Q&A: A plausibility-based doubt and a probability-based doubt
A plausibility-based doubt and a probability-based doubt
Question
The Rivash (responsa 372) discusses a double doubt, and distinguishes between a doubt such as whether a woman giving birth will bear a male or a female, where we know with certainty that the chances are 50%-50%, and a doubt such as whether a woman committed adultery while under her husband, where we also estimate that the chances of each side are evenly balanced, but in the sense of plausibility—because we are not sure about the different considerations, we tend to say that overall the two possibilities appear equal to us.
The practical difference is that in the first case we can apply "combine the minority with the half" and say that when we factor together the chances that the fetus will survive the birth (which are 50%) with the chances that it is male (which are lower—say 20%), the chances that a surviving male will ultimately be born are less than 50%, and halakhically that possibility is a "minority" and is disregarded. But in the second case we would not combine the chances that she committed adultery not under her husband (50% in our estimation) with the chances that she was coerced (say 20%); rather, we would say that the possibility that she committed adultery willingly under her husband still stands before us and we must take it into account, even though in our estimation here too it is more likely that this is not the reality.
I do not understand the distinction. After all, every doubt is ultimately a situation in which we have partial information, so what difference does it make whether the information is partial because we are not sure about the sample space and the various considerations, or because we truly have no ability to obtain certain data because we cannot see the fetus's chromosome. Both are simply different kinds of lack of information, and by logic our caution and distancing from sin should be the same in both.
Perhaps he means that the first doubt is in the objective reality, and therefore reality causes the prohibition to be nullified, like a mixture where we view the mixture as a new thing that is determined to be permitted. But that is not clear to me, because in objective reality there are no probabilities, only one reality, and the lack of information is from our point of view in that doubt too. Therefore when he says, "For just as we might say, combine the minority with the half and permit it—how do we know that it is half and half?" one has to ask: what does it mean that in reality it is not half and half? That it is 30%-70%? But reality is 100% one of the possibilities!
I would appreciate help in understanding his reasoning.
Answer
You are distinguishing between a doubt in which there are two positive sides and a doubt of negative sides (absence of information). For example, if I toss a fair coin, I know that the probability of each result is 50%. But if I toss a coin about which I have no information at all, I will still assume the probability is 50%, but that is a methodological assumption (because I have no reason to prefer result A over B), not probability in the positive sense. A similar distinction exists in Jewish law between a doubt where a prohibition was established and a doubt where a prohibition was not established. A doubt where a prohibition was established refers to a case of one piece out of two pieces—meaning, two pieces are before me and I know that one is forbidden fat and one is permitted fat, but I do not know which is which. There there is a positive reason for doubt, and the probability for each piece is 50%. A doubt where a prohibition was not established is a doubt about one piece whose identity I do not know. There too I assume 50%, but that is because of lack of information. The practical difference is with a provisional guilt-offering, which is brought only for a doubt where a prohibition was established.
I did not understand your claim about "combine the minority with the half." The chance that a male will survive is lower, but the same is true for a female. We still remain at 50%. I have not seen the Rivash inside.
Off the cuff, I can tell you this: your assumption that the cases are equivalent is mistaken even probabilistically. In a doubt where a prohibition was not established, there really is not 50% for each side. We assume equality between the possibilities because there is no reason to prefer one over the other, and therefore in practice we treat it as 50%. From this it follows that even if another consideration joins one of the sides, since we still have no information and there are still two possibilities before us, we will still assume that there is no basis to prefer one over the other, and therefore here too it will still be 50% for each side. But if the 50-50 distribution is the result of positive knowledge, actual probabilistic equivalence between the cases, then of course if there is an additional consideration it will change the balance of probabilities.
Discussion on Answer
I explained it. A negative doubt is not built on 50-50, but on absence of information, and because it is a doubt I assign equal weight to both sides. As a result, I view such a situation as if it were 50-50. Therefore added information does not change the basic way I relate to it. I still have two sides, with no way to prefer one over the other, and so de facto it is still 50-50. Let's take an example.
Suppose I have no information at all about the distribution of women versus men in the world. Now I have a doubt whether the person standing in front of me is a man or a woman. I will assume 50-50. Not because I know something, but because I do not know something, and I have no way to prefer one possibility over the other. Now I receive additional information—for example, that the person in front of me comes from an area where the percentage of women is 10% higher than in the rest of the world. But in the world the distribution might be 30-70, or any other distribution. So it makes sense to say that in such a situation I still view the distribution as 50-50.
You are right that in the new situation there is a greater chance that this is a woman than in the rest of the world. In other words, in the new situation the chance that this is a woman is higher than it was in the previous situation (before the information was added). But because I have no information at all, I still cannot assume anything about such a case. So here too I will assume 50-50. This is not a claim that the probability in the two situations is actually equal, because it is not. But even so, it makes sense to view all these situations as 50-50.
We know the probabilities are not equal, but we still treat them as equal? Can you explain why? Maybe just as a rule of thumb because it is convenient?
Not because of rules of thumb. It is pure logic. You have no way of knowing what the percentage is, so if there are two possibilities and no way to prefer one over the other, then from our standpoint this is a doubt. That is true in all these cases. What would you do differently? So what if there is a difference in probabilities? You do not know the distribution in this case or in that case; you only know that there is a difference. What are you going to do with that information?!
Think about the example I gave earlier: you are uncertain about the person in front of you, and assume it is 50-50 whether this is a man or a woman. In addition, you are uncertain about a fetus in a woman's womb, and let us say you have no information about the distribution between male and female, so again you assume 50-50. And let us assume for the sake of discussion that male mortality is higher. If so, the probabilities in the first case and the second are not equal. The percentage of women is higher than the percentage of female fetuses. And still, clearly, in both cases we will view it as 50-50, because in both cases I do not know what the distribution is. What else can you do when you do not know the distribution?! So what if the percentage of women is higher?
If I know that where I am, the percentage of men is 10 percentage points higher than the percentage elsewhere, then I know the percentage cannot be less than 10. So I take the average of all the possible percentages that remain and get 55%. If I treat other similar questions this way, then on average I will hit man (or the other object the question is about) in 55% of cases. You can rely on lack of knowledge being exactly 50-50, after all a monkey that guesses randomly will hit exactly 50% over large numbers.
I cannot understand what is unclear. I was speaking about a situation where the basic percentages are unknown and one assumes 50-50 by default. That is all. I have nothing to add.
Only the minority of males matters, because this was said with respect to levirate marriage, where the sides are whether there is a male son or not.
I did not understand why you cannot combine a positive probability with a negative one. Suppose I am in a state of negative equivalence; clearly, if I add positive information in favor of one side, it should make me rely on that side more than I would have before.