BAVA KAMA 42 - Ari Kornfeld has generously sponsored the Dafyomi publications for this Daf for the benefit of Klal Yisrael.

QUESTION: Rebbi Eliezer gives two explanations for why a verse is necessary to teach that the owner of a Tam that killed a person is exempt from paying Chatzi Kofer. If the animal is Chayav Misah, obviously the owner is exempt from Kofer because Kofer of a Tam must be paid "mi'Gufo," and after the Gemar Din the animal is worthless. Rebbi Eliezer explains that the verse is necessary for when the goring of the Shor Tam is witnessed by only a single witness, or when there were no witnesses but the owner admits that it gored and killed a person. The Gemara points out that this is a weak answer since, according to the opinion that Chatzi Nezek is a Kenas, the owner would be exempt from paying Chatzi Kofer because he was Modeh b'Kenas, he admit to a Kenas, and one who admits to a Kenas is exempt from paying it. The Gemara explains that Rebbi Eliezer must maintain that Kofer is a Kaparah, an atonement for the owner, and therefore the owner is not exempt from Kofer as a result of his admission, even if the Shor is only a Tam.
Rebbi Eliezer suggests further than the verse is necessary in a case where the animal intended to kill another animal and ended up killing a person. In such a case the animal is not Chayav Misah and therefore it would be possible for the owner to pay Chatzi Kofer.
The Gemara cites two Amora'im who argue about the sequence of Rebbi Eliezer's answers. Both Amora'im agree that the first answer mentioned above is a weaker answer. Rav Tavyumi says that Rebbi Eliezer first said the weaker answer, and then when he thought of a better answer he taught that answer. Rav Kahana argues and says that Rebbi Eliezer first said the better answer, and then he added the weaker answer as additional support to his words.
Why should Amora'im argue about such a seemingly insignificant point? What relevance is there to the sequence in which Rebbi Eliezer suggested his two valid answers?
(a) The BA'AL HA'ME'OR explains why Rebbi Eliezer would have ignored the better answer and suggested the weaker answer first. He merely wanted to test Rebbi Akiva to see if he would reject the weaker answer by himself. This demonstrates that Rebbi Akiva was a pupil of Rebbi Eliezer and of lesser stature, such that Rebbi Eliezer would find it necessary to test him.
Rav Kahana, who says that Rebbi Eliezer presented the better answer first, maintains that Rebbi Akiva was more like a "Talmid Chaver," a colleague, to Rebbi Eliezer, and therefore Rebbi Eliezer would not have tested him. Rather, Rebbi Eliezer added the second answer as an additional possibility, albeit a weaker one, to answer his question.
It is still not clear what, if any, practical difference exists between the opinions of Rav Tavyumi and Rav Kahana. What difference does it make whether Rebbi Eliezer was greater than Rebbi Akiva or whether they were of equal stature?
The answer might be as follows. The Gemara in Kesuvos (84b) teaches that although the Halachah follows the view of Rebbi Akiva when he argues with another contemporary Tana, the Halachah does not follow his view when he argues with his rebbi, such as Rebbi Tarfon.
Whom does the Halachah follow when Rebbi Akiva argues with Rebbi Eliezer? This is the subject of the dispute between Rav Tavyumi and Rav Kahana. According to Rav Tavyumi, the Halachah follows the view of Rebbi Eliezer, since Rebbi Akiva was a student of his. According to Rav Kahana, the Halachah may follow Rebbi Akiva, because he was a Talmid Chaver of Rebbi Eliezer. (See Bava Metzia 59b and the incident of "Tanuro Shel Achna'i.")
(b) The RAMBAN (in Milchamos) explains that the Gemara implies that according to Rav Tavyumi, Rebbi Eliezer changed his mind about the weaker answer and rejected it as a possible answer. According to Rav Kahana, on the other hand, the weaker answer remains a valid answer. The reason why Rebbi Eliezer changed his mind about the weaker answer, according to Rav Tavyumi, is that although the Shor is not killed unless two witnesses testify since -- even when one witness testifies -- the Shor is worthy of being killed had two witnesses been there, it is obvious that the owner of the Shor Tam should be exempt from Kofer (because it is similar to a Shor ha'Niskal and cannot pay "mi'Gufo," according to the testimony of the single witness or of the owner). (This is why Rebbi Eliezer changed his mind according to the Ramban. Alternatively, Rebbi Eliezer changed his mind because Kofer is Mamon and not Kaparah. Therefore, the owner of a Shor Tam would be exempt from paying if he would be incriminated only by his own word or by the word of a single witness, because of Modeh b'Kenas. Accordingly, there is a clear practical difference between Rav Tavyumi and Rav Kahana: a case where a Shor Mu'ad kills a person, but the killing is known only from the owner's testimony or from the testimony of a single witness.)
(See also PNEI YEHOSHUA who suggests a different approach.)