Gemara

(Rav): If exactly half of Yisrael were Tehorim, half is like a majority, so all bring Pesach Rishon. Tehorim bring by themselves [in Taharah, since they are like a majority] and Temei'im by themselves (like a majority, which is not detained to Pesach Sheni);

(Rav Kahana): Half is not like the majority;

Version #1: Tehorim bring Pesach Rishon, and Temei'im bring Pesach Sheni [like a minority that is detained];

Version #2: Tehorim bring Pesach Rishon, and Temei'im do not bring either Pesach.

Chulin 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.

Question (Beraisa): If half the Kaneh was already cut, and one 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, [Rav must agree that] it should require a visible majority!

Answer #2: Rather, Rav and Rav Kahana agree that half is unlike a majority. Rather, they discuss Korban Pesach.

Shabbos 122a (Mishnah): If Yisraelim and Nochrim are in a city, and there was a bathhouse open on Shabbos:

If the majority [of the residents] are Nochrim, one may bathe in the bathhouse immediately [on Motzei Shabbos];

If the majority are Yisraelim, one must wait the time needed to heat the water [after Shabbos].

(Beraisa): If a Ner was lit at a party:

If the majority are Nochrim, one may use the light;

If the majority are Yisraelim, or even half, it is forbidden.

Sukah 35a (Mishnah): If Chazazis (blisters) came on the majority of an Esrog, it is Pasul. On the minority, it is Kosher.

Rishonim

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

Ran: 'Visible' does not mean a sizable majority that is evident without close scrutiny. Rather, it excludes a Halachic majority, i.e. exactly half. A visible majority of the Kaneh cut [before Shechitah] is Tereifah. Surely, anything more than half makes it Tereifah!

Rif and Rosh (Shabbos 46b and 16:14): If half the residents are Nochrim, one must wait before bathing in the bathhouse on Motzei Shabbos, because half is like a majority.

Ran (DH v'Im): Half is like a majority [regarding a fire] because we cannot know for whom they lit.

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

Rosh (Chulin 1:24): If one cut half the Kaneh, paused, and completed the Shechitah, it is Kosher. Half is not like a majority. The first cut has no significance.

Rosh (2:1): If one cut [through Shechitah] half of one Siman in a bird, or one and a half Simanim in an animal, the Shechitah is invalid.

Rashi (79b DH u'Temei'in): Version #2 holds that the Temei'im do not bring Pesach Rishon, for they are unlike a majority. They do not bring Pesach Sheni, for they are not a minority.

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. Rashi agrees.

Taz (2): Surely, regarding Tereifah a 'visible majority' is anything more than half. The same applies to Kosher Shechitah. Nevertheless, if there is not a loss one should be stringent like those who require a sizable majority.

Shulchan Aruch (201:2): If in a spring, rainwater is the majority over water that flowed [from underground sources], it is Metaher only if the water is gathered, like a Mikveh.

Shach (YD 201:10): This connotes that if it is half-half, it is like a spring. The Beis Yosef brings so from the Ra'avad in Ba'al ha'Nefesh. Our text of the Ra'avad says oppositely! This requires investigation. A majority of rainwater disqualifies mid'Oraisa.

Pri Megadim (Sha'ar ha'Ta'aroves DH v'Li): Mid'Oraisa, [Min b'Mino] is forbidden unless there is a majority of Heter. We hold that half is not like a majority. Surely regarding the Sanhedrin we require a majority. However, if an Isur mid'Rabanan became mixed one in one, Iy Efshar Letzamtzem (they cannot be exactly equal), or a big piece and a small piece and it is not known which is Heter, this requires investigation. Perhaps this is not called Ischazek Isura (we know that there was Isur here), for if the big piece was Heter, there is no Isur at all. (It was Batel.) We permit not only if one fell to the sea. Rather, one may even throw one away or give it to a Nochri, and eat the other. The Shach (YD 110 Dinei Sefek-Sefeka 21) forbids an Isur mid'Rabanan mixed one in one, and the Rambam (Hilchos Rotze'ach 9:8) says that Iy Efshar Letzamtzem b'Yedei Adam (even if people try to make them equal). This shows that even a big and small piece are forbidden.

Shulchan Aruch (OC 298:7): If one was outside a city on Motza'ei Shabbos and saw a fire, if most residents are Nochrim, he may not bless on it. If most are Yisre'elim, or even half-half, he blesses on it.

Bach (298:4 DH Hayah): Half is like a majority to be lenient or stringent. Therefore, above (regarding a bathhouse) half-Yisrael is like Rov Yisrael and it is forbidden, and here half-Yisrael is like Rov Yisrael and it is permitted.

Shulchan Aruch (648:11): If an Esrog was half [Chazazis] - half [normal] in one place, some are Machshir and some disqualify.

Taz (16): The Tur brings that his brother R. Yechi'el is Machshir, for Efshar Letzamtzem. The Ra'avad says that whenever a majority disqualifies, half is Pasul, for Iy Efshar Letzamtzem. I disqualify due to the contradiction in inferences of the Reisha (Chazazis on the majority is Pasul) and Seifa (on the minority, it is Kosher). Therefore, we are stringent.

Note: The Ran (Sukah 17a DH Alsah) asked the contradiction, and answered that the Seifa explains the Reisha. It is Kosher only if the Chazazos are contained within the minority. I.e., the Reisha refers to Chazazos on the majority of the Esrog, even if their area is the minority.

Gra (11): This is like Parutz k'Omed (the breaches in a wall equal the part that stands). We hold that Iy Efshar Letzamtzem, but b'Yedei Adam Efshar Letzamtzem. Half is not like a majority regarding Terefah. This is like Terefah. The stringent opinion holds that Iy Efshar Letzamtzem even b'Yedei Adam. Shechitah requires a visible majority. This is a tradition from Sinai. Also Parutz k'Omed is a tradition.

Shulchan Aruch ha'Rav (20): Half is like a majority only for walls of Shabbos and Sukah and for Schach, due to a tradition from Sinai. In the rest of the Torah, we require the majority to be Kosher. Some disagree, and disqualify only when there is a majority of Pesul. The Halachah does not follow them, but in pressed circumstances, if one cannot find other Minim, one may rely on to bless on these, for in pressed circumstances some bless even on totally Pasul Minim.

Mishnah Berurah (44): One should follow the stringent opinion, for the Mechaber put it last.

Aruch ha'Shulchan (329:3): If a Yisrael and Nochrim were in a Chatzer, and most of them went to another Chatzer, and one of those entered a house and it caved in, we excavate it even on Shabbos. Since a Yisrael was Muchzak among those who separated, we do not follow the majority [to say that a Nochri entered the house]. For Piku'ach Nefesh, half is like the majority.

Aruch ha'Shulchan (586:3): The Rosh and Tur say that if most of the circumference of the Shofar was cracked, it is Pasul. Automatically, also half is Pasul. All agree that for Isurei Torah, half is like a majority to be stringent. Some say that half is like a majority even to be lenient, but the Halachah does not follow them. Also, we hold that Iy Efshar Letzamtzem.