Gemara

Nazir 37a - Question (Abaye): What is the source that "Mishras" teaches that Heter joins with Isur? Perhaps it comes only to forbid Ta'am k'Ikar (a food that absorbed the taste of Isur, like in the coming Beraisa)!

(Beraisa): "Mishras" forbids Ta'am k'Ikar. A Nazir is liable for drinking water that absorbed taste from grapes. We learn from here to all Isurim.

Isurei Nazir are not permanent, they do not forbid all benefit, and there is a Heter for them. Still, the Torah forbids Ta'am k'Ikar. Kil'ayim (crossbreeds) in a vineyard are permanently forbidden, all benefit is forbidden, and the Isur cannot be permitted. All the more so, Ta'am k'Ikar is forbidden!

Answer: R. Akiva learns from "Mishras" that Heter joins with Isur. (The Tana of this Beraisa disagrees.)

Zevachim 78a (Reish Lakish): If one ate Pigul and Nosar that were mixed together, he is exempt. Surely, one of them was the majority, and the other is Batel. This teaches that if an Isur gives taste to a food of a larger volume than itself, it does not forbid it (to Mechayev lashes) mid'Oraisa.

Avodah Zarah 67a (R. Avahu): If the Isur gives taste and is intact, if it has k'Zayis (an olive's worth of Isur) bi'Chdei Achilas Pras (in the volume of a half loaf), it is forbidden, and one is lashed for eating (a k'Zayis of Isur in) the mixture;

Chulin 98a: R. Chiya bar Aba and R. Shmuel bar Rav Yitzchak argued about whether an Isur (that cannot be tasted) is Batel among 60 or 100 parts of Heter.

Both opinions learn from (Bitul of) the Zero'a (foreleg) of Eil Nazir. The Zero'a is cooked with the ram. The Zero'a is forbidden to Zarim, but the rest of the ram is permitted. I.e., the Ta'am of the Zero'a is Batel in the ram. There is 60 times as much meat and bones in the ram as in the foreleg, and 100 times as much meat. (One opinion ignores the bones, for they do not give or absorb Ta'am.)

(Beraisa): This is the case of Isur becoming permitted.

Question: Why don't we learn to other Isurim?

Answer #1 (Abaye): The Beraisa is like R. Yehudah, who says that Min b'Mino (a mixture within one species) is never Batel. Only the Zero'a is Batel b'Mino. It is a Chidush (we Mevatel an Isur l'Chatchilah), so we may not learn a leniency from it;

We do learn from it that an Isur is Batel in 60 or 100 parts of Heter. This is a stringency. Mid'Oraisa, an Isur is Batel in any majority!

Answer #2 (Rava): The Beraisa teaches that Ta'am is Batel only regarding the Zero'a. For other Kodshim, Ta'am k'Ikar. We don't learn to other Kodshim, for the Zero'a is a Chidush. We learn from Chatas - "whatever touches its flesh will become forbidden", even if the other food absorbed only a tiny amount.

Rishonim

Rambam (6,7): If wine mixed with honey, and it all tastes like wine, and there is not a Revi'is of wine within a Pras of mixture, it is forbidden mid'Rabanan. If he ate it, he is lashed mid'Rabanan.

Ra'avad: Ba'al ha'Halachos (BaHaG) forbids mid'Oraisa, but one is not lashed for it. I brought proofs for BaHaG elsewhere.

Kesef Mishneh: The Rambam rules that Ta'am k'Ikar, therefore one is lashed for it. The Ra'avad holds like the Poskim who forbid only due to Chetzi Shi'ur mid'Oraisa, and one is not lashed for this.

Rosh (Avodah Zarah 5:30): Ta'am k'Ikar is mid'Oraisa for eating and Hana'ah.

Rosh (Chulin 7:31): Rashi says that Rava holds that Ta'am k'Ikar is mid'Oraisa only regarding Kodshim. We learn from Zero'a not to permit Chulin mixtures unless there are 60 or 100 parts Heter. This is mid'Rabanan. The Gemara in Nazir says that we learn Ta'am k'Ikar from Mishras. This is an Asmachta. Really, we cannot learn, for Nazir and Gi'ulei (forbidden tastes absorbed in pots of) Nochrim are Shenei Kesuvim. What forced Rava to do so? The ram does not absorb true taste for the Zero'a, for it is only one part in 60 or 100! Rather, we must say that regarding other Kodshim, any amount forbids.

Rebuttal (Rosh): This is difficult. Reish Lakish exempts one who are Pigul, Nosar and Tamei, due to Bitul. We rely on Bitul when Menachos get mixed, and to allow cooking Shelamim in a pot that was used for Chatas. Rather, mid'Oraisa we follow the majority. Chachamim did not decree in the Mikdash (to require 60). This shows that only taste forbids. R. Tam says that Ta'am k'Ikar of Chulin is mid'Oraisa. This is why one is Yotzei Matzah even if the majority was rice, as long as the wheat can be tasted. R. Yochanan holds that Ta'am k'Ikar mid'Oraisa. It cannot be an Asmachta, for we ask how he learns from Mishras, since it is needed for Heter Mitztaref l'Isur. We cannot learn from Rava. He merely rejected Abaye's opinion. Some defend Rashi. A dough has taste and substance (wheat flour). It is unlike Chelev that dissolved into food, which is mere taste. This is no defense. The wheat is not recognizable in the dough. Rava himself holds that Ta'am k'Ikar mid'Oraisa in Nazir.

Rosh: R. Chaim Kohen says that when there is k'Zayis bi'Chdei Achilas Pras, this is considered substance. Real taste of the Zero'a is Batel, e.g. if the leg with the Zero'a is outside the liquid, the leg absorbs true taste. According to R. Tam we must be stringent about a Safek whether or not there was 60 times as much Heter as Isur. According to Rashi, we may be lenient.

Ran (Chulin 35a DH Garsinan): The Gemara said that requiring 60 (or 100) parts of Heter is a stringency. Mid'Oraisa, an Isur is Batel in any majority. Really, an Isur that gives taste is not Batel. K'Zayis bi'Chdei Achilas Pras is mid'Oraisa, Min b'Eino Mino. Min b'Mino is Batel mid'Oraisa in a majority, and mid'Rabanan we require 60. What we learn from the Zero'a is an Asmachta. It is a stringency to require 60 parts of Heter when the Isur does not give taste. Mid'Oraisa, a majority of Heter suffices. Chachamim chose 60 because some Isurim, e.g. onions, give taste to 60 parts. Some Isurim give taste to only 10 parts. Chachamim fixed a uniform Shi'ur for Bitul. Rava said that the Beraisa permits Ta'am k'Ikar for the Zero'a, but not for other Kodshim. Really, here there is no Ta'am, because there is 60 or 100. Rava means that R. Yehudah, who normally forbids Min b'Mino, agrees here that a mere absorbed taste is Batel. We cannot learn to a dissolved Isur.

Ran (ibid.): 'Ta'am k'Ikar is forbidden for other Kodshim' connotes that mid'Oraisa, it is permitted for Chulin. The Gemara in Nazir connotes that it is mid'Oraisa. Rashi proved that it is mid'Rabanan. In Avodah Zarah, we forbid taste without substance, but do not lash for it. If Ta'am k'Ikar were mid'Oraisa, we would lash! R. Tam learns from Nazir that Ta'am k'Ikar is mid'Oraisa. Here, the Gemara means that Min b'Mino, Ta'am k'Ikar is mid'Oraisa only for Kodshim, according to R. Yehudah. Even though it is mid'Oraisa for Min b'Eino Mino, one is not lashed for it. Chachamim learn from a Kal va'Chomer from Nazir, and one is not lashed due to a Kal va'Chomer. R. Akiva learns from Giulei Nochrim, which is only an Isur Aseh (to Kasher Nochri Kelim).

Rebuttal (of R. Tam - Ran): If Ta'am k'Ikar, surely one is lashed! The Isur itself is mid'Oraisa. Ta'am k'Ikar merely reveals that the mixture is like the Isur itself, so one is lashed, like for the Isur itself!

Poskim

Shulchan Aruch (YD 98:2): If (Heter and Isur) were mixed Min b'Mino and we cannot tell (the quantities of each), if we know that the majority was Heter, it is permitted. If we do not know that the majority was Heter, it is forbidden.

Taz (3): The Poskim conclude that Min b'Mino is Batel mid'Oraisa in the majority. It is forbidden mid'Rabanan if there is not 60 times as much Heter. Therefore, if there is a Safek whether or not there are 60, one may be lenient. Min b'Eino Mino is Asur mid'Oraisa, so one must be stringent about a Safek.

Shach (7): Toras Chatas says that Isur v'Heter is stringent about a Safek regarding Ta'am k'Ikar of milk and fowl, since Ta'am k'Ikar is mid'Oraisa. This is unreasonable. Milk and fowl itself is mid'Rabanan! Isur v'Heter said so only because he forbids milk and fowl mid'Oraisa. This is not the Halachah.

Beis Yosef (OC 442 DH v'Ika l'Meidak): All forbid k'Zayis bi'Chdei Achilas Pras mid'Oraisa. The argument is about Ta'am k'Ikar when there is not k'Zayis bi'Chdei Achilas Pras.