1) HALACHAH: FOUR TEFACHIM OF INVALID SECHACH
QUESTIONS: The Amora'im argue about how wide a strip of invalid Sechach must be to invalidate a Sukah (when it divides the Sukah into two sections, each of which lacks three walls or an area of seven by seven Tefachim).
(a) What is the Halachah in this case? Does a strip of invalid Sechach which is four Tefachim wide invalidate the Sukah, or does it invalidate the Sukah only when it is four Amos wide?
(b) What is the Halachah when the strip of invalid Sechach is less than four Tefachim wide? Although it is clear that such a strip does not invalidate the Sukah, is it considered valid Sechach? May a person sleep and eat directly below it, or must he make sure to sit below the valid part of the Sechach?
(a) The Rishonim disagree about the Halachah in this case.
1. The BEHAG rules leniently and says that the invalid strip must be four Amos wide in order to invalidate the Sukah. The BA'AL HA'ITUR (cited by the ROSH 1:32) cites the Yerushalmi that supports this ruling. He explains that the Gemara here also provides support for this ruling: according to the Talmidim of Neharde'a, Rav and Shmuel argue about this point (Rav says that the invalid Sechach must be four Amos wide to invalidate the Sukah, and Shmuel says that it must be only four Tefachim), and the Halachah follows Rav whenever he and Shmuel argue with regard to a matter of Isur.
2. The RIF, RAMBAM, and others rule like Shmuel who says that even a strip of four Tefachim of invalid Sechach invalidates the Sukah. The RAMBAN (cited by the Rosh) points out that the way in which the principle of "Dofen Akumah" is applied throughout Shas supports this view. According to the opinion that a strip of four Amos of invalid Sechach invalidates a Sukah even when it is in the middle of the Sukah, "Dofen Akumah" never has any application, because there is no difference between invalid Sechach next to the wall and invalid Sechach in the middle of the Sukah. This is also the ruling of the SHULCHAN ARUCH (OC 632:1).
(b) There are two opinions in the Rishonim with regard to whether one may sit below the invalid Sechach when it is less than four Tefachim wide.
1. RASHI (18a, DH she'Ein) writes (but does not cite proof) that one may sit below the invalid Sechach when it is less than four Tefachim wide.
2. The RA'AVAD (on the Rif, 14a) argues with Rashi's ruling. He rules that one may not eat under the invalid part of the Sechach when its width is less than four Tefachim but more than three Tefachim.
HALACHAH: Although the SHULCHAN ARUCH (OC 632:1) cites only the opinion of Rashi, the MISHNAH BERURAH (632:3) cites the opinion of the Ra'avad and says that it is best to avoid eating under invalid Sechach that is three to four Tefachim wide.
2) THE AMOUNT OF INVALID SECHACH THAT DISQUALIFIES A SMALL SUKAH
QUESTION: The Gemara teaches that in a small Sukah (that has an area of seven by seven Tefachim), even three Tefachim of invalid Sechach disqualifies the Sukah. RASHI explains that a width of three Tefachim is too large to apply to it the principle of Lavud, and thus that part of the Sechach is considered separated from the rest of the Sukah. It does not combine with the rest of the Sechach to form the minimum area of seven by seven Tefachim that is necessary for the Sukah to be valid. Less than three Tefachim of invalid Sechach, however, does not disqualify a small Sukah.
Why does less than three Tefachim of invalid Sechach not disqualify a small Sukah? Even one Tefach of invalid Sechach diminishes the area of the small Sukah to less than seven by seven Tefachim!
The Gemara earlier (9b) teaches that invalid Sechach that is not mixed together ("Chavtan") with valid Sechach does not become Batel to the valid Sechach and remains invalid (see Tosfos to 15b, DH v'Ha, and Insights to Sukah 15:3). Lavud does not apply because the area is not open space. Whenever something occupies the space, the principle of Lavud cannot join together the two sides of that space. Accordingly, even when there is only one Tefach of invalid Sechach, it should not become Batel to the rest of the Sechach, and thus there remains less than seven by seven Tefachim of valid Sechach.
(a) A Sukah does not need to be completely covered with Sechach in order to be valid. It merely needs to have more shade than sunlight from valid Sechach. In the case of a small Sukah with less than three Tefachim of invalid Sechach, the invalid Sechach is a minority while the valid Sechach is a "Rov," and thus the Sukah has more shade than sunlight from valid Sechach. (The strips of invalid Sechach are not worse than small gaps in the valid Sechach, which are ignored as long as there is more shade than sunlight.)
However, if a small Sukah is valid when it has a majority of shade from valid Sechach, then even when there are a full three Tefachim of invalid Sechach, the Sukah should be valid! Why does the Gemara say that a width of three Tefachim of invalid Sechach disqualifies a small Sukah?
Apparently, Rashi means that when there are three Tefachim or more of invalid Sechach, or three Tefachim or more of open space, it is not considered merely a gap in the Sechach that can be ignored, but rather it is considered a self-contained area that cannot be viewed as part of the Sechach at all. (M. Kornfeld)
(b) RABEINU CHANANEL explains the Gemara in an entirely different manner. He explains that in a small Sukah, the amount of invalid Sechach that invalidates the Sukah is not three Tefachim, but rather four Tefachim (or, more accurately, anything more than a majority of the area of the Sechach, or more than 3 1/2 Tefachim). He explains that as long as there is more shade than sunlight from the valid Sechach, the minority of invalid Sechach cannot disqualify the Sukah. It is not like open space, which invalidates a Sukah with merely three Tefachim.
According to Rabeinu Chananel, when the Gemara asks that in a small Sukah, the amount of invalid Sechach and amount of open space needed to invalidate the Sukah are the same, it does not mean that both amounts are defined by the same number of Tefachim. Rather, the Gemara means that both Pesulim are measured in the same units -- that is, in Tefachim and not in Amos. Since both are measured in Tefachim, they are considered to invalidate the Sukah with the same measure. In contrast, in a large Sukah, invalid Sechach disqualifies the Sukah only when there are four Amos of invalid Sechach, which is a different unit of measure than the one with which the Pesul of open space is measured (three Tefachim).
However, according to Rabeinu Chananel's approach, why does the Gemara attempt to prove from the Mishnah that different amounts combine? The Mishnah says that materials of different types combine to form the minimum amount necessary to be Mekabel Tum'ah. In that Mishnah, all of the Shi'urim are measured in Tefachim, and thus they are considered to be the same "Shi'ur" according to Rabeinu Chananel. There is no proof from the Mishnah that different Shi'urim combine.
The answer is that the Mishnah there (Kelim 27:2) also mentions the Shi'ur of three by three Etzba'os (the minimum size of a cloth fabric that becomes Tamei with Tum'as Mes). Even though the Gemara here does not quote that part of the Mishnah, Rabeinu Chananel himself quotes it. He explains that when the Mishnah says that a Beged joins with a Sak, it means that even a Beged, which is measured in units of Etzba'os, joins a Sak to make a Shi'ur of four by four Tefachim. Accordingly, the Gemara has a valid proof from that Mishnah that even Shi'urim which are measured in different units still can combine with each other.