1) FOOD PARTIALLY COOKED BY A JEW AND PARTIALLY BY A NOCHRI
OPINIONS: The Gemara quotes Rebbi Asi who states in the name of Rebbi Yochanan that the prohibition against eating food cooked by a Nochri does not apply to food that was partially cooked first by a Jew to the degree of "k'Ma'achal Ben Derusa'i." "Ma'achal Ben Derusa'i" refers to food that is a third cooked (RASHI, Shabbos 20a, DH Ben) or half cooked (RAMBAM, Hilchos Shabbos 9:5), which was the way the bandit named Ben Derusa'i, who was always on the run, would eat his food, since he never had time to fully cook his food.
Does this allowance to eat food cooked by a Nochri apply when the Nochri cooked it first to the extend of "Ma'achal Ben Derusa'i," and then a Jew completed the cooking, or is such food prohibited?
(a) The ROSH (2:32) quotes an opinion that says that such food is prohibited, based on the following logic. The reason why it is permitted to eat food that was cooked by a Jew "k'Ma'achal Ben Derusa'i" and was then cooked by a Nochri is because of the principle of "Ein Bishul Achar Bishul." Once the Jew cooked it "k'Ma'achal Ben Derusa'i," it is considered cooked and it cannot be cooked again. This is why the further cooking of the Nochri does not affect the food.
The opposite also applies. Once the food has been cooked by the Nochri to the extent of "Ma'achal Ben Derusa'i," it can no longer be cooked again, and it is considered to have been cooked solely by the Nochri and is therefore prohibited because of "Bishul Akum." The RAN attributes this opinion to RABEINU CHANANEL and sides with it.
(b) The Rosh disagrees with this opinion and rejects the logical proof. He writes that our Gemara is teaching a leniency; the prohibition of "Bishul Akum" does not apply when food was first cooked, to some degree, by a Jew. The intention of the Gemara is not to teach a stringency and to say that food cannot be considered cooked by a Jew after it was already cooked "k'Ma'achal Ben Derusa'i" by a Nochri. This is apparent from the words of the Gemara itself. The Gemara states that if a Nochri does something to cause a food to cook faster ("Kerovei Bishula"), then even though the food was not cooked "k'Ma'achal Ben Derusa'i" when the Nochri sped up the cooking process, there is no problem of "Bishul Akum" since the food would have become cooked eventually even without the intervention of the Nochri. The Gemara later (38b) discusses a case in which a Nochri placed the food on the coals and the Jew stoked the fire, causing the food to cook faster. Rav Nachman bar Yitzchak states that if we are lenient in the first case (when the Nochri completed the cooking), then we certainly should be lenient in this case, when the Jew completed the cooking. The Rosh points out that Rav Nachman bar Yitzchak cannot be discussing a case in which the food had not yet reached the state of "k'Ma'achal Ben Derusa'i," because if it had, then the food should be prohibited because it reached "k'Ma'achal Ben Derusa'i" because of the cooking of the Nochri!
The KEHILOS YAKOV answers the question of the Rosh on the view of Rabeinu Chananel. Rabeinu Chananel understands that as long as there is only one action of cooking, the Jew and the Nochri are considered partners in the cooking. The Gemara is discussing cases in which the second party increases the heat of the fire, causing the food to cook faster ("Kerovei Bishula"). As long as a Jew is involved in the cooking process in any manner, the food is considered to have been cooked by a Jew. However, the case of "k'Ma'achal Ben Derusa'i" refers to food that was cooked, removed from the fire, and then cooked again further. That case involves two separate actions of cooking which are not related to each other. Therefore, when a Nochri cooks food until it is "k'Ma'achal Ben Derusa'i" and then removes it from the fire, there is nothing that can be done by a Jew to make that food permitted. (Y. MONTROSE)
2) FOODS SALTED BY A NOCHRI
QUESTION: The Gemara discusses whether fish that was salted by a Nochri is forbidden because of "Bishul Akum." Chizkiyah maintains that it is not forbidden, while Rebbi Yochanan argues that it is forbidden. Rebbi Yochanan apparently maintains that salting the fish is considered like cooking the fish (see Rashi, DH v'Rebbi Yochanan).
The MAGEN AVRAHAM (OC 527:3) comments that a food that was prepared for eating by being salted cannot be used to make an Eruv Tavshilin (see Background to Eruvin 80:6). Although a salted item is "k'Rose'ach" (considered to be brought to boiling), that means only that it is considered to absorb other foods into itself and to expel from itself into other foods, as is the case with boiling items. However, it is not considered cooked, a requirement of Eruv Tavshilin.
Does Rebbi Yochanan, who maintains that a fish salted by a Nochri is considered as though it was cooked by a Nochri, argue and maintain that a salted food item may be used for an Eruv Tavshilin?
(a) The SEDER YAKOV explains that Chizkiyah agrees with the logic of the Magen Avraham that salting is not considered cooking. Rebbi Yochanan also agrees that salting is not considered cooking with regard to using the food for an Eruv Tavshilin. However, he prohibits food salted by a Nochri because he understands the Isur of "Bishul Akum" differently. One reason for the Isur is in order to prevent Jews from becoming too familiar and friendly with Nochrim by eating their prepared food, which might then lead to intermarriage (see Rashi to 38a, DH mid'Rabanan). The fact that the salted fish was not technically cooked by the Nochri does not diminish the risk of breeding familiarity. This is why Rebbi Yochanan does not permit eating a salted fish of a Nochri, even though he does not consider salting to actually be a form of cooking.
(b) The Seder Yakov gives another, similar answer based on the question of the RAN (DH Garsinan). The Ran is unsure whether a heavy salting, such as one does to food before taking it with him on a long journey, is considered Bishul, cooking, with regard to the laws of Shabbos. Even though he is in doubt whether heavy salting is considered Bishul with regard to cooking on Shabbos, such a process which certainly can pose a problem of "Bishul Akum." The Magen Avraham might be discussing only an ordinary salting process, and not a heavy salting, and thus he might agree that food that was heavily salted may be used for an Eruv Tavshilin.
(c) The Seder Yakov suggests a third answer. RASHI in Berachos (38b, DH she'Achal Zayis) writes that we see that Rebbi Yochanan maintains that the blessing recited for cooked fruit is "Borei Pri ha'Etz," as the Gemara there relates that Rebbi Yochanan recited this Berachah on a salted olive, and salting is considered like boiling.
The NODA B'YEHUDAH (Yoreh De'ah, Tinyana #43), using the logic of the Magen Avraham as mentioned above, questions the words of Rashi. What does salting have to do with cooking? How do we see from the fact that Rebbi Yochanan recited "Borei Pri ha'Etz" on a salted olive that he would have recited the same Berachah on a cooked olive? (See the answer of Noda b'Yehudah there.) It must be that Rashi indeed understands Rebbi Yochanan to be saying that salting is considered like cooking, and the argument between Chizkiyah and Rebbi Yochanan involves the basic understanding of whether salting is like cooking or not. Chizkiyah maintains that salting gives food a Halachic status of being boiled, but not cooked, which explains why a salted food is not prohibited because of "Bishul Akum." Rebbi Yochanan, on the other hand, maintains that salting is considered cooking, and therefore a salted food is prohibited because of "Bishul Akum." Accordingly, the Magen Avraham states that a salted food cannot be used for an Eruv Tavshilin because we do not rule in accordance with the view of Rebbi Yochanan. (Y. MONTROSE)