Terumos, Chapter 4, Mishnah 8. Before we turn to our mishnah, let us review and summarize the thresholds for nullification that we established in the previous mishnah.
The thresholds for nullification in mixtures:
By Torah law: In an ordinary mixture of a substance with the same kind of substance, two items of the same type that became mixed together, a simple majority of permitted matter over the forbidden matter is enough. Once the mixture consists mainly of permitted material, the forbidden item has lost its status and the entire mixture is permitted.
Dry with dry (rabbinically): In a mixture of dry items a simple majority suffices. If three chicken legs became mixed together, one of them forbidden and two permitted, one may eat the entire mixture, provided that he does not eat all three at once.
Liquid with liquid: In a liquid mixture, where all the components blend into one another so that every cup drawn from it contains both forbidden and permitted matter, the Sages raised the bar and generally required sixty parts of permitted matter against one part of forbidden matter, even though by Torah law a simple majority would have sufficed.
Medumah (a mixture containing Terumah): Here the Sages raised the bar even higher and required one hundred parts of permitted matter against one part of forbidden matter. This is the opinion of Rabbi Eliezer, and this is the halachah.
In our mishnah we will see that there are certain kinds of mixtures in which the Sages chose to be lenient in the way the hundred permitted parts are counted. These leniencies were built on the recognition that a requirement of one hundred times is very far removed from what the Torah originally demanded, which is a simple majority. We have three cases before us, all of them mixtures of figs, and in all of them the calculation of one hundred permitted figs against the single fig of Terumah is a very lenient calculation, at least according to Rabbi Yehoshua and Rabbi Akiva.
The first leniency - black figs and white figs:
The language of the mishnah: "Rabbi Yehoshua omer: te'enim shechoros ma'alos es halevanos, u'levanos ma'alos es hashechoros" - Rabbi Yehoshua says: black figs raise up the white ones, and white figs raise up the black ones.
The case: A vessel contains about one hundred figs, roughly fifty dark and roughly fifty light, and a dark fig of Terumah fell in among them. Even though it is known that the fig that fell is dark, and there are only fifty dark figs in the mixture to offset it, Rabbi Yehoshua permits joining the light figs to the dark ones; and since the mixture contains one hundred figs in total, they all count as the required hundred, and the entire mixture - one hundred and one figs - may be eaten.
The Bartenura explains the logic behind this: with figs it is normal to mash them, mix them and press them together. Here too, had the owner of the mixture wished, under ordinary circumstances he could have taken the dark and the light figs together and processed them, and then he would have had before him a mixture of one hundred and one - one hundred permitted and one forbidden. It turns out that there is a genuine combination here. According to the Bartenura this is not comparable to a case of fifty apples and fifty oranges into which one item fell: dark and light figs really are considered joined together into a total of one hundred, since had he wished, he would normally have combined them afterward. This is the straightforward reading of the Bartenura, and this is where we will leave it.
In this light the words of the mishnah are understood: "te'enim shechoros ma'alos es halevanos" - if a white fig of Terumah fell in, one may include the fifty dark figs in the count as well in order to complete the hundred permitted figs; "u'levanos ma'alos es hashechoros" - if a dark fig fell in, one may include the light figs in the mixture in order to reach one hundred and to nullify the forbidden status in a mixture of one hundred and one.
The second leniency - large and small cakes of pressed figs:
The language of the mishnah: "Igulei develah, hagedolim memal'im es haketanim vehaketanim memal'im es hagedolim" - cakes of pressed figs: the large ones fill up the small ones and the small ones fill up the large ones.
The case: A vessel contains one hundred cakes of pressed figs in two sizes - fifty large and fifty small - and an additional cake of Terumah fell in among them. Even if it is known that a small cake fell in, and even though it would theoretically be possible now to separate the vessel's contents into large cakes and small cakes, Rabbi Yehoshua says that there is no need to do so: the small ones and the large ones combine together to make up the count of one hundred that permits the entire mixture.
But there is a much greater novelty here than appears at first glance, and this is not a mere repetition of the first case. When we come to calculate whether there are a hundred permitted parts against one unit of Terumah, there are two possible methods of calculation: by counting units, or by weight - the total mass of the permitted food against the forbidden. The novelty of the Mishnah is that both methods are valid: a hundred units by count nullify one unit that fell in, without regard to their weight or volume; and likewise a hundred units of weight nullify one unit of weight that fell in.
Two illustrations of this:
Nullification by count, even though by weight there is no hundred: Fifty large cakes, each weighing two ounces, and fifty small ones, each weighing one ounce - a total of a hundred and fifty ounces in the mixture. A Terumah cake weighing two ounces fell in: from the standpoint of weight there are here two units of forbidden food against a hundred and fifty units of permitted food, a ratio of only one to seventy five, less than what is required. Even so we permit it, because the hundred units by count nullify the one forbidden unit.
Nullification by weight, even though by count there is no hundred: Fifty small cakes, each weighing one ounce, and twenty five large ones, each weighing two ounces - only seventy five cakes in the vessel, but a hundred ounces by weight (fifty plus fifty). A small Terumah cake weighing one ounce fell in: by count there are only seventy six here, but by weight there is one ounce of forbidden Terumah against a hundred ounces of ordinary produce, and therefore we rely on the total weight to nullify the Terumah that fell in.
It emerges that the Mishnah took both cases together: the large ones serve to nullify the small ones, even though the total weight does not have the required measure; and the small ones in the mixture serve to nullify a large piece of Terumah that fell in, even relying on count alone.
The third leniency - round cakes and rectangular ones:
Fig cakes came in two basic types: the round one ('iggul') and the square one ('malben'), something like small bricks. The round ones combine to offset a rectangular one that fell into the mixture, and the rectangular ones count against a round one that fell in - very similar to the first case in the Mishnah.
Indeed, many commentators found it difficult to identify the novelty of this last case beyond the first case of individual dark and light figs. It may well be that the novelty is that while mixing and pressing individual figs together is entirely ordinary, mixing and pressing whole cakes together is far less common. Either way, Rabbi Yehoshua is lenient in all of these cases.
The view of Rabbi Eliezer:
"Rabbi Eliezer oser" - in all three cases one may not rely on these scenarios to permit the mixture. The foundation of his approach is the scope of the question mark: when it is known that half the mixture is black and half is white, then when one falls in, even if it is not known whether it was black or white, it is known that it has with it only fifty black ones or fifty white ones, and either way there is no hundred against it. Rabbi Eliezer therefore rules stringently in every case, whether round and square or large and small, and requires a hundred of the very same type in order to permit the mixture.
The view of Rabbi Akiva - a middle position:
Rabbi Akiva takes an intermediate position, and it depends on knowledge: if it is not known which fig fell in - it may have been dark and it may have been light - then the entire doubt encompasses the dark ones and the light ones alike, and it is not known where that Terumah is: perhaps among the fifty light ones and perhaps among the fifty dark ones. In this case Rabbi Akiva permits counting all one hundred together. But if it is known in advance, because he saw a dark fig fall in, one may not use the light ones to nullify it, since it is well known that the light ones have no connection to this mixture.
It emerges, according to Rabbi Akiva, that the practical implication is this: if there were fifty dark ones and fifty light ones in the vessel, and a light Terumah fig fell in among them - one may not use the dark ones to nullify it, and one must sort the vessel into two containers: the light ones into one container and the dark ones into the second. The fifty one light ones are a forbidden mixture, and he will have to sell it to a kohen at the price of fifty, since one of them belongs to the kohen; and the fifty dark ones remain completely permitted and may be eaten. This is how Rabbi Akiva would learn the Mishnah, and the halachah follows his view.
It is worth noting that the Tiferes Yisrael understands that Rabbi Akiva is not disagreeing at all, but rather explaining the plain meaning of Rabbi Yehoshua's words. In his view, it is inconceivable that Rabbi Yehoshua would permit using the dark ones to offset the light ones, since these are apples and oranges. Rather, Rabbi Akiva is simply clarifying Rabbi Yehoshua's intent: his statement applies specifically where it is not known which ones fell in, and in that case the dark ones and the light ones can be combined together for the nullification. The same law applies to large and small, and to round and square.
An important closing note:
The example we have used throughout - fifty dark and fifty light - cannot be understood as exactly fifty of each type, because then one could simply count and see which type now has fifty-one, and thereby discover what fell in. The intention is that the ratio is roughly half and half, and therefore the doubt remains in place, and there is no way to resolve it by taking the mixture apart and counting the individual pieces to determine what fell into it.