The Halakhic Meaning of Probability Multiplications: B. Sfiq Sfiqa (Tur 613)

With God’s help

In the previous column, I discussed the permission of 'Tari Rubi' in Agunot, as an example of the meaning of the product of probabilities in Halacha. I opened the column with a tree of possibilities that represents a situation of doubt in Sfiqa, because this is the fundamental discussion of the product of probabilities in Halacha. In this column, I will return to dealing with doubt in Sfiqa itself, and it seems that some of the ideas presented in that column are also relevant here. For some reason (providence?), on the very day I posted the previous column, I received question which dealt with the inverted doubt of Sfiqa, which is almost the essence of the meaning of the attribute of the Machpelah in the law of doubt of Sfiqa. As I wrote to the questioner, in this column I will also deal with it, and in my remarks I will explain why I disagree with the words of theShch He mentioned (although, as I will mention, Reka also agrees with them).

What is a supplier?

Halacha, by its very nature, deals with attaching norms to factual situations: in a given factual situation X, we must do or not do Y. In order to act according to Halacha, we must know the factual situation in its entirety, as well as the normative instruction that concerns it. This picture can be represented as follows:

Drawing A: Ordinary Law

The circle describes the situation before me, X, about which I am supposed to know what to do, and at the end of the branch appears the halachic ruling that pertains to it, Y. Halachic is the line that leads from the factual situation to the norm.

The problem of doubt arises in situations where we do not have clear knowledge of situation X (doubt in reality) or of the normative instruction concerning such a situation (we do not know what Y is. This is a sfiqa dedina). These are situations of uncertainty, and we must know how to act in them. Here the laws of sfiqa apply. In every situation of doubt, we have several options of which we do not know which one is correct. For the sake of simplicity, we will only deal here with sfiqa between two options (this is usually the case). If there is a different weight for the two options before us, this is a majority law discussed in the previous column. If their weight is equal – this is a situation of doubt (equitable) and the laws of sfiqa apply here. The main instruction in this area is sfiqa da'oraita lehumra and sfiqa darbanan lekula.

The tree of possibilities for a state of doubt (I will not distinguish here between a hypothetical doubt and a doubt in reality) is the following:

Drawing in: Regular supplier

The circle above represents the factual situation about which I need to decide (what kind of meat is in front of me). In a factual doubt, I have two equally weighted options: A – milk, B – fat. I attach a chance of 0.5 (50%) to each. As we saw above, the path from the question (the situation) to the answer (the normative instruction) is a path on the tree of options. Each such branch is essentially a situation as described in drawing A, except that here we have two possible branches. Note that even if the doubt were in the law (in the instruction), the drawing is the same drawing, except that the circle was the situation and the branches already describe a doubt between two instructions (and not between two situations). But the analysis is the same analysis, and therefore from now on I will refer to a factual doubt.

When I know it is milk, I actually go from the circle in drawing B up along branch A – at the end of which there is an answer: not kosher. If I know it is fat, I go from the circle along branch B – and at the end there is the opposite answer: kosher. Each of these two is a simple tree as in drawing A. But if I don’t know which of the two options is correct, I attach a 0.5 chance (50%) to each branch. Now there are 50% that the piece is kosher and 50% that it is not. Because of the laws of spikut in the Da’ara’ita, I have to be stricter, and in the Darbanan, I can be lenient.

Important reminder: Positive and negative doubt

In the previous column, I argued that there can be two different states of doubt: positive doubt and negative doubt. Positive doubt is a doubt that gives equal weight to both options based on the information I have about them. For example, in a fair coin, there is an equal chance of 'heads' and 'tails', because I know that the coin is fair. But there is also negative doubt, which is a situation in which I give equal weight to both options due to a lack of information (ignorance). For example, a coin whose structure I have no information about (whether it is fair or not, and how not). Here too, if I had to bet, I would give equal probability weight to both options (because I have no way of preferring one of them), and therefore here too I see the situation as equal doubt. But the equivalence in such a situation is a result of ignorance (and not because of information). Therefore, it is negative doubt. In the book Listen carefully. Sha'a Pih explicitly stated this distinction (in the explanation of the Rivash).

I will mention that in the previous column I discussed a similar situation with respect to majority law. There too, there are situations where I have a numerical measure of the majority (80%, 55%, 99%, and so on), and then it is usually a positive majority (based on information). But in most cases I do not have such a numerical measure. At most, I can say that it is a large or small majority (in the previous column I defined three types of majority: overwhelming, absolute, and ordinary). This is a negative majority, since it is not based on clear information. I have general information that the chances are for one of the two options, but nothing beyond that.

What is a Spicah?

What happens if I have doubts about which animal this piece of meat came from? If it is from a pig – then it is not kosher even if it is fat, and if it is from a cow – then it is kosher if it is fat but not if it is milk. This is a situation of doubt spiqa, and the tree of possibilities that describes it is the following:

Drawing C: Speck Spicka for Material

The upper circle is the factual situation before me, or the question: What is the piece of meat before me, milk (a) or fat (b)? The second circle is located under the branch of fat (b), which then raises the second question (another node that also represents a questionable factual situation): Is this cow fat (b2) or pork fat (b1)? As mentioned, if I have complete information, I must walk along the tree of possibilities with my information. For example, if I know that this is pork fat, then the path is right at the upper node and left at the lower node. At the end of the path is the clear result of such a situation: not kosher. The same goes for cow fat (right, right: kosher). Milk (left. And here it doesn't matter whether it's pork or cow, and the result is not kosher).

What happens if I don't know the answer? Again I have to do probability calculations. There are three edges to our tree, but not all of them have the same probability. Each node I passed along the path represents a balanced doubt (negative or positive).[1] Therefore, as I progress along the route, each crossing of an intersection multiplies the result by 1/2. For example, a route that passes through both intersections (ending in 1 or 2) has a probability of 1/2X1/2=1/4. On the other hand, a route that passes through only one intersection (a) has a probability of 1/2. If so, there is a 1/4 chance that it is cow fat and a 1/4 chance that it is pork fat and a 1/2 chance that it is milk (it doesn't matter whether it is pork or cow). Therefore, in such a situation, there is a 3/4 chance that it is not kosher and a 1/4 chance that it is. This is a spiky doubt (most of the chances are for the meat and should be taken more seriously).

It is important to understand that this is just an example to illustrate the logic. Usually, a doubt is a situation that is depicted in such a tree but not in a koll, that is, a graph at the bottom of which are three outcomes of 'kosher' and one of 'not kosher' (where the chance is 3/4 for permission and 1/4 for prohibition). Later, we will deal with such an example from Tractate Ketubot. The reason why a doubt is usually in favor of permission is that if it is a doubt for a stricter issue like ours, it has no consequences, because even in one doubt, one must go to a stricter issue, so of course, when most of the chances are in favor of the stricter issue, one must be stricter.

The same option tree shown in drawing C could be drawn in a different, more complete way:

Drawing D: Spicca Spicca Full view

Here I presented the doubt as to whether it is a pig or a cow even under the possibility that it is milk (a2 – cow, a1 – pig), even though it does not matter halachically there. The advantage of such a presentation is that now each outcome has the same chance (because on the way to it we passed two intersections). In such a situation, there is no need to do arithmetic and multiplications, and one can simply count the edges and arrive straight at the result: there are three outcomes of 'not kosher' and one of 'kosher', so the chance is 3/4 for the more severe and therefore one must be more severe.

Note that if the distribution of the results were 2-2, as in the following drawing, it would not be a clear-cut doubt:

Drawing E: A tree resembling a spicule with no multiplication but parallels

For example, I have a doubt whether it is a goat or a cow, but in both cases there is a doubt whether it is milk (which is forbidden) or fat (which is permitted). Therefore, here the chance that it is not kosher is 1/2, and this is a situation of one doubt and not a doubt of a spicah. This is completely analogous to the case of 'Tari Rubi' which is not a 'Ruba Druba' from the previous column. I explained there that here there is no multiplication and therefore it is actually a situation of a single spicah. Here too there is no multiplication and therefore this tree describes a situation of a single doubt and not a doubt of a spicah.

Doubtful and majority: Most 'sides'

The commentators discuss the question of why we allow ourselves to be lenient in cases of doubt, even in the prohibitions of the Torah (see, for example, a review here). The Rambam's method is that doubt from the Torah to the Chumara is itself a rabbinical law. According to his method, one could say that doubt from the Spicah is doubt from the Spicah and therefore is lenient. But according to the majority of the Rishonim, who generally believe that doubt from the Torah to the Chumara is itself from the Torah, the most obvious explanation for the lenient interpretation of doubt from the Spicah is that it is a law of following the majority. When we have a tree of doubt from the Spicah to the Kula (3 'kosher' options), then the majority of the chances (0.75) are that it is kosher, and we follow the majority. This is how the Rashba really understands it (Theology of the House Verse 4, Chapter 1). The question that arises here is why other commentators do not accept this. Ostensibly, there is a majority here and the law of following the majority can explain everything. One could ask this more forcefully: Why do we even need the concept of doubt of spicah, if we already have the law of majority? Think of a situation where there was a positive doubt here, meaning that we have information that every such doubt is truly 50%. In such a case, in doubt of spicah there is a clear majority of chances for permission (75%), and then we would not need the law of doubt of spicah at all. Here there was a law of following the majority.

The conclusion that emerges from this is that the law of doubt is only relevant in negative doubts. When the doubts are positive, there is only the law of majority. In negative doubts, there is a special innovation of the Torah that, although there is no majority of chances here, it can be seen as a majority. Indeed, we saw in the previous column that in most situations of doubt (and majority), it is a negative doubt. In such a situation, we do not know what the chances are for each of the two options, and out of this ignorance, we assume equivalence between them. In such a doubt, there is no need to say that there really is a majority of chances in favor of the permit, and therefore we need the law of doubt in addition to the law of majority. In a doubt, there is one side against three others, and we have no information about the true chances of each option (side), since it is possible that the individual side has a chance that is equal to the other three or even higher than them. But since we do not know the chances of each side, the halakha says that we must assign equal weight to each of them.

In a normal doubt we assume equivalence, and these are the laws of spikit. As if there is an equal doubt of 50% for each side. And in a spikit doubt, which as mentioned only exists in negative spikits, we again count sides. Here there are three sides in favor of the permit, and therefore we permit. In other words, we really follow the majority here, but it is not a law of following the majority because there is no majority of chances here. Rather, it is an innovation of the Torah that when there is no information, we follow the majority of sides: in a normal doubt it is one side against one, and in a spikit doubt it is one side against three, and therefore we permit.

The concept of 'sides' is halakhic, not probabilistic.

It is very important to understand the logic (or lack of logic) underlying this picture. To do this, we must return to the example of milk and fat in a pig and a cow. The fact that in a normal doubt between a cow and a pig we assume that there is 50% that it is a cow and 50% that it is a pig is just a rule. We have no indication that the doubt is reasonable, but in the absence of other information, this is our assumption. This is a negative doubt. Now we have another doubt added to us as to whether it is milk or fat, and it is also of course a negative doubt. Now we have reached a state of doubt spiqa (because we have no information about the distribution between a cow and a pig and between milk and fat), and as we have seen, we count sides and follow the majority of sides (which is not the normal rule of following the majority).

But in such a situation, there is no need to talk about four sides. The choice to split the question of spica and see four sides here is somewhat arbitrary. To the same extent, we could also say that there are only two sides of the question here: kosher or not kosher, and this itself is a question with two sides, and apparently in the absence of information we must give equal chance to both, and therefore there is a doubt of the grave. Similarly, we could also say that there are three sides here (as in drawing C) and then there is a majority of 2/3 and not of 3/4. The assumption that branch A has twice the weight of the other two is based on a probabilistic view that has no real basis (it is a negative doubt). This is the reason why I proposed the full presentation in drawing D. There, the question of spica has four sides, and then following the majority for the permit seems more natural without resorting to probabilistic assumptions about different weights of the ends. All paths, or their ends, have the same weight.

But there are other options for defining sides. Not just two or three as we have just seen, but to the same extent the tree of possibilities can be split into more sides (seven, ten, a hundred). For example, we could discuss whether it is a warthog, a regular pig, or a giraffe? And on the kosher side we could discuss whether it is a Dutch or Australian cow, or perhaps a goat. We could discuss whether it is the soft part of the milk or the hard part, and so on. If we give each such side equal weight (in the absence of information) we can arrive at any halakhic result you wish. Note that there is nothing statistically unreasonable here. It is no less logical than choosing to present a spiky spiky tree as a four-sided tree.

The conclusion is that the division of a negative doubt into four sides is an arbitrary decision, and has no factual or statistical basis. In positive doubts, such a division has a probabilistic basis, but as we have seen in positive doubt, there is no need to reach the laws of doubt, since there we have the usual majority rule that we follow. In doubt, it is always a negative doubt. While this may seem like a probabilistic calculation, it is based on a non-probabilistic foundation. The division of the sides in the given problem is a law, that is, a halakhic consideration, and it is what underlies the probabilistic calculation. After we have sides, we give them equal weight and then count them as if they were probabilities. The conclusion is that the division into sides is the focus of the discussion of the laws of doubt, and this is a halakhic question, not a probabilistic one. But from here on, from our perspective, it is a probabilistic calculation (similar to what we saw in the previous column regarding majority and 'tari rubi' in agunot).

This brings us to two rules in the laws of doubt that are related to the picture I have described so far.

There is no doubt that

The Gemara in Ketuvot 9a discusses the husband's claim of "I found an open door" regarding his wife:

Rabbi Elazar said: I found the one who says "Open Door" faithful to the prohibition against him.

The husband claims that she is not a virgin, and if she is married under his authority, she is forbidden to him and the husband. RA says that the husband is faithful and she is forbidden to him. The Gemara makes it difficult:

And my mother? He is a doubter! There is no doubter under him, and if you say under him – there is no doubter under rape, there is no doubter under will..

It is explained in the Gemara's difficulty that the claim of "open door" is subject to a situation of doubt spiqa: doubt whether it happened under his authority or not (if the wife did not marry under his authority, she is not prohibited), and even if so – doubt of rape, doubt of consent (only if the wife married under his authority willingly, she is prohibited). If so, it seems that the husband cannot be faithful to the prohibition, since in doubt spiqa they go to the kolah and rule that it is permitted. It is difficult for RA.

The Gemara settles it in two ways:

Not Zricha – in the wife of a priest. And I've a mother: in the wife of a Jewish woman, and such as a Dakbil in which her father is a child of a priest who is less than three years and one day old.

If it is a priest's wife, there is no difference between rape and consent (in both cases, she is forbidden from it), so it is only one doubt. A second excuse is that her father received her Kiddush when she was less than three years old, and in such a situation it is impossible for her to marry before the Kiddush because her virginity would have returned.

In the Toda, "Vai Ba'it Ima," there, it is stated:

I am afraid of being raped, and I am afraid of being willing, and I am afraid of being willing, and I am afraid of being small, and I am afraid of being tempted by a small thing, and so on.

Thos makes it difficult for the second okimata (whose father consecrated her when she was less than three and a day old), that there is still a doubt here: doubt about rape or consent, and even if consent is possible, it is possible that it was while she was still a child and it is still considered rape (seduction of a child is considered rape because she does not have the knowledge).[2]

The tree in question is similar to the tree in drawing C or D (which is a full representation of the same doubt as C), but this time for the voice (there are three possibilities of the other):

In our case, the first doubt is a doubt of rape (a) and a doubt of desire (b), and if it was desire, there is another doubt whether it was small (b1) or large (b2). If we use the full presentation of drawing D (in which the second doubt also appears under option a), we get four options at the ends: small rape (a1), large rape (a2), small desire (b1) and large desire (b2). Of these, in three it is permitted and in one it is prohibited.

Thos answers:

Vil Desham is a rape.

The possibility that it was willful when it was small (B1) and the possibility that it was rape when it was large or small (A1 and A2) are the same possibility. These are all rape possibilities, and rape is a single possibility. So we actually only have one doubt, as in drawing B above (option A represents the three rape possibilities), rape or willful:

Therefore, we must prohibit it as a common doubt.

The commentators take the principle of Thos' and turn it into a general principle in the law of doubt: When there is a clear doubt, we view all of these possibilities as one possibility. This is of course based on what we saw above, that the division into possibilities (sides) is a halakhic matter, not a probabilistic one. Thos's halakhic consideration is that all of the possibilities that lead to rape should be considered one possibility.

But if we see all the possibilities of permissibility as one possibility, is there no doubt in the world? Apparently, we can turn every doubt into one doubt. For example, the doubt at the beginning of the question: doubt in rape, doubt in will, doubt under it, doubt not. We can say that the three possibilities of permissibility (rape under it and not under it, and will not under it) have one name: all of them are permissibility. The name permissibility is clear. But this is a mistake. Thos' rule deals with reasoning and not with law. When the reasonings are of the same type (=rape), it is considered a clear doubt. But when the law is shared (=permissible), it does not unite on one side. When talking about rape, this is the reasoning that the woman is not forbidden, and different types of rape are the same reasoning: rape.

This is similar to what we saw above, where the doubt is not artificially divided into additional sides (perhaps a pig and perhaps a warthog). If we were to separate seduction and minor rape from major rape, we could equally break down the possibilities of rape into several more different options: rape by holding hands, with or without tying, deception, and so on. All of these are types of rape and there is no point in making different sides of it: one name for one-sided rape. When we talk about one-sided permission, this is a union between different reasonings just because they lead to the same law. Such a union is not done. The basis for this principle is that our doubt deals with reasoning and facts (in the circle and the branches that come out of it) and not with the law (which is only a result of these options). Therefore, the rule of 'one-sided doubt name' deals with a situation in which what is written on the side of the relevant branches (the reasonings) is from the same name. But in the case where the norms written at the end of several of the paths are similar (all 'permitted' or all 'kosher', and so on) - this is of course not unified. If we were to unify this, there would be no doubt in the world.

From this picture you can understand that the principle of Thos, 'where there is a clear doubt', is a consequence of the description I gave above (Rabbi Z"N Goldberg emphasized this in a similar way in his article here). Of course, if we are talking about positive spikahs, this entire discussion is irrelevant. First, we saw that in such spikahs, questions of spikah are not even entered into. The ruling is determined statistically. Second, in such spikahs, there is a natural division of the parties and there is no need to hesitate according to their names: A party in a positive spikah is one whose chances are 50%. If there is a sum of several parties that together give 50%, they will be combined regardless of their names. If we suppose I know that the chance that the woman seduced as a minor or was raped is 50% and the chance that he did it to her will as a major is 50%, then I am a reasonable spikah, and I don't care that the first party is a combination of several parties. I am a reasonable spikah because there really is a 50% that is permitted and a 50% that is prohibited. This is a statistical consideration and not a halakhic one. But as mentioned, only negative spicahs are relevant to our case, and there we need the rules of spicah doubt and the names of the doubt determine the relevant parties.

The doubt turns upside down

Some first and last scholars wrote that in the rules of doubt spiqa there is an additional principle, that doubt spiqa must be 'reversible'. According to them, a doubt spiqa that is not reversible is not doubt spiqa. I should note that this rule is not agreed upon by all the jurists (theShch Embrace it warmly, but from the R.T.flower (There are disagreements about it), but it is accepted by many. Here we will examine it according to the methods by which it exists.

One source for this is in Tos' Yemin Ketubot 9:2. We begin by saying that a woman who married a non-virgin had a ketuba (not two hundred). Therefore, in the husband's claim of "open door" there is a discussion of both her prohibition against her husband and her financial situation (the amount of the obligation in the ketuba). In Tod's 'Ei Lameitav', there, he cites from the Gemara that with regard to financial matters, the husband was faithful to lose her a ketuba. And he immediately makes it difficult to understand why the husband is faithful to lose her financial situation, since this is a question of doubt: whether he is knowledgeable in open door (knows how to determine that his wife is not a virgin), and even if he is, he is doubtful about rape, doubtful about consent. In fact, there is doubt whether she is in the oleh at all (because perhaps he does not know how to determine and she is actually a virgin), and if in the oleh there is doubt whether it was rape or consent. Tos' and the commentators there resolve this in different ways.

But the Torah there would put it this way:

And here he did not consider doubt sufficient, since they could not turn around and say, "In the head, doubt in rape, doubt in will, doubt in the will, doubt in the will, etc." And he did not consider anything but one sufficient.

He claims that a spiky doubt must be one that can be interchanged between the nodes in drawings C and D. For example, in the example above, one can be satisfied with whether she is under him or not, and then be satisfied even if it is under him – whether by rape or by will. And one can interchange the nodes and be satisfied with whether by rape or by will, and then also if by will – doubt whether she is under him or not. Only when one can interchange the nodes is it a spiky doubt. If the nodes are not interchangeable, it is considered one spiky doubt and should not be made light of. In the case of To'i, this is a spiky doubt that does not reverse, since one cannot be satisfied with whether she is under him or by will, and then be satisfied with whether he is familiar with an open door. The reason for this is that if he is not familiar with an open door, then she is not a virgin at all and in any case there is no point in being satisfied with how she was married (by rape or by will). She was not married. As mentioned, the commentators and jurists expanded the rule to become a general principle in the laws of spiky doubt.

The logical explanation for this rule is not clear. The Rema of Pano suggests that since the sides of the doubt can only be raised in one order and not the other, then we have a doubt in this itself, whether to be satisfied with the one order and then go to the Kola or with the second order and then we must be stricter. And since this is the body of the Torah, we must be stricter in it. But this explanation is illogical for at least two reasons:

• This in itself creates a doubt of the Spicah: Should we settle for this order or for the second, and if we settle for the first order, there is still the only remaining doubt. So why don't we ease up on such a doubt of the Spicah? One can similarly wonder about the methods of the first who believe that doubt from the Torah to the grave is more serious than the Torah: why don't we also be stricter on the doubt of the Spicah, since this is in itself a doubt of the Torah (doubt that we have a case of doubt from the Torah to the grave). The latter have already discussed this (see, for example, inStraight hair Chapter 1, Chapter 3 and Chapter 9). This can be explained by the fact that if we have a doubt about whether to be content, we are not obligated to be content. The law of contention from the Torah to the grave does not impose this on us.
• It is enough for us that there is one order of sufficiencies that creates a double doubt to facilitate. What do I care if there is another order in which the law is made more difficult?! It is like a simple matter. It is enough for me to have one formulation to prove the law, even if we refute the 'perpendicular' formulation (see in the column 537).

We will return to the explanation of this rule after discussing the relationship between the two rules.

The relationship between the two rules: reversal and the name of a sharp doubt

The poskim cite another example of a non-reversible spiqa, regarding damage to a slaughtering knife (see Shch In general, there is no doubt that the letter 13 is in the name. The store). The Gemara in Chulin 10b states that if someone slaughters an animal and finds after slaughtering that the knife is damaged – the slaughter is invalid and the animal is considered carrion. Some Rishonim have questioned why we should not allow the animal to be slaughtered without a doubt: It is doubtful whether the knife was damaged during or after slaughtering (when the knife hit the joint), and even if the knife was damaged during slaughter – perhaps it happened after most of the neck had already been cut (in which case the slaughter is not invalid). The answer is that this is a doubt of a doubt of a doubt that cannot be reversed, because if we start with the second doubt – whether the knife was damaged before most of the neck was slaughtered, we cannot of course continue and say that even if it was damaged before most of the neck was slaughtered, it may have been damaged after slaughter.

In this example, you can see that it is also possible to argue that this is a clear doubt. Whether it was damaged after the slaughter or after most of the slaughter, the clear doubt is: damaged after most of the slaughter, the animal is not carrion. To the same extent, it was possible to add many more aspects: perhaps it was damaged after 51% of the slaughter, after 54%, after 80%, after 98%, and completely after the slaughter. All of these are the same aspect: that the knife was damaged at a stage when the slaughter was already kosher.

This is an initial clue to some scholars' questioning of why these two rules should even be seen as different: on the face of it, it seems that a sharp doubt and a non-reversible doubt are the same rule. To reinforce this, see here Shrek'a understood the excuse of the Toss in the 9th chapter of the 1st century AD that spoke of a single rape, which in essence meant to say that this is a doubt that cannot be reversed. If we start from the doubt that she was raped by rape or willingly, we cannot continue and say, "Ethal willingly, perhaps she was raped in her infancy." After all, in her infancy, rape is not considered willing, since a small temptation is rape.

From these two examples, it could be concluded that these two rules are identical. And indeed, there is a tendency among many to see them as two sides of the same rule. If this is indeed the case, there is no question why the doubt of a certain kind should be reversible. The explanation for this is like the explanation for the rule of the name of the doubt of a certain kind that we saw above, namely, the reversal is required so that we really have different doubt sides that create a majority of sides to the kola (for example, Rabbi Shlomo Levi argues at the beginning of His article(He later denies it.) But as we now see, this is probably not true.

But already in the example of Thos' in the inscriptions, it can be seen that the two rules are not identical. Unlike what Reka assumed, it is certainly possible to be satisfied with the side that was taken willingly, lest she was taken away by a minor. It is true that a minor act of willful taking is considered rape, and therefore at the end of the path here, 'permitted' will be written on both sides (see, for example, above in drawing D, where under branch A there is a bifurcation of a doubt on both sides of which the same law applies). At most, it can be argued that there is no point in being satisfied because there is a clear doubt (after all, this and that are cases of rape).[3] That is, the example of Tos' Ketubot is an example of a question of certainty that is reversed, but there is a clear rape (all the reasons for the permission are the same reason: rape) and therefore stricter. This is not similar to the case of the slaughter, where it is not possible to be satisfied with the reverse order at all, meaning that it is neither reversed nor there is a clear question. I will note here that inquestion I mentioned at the beginning of the column, an example of theShch The supplicant spiqa is not reversed in slaughter, and even there he claims that it is not reversed because there is no nepma for the second supplicant on the side of the first supplicant. But as I argued here, the fact that there is no nepma does not mean that the supplicant spiqa is not reversed. For such a structure to be considered non-reversible, it must be impossible to satisfy the second supplicant at all (as we saw in the case of the t'i in the ketubah).

So far we have seen that there is a possible situation where there is a clear doubt but the doubt is reversed (this is the case of Tos' Ketubot). Is there a possible situation where there is a clear doubt but the doubt is not? Slaughter is not such a case, because we have seen that there is neither a clear doubt nor a clear doubt (the doubt is whether it was spoiled after the slaughter was already kosher). But the case of To"i in Ketubot 9 A"b, which is the first case we have seen of a doubt that is not reversed, and there it seems that there is a clear doubt. If the husband does not know how to diagnose virginity - then the permission is by virtue of the fact that she was not married at all. But if he knows how to diagnose but that she was married by rape - then the permission is for a different reason: that she was married by rape. These are two different names for the sides of the permission that lead to the same norm. We saw above that there is a clear doubt that is said about the reason (the sides) and not about the norm itself.

From what we have seen here, it clearly emerges that these two rules are not identical, and in fact, as we have seen, they are independent: there is a doubt that we will be strict about because it is not reversed even though the names of the parties to the permit are not distinct (To"i in Ketubot 9 a"b), and there is a doubt that we will be strict about because the names of the parties are distinct even though it is reversed (Tos' Ketubot 9 a"a). This also emerges from the fact that there are methods that do not require the reversal of a doubt that is distinct, but do accept the rule that if there is a distinct doubt, it is not a doubt that is distinct.[4]

I will now recall that the rule in the event of a clear doubt is logical and necessary, as explained above. However, if we are indeed right in that these two rules are independent, we must once again seek a separate explanation for the rule that requires reversal. So far, we have not found a satisfactory explanation for it.

Explanation of the rule of doubt is reversed

We saw in the previous and current columns that, contrary to popular belief, the rules of doubt and majority are based on a probabilistic perspective. The rules of doubt are only necessary in a situation where there is negative doubt (and these are the vast majority of situations and examples in halakhah). In such situations, we do not really have numbers that describe the probability, and the decision that such a doubt is reasonable is a halakhic decision, not a probabilistic one. But from here on out, from our perspective, we have a probabilistic question that is described by a tree in which each node is a reasonable (negative) doubt, and we must decide it through probabilistic lenses. The probability of each outcome is the product of the probabilities of the nodes we passed along the way. If so, we must examine the tree of doubt and ask ourselves what the meaning of the fact that it does not invert. I can offer two explanations for this, and I will describe them now.

First explanation: Between counting sides and probability

The first explanation follows the path of the Rama of Pano mentioned above, but in a slightly different conceptual framework. Our starting point is that these are negative spikahs. If so, we are actually counting sides and not chances. The fundamental rule that underlies the permissibility of a spikah doubt is that a majority of sides is also a majority, even though it does not necessarily indicate a probabilistic majority. In a situation of ordinary doubt, there is one side in each direction, and although we have no way of examining whether there is 50% for each of them, the halakhah tells us to treat this as a balanced doubt and to be strict (in the Da'ara'ita). In a spikah doubt, we have three sides for permissibility versus one for prohibition, and therefore we follow the majority of sides and make it easier. What happens in a spikah doubt that does not reverse?

The Rema of Pano suggested that in such a situation we are in doubt whether to present the order of spikah in such a way that a spikah doubt is created (and then one can be lenient), or in the opposite way where it is a single doubt (and then one must go to the humara). He argued that this doubt itself is a doubt from the Torah and must be made stricter. But we saw above that this is a problematic explanation. Here I will offer a small variation on his explanation. We will treat this situation as a doubt as to whether to draw a tree of possibilities for a spikah doubt (i.e., three sides for permission and one for prohibition) or for a regular doubt (then there is a side for lenience and a side for prohibition), which for the sake of comparison we will present as two sides for permission and two for prohibition (this is a more detailed presentation of the situation – similar to what we saw in drawing D above).

To illustrate the analysis, we will take the example of a non-reversible spicah doubt in which there is no sharp doubt (this is the case we must explain), namely the case of To'i in Ketuvot 9:2: doubt whether he is knowledgeable in an open door or not, and even if he is knowledgeable – doubt whether in rape or in will. In such a situation, we have two options for drawing this structure. The first option is to present it according to the order of spicahs that I described here, and then we will get the following tree:

When A represents the possibility that he is not knowledgeable (i.e. the door was not open), then there is no question of rape or will and it is permissible. B represents the possibility that he is knowledgeable (i.e. the door is indeed open), then it depends on whether it is rape (in 1) or will (in 2). There are three sides to the permission here and one to the prohibition.

A second option is to reverse the order of the sufficiencies, and start with the doubt of whether it was by force or by will. We saw that in such a situation we cannot continue to be satisfied with whether he is knowledgeable or not (because the first doubt assumes that the door was indeed open). Therefore, in such a case we will get a tree of one doubt (by force or by will). We will also represent it in the form of a double tree, but the norms at the bottom are divided equally between the sides of permission and prohibition:

Where A represents the possibility that it is rape – then in any case it is permitted, and B represents the possibilities that it is desire – then in any case it is prohibited. There is really no possibility here that he is knowledgeable (1) and not knowledgeable (2), and I split each branch in two only so that the basis for comparison with the previous tree is the same.

Now I propose, following the Rema from Panu, to treat these two trees as two equal options, and therefore we must count all the sides written at the bottom of both trees. We have eight options, each with a probability of 1/8 (and not four, each with a probability of 1/4 as in a regular spiky spiky). Here I accept a total of five options for permission and three for prohibition. There is a majority of options for permission here, but a smaller majority than in a regular spiky spiky. Here it is 5/8 and there it is 3/4. Now we must remember again that we are dealing here with a comparison of sides and not with a comparison of chances, since this is a negative majority (there is no way to quantify the chance that the person is ahead or behind, and whether he is knowledgeable or not knowledgeable). But we have seen that a majority of sides does not necessarily reflect a probabilistic majority, and only the halakha instructs us to treat it as an indication of a probabilistic majority. My suggestion is that in order for us to accept a majority of sides as a probabilistic majority, the halakha requires an overwhelming majority of sides, at least 3/4 (or 6/8). If so, then a smaller majority of parties, for example 5/8, is not sufficient, because there is a concern that there is no probabilistic majority here and therefore, according to the Torah prohibition, it is not relied upon.

This is a very similar explanation to the one I proposed in the previous column for 'Tari Rubi'. There I explained that since this is a negative majority and we have no way of checking the numerical probability, we require a doubling to ensure that there is a probability of an absolute majority here. The same is true of the proposal described here for a Spika doubt. This is seemingly a possible explanation why a Spika doubt that does not reverse is considered one doubt, meaning that we cannot make it easier.

But on further inspection, this analysis seems unlikely. The reverse order of the spiky starts with the question of whether it was rape or consent and completely ignores the possibility that it was not consensual at all, meaning that the husband was simply mistaken because he is not knowledgeable. But clearly there is such a possibility, it just does not appear on this tree due to the limitations of the description. Therefore, it is not reasonable to treat this tree as an equivalent possibility to the tree of the regular spiky doubt that presents all the possibilities. The tree that faithfully represents the case is only the regular tree. The other tree is fictitious and there is no point in considering it. Therefore, there is no point in counting the edges of the two trees as if they were two possibilities for describing the case.

The conclusion is that only the regular tree correctly describes the case in these situations. The second tree is not real and there is no point in considering it. It is true that the tree in question in such a case has the property of not being inverted, and this is probably the factor that prevents us from using it for the voice. Now we must seek an explanation for why the fact that the tree does not invert weakens the permission.

Second explanation: Dependence between events

This explanation starts from the premise that even though every doubt at each intersection is not really a probabilistically equivalent doubt (this is just a methodological assumption, since it is a negative doubt), we still see it from here on out as a probabilistic calculation. The sting, according to my proposal here, is precisely at this point. In probabilistic calculations, if the order of occurrence is important, this means that there is a statistical dependence between them.

Let's say we are dealing with cot death of infants (see the column on this) 144). The chance that baby A will die of SIDS is P(A), which is about 1/8,000. What is the chance that two babies will die of SIDS? If these are two independent events, then the chance of this is the product of the odds: P(A)*P(B). But if the two babies are from the same family, then there is room to argue that there is some genetic factor that affects infant mortality, and therefore if we saw that one baby died, the chance that the other baby will die is already higher. This is a situation in which there is a dependence between the two events (there is a factor that affects both). In such a situation, the chance that they will both die is not the product of the probabilities of each (this is only when there is no dependence), but: P(B/A)*P(A), where PB/A) is the conditional probability. If there is a dependence between the events A and B, this means that the chance that B will happen, P(B), is different from the chance that B will happen assuming that A has happened, that is, from P(A/B). In our case, the conditional probability is greater than the absolute probability P(B), and therefore the product is also greater. The conclusion is that if there is a dependence between the two events, this means that the probability of the second changes (in the previous case, increases) if the first occurs.

Now let's return to the non-reversible doubt of Sfiqa. In order for the woman to be forbidden, two events must occur: 1. That the woman is not a virgin (and this is only if the husband is knowledgeable in the open door (knows how to diagnose virginity) – and he will diagnose that she is indeed not a virgin). 2. The opening that the husband diagnosed is the result of a voluntary pretext and not rape. What is the chance that these two will happen (that she is in the open door and that the pretext is voluntary)? Ostensibly, this is the product of the chances of each of the two events: P(A)*P(B). But this is only when the doubt of Sfiqa is reversed, that is, when there is independence between the events. But if the order of occurrence of the events is important, that is, event B can only happen if event A happened first, this means that there is a dependence between them. In such a situation, the chance that she is forbidden is P(B/A)*P(A). But as mentioned, when there is dependence, the conditional probability can be greater than the absolute probability P(B). It can also be smaller (there are also types of pendants like that), but there is also the possibility that it is larger.

The conclusion is that if the doubt of the spiqa does not reverse, there is a dependency between the events (the nodes in the tree), and therefore the chance that the woman is forbidden may be greater than we thought. In such a situation, there is no certainty that the chance that she is forbidden is small enough, and it is not even clear that there is a majority of the chances that she is permitted. Therefore, in such a doubt of the spiqa, the woman is not permitted. This is a possible probabilistic explanation for why the doubt of the spiqa that does not reverse does not permit the woman.

Two reasons for the failure of Speka Spica to turn around: program and statistics

When we look at the example that accompanies us (doubt whether the husband is knowledgeable and whether the wife was married willingly), this explanation does not seem to hold water. The fact that it is impossible to reverse the sides of the doubt is not a matter of statistical dependence but of the content of the events in question. In short, it does not seem that the chance that the wife was married willingly is actually greater because of the lack of reversal. If the husband knows how to diagnose virginity, this does not change the ratio of odds between a voluntary basis and a rape basis, but only allows us to ask the question (because if the wife was not married, there is no point in discussing whether the wife was married willingly or by rape). Therefore, in this particular case, it is not reasonable to say that the lack of reversal changes the odds. The lack of reversal stems from a content, not a statistical, issue.

This is also the reason why I disagree with the statement of theShch And the Reka who understood that if the second doubter does not have a NFM, it is called a non-reversible doubter. I argued here that, contrary to their opinion, I think that as long as it is possible to be satisfied with it, it is considered a non-reversible doubter. A non-reversible doubter is only in the case where it is not possible to be satisfied with the second doubter at all. Now you can understand the explanation for this. The non-reversibility has meaning only if it indicates statistical dependence, that is, when the order of the satisfaction changes the probability for each of the possibilities (not the norms but the facts, I mentioned several times that the satisfactions are on the facts and not on the norms). In cases ofShch And the probability of the facts does not change, and only the norms change (their claim is that there is no halachic nefm for the different facts, but they are still different). In light of the explanation I have proposed here, it is clear that this is a reversed sifika doubt.

The conclusion is that only if the reason for the non-overturning is statistical does the permission based on a doubt of a spicah become more dangerous. But with regard to a doubt of a spicah that is not overturned (in facts and not in law), then even if the reason for the non-overturning is not statistical dependence, it is still possible that because in a structure of this type there can be dependence between the events, the halakha establishes a sweeping rule: in such a doubt of a spicah, it is possible to rule for permission. Hence, in spicahs that are not overturned, the permission of a doubt of a spicah is in principle canceled. Now we will not go into the question of whether the lack of reversal indicates statistical dependence or not, without further ado.

And it is still clear that there must be situations in which the non-overturning is due to statistical dependence, otherwise this whole process has no meaning. If indeed there are such cases, then we can say that the laws of spik spik are canceled when it does not turn over, even if it does not stem from dependence. Are there really such examples? Let's look at the spik spik from Tractate Chulin regarding the slaughter knife. We saw that it does not turn over (but also from one place), and there it seems that the lack of turning over is truly due to a dependence between the events. If I know that the knife was damaged before most of the slaughter, then it certainly was not damaged after it. As I advance along the timeline, I leave behind everything that comes after the point in time in question. In such a situation, it seems that the non-overturning really indicates statistical dependence. But as mentioned, this is a spik spik that does not turn over, which is also a sharp spik. The question is: is there an example of a spik spik that does not turn over that is not from one place, and in which the non-overturning also stems from statistical dependence? For the explanation I proposed to be correct, it would be better to find such an example (otherwise the law of reversal would be unnecessary, because we already have an explanation for the law of 'clear doubt' and it certainly exists). Right now I can't think of an example, but I see no reason why there shouldn't be one.

[1] It is necessarily equal, because if it is not equal, majority law applies here. We discussed this in the previous column.

[2] There is an assumption here that the marriage ceremony took place after she reached the age of marriage. That is, more than nine years passed between the consecration ceremony that took place before the age of three and the marriage.

[3] Indeed, ultimately, the Ra'a rejects this explanation, since according to it, even if Kiddshah is less than Bat 12 (and not just less than Bat 3), we would have to prohibit it, since it is not a SS that is reversed according to the same calculation. It is proven from its own merits that the two rules are not identical.

[4] This is also what Rabbi Shlomo Levi concluded inHis article The above.