1) "DAVAR SHE'YESH LO MATIRIN" AND "MIN B'MINO"
The RAN discusses at length the topic of "Davar she'Yesh Lo Matirin Eino Batel," a prohibited object which has a way to become permitted is not annulled when it becomes unidentifiable in a mixture of permitted items. The Ran cites various sources to prove that this rule applies only when a "Davar she'Yesh Lo Matirin" becomes mixed with another object of the same type ("Min b'Mino"), but when it becomes mixed with an object of a different type ("Min b'she'Eino Mino") it is Batel even though it is a "Davar she'Yesh Lo Matirin."
Based on this premise, the Ran questions the ruling of the RIF. The Rif rules that bread baked in an oven with roasting meat may not be eaten with milk, even though only a minuscule amount of meat was absorbed into the bread. His reasoning is that the bread is a "Davar she'Yesh Lo Matirin" since one can simply eat it without milk, and therefore one may not rely on the principle of Bitul. That case, however, is a case of Min b'she'Eino Mino (the taste of meat is mixed with the bread). In such a case the meat should be Batel even though it is a "Davar she'Yesh Lo Matirin."
The Ran suggests three answers to this question on the Rif. The most widely-discussed answer is his first answer, in which he explains the underlying reason for why the Chachamim enacted that a "Davar she'Yesh Lo Matirin" is not Batel.
He first addresses the question of why the Chachamim were stringent in the case of a "Davar she'Yesh Lo Matirin" only when the item becomes mixed with a like object (Min b'Mino). If the logic behind "Davar she'Yesh Lo Matirin Eino Batel" is -- as RASHI in Beitzah (3b, DH Afilu b'Elef) says -- because a person should not rely on the Bitul of an Isur when he can eat the item when it is completely Mutar, then even when the Isur becomes mixed with a different type (Min b'she'Eino Mino) it should also not become Batel.
The Ran explains that Bitul occurs only when there are two opposing entities (Min b'she'Eino Mino), and the majority entity overpowers the minority. When the two components of the mixture are of the same substance (Min b'Mino), the Tana'im argue (Menachos 22a) about the status of the mixture. Rebbi Yehudah maintains that a substance cannot be Mevatel an identical substance. The Rabanan maintain that it can; even though the two substances are physically the same, Halachically they oppose each other since one is Heter and one is Isur. However, if the object of the Isur also has some quality of Heter (for example, it is a "Davar she'Yesh Lo Matirin" and will eventually become Mutar) and, in addition, the two components of the mixture are made of the same substance (Min b'Mino), since the two are so similar one cannot be Mevatel the other. This is why the Chachamim enacted that a "Davar she'Yesh Lo Matirin" is not Batel in a mixture of Min b'Mino.
The Acharonim discuss at length various aspects and applications of the Ran's explanation. (The Ran's explanation is cited by the TAZ YD 102:5, SHACH YD 299:1, ARUCH HA'SHULCHAN YD 102:26, and others.)
1. The Acharonim discuss the application of the principle of "Davar she'Yesh Lo Matirin" in the case of a Safek Isur d'Rabanan. The Gemara in Beitzah (4a) says that just as the Chachamim enacted that a "Davar she'Yesh Lo Matirin" is not Batel in a mixture, they also enacted that a Safek Isur d'Rabanan which is a "Davar she'Yesh Lo Matirin" is not permitted by the ordinary principle of "Safek d'Rabanan l'Kula," but rather it remains prohibited. The Acharonim (see TESHUVOS REBBI AKIVA EIGER 1:189) point out that the Ran's explanation does not explain why the stringency of "Davar she'Yesh Lo Matirin" applies to a Safek d'Rabanan. A Safek d'Rabanan is an item which might be Asur mid'Rabanan; it does not involve any mixture of items. There is no question of Bitul, which obviously applies only in a case of a mixture. A Safek Isur d'Rabanan, even one which is a "Davar she'Yesh Lo Matirin," should be permitted. (This is because in the case of a Safek, in order to be lenient there is no need for an opposing element to be Mevatel the Isur, for there is no definite Isur. Rather, the Heter comes automatically by virtue of the Safek d'Rabanan.)
Apparently, the Ran agrees that the Chachamim were stringent in the case of a Safek Isur because of Rashi's logic, that one should eat the item when it is completely Mutar and not rely on the leniency of being lenient in the case of a Safek. However, if it is true that the Ran agrees with the logic of Rashi in such a case, why does he not use the same logic to prohibit a "Davar she'Yesh Lo Matirin" that became mixed with an object of Heter of a different substance (Min b'she'Eino Mino)? The Ran should agree that the Chachamim were stringent in the case of a "Davar she'Yesh Lo Matirin" because of this logic, even when it is Min b'she'Eino Mino!
The Acharonim (see EINAYIM LA'MISHPAT) answer that the Ran maintains that Rashi's logic cannot be applied to a mixture. In the case of a mixture, the minority becomes entirely Batel, as if it was not present. Accordingly, there is no point in waiting until the Isur becomes Mutar; the Isur already is Mutar because of the Bitul! This is why the Ran needs the other line of reasoning, that the Bitul itself cannot occur when the Isur is a "Davar she'Yesh Lo Matirin" because the mixture is like a mixture of Min b'Mino. (The REMA in TORAS CHATAS (see TAZ YD 101:12 and 102:5) gives a similar reason to explain why, according to Rashi's logic that one should eat the item when it is completely Mutar, a "Davar she'Yesh Lo Matirin" is Batel in a mixture of Min b'she'Eino Mino. He explains that in a mixture of Min b'she'Eino Mino the Isur is completely Batel and it loses its independent identity. In contrast, in a mixture of Min b'Mino, although the Isur becomes Batel mid'Oraisa in the Heter, the Isur still maintains its original identity, and that is why the Chachamim were stringent and required one to eat it b'Heter and not rely on Bitul.)
2. The NODA B'YEHUDAH (2:186) questions the Ran's hypothesis that when two substances which are both Mutar become mixed with each other (Heter b'Heter) the mixture is considered Min b'Mino. The Mishnah in Kil'ayim (9:1) states that only sheep's wool can become Kil'ayim. When sheep's wool becomes mixed with camel's wool, if the majority of the mixture is camel's wool, the entire mixture may be woven with linen. According to the Ran, however, the sheep's wool should not be Batel because the mixture is one of Heter b'Heter, and the mixture of the sheep's wool in the camel's wool is a "Davar she'Yesh Lo Matirin" because it can be used by itself without linen. The Acharonim ask a similar question from the Mishnah in Nega'im (11:2).
The Noda b'Yehudah answers that according to the Rambam (Hilchos Kil'ayim), the Mishnah discusses threads spun from a mixture of sheep's wool and camel's wool. It does not refer to separate threads of sheep's wool and separate threads of camel's wool which became mixed together. Hence, even if the fibers of the sheep's wool is not Batel in a mixture with camel's wool, when the thread that is made from that mixture is spun and completed, it attains a new title -- "thread" and not "fibers," and this title applies only to the majority of fibers that constitute the thread and not to the minority of fibers. Therefore, as a thread, it can be called a thread of only camel's wool. As fibers, it can be called fibers of sheep's wool since the fibers are not Batel, but fibers that are not spun into thread are not Asur b'Kil'ayim. Therefore, the Isur of Kil'ayim cannot apply to such thread because there is no spun threads of the sheep's wool. (The same approach answers the question from Nega'im.) (The Noda b'Yehudah cites the TASHBETZ 2:4 who gives a similar answer.)
3. The ROSH cites the Yerushalmi which questions the very notion that a Neder is a "Davar she'Yesh Lo Matirin." The Yerushalmi says that when a Chacham annuls a Neder, the Neder is retroactively removed such that there never was an Isur. Why, then, should a Neder be considered a "Davar she'Yesh Lo Matirin"? The object of the Neder is not an object which is presently Asur but which can become Mutar. Rather, there is a Safek whether the object is Asur now or Mutar now (since perhaps a Chacham will annul the Neder in the future and thereby render the object of the Neder permitted retroactively).
The Yerushalmi answers that when a Chacham annuls a Neder, his Hatarah does not uproot the Neder retroactively in an absolute manner, but rather his Hatarah uproots the Neder "mi'Kan ul'Haba l'Mafrei'a": at the time that the Chacham is Matir the Neder, the Neder becomes annulled from that time onward, and although it indeed was Asur until that point, it is viewed from now on as if there never was a Neder. Until the time of the Hatarah, the object certainly was prohibited. The Chacham, by being Matir the Neder, removed the Isur from now on as if it was never there. Therefore, a Neder is a "Davar she'Yesh Lo Matirin." (The Ran and other Rishonim have the opposite Girsa in the Yerushalmi, as the RASHASH and the commentators to the Yerushalmi point out.)
According to the Rosh's explanation, however, the Yerushalmi's question is not clear. How does the Yerushalmi understand the reason why the Chachamim were stringent in the case of a "Davar she'Yesh Lo Matirin"? If the Rosh learns like Rashi that the logic for the stringency is that one should eat the food when it is completely Mutar and not rely on the Bitul of an Isur, that logic still applies even if Hataras Nedarim works completely retroactively! According to Rashi's logic, the person who made the Neder simply may have the Neder annulled before he eat the item, so that the item is in a state of complete Heter, and is not a Safek Isur, at the moment he eats it! The Ran's logic, in contrast, certainly applies; if there is a Safek that the object of the Neder might be permitted right now, there is even more of a reason to say that the object is considered Heter b'Heter (like Basar b'Chalav).
The answer is that the Rosh apparently disagrees with a basic point of the Ran. The Ran says that the stringency of "Davar she'Yesh Lo Matirin" applies not only to a prohibited object which could become Mutar later, but even to a prohibited object which has a way to be used at present in a permitted manner (such as meat, which may be eaten with other meat but not with milk). The Rosh clearly implies this is not so -- "Davar she'Yesh Lo Matirin" applies only to a prohibited object which can become Mutar. (See the Ran's explanation of the Gemara in Yevamos 81b with regard to Terumah.) The reason for the Rosh's view is that the Chachamim are stringent only in the case of a "Davar she'Yesh Lo Matirin" when such a stringency does not limit the person's use of the object by anything other than time. If the person can use the Isur tomorrow in a permitted manner in exactly the same way in which today it is prohibited to be used (due to the Safek), the principle of "Davar she'Yesh Lo Matirin" requires that he wait until tomorrow. If, however, the object is permitted to be used right now in a different manner from the way the person wants to use it, the stringency of "Davar she'Yesh Lo Matirin" does not apply and his usage of the item is not limited to the permitted manner.
The Rosh maintains that the Isur of Basar b'Chalav, a mixture of meat with milk, is not a "Davar she'Yesh Lo Matirin," because if a person seeks to eat milk together with bread which has absorbed the taste of meat, he is not required to eat his bread with something else since he wants to eat it with milk. Similarly, in the case of a Neder, if the Hatarah (annulment) would be completely retroactive, a person would be allowed to eat the mixture and would not be required to annul his Neder, because he wants to eat the mixture without having to annul the Neder he created (as he made the Neder for a reason). The Yerushalmi answers that Hataras Nedarim is "mi'Kan ul'Haba l'Mafrei'a." The Neder, in essence, is uprooted only from now on. Therefore, when he is Matir the Neder, he does not entirely uproot and forfeit the Neder he made. Consequently, since Hatarah of the Neder will not cancel the original Neder that he made (but only from now on), instead of eating the item in a state of Safek Isur now, he is required to annul the Neder and then eat the mixture.
2) DERIVATIVES OF OBJECTS PROHIBITED BY A NEDER
QUESTION: In the Mishnah (51b), Rebbi Yosi says that one who makes a Neder to prohibit himself from "Chalav" (milk) is also prohibited from "Kum" (whey), because Kum is also referred to as "milk." The Gemara asks that the statement of Rebbi Yosi here contradicts his statement in the Mishnah later (53b), where he says that one who makes a Neder to prohibit himself from "Adashim" (lentils) is permitted to eat "Ashishim." The RAN explains (based on the Yerushalmi) that Ashishim are lentils which have been cooked, ground, and roasted in honey. Why does Rebbi Yosi permit one who prohibited himself from lentils to eat Ashishim, while he forbids one who prohibited himself from milk from eating Kum?
The Gemara answers that in the place where Rebbi Yosi lived, the people called whey "milk" (milky whey), but they did not call Ashishim "lentils."
Although this answers explains why Ashishim are not forbidden as a result of being called lentils (since they are not called lentils), Ashishim should still be forbidden for a different reason: Ashishim are made from lentils! Since he may not eat lentils, he should be prohibited from eating Ashishim which contain lentils.
Although the Chachamim in the Mishnah rule that one who prohibits himself from meat is permitted to eat the sauce that comes from the meat, Rebbi Yehudah prohibits the sauce. Rebbi Yehudah maintains that the Neder prohibits not only the particular object which the person mentions in the Neder, but it also prohibits any derivative of that object. The Ran's Girsa in that part of the Mishnah is not "Rebbi Yehudah" but "Rebbi Yosi" (see the first two lines of the Ran on 52a, and Rashash). Accordingly, Rebbi Yosi should be consistent with his own opinion and certainly should prohibit Ashishim, which not only derive from lentils but is made entirely of ground lentils.
ANSWER: The Gemara later (end of 52b) quotes Rava who differentiates between something derived from the object of the Neder before the Neder was made and something derived from the object of the Neder after the Neder was made. Rebbi Yosi prohibits only what was derived from the object after it was prohibited by the Neder. When Rebbi Yosi says that one who makes a Neder to prohibit himself from lentils is permitted to eat Ashishim, he means that one is permitted to eat any Ashishim produced before his Neder, since they are not called "Adashim" but "Ashishim."