MISHNAH: DOFEN AKUMAH
An open space of three Tefachim between the wall and the Sechach invalidates the Sechach.
A skylight in a room is invalid if it four Amos away from the wall.
A courtyard (three adjoining homes with a covered porch in front of each) will be invalid if the ceiling of the porch is four Amos.
Similarly invalid is Sechach which is surrounded with four Amos of invalid Sechach.
DETAILS OF DOFEN AKUMAH
Question: Why did we need three illustrations of Dofen Akumah?
Answer: Each case has a potential leniency which might cause us to limit the application of Dofen Akumah.
The case of the skylight might be lenient in that the walls were constructed to enclose a dwelling (unlike the walls of the homes in the courtyard which were not made to enclose a dwelling).
The Sukah with Sechach Pasul around the side might be lenient since those surrounding materials are fit for Sechach (their Pesul is owing to Ta'aseh ve'Lo min he'Asuy).
We thus are taught that Dofen Akumah applies in all cases, even if the material of which the wall was made is not fit to be used as Sechach at all.
(Rabah, reporting the view of Rabanan d'vei Rav): A three- Tefach break in the Sechach will invalidate the Sukah and four Tefachim of Sechach Pasul will interrupt the Sukah.
Question (Rabah): But our Mishnah allows Sechach Pasul up to four Amos (while three Tefachim is only an open space)!?
Answer (the Rabanan citing Rav and Shmuel): The Mishnah speaks of the distance to the side of the Sukah (Dofen Akumah).
Question (Rabah): But what of the case of a Sukah with less than four Tefachim of Sechach Pasul next to less than three Tefachim of space.
Surely it should be Kosher.
If filling in that space with Sechach Pasul (which is more lenient than space, since the Shiur of Sechach Pasul is four Tefachim, while that of space is three) will invalidate the Sukah, then why should the space (which is stricter than Sechach Pasul) not likewise invalidate it?
Answer (Rabanan): With that reasoning, you (Rabah) should ask yourself the same question (since according to you too, the Shiur of air is even more stringent vis-a-vis Sechach Pasul than it is according to us)!
Question (Rabah): According to my opinion, four Amos is the fixed Shiur of Sechach Pasul that disqualifies a Sukah while the Shiur of space (three Tefachim) is unrelated to the Shiur of Sechach Pasul and they do not combine to disqualify the Sukah.
According to you, however, in whose opinion the four Tefachim of Sechach Pasul in the middle of the Sukah is not a given Shiur, but a Sevara (based on four Tefachim as a Makom Chashuv, ie. there is a significant area of four Tefachim without Kosher Sechach) then what is the difference if those four Tefachim are Sechach Pasul or space?
Hence Rabah's original question returns, if Sechach Pasul invalidates the Sukah, the certainly space should?
Answer (Abaye): The case of a minimum Sukah, where the Shiurim are the same (since three Tefachim of either space or of Sechach Pasul will invalidate the Sukah) then space and Sechach Pasul should combine even by a larger Sukah?
Question (Rabah): The invalidation of a minimum Sukah is not the result of the Shiur of Sechach Pasul or of space, but because there is lacking a Shiur Sukah!
COMBINING DISTINCT SHIURIM (as learned in Sura)
Question: Rabah's premise, that when Shiurim are distinct (such as four Amos for Sechach Pasul and three Tefachim for space) they do not combine, seems counter to the Mishnah in Kelim!?
The Shiur Tum'ah for cloth is three by three Tefachim; for sackcloth, four by four; for leather, five by five; and for reed mat, six by six.
Each of these will combine with the next in line (if sewn together) to become subject to the more lenient Shiur of Tum'ah (e.g. cloth with sackcloth combine to be subject to a Shiur of four Tefachim; sackcloth with leather combine to a Shiur of five; and leather and platted reeds to a Shiur of six Tefachim.
Answer: They combine to become Tameh because sometimes they all have the same Shiur; whereas Sechach Pasul and space never have the same Shiur.
All of the above materials have the same Shiur (of one Tefach) when they are sewn as a patch into the saddle-cloth of a donkey.
Question: Of what use is a cloth of one Tefach!?
Answer (Resh Lakish citing R. Yanai): It can serve as a patch on the donkey.
COMBINING DISTINCT SHIURIM (as learned in Neharda'a)
(R. Yehudah citing Shmuel): Sechach Pasul invalidates beyond four Amos, when its area is four by four Tefachim.
(Rav): The Pesul of Sechach Pasul is always four Amos (the position of Rabah in the Sura version).
Question: What of the Mishnah, which allows a plank of four Tefachim!?
This is understandable according to Rav, since Sechach Pasul always invalidates with four Amos.
But for Shmuel, this should be Sechach Pasul!?
Answer: That plank was at the side of the Sukah, within four Amos of the wall.
Question (on Rav): What of the first version of the Beraisa discussed on 14a (that the Shiur for planks is four Tefachim)?
R. Meir holds that two planks combine (like cloths).
According to the first Lashon there, Rav holds that in the middle of the Sukah, four Tefachim invalidate the Sukah, then of what case is the Beraisa speaking?
If each plank is four Tefachim wide, then why should it be necessary to combine two planks?
If they are less than four Tefachim then for Rav (who establishes the Machlokes between R. Meir and R. Yehudah in the Mishnah on 14a by planks of four Tefachim) anything less than four Tefachim is Sechach Kosher, no matter how many combine to form the Sechach!?
Answer (Rav): The Beraisa speaks by planks of four Tefachim, that are placed at the side of the Sukah which will invalidate it if they total four Amos.