The Halakhic Meaning of Probabilistic Multiplication: A. “Trei Rubei” and “Ruba de-Ruba” (Column 612)

A few days ago I had an interesting discussion with my friend Rabbi David Bas about “trei rubei” (two majorities) in the context of permitting an agunah. Sources were cited there that used this term in a way that seemed problematic to me, and I thought it worthwhile to try and present the matter in an orderly fashion to clarify the whole issue. Similar questions arise regarding a safek sefeika (double doubt), which I intend to address in the next column. Here I will first note several important points about safek sefeika that I will need for this column. My thanks to Rabbi Bas for his class on the subject and for our discussion that followed.

Doubt and Double Doubt

We are familiar with the halakhic difference between a safek (doubt) and a safek sefeika (double doubt). A safek is a situation in which we have several possibilities and do not know which of them is correct. When the sides are evenly balanced, the laws of doubt apply. A safek sefeika is a compounded doubt: I have one doubt, and on one of its branches another doubt arises. The structure is that of a tree (for simplicity, let’s speak of a binary tree), as in the diagram below:

Let us take a schematic example (I am ignoring many additional parameters, for illustration only). A woman is found not to be a virgin after marriage (the husband claims, “petach patuach matzati”—“I found an open orifice”). We may doubt whether this occurred before the betrothal (A), in which case she is not prohibited to her husband, or after the betrothal (B), in which case she becomes prohibited to him. But one may also doubt whether this resulted from intercourse or from an injury (mukat etz, trauma). The second doubt arises only on the assumption that she had intercourse after the betrothal, for if she had intercourse before, it does not matter for the prohibition to the husband. Therefore the second doubt is depicted in the diagram above as branching from the “after betrothal” branch (B), with B1 representing the possibility of mukat etz and B2 the possibility of intercourse as a married woman (be’ulat ish). In principle, one could have drawn a similar split beneath the other branch as well (whether, before the betrothal, she had intercourse or was injured), but there, as noted, it would not affect the ruling.

When we come to rule, if we had only a single doubt—clearly she had intercourse, but it is doubtful whether it was before or after the betrothal (A or B)—we would prohibit her to the husband because a biblical doubt is treated stringently. But since we have an additional doubt (it is unknown whether it was intercourse or trauma), this is a case of safek sefeika as the tree above shows. We see that only at one leaf of this binary tree is she prohibited to the husband (only in option B2). In options B1 and A she is permitted. In such a case we rule that she is permitted.

For the sake of discussion we assume each split is equiprobable, i.e., each branch has a 50% chance. In the case of a single doubt, the chance that she is prohibited is 50%, and then we must be stringent and prohibit due to doubt. But in the case of a safek sefeika one can say that the chance the woman is prohibited is the product of the probabilities along the path to leaf B2, i.e., 1/4 (1/2 × 1/2).

Positive and Negative Doubt

For what follows I will note that when we say a doubt is “balanced,” this does not necessarily mean we have positive information about the distribution between the two options. For example, in the woman’s case we have no information about the chance that she had intercourse before or after the betrothal, nor whether she had intercourse or suffered an injury. In such a case we have two possibilities and no information about them; we assume they are balanced and view these states as equally likely doubts. The decision that this is a balanced doubt stems from lack of knowledge—that is, from our ignorance (“veil of ignorance”). This is what I will call here a “negative doubt.” By contrast, if we know that the number of women with injuries equals the number who had intercourse, then this doubt would be balanced because of information (and not due to ignorance). That is a “positive doubt.”

This is analogous to the difference between a doubt about a single piece of meat (is it forbidden fat or permitted fat) and a doubt about one piece taken from two pieces (one forbidden fat and one permitted fat). In the second case it is a positive doubt because I know there is one of each. In the first case it is a negative doubt (perhaps if half the stores nearby sell only permitted fat and half only forbidden fat it would more closely resemble a positive doubt). Another example is a coin toss. If we know the coin is fair, the distribution is equal—probability one half for each outcome. That is balance based on information (a positive balance). But if we know nothing about the coin, we will still assume an even distribution because we have no other information and thus no way to prefer one possibility over the other. In the absence of information all possibilities are given equal weight, but that balance is based on ignorance, not information. This distinction is critical for the issue of safek sefeika that I will address in the next column (see also Column 226).

Safek Sefeika and Ruba de-Ruba

In cases of balanced doubt, the difference between a safek and a safek sefeika is clear. A safek is an even balance (positive or negative) between two possibilities. Each has a 50% chance, and therefore we prohibit. By contrast, when we have a safek sefeika (see the tree above), the balance between permitted and prohibited is no longer even. The chance that the woman is prohibited drops below one half, and therefore the ruling in such a case is that the woman is permitted.

We may now ask what happens when at each such node the split is not even. Suppose, for some reason, the chance that the woman had intercourse before the betrothal is greater than the chance that she had it after. In such a case the woman is already permitted with a single doubt. This is what halakhah calls “following the majority” (ruba). In such situations, ostensibly there is no special meaning to ruba de-ruba[1]—i.e., to a tree like the one above with two nodes, where each node represents a majority rather than an even split—since two such nodes simply produce a larger majority.

For example, the chance that a woman had intercourse before the betrothal is 80%, but even if she had it after (note: this node is under branch A) there is an 80% chance she is mukat etz. What is the chance she is permitted? 0.8 + 0.2×0.8 = 0.96. That is, the majority for permission grows even larger. But in halakhah, for the rule of following the majority, the strength of the majority does not matter; fundamentally we permit based on a 51% majority just as on a 99% majority (I am ignoring questions of a “noticeable minority,” etc., which fundamentally have no impact). For example, in a city with one hundred stores, fifty-one of which are kosher, a piece of meat found in the market is permitted to eat, just as in the city of ten stores of which nine are kosher. If so, what we saw regarding safek sefeika does not hold for ruba de-ruba. The status of ruba de-ruba should be like a single majority. In the usual halakhic contexts where a single majority permits, then of course ruba de-ruba will permit as well (by the rule of majority). But in contexts where we prohibit even with a majority (e.g., in monetary law according to opinions that do not follow majority, or in lineage because “a higher standard was made for lineage”), then ruba de-ruba will also not suffice, for there is still only a majority for permission, and a majority is insufficient.

Incidentally, there may be significance to ruba de-ruba where there is a majority leading to stringency (e.g., most women are injured after betrothal) that is offset by a further majority toward leniency (most are injuries rather than intercourse). In the tree, that split would be after branch B (as in the earlier diagram), not after node A as in the previous example. In such a case the product lowers us and may sometimes fall below 50%, which might in some cases lead to leniency. For example, if in both nodes the majority is 60%, then the chance of prohibition is 0.24, which is a majority for permission.

Another important note: in majority (rov) there is no difference between the negative and positive cases. If I know the chance of permission, then I have information (it cannot be set from ignorance). True, one might know that there is a majority but lack concrete information about its magnitude (what the chance of permission is). This is the analogue, in the law of majority, of a negative doubt. In such a case, of course, one cannot multiply probabilities, and simply follows the sides (the possibilities).

Leniencies and Stringencies in Permitting an Agunah: A General Look

There are quite a few leniencies in permitting agunot, due to concern for the woman’s distress when she cannot remarry. Therefore they were lenient to accept the testimony of a single witness, even though it concerns a matter of sexual prohibition that normally requires two witnesses. They permitted testimony through hearsay, testimony by those ordinarily disqualified (a woman, a slave, a minor), and other such leniencies. These leniencies create the impression that permitting an agunah is a field prone to formal tricks to release the woman, due to her great distress. This naturally raises questions when people see judges refusing to permit agunot easily, even in cases where it is fairly clear that the husband is dead (such as with Ron Arad).

Many have wondered how the Sages can “uproot” a Torah law, and in such a stringent area as permitting a married woman to remarry. When a single witness appears, by Torah law she remains prohibited as a married woman (for two witnesses are required), and the Sages were lenient and allowed her to marry. This is also a leniency involving action (kum va’aseh), which, on the common view, the Sages cannot do.[2] That is one difficulty. Another is an inherent tension in the laws of permitting agunot.

As is known, there are also pronounced stringencies in permitting agunot—for example, requiring identification by the face and nose specifically (and not suffice with a general assessment that it is the husband), and the rule of “waters without end” (mayim she’ein lahem sof; see Yevamot 121a–b). In a body of water whose boundaries cannot be seen (sea, ocean), even if a person drowned in its midst, we do not permit his wife, because of the concern that he survived and emerged on some shore without being seen (in waters with visible limits, had he emerged, we would have seen him). Think of someone who drowned in the middle of the Atlantic Ocean. It is clear to all of us that he died, and yet we do not permit his wife (although strictly speaking this is a stringency; fundamentally she is permitted, and therefore, practically, if she married, she would not be forced to divorce). This is an enormous and seemingly illogical stringency. Where did the consideration of the agunah’s distress go? How does this stringency fit with the leniency of believing a single witness or hearsay?

The simple explanation of this dissonance is that the Sages did not rule leniently where they had a real concern that the husband was alive. It is inconceivable that we would permit a woman to remarry if there is a possibility that the husband will return alive and a catastrophic situation of mamzerut and prohibitions to husband and paramour will result. All the leniencies in permitting agunot deal with cases where there is no real concern that the husband is alive—i.e., situations in which, factually, we are convinced that he died—but where there are formal halakhic impediments to permitting her. For example, if there is conclusive circumstantial evidence that the husband died, still, fundamentally, in matters of sexual prohibition we require two witnesses. This is a case where we have no real concern that the husband is alive, and the impediment is only formal. In such a case, and only in such a case, the Sages were lenient and allowed her to marry even without two witnesses.

This resolves the tension found in the laws of permitting agunot. The leniency stems from the woman’s distress, but it can operate only to remove formal impediments to permission. The other side of the coin is the great stringency of a married woman mistakenly permitted to the public. Where such a concern exists, we are exceedingly stringent; there is no contradiction to the leniencies that apply to the other situation.

This also resolves the first difficulty mentioned. The leniencies described here are not leniencies of action (kum va’aseh), for they do not permit a married woman to marry; they permit an unmarried woman to marry (since, factually, we know she is unmarried). Permitting a married woman to marry would be a leniency of action in sexual prohibitions, but, as noted, that is not what is being done here. The Sages only waive the requirement of two witnesses; that is a leniency of inaction (shev ve’al ta’aseh) in the law of evidence. Incidentally, this may be what Maimonides intended in Hilchot Gerushin 13:29, where he wrote:

“Let it not be difficult in your eyes that the Sages permitted the severe sexual prohibition by the testimony of a woman or a slave or a maidservant or a gentile speaking innocently, or by hearsay, or in writing, and without interrogation and examination as we explained; for the Torah was particular about the testimony of two witnesses and other laws of testimony only in matters which cannot be determined except through the witnesses and their testimony, such as testifying that ‘so-and-so killed so-and-so’ or ‘so-and-so loaned so-and-so.’ But in a matter that can be ascertained without this witness, and the witness cannot evade exposure if it is not true—such as one who testifies that so-and-so died—the Torah was not particular about it, for it is far-fetched that a witness would lie about this. Therefore the Sages were lenient in this matter and believed a single witness, a maidservant, writing, and without interrogation and examination, so that the daughters of Israel not remain agunot.”

As Rabbi Bas rightly noted, this is almost explicit in the passage in Bekhorot 46b, which itself grapples with how the far-reaching leniencies fit with the stringencies found in the laws of agunah. It explains that they were lenient in accepting otherwise disqualified witnesses (hearsay, a woman, a minor), but not regarding the actual identification of the body (face and nose). The Talmud there offers another explanation as well, but as I understand it, that alternative pertains to the law of “recognizing the firstborn” (yakir) and not to the understanding of the laws of agunah per se.

I have not entered here into the question of the presumption that a woman is careful and would not remarry unless certain (deyka u’minseva), nor the Rambam’s apparent contradictions as to whether some of these leniencies are biblical—along with other considerations raised in the Talmud and commentaries. My aim here is only to sketch the outlines of the issue. I mainly wanted to show the basic motivation: there is room for great leniencies in permitting agunot, but only when the factual situation is clear. If it is not clear, the woman’s distress yields no leniency; on the contrary, there we find far-reaching stringencies.

“Trei Rubei” in Permitting an Agunah[3]

One of the major leniencies in permitting an agunah is the leniency of “trei rubei.” What stands out about it is that it has no clear source in the Talmud and early authorities. It first appears in Responsa Qohelet Ya’akov §9, by R. Yaakov of Karlin, a student of R. Chaim of Volozhin (cited also in Pitchei Teshuvah, Even Ha-Ezer §17):

“I also recall that in my youth, when I was in the home of my honored teacher, our master, the Gaon R. Chaim of blessed memory of Volozhin, a similar question was asked: a man fell from a very high bridge onto ice, and from the ice into the water, and was lost and not found. He inclined to permit on the basis of trei rubei. That is, a fall from a height of two stories kills in most cases, as is proven from the law of the stoning platform, etc., and from the laws of a mortally wounded animal that fell—a law from Moses at Sinai—together with the weakness of the case of ‘waters without end.’ He elaborated in a responsum, and it seems the woman was then permitted with the agreement of all the sages of Vilna. I know that reasons to be stringent can also be found in some responsa of later authorities; but there is less to fear in ‘waters without end,’ which is rabbinic, and one may rely on lenient views in this matter in a case of agunah.”

A man fell from a high bridge onto ice in the water, and from the ice into the water and drowned. The case involved “waters without end” (it was a long river), and ostensibly the woman is prohibited. Nevertheless he permits the woman based on trei rubei: there is a majority that the person died from a fall from such a height onto ice (from the stoning platform it is proven that a height of two stories kills), and an additional majority that even if he somehow came out alive, he likely died when he fell into the water and drowned (most who fall into such waters die).

Following this source, many decisors to this day often use the trei rubei leniency. One event that sparked such discussion was a case of an air force pilot whose plane was hit by a missile and crashed. The pilot and the plane were not found. The pilot flying in the adjacent plane tried to check for ejection and searched the crash area but did not find him. Rabbi Ovadia Yosef, in Yabia Omer VI, Even Ha-Ezer §4 (“Ma’aseh ba”), used trei rubei to permit the pilot’s wife, and wrote:

“Here, since each of the two pilots was intent on tracking his fellow’s actions and reporting them to control, it is uncommon that there would be ejection by the pilot upon being hit, and the other pilot would not notice;

and even if we fear the minority and say that perhaps the pilot nevertheless succeeded three times to eject from his plane and the pilot who testified did not notice, in any case since he fell into the sea, and the helicopter that hurried to the place to save him did not find him, behold it is as one who fell into ‘waters without end,’ most of those who drown die. We thus have two majorities (trei rubei) for death: according to the majority he was killed in the crash, and if you say he fell into the sea, most who drown die. And with trei rubei one can say we rule leniently. As written by the Gaon R. Yaakov Av Beit Din of Czazmir in Responsa Beit Yaakov…

“And even in our case where the plane exploded due to a missile and fell into ‘waters without end,’ the two majorities came by two incidents, and in such a case they are fit to join to be like trei rubei and to be lenient even ab initio. And it is known what was written in Responsa Qohelet Ya’akov of Karlin (§9, 26a), regarding one who fell from a high bridge over the sea whose waters had frozen and become ice, and from the ice he rolled and fell into the waters between the ice masses. And the Gaon R. Chaim of Volozhin ruled to permit his wife to marry, on the basis of trei rubei: that falling from a height of two stories is a majority to death, as taught in Sanhedrin (45) that the stoning platform was two stories high. And in combination with his falling into ‘waters without end,’ we have two majorities, and therefore his wife may marry ab initio. And all the sages of Vilna agreed. See there. And see also in his work Chut HaMeshulash (§6, 12b). And indeed Pitchei Teshuvah (§17, n. 133) brought the words of Qohelet Ya’akov and wrote, ‘and see Tosafot Yevamot 121 s.v. velo hi.’ It needs investigation. According to what we have said, the resolution is clear for several reasons, as above…”

And so we find in other decisors.

However, the Chazon Ish (Even Ha-Ezer §31 n. 7; addressing the agunot of Palmach fighters from the explosion of the A-Ziv Bridge during the Night of the Bridges in 1946) disagreed with the “trei rubei” thesis and wrote, among other things:

“In Pitchei Teshuvah n. 133 he brings the words of Qohelet Ya’akov, who discussed that where there are two majorities one may marry ab initio.

“And Pitchei Teshuvah raised a difficulty from Tosafot: for in the case of a Torah scholar (tzurba merabbanan) there are two majorities. One can distinguish: in Qohelet Ya’akov’s case there are two majorities to death; but in Tosafot—one majority to death and one majority attesting to death. Nonetheless, we have no license to innovate what is not mentioned in the Talmud; there is no difference between a single majority and two majorities. And in Pitchei Teshuvah n. 138 he brings the words of the Chatam Sofer that nowadays with the post we can inform the home more readily.”

His claim is that this leniency is not mentioned in the Talmud and should not be used. It is hard to escape the feeling that the leniency seems problematic in itself. I assume that even the Chazon Ish would agree to logical leniencies, even if they do not explicitly appear in the Talmud.

Rabbi Ovadia, in the responsum cited, addresses the Chazon Ish’s view and writes:

“And though the Gaon Chazon Ish, Even Ha-Ezer (§31:7), after dismissing Pitchei Teshuvah’s difficulty on R. Chaim of Volozhin’s ruling permitting an agunah by two majorities—from Tosafot Yevamot (121)—by saying that the Tosafot’s case is different, for it counts as one majority to death since the second majority merely attests and proves the first majority (that he is among the drowned and not among survivors), nevertheless he concludes that we may not innovate what is not explicit in the Talmud to be lenient, and there is no difference between a single majority and two majorities. But his words are not compelling; on the contrary, since the stringency of ‘waters without end’ is itself an extra stringency—fearing an extremely uncommon minority—we say, ‘let us not add further to it,’ and they did not decree in the case of two majorities…”

His conclusion is that the stringency of “waters without end” is itself an extra stringency, and therefore it should be confined only to what is explicitly included in the decree. His starting point is the reverse of the Chazon Ish’s starting point. The Chazon Ish assumes the leniency itself is problematic, and therefore rejects it since it does not appear in the Talmud. Rabbi Ovadia assumes that the stringency of “waters without end” is the strained one, and therefore any distinction that limits it is acceptable (the principle that “you have only its novelty and no more”).

Before continuing, I will present another use of the trei rubei leniency.

The Plane Crash in Rabbi Herzog’s Ruling

Decisors applied this leniency also to another case of a plane crash. In 1944 a South African plane crashed in an aerial battle, and a question arose regarding the pilot’s wife. Rabbi Herzog, in Heikhal Yitzhak, Even Ha-Ezer I §29, discusses this and writes (there §10):

“Now, concerning our case: since there is a compelling presumption that the airplane broke and fell, even if it fell on land this would be included in ‘ruba de-ruba to death,’ and all the more so from the case of a ship lost at sea, of which it is said in Bava Batra 73b that most die. The Rivash and the Noda BiYehuda, and this is known; and the Tosefot Yom Tov and other great authors discussed that the very fact that we impose on them both the stringencies of the living and the stringencies of the dead when a ship is lost is itself not biblical; this is ancient, and I have elaborated elsewhere. It therefore follows that the very fact of the plane breaking and falling from a great height is a majority to death, and the fall into the sea adds another majority, and one can judge this as two majorities.”

Note that his consideration is slightly different from the previous one. He argues that if the pilot fell on land there is a majority that he died because of the height, and if he fell into the sea, there is again a majority that he drowned; this is a case of trei rubei, and therefore one should permit.

Here I arrive at my main point.

The Difference Between the Considerations

Although this seems similar to Rabbi Ovadia’s plane-crash case, and the considerations seem similar, if you examine closely you will find Rabbi Herzog’s consideration differs from Rabbi Ovadia’s. Rabbi Ovadia’s consideration matches precisely the tree shown above at the start of this column. The pilot likely died from the missile strike (that’s the first majority), and even if he did not die then, he likely drowned in the sea (that’s the second majority). By contrast, Rabbi Herzog reasons differently: the pilot likely died if he reached land (the first majority), and if he reached the sea there is also a majority that he died (the second majority). If you try to draw the tree for this consideration, you will see it differs from Rabbi Ovadia’s:

Rabbi Ovadia’s consideration	Rabbi Herzog’s consideration

For Rabbi Ovadia, option A is that he crashed and died from the impact within the plane itself (the possibility of falling to land does not arise there, apparently because in that case the plane was not found, so he certainly fell into the sea[4]). The only question is when, if at all, did the pilot die. Option B is that he fell into the sea. Option B1 is that he drowned in the sea, and option B2 that he remained alive there. In this case only the leaf B2 leads to the conclusion that the woman is prohibited. Opposite it there are two leaves of permission: the leaf of branch A (which is balanced against two possibilities—if it were mere doubt this would be 50%) where the woman is permitted; and together with B1 (that he drowned in the sea) we have in total three leaves of permission (in a pure doubt this would be 75%). In our case both nodes have a majority toward permission. Suppose it is 80%. What is the chance the woman is permitted now? For her to be prohibited, the plane must fall into the sea (20%) and the husband must nevertheless remain alive (20%). That is, the chance she is prohibited is 4% (0.2 × 0.2). This is a consideration of ruba de-ruba (like a safek sefeika, only with majorities), and there is room to permit, because indeed the chance of permission (96%) is greater than with a single majority (80%). The chance of error in permitting with ruba de-ruba drops to a quarter of the chance of error with a single majority. This is a case of multiplication, since the chance of prohibition is the product of the probabilities along the path ending at B2.

By contrast, in Rabbi Herzog’s consideration there is no multiplication of two majorities, and therefore it is not ruba de-ruba but perhaps should be called merely “two majorities” (trei rubei). In the tree describing his consideration, option A is that the pilot reached land and option B that he reached sea. Option A1 is that he died in the crash and A2 that he did not (he reached land alive—a slim chance). Option B1 is that he drowned in the sea and B2 that he remained alive there (again, slim). If you count the leaves of permission and prohibition you will find there are two on each side. If he is alive on land or in the sea she is prohibited, and if he died on land or in the sea she is permitted. In the case of nodes with an even split (pure doubt), this would not be a safek sefeika but a single doubt. In our case, each node is a majority and not a doubt (since the options are not equal). In such a case, if we suppose both majorities are 80%, then the chance the woman is permitted at the end of the calculation is still 80%, as with a single majority. Therefore, if a single majority does not suffice to permit (because of the rule of “waters without end”), this consideration also cannot permit her.

Suppose it is known that the plane fell into the sea; now we have only the majority of “waters without end.” That is a single majority. But even that could be formulated as “two majorities”: perhaps he fell into the Mediterranean Sea—then either he remained alive (slim) or not (very likely); or he fell into the Red Sea—with the same two options. Any single majority can be rephrased as “two majorities”; it is only a matter of terminology. It is like asking whether I would prefer to be under ten shells in a basketball court, or under two bombardments of five shells each. Clearly, in terms of the chance of being hit, it is the same.

The conclusion is that despite the similarity between the considerations, Rabbi Herzog’s consideration is, in my view, mistaken. His case may perhaps be called “two majorities,” but it is not a case of ruba de-ruba (multiplication), and therefore it is entirely equivalent to a single majority and cannot in itself permit the woman.

The Chazon Ish’s Reasoning

We saw that the Chazon Ish does not accept the leniency of “trei rubei,” and not only in Rabbi Herzog’s case where it is indeed problematic; he also disagrees with Rabbi Ovadia and Qohelet Ya’akov in cases that are ruba de-ruba, i.e., multiplication. For him, one may not use this leniency even where there is ruba de-ruba, i.e., even when the chance of permission is higher.

His reasoning seems straightforward. We saw that in the case of multiplication the chance of permission is 96%. That is still identical to a case of a single majority of 96%. But if I had only a single majority and not ruba de-ruba, even if that majority were 96% (as in “waters without end”), we would not permit the woman, since a single majority is insufficient. So why, when the same chance is reached by multiplying two nodes, should it be acceptable? The Chazon Ish apparently holds that if in agunah matters we do not follow majority, we should not follow ruba de-ruba either.

This, of course, raises the question: what do the others hold? Ostensibly, the Chazon Ish is correct. What difference does it make if we reach 96% permission by multiplication or without it? If we do not follow majority, then ruba de-ruba should not help either. I can think of two possible explanations for their view, which I will now present.

A. The Hand of Providence

If we assume that everything that happens in the world results from divine consideration and not natural processes, we might offer the following explanation. If two separate “lotteries” must be passed in order to survive (both to fall into the sea and to survive in the sea), there is less chance of survival—not because of probability but because divine involvement is required to save him against the odds in two distinct cases. The Holy One must ensure that he does not crash with the plane but falls into the sea (despite the slim chance), and also ensure he does not drown but survives (again a slim chance). According to this, there is a difference between reaching a given probability X through multiplication (two cases) and having a single majority of the same magnitude (X).

As is known, many claim that statistics are merely a covering beneath which hides divine providence (Eli Marzbach wrote this more than once). In my view, generally there is no divine involvement in the world (see Column 280, 463, and more), and therefore it is incorrect to assume its presence in any given case. I have explained more than once that claims of involvement within the framework of nature are also mistaken. Any divine involvement is a deviation from nature, a miracle. Likewise, statistics are not a cover for divine providence but are the result of natural processes (I have explained that there is no true randomness in the world, perhaps apart from quantum theory, which is irrelevant here; the use of statistics is always due to missing information). Therefore, I do not accept the claim that what happens in the world—i.e., whether the pilot survived or died—is a product of providence. Statistics reflect nature and its laws, not divine intervention. You can understand that I do not really accept this explanation. It may explain the later authorities’ view, for they likely did not hold my view about providence; but in practice, in my understanding, it is incorrect. If this is the explanation for the “two majorities” leniency, in my opinion one cannot permit an agunah on that basis.

B. A Prelude: Ordinary Majority vs. Overwhelming Majority

Consider a case where the majority in question is not 80% but 51%. Do you think anyone would entertain permitting a woman to remarry based on such a majority? Suppose there is a ruba de-ruba (multiplication) where the probabilities are as follows: the chance of falling into the sea is 49% (and 51% to crash with the plane), and the chance that, if you fell into the sea, you would not drown is also 49%. The chances the pilot is alive (and thus the woman is prohibited) are the product: 0.24. That is, the chance the husband is alive is around a quarter. Would anyone in such a case dream of permitting the woman? The stringency of “waters without end” is in essence a stringency in a situation where there is virtually no chance that the husband survived. Fundamentally we would permit the woman, and only due to the stringency of a married woman do we prohibit. What would you say about a case where the chance that the husband drowned is indeed a majority, but only 51%? Would you think there is room to permit the woman to remarry in such a case? Formally, yes—for as we saw, fundamentally in all areas of halakhah we follow any majority whatsoever. And yet it is hard to believe that anyone would consider permitting the woman or even think her prohibition is only rabbinic. With nearly a fifty-fifty chance that her husband is alive, can we allow her to remarry?!

This implies that the majority discussed in the context of agunot must be an overwhelming majority (i.e., the minority is not noticeable). To permit the woman we need a situation like “waters without end,” where the chance that the husband survived is negligible. In such a case, fundamentally, there would be room to permit, but even then we are stringent and prohibit due to the gravity of the prohibition. Indeed, that is the case with “waters without end.” It is not a majority close to 50% but something very close to 100%; even so, there remains a small doubt due to which we are stringent. We could formulate it thus: if a person fell into “waters without end,” it is in effect clear to all of us that he is dead in terms of factual assessment. Halakhah has very strict standards and therefore, legally, we do not accept a mere majority and we fear even an extremely small minority. But that stringency only says that, in principle, the husband died—only that perhaps we are mistaken and he did not. By contrast, in a case of a 51% (or even 80%) majority, it is not correct to say “the husband died but perhaps we are mistaken”; rather, it is more correct to say “it is not known that he died.”

To understand this better, think of Ron Arad, the missing navigator, absent for decades. It is clear to any reasonable person that he is no longer alive—though, of course, theoretically there is some chance he is. And indeed it took many years before the military rabbinate was prepared to rule that his wife may remarry. Such a situation is not one of doubt or majority; here we say Ron Arad is dead, except that, as always, there is a small chance that we are mistaken. But if the situation were that there is a 51% or even an 80% chance that he is dead, no one would say “he is dead, but there is a concern that we are mistaken.” There it is more accurate to say “it is not known that he is dead.” In practice, if you ask a person on the street for his opinion about Ron Arad, most would laugh at rabbis who “fear” that he might be alive. This is precisely a case where we can say the person is dead, but there is a small chance we are mistaken. That is an overwhelming majority; and indeed halakhah nevertheless is stringent. But in cases where this is not merely rabbinic fastidiousness but where a reasonable person recognizes that there is a real chance he is alive, then it is an ordinary majority and irrelevant for permitting agunot.

It is natural to test this against the cases of “waters without end” and “waters with visible limits.” In “waters without end,” it is an overwhelming majority, but, legally, that is not enough. By contrast, in waters with visible limits—i.e., a body of water whose entire boundary we can see—there we permit the woman to remarry: a person who drowned in such a pool and we did not see him emerge—his wife is permitted (for if he did not emerge, he clearly drowned). But even there there is some chance of error: perhaps we did not see, or did not pay attention, or perhaps there is some hidden border of the pool we did not notice. And yet there we permit the woman. The reason is that in such a case we are not dealing with an overwhelming majority but with certainty. True, there is no such thing as 100% certainty in life (not even with the testimony of two witnesses—see Column 226), but for our purposes this is deemed certainty. It is conclusive circumstantial evidence (see that column). “Waters without end” is a case a bit less clear, and therefore there it is only an overwhelming majority, not certainty.

We can now explain the halakhic situation.

B. Explaining the Leniency of Ruba de-Ruba

We saw that halakhah does not suffice with the majority in “waters without end,” but suffices in “waters with visible limits.” Waters with visible limits amount to certainty; “waters without end” is an overwhelming majority. If we are dealing with an overwhelming majority, then multiplying by another majority really improves our position. Note that with an 80% majority there is a 20% chance of error. When we multiply two such majorities, the chance of error drops to 4% (96% certainty). That is a very significant improvement: we have reduced the chance of error by 80%. This is in contrast to two majorities of 51%, where we saw that the chance of error is a quarter.

Of course, 80% is still not truly overwhelming; 20% error is a noticeable minority. Think of “waters without end”—that seems much closer to 99%. But if you are speaking of two 99% majorities, you get a vanishingly small chance of error—practically absolute certainty (as in waters with visible limits). The mathematical rule is that the second majority reduces the chance of error by its measure. Thus, when both majorities are 80%, the single-majority error is 20%, and with two majorities it falls to 4% (an 80% reduction). When the majority is 90%, the error is 10%; the second majority reduces it by 90%, leaving a 1% error; and so forth.

If so, it is true that with a single majority one can reach 96% certainty (as with “waters without end”), and ostensibly the Chazon Ish is correct that there is no reason to distinguish between that and a 96% reached by multiplication (ruba de-ruba). But here a critical point enters. Usually we have no metric for determining the size of the majority in question. That is, it is a negative majority in the terminology above. Can anyone quantify the chance that one who fell into the sea survives? It also depends on how he fell, how far from shore, his swimming skills, maritime traffic, etc. And even if all these were known, we would still not be able to fix a numeric probability, only say that the chance is very small, or very, very small, etc. In most such cases one cannot set precise quantitative metrics, certainly not universal ones (for all falls into the sea). This is likely the reason that our way to ensure the chance is small enough is to demand a multiplication of two majorities (ruba de-ruba). The requirement of multiplication serves as a surrogate for setting a sufficiently small numeric chance of error in permitting. If we multiply two majorities when it is clear that at least one of them is overwhelming, we may assume that, generally, the result will be sufficiently overwhelming to permit the woman (and in effect, amount to certainty).

Why did I write that it suffices if only one of the majorities is overwhelming? Because the chance that the plane fell into the sea or onto land is not necessarily an overwhelming majority and may even be balanced; yet it still significantly improves the chance found in the standard case of “waters without end.”[5] Suppose that in “waters without end” there is an overwhelming 99% chance that the husband drowned. If there is a 50% chance he fell into the sea and 50% onto land, the chance he died rises to 99.5% (cutting the error in half). Of course, if there is a majority—or even an overwhelming majority—that he fell on land, the chance of permission is even higher, but it is not clear that we need that, and still less that we need that too to be overwhelming.

In short: unlike other areas of halakhah, where we follow any majority (even 51%), in permitting agunot, due to the fear that “the dead will show up alive,” we deal only with overwhelming majorities. And even that does not suffice to permit until another node is added to the tree (ruba de-ruba). When I have a positive majority (i.e., I know its size), then indeed multiplication has no significance—ruba de-ruba is no better than a single majority of the same size. The question is whether the resulting majority is sufficiently overwhelming. That comparison led directly to the Chazon Ish’s view, thus making the opposing view seem forced. My explanation for the opposing view is that usually our majority is negative (as in “waters without end”), and the way to ensure it is sufficiently overwhelming is to require multiplication. That occurs only in ruba de-ruba (where the chance is reached by multiplication), not in cases like Rabbi Herzog’s “two majorities” (which is not ruba de-ruba). There it is like a single majority; however overwhelming (like “waters without end”), it does not suffice to permit the woman. Such a consideration seems to have no place in permitting agunot.

A Note on Waters with Visible Limits

From this perspective we can view the permission in waters with visible limits as a case of multiplying two majorities. When a person drowned in water, the most likely outcome is that he died. And even if he did not die, there is a very small chance that we would not have seen him emerge (because the boundaries are visible to us). Therefore here there is certainty no less than with two witnesses. Incidentally, according to this, the law of waters with visible limits itself could be the source for permitting by two majorities. In “waters without end” there is only a majority for drowning—but there we lack the second majority that, had he not drowned, we would have seen him emerge. If there is another majority (as in the ice case in Qohelet Ya’akov), that too becomes like the case of waters with visible limits.

One can debate how well-founded it is to distinguish between the question of whether he drowned and the question of whether we saw him emerge, but this is not the place.

Summary: Three Types of Majority

We distinguished three types of majority regarding permitting agunot:

• An ordinary majority (51%) suffices for prohibitions but has no significance in permitting an agunah. Simply, she remains prohibited fundamentally.
• An overwhelming majority, like “waters without end”: fundamentally, even in agunah matters the woman should be permitted, but a stringency was adopted not to suffice with it.
• An absolute majority is treated as certainty, like “waters with visible limits.” There the woman is permitted fundamentally and in practice.

My claim is that multiplying two overwhelming majorities (and perhaps even where one is overwhelming and the other ordinary) moves us into the category of absolute majority—i.e., certainty—and this is the leniency of ruba de-ruba. “Two majorities” like in Rabbi Herzog’s case is essentially a single overwhelming majority; there one might have had room to permit fundamentally, but in practice halakhah is stringent about it.

Two Further Examples: Between Particular and General Discussion

In our conversation, Rabbi Bas brought two further examples. R. Yitzchak Elchanan Spektor wrote a leniency based on a “two majorities” consideration regarding a ship that sank at sea: most passengers were gentiles and most who drown die—and for him that is trei rubei. Rabbi Goren (Meshiv Milchamah, vol. 3) likewise relied on this leniency in several responsa, such as regarding the fallen of Kfar Etzion and the fallen from the ZIM ship “Metzuda.” Concerning Kfar Etzion he wrote a trei rubei leniency on the basis of the following: most Jews there were single and most were killed. In both cases it is important to distinguish between a case in which we are dealing with a specific individual’s wife, and a case in which we permit all the wives of the missing as a group (regarding the ZIM missing, Rabbi Goren was careful to address them one by one and based the leniency on that; below I will explain why I think this is problematic).

But I think there is a mistake here. When a single person comes before us and we rule regarding him, in his case there is no question whether he was married or single. He is certainly married—otherwise there would be no need to discuss permitting his wife (his wife came before us and thus the question arises). Therefore, regarding whether he survived or not, there is only one relevant majority and not “two majorities”: most people did not survive. That majority applies to singles and to married men alike. But the man we are discussing is certainly married; therefore the only relevant majority is that most did not survive. Beyond this, I think that in such a case the majority in question is an ordinary majority and not overwhelming (there were not a few who were taken captive in Kfar Etzion). Therefore, in my opinion, it is quite implausible to permit his wife (even if there were “two majorities,” that would not help when the majority is not overwhelming). Perhaps if much time has passed and it is already known who was taken captive (or the captives returned and he did not, nor did he make contact), then we are in an entirely different situation. But then another majority must be added: that if he survived he certainly would have made contact with his family. The same applies to R. Yitzchak Elchanan’s consideration. If the man is before us he is certainly Jewish (otherwise there would be no question about his wife). Therefore, for him there is only one relevant majority: that most who drown die.

It seems their consideration is relevant specifically to all the people under discussion together, and therefore it is structured in reverse: assuming someone survived (itself a minority, since most drowned or were killed), there is a majority that he is single/gentile and not a married/Jewish man—hence not the husband of the particular woman before us. Ironically, it is this group-level consideration that has more room, because here there are indeed two majorities. Yet even from this vantage point there is a problem: if we permit all the women, there will be among them some for whom the permission is erroneous (the minority of those who are married and whose husbands survived). If there are one hundred women and we permit them all on a 0.1% error rate, there is a decent chance that one of them was mistakenly permitted (see Column 226). I do not believe any decisor would be willing to issue a permission that is certainly mistaken in some of the cases (like the “certain swindler” in Tosafot at the beginning of Bava Metzia, where one does not adjudicate in such a situation).

But beyond all that, I think the consideration I described from these two decisors is based on mis-formulating the question. The question before us is about a specific person, and for him there is only one majority. Their formulation would be relevant if a rumor had reached us that someone survived, and the question were whether he is a gentile/Jew and married/single. In such a case there would be room to discuss a “two majorities” consideration as I described. But when the question concerns a specific person—did he survive or not—this is only a single (non-overwhelming) majority. To the best of my understanding—at least if we consider only the “two majorities” consideration presented by these decisors (there were other considerations)—there is an error in the leniency. This is not a case of ruba de-ruba, but resembles what I deemed (mistakenly, in my view) the leniency of Rabbi Herzog above.

Are Agunah Leniencies Based on Statistics? The Place of Formalism

Here we come to another very important point. Anyone experienced in statistics knows that the most important part of statistical calculation is to formulate the question correctly. A wrongly formulated question yields a mistaken calculation. We saw examples of this in several columns (such as 145, 402, 506, and more), most of which deal with confusion over conditional probability. Thus, for example, there is a difference between the question “if reality is X, what is the chance the judge will rule X?” (a metric of the judge’s quality) and the question “if the judge ruled X, what is the chance reality is X?” (a metric of the ruling’s quality). We saw that these are two entirely different questions, with entirely different probabilities. So too we saw the difference between “what is the chance that two infants will die of SIDS” and “if two infants died, what is the chance it was SIDS?” (the debate around Munchausen by proxy). Note that in our case the question discussed by the decisors was: assuming someone survived, what is the chance it is the particular person we are discussing. But the correct question when we come to permit a specific woman, where no survivor is known, is the reverse: what is the chance that the specific person under discussion survived. That is an entirely different question.

This brings me to the broader question of whether these leniencies are based on statistics. At first glance, it would seem not, because probabilistically ruba de-ruba is only a calculation that yields a majority, and there is no difference between it and a single majority of the same size. I thought so as well (and wrote as much to Rabbi Bas), but after the analysis here I recant. Permitting an agunah is indeed based on statistical considerations. What confuses is that the majority in question is usually a negative majority (we lack a numeric metric), hence the insistence on multiplication and not on numbers. But, as I explained, the purpose of the multiplication is to create an overwhelming majority that can be viewed as statistical certainty.

We must understand that, in permitting agunot, the factual situation is of critical importance. We do not want a formal permission to marry when, factually, it may turn out that the husband is alive and “walks in.” In such a situation no decisor would permit the woman. But if the facts are decisive, then it is statistics, not formal halakhic tricks, that determine our assessment of the facts. I explained above that all the leniencies in permitting agunot always deal with cases where it is factually clear that the husband is dead, and the obstacle to permitting is merely formal. Therefore, a formal leniency in a case without real statistical backing (i.e., where there is no factual certainty) is out of the question. In the cases described here—at least as far as the “two majorities” consideration raised by the decisors—the husband’s death is far from clear. The relevant majority is not overwhelming, and certainly not absolute, and it is not a case of ruba de-ruba. A “two majorities” leniency like those of Rabbi Herzog, Rabbi Goren, and R. Yitzchak Elchanan Spektor is a formal leniency lacking statistical grounding—but we cannot permit a woman based on formalism when there is a real chance that her husband will return.

Of course, I speak only about the “two majorities” aspect I described. In all the permissive responsa I referenced there were additional considerations that could show that, factually, there is no chance the husband is alive (e.g., that for a long time he made no contact with his wife), and then in practice there is room to permit the woman. I do not intend to cast aspersions on the permissions, but only to critique the “two majorities” argument as employed there. If the factual situation is indeed clear, then there is no need to create the fiction of “two majorities” (which is not ruba de-ruba, i.e., without multiplication) in order to permit. If there is a consideration showing an absolute majority that the husband died, one can rely on that directly (as in waters with visible limits). I think formal tricks are unhelpful here, because we do not have a formal halakhic obstacle to solve; the Sages already solved everything for us.

We must understand that, contrary to the common image, in the laws of permitting agunot there are in fact no formal problems; therefore, there is no room for formalistic considerations. Here, only statistics matter. The recourse to formalism is due to the agunah’s distress. But the formal leniencies in this field were already instituted by the Sages (and only they could, since formal authority of the Sanhedrin or Talmud is needed). They already were lenient in the laws of witnesses (a single witness, even a disqualified one; hearsay; etc.) and agreed to accept an absolute majority (as in waters with visible limits). These leniencies suffice for any practical need; there is no need for decisors to add more. What is incumbent upon us decisors is only to use what they permitted. In practice, our entire task is to try to show that there is an absolute majority for permission (or at least an overwhelming majority, which fundamentally also suffices). If we succeed in showing that, then factually we are convinced the husband died, and therefore, in principle, the woman is permitted and there is no need to resort to formalism (for the Sages already allowed us to use an absolute majority, even circumstantial—i.e., without two witnesses—as evidenced by the law of waters with visible limits). And if we did not succeed in showing such factual certainty, then there is a real chance the husband is alive—and, as I explained, in such a case no formal consideration will help.

So much for “two majorities” in permitting an agunah. But the term “trei rubei” already appears in the Talmud in connection with lineage. R. Chaim of Volozhin’s innovation was not the very logic of “two majorities,” but the willingness to apply it also to the issue of agunot. To conclude this column, I wish to examine the rule of “two majorities” in the Ketubot passage in light of all we have seen here.

“Trei Rubei” in Lineage

The case of “two majorities” appears in Ketubot 15a, regarding the wagons of Tzippori. Due to the complexity, I will not quote the Talmud or enter the views of the commentators. For our purposes I will only describe the conclusion according to the simple reading. The Mishnah there concerns a young girl who was raped by an unknown man, and the question is whether she is disqualified from marrying a priest (if the rapist was disqualified for priesthood, he disqualifies the raped girl). R. Yochanan ben Nuri says there that we follow the majority of the city: if they marry into the priesthood, she too is eligible. But the Talmud explains that a majority is not enough to decide the doubt; however, there were “two majorities,” and therefore they permitted the girl. The Talmud explains that there were caravans of people from other places (in addition to the city’s residents), and that day there was a group with a majority of qualified people. The Talmud says two majorities are required to permit: the city’s majority alone is insufficient, and the caravan’s majority alone also insufficient. But when both are present, she is permitted because there are “two majorities” to leniency.

On its face this is a case of “two majorities” and not ruba de-ruba. The relevant tree looks exactly like Rabbi Herzog’s model above. If the rapist was from the city, there is a majority to permit; and if from the caravan, there is a majority to permit. There is no multiplication here but two parallel tracks. As we saw, such a case is, de facto, the same as a single majority. So if a single majority is insufficient for that girl, why does the Talmud say that “two majorities” suffice? This seems to challenge my thesis that distinguishes between “two majorities” and ruba de-ruba (i.e., multiplication).

Before addressing this, we must understand why each majority alone (city or caravan) is insufficient. The Talmud explains that the city’s majority is insufficient because it is a “fixed” case, and in fixed cases we do not follow majority (see Column 237 and also here). But the caravan’s majority is mobile, and in principle that would suffice. The Talmud explains that we nevertheless do not follow it because of a decree lest people confuse it with the city’s majority (people will not distinguish between mobile and fixed).

Further, the Talmud states that the requirement of “two majorities” was said only for concerns of lineage and not for other prohibitions (“they made a higher standard for lineage”). For other prohibitions—such as the case of the ten butcher shops—a mobile majority suffices (but a fixed case does not, for legally it is treated as half-and-half), and in any event a single majority is enough. Based on that passage, R. Chaim of Volozhin proposes applying it to permitting agunot as well. It seems his claim is that in agunah matters too the Sages adopted a higher standard such that a single majority does not suffice (as seen in “waters without end”), and therefore one could argue that there too, with “two majorities,” we would permit, as in lineage.

We can now understand why we should not learn from there to our issue regarding the type of tree that permits. There, indeed, a tree like Rabbi Herzog’s (two majorities without multiplication) works—even though statistically it is like a single majority. The reason is that there the requirement of a majority is itself only formal. By law a single majority would suffice, even if it is not overwhelming (the simple reading of the passage suggests any majority suffices there). If so, the requirement of “two majorities” is only to avoid permitting based on a fixed case. It is a formal and not a statistical requirement, and therefore, for that, “two majorities” suffice even without multiplication. The aim of “two majorities” there is not to improve the statistics but to serve as a warning flag or marker, so that people will not permit based on the city’s majority (a fixed case). Moreover, note that, legally, a case of a caravan majority plus a city majority is actually worse than a caravan majority alone. Suppose the caravan has an 80% majority; if we had only the caravan, we would have an 80% permission. But if there is also the possibility that the rapist was from the city, then legally the city is treated as half-and-half (fixed case), so the overall majority of potential rapists becomes weaker. Suppose the city’s size equals that of the caravan; then the overall majority becomes about 65% and not 80% (legally, due to the fixed-case rule). This further indicates that, in the lineage passage, the requirement is only formal, not statistical.

By contrast, we saw that regarding permitting an agunah, the problem is not formal but statistical. The higher standard adopted for “waters without end” stems from a real concern (even if remote) that the husband is alive—not from a decree lest people permit in some other case. The aim of “two majorities” there would be to improve the statistical (factual) situation. Therefore, there it is very difficult to innovate—and without a Talmudic source—that a formal trick (“two majorities” without multiplication) would help. Only something that truly changes the assessment of the facts themselves—i.e., brings us to near-certainty that the husband died—can help (as with waters with visible limits, which is like ruba de-ruba and indeed changes our assessment of reality). Therefore, in permitting agunot, unlike lineage, in my view only ruba de-ruba helps, not “two majorities.” Indeed, the case addressed by R. Chaim of Volozhin, which is the basis of his innovation, was a different kind of case than Ketubot. His was ruba de-ruba. Extending it to mere “two majorities” is an expansion without logical or halakhic basis. Moreover, as far as I have seen, even the decisors who use “two majorities” do not bring the Ketubot passage as the source for this innovation, and not for nothing does the Chazon Ish see it as an innovation with no Talmudic source.

To sharpen this further: imagine a city where the minority are qualified. Legally, due to the fixed-case rule, this is treated as half-and-half (for the fixed-case rule applies even to leniency, as stated later in Ketubot regarding one who throws a stone into a group), and therefore if we add to this a caravan majority, it will be permitted. But in such a case it may be that, statistically, there is no overall majority of qualified people at all. If in the city 90% are disqualified and in the caravan 60% are qualified, then factually there is a minority of qualified people, and yet the simple reading of the passage suggests we would permit that girl. Would anyone entertain permitting an agunah in such a situation? Statistically, most chances are that her husband is alive and might show up and overturn everything. It is very implausible to permit in such a case.

We must also recall that, as we saw, the agunah equation has two sides: on the one hand, there is the woman’s distress—a reason to be lenient; on the other, the gravity of a married woman—a reason to be stringent. In the case of the raped girl, both the distress is lighter (she is only disqualified from marrying a priest, not prohibited to the entire world as a married woman is), and the prohibition is lighter (not the sexual prohibition of a married woman, but the negative commandment of a chalalah to a priest). It is therefore difficult to learn from the formal leniency there and extend it to the grave matter of a married woman (especially since, with a married woman, reality matters, not formal halakhic categories).

A Note on Maimonides’ View

Incidentally, while I have not seen commentators note this, Maimonides’ wording in Hilchot Issurei Bi’ah 18:14 (quoted also in Shulchan Aruch, Even Ha-Ezer §6:17–18) slightly shifts the picture regarding the Ketubot passage:

“When does this apply? When the place where she had intercourse was a crossroads or corners in the fields where everyone passes, and most of those passing there were qualified, and most of the city from which those passers came were qualified; for the Sages made a higher standard in lineage and required two majorities. But if most of the passers disqualify her (gentiles or mamzerim, etc.), even though most of the place they came from are qualified; or if most of the townspeople are disqualified, even though most of the passers are qualified—we are concerned that perhaps she had intercourse with one who disqualifies her, and she should not marry a priest ab initio; if she married, she (does not) leave.”

He explains that the “two majorities” in the Ketubot passage are the majority of the city and the majority of the caravan that separated from that same city (and not just some other passers-through, as the simple reading of the Talmud and early commentators suggests). That begins to resemble ruba de-ruba more. There is a majority in the city of qualified people, and from it separated a caravan in which there is also a majority of qualified. Still, strictly speaking, regarding the chance that this particular person is qualified, it does not change much; the caravan separated from the city’s qualified and disqualified alike, so the separation need not change the ratio of qualified to disqualified among those who could have come upon her. Probabilistically, it is still equivalent to a single majority—but since the requirement there is formal (to avoid permitting by the city’s majority), this suffices.

There is a small point of difference that brings Maimonides’ picture a bit closer to ruba de-ruba. Suppose we had, in that situation, only the city’s majority and knew nothing about the caravan’s distribution. It is possible the caravan had a majority of disqualified people. The separation from the city is not necessarily a representative sample of all the city’s residents and need not preserve the city’s proportions. True, in the absence of information we presume uniform separation (as in the case of the ten shops), yet there is an additional presumption here that is not necessary. By contrast, if we know that, in the caravan itself, there is also a majority of qualified people, then the above doubt is removed. Therefore, the information about the caravan’s majority does add something beyond merely knowing the city’s majority. Still, this is not really a case of ruba de-ruba, for there is no multiplication.

Perhaps Maimonides makes this reading of the passage because of the difficulty I raised above. In truth, a city-majority together with a caravan-majority is legally worse than a caravan-majority alone (since the city is treated as half-and-half due to the fixed-case rule). But if the caravan separated from the city’s inhabitants themselves, this is not the case. The relevant statistical majority is that of the caravan; the city’s distribution is irrelevant, and thus the city’s majority has no effect. It is needed only because, had we permitted based on the caravan’s majority alone, people might have permitted based on the city’s majority as well.

[1] I am deliberately not using the term “trei rubei” here, which will be discussed below. We will see that it is not necessarily the same as ruba de-ruba. Ruba de-ruba is the analogue of safek sefeika when each node in the tree has a majority in the same direction (toward leniency) rather than an even split.

[2] See, for example, Maharsha, Chiddushei Aggadot, Yevamot 121b.

[3] For a survey, see Rabbi David Marciano’s article here.

[4] I assume that because Israel is small, not finding the plane indicates it fell into the sea. In the case of the South African pilot, the area was much larger and there was greater uncertainty about the crash location, so not finding the plane does not necessarily mean it crashed into the sea; it could also be on land somewhere out of sight.

[5] This resembles the explanation I suggested in my article here regarding a piece of evidence that needs to be added to self-incrimination.