I was looking for some answers on two questions which bothered me:
1- The gemara states that the hide of a tamei animal can't be used for tefilin, and certainly not if the tamei animal is treifa or neveila ? Question: What else could a tamei animal be besides tamei or neveila ?
2- The gemara states that a tahor animal who died at the hands of Hash-m (natural death) is more logical to be used for tefilin that the one who died by the hands of a shochet. Question: If so, it would be a hidur mitzvah to use the hide of a tahor animal neveila or treifa, rather than the hide from a kasher animal. Yet, not knowing hilchot tefilin, I was assuming that the hidur mitzvah would be the opposite. And if so, why ?
1) Excellent question! In fact, it appears that the true intention of the Beraisa is to teach that Kol sh'Ken Tefilin may not be written on a Behemah Teme'ah which is a Tereifah . The Beraisa mentions Neveilah here only because it mentioned Neveilah along with Tereifah in the first part of the Beraisa which was discussing a Behemah Tehorah.
(We actually do find an instance where Shechitah is relevant for a Behemah Teme'ah. In Chulin 116b it says that if a Jew slaughters a Behemah Teme'ah for a gentile, the flesh of that animal will then be Metamei Tum'as Ochlin even though it is still moving. Nonetheless, the Peshat in our Gemara seems to be what we suggested above.)
2) There is certainly no Hidur in this, for Rebbi Yehoshua ha'Garsi only said this in order to refute the statement of the Tzeduki. (If this were not the case, a Neveilah which died at the hands of man would be Pasul for Tefilin since there is no Kal v'Chomer for such a Neveilah!) Obviously, Rebbi Yehoshua ha'Garsi said this argument only to refute the Tzeduki who did not accept the Derashos of Chazal and wanted proof based on an explicit verse or on logic.
According to Chazal, though, all Neveilos or Tereifos of Behemah Tahorah are included in the Derashah of "Min ha'Mutar b'Ficha," since they come from a species of animal that is permissible. It does not matter whether the specific animal was properly slaughtered or if it is a Neveilah. Therefore, though, there is no Hidur in using a Neveilah.