Gemara

59a (Beraisa): The Simanim of Tahor animals are "the hooves are split..." - any animal that chews the cud surely has no upper teeth, and it is Tahor.

This means that if an animal has no upper teeth, it surely chews the cud and has split hooves, so it is Tahor.

(Rav Chisda): If one finds an animal whose hooves are cut off, if there are upper teeth, it is Tamei. If not, it is Tahor. He must know that it is not a young camel.

(Beraisa - Tana d'Vei R. Yishmael): "The camel, for it chews the cud" - the Torah lists all the Tamei animals that chew the cud. All others that chew the cud are Tehorim. "The pig, for it has split hooves" teaches that pigs are the only Tamei animals with split hooves.

(Rav Chisda): If the hooves and mouth are cut off, he checks the meat (under the tailbone). If it tears lengthwise and widthwise, it is Tahor. If not, it is Tamei. He must know that the animal is not an Arod (wild ass?).

We have a tradition from Moshe from Sinai that there are no other exceptions.

(Beraisa): These are the Simanim of Chayos, whose Chelev is permitted: any animal with horns and hooves;

The horns must branch out. This is a sure Siman that it is a Chayah. If not, the horns must be shelled, grooved and pointed. The grooves must be Muvla (very close to each other).

The argument about goats of Karkuz depended on this;

Rav Achai forbade the Chelev. Rav Shmuel brei d'R. Avahu (permitted it and) ate some of the Chelev.

80a (Beraisa): A Koy is a wild ram;

Some say, it is the child of a male goat and a female deer;

R. Yosi says, it is a species unto itself. Chachamim could not decide if it is a Chayah or Behemah;

R. Shimon ben Gamliel says, it is a Behemah.

Question (Aba brei d'Rav Minyamin bar Chiya): Are forest goats Kosher for Korbanos?

Answer #1 (R. Zeira): Yes;

(R. Yitzchak): The Torah lists only 10 Kosher animals. Since forest goats are not listed among the Kosher Chayos, they must be (Behemos, i.e.) goats.

Question #1: (Rav Acha brei d'Rav Ika): Perhaps the forest goat is Ako (one of the Kosher Chayos that we do not recognize)!

Question #2: (Rav Acha brei d'Rava): Perhaps it is Te'o or Zemer (other Kosher Chayos)!

Support (for Answer #1 - Rav Huna bar Chiya): Chachamim and R. Yosi argue only about a wild ox. All agree that forest goats are goats;

(Mishnah): A wild ox is a Behemah;

R. Yosi says, it is a Chayah.

Chachamim learn from the Targum of Te'o, i.e. 'Torbela (ox of the forest)';

R. Yosi says, since it is listed among the Chayos, it is a Chayah.

Answer #2 (R. Chanan): Ameimar permitted Chelev of the forest goat. (Surely it is a Chayah.)

Rishonim

Rambam (Hilchos Ma'achalos Asuros 1:10): We have a tradition for Simanim of Chayos. If it has split hooves and chews the cud and the horns branch out like a ram's, or if the horns are shelled like a bull's horn, grooved like a goat's horn and the grooves are Muvla, and pointed like a deer's horn, it is a Tahor Chayah.. If it is missing one of these, the Chelev is forbidden.

Rambam (11): If one recognizes one of the seven Chayos listed in the Torah, even if does not have horns, the Chelev is permitted and one must cover the blood.

Poskim

Shulchan Aruch (YD 79:1): If an animal has no teeth in the upper jaw, nor Nivim, it is Tahor. Nivim are tooth-like protrusions of dangling flesh in the gums. Some say that they are two teeth, one at each end of the jaw. One must know that it is not a young camel.

Shulchan Aruch (ibid.): If one cannot check the hooves and mouth, for they are cut off, he checks the meat under the tailbone. If it tears lengthwise and widthwise, it is Tahor. He must know that it is not an Arod.

Shulchan Aruch (80:1): Chachamim gave Simanim, based on tradition, to show that an animal is a Chayah, so its Chelev is permitted. If the horns branch out, surely it is a Tahor Chayah. If not, if the horns are shelled, grooved and pointed. The grooves must be Muvla. If it is missing one of these, the Chelev is forbidden.

Shach (1): The Beis Yosef explains the Simanim of the horns. Since now we have only what we received from tradition, like we say below (82:3) regarding Simanim of birds, I was brief.

Pri Megadim (Sifsei Da'as)): In 82:3 we require a tradition like Rashi says, due to the swamp chicken, which initially was established to be Tahor and afterwards people saw that it is Dores (and is Tamei). There are many opinions about what is 'Dores', unlike the Simanim of Chayos and Behemos. The Shach did not say that we do not eat animals without tradition. Surely, for Behemos one need not be an expert to recognize split hooves and chewing the cud! Even if we have a tradition that an animal is Tahor, we must recognize if it is a Chayah or Behemah. The Shach means that we may not eat the Chelev without a tradition, for we are not experts regarding grooved horns that are Muvla and other Simanim.

Shach (2): Surely we require that it has split hooves and chews the cud, like the Rambam says.

Shulchan Aruch (2): If one recognizes one of the seven Chayos listed in the Torah, even if does not have horns, the Chelev is permitted and one must cover the blood.

Chachmas Adam (36:1): Nowadays we eat only what we have a tradition for it. Therefore, the only Chayah we may eat is deer.

Aruch ha'Shulchan (10): In practice, we do not rely on Simanim, since we are not experts. We may eat a Chayah only with a tradition, like we say regarding birds.

Zichei Tzedek (Nowadays we are not experts in Simanim, so we have only what we received a tradition for. Here in Bagdad, we have a tradition for deer, hart and Yachmor (fallow deer?).

Kaf ha'Chayim (5): We need a tradition only for the Chelev. One need not be an expert to know that it chews the cud and has split hooves. Beis Yitzchak says so. The Shulchan Aruch connotes like this. It says that if we do not have the Simanim on the horns, the Chelev is forbidden. This implies that the meat is permitted.

Chazon Ish (YD 11:4): The simple reading of the Shach is like the Rema says about fowl, that we do not eat new species based on Simanim. It was clear to them that this is the custom. The Shach lived in a generation of great Chachamim, i.e. the Beis Din of the Chelkas Mechokek and R. Yehoshua Heshel. All of Yisrael followed them.

Chazon Ish (4): The Rishonim and Acharonim were very stringent not to breach fences made by previous Chachamim. The Chachmas Adam (36:1) said that we do not eat Behemos or Chayos without a tradition. This is like the simple reading of the Shach, unlike the Pri Megadim. His explanation is unlike the Shach's words. If the Shach meant that we should not accept new species without a tradition, rather we must conduct with the stringencies of Behemos and Chayos, he should have written 'it seems to me.' The Shach connotes that he writes a known custom. Surely it was not common to accept a new species and accept the stringencies of Behemos and Chayos! Rather, the Shach knew that the only Behemos we have are (cattle,) sheep and goats, and that Chachamim do not accept new species. He mentioned Siman 82 to teach that our custom regarding animals is like for birds. He did not come to learn from birds. All of this is a fence. Our Chachmah has decreased, so it is better to rely on tradition. Also, if we accept new species, it is a complicated issue regarding the Chelev, Kisuy ha'Dam and description of the limbs that is relevant for some Treifos. We need not seek the reasons (to require tradition). Do not say that the Chachmas Adam means that we do not accept Chayos (based on Simanim, without a tradition), but we may accept Behemos. The Shach said 'therefore, I did not elaborate about the Simanim.' If we distinguish between Chayos and Behemos, we need the Simanim! Rather, we never accept new species. Even according to the Pri Megadim, nowadays we require tradition, for the Chachmas Adam ruled like this and his Sefer was accepted in all of Lita (Lithuania).

Shevet ha'Levi (10:114): The Chazon Ish is a stringency regarding Lita. Even though the Shach was accepted throughout the world, the Pri Megadim and others who commented on him are like 100 witnesses that there was no such tradition in their lands. The Rema did not mention such a stringency, so surely in his land they did not require a tradition. Since other lands surely permitted any animal with Kosher Simanim, this overrides the Safek of Lita. I do not address the question in Eretz Yisrael, which was the locale of the Chazon Ish.

Shulchan ha'Levi (R. Yisrael Belsky Shlita) Birurei Halachah 19: This was posthumously added to Sefer Chazon Ish, from letters that he wrote. His only source to forbid is his understanding of the Shach. The other Acharonim (Kreisi u'Pleisi 80:2, Kaf ha'Chayim) understood like the Pri Megadim. The Chachmas Adam wrote Stam, like the Shach. The Acharonim understand that he forbids only the Chelev. Sefer Pri Megadim spread throughout Yisrael. R. Yakov Kaminetsky said that before being accepted to be a Rav in Lita, one was tested on every line in the Pri Megadim, part 1 of Yoreh De'ah 1. The spread of Sefer Chachmas Adam does not show any retraction from the Pri Megadim's understanding. The Chachmas Adam explicitly says that we eat deer. YD 28:4 says that we eat Bufalo (i.e. bison. It is not what is called buffalo nowadays). The Rema says that we are stringent to cover its blood without a Berachah. No one mentioned that nowadays we do not eat it. The Chachmas Adam omitted the Simanim for Chayos, for it is not relevant to us. This implies that the Simanim of chewing the cud and split hooves, which he wrote, are relevant to us. We may eat any animal with the Simanim. He says 'we do not eat Chayos (other than deer)', i.e. like the law of Chayos, to permit the Chelev.

Shulchan ha'Levi: Others argued with the Chazon Ish, and they never agreed. This is why the Chazon Ish himself did not put this in his Sefer. His letters connote that he wanted Bnei Yisrael to require a tradition. The Sugya is unlike this. One who forbids like the Chazon Ish erred in a clearcut matter, and he must pay for any loss he caused. Perhaps in Bnei Berak, the Chazon Ish's locale, one should be stringent. Perhaps the Chazon Ish would agree about Bufalo, for there was never a custom to refrain from it. Since there are only the Kosher species, and Bufalo is much closer to an ox than to anything else, surely it is a species of ox. Initially ha'Gaon R. Y.S. Elyashiv, Shlita was concerned, but when he found that all dairy cattle in Eretz Yisrael are from them, and most of the hides used for parchments for Sifrei Torah, and much of the organ meat, he retracted.

Note: Rav Yitzchak Herzog, the first Chief Rebbi of Eretz Yisrael, suggested that perhaps requiring a tradition transgresses Bal Tosif (adding to Mitzvos) - Pnei Shor YD 1:20, cited in an article by R. Ari Zivotofsky Shlita. To avoid a rift, he did not allow Zebu (Indian cattle) into Eretz Yisrael. Recently, much of the beef consumed in Eretz Yisrael is slaughtered in South America, where cattle are bred with other species including Zebu. Many Poskim in Eretz Yisrael say that even if one is stringent about Zebu, the South American cattle resemble European cattle for which we have a tradition (from an article by R. Yirmiyahu Kaganoff Shlita).