In parshas ki savo we read that Moshe divided the Binai Yisroel to receive brachos and klalos. He set six tribes on Har Grizim and six tribes on Har Aival, with the Kohanim, Leviyim, and the Aron in the valley. He told the Kohanim and leviyim to turn towards har Grizim and recite a blessing (i.e. Blessed is the man who does not make an idol…) and everyone should answer "Amen". Then the Kohanim and Leviyim would turn towards har Aival and recite a klala (such as Accursed is the man who makes an idol…) and everyone would say "Amen". The Gemarah in Sota 37b derives from this parsha the concept of arvus (responsibility)- that each member of klal Yisroel is responsible for the other members' obligations.
One common application of the concept of arvus is found in the Gemarah in Rosh Hashana 29a. Generally one can only perform a mitzvah and make a bracha on it when he is obligated in it. Once he fulfills his obligation in the mitzvah he can no longer recite a bracha over its performance since he is no longer obligated in it. The Gemarah says that one can perform a mitzvah and recite a bracha on behalf of another person and fulfill his obligation in the mitzvah for them even though he has already fulfilled his own obligation. Rashi explains that this is because all of Klal Yisroel are araivim one to each other regarding their obligation in mitzvos. The Ran there adds that since we are all araivim for each other, even though one person has already performed his mitzvah, as long as another person has not yet fulfilled his mitzvah it is considered as if the first one has not yet completely fulfilled his obligation in the mitzvah. Therefore he can perform the mitzvah with on his friend's behalf even with a bracha.
The Gemarah in Brachos 20b discusses whether a woman is obligated in the mitzvah of birchas hamazon midoraisa (from the Torah) or only midrabanan (from the Rabanan). The Gemarah says that if she is only obligated midrabanan she cannot recite brchas hamazon for a man who is obligated midoraisa. The R'Osh explains that this is because women are not included in arvus with men. There is a machlokes regarding the correct intent of this R'Osh.
The Dagol Mirvavah (written by the Nodeh B'Yehuda) takes the R'Osh literally; that women are not included in arvus with men. Rabbi Akiva Aiger understands that the R'Osh was only referring to the mitzvah of birchas hamazon. The R'Osh was discussing the opinion that said that women were not obligated in the mitzvah of birchas hamazon midoraisa. According to that opinion the R'Osh explained that women would not be included in the arvus with men who were obligated in the mitzvah of birchas hamazon midoraisa and thus would not be able to recite it on their behalf. However regarding all other mitzvos that women are obligated in, they would be included in the arvus with men.
One application of this machlokes is regarding Kiddush on Friday night. Once one has davened ma'ariv he has already fulfilled his obligation of kiddush midoraisa, since he mentioned 'mikadesh hashabbos' in his davening. However he is still obligated to recite kiddush again over a cup of wine midrabanan. Women are obligated in the mitzvah of kiddish midoraisa. When a man comes home Friday night after davening he is no longer obligated in the mitzvah of Kiddush midoraisah. His wife on the other hand (if she has not davened) is obligated in the mitzvah of kiddish midioraisa. The Dagol Mirvava says that in the view of the R'Osh, the husband would not be able to be motzi his wife in kiddish since she is not in arvus with him. Only if he had not davened, and thus was still obligated in the mitzvah midoraisa would he be able to be motzi her without arvus since he is obligated in the mitzvah on his own.
Rabbi Akiva Aiger argues that regarding the mitzvah of kiddush men and women are both included in arvus and therefore even if a man has already fulfilled his obligation he can still recite kiddish on behalf of a woman who is obligated midoraisa.
Additionally Rabbi Akiva Aiger points out that if the woman would merely say "Gut Shabbos" she would have fulfilled her obligation of kiddush midoraisa as well. Therefore even according to the Dagol Mirvava that they are not included in arvus together, after she says gut Shabbos, a man would be able to recite kiddish on her behalf.
Posted By KH to KollelH blog at 9/19/2011 11:47:00 PM