Gemara

(R. Elazar): If the thigh and its cavity were removed, the animal is a Neveilah.

(Rava): We say that the cavity was removed if when it crouches, one can see that something is missing.

58b (Mishnah): If an animal has five or three legs, this is a blemish.

(Rav Huna): This refers to the forelegs, but a missing or extra hind leg is a Tereifah.

This is because anything extra is considered as if it were removed.

Bechoros 40a (Rav Huna): The Mishnah calls five or three legs a Mum. It refers to an extra or missing foreleg. If there is an extra or missing hind leg, it is Tereifah. Anything extra is as if it were missing.

Rishonim

Rambam (Hilchos Shechitah 3:19): If the thigh and its cavity were removed, so that when it crouches, one can see that something is missing, it is a Neveilah.

Rambam (8:24): Any limb that we say that if it were removed it is Tereifah, if it was born missing that limb, it is Kosher.

Rosh (3:54): Rashi says that anything extra is as if it were removed, so it is as if it has only one leg, and it is Tereifah, like an animal missing a leg. The only exceptions to this rule are where Chachamim explicitly said so. The Ramban says that the Gemara said 'it is as if it were removed', not 'it is as if it is missing.' If there was an extra leg below Tzomes ha'Gidin, it is Kosher. Rashi's Perush is primary.

Ran (19b DH k'Natul): The Rashba says that 'anything extra is as if it were removed' does not mean as if it were removed with what it was attached to, rather, as if it were removed by itself. If so, why is an extra leg Tereifah? This is when it is connected at the hip. If it were removed, the other leg would fall out. The same applies if it was attached at Tzomes ha'Gidin, and removing it would cut Tzomes ha'Gidin of the other leg. If it could be removed without harming the other leg, it is Kosher. The Rashba said so l'Halachah, but not to be lenient in practice.

Rashi (Bechoros 40a DH k'Natul): If the thigh and its cavity were removed, the animal is a Neveilah. We hold that if an animal's leg was cut from the Arkuvah (the lowest joint of the leg) and above, it is Tereifah. Also, if the hip was dislocated, it is Tereifah.

Ma'adanei Yom Tov (Bechoros 6:60): Rashi should have said that if the thigh and its cavity were removed, it is Tereifah. This is clear from the proofs he brings.

Pri Megadim (YD 55 Sifsei Da'as 8, DH Od): Rashi wrote unlike he wrote in Chulin. He forbids due to three reasons. The Ma'adanei Yom Tov asks (why he calls it Tereifah, for) really, it is a Neveilah! (Note: our text of Ma'adanei Yom Tov asks oppositely.) We distinguish when it was born this way, or resulted from a wound. The Rashba (98) distinguishes and says that an animal with an extra limb can live 1000 years, but even so it is Tereifah. The same applies here. (If it was born this way, it is only Tereifah - PF.)

Mordechai (636): Ra'avan says that if the extra limb has a Heker (it is clear which is extra), e.g. one is in the proper place for the leg, and one is not, it is Kosher The extra limb is not considered a leg. If it does not reach to the ground, and it does not walk on it, it is not a leg. It is dangling flesh. It is Kosher. If (extra) legs come out of the Arkuvah, even if they reach to the ground, it is Kosher, for their removal would not make a Tereifah. A case occurred in which a chicken had an extra limb like a third leg near the tail. It did not cling at all to the other legs, and Rabanan permitted it. My Rebbeyim argued about a chicken that had an extra toe on the Shok in the place of Tzomes ha'Gidin. It had a bone and nail. Some forbade, for it is as if it was removed. Some permitted, for only an extra leg is Tereifah, but not an extra toe. Rabbeinu Yehudah permitted, for we say that anything extra is as if it were removed only if it is like something it stands on, such as a leg, but not something that is not a leg, e.g. a wart with a bone. It does not make an animal Tereifah.

Poskim

Shulchan Aruch (YD 55:4): If an animal has three (hind) legs, it is Tereifah, for anything extra is as if it were removed. It is as if it were removed a leg. This applies when the extra leg is from the end of the bottom bone and above.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): The Tur says that this is whether it is in the place of Tzomes ha'Gidin or below. This is astounding. At the beginning of the Siman he wrote that there are three Perakim (segments, or joints) in an animal's leg. The bottom is called Arkuvah (knee), which is normally sold with the head. The next is called Shok (thigh). Tzomes ha'Gidin is at its bottom, near the Arkuvah. The Tur also says that anywhere one cuts on the bottom Perek is Kosher. Since Tzomes ha'Gidin is at the bottom of the Shok, near the Arkuvah, why does he forbid if the extra leg is connected below Tzomes ha'Gidin? This is on the Arkuvah. Even if one cuts there, it is Kosher! Even though he wrote that Tzomes ha'Gidin is at its bottom, near the Arkuvah, perhaps he does not mean the very end of the bone. It is slightly above (and if one cuts on the Shok, even below Tzomes ha'Gidin, it is Tereifah). Alternatively, even if it is at the very end, since Rashi forbids if one cuts in the joint between the Shok and Arkuvah, we can say that if the extra leg is connected below Tzomes ha'Gidin, it is Tereifah. At first the Tur wrote that the extra leg is at the end of the bottom bone or above, to include in the joint. Do not say that 'from the end of the bottom bone' means from the end of the leg, and includes the Arkuvah. Since cutting on the Arkuvah does not make a Tereifah, also an extra leg there is no worse than if it were cut. However, if the extra leg was connected to the Shok above Tzomes ha'Gidin, even according to the Rif and Rambam who are Machshir if it was cut there, forbid, since anything extra is as if it was missing from the beginning, and it was missing a Shok (from the beginning, which is Tereifah). According to this, perhaps it is Tereifah even if the extra leg is connected to the Arkuvah, for it is as if the entire Arkuvah was removed, and we forbid if it was cut in the joint between the Arkuvah and the Shok. However, we forbid there only due to Safek, so the Tur should not have said Stam that it is Tereifah. Therefore, I prefer my first Perush. Also R. Yerucham, the Mordechai and Or Zaru'a say so. The Roke'ach (405) brings the case that the Mordechai's Rebbeyim argued about, and says that Rivak forbade, since it is as if the extra finger was cut there.

Taz (4): Toras ha'Bayis ha'Aruch (39) says that the Ramban says that the Gemara said 'it is as if it were removed', not 'it is as if it is missing.' It should say 'were removed and its place', not 'from its place.' (This is the Rosh's text.) An extra matter makes a Tereifah only if when man removes such a matter, this makes a Tereifah. The Rosh (and Ran, who also brought Rashi and the Ramban) and Tur connote that Rashi would forbid an extra leg even below Tzomes ha'Gidin. Here, the Tur stipulates that the extra leg is from the end of the bottom bone or above, whether this is in Tzomes ha'Gidin or below. Since he holds like Rashi, he should forbid even below Tzomes ha'Gidin, like he concludes, i.e. even on the Arkuvah! Therefore, I say that the Tur's words 'from the end of the bottom bone' mean from where the bottom bone begins, close to the bone in which the hooves are attached, and above until the torso. Anywhere there is an extra (leg), it is Tereifah. The Beis Yosef did not explain the Tur like I said, for he held that surely, an extra leg below Tzomes ha'Gidin on the Arkuvah, is Kosher. If the Tur meant like the Beis Yosef said, why did he write Stam 'below'?

Yam Shel Shlomo (3:18): Rashi always forbids an extra liver, the Ramban distinguishes, and the Rashba forbids only if a hole in that place makes a Tereifah. One should be stringent for all three opinions. We should forbid an extra leg unless it is below the Arkuvah. That is always Kosher.

Rebuttal (Taz 4): The Rosh and Tur hold like Rashi, and forbid an extra leg in any place. However, l'Halachah we should not rule like this, for all the Acharonim that the Beis Yosef brings are Machshir if it is below the Arkuvah, like the Ramban. However, regarding the liver, they forbid, like Rashi! The Maharshal says that even Rashi is lenient about the liver, but this is wrong. The Prishah says that we are lenient about an extra leg, because for many limbs we do not say that it (an extra limb) is as if it was totally removed. This is not like any opinion! Rashi is stringent about all limbs, and the Ramban is stringent! We follow the Shulchan Aruch's ruling about an extra leg, but we should be stringent about an extra liver even if it is not in a place where removing it makes a Tereifah, for so hold the Ramban, Rashba and Maharshal.

Be'er Heitev (7): Some say that an extra leg is different, for it has many limbs. All the Poskim connote unlike this. The Pri Chodosh (12) says that we should forbid an extra leg from the Arkuvah and below, even if it will be a big loss. However, if the extra leg does not reach to the floor and it does not walk on it, it is merely dangling flesh, and it is Kosher. The same applies if there is an extra toe on the foot with a bone and nail, or a wart with a bone.

Beis Yosef (DH v'Im): If an animal was created missing a leg, the Rambam (8:24) is Machshir, and the Rashba holds that it is as if it were removed. He says that if it was created missing the top or middle bone, it is Tereifah, but if it was created missing the Arkuvah sold with the head, it is Kosher, for even if it were cut, it would be Kosher.

Gra (14): Even Rashi says that anything extra is as if it were removed only if it is the proper size and form.

Rema: This is only if the extra leg has the form of a leg and stands in the place of the legs. If not, it is not called extra, and it is merely (a) dangling (limb), and it is Kosher. If it was created missing a leg, it is Tereifah.

Gra (15); Ra'avan says that anything extra is as if it were removed, for we say about each of them 'it is the extra one.' This does not apply to if it does not have the form of a leg or if it is not in the place of a leg.