Hullin, Chapter Two, Mishnah One
Our mishnah returns to the subject of slaughtering. Slaughtering involves cutting the trachea and the esophagus. Our mishnah deals with a case where the slaughterer did not fully cut both organs.
1) If one cut one [of the organs of the throat] in the case of a bird, or both organs in the case of cattle, the slaughtering is valid.
2) The greater part of an organ is equivalent to [the whole of] it.
a) Rabbi Judah says: he must cut through the veins.
3) [If one cut] half of one organ in the case of a bird, or one and a half organs in the case of cattle, the slaughtering is invalid.
4) [If one man cut] the greater part of one organ in the case of a bird, or the greater part of each organ in the case of cattle, the slaughtering is valid.
Section one: There is a difference between slaughtering birds and slaughtering cattle (cows, sheep and goats). In order for the cattle to be valid both organs must be cut, whereas it is sufficient for one of the organs to be cut for a bird to be valid.
Section two: If he cuts through most of the organ, the animal is valid. Rabbi Judah says that he must at least cut through the veins. The Talmud states that this applies only to the bird.
Section three: Cutting through half of an organ is not sufficient. Therefore, if one cuts through half of one organ of a bird or one and a half organs of an animal, the slaughtering is not valid.
Section four: However, cutting through the greater part of the organ counts as cutting through the organ. Therefore, if he cuts through the greater part of one of the birds organs, or the greater part of both of the animals organs, the slaughtering is valid