The Mishnah (11:5) considers a case where a kohen's baby was mixed up with a shifcha's baby. The Mishnah discusses the various limits placed on each of the children due the doubts regarding their status. Even after the children grow up and eventually free one another, a doubt remains whether each of them is a kohen or yisrael. Consequently, the Mishnah addresses the further legal consequences of their questionable status. One of those listed relates to a bechor – a first born kosher animal.
Ordinarily, a bechor is brought to the Beit HaMikdash, given to a kohen and offered as a korban. The kohen that offers the korban can consume it. If however it develops a mum (blemish) that prevents it from being offered as a korban, it is still given to a kohen. Nevertheless, when slaughtered it can consumed by anyone. The Mishnah explains that if one them owned a bechor, since there is a doubt whether he is a kohen, he waits till it develops and mum and he can keep it.
The Gemara (Temura 8a) asks at what point does a kohen have rights to the bechor. Is it as soon as he is given the bechor or only after it is offered? The Gemara brings our Mishnah as part of the discussion and asks, what time does the law relating to the bechor in our Mishnah apply? Only after the destruction of the Beit HaMikdash or even during the time korbanot can be offered? The Gemara first suggests that if the case only relates to nowadays, then the law would be the same for anyone that owns a bechor and not just in the unique situation in our Mishnah. Everyone needs to wait for the bechor to develop a mum before a bechor can be consumed. Consequently, it must refer to the times of the Beit HaMikdash. If so, the Gemara reasons that the bechor must be considered as belonging to the kohen, otherwise the gizbar (treasurer of the Beit HaMikdash) could forcibley take the bechor for it to be offered. Ultimately, the Gemara rejects this case as a proof, since the Mishnah could be referring to nowadays only. The novelty of the Mishnah is that while normally, once the bechor develops a mum it would still need to be given to a kohen, in this case, since each of the individuals may be the kohen, we cannot force him to give it to another kohen based on a doubt.
The Tosfot Yom Tov, citing Rashi, explains that the rule in our Mishnah applies even in the times of the Beit HaMikdash. Since he may be kohen we cannot force him to give it to another kohen. If he would offer the korban, he would lose it, as he might not be a kohen and therefore not able to consume the korban. The Tosfot R' Akiva however cites the above Gemara that argues if the kohen only merits the korban after it is offered (as the Tosfot Yom Tov reasons in Maaser Sheni 1:3) then the gizbar could take the korban to be offered. If so, why could they retain the korban till it develops a mum? Furthermore, he cites the Korban HaEida that questions how we can delay offering the korban, violating the prohibition of "ba'al te'acher" (do not delay) when doing so, based on financial motivations.
The question of the Korban HaEida also applies to the Gemara. Recall that the Gemara reasoned that if the bechor already belonged to the kohen prior to its offering, then the ruling of the Mishnah would make sense even in the times of the Beit HaMikdash. The Gemara did not appear to be bothered by the potential issue of ba'al te'acher. Why?
The Shita Mekubetzet (8a, s.v. chelek) provides two answers. The first is that the individual in our Mishnah can wait for the bechor to develop a mum, but for no longer than a year so that the prohibition is not violated. The Shita Mekubetzet also suggests that prohibition of ba'al te'acher only applies in the case where it is clear it belongs to an Yisrael. Since our case, the status of the owner is in doubt, the prohibition would not be violated if we wait till it develops a mum.
The Chazon Yechezkel (Rosh Hashana 1:2) notes that the Rambam (Maaseh Korbanot 14:13) rules that ba'al te'acher only applies to korbanot that are offered voluntarily. The Chazon Yechezkel therefore understands that the regular ba'al te'acher that applies to offering a korban does not apply to a bechor since its kedusha is automatic. Since however it must be consumed within a year ("tochlena shana be'shana") the prohibition however would be violated if it was not consumed within the year (Bechorot 1:13). In our case however, no prohibition would be violated. If he is an Yisrael, we already explained that the prohibition of baal te'acher does not apply. Furthermore the mitzvah to consume the korban with a year does not apply to him since, as an Yisrael he is not allowed to eat the korban. Even if he is a kohen, since the doubt regarding his status prevents him from consuming it, the prohibition would not be violated since the mitzvah only applies one that can consume the korban.
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