If one holds that Jews living outside of Israel should sit in a sukkah because of Sfeika Dyoma (We aren't sure what day it is), then why shouldn't we shake lulav and esrog? Worst comes to worst we are just shaking a bunch of leaves?
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1judaism.stackexchange.com/questions/3244, written as an answer to another question, answers this one also.– msh210 ♦Oct 23, 2011 at 4:24
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judaism.stackexchange.com/questions/3240/…– הנער הזהOct 7, 2014 at 4:06
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Lulav is only Derabanan on the last days judaism.stackexchange.com/questions/28194/…. Moreover, it's only a Zecher leMikdash where they for sure did it for exactly 7 days because they knew the date; no need to add Safek where there never was any.– Double AA ♦Oct 7, 2014 at 4:16
1 Answer
Sitting in the sukkah can be considered a "passive" action, for without saying a berakhah (ברכה) all we're doing is simply sitting and eating outside. However, shaking the Lulav is an "active" or "positive" action which is very specific to Sukkot, and as such, it is not done for the same reasons we don't say a berakhah when sitting in the sukkah on Shmini Atzeres. (In other words, "Worst comes to worst we are not just shaking a bunch of leaves.")
In addition, the Talmud (Sukkah 46b) rules that the Lulav and Esrog are muktzah on Shmini Atzeres.
(Answer is loosely based on http://www.askmoses.com/en/article/103,21341/Do-we-eat-in-the-Sukkah-on-Shmini-Atzeret.html )
For more information on why we don't say a berakhah when sitting in the sukkah on Shmini Atzeres, please see:
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1מיתב כולי עלמא לא פליגי דיתבין. בלולב לא רצו לתקן כלל שיטלנו הלולב מספק לפי שהוא י"ט ומוקצה לטלטול ומינכר' למילתיה שנוהג בו מנהג חול אבל סוכה פעמים שסוכתו עריבה עליו אוכל בה אפי' בי"ט– samOct 7, 2015 at 19:13