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.

Pesachim 9b ~ Talmudic Probability Theory

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:

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.

Niddah 24a ~ Halachic Reality and Anatomic Reality: Treif People and Treif Animals

In tomorrow’s page of Talmud we read of a dispute about survivability of an infant with a birth defect. According to Rav Zakkai, an infant lacking legs from the knees downward cannot live, and is classified as a treifah. Rav Yannai declared that such a child could live, but agreed that it is classified as a treifah. Rav Yannai believed that only a birth defect that included the urinary opening was severe enough to be incompatible with life. In both cases the child is declared a treifah, but what is in dispute is the prognosis of such a treifah.

נדה כד,א

בין רבי זכאי לרבי ינאי איכא בינייהו טרפה חיה מר סבר טרפה חיה ומר סבר טרפה אינה חיה

The difference between the opinion of Rabbi Zakkai and that of Rabbi Yannai is whether a tereifa [can survive beyond twelve months]. One sage, [Rabbi Yannai,] holds that a tereifah can survive [beyond twelve months. Therefore, although one whose legs were removed until above the knee has the status of a tereifa, if a woman discharges a fetus of this form she is impure. Only if the fetus lacks legs until his orifices is the woman pure, as such a person cannot survive.] And one sage, [Rabbi Zakkai], holds that a tereifah cannot survive [beyond twelve months]. …

Treifah Animals

We have previously met the concept of treifah when we studied Chullin and the laws of rural slaughter called shechitah. Here is a reminder, from the start of the third chapter of Chullin (42a) where we took a deep dive into animal anatomy.

אלו טרפות בבהמה נקובת הוושט ופסוקת הגרגרת ניקב קרום של מוח ניקב הלב לבית חללו נשברה השדרה ונפסק החוט שלה ניטל הכבד ולא נשתייר הימנו כלום הריאה שניקבה או שחסרה ר"ש אומר עד שתינקב לבית הסמפונות ניקבה הקבה ניקבה המרה ניקבו הדקין הכרס הפנימית שניקבה או שנקרע רוב החיצונה רבי יהודה אומר הגדולה טפח והקטנה ברובה המסס ובית הכוסות שניקבו לחוץ נפלה מן הגג נשתברו רוב צלעותיה ודרוסת הזאב רבי יהודה אומר דרוסת הזאב בדקה ודרוסת ארי בגסה דרוסת הנץ בעוף הדק ודרוסת הגס בעוף הגס זה הכלל כל שאין כמוה חיה טרפה

These wounds constitute tereifot in an animal,rendering them prohibited for consumption:

1. A perforated esophagus, where the perforation goes through the wall , 

2. or a cut trachea.

3. If the membrane of the brain was perforated, 

4. or if the heart was perforated to its chamber; 

5. if the spinal column was broken and its cord was cut; 

6. if the liver was removed and nothing remained of it…

7. a lung that was perforated

8. or a lung missing a piece….

9. If the abomasum was perforated

10. or the gallbladder was perforated, 

11. or the small intestines were perforated, it is a tereifa…

This is the principle: Any animal that was injured such that an animal in a similar condition could not live for an extended period is a treifa, the consumption of which is forbidden by Torah law. 

The original meaning of the term treif in the Torah is torn, and it describes a domestic animal that was attacked by a wild animal and suffered an injury that led to its death.

“וְאַנְשֵׁי־קֹ֖דֶשׁ תִּהְי֣וּן לִ֑י וּבָשָׂ֨ר בַּשָּׂדֶ֤ה טְרֵפָה֙ לֹ֣א תֹאכֵ֔לוּ לַכֶּ֖לֶב תַּשְׁלִכ֥וּן אֹתֽוֹ׃
You will be holy people to Me: you must not eat flesh torn by beasts in the field; you shall cast it to the dogs.”

— Exodus 22:30

But the rabbis of the Talmud greatly expanded this category - hence the list in this Mishnah in Chullin. Elsewhere in Chullin (57b) there is a dispute as to the prognosis of living animal that has been declared treif. According to Rav Hunna, if an animal is treif, by definition it cannot live for longer than a year (אמר רב הונא סימן לטרפה י"ב חדש). But there are other opinions. The great editor of the Mishnah, Rabbi Yehudah HaNassi held that a treifa is destined to die within 30 days, while a berasia states that a treif animal cannot give birth (leaving open the question about male animals).

““Jason Marcus, chef and owner of the new Traif restaurant on S. Fourth Street in Williamsburg, says the name is just cheeky, not a slap at his mostly Kosher eaters.”

“But [the name] really represents our philosophical view of how restaurants should be free of rules. We’re just people who live for good food.””

It is generally agreed upon that list in Chullin detailed the kinds of lesions that would be fatal within a year. And that’s when the problems begin. Some of them are certainly likely to be fatal. For example a perforated esophagus (נקובת הוושט) leads to mediastinitis, an inflammation of the chest cavity. And that is commonly fatal. If the animal swallows something sharp it can pierce not only the esophagus, but the membranes that surround the heart, called the pericardium. Way back in 1955 - at the start of the era of antibiotics - The Australian Veterinary Journal published a case series of twenty-one dairy cows that developed traumatic pericarditis. “Fifteen cases were treated with sulphonamide [an antibiotic] and six were not. ” The six animals untreated cows all died, and even among the cows treated with antibiotics, almost half died. So yes, some lesions recorded in the Mishanh (and later refined in the talmudic discussion which follows) are indeed fatal.

The Case of Serachot

But other lesions that render an animal treif are certainly not fatal. Take for example lung adhesions, called סרחות (serachot, or sircha in the singular), which are discussed elsewhere in the tractate Chullin (46b et. seq). These adhesions are fibrous tissues that may run between different lung lobes, or between the lungs and the rib cage. They are common and are caused by a number of conditions, including trauma or previous infections. Many kinds of serichot render an animal treif. But lung adhesions are certainly not lethal. Animals and humans live quite happily with them. In fact this doctor recently told me that the presence of lung adhesions does not prevent lungs from being donated and used for a lung transplant. Now, if they are used in that delicate situation, they most certainly do not have a fatal defect, or anything even close.

the case of the missing liver (and the missing heart)

Opening paragraph of the famous responsa on “the chicken that had no heart”. From שו׳ת חכם צבי, Amsterdam 1712.

Equally puzzling to the modern reader is the sixth category in the Mishna’s list: ניטל הכבד ולא נשתייר הימנו כלום - if the slaughtered animal was found to have no liver. Here’s the thing: an animal cannot live without a liver. If a healthy looking cow - or indeed any cow -was well enough to be slaughtered, it must have had a liver. So this is not an example of a treif animal - it’s an example of one that could not possibly have existed. But don’t take my word for it.

In 1709 the great rabbi of Hamburg, Zevi Ashkenazi, (better known as the Chacham Zevi, after the name of his responsa) was asked the following question. A young woman had opened a slaughtered chicken to remove the unwanted entrails, while her cat sat at her feet “waiting patiently for anything that may fall to the ground.” To her great surprise, the young woman found that the chicken did not have a heart, and so assumed the bird was treif. Not so, claimed her mother, who apparently owned the chicken. The cat must have eaten it, when it was thrown to the ground together with the entrails. The young women was however quite adamant, and insisted she had never fed anything that resembled a heart to the cat. The bird had been perfectly healthy before it was slaughtered, eating and drinking like any other healthy chicken, (וגם בעודנה בחיים חיותה היתה חזקה ובריאה ובכל כחה לאכול ולשתות). The question of the kashrut of the bird was brought to the local rabbis, who declared it to be treif, on the basis that while alive, it had no heart.

The Chacham Zevi was asked to weigh in on the matter. “It is absolutely clear to any person who has a wise heart” he wrote, apparently enjoying the play on words, “or who has a brain in his skull, that it is impossible for any creature to live for even a moment without a heart…Clearly, the heart fell out when the bird was opened, and that cat ate it…It is obvious that the chicken is permitted” Strike one for common sense. You would think. But not so fast. This answer of the Chacham Zevi engendered one of the great halachic disputes of the eighteenth century. In one corner, the Chacham, and in the other at least four leading rabbinic figures who vehemently opposed this ruling: Naphtali Katz of Frankfurt, Moses Rothenburg, David Oppenheim (who was the Chief Rabbi of Prague, no less) and Jonathan Eyebeschuetz (who spent much of his later life fighting halachic battles against Rabbi Yaakov Emden, who was the son of the Chacham Zevi). It got nasty, but that’s a story for another day.

Halachic Reality

No bird or animal can live without a heart, and none can do so without a liver. So there can be no case, like the one in the Mishnah, in which a healthy living animal was slaughtered and found to be without a liver.

Some of the categories of treifot overlap with conditions that are indeed incompatible with life. Others are perfectly innocuous and compatible with a long and healthy life. And a few make no sense given what we know about animal physiology. But none should be thought of as describing an anatomical reality. They describe instead a halachic reality, a reality that reflected a world some 1,500 years ago. And while our understanding of physiology has changed, these halachic classes remain a fixed part of Jewish tradition. Here is the great Maimonides, who was obviously troubled by the chasm that sometimes exists between halacha and facts.

רמב’ם משנה תורה הלכות שחיטה י, יג וְכֵן אֵלּוּ שֶׁמָּנוּ וְאָמְרוּ שֶׁהֵן טְרֵפָה אַף עַל פִּי שֶׁיֵּרָאֶה בְּדַרְכֵי הָרְפוּאָה שֶׁבְּיָדֵינוּ שֶׁמִּקְצָתָן אֵינָן מְמִיתִין וְאֶפְשָׁר שֶׁתִּחְיֶה מֵהֶן אֵין לְךָ אֶלָּא מַה שֶּׁמָּנוּ חֲכָמִים שֶׁנֶּאֱמַר (דברים יז יא)"עַל פִּי הַתּוֹרָה אֲשֶׁר יוֹרוּךָ

Each one of these lesions that were declared treif remain so even if modern medicine can demonstrate that some of them are not actually fatal, and that it is indeed possible to live despite them. Rather we must follow these rabbinic categories, as the Torah states“ You shall act in accordance with the instructions given you and the ruling handed down to you; [you must not deviate from the verdict that they announce to you either to the right or to the left.]

In more recent times, Rabbi Avrohom Yeshaya Karelitz, better known as the Chazon Ish, also addressed this question. “We see today” he wrote, “that very often surgeons operate on the abdomen of a person [with an injury like one found in a treif animal], and he is completely cured, and lives a long life.” But this does nothing to change the way we view the categories of treif. These depend solely on what was decided by the rabbis of the Talmud, and no modern findings can change them.

[Repost from here.]

Niddah 18a ~ Talmudic Probability Theory

נדה יח, 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 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 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.
”

— Gabbay and Koppel 2010

In other words, what happens is that if there are more kosher stores than there are trief, 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 way back in February 2015 on page 9a of 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:

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 (b.1928) is the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also has 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 Rosh Yeshiva, Rabbi Rabinovitch שליט׳א.

“(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.

[Repost from here.]