Talmudic reasoning

Bava Kamma 27 ~ Bizarre Talmudic Scenarios

On this page of Talmud we read of a very bizarre case:

בבא קמא כז, א

 אָמַר רַבָּה: נָפַל מִן הַגָּג וְנִתְקַע — חַיָּיב בְּאַרְבָּעָה דְּבָרִים, וּבִיבִמְתּוֹ לֹא קָנָה. בְּנֵזֶק, בְּצַעַר, בְּשֶׁבֶת, בְּרִפּוּי. אֲבָל בּוֹשֶׁת לָא מִיחַיַּיב, דְּאָמַר מָר: אֵין חַיָּיב עַל הַבּוֹשֶׁת עַד שֶׁיִּתְכַּוֵּון

Rabba said: One who fell from a roof and was inserted into a woman due to the force of his fall is liable to pay four of the five types of indemnity that must be paid by one who damaged another, and if she is his yevama he has not acquired her in this manner. He is liable to pay for injury, pain, loss of livelihood, and medical costs. However, he is not liable to pay for the shame he caused her, as the Master said: One is not liable to pay for shame unless he intends to humiliate his victim…

As weird Talmudic cases go, this is among the weirdest. It is entirely impossible, and not least because this would happen. Did the rabbis of the Talmud really believe that such a case could occur? To answer this, let’s consider come other rather implausible cases from across the Babylonian Talmud.

 The AMAZING Shechita Knife

We begin with a fanciful question that is somewhat analogous to falling intercourse case. What happens if a person throws a knife across the room, but in doing so the flying knife somehow manages to cut the neck of an animal in just the correct fashion to perform a kosher shechita (ritual slaughter). Is the meat of this slaughtered animal kosher?

חולין לא, א

דתני אושעיא זעירא דמן חבריא זרק סכין לנועצה בכותל והלכה ושחטה כדרכה ר' נתן מכשיר וחכמים פוסלים הוא תני לה והוא אמר לה הלכה כר' נתן

Oshaya, the youngest of the company of Sages, taught a baraita: If one threw a knife to embed it in the wall and in the course of its flight the knife went and slaughtered an animal in its proper manner, Rabbi Natan deems the slaughter valid and the Rabbis deem the slaughter not valid. Oshaya teaches the baraita and he says about it: The halakha is in accordance with the opinion of Rabbi Natan that there is no need for intent to perform a valid act of slaughter.

 The Fish That Pulled a Plough

The Bible (Deuteronomy 22:10) forbids a farmer to plough his land using an ox and a donkey together. While no reason for this law is given, we might suppose it has something to do with the concern that doing so might cause unnecessary pain to the smaller (or perhaps the larger?) animal. Regardless of the reason, later in our tractate the Talmud explains that this law applies to any kind of work and any two different species of animal. Then comes this fantastic question: “What is the law if someone pulls his wagon using a goat and a fish?” 

בבא קמא נה, א

בעי רחבה המנהיג בעיזא ושיבוטא מהו מי אמרינן כיון דעיזא לא נחית בים ושיבוטא לא סליק ליבשה לא כלום עביד או דלמא השתא מיהת קא מנהיג

The Sage Rachava raised a dilemma: With regard to one who drives a wagon on the seashore with a goat and a shibbuta, a certain species of fish, together, pulled by the goat on land and the fish at sea, what is the halakha? Has he violated the prohibition against performing labor with diverse kinds, in the same way that one does when plowing with an ox and a donkey together, or not?

This turned out to be such a hard question that the Talmud could not answer it. The Rosh concludes though that just to be sure, best not to hitch up your wagon to a fish, if you also intend for it to be pulled by a goat (ולא איפשיטא ואזלינן לחומרא). Don’t say you weren’t warned.

The Bird that built her nest on a person’s head

The Bible also demands that the mother bird must be shooed away before collecting the eggs upon which she is brooding. But what happens if a bird makes her nest in a person’s hair? Must this mother be driven away before her eggs are collected? (Chullin 139b)

חולין קלט, ב

אמרי ליה פפונאי לרב מתנה מצא קן בראשו של אדם מהו? אמר (שמואל ב טו, לב) ואדמה על ראשו

The residents of Pappunya said to Rav Mattana: If one found a nest on the head of a person, what is the halakha with regard to the mitzva of sending away the mother? Is the nest considered to be on the ground, such that one is obligated in the mitzva? Rav Mattana said to them that one is obligated in the mitzva in such a case because the verse states: “And earth upon his head” (II Samuel 15:32), rather than: Dirt upon his head, indicating that one’s head is considered like the ground. 

(And just to be clear - the Talmud is not discussing a case like this one, in which a woman allowed an abandoned fledgling to nest in her long har for 84 days, though it is, I will admit, a very touching story.)

One hypothetical too many

Sometimes, even bizarre questions can go too far. If a baby pigeon is found within 50 cubits of a coop, it is presumed to belong to the owner of that coop. If it is found further away than 50 cubits, it belongs to the finder. Ever keen to push the limits of rabbinic law, Rabbi Yirmiyah asked “if one foot of the pigeon is within the fifty cubits and one foot is outside, to whom does it belong?” This apparently was one question too many. The rabbis (rather unfairly in my opinion) expelled Rabbi Yirmiyah from the Yeshivah for asking it.

בבא בתרא כג, ב

בָּעֵי רַבִּי יִרְמְיָה רַגְלוֹ אַחַת בְּתוֹךְ חֲמִשִּׁים אַמָּה וְרַגְלוֹ אַחַת חוּץ מֵחֲמִשִּׁים אַמָּה מַהוּ וְעַל דָּא אַפְּקוּהוּ לְרַבִּי יִרְמְיָה מִבֵּי מִדְרְשָׁא

Rabbi Yirmeya raises a dilemma: If one leg of the chick was within fifty cubits of the dovecote, and one legwas beyond fifty cubits, what is the halakha? The Gemara comments: And it was for his question about this far-fetched scenario that they removed Rabbi Yirmeya from the study hall, as he was apparently wasting the Sages’ time. 

The Role of Bizarre cases

In a 2004 paper published in Thalia: Studies in Literary Humor, Hershey Friedman, a Professor of Business at Brooklyn College, suggested that “whether a situation is possible or not is immaterial when the Talmud is trying to establish legal principles.”  

Purely theoretical (at least in their days) cases are discussed because the sages felt that principles derived from these discussions would clarify the law and thus provide a more thorough understanding of it. Discussions of theoretical cases in the Talmud have allowed scholars of today to use the Talmudic logic and principles to solve current legal questions 

The theoretical questions make a legal point, and it is that legal point that is the real object of the discussion. Whether or not the case could actually happen is immaterial. Friedman also suggests that these unusual cases serve to keep the material interesting, and also act as brain teasers, which don’t necessarily make a legal point but serve to sharpen the minds of both the students and the teachers who ask them.

There was a time, not many years ago, when a lawyer could feel reasonably confident as he approached oral argument in the United States
Supreme Court if he had thoroughly absorbed the record in his case and
had obtained a working knowledge of all relevant cases. No longer. Today, an advocate must, more than ever before, prepare himself for a
stream of hypothetical questions touching not only on his own case but on
a variety of unrelated facts and situations.
— E. Barrett Prettyman Jr., The Supreme Court's Use of Hypothetical Questions at Oral Argument, 33 Cath. U. L. Rev. 555 (1984). 555.

 Hypothetical Cases in the US Legal System

It may help to understand the role that these weird cases have in the Talmud by understanding that hypothetical cases have an important role to play in many legal systems, including that of the United States. Consider this series of questions that were asked in the famous 1984 case of California vs Carney. Police officers entered a motor home without a search warrant, and found marijuana. The question before the court was whether this motor home was a more like a car, which should not require a search warrant, or more like a home, which would. There were various appeals, and the case ended up being heard in front of the US Supreme Court, which is where the following hypothetical questions were raised:

Q: Well, what if the vehicle is in one of these mobile home parks and hooked up to water and electricity but still has its wheels on?

Q: Suppose somebody drives a great big stretch Cadillac down and puts it in a parking lot, and pulls all the curtains around it, including the one over the windshield and around all the rest of them. Would that be a home?

Or how about this exchange back in 1982, (and the subject of which is once again a hot topic of debate in the US). In Board of Education v.Pico, the question before the Court was whether a public school board could remove books which it found to be objectionable from the shelves of junior and senior high school libraries, in order to promote the community's "moral, social, and political values." 

Q: Suppose they [the Board] barred the St. James version of the New Testament, and the Constitution of the United States, and the Declaration of Independence?

Q: Suppose some of these books were assigned as outside reading, and the children were told, you can get it in the public library?

Q. Suppose you had a book, counsel, that had been the subject of criminal proceedings, and conviction of someone in connection with that book had been sustained, a criminal conviction. Would you say that the book comes under this broad authority you suggest? 

(By the way, the Court, in a 5-to-4 decision, held that as centers for voluntary inquiry and the dissemination of information and ideas, school libraries enjoy a special affinity with the rights of free speech and press. Therefore, the School Board could not restrict the availability of books in its libraries simply because its members disagreed with their content. That might be useful to remember.)

There are many, many similar examples. Here is one of my favorites. It comes from United States v. Ross, in which the defendant’s lawyer argued that a small brown paper bag should not have been searched for narcotics because it was protected under the Fourth Ammendment, the right against unlawful search. Here is one of the hypotheticals:

Q: Suppose what they were hunting for was, say, a waffle iron, a stolen waffle iron, or something else that couldn't go in the paper bag. You might have probable cause to search the car for the waffle iron, but if you got to the paper bag, you wouldn't be searching it, would you?

Commenting on this case, the late American lawyer Elijah Barrett Prettyman Jr. (d. 2016) wrote that “one cannot help but be impressed with how far removed some hypotheticals are from the facts before the Court. In Ross, the brown paper bag case, one Justice had the police hunting for a waffle iron.”

Why we need Hypotheticals

Hypothetical cases are really important when the Supreme Court is trying to figure out the difficult cases that come before it. And they are all difficult cases, because if they were easy, the Supreme Court wouldn’t be considering them. The rabbis of the Talmud needed to do the same, which is why they often consider outlandish, implausible or downright fanciful cases to ponder.

The best way to think about these cases is to add in the following missing words: “Hypothetically, what would happen if…” Then, as if by magic, they cease to be silly and start to be really important.

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Ketuvot 15a ~ Talmudic Probability Theory

תלמוד בבלי כתובות דף טו עמוד א 

  א"ר זירא: כל קבוע כמחצה על מחצה דמי ..מנא ליה לר' זירא הא? ...מתשע חנויות, כולן מוכרות בשר שחוטה ואחת בשר נבלה, ולקח מאחת מהן ואינו יודע מאיזה מהן לקח - ספיקו אסור, ובנמצא - הלך אחר הרוב, 

R. Zera said: Any doubt about something that is fixed in its place is considered be a fify-fifty chance... Where does he learn this from ? [From a Baraisa which teaches the following. Consider a town in which] there are nine shops, all of which sell kosher meat, and one store that sells meat that is not kosher. If a person bought meat from one of these [ten] stores but he cannot recall from which, his doubt means that the meat is forbidden. But if he found a piece of meat [in the street and he cannot tell from which store it came] he may follow the majority [and assume the meat is kosher]...

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principle is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.
— Gabbay and Koppel 2010

In other words, what happens is that if there are more kosher stores than there are treif, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher

We encountered another example of talmudic probability theory only a week ago, on Ketuvot 9a. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

  1. She was raped before her betrothal.

  2. She was raped after her betrothal.

  3. She had intercourse of her own free will before her betrothal.

  4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gabbay and Koppel explain the case - using formal logic:

 
 

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gabbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch (1928-2020) was the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also had a PhD in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seemed to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justinian around 533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the late Rosh Yeshiva, Rabbi Rabinovitch.

(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.
— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.
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Yevamot 54a ~ Bizarre Talmudic Scenarios

On this page of Talmud we read of a very bizarre case:

יבמות נד, א

 אָמַר רַבָּה: נָפַל מִן הַגָּג וְנִתְקַע — חַיָּיב בְּאַרְבָּעָה דְּבָרִים, וּבִיבִמְתּוֹ לֹא קָנָה. בְּנֵזֶק, בְּצַעַר, בְּשֶׁבֶת, בְּרִפּוּי. אֲבָל בּוֹשֶׁת לָא מִיחַיַּיב, דְּאָמַר מָר: אֵין חַיָּיב עַל הַבּוֹשֶׁת עַד שֶׁיִּתְכַּוֵּון

Rabba said: One who fell from a roof and was inserted into a woman due to the force of his fall is liable to pay four of the five types of indemnity that must be paid by one who damaged another, and if she is his yevama he has not acquired her in this manner. He is liable to pay for injury, pain, loss of livelihood, and medical costs. However, he is not liable to pay for the shame he caused her, as the Master said: One is not liable to pay for shame unless he intends to humiliate his victim…

As weird Talmudic cases go, this is among the weirdest. It is entirely impossible, and not least because this would happen. Did the rabbis of the Talmud really believe that such a case could occur? To answer this, let’s consider come other rather implausible cases from across the Babylonian Talmud.

 The AMAZING Shechita Knife

We begin with a fanciful question that is somewhat analogous to tomorrow’s falling Yibbum case. What happens if a person throws a knife across the room, but in doing so the flying knife somehow manages to cut the neck of an animal in just the correct fashion to perform a kosher shechita (ritual slaughter). Is the meat of this slaughtered animal kosher?

חולין לא, א

דתני אושעיא זעירא דמן חבריא זרק סכין לנועצה בכותל והלכה ושחטה כדרכה ר' נתן מכשיר וחכמים פוסלים הוא תני לה והוא אמר לה הלכה כר' נתן

Oshaya, the youngest of the company of Sages, taught a baraita: If one threw a knife to embed it in the wall and in the course of its flight the knife went and slaughtered an animal in its proper manner, Rabbi Natan deems the slaughter valid and the Rabbis deem the slaughter not valid. Oshaya teaches the baraita and he says about it: The halakha is in accordance with the opinion of Rabbi Natan that there is no need for intent to perform a valid act of slaughter.

 The Fish That Pulled a Plough

The Bible (Deuteronomy 22:10) forbids a farmer to plough his land using an ox and a donkey together. While no reason for this law is given, we might suppose it has something to do with the concern that doing so might cause unnecessary pain to the smaller (or perhaps the larger?) animal. Regardless of the reason, the Talmud explains that this law applies to any kind of work and any two different species of animal. Then comes this fantastic question: “What is the law if someone pulls his wagon using a goat and a fish?” 

בבא קמא נה, א

בעי רחבה המנהיג בעיזא ושיבוטא מהו מי אמרינן כיון דעיזא לא נחית בים ושיבוטא לא סליק ליבשה לא כלום עביד או דלמא השתא מיהת קא מנהיג

The Sage Rachava raised a dilemma: With regard to one who drives a wagon on the seashore with a goat and a shibbuta, a certain species of fish, together, pulled by the goat on land and the fish at sea, what is the halakha? Has he violated the prohibition against performing labor with diverse kinds, in the same way that one does when plowing with an ox and a donkey together, or not?

This turned out to be such a hard question that the Talmud could not answer it. The Rosh concludes though that just to be sure, best not to hitch up your wagon to a fish, if you also intend for it to be pulled by a goat (ולא איפשיטא ואזלינן לחומרא). Don’t say you weren’t warned.

The Bird that built her nest on a person’s head

The Bible also demands that the mother bird must be shooed away before collecting the eggs upon which she is brooding. But what happens if a bird makes her nest in a person’s hair? Must this mother be driven away before her eggs are collected? (Chullin 139b)

חולין קלט, ב

אמרי ליה פפונאי לרב מתנה מצא קן בראשו של אדם מהו? אמר (שמואל ב טו, לב) ואדמה על ראשו

The residents of Pappunya said to Rav Mattana: If one found a nest on the head of a person, what is the halakha with regard to the mitzva of sending away the mother? Is the nest considered to be on the ground, such that one is obligated in the mitzva? Rav Mattana said to them that one is obligated in the mitzva in such a case because the verse states: “And earth upon his head” (II Samuel 15:32), rather than: Dirt upon his head, indicating that one’s head is considered like the ground. 

(And just to be clear - the Talmud is not discussing a case like this one, in which a woman allowed an abandoned fledgling to nest in her long har for 84 days, though it is, I will admit, a very touching story.)

One hypothetical too many

Sometimes, even bizarre questions can go too far. If a baby pigeon is found within 50 cubits of a coop, it is presumed to belong to the owner of that coop. If it is found further away than 50 cubits, it belongs to the finder. Ever keen to push the limits of rabbinic law, Rabbi Yirmiyah asked “if one foot of the pigeon is within the fifty cubits and one foot is outside, to whom does it belong?” This apparently was one question too many. The rabbis (rather unfairly in my opinion) expelled Rabbi Yirmiyah from the Yeshivah for asking it.

בבא בתרא כג, ב

בָּעֵי רַבִּי יִרְמְיָה רַגְלוֹ אַחַת בְּתוֹךְ חֲמִשִּׁים אַמָּה וְרַגְלוֹ אַחַת חוּץ מֵחֲמִשִּׁים אַמָּה מַהוּ וְעַל דָּא אַפְּקוּהוּ לְרַבִּי יִרְמְיָה מִבֵּי מִדְרְשָׁא

Rabbi Yirmeya raises a dilemma: If one leg of the chick was within fifty cubits of the dovecote, and one legwas beyond fifty cubits, what is the halakha? The Gemara comments: And it was for his question about this far-fetched scenario that they removed Rabbi Yirmeya from the study hall, as he was apparently wasting the Sages’ time. 

The Role of Bizarre cases

In a 2004 paper published in Thalia: Studies in Literary Humor, Hershey Friedman, a Professor of Business at Brooklyn College, suggested that “whether a situation is possible or not is immaterial when the Talmud is trying to establish legal principles.”  

Purely theoretical (at least in their days) cases are discussed because the sages felt that principles derived from these discussions would clarify the law and thus provide a more thorough understanding of it. Discussions of theoretical cases in the Talmud have allowed scholars of today to use the Talmudic logic and principles to solve current legal questions 

The theoretical questions make a legal point, and it is that legal point that is the real object of the discussion. Whether or not the case could actually happen is immaterial. Friedman also suggests that these unusual cases serve to keep the material interesting, and also act as brain teasers, which don’t necessarily make a legal point but serve to sharpen the minds of both the students and the teachers who ask them.

There was a time, not many years ago, when a lawyer could feel reasonably confident as he approached oral argument in the United States
Supreme Court if he had thoroughly absorbed the record in his case and
had obtained a working knowledge of all relevant cases. No longer. Today, an advocate must, more than ever before, prepare himself for a
stream of hypothetical questions touching not only on his own case but on
a variety of unrelated facts and situations.
— E. Barrett Prettyman Jr., The Supreme Court's Use of Hypothetical Questions at Oral Argument, 33 Cath. U. L. Rev. 555 (1984). 555.

 Hypothetical Cases in the US Legal System

It may help to understand the role that these weird cases have in the Talmud by understanding that hypothetical cases have an important role to play in many legal systems, including that of the United States. Consider this series of questions that were asked in the famous 1984 case of California vs Carney. Police officers entered a motor home without a search warrant, and found marijuana. The question before the court was whether this motor home was a more like a car, which should not require a search warrant, or more like a home, which would. There were various appeals, and the case ended up being heard in front of the US Supreme Court, which is where the following hypothetical questions were raised:

Q: Well, what if the vehicle is in one of these mobile home parks and hooked up to water and electricity but still has its wheels on?

Q: Suppose somebody drives a great big stretch Cadillac down and puts it in a parking lot, and pulls all the curtains around it, including the one over the windshield and around all the rest of them. Would that be a home?

Or how about this exchange back in 1982, (and the subject of which is once again a hot topic of debate in the US). In Board of Education v.Pico, the question before the Court was whether a public school board could remove books which it found to be objectionable from the shelves of junior and senior high school libraries, in order to promote the community's "moral, social, and political values." 

Q: Suppose they [the Board] barred the St. James version of the New Testament, and the Constitution of the United States, and the Declaration of Independence?

Q: Suppose some of these books were assigned as outside reading, and the children were told, you can get it in the public library?

Q. Suppose you had a book, counsel, that had been the subject of criminal proceedings, and conviction of someone in connection with that book had been sustained, a criminal conviction. Would you say that the book comes under this broad authority you suggest? 

(By the way, the Court, in a 5-to-4 decision, held that as centers for voluntary inquiry and the dissemination of information and ideas, school libraries enjoy a special affinity with the rights of free speech and press. Therefore, the School Board could not restrict the availability of books in its libraries simply because its members disagreed with their content. That might be useful to remember.)

There are many, many similar examples. Here is one of my favorites. It comes from United States v. Ross, in which the defendant’s lawyer argued that a small brown paper bag should not have been searched for narcotics because it was protected under the Fourth Ammendment, the right against unlawful search. Here is one of the hypotheticals:

Q: Suppose what they were hunting for was, say, a waffle iron, a stolen waffle iron, or something else that couldn't go in the paper bag. You might have probable cause to search the car for the waffle iron, but if you got to the paper bag, you wouldn't be searching it, would you?

Commenting on this case, the late American lawyer Elijah Barrett Prettyman Jr. (d. 2016) wrote that “one cannot help but be impressed with how far removed some hypotheticals are from the facts before the Court. In Ross, the brown paper bag case, one Justice had the police hunting for a waffle iron.”

Why we need Hypotheticals

Hypothetical cases are really important when the Supreme Court is trying to figure out the difficult cases that come before it. And they are all difficult cases, because if they were easy, the Supreme Court wouldn’t be considering them. The rabbis of the Talmud needed to do the same, which is why they often consider outlandish, implausible or downright fanciful cases to ponder.

The best way to think about these cases is to add in the following missing words: “Hypothetically, what would happen if…” Then, as if by magic, they cease to be silly and start to be really important.

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Pesachim 9b ~ Talmudic Probability Theory

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Our new tractate Pesachim, deals with all things Paschal. (Well, nearly all). What happens if there were nine piles of matzah and one pile of forbidden leavened bread known as chametz, and along came a mouse and took a piece from one of the piles and carried it into a house that had already been searched for chametz. Must the house be searched a second time? To find an answer, the Talmud quotes a Baraiasa that deals with an analogous question.

פסחים ט, ב 

דִּתְנַן: תֵּשַׁע חֲנוּיוֹת, כּוּלָּן מוֹכְרִין בְּשַׂר שְׁחוּטָה, וְאַחַת מוֹכֶרֶת בְּשַׂר נְבֵלָה, וְלָקַח מֵאַחַת מֵהֶן, וְאֵינוֹ יוֹדֵעַ מֵאֵיזֶה מֵהֶן לָקַח — סְפֵיקוֹ אָסוּר.

With regard to nine stores in a city, all of which sell kosher meat from a slaughtered animal, and one other store that sells meat from unslaughtered animal carcasses, and a person took meat from one of them and he does not know from which one he took the meat, in this case of uncertainty, the meat is prohibited.

וּבַנִּמְצָא — הַלֵּךְ אַחַר הָרוֹב

And in the case of meat found outside, follow the majority.

What this boils down to is this. If most stores in the city sell kosher meat then a piece of meat that is found in the city (that is “outside”) is assumed to be kosher, since the majority of the stores sell only kosher meat. But if a person bought meat from one of the ten stores, but he cannot recall whether or not it was from a kosher store, the meat may not be eaten. In this latter case, we assume that there were simply an equal number of kosher and non-kosher stores. There is a 50-50 chance that the meat comes from a non-kosher store, and it may not be eaten.

By analogy, if the mouse took the morsel from one of the piles, the legal status of the morsel is that of an equally balanced uncertainty concerning whether it was taken from a pile of matzah or a pile of chametz. Consequently, the owner is required to go back and search the house all over again.

Talmudic Probability

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principal is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.

In other words, what happens is that if there are more kosher stores than there are non-kosher, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher

We encountered another example of talmudic probability theory when we studied the tractate Ketuvot. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

  1. She was raped before her betrothal.

  2. She was raped after her betrothal.

  3. She had intercourse of her own free will before her betrothal.

  4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gubbay and Koppel explain the case - using formal logic:

 
Detail from Gabbay paper.jpg
 

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gubbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch, who died in May of this year at the age of 92 was the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also had a PhD. in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seems to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justine c.533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the late mathematician and Rosh Yeshiva, Rabbi Rabinovitch.

(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.
— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.
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