Terumos, Chapter 5, Mishnah 4. Before us is a fascinating and challenging Mishnah, one that demands focused attention from beginning to end. Let us say it slowly and clearly, and anyone who does not lose even a moment of concentration will find the matter perfectly clear.
The question of the Mishnah:
One se'ah of tamei Terumah fell into a hundred se'ah of tahor Terumah. Is this mixture, consisting of a hundred and one units, permitted to a kohen to eat?
The classic model: a prohibited item that falls into a permitted one:
The only case from which we can learn anything relevant here is Terumah that fell into chullin. We have a rule that if one unit of Terumah fell into a hundred units or more of chullin, the Terumah is no longer considered significant, and the status of the entire mixture - all hundred and one units - is permitted, chullin that may be eaten. There may be a monetary requirement to remove one unit and return it to a kohen, since money belonging to the tribe of kohanim must come back to them, but the mixture itself does not become forbidden when the Terumah is one part in a hundred and one.
Here, then, we have the classic makeup of a prohibited substance and a permitted substance mixing together: the prohibited item is the Terumah, and the permitted item is the chullin. A hundred units of permitted food nullify the one unit of Terumah, and the halachic status of the entire mixture is permitted - for a kohen and for a non-kohen alike.
The ordinary rule in every mixture is that we seek to clarify one question only: is the mixture permitted or forbidden. The classic situation is a prohibited item that falls into a permitted one, and the question is what threshold makes the mixture permitted. The thresholds vary according to the nature of the item that fell in:
Sometimes a simple majority is enough.
Sometimes sixty times as much is required.
Sometimes a hundred times.
And sometimes two hundred times.
The model to engrave in your memory is therefore this: a hundred units of chullin, completely permitted produce, into which falls one unit of Terumah, which is forbidden to a Yisrael - a hundred and one in all.
The difficulty in our case:
In the case of our Mishnah, we are asking that the tahor Terumah serve as the permitted substance into which the prohibited substance falls. And this is puzzling: just a moment ago, in the basic paradigm, tahor Terumah itself served as the prohibited substance and not as the permitted one. The question, then, is whether Terumah can be considered permitted to such a degree that it can permit another prohibited substance that falls into it, namely tamei Terumah.
Beis Shammai say that the kohen may not eat from the mixture, since a status of prohibition can be removed from a mixture only when the substance into which the prohibited item fell is fully permitted. And who says that Terumah counts as permitted? Terumah is a restricted food, a flawed permissibility and not pure permissibility. From the fact that chullin nullifies Terumah in a ratio of a hundred to one, there is no proof that Terumah, which is itself a prohibited substance, can nullify another prohibited substance - tamei Terumah.
The wording of the Mishnah:
"Se'ah Terumah teme'ah" - a single unit of Terumah that became tamei, which the kohen may not eat but may only burn. "Shenaflah le'meah se'ah Terumah tehorah" - Terumah that the kohen does eat, but a non-kohen who eats it is liable to death at the hands of Heaven. It turns out that it is a prohibited item relative to a non-kohen, and a permitted item relative to a kohen.
"Beis Shammai osrin" - all one hundred and one units become forbidden, and there is nothing at all to be done with them. "U'Veis Hillel matirin" - the kohen eats all one hundred and one units together.
The argument of Beis Hillel:
"Ho'il u'tehorah asurah le'zarim u'teme'ah asurah la'kohanim" - we have before us a case from which we are learning, and a case to which we wish to apply the lesson. The case from which we learn is ordinary Terumah, which is prohibited, that fell into chullin, which is permitted, and we permit the entire mixture. From here we wish to learn to Terumah teme'ah that fell into Terumah tehorah. Say Beis Hillel: just as Terumah tehorah is forbidden to a non-kohen, so Terumah teme'ah is forbidden to a kohen - the two cases parallel one another, a prohibited item falling into a permitted one.
"Mah tehorah olah af teme'ah ta'aleh" - just as in the first case the Terumah tehorah was nullified in a mixture of permitted food, so too the Terumah teme'ah should be nullified in the mixture before us. Put simply: through the eyes of the non-kohen, there are a hundred units of chullin into which a unit of Terumah fell, and the mixture is permitted; and through the eyes of the kohen, there are a hundred units of Terumah tehorah - which for him is permitted food - into which a unit of Terumah teme'ah fell, and one hundred and one is enough to nullify it. This is exactly the same thing.
The response of Beis Shammai:
"Lo, im he'elu ha'chullin ha'kalin ha'mutarin le'zarim es ha'tehorah, ta'aleh Terumah ha'chamurah ha'asurah le'zarim es ha'teme'ah?" - this is not a legitimate kal vachomer. To nullify a prohibition you need something permitted, and when we are dealing with chullin, that is a pure and unqualified permitted status, and it has the power to remove the label of prohibition from the mixture in a ratio of one hundred and one. But Terumah tehorah is not a pure permitted status, rather a qualified one, permitted to a kohen alone and not permitted across the board.
The phrasing of the rhetorical question is saying: it cannot be that Terumah, which is stringent in that it is forbidden to non-kohanim, should have the power to serve as the permitting agent and nullify the significance of Terumah teme'ah that fell into the mixture. The flaw in the kal vachomer lies in its direction: the lesson is being drawn upward, from a stronger permitting agent - chullin - to a weaker permitting agent - Terumah tehorah. If the pure permitted status of chullin makes the problem of the Terumah disappear in a ratio of a hundred to one, who says that a flawed permitted status, which is a lesser form of permission, will do the same?
The reply of Beis Hillel found in the Tosefta and the Yerushalmi:
The Mishnah does not record the response of Beis Hillel, but goes on to say "L'achar she'hodu" - after Beis Shammai conceded to Beis Hillel. It emerges that Beis Shammai changed their minds and came to see eye to eye with Beis Hillel, and the question is why. The Tosefta here, and likewise the Yerushalmi, explain that there was a reply from Beis Hillel, and that reply convinced Beis Shammai.
Said Beis Hillel: the question to examine is not which is the stronger permitting agent - chullin or Terumah tehorah - but rather the severity of the prohibition being nullified. With Terumah, the severity of the prohibition is death at the hands of Heaven: a non-kohen who ate Terumah is punished with the severe penalty of dying before his time. By contrast, a kohen who eats Terumah teme'ah is merely nullifying a positive commandment, and that restriction is far lighter.
Therefore: do not look at the strength of what permits, but at the strength of the prohibition you are nullifying. If a severe prohibition like Terumah, which carries with it death at the hands of Heaven, is nullified in a mixture, then all the more so a lighter prohibition, impure Terumah, which for a kohen involves nothing more than the failure to fulfill a positive commandment, should also be nullified.
The underlying principle:
At first glance this answer does not seem to address Beis Shammai at all, since their claim was that the focus belongs on the permitting element, on the hundred, and that the kal vachomer runs in the opposite direction. But the words of Beis Hillel express a sweeping principle, and since Beis Shammai conceded to it, it must be understood as a general rule that applies in other contexts as well: in all the laws of nullifying a prohibition in a mixture, we do not focus on the strength of the permitting element, of the hundred that make the prohibition disappear, but solely on the strength of the prohibition itself.
An illustration: there is a whole range of forbidden foods in the world, and by Torah law they are nullified in a simple majority, but the Sages established a scale of thresholds - a majority, sixty, a hundred, two hundred - all according to the severity of the prohibition. With kilei hakerem and orlah, for example, two hundred times the amount is required, because these are not merely forbidden to eat but forbidden to derive any benefit from, and on account of the prohibition of benefit the Sages doubled the measure from the hundred of Terumah to two hundred. So the focus is not on the permitting element but on the prohibited element.
Given this, the kal vachomer of Beis Hillel stands up beautifully: the severe prohibition of Terumah, which carries with it death at the hands of Heaven, is nullified against a hundred of unconsecrated produce - all the more so should impure Terumah, which involves nothing more than the failure to fulfill a positive commandment, be nullified against a hundred units of pure Terumah. This is the answer of Beis Hillel that Beis Shammai accepted.
Must anything be removed from the mixture?
The Mishnah is not finished yet. In the earlier cases, in the mishnayos we have learned, the non-kohen farmer bore a financial obligation to remove the equivalent of what had fallen in and give it to a kohen, and even though it is no longer Terumah, we treat it as Terumah. The central point was that a person may not take for himself money belonging to the tribe of kohanim. Here, by contrast, it is the kohen himself who eats everything, and so we must clarify whether anything needs to be removed from the mixture at all.
"Rabbi Eliezer omer: torem v'tisaref" - Rabbi Eliezer is consistent with the approach of the Tanna we saw in the previous mishnayos: there must be a negative financial incentive. Otherwise the mixture would turn into a kind of laundering machine for impure Terumah, and every batch of impure Terumah would find its way to fall into something and disappear. For a kohen who holds a hundred units of pure Terumah and one unit of impure Terumah that is destined to be burned would come out ahead: once the impure falls into the pure, he eats a hundred and one instead of a hundred. Rabbi Eliezer therefore rules that one unit be removed and treated as though it were literally the one that fell in, even though physically that is obviously not the case, and that it be burned. This way the kohen gains nothing, and there is no room to allow him to let impure Terumah fall in "by accident." It is worth noting that Rabbi Eliezer came from the study hall of Beis Shammai, and his ruling was stated after they had conceded.
"V'chachamim omrim: avdah b'mi'utah" - the prohibition was lost in its own smallness. Here too the language focuses on the prohibited portion: the impure Terumah that fell in has disappeared halachically, and therefore we have here a hundred and one units of Terumah, and the kohen eats them all. True, he comes out one unit ahead, but it happened by accident. Of course, one may not nullify a prohibition intentionally, and the kohen may not deliberately toss impure Terumah into the mixture; but once it has happened, all hundred and one units are permitted. Just as we do not say about a drop of milk that fell into chicken soup that the soup has now increased and we have profited - rather, an accident occurred, and we do not penalize the kohanim with an additional fine.
The halachah: The halachah follows Beis Hillel and the Sages - if a se'ah of impure Terumah fell into a hundred of pure Terumah, the mixture is permitted, and the kohen eats all hundred and one units as pure Terumah.