Gemara

30b (Mishnah): The Vlad (offspring) of any animal invalid for a Korban is Kosher for a Korban;

R. Eliezer says, the Vlad of a Terefah may not be offered.

31a: According to the opinion that a Terefah can (become pregnant and) give birth, the case is, it became Terefah and then became pregnant;

R. Eliezer forbids Zeh v'Zeh Gorem, and Chachamim permit;

According to the opinion that a Terefah cannot (become pregnant and) give birth, the case is, it was pregnant and then became Terefah;

R. Eliezer holds that Ubar Yerech Imo, and Chachamim hold that Ubar Lav Yerech Imo.

12a (Mishnah - R. Eliezer): If Se'or (a fermenting agent) of Chulin and of Terumah fell into a (Chulin) dough, and there was not enough of either one to ferment the dough, and together they fermented it, the dough is like the last one that fell in. (If it was Terumah, only Kohanim may eat it, in Taharah);

Chachamim say, no matter which fell in last, it is like Chulin unless there was enough (Se'or of) Terumah to ferment it without the Chulin.

Avodah Zarah 49a (Abaye): R. Eliezer permits (when the Chulin fell in last) only if the Terumah was removed before the Chulin fell in. If not, it is forbidden. (He forbids Zeh v'Zeh Gorem, i.e. something that results from two (or more) causes, one of which is forbidden. Chachamim permit this.)

Rejection: Perhaps also R. Eliezer permits Zeh v'Zeh Gorem (e.g. if they fell in at the same time). He forbids here only because we attribute the entire action to the final cause, whether or not the first cause was removed.

Shabbos 42a (Mishnah): If one removed a frying pan or pot from the fire, one may not put spices into it (on Shabbos);

One may put spices into a bowl (a Kli Sheni).

(Rav Yosef): Salt is like spices. It is cooked only in a Kli Rishon (one that was on the fire), but not in a Kli Sheni (a Kli into which one poured from a Kli Rishon).

Version #1 - Objection (Abaye - R. Chiya's Beraisa): Salt is unlike spices, it is cooked even in a Kli Sheni.

The Beraisa also argues with Rav Nachman, who says that salt needs much cooking, like ox meat. (It gets cooked only in a Kli Rishon.)

Version #2 - Objection (Abaye - R. Chiya's Beraisa): Salt is unlike spices. It is not cooked even in a Kli Rishon (off the fire).

This supports Rav Nachman:

Rishonim

Rif (Shabbos 20a): The Halachah follows Version #2.

Rambam (Hilchos Shabbos 22:6): One may not put spices into a boiling pot even if it is off the fire, but one may put salt into it, for salt cooks only on a big fire.

Rosh (Shabbos 3:17): The Halachah follows Version #2, which supports Rav Nachman. This connotes that ox meat does not cook in a Kli Rishon, so one may put raw meat in a Kli Rishon off the fire. However, perhaps old salted meat is forbidden, for it cooks quickly. (Perhaps we forbid) even fresh salted meat. Granted, the meat does not cook, but the salt and blood inside cook and are absorbed in the meat, so it is forbidden.

Poskim

Shulchan Aruch (OC 318:9): A Kli Rishon (one that was used on the fire) cooks, even after it was removed from the fire, as long as it is Yad Soledes Bo. Therefore one may not put spices into it, but one may put salt in it after it was rmvd from the fire, because salt needs to be cooked a great time, like ox meat. Some forbid putting salted meat in this pot, even ox meat.

Rema: Even without salt it is forbidden, just in any case that is forbidden due to blood. Some forbid putting salt even in a Kli Sheni as long as it is Yad Soledes Bo. One who is stringent will be blessed. If one transgressed and put salt, even in a Kli Rishon, even if it is on the fire, and he transgressed, the food is permitted, for the salt is Batel to the food.

Beis Yosef (DH v'Chasuv): Shibolei ha'Leket (86) says that if one put salt in a boiling pot on Shabbos b'Mezid, the only reason to forbid is Davar she'Yesh Lo Matirim (it will be totally permitted in another situation, in this case after Shabbos). We do not forbid like Se'or and spices, which are for the sake of (adding) taste. When they did not add taste, they are Batel, like other Isurim. We hold that Zeh v'Zeh Gorem is permitted. This is the ultimate Zeh v'Zeh Gorem, for presumably it was salted before Shabbos, just not enough. Davar she'Yesh Lo Matirim does not apply here, because it was Batel once it dissolved, before there was an Isur Shabbos.

Taz (15): This is astounding. We find Zeh v'Zeh Gorem only when in the same time that the Isur acts, also the Heter acts, e.g. Se'or of Chulin and of Terumah that ferment together. This applies to every Zeh v'Zeh Gorem mentioned in Yoreh De'ah. Here, it was initially cooked with permitted salt. It did its action, just it did not suffice to flavor the food. Afterwards, taste came through forbidden salt. This is not Zeh v'Zeh Gorem. The Isur caused it to have taste! This is like a food with spices, but they do not suffice for it. He added Isur spices, and improved it. Will we permit this?! This is the ultimate case of eating Isur! Zeh v'Zeh Gorem applies only if they were mixed before they gave taste, i.e. before they were added to the pot. Then, they give taste together. Here, it is like putting spices in a Kli Rishon for the first time on Shabbos. Here, one overtly tastes the Isur added to the Heter. One should not eat it until after Shabbos.

Magen Avraham (31 and Mishnah Berurah 73): Even though normally, something added for taste is not Batel, here is different. It is Zeh v'Zeh Gorem, for presumably it was salted also before Shabbos. Even though Davar she'Yesh Lo Matirim is not Batel, here it was Batel before it was forbidden. It dissolved immediately, and it does not cook immediately, for it must cook like ox meat.

Machatzis ha'Shekel (DH deha'Melach): Even though Davar she'Yesh Lo Matirim is Batel Min b'Eino Mino (when mixed with a different species), since the salt was added to improve the food, regarding this it is considered Min b'Mino. Tosfos (Beitzah 39a DH Mishum) says so, and we rule like this (Rema YD 120:1).

Magen Avraham (31): We must say that enough salt was not added on Shabbos to flavor it without the salt from before Shabbos. If there was, it is forbidden, like we say in YD 87:11 (Rema, regarding curdling agents), and the Mishnah (Orlah 2:11) and Avodah Zarah 73b.

Machatzis ha'Shekel (DH she'Im): He connotes that if there was enough Isur to flavor it by itself, even if there was also enough Heter to flavor it by itself, we forbid. Really, Poskim argue about this. See Tosfos Pesachim 27a DH Ad, Avodah Zarah 68 DH l'Rebbi.

Kaf ha'Chayim (110): The Acharonim disagree with the Taz, and rule like Shibolei ha'Leket, the Rema and Magen Avraham.

Gra (31): The Gemara applies Zeh v'Zeh Gorem to Se'or, even though it is added for taste. See 320:2. (The Shulchan Aruch says that if grapes are in a barrel of wine and they split open on Shabbos, the juice that exudes is permitted, for it is Batel in the wine in the barrel.)

Magen Avraham (31): There is no concern for salt made through cooking water, since Ein Bishul Acher Bishul (once something was cooked, there is no Isur to cook it again).

Kaf ha'Chayim (114): This is like the Darchei Moshe (4), but he added that one who is stringent will be blessed. Also other Acharonim added this. Eliyahu Rabah says that even though Yesh Bishul Acher Bishul regarding (wet) foods that cooled off, we must say that salt is dry.

Note: One might have thought that salt is considered wet become it comes from water, or because it dissolves.

Sha'arei Teshuvah (28): If one erred and poured boiling water from a Kli Rishon into a Kli Sheni with sugar, this is not Bishul, since the sugar was already cooked in the processing, and Ein Bishul Acher Bishul for dry foods. Even though the sugar dissolves, it is considered dry. One may put sugar even into a Kli Rishon, like the Magen Avraham says about salt. Machazik Berachah was hesitant about this, and concluded that it is good to pour the coffee into the cup, and afterwards to add sugar. Obviously, this is only if the Kli Rishon is Yad Soledes Bo.

Bi'ur Halachah (DH v'Im): If one added spices to a Kli Rishon, even if some spices were added also before Shabbos, it is not clear if it is permitted b'Di'eved. This is Davar she'Yesh Lo Matirim. R. Simchah is lenient about salt, because it dissolves immediately and does not cook for a long time, so it is Batel before it is forbidden. Spices are different. Or, perhaps also they are Batel before they are cooked. This requires investigation.