The Rishonim discuss whether or not a person may do an aveira so the public can benefit. Tosfos in Shabbos, daf daled amud alef brings proof that it is permissible from the gemara in Gittin, daf lamed ches that tells us that Reb Eliezer freed his slave to complete a minyan despite the fact that there is a lav against freeing slaves. The gemara explains that it was done because this was a mitzvah for the rabim (public.) Tosfos tells us we learn from here that when it comes to a mitzvah for the public one may do an aveirah.
Reb Elchanan in Kovetz Hearos explains that the reasoning for permitting an aveira is due to the concept of arvus, every Yid is responsible for other Yidden. If one Yid is faced with an asseh and the other Yid is faced with a lav, the result is that the asseh will be docheh the lo saaseh since the person being oveir the lav is also required to do the asseh of his friend because of arvus.
The question arises whether one has a mitzvos aseh kiyumis where he could be mekayem the mitzvah but it is not mandatory; for example, if a person does not have a minyanhe could still daven on his own. Do we say even in the case of mitzvas kiyumis that asseh docheh lo saaseh or not? This question would be applicable in a case where a person is patturfrom a mitzvah but still wants to do the mitzvah. Do we say asseh docheh lo saaseh, or do we say he is really pattur from the mitzvah, therefore the asseh is not docheh the lo saseh. Would a person need to daven with a minyan if it would translate into him losing money? (Note: this is not a case where he won’t earn money.) No, a person would not. How could Reb Eliezer free his slave in order to daven with a minyan when he was losing money by freeing his slave? If you hold that a mitzvah kiyumis is docheh a lo saaseh, then it is obvious that he should free the slave and be mekayem the mitzvah of minyan. If you hold that asseh is docheh lo saaseh only by a mitzvah chiyuvis, he should not have freed his slave because he was losing money by doing so.
There is another tirutz that differs from the Kovetz Hearos mentioned above. The reason we tell someone to do an aveirah so the public can do a mitzvah is not due to a din of arvus but rather a din in kadima. When a scenario comes along where a private person is up against a tzibbur, we say that the tzibbur has kadima. This would make sense why Reb Eliezer freed his slave despite the fact that he personally had no chiyuv to daven with a minyan since he was losing money.
The difference between these two tirutzim would be whether the person has to do teshuva for doing an aveirah. According to Reb Elchanan in Kovetz Hearos he would not need to do teshuva since his aveira is nidcheh for the mitzvah. According to the din of kadima the person lemaaseh did an aveirah and would require teshuva.
We see from this discussion how important it is to help the klal and how much appreciation we need to have for those who do so; they sometimes sacrifice their own mitzvos and or do aveiros le’tovas the klal. May they be zocheh to all the brachos we say before mussaf on Shabbos.
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