1) A SUKAH WITH ALTERNATING VALID AND INVALID SECHACH
QUESTION: The Beraisa states that according to both Rebbi Yehudah and Rebbi Meir, a Sukah that was covered with wooden boards that are each four Tefachim wide is invalid. Rebbi Meir says that the Sukah is invalid even when the boards are only three Tefachim wide. Rebbi Meir agrees, however, that when one places an equal width of valid Sechach between each board, the Sukah is valid.
The Gemara asks that this Beraisa contradicts the view of Shmuel. Shmuel rules that the amount of invalid Sechach that disqualifies a Sukah when placed in the middle of the roof (as opposed to at the sides) is four Tefachim. According to Shmuel, why does the Beraisa say that the Sukah is valid in the case of boards that are four Tefachim wide but are separated by strips of valid Sechach? The very presence of the four-Tefach-wide boards should invalidate the Sukah.
The Gemara answers that the Beraisa refers to a case of a Sukah which is exactly eight Amos long, on which one placed alternating strips of invalid and valid Sechach: He first placed a four-Tefach-wide board (invalid Sechach) atop the Sukah at each of the two far sides, followed by four Tefachim of valid Sechach, next to which he placed another strip of invalid Sechach, and so on until he reached the center. In the center of the Sukah's roof, he ended up with two pieces of valid Sechach with a total width of eight Tefachim. The Sukah is valid in this case because the three strips of invalid Sechach on each side are disregarded due to the principle of "Dofen Akumah" (which works for a width of up to 24 Tefachim) and eight Tefachim of valid Sechach are left in the middle.
The Gemara concludes that this mechanism works only in the case of a Sukah which is exactly eight Amos, or 48 Tefachim, wide. Each half of the Sukah (24 Tefachim of its width) is covered by three pairs of alternating invalid and valid strips of Sechach, each of which is four Tefachim wide.
If the Gemara's goal is to find a case of a Sukah with alternating strips of invalid and valid Sechach such that the final two (valid) strips end up next to each other in the center, then why does the Gemara say that there must be three sets of invalid and valid strips?
It is obvious that if there would be four sets, the Sukah would not be valid, because the valid strips of Sechach in the center would be too far away from the walls of the Sukah (28 Tefachim) for "Dofen Akumah" to work. Similarly, the Beraisa cannot be discussing a case in which there is only one set of invalid and valid Sechach on each side of the Sukah, because in such a case no two boards would have between them "a single board's width of valid Sechach," the Beraisa's description of how to validate the Sukah.
Why, though, does the Gemara not say that the Beraisa refers to a case in which there are two sets of invalid and valid Sechach? (Each set of two strips is eight Tefachim wide, and thus if two sets cover each half of the Sukah's width, then the Sukah's total width is 32 Tefachim.) In such a case, "Dofen Akumah" still applies, and the eight Tefachim of valid Sechach in the center should be considered a full-fledged Sukah. Why does the Gemara insist that the Beraisa refers only to a Sukah that is 48 Tefachim (eight Amos) wide? (MAHARSHA)
(The ARUCH LA'NER suggests that the reason why the Gemara does not say that the Beraisa is discussing a Sukah that is 32 Tefachim wide is because 32 Tefachim is not a round number of Amos (32 Tefachim = 5 1/3 Amos, or 5 Amos and 2 Tefachim). This answer, however, is rather forced.)
ANSWER: In order to answer this question, we first must ask another question. Why does the Gemara say that the Beraisa is discussing a Sukah which is "exactly" eight Amos (48 Tefachim) wide, and, as Rashi explains, "not more and not less"? The Sukah could also be one Tefach less than eight Amos (7 5/6 Amos, or 47 Tefachim) and still be valid: Since one starts to cover the Sukah with alternating Sechach by placing a four-Tefach strip of invalid Sechach at each side of the Sukah, the missing Tefach will be lost from the valid Sechach at the center. Consequently, after "Dofen Akumah" is applied, the Sukah will have only seven Tefachim of valid Sechach, but we have learned that a Sukah with seven Tefachim of Sechach is a valid Sukah!
(One cannot answer that the Gemara does not mention a Sukah with a width of 47 Tefachim because in such a case the width of the valid Sechach in the center (seven Tefachim) does not equal the width of the invalid Sechach on either side (each of which is four Tefachim), as the Beraisa stipulates. This answer is not valid, because even in the case of a 48-Tefach-wide Sukah, the amount of valid Sechach in the center (eight Tefachim) is not equal to the amount of invalid Sechach at its sides.)
What is the Gemara's intention when it limits the case to a Sukah that is exactly eight Amos wide?
The answer is as follows. Rashi (end of 17b, DH u'Modeh Rebbi Meir) explains that when the Beraisa says, "Rebbi Meir agrees that the Sukah is valid if there is an equal width of valid Sechach between each board," the Beraisa means that Rebbi Meir says the Sukah is valid "also" when the boards are four Tefachim wide. That is, Rebbi Meir refers to boards of both widths -- boards of four Tefachim and boards of three Tefachim. Accordingly, whatever Sukah he permits with boards of four Tefachim he must also permit with boards of three Tefachim.
If the Sukah is one Tefach less than eight Amos (47 Tefachim instead of 48 Tefachim), it is true that it will be a valid Sukah when the boards are four Tefachim wide. However, it will not be a valid Sukah when the boards are three Tefachim wide. When each board is three Tefachim wide, how many sets of invalid and valid Sechach will fit on each side of the roof of a 47-Tefach-wide Sukah? Starting from the edge at each side, there will be three pairs of invalid and valid strips of three Tefachim (a total of 18 Tefachim), plus another invalid board of three Tefachim, plus valid Sechach of 2 1/2 Tefachim (for a total of 23 1/2 on each side). Accordingly, some of the valid Sechach is lost, but not any of the invalid Sechach, and thus a majority of invalid Sechach (six Tefachim) is left around the valid Sechach (five Tefachim), which disqualifies the Sukah. The principle of "Dofen Akumah" is unable to cut out the invalid Sechach (as it does in the case of four-Tefach-wide boards), because there are only five (and not seven) Tefachim of valid Sechach in the middle.
Therefore, the Gemara says that the Sukah must be exactly 48 Tefachim wide. Only in such a case does each side have 24 Tefachim which contain four full sets of invalid and valid Sechach, and enough valid Sechach to make the Sukah acceptable (as the Mishnah and Gemara explain on 15a).
For the same reason, the Beraisa cannot be discussing a Sukah that is 32 Tefachim wide. Although such a Sukah indeed would be valid with alternating strips of invalid and valid Sechach that are four Tefachim wide, it would not be valid with strips of Sechach that are only three Tefachim wide. In such a case, each half of the Sukah would be 16 Tefachim wide, which would contain two sets of invalid and valid Sechach (each set is six Tefachim wide), plus one invalid board (three Tefachim), plus one Tefach of valid Sechach. The Sukah would have a total of two Tefachim of valid Sechach in the middle, leaving it with a majority of invalid Sechach and without the minimum Shi'ur of a valid Sukah in the middle. (M. Kornfeld)
2) CAN "PI TIKRAH" OF THE SECHACH FORM A THIRD WALL
QUESTION: Rava and Abaye argue whether a Sukah can be formed with the principle of "Pi Tikrah Yored v'Sosem." Rava says that a Sukah that has Halachic walls due to "Pi Tikrah" is invalid. Abaye says that such a Sukah is valid.
Rava challenges Abaye's opinion from the case of a Sukah which has only two parallel walls. In such a case, Abaye should rule that the Sukah is valid, because the edge of the beam ("Pi Tikrah") above one of the open sides between the two walls should "descend" and form the third wall ("Yored v'Sosem"). Rashi explains that Rava's question is that the edge of the Sechach that protrudes over the third side of the Sukah should be considered a "Pi Tikrah." (When the Sechach rests on an Achsadrah, the Sechach cannot be a "Pi Tikrah" because it cannot be seen from within the structure, but when it is alone on the top of a Sukah it should be a "Pi Tikrah.")
Abaye answers that in the case of two parallel walls, "Pi Tikrah Yored v'Sosem" cannot function, because people constantly walk through the area, and it is considered like a Mavoy Mefulash.
What is Rava's question on Abaye in the first place? Rava himself (19a) says that he follows the view of Rav, who says that "Pi Tikrah" normally does work to enclose the area inside of an Achsadrah (for example, to permit one to carry in it on Shabbos). He does not apply "Pi Tikrah" in the case of a Sukah adjacent to an Achsadrah because the beams of the Achsadrah are made only to serve what is inside (the porch that they cover), but not to serve what is outside (such as the Sukah adjacent to the Achsadrah). The Sechach on the Sukah, though, certainly was made to serve the inside of the Sukah, and thus even Rava should agree that "Pi Tikrah" works in such a case. Why, then, does Rava ask this question on Abaye? It is also a question on his own opinion. (PNEI YEHOSHUA)
(a) The PNEI YEHOSHUA answers that Rava knew the answer that Abaye would give him (that such a Sukah is similar to a Mavoy Mefulash and thus "Pi Tikrah" does not apply). Since Rava knew the answer, the question did not bother him according to his own opinion.
If, however, he knew the answer to the question, then why did he pose the question to Abaye? Rava reasoned that Abaye, who rules leniently and says that "Pi Tikrah" works even to make a partition to enclose what is outside of the beam, also would rule leniently in the case of a Mavoy Mefulash and say that "Pi Tikrah" works there as well. Since Abaye maintains that "Pi Tikrah" is like a solid wall (and is not based on a Halachah l'Moshe mi'Sinai, which has certain limitations and prerequisites; see Rashi 19a, DH d'Mechitzos), it should be able to form a wall even where people frequently tread.
The Gemara answers that Abaye differentiates between partitions that are made to enclose what is inside of them (in which case "Pi Tikrah" works even to enclose what is outside of the partition), and partitions through which people walk (in which case "Pi Tikrah" does not work). (See TOSFOS to Zevachim 77b, DH Odu Li, who applies a similar line of reasoning.)
(b) TOSFOS (DH Sikech), the RAN, and the RITVA understand that the question of Rava is not that the edge of the Sechach should be "Yored v'Sosem" to make a third wall. Sechach cannot be "Yored v'Sosem," because it is placed on the Sukah only as a temporary ("Arai") roof. Rather, Rava's question is that the side of the beam of an Achsadrah which is adjacent to the open side between the parallel walls should be "Yored v'Sosem" to form the third wall, according to Abaye. According to Rava's own opinion, the Achsadrah cannot be "Yored v'Sosem" to enclose the Sukah which stands outside of the Achsadrah.
If this is Rava's question, then why indeed should "Pi Tikrah" not work in such a case according to Abaye? Abaye certainly should maintain that it works in such a case, because Abaye himself states that "Pi Tikrah" works to make the beam of an Achsadrah into a wall even when there are no other walls to the Sukah.
The answer is that according to these Rishonim, "Pi Tikrah" of an Achsadrah can form only one wall of the Sukah, but it cannot form two walls (see TOSFOS DH Achsadrah). The case in which Rava and Abaye argue is a case of an Achsadrah that has two adjacent (perpendicular) walls, and in order to make a Sukah one needs to make only the third wall. Rava asks that according to Abaye, even if the two walls are opposite each other (parallel) and not adjacent, the beam of the Achsadrah should work through "Pi Tikrah" to make the third wall (as Tosfos explains in DH Sikech).
The RITVA adds that according to this interpretation, Rava's question on Abaye is understandable even according to the second version of their dispute (on 19a). According to that version, Abaye and Rava argue about the case of an Achsadrah that has Petzimin (pillars beneath the beams). In that case, Abaye says that "Pi Tikrah" works to enclose the third wall, while Rava says that it does not work. When there are no Petzimin, they agree that "Pi Tikrah" does not work to enclose it.
According to Rashi's explanation of "Petzimin," the argument between Rava and Abaye is unrelated to "Pi Tikrah." The Petzimin are pillars that are within three Tefachim of each other, and Rava and Abaye argue about the application of the principle of "Lavud." In such a case, Petzimin certainly are able to enclose the third wall of a Sukah which has only two parallel walls. Accordingly, this version of the dispute maintains that Rava never asked his question on Abaye's opinion.
However, the Ritva points out that according to Tosfos, Rava could have asked this question on Abaye's opinion, even according to the second version of the dispute. According to Tosfos and the other Rishonim, Petzimin are not pillars that are within three Tefachim of each other, but rather they are posts at the end of each of the parallel walls of a Sukah. If the parallel walls are set next to an Achsadrah (with a beam that crosses from one wall to the other, and with Petzimin below the beam), then Abaye says that since there are Petzimin, "Pi Tikrah" in the presence of Petzimin works to close the additional, third wall. Rava asks Abaye that if "Pi Tikrah" works when there are Petzimin, then it should also work when there are no Petzimin, but merely two parallel walls adjacent to an Achsadrah. Abaye answers that the normal rule of "Pi Tikrah" cannot apply here because the area is Mefulash, open on both ends with people passing through. Only in the presence of Petzimin is "Pi Tikrah" able to form the third wall.
(In fact, this also explains why Rava, in the first version of the dispute, agrees that "Pi Tikrah" works only where there are Petzimin: in the presence of Petzimin, the roof of the Achsadrah certainly forms a "Pi Tikrah" even though it is made to serve the inside of the Achsadrah. (The wall is not formed through "Lavud" of the Petzimin.) Although the Gemara later (19a) uses the word "Lavud" with regard to the Petzimin, it uses the word there figuratively to mean that the wall becomes "solid." The way that the wall is formed, however, is not by way of the principle of "Lavud" but by way of "Pi Tikrah," because the Petzimin are at a distance of more than three Tefachim from each other.)
(c) RABEINU CHANANEL and SHITAS RIVAV (on the Rif) appear to have an entirely different approach to the Sugya. They explain that even in the case of an Achsadrah adjacent to a Sukah, it is the Pi Tikrah of the Sechach that is "Yored v'Sosem." The walls of the homes around the Achsadrah, and the "Pi Tikrah" of the Achsadrah, cannot enclose the Sukah (even according to Abaye), because they are made to serve what is inside of them, and not the Sukah that is outside of them. Only when combined with "Pi Tikrah" of the Sukah do such walls suffice, according to Abaye. Rava, on the other hand, maintains that even with "Pi Tikrah" such walls cannot be used. (However, in the presence of Petzimin -- that is, poles at the four corners of the Sukah that are not part of adjacent houses and therefore are considered part of the Sukah -- even Rava allows "Pi Tikrah" to enclose the Sukah.)
Accordingly, Rava's question on Abaye is that if "Pi Tikrah" of the Sechach works, then it should enclose the Sukah even when it is not adjacent to an Achsadrah, such as in the case of a normal Sukah whose third wall has collapsed. Abaye answers that "Pi Tikrah" alone does not suffice to form the walls of a Sukah when it is not near an Achsadrah (that is, when it is not surrounded by walls of other houses), since it is similar to a Mavoy Mefulash. (This appears to be the intention of Rabeinu Chananel. -M. Kornfeld.)