Masechet Horayot opens with the law of par helem davar shel tzibur. We learnt that if the Sanhedrin makes a mistake in a ruling of prohibition that is punishable with karet and this led to its violation by a majority of Israel, then this special bull offering must be brought. In the fifth Mishnah we saw that debate regarding who bring this offering. According to R' Meir, the obligation is the Sanhedrin's. R' Yehuda however rules that obligation is with every tribe, and each must bring their own offering. Finally, R' Shimon rules that both the Sanhedrin and each tribe are obligated. We shall focus on the opinion of R' Yehuda.
The Mishnah continues that if only seven of the tribes sinned as a result of following the ruling, then all the tribes must still bring their own. Interestingly, the Gemara notes that the requirement for the other tribes to bring their own offering exists even if it was only one of the tribes that followed the ruling. Why then did the Mishnah mention seven tribes? Rashi explains that the case was necessary for the opinion of R' Meir that only obligates the Sanhedrin when the prohibition is violated by either most of the general population or a majority of the tribes.
The Rambam however disagrees, explaining that in a case where one or two of the tribes followed the ruling, the other tribes would only be obligated if those one or two tribes constituted a majority of the population. The Tosfot Yom Tov explains that the capacity of one tribe to obligate the rest is implied by the last case in the Mishnah. There, R' Yehuda rules that if one tribe's Beit Din ruled erroneously, then that tribe would be obligated to bring the offering, while the other tribes would be exempt. The implication is that the exemption is because it was the tribe's Beit Din that ruled. Had it been the Sanhedrin's ruling, then the other tribes would be obligated as well.
The Rambam's position gives pause for thought considering that R' Yehuda treats each tribe independently and we do not find him sharing R' Meir's requirement of most of the population. The Lechem Mishneh (Shegagot 13:1) explains that while it true that each tribe is treated independently to generate their own independent obligation, the novelty that they can obligate the other tribes also, requires a majority of the population (or tribes). Consequently, when our Mishnah brings the case of seven tribes, it specifically choosing one where there was a majority of the tribes such that other tribes would be obligated as well.
A novel approach is found in the Rashash, that appears to bridge the opinions of Rashi and the Rambam. The Rashash also maintains that our case of seven tribes was also critical for the opinion of R' Yehuda, for it is only in that case that all the tribes would each bring their own offering. Where the Rashash differs is where a single tribe sinned that was not the majority. In that case he explains that the other tribes would be required to bring one offering together.
The Rashash maintains that this understanding explains the Beraita cited in the Gemara. The Beraita teaches that according to R' Yehuda if two tribes sinned, they would bring two bulls. Rashi explains that the case of two tribes was necessary because one might of thought that in that case only one bull would be brought for both tribes. The Maharsha however asks that according to Rashi in that case, even the tribes that did not sin also brought offerings, so why would we have thought that the two tribes would have only brought one bull? The Rashash answers that in the case where only two tribes sinned (that did constitute the majority) then the remaining tribes would bring one offering together. Consequently, one might have thought that those that sinned would also be grouped together and bring one offering. That is why that Beraita needed to teach that those that sinned, despite being in the minority, are still treated independently.
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