachanah of an egg laid on Yom Tov. A chick is never Asur mid'Oraisa on the day of birth. Therefore, we do not decree to forbid chicks born on Yom Tov lest one permit a chick born on Yom Tov that is on Motza'ei Shabbos; according to the opinion that permits Nolad and Muktzah on Yom Tov, it is permitted.

Mishnah Berurah (37): Some say that the Hachanah is mid'Oraisa because had it not been born there would be no way to permit eating the chick. Some say that it is only mid'Rabanan because it was Muktzah on Shabbos, therefore Shabbos cannot prepare for Yom Tov.