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Answer: Regarding sukka, it seems to be an explicit gemara (Sukka 8b; see Shulchan Aruch, Orach Chayim 635:1) that a non-Jew can make a kosher sukka as long as he did so to provide shade. The Rama (OC 649:1) confirms that the same is true for tying up the hadasim and aravot with the lulav. However, it is somewhat more complicated. The Magen Avraham (649:3) infers from the Rama’s wording that such a sukka or lulav is only kosher after the fact, but that, l’chatchila, a non-Jew should not do it.
The Magen Avraham traces stringency to the halacha for tefillin and tzitzit. The gemara (Gittin 45b) derives from the proximity of "you shall tie" and "you shall write" (Devarim 6, regarding tefillin and mezuza, respectively) that only those who are connected to the mitzva of tefillin can write sacred scrolls. Tosafot (ad loc.) cites Rabbeinu Tam as using this source to disallow a woman tying on tzitzit or assembling arba’a minim because they are exempt from these mitzvot. The Magen Avraham accepts Rabbeinu Tam, but only partially. Rabbeinu Tam seems to disqualify people even b’dieved, whereas the Magen Avraham, working within the Rama, who fundamentally accepts non-Jews for sukka and lulav, recommends avoiding them l’chatchila.
Tosafot, for their part, reject Rabbeinu Tam’s extension of the disqualification to women, as even non-Jews are disqualified even for tzitzit only due to a pasuk, as well as his extension to other mitzvot, based on the gemara that allows a sukka made by a non-Jew. Almost all poskim (see Be’ur Halacha to 14:1; Chochmat Shlomo (Kluger) ad loc.) fundamentally reject Rabbeinu Tam/Magen Avraham. However, due to the weight of these great authorities, many poskim recommend avoiding using a non-Jew for sukka/arba’a minim (ibid.; Mishna Berura 649:14).
If the disqualification from writing holy texts does not extend to other mitzvot, then why does the Rama (14:1; the Shulchan Aruch ad loc. is lenient) cite two opinions regarding women’s ability to tie tzitzit and prefer avoiding it? Acharonim distinguish between the mitzvot. The Be’ur Halacha (to 14:1) mentions those who disqualify women from tzitzit based on a pasuk. The Chatam Sofer (Shut, Yoreh Deah 271) distinguishes that the fact the Torah describes tying tzitzit as its own positive mitzva explains requiring someone to whom the mitzva of tzitzit is incumbent. In contrast, building a sukka is just a hechsher (preparation for a) mitzva, and thus anyone can do it. Chochmat Shlomo’s (ibid.) distinction is that tzitzit must be tied on lishma (for the purpose of the mitzva), but since sukka does not require it (just that it be done for shade), it does not need to be done by one who is obligated in the mitzva.
The indications regarding lulav are similar to those for sukka (see Rama OC 649:1 and Mishna Berura 649:14(, with two notable differences. On one hand, there is no gemara saying arba’a minim assembled by a non-Jew is kosher. On the other hand, the whole need for the binding together of the lulav, hadasim, and aravot is of a low or possibly technical level (see Sukka 33a; Shulchan Aruch, OC 651:1; development of this point is beyond our scope).
Now for our recommendations. If your neighbor would ask us, we might tell him that since the only part of the sukka that there is a question about is the s’chach, he could have his worker build the frame and walls and let you happily put up the s’chach (or even just lift it up – see Shulchan Aruch, OC 626:2). Similarly, he could let you assemble the arba’a minim without needing to teach you. However, since it sounds (it may be worth checking better) like he does not want you to do it and the case for chumra is quite tenuous and only l’chatchila, we recommend to leave things as they are.
Bemare Habazak - Rabbis Questions (652)
Rabbi Daniel Mann
635 - Ask the Rabbi: What May One Do before Havdala after Yom Kippur?
636 - Ask the Rabbi: Having a Non-Jew Build a Sukka and Assemble Arba’a Minim
637 - Ask the Rabbi: Depriving a Tree of Water
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