The Mishnah (3:2) teaches that if one set traps for fish or animals prior to yom tov, they may not take the animals found in the traps on yom tov unless they are certain the animals were already caught prior to yom tov. We shall try to understand this Mishnah.
The Gemara (34a) explains that the Chachamim maintain that safek muchan -- a doubt regarding whether something is muktzeh -- is forbidden. In other words, anything one uses on yom tov must be muchan -- designated for that purpose prior to Yom Tov. Consequently, if the animal was indeed trapped on yom tov it would be muktzeh. Even though we are not sure if that was indeed the case, it is still forbidden to take that animal.
The ruling in the Mishnah requires thought considering that muktzeh is a rabbinic prohibition. In general, we rule leniently when there is a doubt regarding a rabbinic prohibition. Why then does the Mishnah rule that the trapped animals, in this case of doubt, are assur?
The Tosfot Yeshanim explains that safek muchan is assur since muktzeh itself is treated more stringently and therefore treated like a Torah prohibition. Why would that be so?
The Chazon Yechezekel (1:7, chiddushin) cites the Gemara that explains that we treat Yom Tov more stringently than Shabbat by ruling like R' Yehuda that (all types) of muktzeh are forbidden. Why so? Since the punishment associated with its desecration is less severe then that of Shabbat, there is a risk that it will be treated flippantly and degraded. It follows then that since muktzeh itself was applied to "protect" yom tov, even a safek muchan is treated stringently.
The Rashba (34b, s.v. safek) however explains that we only rule stringently, when it relates to a melacha (like trapping). The Rashba (Avodat HaKodesh 5:1) explains that were it not the case, one runs the risk of also permitting a real melacha.
The Rambam (Shevitat Yom Tov 2:6) however rules that "all" safek muchan is prohibited. The Magid Mishnah explains that this means that doubts relating to any type of muktzeh are prohibited since it is a davar she'yesh lo matirin. In other words, the matter will eventually become permitted, so there is no reason to permit it now in this case of doubt.
The Mishnah Berurah (Shaar HaTzion 497:10) explains that a practical difference between these two opinions would be if, for example, the fish would spoil if one waited till after Shabbat. If the reason why we rule stringently in our Mishnah is because it is a davar she'yesh lo matirin then that reason no longer applies. If however it is simply because the Chachamim rule stringently in a case of safek muchan, then whether or not the fish or meat will spoil is not a consideration. The Mishnah Berurah rules that one can rely on the opinion that the reason is because it is a davar she'yesh lo matirin, in order to permit a case of safek muchan if it will spoil if left until after Shabbat.
Interestingly, the Aruch HaShulchan (YD 102:19) explains that even if the food will not spoil, but simply not taste as good, that would be reason enough to define the case as a davar she'yesh lo matirin.1
1 The Aruch HaShulchan notes that the Kreiti UPleiti maintain a different position, that food would have to substantial spoil for it not to be defined as a davar she'ish lo matirin.
The understanding above assumes that the reason why a davar she'ish lo matirin is forbidden, is because one can simply wait. The Aruch HaShulchan however explains (102:1) that there is another understanding. Regarding cases of mixtures that involve a davar she'ish lo matirin, the Aruch HaShulchan explains that bitul is based on the assumption that the heter and issur are different from one another. In a case of davar she'ish lo matirin since they will soon be the same, they are not different enough to affect bitul. Whether it will spoil before the time it is mutar is irrelevant. According to that understanding, the Aruch HaShulchan explains that the Chachamim simply did not apply the stringency of davar she'ish lo matirin in those cases.
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