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Teshuva #5 Question: Is it permitted to put ketchup during Shabbos onto hot meat which is in a kli sheini? [To forbid doing so would involve] 2 chumras: 1) to act according to the Maharshal that a hot solid food has the status of a "kli rishon". 2) To hold that there is cooking on an already cooked item that is liquid. Additionally, maybe ketchup is not considered a liquid since it is very thick? Similarly, is it permitted to put salt which was not cooked onto a hot solid food in a kli sheini, since according to the letter of the law this is permitted even in a kli rishon?
Answer: It would seem that the halacha is like you suggest that one is allowed to put ketchup on hot meat which is in a kli sheini, for even if we would [sometimes] say that we will keep 2 stringencies, that is when the doubt is real for each chumra [that we should really hold that way as opposed to doing so out of respect for a minority opinion] for even though it is a double doubt [which should make it permitted] maybe by Shabbos there is room to be strict anyways because of the gravity of Shabbos. Though here, the chumra that there is bishul on already cooked items is just a chumra that the great poskim held that even by liquids there is no cooking an already cooked item and this is the ruling of the Rama and that is our custom; that only when it has totally cooled off are we machmir. Also the chumra of a hot solid item being considered a kli rishon even when it is in a kli sheini, many argue against this among them the Rama who is a main opinion in halacha. The Taz there is only machmir in combination with the "pressure of the knife" which does not exist when one puts ketchup on food. The Chasam Sofer brought in the Pischei Tshuva Seif katan 7, is lenient like the Rama in a case of great financial loss and he likewise writes that his Rebbe the Gaon [Rav Nissim Adler?] also ruled leniently in a time of need. So too here, even though the Shach (105,8) holds like the Maharshal as does the Magen Avraham (318,45), we don't view it as an equal doubt, rather since it's not clearly permitted [since some argue] we are machmir, therefore you are right that even by an Issur of Shabbos one does not need to keep this double chumra.
That which you were unsure about ketchup if it is considered a food [as opposed to a liquid] since it is thick even though it pours out of its container, it would make sense that this is not dependent on the laws of how foods and liquids are defined with regards to Tumah, rather [it depends on] if it flows or if it is a mass that holds its form even without a container to give it that form, which would make ketchup a liquid.
Salt is obviously permitted to put on hot meat in a kli sheini like you thought, for according to the letter of the law it doesn't cook even in a kli rishon, just it is good to be strict. Therfore by a davar gush which many argue against [and hold it is like a kli sheini] one does not need to be strict, though by meat in a kli rishon it is good to be strict.
Teshuva #6: Question: Is it permitted to put butter on a hot solid item that is in a kli sheini?
Answer: Butter in this country [America] is made from milk which is cooked first, called pasteurized, if the cooking is to the degree of yad soledes then the butter is made from cooked milk and once it solidifies it is like a dry/solid food which is no longer a problem of cooking, as it says in the Magen Avraham (318,40) regarding congealed fat which now has the status of a dry/solid food and is no longer a problem to cook and even if it liquifies we don't care like it is explained in the Levush[ei Srad?] and so too it is in the Mishneh Berurah (318,100) and that is exactly the situation with butter in this country. Though if the heating is less than yad soledes it should be forbidden since it was never cooked, though since it sounds like the temperatures are 160 degrees [F] or 168 degrees[F], based on what we ourselves tested 160 degrees is certainly yad soledes and it[the milk and consequently the butter] is permitted, though if there would also be [pasteurization done at] less than 160 degrees it would be a safek [I assume he means the doubt of whether this butter/milk was pasteurized at above 160 or less].

Teshuva#7 Question: Food which is mostly solid/dry and it is a bit liquid/moist should one be machmir that there is bishul achar bishul if it cooled down completely?
Answer: I don't understand the difference between mostly liquid and only a bit of liquid since the little bit of liquid will [be able to] cook [again] after it cooled down, which therefore leaves me no choice but to understand the question in terms of the Pri Megadim (253,13) on the Taz, who discusses if the majority of a food will improve from the cooking while the minority of the food will get worse, which should be forbidden since many argue with the Rama (318,4) who writes that specifically when the cooking will improve the item there is bishul after bishul. Though in the Pri Megadim (253,41) on the Magen Avraham, in the case of pouring water from a kli rishon onto tea or coffee that one already poured hot water on before Shabbos, [he rules that] having already cooked a dry item [the tea or coffee] there is no more cooking after that cooking, and even though there remains little drops of cold water [which will cook] we rule based on the majority, so we see that he holds that when the rules of dry/solid vs liquid are in conflict we follow the majority, and in Siman 259,3 on the Taz the Pri Megadim writes to prove that certainly we rule based on the majority between solid and liquid, for even in the same pot we rule that way; when the cooking benefits part of the food and ruins the other part [so certainly between solid and liquid we follow the majority] for if not, the Pri Megadim that we quoted earlier would not be right and we would permit a case where there is a minority of liquid. Though maybe the reasoning is that really the law should be by liquids as well that once it is cooked it is no longer a problem to cook. Since this halacha is not clear it is fitting to be machmir, though in a situation of great need maybe you could permit it.

Teshuva#15 Question: Tea leaves are not cooked when they are made, they are just dried with heat, is it permitted to prepare tea in a kli shlishi?
Answer: In my humble opinion it does not seem correct to say that there are things which cook in a kli shlishi. For we only find that in a kli sheini there are things which cook and therefore since we don't know what those things are, we should forbid everything, though in a kli shlishi we don't find such a thing. That which the Aruch HaShulchan writes in 318,28 that we see with our eyes that the tea leaves cook even in a kli shlishi is not understood, that which we see the water turn red has nothing to do with cooking since this happens in cold water as well, over time, and even more so in "hot water" which is not yad soledes [it turns red as well,] so this is not cooking. Which leads us to the conclusion that what our eyes see is worthless [with regards to determining whether or not something is cooking] for [by] salt we see that when put into something which is a little hot, you can taste the salt just like when you put it into water that is ys'b, either way the halacha is like Rav Nachman (Shabbos 42b) that even in a [kli] rishon it does not cook. Even according to Rav Yosef it only cooks in a kli rishon even though we don't taste the difference between salt cooked in a kli rishon vs cooked in a kli sheini at all. Even according to the first version of Abaye on the Braisa that Rav Chiya taught, that it also cooks in a kli sheini it sounds like it is specifically a kli sheini and not less [less here refers to heat/potency and he means not kli shlishi, revii, etc.], so we see that "cooking" is something else, which is the purpose that is achieved with the salt through cooking it, that is more than [just] the "salty taste" for this can be achieved even in a completely cold food, for uncooked salt [meaning salt that is not being used with fire to improve food] also gives a salty flavor as is known from salting meat. Salting itself when done on cold food can create a status of cooked with regards to the laws of Issur veHeter, though this is not the purpose of "cooking". Chazal calculated that it is specifically in great heat [that salt achieves this purpose] and so the law became that salt is like ox meat [with regards to how much heat is necessary to achieve a bishul effect on it]. So too with the tea leaves, that which the water changes color with a bit of heat is not "bishul", therefore it might be that tea leaves are "hard to cook" [meaning it takes a lot of heat to achieve the bishul effect] and if so, we should not say on our own anything other than what we received a tradition on, and therefore it is like all other regular things; in a kli sheini it is a doubt [whether it is permitted or not] and in a kli shlishi you can permit it.
Teshuva # 16 Question: In the production of instant coffee, the coffee [beans] are cooked and then dried in very hot air. Should this cause it to be viewed as "fully cooked", or maybe since it doesn't taste the same when put into cold water vs when it is put into hot water we should say that it is like the Aruch haShulchan writes (318,28) about tea that even if you cook it, the cooking is not to cook it [meaning to render it ready to eat] rather it is to make it into food that can then be cooked, and therefore [the issur of] cooking still applies to it?
Answer: It is obvious that if it is true that instant coffee is cooked in water after the beans are roasted it has the status of "cooked" if it is cooked fully. It makes sense that it is true since the coffee dissolves totally, and also the heat of 800 [degrees] which causes it to dry to the point that it becomes powder is also cooking, and therefore there is no cooking after it has been cooked since it is dry/solid. That which it tastes different in cold and hot water is not significant as I proved by salt (earlier in teshuva 15), and that which you were unsure based on what the Aruch hashulchan said that even if it is true that they cook the tea it's only a preparatory cooking..., I don't understand his intention, for if there is a proper cooking of the tea, then it is considered cooked, even if it is only done to make it into a food that can be cooked? Maybe his reasoning is that we see from this, that this cooking is not enough, it just helps that now it will cook when each individual cooks it in their own house, like the turmus which needed to be cooked 7 times, is not considered fully cooked until it is cooked enough, and this does not apply to the coffee which is completely cooked. That which the Shaarei Teshuva brings (318,20) from the Machazik Bracha in the name of the Zerah Emes that it is forbidden to pour from a kli rishon onto sugar even though it is already cooked, the logic is not that it is not the normal way to eat it, for why does that matter? Certainly also it is normal to eat it "as is", both for children and adults, just since they don't want to eat something too sweet, just like they wouldn't eat too much honey as it says in the pesukim [see Mishlei 25:16] and even more so sugar which is even sweeter, so it must be for a different reason maybe he was unsure if it was cooked or maybe because people don't know [that sugar is cooked in the production], though everyone knows that instant coffee is cooked, and also I believe that the minhag is to permit sugar like the Panim Meiros brought in the Shaarei Teshuva there. So too is implied by the Mishnah Berurah (318,71).
In truth I am personally machmir both with instant coffee and with sugar to put it into a kli shlishi, although the real halacha is not to be machmir.
Teshuva #17 Question: Is it permitted to put salt that is not cooked into a kli shlishi?
Answer: In my humble opinion I wrote before (Teshuva #15) that we don't find things that cook in a kli shlishi, and by salt we find explicitly in the gemara according to the version which Abaye said Rabbi Chiya taught salt strictly that it even cooks in a kli sheini therefore it is certainly not applicable to be strict from a kli shlishi [and on, meaning other keilim even more removed from the original heat source] and specifically when the main halacha is that it is permitted even in a kli rishon.
Teshuva #18 Question: Is it permitted to put onions or lemons into a kli sheini as a spice?
Answer: Onions are explicit in the Aruch Hashulchan (318,44) that it has the same status as spices which are permitted in a kli sheini, and the Taz (318,14) forbids putting onions in a kli sheini though his proof; from the fact that they sweeten, is not a proof as the Aruch hashulchan writes, [the reason the onions turn sweet] is from the sauce and the fat [not the cooking] and based what I wrote earlier in #15 [that cooking is not defined necessarily by a change in the food or liquid for tea changes the color of the water even without heat etc.] it's totally no proof. Though maybe onions are not included in "spices" because they are obligated in tithes. See Tosafos in Niddah (50a s.v. All that are obligated) who says that garlic and onions are also spices, even though they are edible on their own which makes them require tithing, they are also used [sometimes only] for flavor. It is also possible that spices of the mishnah in Shabbos are referring to spices that are only useable to give flavor to food [and are never edible on their own] and therefore garlic and onions would not be included since they are edible [on their own] and therefore one should be strict like the Taz [not to put them in a kli sheini] though not for his reason, rather because there are things which cook in a kli sheini which are not identified by us, though in a kli shlishi they are permitted. Lemons should be forbidden to place in a kli sheini and permitted in a kli shlishi.
Regarding uncooked coffee or cocoa which are not eaten [on their own] and are only used as a spice [to give flavor to the water] it would seem that they are certainly included in the general term "spices" and if so, it is explicit in the mishnah that they don't cook in a kli sheini and can be placed in one intentionally. It is astonishing that the Aruch Hashulchan is strict not to put these items into a kli sheini or even into a kli shlishi according to his opinion, and so too it is implied in the Mishnah Berurah (318,39) who also forbids [putting] coffee in a kli sheini based on the logic that there are things [unidentified by us] which can cook in a kli sheini, yet the mishnah explicitly says that they don't cook in a kli sheini!? Granted tea leaves might not be considered spices since they are not put in the water for taste rather for color so they would not be included in the spices of the mishnah and therefore it is possible to be unsure [if they can cook in a kli sheini], but coffee and cocoa which are spices, certainly are included in the spices of the mishnah and there is no doubt? This needs further study.
In practice, one should either way be strict [not to put them] in a kli sheini and they are permitted in a kli shlishi.

Teshuva #19 Question: Is it permitted to pour water from a kli rishon into a kli that has remaining water droplets which have cooled down inside it?
Answer: There is a difference between water droplets [and a larger amount of water, because water droplets] don't give any additional pleasure [when heating them up] and they even don't increase the quantity of water, for even [heating up] cold water which is forbidden cooking on the Torah level, [here] would be not be forbidden according to the Aruch and many poskim who hold that a melacha which you don't want to happen even if it for sure will happen, it is permitted to go ahead and do, so by water droplets which is less than the amount [required to make the melacha of cooking a significant action and therefore forbidden on the Torah level] it could be that we shouldn't be strict, though anyway I wrote in the Iggros Moshe (1,93 end) that it should be forbidden. Therefore, when boiling hot water cooled down which really the halacha is that there is no cooking of something which was already cooked for most poskim hold this way, certainly it is not applicable to be strict by droplets. Though if there is a large amount of water that was cooked and it cooled down it should be forbidden to pour a lot of hot water onto them in a way that the cold water will also turn hot since that is a melacha that he does want and is guaranteed to happen.
Teshuva #20 Question: If one poured water from a kli rishon into a kli which had droplets that had never been cooked before in it, is it permitted post facto (after it had been done), for the Tiferes Yisrael writes that cooked water is not nullified?
Answer: When one rinses a cup, with water that was never cooked before, and droplets are left behind [in the cup] I wrote before that it should be forbidden [to go ahead and pour the hot water in and cook the droplets], it is obvious that it is permitted [post facto] since there is no clear prohibition as we said before. The logic of the Tiferes Yisrael that it is not nullified if there is 60 times the amount of permitted water to the forbidden water, is talking about when there is enough water that you gain a benefit from the increased quantity of forbidden water, [so there isn't 60 times more permitted water than forbidden. And that which] it is not permitted using nullification to the majority [(whatever presence is the majority determines the status, in this case the permitted water which is the majority would nullify the forbidden water by turning all the water into the same status as the majority)] the logic is that it has a way to become permitted [(since after Shabbos the cooked water will become permitted, on Shabbos while it is forbidden one cannot nullify it in any way {since it is better to wait for the water to become permitted than to drink it while it is forbidden through nullification, Rashi in Beitzah 3b d"h Afilu})].
Teshuva#21 Question: It says in the Mishnah Berurah at the end of Siman 318 in the name of Tosafos Yerushalayim in Siman 253 that a mulyar [(this was a metal pot with a compartment on the outside of it in which coals would be put to heat up the contents of the pot)] whose coals are not cleared out should not be drunk from on Shabbos since it is made of parts and one will be worried that the [heat will] break the parts [by melting the glue or whatever was used to connect the parts], so he will add water [which will absorb the heat and this will prevent the parts of this urn from breaking instead] and he writes that therefore it is forbidden to take water from the urn on Shabbos if it still has its coals. I heard from Rebbe Shilt"a that by our electric hot water heaters it is permitted and I wanted to know his reason?
Answer: It is simple, since this worry is not applicable by our urns.