Gemara

Chulin 27a - Mishnah: If one is Shochet one Siman of a bird, or two Simanim of an animal, this is Kosher. (Cutting) the majority of a Siman is like (cutting) the whole Siman.

The majority of one (Siman) of a bird, or the majority of two of an animal is Kosher.

Question: 'If one is Shochet' connotes b'Di'eved. Regarding an animal, there are no other Simanim to cut!

Answer #1: One Siman of a bird is b'Di'eved.

Answer #2: Cutting only the majority of each Siman is b'Di'eved.

Yoma 32b - Question: Our Mishnah says Kortzo. How much did the Kohen Gadol Shochet?

Answer (Ula and R. Yochanan): He cut the majority of both Simanim.

Reish Lakish agrees with them.

Question (Reish Lakish): The Reisha (in Chulin) already taught that the majority of a Siman is like the full Siman. Why does the Seifa need to say that the majority of one Siman in a bird or the majority of two in an animal suffices?

Answer (Reish Lakish): The Mishnah in Yoma teaches that the Kohen Gadol is Koretz, and someone else completes the Shechitah. One might have thought that if the Shechitah is not completed, it is invalid. Therefore, the Seifa says that the majority of one Siman in a bird or the majority of two in an animal suffices.

Question: How could Reish Lakish even suggest that incomplete Shechitah is invalid? If so, the one who completed it did part of the Avodah!

Beraisa: All Avodah of Yom Kipur is valid only if done by the Kohen Gadol.

Answer: Reish Lakish meant, one might have thought that if the Shechitah is not completed, it is invalid mid'Rabanan;

The Seifa teaches that it is Kosher even mid'Rabanan.

Question (Mishnah): If Shechitah of the majority is Kosher even mid'Rabanan, why do we complete the Shechitah?

Answer: It is a Mitzvah to complete the Shechitah.

Chulin 28b: A goose was found in Rava's house. Its neck was full of blood.

Rava: How can we try to permit the goose?

Suggestion: We will be Shochet it and then check (if the Simanim were already cut).

Rejection: Perhaps the knife will cut where there was already a hole, and we will not know that there was a hole there before the Shechitah!

Suggestion: We can (bare the Simanim where they will be cut and) check the Simanim (there) and then be Shochet it.

Rejection: Rabah taught, one cannot check the Veshet (esophagus) from the outside (for it is red, perhaps there is a tiny hole that was filled with a drop of blood and is not noticeable), only by turning it inside out (the inside is not red).

Rav Yosef, son of Rava: We can check the Kaneh, be Shochet the Kaneh, and then turn the Veshet inside out to check it!

This shows that it suffices to cut the Kaneh alone.

Rishonim

Rif (Chulin 6a): The majority of a Siman is like (cutting) the whole Siman. The majority must be visible. This is b'Di'eved; l'Chatchilah one must totally cut the Siman.

Ran: A 'visible' majority does not mean a sizable majority that is evident without close scrutiny. Rather, it excludes exactly half, which (according to one opinion) is a Halachic majority. The Gemara also says that if a visible majority of the Kaneh (windpipe) was cut (before Shechitah), it is Tereifah. Surely, anything more than half makes it Tereifah!

Rambam (Hilchos Shechitah 1:9): Both the Kaneh and Veshet must be cut. A proper Shechitah is to totally cut both, whether in a bird or an animal. If the majority of one Siman of a bird, or the majority of two of an animal was cut, it is Kosher.

Rosh (Chulin 2:2): The Gemara said that cutting one Siman of a bird is b'Di'eved; alternatively, cutting only the majority of a Siman is b'Di'eved. Presumably, we are stringent like both answers. The Rif says that the majority of a Siman is b'Di'eved (but does not require both Simanim of a bird l'Chatchilah). Since the answers argue about a mid'Rabanan law (mid'Oraisa, one Siman suffices for a bird), we are lenient about a Safek.

Pri Megadim (YD 21:1): The Rif understands that the first answer agrees that l'Chatchilah, the entire Siman must be cut. They argue only about whether or not one Siman is l'Chatchilah for a bird, and we are lenient.

Poskim

Shulchan Aruch (YD 21:1): Both the Kaneh and Veshet must be cut. A proper Shechitah is to totally cut both, whether in a bird or an animal. If the majority of one Siman of a bird, or the majority of two of an animal was cut, it is Kosher, as long as it is found to be the majority

Beis Yosef (DH uv'Perek Kama): Rashi (Chulin 21b DH v'Eino) and the Mordechai say that there is no Mitzvah to Shochet the second Siman of a bird. The Rambam, Rashba and Rosh say that l'Chatchilah he must Shochet both; this is correct:

Question: For a bird, it should suffice to cut one entire Siman and the majority of the other, for this fulfills both opinions!

Answer: The Magid Mishnah (of R. Yehudah Kaltz, Hilchos Shechitah 1:9) says that the Rambam holds that both answers in Chulin are true. Therefore, l'Chatchilah he requires totally cutting both Simanim.

Note: The Gra (1) says that the Shulchan Aruch is stringent like both answers. This connotes that we are stringent because of a Safek. Perhaps he holds that the first answer could agree that l'Chatchilah, the entire Siman must be cut (like the Pri Megadim above), or that the second answer could agree that l'Chatchilah, both Simanim of a bird must be cut.

Sefer ha'Terumos (brought in Gra 2): We learn from the goose (Chulin 28) that one Siman suffices l'Chatchilah!

Rebuttal (Rosh, ibid., brought in Gra 2): Since it was impossible to Shochet both Simanim (for then the Veshet could not be checked), that was like b'Di'eved.

Beis Yosef (DH v'Chosav ha'Rashba): The Rashba, Ran and Tur explicitly say that a 'visible' majority means anything more than half. This is also Rashi's opinion.

Taz (2): Rashi (29a DH Rov) defines a visible majority to be an absolute majority which is recognizable. This connotes a sizable majority. Similarly, the Mordechai does not allow inserting a feather through the mouth to test whether or not the majority of the Siman was cut because we require a visible majority, i.e. a sizable majority. The Agudah and Mahari Veil agree. However, since regarding Tereifah a 'visible majority' surely means anything more than half, the same applies to Kosher Shechitah. Nevertheless, if there is not a loss one should be stringent to be concerned for those who require a sizable majority.

Shach (1): 'Kaneh' refers to the membrane inside the rings of the Gargeres (windpipe); the majority of the Kaneh must be cut.