I promised to write about sfek sfeka and probability. I'm a bit hesitant, though, because Moshe Koppel has already written on this and I have little to add to what he has already written. Nevertheless, here goes.

Let's begin with an example, a bit tacky but in some ways canonical. A man finds that his bride is not a virgin. If she has willingly betrayed him subsequent to their betrothal, she would be forbidden to him. Barring any evidence to the contrary, we can assume, however, that it is not the case that she has willingly betrayed him subsequent to betrothal. We can put it this way: she may or may not have willingly betrayed him (she might have been raped) and even if she did, it may or may not have been subsequent to their betrothal. This is a sfek sfeka.

More generally, imagine that some ruling would follow from conditions A and B both holding. We say that A may or may not hold, and even if it does, B may or may not hold. Hence the ruling is not invoked on grounds of sfek sfeka.

Broadly speaking, the mechanism by which this works can be understood in two different ways. The first is related to principles of safek. Safek d'oraissa l'chumra; safek d'rabanan l'kula. The Ra'ah in Bedek haBayis argues that, even in a Torah matter, the question regarding A is resolved l'chumrah only mi-d'rabanan and hence the (logically) subsequent question regarding B is resolved l'kula. Reb Shimon Shkop in Shaarei Yosher (Shaar 1, Ch. 19) argues strongly for this interpretation. The Rov arguing along similar lines put it more abstractly: when the stringent side of a safek is itself only a safek, there is in fact no leidas hasafek. This is not the format of a safek to which the principle of safek d'oraissa l'chumra can be applied. This understanding has a number of advantages, one of which is that a number of lenient rulings by early poskim can best be explained on the basis of the assumption that sfek sfeka really does entail no leidas hasafek. (Moshe Taragin discusses some such rulings in the sites listed here.)

The second way of understanding sfek sfeka is more closely related to some simple probabilistic notions. To make things interesting, let's first consider a vastly overstated version. We can think of each of the questions "does A hold?" and "does B hold?" as being akin to a fair coin toss. That is, if A has probability 1/2 and B has probability 1/2 and A and B are independent of each other, then the probability that A and B both hold is 1/4. The principle of sfek sfeka simply entails choosing the more probable conclusion that it is not the case that 'A and B' holds. This has the virtue of elegance and obvious generalizability. It is also true that Tosfos actually posits that 1) if either A or B have probability greater than 1/2 then sfek sfeka fails (Kesubos 9b, s.v. eeba'is) and 2) A and B must be independent of each other (sfek sfeka mis'hapeches, see the outer margin on Kesubos 9b).

But, as I mentioned, this formulation is way too optimistic and collapses under Rivash's devastating attack (Resp. 372). Rivash asks why it is necessary for both A and B to hold with probability 1/2. It should be enough that A have probability 1/2 and B hold with any probability less than 1. In any such case, the probability of 'A and B' holding is less than 1/2. In fact, there is just such a case in Hullin 77b. We may assume that a fetus of indeterminate nature is not a viable male since we can formulate the question as follows: is the fetus male? (A) and, if so, is it viable? (B). A holds with probability 1/2 and B holds with probability below 1. Why isn't sfek sfeka defined in a sufficiently general fashion to include this case?

Rivash answers that for sfek sfeka to hold it is not required that A and B hold with probability 1/2 but rather (pay careful attention here because this is the part where the modern mind has to unlearn some habits) that they are not assigned any known probability at all. When either A or B holds with some known probability, sfek sfeka might not be applicable. There are two ways this can happen. First, if A holds with known probability of 1/2 (as in the case of male/female), then it is sufficient that B hold with probability less than 1 in order to rule that 'A and B' does not hold. However, this ruling has nothing to do with sfek sfeka but rather with a different principle involving following the greater propensity (ruba d'leisa kaman). Second, if A does not have known probability (so far so good), but B holds with probability greater than 1/2, then sfek sfeka fails.

Let's apply the Rivash's idea to a more modest formulation of sfek sfeka. As long as we know nothing about the probabilities of A and B, we can assume that 'A and B' does not hold simply because there are four possibilities: A and B, A and not-B, not-A and B, not-A and not-B. We rule in favor of the three cases against the one. Note that there is no assumption that the four cases are equiprobable but rather only that nothing specific is known about their probabilities. This more modest formulation also involves a more modest understanding of the requirement that A and B be independent of each other (sfek sfeka mis'hapeches). We simply require that A does not entail B (or vice versa) because then there would not be four cases as enumerated above but rather only three, since 'A and not-B' would not be possible. In such cases the 3/4 majority that lies at the root of sfek sfeka would be reduced to 2/3, which -- at least according to Tosfos -- is inadequate for invoking sfek sfeka.

Abandoning the idea that every event has some probability assigned to it is crucial for getting this. But, admittedly, it invites a kind of fuzziness. Sfek sfeka may or may not be applicable in a given case depending on how it is formulated. Returning to our hapless bride, imagine that the only question is whether she willingly betrayed her betrothed (the other question having been resolved or rendered irrelevant). We might formulate it this way: "did she consent?" (A) and, if so, "did she do so subsequent to the age of consent?" (B). Formulated this way, we have a sfek sfeka. Alternatively, we might simply ask: "was she raped?" (making no distinction between assault and statutory rape), in which case there is no sfek sfeka. Tosfos raises this question and chooses the second option but it's clear that there is ample room for ambiguity in such cases.

Rav Chaim Ozer raises a brilliant objection against this whole approach but this post is already way too long, so I'll save that for another time.

P.S. I made my annual visit to The Rebbe tonight (I'm as irritated as you are when Lubavitchers refer to their Rebbe as The Rebbe; I'm just being coy). One good thing is that as a result of the weird anti-merit system at work there, our modern dress actually got us in faster; the katchalappers had to stew for a while. As is well known, this Rebbe has hadras panim, but certainly not the gift of gab (unlike his father and two uncles who preceded him). We spent a very meaningful ten seconds in there.

This is my final post until after Rosh Hashanah. A kesivah vechasimah tovah to all.