1) "MEFAKPEK" AND "TA'ASEH V'LO MIN HE'ASUY"
QUESTION: Rebbi Yehudah and Rebbi Meir in the Mishnah disagree with regard to the opinions of Beis Shamai and Beis Hillel.
According to Rebbi Yehudah, Beis Shamai and Beis Hillel argue about a case of a Sukah that has wooden boards as its roof. Beis Shamai (as the Gemara explains) maintains that it does not suffice to pick up the boards and place them back down ("Mefakpek"). Rather, one must remove at least every other board and replace it with valid Sechach ("Notel Achas mi'Beintayim"). According to Beis Hillel, it suffices to pick up the boards and place them back down.
Rebbi Meir maintains that Beis Shamai and Beis Hillel do not argue. They both require that every other board be replaced with new Sechach.
The Gemara quotes Rav who explains that Beis Hillel, according to Rebbi Yehudah's understanding, maintains that there is no "Gezeiras Tikrah" at all -- even a board that is four Tefachim wide is valid Sechach. The only reason why boards may not be used as Sechach for the Sukah is because they were not placed there in order to create shade but in order to construct a house. Consequently, the Sukah is "Min he'Asuy" and the Torah's condition that it be "Ta'aseh v'Lo Min he'Asuy" has not been fulfilled. Therefore, Beis Hillel says that there are two ways in which one may make the boards valid: he may pick up the boards and place them back down, or he may remove every other board and place new, valid Sechach in its place.
Rav's explanation is difficult to understand. If the problem with the boards is that they were placed on the Sechach for a purpose other than to create shade (and the Sechach is "Min he'Asuy"), then why does it help to remove every other board? The boards that were not removed are still invalid Sechach, and one who sits under those boards does not fulfill the Mitzvah of Sukah.
The answer, as RASHI implies (DH v'Hachi ka'Amri), is that the removal of even half of the boards is considered a significant action (a "Ma'aseh Ma'alya"), so that the entire roof is considered to have been properly made, even the boards that were not removed.
However, if the removal of half of the boards is considered a significant action to validate the entire roof of the Sukah, then why does Beis Hillel, in his second option, say that one should be "Mefakpek," pick up and place back down all of the boards on the roof? It should suffice to lift up every other board and place it back down, because an action done with half of the boards suffices to make all of the boards no longer considered "Min he'Asuy."
(When Beis Hillel in the Mishnah says that one may be "Mefakpek" the boards, he clearly does not mean that one may pick up every other board, because had that been his intention, he would not have added that one may be "Notel Achas mi'Beintayim" (replace every other board); even returning the same board that one picked up suffices. It is clear that Beis Hillel maintains that one either must pick up every board and place it back down, or replace half the boards with new Sechach.)
ANSWER: Apparently, the act of lifting the board and placing it back down is not considered significant enough to qualify as an effective fulfillment of the condition of "Ta'aseh" when one makes Sechach. (The reason for this is because the primary act -- placing the boards upon the roof -- was already done, and when it was done it was not done for the sake of shade.) Only when one is "Mefakpek" all of the boards is that act considered significant enough to fulfill the requirement of "Ta'aseh." In contrast, the complete removal of the boards and the placement of new, valid Sechach in their place is a significant act and fulfills the requirement of "Ta'aseh" even when only half of the boards are removed and replaced.
2) "BITUL TIKRAH"
OPINIONS: The Gemara concludes that, according to Shmuel, both Rebbi Yehudah and Rebbi Meir agree that there is a "Gezeiras Tikrah" and, therefore, a Tikrah (wooden board) that is four Tefachim wide is invalid Sechach. The point of dispute between them is whether or not one may "cancel a wooden board" ("Bitul Tikrah") by being "Mefakpek" the board. What does "Bitul Tikrah" mean? How does one cancel or nullify a wooden board?
(a) RASHI explains that even though the "Gezeiras Tikrah" normally invalidates a wooden board that is four Tefachim wide, it does not apply when a person picks up the board and places it back down. Through his action, he demonstrates that he is aware that there is a Pesul of "Ta'aseh v'Lo Min he'Asuy," and thus the Gezeirah is not necessary. (See Rashi 12b, DH Amar Lach, where he writes similarly that when a person performs an action that shows that he knows that there is a Pesul of "Min he'Asuy," the Gezeirah of Tikrah does not apply.) This is what the Gemara means when it says that the act of "Mefakpek" can "nullify" the Tikrah -- the act of "Mefakpek" exempts the board from the "Gezeiras Tikrah."
How, though, does the act of "Notel Achas mi'Beintayim" (one removes every other board and replaces it with valid Sechach) validate the Sukah? Everyone, even Rebbi Meir, agrees that such an act validates the Sukah. According to Rebbi Meir, who maintains that a mere display of one's knowledge of the principle of "Ta'aseh v'Lo Min he'Asuy" does not remove the "Gezeiras Tikrah," the act of "Notel Achas mi'Beintayim" also should not work, because four Tefachim of invalid Sechach remain between each row of valid Sechach!
Indeed, Rashi (on the Mishnah) explains that according to Rebbi Meir "Notel Achas mi'Beintayim" does not remove the "Gezeiras Tikrah"; replacing every other board with valid Sechach should not validate the remaining boards. Rather, "Notel Achas mi'Beintayim" together with the principle of "Dofen Akumah" makes the Sechach in the middle of the Sukah valid (because two rows of valid Sechach now lie side by side and together total more than seven Tefachim).
(b) The RA'AVAD and RAMBAN (in his first explanation) point out that the RIF does not learn like Rashi. Even though the Rif rules like Shmuel (who says that both Rebbi Meir and Rebbi Yehudah agree that the "Gezeiras Tikrah" applies to a board four Tefachim wide), he writes that the argument between Rebbi Yehudah and Rebbi Meir in the Mishnah is whether "Mefakpek" removes the problem of "Ta'aseh v'Lo Min he'Asuy." Why does the Rif not explain that the argument is whether "Mefakpek" removes the problem of the "Gezeiras Tikrah"?
The Ra'avad and Ramban explain that the Rif understands that both Rebbi Yehudah and Rebbi Meir agree that one may cancel the "Gezeiras Tikrah" by lifting up the board and placing it back down, for this act shows that he knows about the Pesul of "Ta'aseh v'Lo Min he'Asuy" (as Rashi explains). When the Gemara says that they argue about "Bitul Tikrah," it means that they argue whether or not one may cancel the Pesul of "Min he'Asuy" that invalidates the Tikrah. The reason why Rebbi Meir says that one cannot validate the Sukah with an act of "Mefakpek," lifting and returning the boards, is because "Mefakpek" is not a significant enough action to cancel the Pesul of "v'Lo Min he'Asuy," but it does cancel the "Gezeiras Tikrah." Rebbi Yehudah maintains that it is able to remove the Pesul of "v'Lo Min he'Asuy" as well.
(Even though the Gemara earlier (11a) says that shaking the Sechach ("Na'anu'a") fulfills the requirement of "Ta'aseh," perhaps the act of "Mefakpek" is a less-significant action. "Mefakpek" might refer merely to the removal of the nails from the boards (without lifting them), as the ROSH and TUR imply (see Korban Nesanel 29:1).)
(c) The RAMBAN concludes, however, that the RIF means something entirely different. The Rif means to say that according to Shmuel, the Tikrah mentioned in the Mishnah is not a board that is four Tefachim wide, or even three Tefachim wide. The Mishnah refers to a board that is even one Tefach wide. The reason why the "Gezeiras Tikrah" applies to such boards is because they were part of an actual Tikrah (ceiling board) until now. The Tana who maintains that the "Gezeiras Tikrah" applies to a board of four Tefachim also rules that the "Gezeiras Tikrah" applies to anything that was part of an actual Tikrah until now.
However, this "Gezeiras Tikrah" -- unlike the "Gezeiras Tikrah" that applies to a four-Tefach-wide board -- can be corrected by "Bitul," by removing the boards and taking away their status of roof-boards. (That is, "Bitul" removes the association of these boards with the "Tikrah" which they originally formed.) What exactly constitutes "Bitul" is the subject of the dispute between Rebbi Yehudah and Rebbi Meir. Rebbi Yehudah rules that even an action like "Mefakpek" dissociates the boards from the Tikrah they once comprised. According to Rebbi Meir, only a significant action, and one which is clearly noticeable to all who enter, such as "Notel Achas mi'Beintayim" (replacing every other board with valid Sechach) is able to remove the status of Tikrah from the boards. Merely lifting up the boards and placing them back down does not work, either because a "Ma'aseh Rabah" (a significant action) is required (RAMBAN and RAN), or because people who see the Sukah afterwards will not know that some action ("Mefakpek") was done to the boards and thus the "Gezeiras Tikrah" still applies to them (RITVA).
This seems to be the opinion of the RAMBAM (Hilchos Sukah 5:8) as well, as the KORBAN NESANEL (29:3) notes.

15b----------------------------------------15b

3) EQUAL AMOUNTS OF INVALID SECHACH AND VALID SECHACH
QUESTION: The Mishnah (15a) states that when the roof of a Sukah is comprised of metal rods (invalid Sechach) and the gaps between the rods are equal in size to the width of the rods, the Sukah is valid as long as proper Sechach is placed in those gaps.
The Gemara challenges the opinion of Rav Huna brei d'Rav Yehoshua from this Mishnah. Rav Huna brei d'Rav Yehoshua says that there must be more valid Sechach than invalid Sechach in order for the Sukah to be valid. In the Mishnah's case, however, the Sukah is valid even though there is an equal amount of valid and invalid Sechach.
Rava answers that in the Mishnah's case, the proper way to put the Sechach on the Sukah so that it will be valid is to place the valid Sechach perpendicularly across the invalid Sechach. In this manner, the roof is comprised of more valid Sechach than invalid Sechach, because the valid Sechach covers not only the gaps between the rods, but it also covers some of the area upon which the rods themselves are resting.
How can the Sechach that rests on top of the rods be considered valid Sechach? That Sechach should be no different from the Sechach of a Sukah underneath a tree, in which case the Sechach is invalid (9b). The Sechach on top of the rods should be invalid and not be included in the measure of valid Sechach. (TOSFOS DH v'Ha)
ANSWERS:
(a) The CHIDUSHEI HA'RAN (Eruvin 15b) suggests that according to the opinion of Rashi, the act of placing the valid Sechach on top of (but not mixed together with) the invalid Sechach suffices to nullify the invalid Sechach. We learned earlier (9b) that when one does an act of "Chavtan" -- he lowers the invalid Sechach (the attached branches of a tree) onto the valid Sechach, he nullifies the invalid Sechach. The Ran appears to understand that "Chavtan" refers to an act of merely laying the valid and invalid Sechach together, but not necessarily mixing them with each other.
However, this approach does not seem to be consistent with Rashi's own explanation of "Chavtan" earlier (9b, DH b'she'Chavtan). Rashi there writes that only when the invalid Sechach is mixed with the valid Sechach, and is no longer apparent to the eye, does it become Batel to the valid Sechach. In the case of metal rods, the rods remain apparent to the eye. How, then, do they become Batel?
(b) Perhaps Rashi does not mean that the invalid Sechach becomes Batel to the majority of valid Sechach ("Batel b'Rov"). Rather, Rashi maintains that the invalid Sechach becomes Batel for a different reason. The Gemara (10a) teaches that when one spreads a sheet underneath the Sechach to beautify his Sukah, the sheet does not invalidate the Sechach, because it is considered Batel to the Sechach. According to Rashi, this applies to any object that does not serve to protect what is below it (the purpose of Sechach), such as a cloak laid out to dry on the Sechach (see Insights to Sukah 10:3).
Accordingly, perhaps Rashi here understands that since there is more Sechach than metal rods, the rods can be considered secondary ("Tafel") to the Sechach. The rods do not disqualify the Sechach, just as a sheet spread out for decorative purposes under the Sechach does not disqualify the Sechach. Only in the case of a tree, where the branches serve the purpose of Sechach and provide protection for the Sukah, is it necessary for the invalid Sechach to be bent down upon the valid Sechach and be rendered unrecognizable. In the case of metal rods, it is not necessary to perform a real act of "Chavatah," because the rods were not placed on the Sukah to create shade or provide protection, but rather to help support the Sechach. (This understanding is evident in the words of Rashi on the Mishnah, where Rashi emphasizes that the rods were placed there only to support the ceiling, which implies that they were not placed there to serve as the ceiling in their own right.) (M. KORNFELD)
(c) RABEINU TAM (cited by Tosfos 9b, DH Ha) explains that when a Sukah has valid Sechach that creates more shade than lets in sunlight, any invalid Sechach that is above or below it that lets in more sunlight than shade cannot invalidate the Sechach (see Insights to Sukah 9:2).
(d) RABEINU CHANANEL explains the Gemara differently. He explains that the Gemara relies on what it said earlier, that there is more space between the metal rods than the exact width of the rods themselves. The Gemara questioned that suggestion and asked that "it is not possible to be Metzamtzem" (this differs from our text of the Gemara, which asks that "it is possible to be Metzamtzem"). The Gemara means that even though the space between the rods is greater than the width of the rods, perhaps one will not fill that space entirely with valid Sechach, and he will be left with a majority (or equal amount) of invalid Sechach.
Rava answers that one places the Sechach on top of the rods crosswise. In this manner, one assures that he will cover all of the space in the gaps between the rods, which are slightly larger than the width of the rods themselves. Accordingly, the amount of valid Sechach is greater than the amount of invalid Sechach.

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