The following posts are a series in the understanding of doing havdalah after motzei Shabbos when 9 Av is Sunday based upon shiurim by Rabbi Yosef Elefant.
The Gemorah in Pesachim (106-107) discusses if one did not say havdalah on motzei Shabbas until when can one still do havdalah. There are two main opinions in the Gemorah if one can do havdalah all of Sunday or through Tuesday night. The halacha is that one can do havdalah through Turesday night. There are three ways to understand the machlokes in the Gemorah.
#1 Everyone agrees the time for havdalah is motzai Shabbos. However one can make up the missed time of the havdalah and the machlokes is how long is the time frame to make it up, to do tashlumin for havdalah.
#2 There is so such concept as making up a missed havdalah rather the machlokes is what is the time frame for havdalah. The opinion that holds Sunday is because since the day follows the night, Sunday is connected to motzei Shabbos (Mishna Berurah (299:17).) The opinion that holds through Tuesday night holds that the first three days of the week are all connected to the previous Shabbos.
#3 Rabbi Akiva Eger (299:6) says a hybrid approach. He says everyone agrees that Sunday is a continuum of motzei Shabbos and is the time for havdalah. The machlokes is if one has a make up time through Tuesday.
The Rambam (Shabbos 29:4) says וְאִם לֹא הִבְדִּיל בַּלַּיְלָה מַבְדִּיל לְמָחָר וּמַבְדִּיל וְהוֹלֵךְ עַד סוֹף יוֹם שְׁלִישִׁי. Why does he say you can do havdalah tomorrow and through Tuesday, just say through Tuesday and that includes Sunday? This Rambam sounds like R.A.E. He is emphasizing Sunday because that is the proper time for havdalah, and then tells you that one can make up the missed havdalah through Tuesday.
There are 4 נ"מ between these approaches in the understanding of the halacha.
#1 Another area where we find the concept of a make up is if one missed a prayer. The halacha of tashlumin by prayer is only if the prayer wasn't missed wantonly (S.A. 108:7.) What if one didn't do havdalah on motzei Shabbos wantonly can one still recite havdalah on the following days?
The Tur (271:8) says if one purposely didn't recite kiddush Friday night the Rambam holds you can make it up Shabbos day but Rav Amram Gaon holds that you can't make it up Shabbos day if you purposely neglected to recite kiddush Friday night. (The Bach explains that Rav Amram compares making up the kiddush Shabbos day to making up a missed prayer, it can only be done if it was missed unintentionally. The Rambam holds that you can make it up because the entire Shabbos is the proper time of kiddush, it is just better to recite it at the beginning.) This would parallel the דיון that we have regarding havdalah. The S.A. (299:6) says 'שכח ולא הבדיל במוצאי שבת מבדיל עד סוף יום ג. The M.B. (#15) says the same is true if one intentionally did not say havdalah on motzei Shabbos. This should depend on what the geder of the havdalah is (it is interesting the M.B. thinks that is the opinion of the Rambam. See also the Meiri Taanit (30b) mentions making up havdalah even if it was intentionally not said.)
#2 Can one make up the havdalah for Yom Tov that was missed? It the next day is a makeup day that halacha is only for Shabbos not Yom Tov but if it is considered the proper time because the day follows the night, it would apply to Yom Tov as well. R.A.E. says according to his understanding that after Yom Tov you have the next day to say havdalah.
#3 The Rosh in Berachot Ch.3 #3 brings a machlokes if one was an onan on motzei Shabbos should they say havdalah the next day. The Maharam holds that he says havdalah the next day. The Rosh disagrees since he was exempt at motzei Shabbas he does not say havdalah Sunday as well. The Minchas Chinuch (31:9) says they argue if extension of havdalah is a makeup or it is considered the proper time. If it is the proper time, then even one who is exempt motzei Shabbos will still say havdalah the following days but if it is a makeup, one who is exempt from the main obligation can not be obligated in the makeup. (It is noteworthy according to this understanding that the Rosh holds even Sunday is a makeup day not like R.A.E.)
#4 Tosfos Pesachim (107a) say since we hold havdalah can be said through Tuesday when 9 Av falls out on Shabbos we do havdalah after the fast is over. The Maharsha points out that Tosfos holds if havdalah can only be said through Sunday since the fast is over at the conclusion of Sunday, one would not be able to say havdalah. The Behag (cited in Ran Pesachim (21b-22a of dafa Rif) holds even if normally you hold havdalah can only be said through Sunday, in the case when 9 Av is Sunday the time frame for havdalah is extended. The Ran brings the Rishonim ask how can the Behag say the time frame is extended if Sunday is over? The Behag holds that you can do havdalah on Sunday as a makeup. Normally Chazal limited the makeup to Sunday because that gives you ample time to say havdalah but in the case when 9 Av is Sunday and there is no opportunity for havdalah before Monday, the makeup time is extended (that is לכאורא the peshat in the Ran's defense of the Behag.) [The Tur (299:6) brings the Behag holds the halacha that one say say havdalah even if he inappropriately ate beforehand only applies if one says havdalah on motzei Shabbos but if one is saying havdalah on subsequent days then one can no longer say havdalah. The Tur asks what is the difference? The Amek Beracha (kiddush and havdalah #5) says the Behag is lishitaso that the additional days are makeup days for havdalah. One can say havdalah even if they ate beforehand inappropriately if it is still the time for havdalah but one is not granted a makeup time if they already misappropriated the havdalah.]
The Ramban in Toras Adam (#111) argues on the Behag and says the complete opposite. He says that even if normally one would hold havdalah can be said through Tuesday, when 9 Av is Sunday you don't do havdalah following the fast since you couldn't day havdalah motzei Shabbos you can't say it on the following days as well. (similar to the Maharam held when one is an onan on motzei Shabbos since there was no obligation in the timeframe of the chiuv , there is no obligation in the makeup time.) The Meiri Taanis (30b) explains there are two ways to understand why Tosfos disagrees with the Ramban. Either he holds the obligation of havdalah through Tuesday is not as a makeup but that is considered the proper timeframe for havdalah or even if it is a makeup it is not like onan where the individual is exempt from the obligation, here there is an obligation of havdalah but one can't fulfill it because of the fast. Therefore the obligation of havdalah can be fulfilled after the fast.
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