The Mishnah discusses what to do when the blood of an offering that requires four applications becomes mixed with the blood of an offering which requires one application. Rebbi Eliezer says to do four applications and Rebbi Yehoshua says to do one application. The Mishnah then proceeds to explain what the underlying machlokes is. Rebbi Eliezer wants to avoid bal tigra and Rebbi Yehoshua wants to avoid bal tosif. Rebbi Eliezer responds that it would only be bal tosif if you do four applications solely with an offering which requires one. Rebbi Yehoshua counters that it would only be bal tigra if you do one application solely with an offering that requires four. Now there doesn't seem to be any reason why this caveat would apply only to bal tigra but not bal tosif or only to bal tosif but not bal tigra. So presumably, at this point they both agree that there would not be bal tosif or bal tigra regardless of whether you do one application or four applications. If so then Rebbi EliezeRebbis position makes much more sense because if you do four applications there is no issur and you were yotzei everything lechatchila. But if you do one application then you regarding the offering that requires four applications you are only yotzei bedieved. Rebbi Yehoshua then adds (perhaps to avoid this) that even if there was an issue of bal tosif and bal tigra it would be better to violate the issur of bal tigra because it is shev ve'al taaseh. This indicates that we cannot say for certain that there is no issue of bal tosif and bal tigra. Now there is no reason to assume that Rebbi Eliezer holds that it is better to violate an issur that is kum v'aseh than to violate an issur that is shev v'al taaseh. So why does he not agree with Rebbi Yehoshua that you should only do one application? Well there are two possibilities: 1)Rebbi Eliezer is sure that the above caveat is true. 2)As Rashi explains in the beginning, Rebbi Eliezer holds that there would never be bal tosif in this situation because the additi onal applications would be considered like water. The problem is that according to both of these possibilities, none of the sevaros discussed in the mishnah are what they are actually arguing about.
Tzvi Hertzberg, Flushing, NY
1) There is in fact a difference between Rebbi Eliezer and Rebbi Yehoshua at the beginning of the Mishnah, as the Tosfos Yom Tov writes in the name of Tosfos. When Rebbi Eliezer said that if he does 1 application instead of 4, he transgresses Bal Tigra, he did not think that Rebbi Yehoshua was going to retort that if he does 4 instead of 1 he trangresses Bal Tosif. This is because it was clear to Rebbi Eliezer that the extra blood is considered like water. However, it transpired that Rebbi Yehoshua does not agree with the latter Sevara.
When Rebbi Eliezer continued the discussion by asserting that Bal Tosif applies only when the blood is by itself, it was necessary for him to say this only because Rebbi Yehoshua did not consider the extra blood as water, but according to Rebbi Eliezer himself there was actually no need to continue to discuss Bal Tosif.
According to this, there indeed is a dispute between the two Tana'im at the beginning of the Mishnah, namely concerning whether we consider the extra blood as water or not.
2) The Ritva (Rosh Hashana 28b, DH v'Od) explains the dispute in the continuation of the Mishnah as follows. Rebbi Yehoshua said that it may be that it is only Bal Tigra that one does not transgress if the blood is mixed with other blood, but Bal Tosif one always transgresses whether the blood of the Korban is by itself or is mixed with the blood of a different Korban. Rebbi Yehushoua then continues and argues that even if one says that both Bal Tigra and Bal Tosif apply only when the blood is by itself, nevertheless there is a different problem involved. This is the problem of "Mar'is ha'Ayin." Even though, mid'Oraisa, there is no problem when it is mixed up, it still looks to the onlooker that one is transgressing Bal Tosif or Bal Tigra. The question now is as follows: If, anyway, the person will appear to others as though he is transgressing one of the above, which is it is preferable for him to appear to be transgressing?
Rebbi Yehoshua answered this question by saying that it is better to appear to be transgressing Bal Tigra, because such a transgression is done only in the passive (Shev v'Al Ta'aseh) manner, but if one is suspected of Bal Tosif, this means people suspect him of actively transgressing, b'Kum v'Aseh.
In contrast, Rebbi Eliezer is not concerened about the problem of Mar'is ha'Ayin, and therefore he says that it is preferable not to take away from the Mitzvah. Consequently, it is better to do 4 applications (see Tosfos, Rosh Hashanah 28b, DH LO (#1)).
Kol Tuv,
Dovid Bloom