Gemara

27a (Mishnah): (Cutting) the majority of a Siman is like (cutting) the whole Siman.

28b (Rav): Exactly half is considered like a majority.

(Rav Kahana): It is not like a majority.

Rav holds that the tradition about Shechitah says that the majority cannot be left uncut. Rav Kahana holds that the tradition obligates cutting the majority.

Question #1 (Mishnah): (If one cut) half of one Siman in a bird, or one and a half Simanim in an animal, the Shechitah is invalid.

According to Rav, this is like cutting the majority, the Shechitah should be valid!

Answer #1: Mid'Oraisa, the Shechitah is valid. The Mishnah teaches a stringency mid'Rabanan, lest one slaughter less than half.

Question (Beraisa): If half the Kaneh was already cut, and he cut a drop more and completed the Shechitah, this is Kosher.

If half is like the majority, if half the Kaneh was cut, it was Treifah from the beginning!.

Answer #1 (Rava): It is Treifah only if a visible majority is cut.

Objection (Abaye): This is illogical! (It some places), a hole of any size makes a Treifah. When only a majority makes a Tereifah, you require a visible majority;

Shechitah always requires a majority. All the more so, it should require a visible majority! (How can Rav say that half suffices?)

Answer #2 (to both questions): Rather, Rav and Rav Kahana did not discuss Shechitah. Both hold that half is not like a majority. Rather, they discuss bringing the Korban Pesach when half of Yisrael are Teme'im.

Rishonim

Rif (Chulin 6a): The majority of a Siman is like (cutting) the whole Siman. The majority must be visible.

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 was cut (before Shechitah), it is Tereifah. Surely, anything more than half makes it Tereifah!

Rambam (Hilchos Shechitah 1:9): If the majority of one Siman of a bird, or the majority of two of an animal was cut, it is Kosher.

Rosh (Chulin 2:1): If one cut half of one Siman in a bird, or one and a half Simanim in an animal, the Shechitah is invalid. If one cut the majority of one Siman in a bird, or of both Simanim in an animal, it is Kosher.

Poskim

Shulchan Aruch (YD 21:1): 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 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 that 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.

Pri Chodosh (3): It seems that Rashi holds that it is not enough to see through measuring that the majority was cut. Rather, it must be evident to all. This is unlike how the Rashba explains Rashi. In Avodah Zarah (37a) Rav Chiya bar Ami says that Tana'im argue about a species of grasshoppers, whether the wings must cover an evident majority (to be Kosher), or if even a bare majority suffices. Rashi explains that the latter is a majority that one must measure, and the former is visible to the eyes. The Shulchan Aruch (YD 309:2) says that something missing from the ear is a proper blemish if it is well evident and visible to the eyes. If not, one may not slaughter (a Bechor based on this blemish) due to Mar'is Ayin (lest people think that there is no blemish). These show that 'visible to the eyes' means a large majority that is visible without measuring. Tosfos (Chulin 21b DH v'Eino and 123b DH Olas) says 'a majority visible to the eyes' to denote a large, evident majority. The Mordechai (578) says that Shechitah requires 'a large majority evident to everyone.' The Agudah requires a majority visible to the eyes; he does not permit a minimal majority. The Gemara connotes like this. In the Hava Amina, Rav holds that half is like the majority, and mid'Rabanan it is Pasul, lest one cut less than half. Likewise, (in the conclusion that we require a majority), mid'Rabanan we should disqualify a bare majority, lest one not cut the majority.

Pri Chodosh (3): The Ran and Rashba proved unlike this, for the Gemara also says that a visible majority makes a Tereifah. Surely, if anything more than half of the Kaneh or spinal cord was cut, it is Tereifah! Surely, this is a Torah stringency, and any majority suffices, even if it is not visible to the eyes. Since there is no concern for misunderstanding, Rava said 'visible to the eyes', even though it is not precise. When we can say that we truly require a visible majority, we should explain simply. The Ran and Rashba also brought a proof from Toras Kohanim, which says that the difference between half and a majority is a hair's width. This is not a proof. That is mid'Oraisa. The stringency to require a visible majority is mid'Rabanan! I say that we must be stringent, unlike the Shulchan Aruch.

Tevu'os Shor (2): R. Yerucham explained that half is Pasul, and said that the Rif requires a majority visible to the eyes. He cannot explain like the Ran and Rashba, that the Rif requires a measurable majority. If so, he did not need to cite the Rif, for he already disqualified half! Rather, he explains the Rif like Rashi, to require a large majority. We must say that they were unsure how to learn the conclusion of the Gemara. Did Rava have a tradition to require a visible majority for Tereifah, and all the more so for Shechitah (and everywhere else in the Torah, except for Mechitzos)? Or, perhaps Rava said so from his own reasoning, to answer the question. The Gemara answered that when we need a Stam majority, half suffices. It does not suffice to make Tereifah, for Rov Acher Kaima Lei (or Lan).

Note: I thought that these words mean that the other half is intact. However, it seems that R. Akiva Eiger understood them to mean that we have a tradition to require a greater majority.

Tevu'os Shor: We concluded that the majority is not like half; so we do not need Rava's answer. We are stringent both ways, to say that even a bare majority makes Tereifah, but a visible majority is needed for Shechitah.

Question (R. Akiva Eiger): We cannot say that from reasoning, Rava requires a visible (large) majority for Tereifah. It would have sufficed to say that a measurable majority is needed! Rather, we must say that he had a tradition. If so, we should rule like him in the conclusion! Perhaps it is difficult to say that Rav and Rav Kahana argue about whether there was a tradition. Perhaps all agree that there was a tradition that Tereifah requires a large majority, and they argue about whether normally half suffices, or we need a measurable majority.

Tevu'os Shor: The Rambam, Rashba, Ran, Tur and Shulchan Aruch all say that a measurable majority suffices. The fact that no Posek is Machshir a mere majority is no proof for them. It is a Safek how to learn the conclusion of the Gemara, so we are stringent. There is no proof from Toras Kohanim. Our Gemara is primary. Since we are unsure about the conclusion of our Gemara, we do not follow the Toras Kohanim. If the Pri Chodosh held that the conclusion of the Gemara never requires a visible majority, just the Poskim saw a need to be stringent, this would be difficult. However, I would not question him, for it is a reasonable decree. However, he holds that the Gemara is stringent mid'Rabanan. It is astounding to say that the Gemara was imprecise to say that a visible majority makes Tereifah. I say that we must be stringent like R. Yerucham, even if this will entail a big loss.

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