Gemara

38b (Beraisa): Refuse of dates is permitted whether it was put in water heated in a big or small pot.

The Tana permits Nosen Ta'am li'Fgam (what harms the taste).

(Rav Sheshes): Cooked oil of Nochrim is forbidden.

Rebuttal (Rav Safra): There is no concern lest they mix in wine, nor for Bishul Akum, and nor for absdorptions, for Nosen Ta'am li'Fgam is permitted!

39a (Mishnah): A piece of (the spice) Chiltis (of Nochrim is Asur).

This is because they cut it with knives that absorbed Isur. Even though Nosen Ta'am li'Fgam is permitted, because Chiltis is (so) sharp, it causes the absorbed fat (in the knife) to be li'Shvach (improve the taste).

65b (Mishnah): The rule is, if Isur is li'Shvach, (what it fell into) is forbidden. If it does improve the taste, it is permitted, e.g. vinegar that fell on grits.

67a - Version #1 (Reish Lakish): Nosen Ta'am li'Fgam does not permit if the mixture tastes bad due to too much or too little salt or spices. Rather, it applies if it is properly seasoned, and the Isur detracts from the taste.

Version #2 (Reish Lakish): Do not say that if a mixture had more or less seasoning it would taste good, therefore Nosen Ta'am li'Fgam does not permit. Rather, if (the way it is) now the Isur detracts from the taste, it permits.

67b: The Halachah follows Version #2 of Reish Lakish.

(Beraisa - R. Meir): Whether an Isur is li'Shvach or li'Fgam, it forbids;

R. Shimon says, li'Shvach forbids, but li'Fgam does not.

(Beraisa): "Lo Sochlu Chol Nevelah la'Ger (... Titnenah va'Achalah)" - it is called (and forbidden like) Neveilah only if a Ger (Toshav) would eat it.

This is R. Shimon's source. R. Meir holds that the verse excludes only a Neveilah that was spoiled from the beginning (before the animal died). R. Shimon does not need a verse to exclude that case. It is like earth.

68a (Ula): R. Meir and R. Shimon argue when it initially improved the taste and later was Pogem, but all permit if it was Pogem from the beginning.

(R. Yochanan): They argue when it was Pogem from the beginning.

Support (Mishnah): If enough Se'or (a fermenting agent) of Chulin fell into a dough to ferment it and it fermented it, and afterwards, Se'or of Terumah or Kilai ha'Kerem fell in, and it was enough to ferment it, the dough is forbidden;

R. Shimon permits it.

The Se'or of Isur is Pogem from the beginning (it overferments the dough), the Tana'im argue about this, like R. Yochanan!

Rejection (R. Zeira): That is li'Shvach from the beginning. Even though it harms the taste, it enables the dough to ferment other doughs.

68b - Support (Beraisa): If (forbidden) wine (or vinegar) fell into lentils (or grits), they are forbidden (even though it is Pogem from the beginning);

R. Shimon permits it.

Rejection: No, the vinegar fell into cold grits, and later they were heated. Initially it was li'Shvach, and later it was li'Fgam.

Rejection of Rejection: All forbid when it was initially li'Shvach!

(Beraisa): If Se'or of Terumah and Se'or of Chulin fell into a dough, and each was enough to ferment it by itself, if the Terumah fell in first, R. Shimon and Chachamim both forbid the dough (to a Zar).

Conclusion: They argue when it was li'Fgam from the beginning.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 14:11): If one mixed bitter things such as wormwood into Yayin Nesech or a pot of Neveilah and ate it when it is bitter, or ate a forbidden food after it spoiled and is no longer considered a food for people, he is exempt. If one mixed something bitter into a pot of meat and milk or wine of Kilai ha'Kerem and ate it, he is liable.

Rambam (16:16): If Se'or or spices of Terumah, Orlah or Kil'ai ha'Kerem fell into a dough or pot of Chulin that was already fermented or spiced:

If there was enough Isur to ferment or spice the Chulin had it been Matzah or unseasoned, the dough or pot is Asur. If not, the Isur is Batel if there is enough Heter (100 parts for Terumah, 200 times for the others).

Lechem Mishneh: Orlah 2:9 teaches this regarding a dough.

Rebuttal (Ra'avad): A fermented dough is unlike a spiced Tavshil (cooked food). One keeps a dough with extra Se'or in order to ferment other doughs, but one does not keep a Tavshil with extra spices in order to spice other Tavshilim.

Rosh (34): We permit hot water, cooked oil and honey of Nochrim. We are not concerned for absorptions in the pots, for we permit Nosen Ta'am li'Fgam. Even Beis Hillel and Beis Shamai, who forbade oil, did so only due to intermarriage. The proof from water is not solid, for even if the absorptions were intact their mixture would be detrimental. We can bring a proof from Chiltis. We forbid it even though we permit Nosen Ta'am li'Fgam. I.e. the knife is a Stam Keli of a Nochri. It is not a Ben Yomo. The sharpness of Chiltis causes the absorbed fat (in the knife) to improve the taste. If it did not cause the taste to improve, it would be permitted. R. Shimon permits an absorption after one day, even though it would forbid if it were intact (67b). This is unlike R. Meir. The Halachah follows him against R. Meir (Eruvin 46b. This is difficult, for the Gemara there did not resolve whom we rule like when they argue - Piplulei Charifta 200). Also, R. Yochanan expounds "Mishras" to teach that Heter joins to Isur (to comprise a Shi'ur to be liable - Pesachim 43b). The Gemara says that he learns, like R. Akiva, from Ge'ulei Nochrim (absorptions in their pots, which forbid) that Ta'am k'Ikar (an absorbed taste of Isur is like Isur itself). Chachamim disagree, for Ge'ulei Nochrim is a Chidush (normally Nosen Ta'am li'Fgam is permitted). R. Akiva says that the Torah forbids only a Ben Yomo, i.e. for it is li'Shvach, but permits when it is not a Ben Yomo and is li'Fgam.

Poskim

Shulchan Aruch (YD 103:1): Anything whose taste is Pagum (dulled) does not forbid what it gets mixed with. Even if the taste is not Pagum due to itself, that it itself has a good taste, but it mars its mixture, it is permitted.

Beis Yosef (DH Kol): The Rosh proves that we permit even if it was initially li'Shvach. All the Poskim agree.

Beis Yosef (DH u'Mah she'Chosav v'Afilu): We learn from the Mishnah (65b) that we permit even if the Isur itself was not Pagum. The Tur says 'e.g. Yayin Nesech that fell into grits.' This should say 'vinegar of Yayin Nesech...' Vinegar itself is good. The Rashba says so, and the Tur's words are from the Rashba.

Taz (1): We learn from Neveilah. It is forbidden only if it is proper for a Ger to eat. Therefore, whenever the Isur is the majority, it is as if it is all Isur, and it is permitted only if it is totally Pagum, like a Neveilah that is not proper to eat. Anything forbidden due to a mixture, whether the majority is Heter and intact Isur was mixed in, or whether the Isur was not intact Isur and so much was mixed in that there is not a majority of Heter, it suffices if it slighty mars the mixture, for he does not enjoy any taste from the Isur.

Gra (1): See 98:1 (we do not permit Nosen Ta'am li'Fgam if later it will be li'Shvach).

Rema: However, important things such as a Briyah (full creation) or similar matters, if they are not Pagum due to themselves, even though they mar the food, they are not Batel even in 1000 parts.

Taz (2): If they are Pagum due to themselves, they lose their importance. Even (to the point that) if they are cooked and we do not recognize them, even they themselves are Batel in a mere majority. The primary stringency of cooking over Kivush (pickling; some texts - b'Yavesh (dry)) is due to taste, and here there is no taste of Isur, for the Isur is spoiled. It is as if there is no intact Isur, just mere tatse. If the matter itself tastes good, even though it is bad for the cooked food, it is not Batel even in 1000, since it was a Briyah or a matter that is not Batel. The entire mixture is like the pure Isur itself without a mixture.

Shach (1): Another important thing is a Chatichah ha'Reuyah Lehiskaved (a piece fitting to honor guests with) or Davar sheb'Minyan (something normally sold by number). The Isur itself is not Batel in 1000, even if it is not Nosen Ta'am. Therefore, we require that it itself is Pagum. See OC 447:10. (The Mechaber permits Nosen Ta'am li'Fgam even regarding Chametz. The Rema says that some are stringent.)

Gra (2): The Isur must be Pagum by itself so it itself will be Batel. Here we discuss an important Briyah, e.g. a Tamei bird that spoiled, like we learn from a spoiled Neveilah. He is exempt even if he ate it by itself, like the Rambam writes.