Gemara

78b (Beraisa): Oso v'Es Beno applies to a mother (and her child), but not to a father;

Chananyah says, it applies to mothers and fathers.

79a (Shmuel): The Halachah follows Chananyah.

This is consistent with another teaching of Shmuel;

i. (Mishnah - R. Yehudah): One may mate offspring of a female horse, or make them work together, even if the fathers were donkeys;

i. A mule born from a (female) horse is forbidden with one born from a (female) donkey.

(Shmuel): This is R. Yehudah's opinion. He holds that we are not concerned for the father's seed. Chachamim say that all mules are one species (and are permitted with each other.)

He calls Chananyah's opinion 'Chachamim.' We are concerned for the father's seed. Every mule is a half-donkey, half-horse.

Question: Is R. Yehudah certain that we are not concerned for the father's seed, or is he in doubt?

Answer #1 (Beraisa - R. Yehudah): If a mule desires to mate, we may not mate it with a horse or donkey, only with its own kind.

If R. Yehudah were certain, he should permit mating it with its mother's species!

Rejection: The case is, we do not know what its mother was.

Objection: There are signs to tell what a mule's mother was!

(Abaye): If it has a thick voice, its mother was a horse. If not, it was born to a donkey.

(Rav Papa): A mule from a (female) donkey has big ears and a small tail. A mule born from a horse is vice-versa.

Answer: The case is, the mule lost its ears and tail, and it is mute. (We cannot determine what the mother was.)

Answer #2 (to Question 5 - Rav Huna brei d'Rav Yehoshua): All forbid a mule with its mother.

This shows that R. Yehudah was in doubt.

R. Aba told his servant 'if you lead my wagon with mules, check that they have the same mother.'

Inference: He holds that we are not concerned for the father's seed, and that we may rely on the signs given.

79b (Beraisa): Oso v'Es Beno applies to Kil'ayim and to a Koy;

R. Eliezer says, it applies to Kil'ayim born from a goat and sheep, but not to a Koy.

Chachamim are concerned for the father's seed. The Torah forbids a Seh and its son, even a partial Seh;

Conclusion: Also R. Eliezer is unsure whether we are concerned for the father's seed.

Kesuvos 111b (Yitzchak, Shimon and Oshaya): The Halachah follows R. Yehudah regarding mules;

(Beraisa - R. Yehudah): If a mule desires to mate, we may not mate it with a horse or donkey, only with a mule like itself.

Rishonim

Rif (Chulin 27a DH Gemara): Some say that the Halachah does not follow Chananyah, for R. Aba told his servant 'if you lead my wagon with mules, check that they have the same mother.' He holds that we are not concerned for the father's seed. Yitzchak, Shimon and Oshaya rule like R. Yehudah regarding mules. The Halachah follows R. Yehudah, that we are not concerned for the father's seed.

Ba'al ha'Ma'or: The Halachah follows Chananyah regarding Isur, but not regarding lashes. R. Yehudah is stringent about mules. The Gemara concludes that he is unsure, for it is a Safek whether the Mitzvah applies to males. Regarding Kil'ayim, Chachamim and R. Eliezer are both concerned for the father's seed. Rav Papa (80a) says that in every case we are stringent - for Oso v'Es Beno, Kisuy ha'Dam, Matanos and mules. R. Aba told his servant to be stringent. Rav Mesharshiya (75b) said 'according to the opinion that is concerned for the father's seed, if a Ben Peku'ah (a fetus found inside a slaughtered animal) mates with a normal female, even Shechitah does not permit the child.' The Halachah follows him. The same applies to a regular male that mated with a Bas Peku'ah. Rav Mesharshiya does not come to disagree with R. Yehudah, rather, to oppose Rabanan who argue with Chananyah. (I.e. they are not concerned for the father's seed at all - PF.)

Rebuttal (Milchamos Hash-m): If all are unsure whether we are concerned for the father's seed, who taught our Mishnah, which says that Oso v'Es Beno applies only to the mother? Really, it is R. Yehudah, and also the Rabanan of R. Eliezer. This is why the Gemara asked whether R. Yehudah is concerned for the father's seed. The Torah taught that we are not concerned for a father and its child. This is his source that we are not concerned for the father's seed. Or, perhaps he is unsure. This Mitzvah is to be merciful, since the child follows the mother, like we say about Shilu'ach ha'Kan, but it does not recognize its father. However, we are concerned for the father's seed for all other laws. Chananyah has no Safek. The Torah is always concerned for the father's seed, even to be lenient. Shamayim has no doubt. If all are concerned for the father's seed, why did Rav Mesharshiya say 'according to the opinion that is concerned'? Indeed, perhaps all are unsure. However, perhaps R. Yehudah holds that surely we are not concerned. Rav Mesharshiya did not want his teaching to depend on this question, so he taught 'according to the opinion..', even though perhaps all agree to it.

Milchamos Hash-m: Here and in Kesuvos, the Gemara ruled like R. Yehudah regarding mules. This implies that Oso v'Es Beno does not apply to fathers. Presumably, the Rif relied on this. Bahag says so. However, the She'altos rules like R. Yehudah that we are not concerned for the father's seed, even to be lenient. He permits a mother with her offspring (from a different species of father). He understands that R. Aba told his servant 'if you lead my wagon with horses or donkeys, and you want to use also mules, check that their mother is from the species they will pull with.'

Note: The others learn that only mules pulled. If so, we infer only that R. Aba is not Vadai concerned for the father's seed. However, perhaps he is unsure if we are concerned for it! (It is difficult to say that he holds that the Torah forbids working only with a Tamei and Tahor species, so two mules are forbidden only mid'Rabanan, so he would be lenient about a Safek.)

Milchamos Hash-m: He argues with Rav Huna (brei d'Rav Yehoshua), even though Rav Huna settled the question and the Gemara said that we learn from him. This is why the Gemara says that we cover the blood of a Koy, for hold that it is a species unto itself, like R. Yosi, for R. Yehudah holds like this. He is not concerned for the father's seed at all. In any case Oso v'Es Beno does not apply to fathers. Some support this from the Mishnah that discusses selling the mother to a Chasan and the day to a Kalah (we assume that they intend to slaughter them the same day). It never discusses males! We establish a Stam Mishnah in Bechoros (45b) like Chananyah, but we rely on our Mishnah, since it is clearcut Halachah.

Rambam (Hilchos Kil'ayim 9:6): Animals born from Kil'ayim, if their mothers are the same, one may mate them together. If their mothers are different it is forbidden, and one who does so is lashed. Similarly, if one mated the child even on its mother's species, he is lashed. One may mate a mule from a female donkey with another such mule, but not even with a donkey. One may not mate a mule from a female horse with a mule from a female donkey. The same applies to all similar cases. Therefore if one wants to mate two mules or pull a wagon with them, he must check the Simanim of the ears, tail and voice. If they are similar, they are the same species, so it is permitted.

Rosh (Hilchos Kil'ayim 5): According to the opinion that is concerned for the father's seed, all mules are the same. It seems that the Halachah follows Chananyah, for Shmuel rules like him, and a Stam Mishnah in Bechoros is like him. However, Bahag, the She'altos and the Rif say that the Halachah does not follow Chananyah regarding Oso v'Es Beno. Presumably this is correct, for R. Aba and all the Amora'im who gave Simanim for mules hold like R. Yehudah, and we rule like him in Kesuvos. The Targum of the verse of Oso v'Es Beno connotes that it applies only to the mother. However, we concluded that R. Yehudah was unsure whether we are concerned for the father's seed. Therefore, Kil'ayim is forbidden with its mother. R. Tam and the Ri rule like this.

Rosh (ibid.): The Rambam says that if animals born from Kil'ayim have different mothers, one who mates them together is lashed. Similarly, if one mated the child even on its mother's species, he is lashed. This is astounding. Since he lashes if they have different mothers, this shows that surely we are not concerned for the father's seed. Since he lashes for mating the child on its mother's species, this shows that surely we are concerned for the father's seed! Had he said regarding both 'it is forbidden' or 'one is lashed mid'Rabanan', this would be fine.

Beis Yosef (YD 297 DH va'Ani): I found a Sefer (of the Rambam) in which a line was drawn over 'if he mated them, he is lashed' (i.e. the Rambam retracted and deleted this). This is difficult, for he says 'similarly, if one mated the child even on its mother's species, he is lashed.' If there are lashes only for the latter, this is not similar! Also, why is he lashed in the latter case? It is a Safek! Perhaps he means that he is lashed mid'Rabanan, like the Gemara says in several places. If so, in both places the text should say 'he is lashed.' However, the Rambam is not wont to call lashes mid'Rabanan 'Malkos' (Stam lashes). Also, we do not lash mid'Rabanan for a Safek! It would be better if the text said 'he is not lashed' in both places, if we found such a text.

Poskim

Shulchan Aruch (YD 297, part 2, 7): Animals born from Kil'ayim, if their mothers are the same, one may mate them together. If their mothers are different it is forbidden, even if their fathers are the same.

Shach (9): Seemingly, this should say 'even if their fathers are different'! One might have thought that since each mule is half-horse and half-donkey, it is permitted. Really, it is forbidden, for perhaps we are not concerned for the father's seed, and they are proper Kil'ayim (a full horse with a full donkey)! If the fathers are the same, we do not hear this Chidush. Perhaps we are Vadai concerned for the father's seed! Perhaps the Shulchan Aruch means that even if the fathers are the same, and it does not appear like Kil'ayim, it is forbidden. This is difficult.

Gra (18): The correct text says 'even if their fathers are different.'

Shulchan Aruch (ibid.): One may not mate the child even on its mother's species.

Shach (10): One is not lashed for this, for perhaps we are not concerned for the father's seed. One who mates the child on its father's species is lashed, for Vadai we are concerned for the mother's seed.

Shulchan Aruch (ibid.): One may mate a mule from a female donkey with another such mule, but not even with a donkey. One may not mate a mule from a female horse with a mule from a female donkey. The same applies to all similar cases. Therefore if one wants to mate two mules or pull a wagon with them, he must check the Simanim of the ears, tail and voice. If they are similar, they are the same species, so it is permitted.

Beis Yosef (DH ha'Ba'im): We concluded that R. Yehudah is unsure whether we are concerned for the father's seed. Chananyah is concerned for the father's seed, and he holds that all mules are the same, i.e. half-donkey and half-horse. However, the Poskim rule like R. Yehudah, like it says in Kesuvos, and R. Aba followed R. Yehudah in practice.

Gra (17): Shmuel is an individual against all those who rule like R. Yehudah.