Scientific Generalization and the Problem of Induction in Halakha and Beyond (Column 729)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

Introduction

In the scientific process we measure some phenomenon over a set of sample cases, and from there infer conclusions about a general law that applies to all cases of those kinds. Thus, for example, we observe several massive bodies falling toward the earth and infer that there is a general law whereby every body with mass is drawn to the earth. In Column 79 and elsewhere I noted that this is the process on which a rov d’leita kaman (a majority not presently before us) is based. Such a majority is in fact a kind of law of nature reached by generalizing from data known to us about a sample.

David Hume pointed out the problem inherent in scientific generalizations. According to him we have no way to justify the assumption that what we observed is indeed a representative sample of a general law. Who says what we saw is not merely a random collection of examples that does not represent a general phenomenon? Perhaps the sample we measured is not representative? Perhaps it is merely accidental or biased by some unique feature? In fact, in light of his words one can wonder who says there are general laws in the world at all? This is what has been called since Hume “the problem of induction.” This is apparently why most early authorities view a rov d’leita kaman as weaker than a rov d’eita kaman (so it also emerges from the plain sense of the sugya in Chullin 11a; see that column). It is a majority based on some speculation, on a generalization. In several places in the past (see, for example, Column 653) I argued that the only solution to the problem of induction is to recognize that we have a sixth sense—an intuition—that helps us discern general laws. According to this proposal, generalizations are not made solely on the basis of observations, as people delude themselves, but by means of insights that go beyond the set of observations (the sample). This is a process that is not purely empirical. These insights help us discern the general law through the examples we observed (the sample)—what Husserl calls “eidetic seeing,” or the contemplation of ideas.

In this column I wish to examine the problem of induction in Halakha, to point out difficulties in the processes of generalization we employ, and the unavoidable influence of our intuition on those processes. The halakhic and Talmudic sugyot that deal directly with generalization are those that concern the “three-times presumption” (chazakah after three occurrences), and these I shall discuss here. In the next column I will continue with samples that look significant but are in fact accidental and therefore cannot serve as a basis for generalization.

The “three-times” presumption

It is accepted in Halakha that if some event repeats itself three times, this indicates a fixed pattern, i.e., a general law. The assumption is that it will continue to behave so in the future. For example, an ox that gored once is a tam (non-forewarned), and its owner pays half the damage. But an ox that gored three times is mu’ad (forewarned), and its owner pays full damages. In other words, the fact that the ox gored three times teaches us that there is a general law here; it is not a coincidence. This is an ox that is by nature prone to gore, and one can expect it to gore in the future as well. This is the biblical source for the “three-times” presumption.

The Sages extended this mode of thinking to other halakhic contexts. For example, a woman who sees menstrual blood on a fixed calendar date in the month (say, on the 6th of each month) three consecutive times establishes a presumption that she will continue to see on that date the next month as well. This is called a monthly veset (fixed cycle). There is also a veset based on intervals (haflagah), which likewise rests on a three-times presumption. Another case is a woman whose three sons died as a result of circumcision; in such a situation we do not circumcise her fourth son, for there is a presumption that if we circumcise him, he too will die. Similarly, regarding a woman whose three husbands died, she should not marry a fourth, since there is concern that the fourth will also die. The assumption in all these cases is that what occurred three times is not accidental but expresses a general law, and therefore it is expected to continue in the same manner in the future.

The explanation for this is found in the words of R. Chaim of Brisk, based on a sugya in Chagigah 3b–4a:

“Our Rabbis taught: Who is deemed insane (shoteh)? One who goes out alone at night, and one who sleeps in a cemetery, and one who tears his garments. It was stated: Rav Huna said—only if he does all of them at once; Rabbi Yohanan said—even if [he does] one of them. What are the circumstances? If he does them in a manner of insanity—then even one [should suffice]; and if he does not do them in a manner of insanity—even all [three] should not [suffice]!—Rather, indeed he does them in a manner of insanity; [yet] as for sleeping in a cemetery—say he did so in order that an unclean spirit rest upon him; as for going out alone at night—say a ‘gandrifas’ seized him; as for tearing his garments—say he is a man deep in thought. But once he has done all of them, it is like an ox that gored a donkey and a camel [on three separate days], and it becomes forewarned regarding all.”

According to Rav Huna, to declare someone a shoteh one needs all three signs: going out alone at night, sleeping in a cemetery, and tearing his garments. Why is one sign not enough? Because each one by itself can be explained otherwise even if the person is not insane. For example, someone who sleeps in a cemetery at night may be seeking to have an unclean spirit rest upon him, and not necessarily because he is insane. And one who goes out alone at night may have been seized by some spirit (“gandrifas”). And one who tears his garments may have been engrossed in thought and not noticed.

So why not declare a person insane when there are two signs? Because each of the two events can be given a different explanation, and again there is no necessity to say the person is insane. For example, if he goes out alone at night and also sleeps in a cemetery, perhaps the first event was because he was seized by a spirit and the second because he wanted an unclean spirit to rest upon him. Well then, why does Rav Huna hold that if all three signs occur he is indeed insane? Seemingly, one could still explain each event with its own explanation. Here, says R. Chaim, that is no longer reasonable. Three such events are not a mere coincidence.

In essence we have two interpretive options for these three events: (1) He is insane, and therefore he did all three things without any rational reason; (2) He is not insane, and each event occurred for some other (supposedly rational) reason. Option (1) is preferable to (2), because it offers a single explanation for all three events. Occam’s razor says that the simpler explanation is preferable. So why is someone who did only two of the things not deemed insane? There too Occam’s razor applies, albeit with a less sharp blade. It seems that there the advantage of a single explanation over two separate explanations is not sufficiently decisive to prefer it and declare the person insane. One needs the preference to be clear—hence three signs are required.

This appears to be the underlying explanation for all “three-times” presumptions. For example, regarding menstrual cycles, one could say that in each month the woman happened to menstruate on the sixth, but each time it was accidental. If it occurred three times, it is unlikely to be accidental; it is preferable to assume a single explanation for all three events: apparently she has a tendency to menstruate on the sixth of the month. The same holds for an ox’s goring. If it gores three times, it is unlikely that each time some different factor caused it; it is more reasonable that it has a goring nature, and therefore we conclude that it is a goring ox, i.e., mu’ad. And so with all three-times presumptions. Indeed, at the end of the Chagigah sugya cited above, the Gemara brings support for Rav Huna from the process of rendering an ox forewarned when it gored different species on different days, leading to the conclusion that it is forewarned with respect to all species (see the second part of Column 308). These were not mere coincidences but manifestations of a general law—or more precisely, of the ox’s nature. Thus the Gemara itself hints at R. Chaim’s explanation.

I would note that in that column I discussed a later halakhic inquiry regarding three-times presumptions: whether they are evidentiary (re’ayah) or habituation (hergel)—that is, whether the three gorings indicate that the ox has a goring nature, or whether the three gorings accustom it and thereby make it goring. Later authorities debated similarly for all three-times presumptions. In the discussion here I will assume they are evidentiary presumptions, not habituation, for that is the straightforward reading.

Note that in all these cases a generalization is made on the basis of a sample. We observed some event three times in varying circumstances, and from this we infer a general law. The assumption is that the three events were not mere coincidence but an expression of a general law that applies in all such situations. The three-times presumption is thus based on a scientific-style generalization from a sample, and the product of the generalization is a general law of nature—just like in the scientific process.

The claim of Mekor Chaim: The logic of generalization

In the book Mekor Chaim, Orach Chaim §467:5, he discusses a case where, on Pesach, three wheat kernels in our dough were found split—i.e., leavened. Among his words he writes the following:

“Therefore it seems to me that we say a three-times presumption only where there is some causative factor such that it is reasonable that it be so and the comparison of the intellect requires it to be so. But in a matter that occurs by chance and there is no such reasonable causative factor—there is no presumption for chance occurrences.”

That is, he maintains that we do not create a presumption merely because any three cases recurred. The generalization is made only if we have a plausible common factor that explains the three cases as expressions of one general law. In the absence of such a factor, the three cases do not yield a general law—that is, we do not perform an induction.

His proof comes from a sugya in Yevamot. The Mishnah there (64a) states:

“If a man married a woman and remained with her ten years and she did not bear—he may not refrain [from procreation]. If he divorced her—she is permitted to marry another, and the second may remain with her ten years. And if she miscarried—he counts from when she miscarried.”

If the woman did not bear children after ten years, they must divorce. But another man may marry her and try again to have children with her. After ten years without children, he too must divorce her. What now? May a third man marry her? The Gemara (64b) states:

“‘If he divorced her—she is permitted, etc.’ A second—yes; a third—no. In accordance with whom is our Mishnah? [It is in accordance with] Rabbi; for it was taught: If the first [child] was circumcised and died, and the second [was circumcised] and died—one does not circumcise the third; these are the words of Rabbi. Rabban Shimon ben Gamliel says: One circumcises the third; one does not circumcise the fourth.”

That is, a third husband is forbidden to marry her, for she has the presumption of an aylonit (a woman who does not bear). This is Rabbi’s view, as seen in the circumcision case: after two circumcisions where the infants died, one does not circumcise the third. According to him a presumption is formed after two occurrences, whereas according to R. Shimon b. Gamliel—after three. Hence the Gemara infers that the Mishnah follows Rabbi.

Further the Gemara notes:

“Moreover, say that they [Rabbi and R. Shimon b. Gamliel] disagree regarding circumcision; do they disagree regarding marriage as well? Yes; for it was taught: If she married the first and he died, the second and he died—she should not marry a third; these are the words of Rabbi. R. Shimon b. Gamliel says: She may marry a third; she should not marry a fourth.”

That is, we see that their dispute applies both to circumcision and to marriage. Now the Gemara wonders about the logic:

“Granted with respect to circumcision—there is a family in which the blood is thin, and a family in which the blood congeals [quickly]; but what is the reason concerning marriage?”

Regarding circumcision we have a common explanation: perhaps the blood in that family is thin, and therefore the bleeding continues after circumcision and the infant dies. If this is a family trait, it is clear why we assume it will occur with the next child as well. But in marriage (i.e., a woman whose three husbands died, and the question is whether a fourth may marry her or must fear that he too will die), there is no common logic to the three husbands’ deaths—thus it is not reasonable to assume that the fourth will die. Without a reasonable explanation common to the three cases, we do not generalize—precisely as Mekor Chaim wrote.

The Gemara then offers two possible explanations for the husbands’ deaths, so that even there we should generalize and create a presumption:

“Rav Mordechai said to Rav Ashi: Thus said Avimi of Hagrunya in the name of Rav Huna—‘the spring [= her bodily fluids] causes it’; and Rav Ashi said—‘her constellation (mazal) causes it.’ What is the practical difference between them? A case in which she was only betrothed and [the husband] died; alternatively, [a case where] he fell from a palm tree and died.”

According to Avimi of Hagrunya, the woman’s bodily fluids caused the husbands to die. According to Rav Ashi, her “star-luck” (astrological fate) caused it. The practical difference would be a case where the man was only betrothed (no consummation) and died—then bodily fluids could not be the cause; or where the husbands died by falling from a tree—again, one cannot attribute it to bodily fluids.

Again we see that only if there is a plausible explanation that threads the three cases together do we perform the generalization.

Indeed, the Mekor Chaim writes the following:

“A clear proof of this is from Yevamot 64b regarding a ‘killer woman’ (katlanit): according to the view that ‘the spring causes it’ there is no din of katlanit in a betrothed woman; similarly there is no din of katlan regarding men. It follows that wherever there is no plausible causative factor—there is no presumption.”

This Gemara is proof for his principle: not every recurrence of three cases constitutes a basis for generalization. Only if we have a plausible common explanation for the three do we generalize. In other words, Hume’s worry about hasty and invalid generalizations receives an answer here: only when a generalization has an a priori rationale do we perform it (needless to say, Hume the empiricist challenges even this; see below).

However, on the face of it, the sugya yields the opposite conclusion. Note that Rav Ashi attributes the presumption to the woman’s mazal. The influence of the stars that determine her fate causes her husbands’ deaths. In effect, Rav Ashi seems to pull out a joker: whenever we lack a plausible natural explanation, we attribute it to the influence of the stars (mazal). If so, he empties Mekor Chaim’s principle of content, since whenever we lack an explanation we can resort to mazal. According to him there is no situation in which a triple recurrence would not lead us to generalize. Moreover, the decisors (Rambam and Shulchan Aruch) rule like Rav Ashi (for they also write that if three fiancés died during betrothal one should not become engaged to her again, and the Gemara explains this according to Rav Ashi). Thus, while according to Avimi the result aligns with Mekor Chaim, the halakha follows Rav Ashi, and on his view we generalize in every triple recurrence—either on the basis of some natural explanation, and if none is found, then on the basis of mazal.

The difficulty with this conclusion

This conclusion is highly implausible. Did Rav Ashi truly intend a universal joker that solves every difficulty—a catch-all explanation for every triple recurrence? If so, we would have to worry in every situation, for one could always stitch together three precedents to create a presumption that threatens us. For example, suppose on the third Sunday of every month a person dies in a traffic accident on a main street. Should we therefore close all main streets in all cities every third Sunday? It is quite clear this is mere coincidence. More generally, it is a fact that from any three cases one can construct a general rule (see Column 482 and much more on Wittgenstein’s “following a rule”; see also the next column). It is thus utterly unreasonable to accept a position whereby every triple recurrence creates a presumption. Mekor Chaim is certainly correct.

How then should we understand Rav Ashi’s view? It seems that when he says “mazal causes it,” he does not mean to pull out a joker. In the ancient world—and thus in the world of Hazal—mazal was a concrete scientific principle, and in their view there are circumstances where one attributes events to mazal and circumstances where one does not. Therefore, even according to Rav Ashi one needs an explanation in order to generalize; mazal is merely one possible explanation. There will be situations where an explanation based on mazal will not seem correct, and there we will indeed not generalize. Not in every case can the recurrence be attributed to mazal. Mekor Chaim certainly understood it this way, since otherwise the Yevamot sugya would be a proof against him, not for him. It is clear that in his view split kernels of wheat in this dough are a coincidental recurrence—i.e., one cannot attribute them to mazal—and therefore we do not apply a three-times presumption there.

Bottom line: generalizing from observed cases to a general law parallels scientific induction. Hume’s problem of induction is present here, since the generalization is speculative. This is precisely why we should not generalize unless there is a reasonable logical basis for it. Of course, Hume will attack even such generalizations, since the fact that something seems reasonable to us does not mean it truly is. This is a speculative hypothesis, and an empiricist like Hume does not view it as a valid claim.

A note about the influence of mazal

Seemingly, what we have seen here contradicts the common view in Hazal and commentators that “Israel is above mazal”—that there is “no mazal for Israel,” meaning they are not subject to the governance of stars and constellations. How does this comport with Rav Ashi’s view (which was also codified) that we attribute events that happen to us to mazal?

It seems that when Hazal say “there is no mazal for Israel,” the meaning is that they are not entirely delivered into mazal’s hands. It does not determine what will happen to them in a strictly deterministic manner. But that does not mean it has no influence on them. In Hazal’s world, mazal is a natural factor that affects us just like physical fields (electric or gravitational). The claim that Israel literally has no mazal, in their world, is roughly like saying there is no gravity for Israel—that the law of gravity does not apply to Jews. It is more reasonable that their intent is that mazal does indeed influence Israel just as it influences the entire world. But Israel has the power to overcome mazal, at least in those situations where we choose freely. In our natural conduct we are no different from ordinary material entities, and we are subject to mazal as to gravity. Just as the law of gravity operates upon us because we have mass, so mazal operates upon us and can even kill us. In those points governed by our free choice, there mazal merely influences but does not determine (on the difference between influence and determination, see Column 175 and much more—everywhere I discussed the topographic map of the soul).

But of course this raises the question: why are gentiles subject to mazal? They too have free choice. It seems the difference between Israel and the nations is specifically in situations where we do not have choice. The nations are fully given over to the influence of mazal, whereas Israel can pray and be saved from mazal’s influence. Wherever a person chooses, there he is not subject to mazal—whether Jew or gentile.

If so, the husbands who married that woman could have prayed and perhaps been saved (and perhaps not; even for Hazal not every prayer is necessarily answered). Yet if we see three such husbands who fell from a tree to their death, apparently there was a mazal at work. In such a case, one does not take risks: although a fourth husband could pray and if answered might be saved, he is not permitted to rely on that and marry her. That is a risk not to be taken. The fact that three husbands died does not mean the fourth will certainly die; it does mean that there is a significant danger, and therefore he is forbidden to take the risk. More generally, it is important to understand that all three-times presumptions in Halakha are not positive conclusions but concerns: once something has happened three times, a well-founded concern is created. Hazal do not assume a positive conclusion that the fourth husband or fourth son will die, that the forewarned ox will certainly gore, or that the woman will certainly see blood on that date. After a triple recurrence, the concern that it will happen again is sufficiently grounded, and we may not ignore it. We will return to this point below.

All this is, of course, within Hazal’s framework. In my judgment there is no such thing as the influence of stars and constellations on anyone. These are ancient folk beliefs—the science of old—that also influenced Hazal’s thinking. Therefore, de facto, I would not prohibit a fourth man from marrying the woman even if three of her previous husbands died. True, the Talmud has formal authority in Halakha, but I have explained more than once that when a halakhic ruling rests on a factual/scientific error, it has no force and is void even without a later court nullifying it (see, e.g., Column 667). See more on this below.

Let us now return to the question of generalization. We saw that we generalize only if there is reasonable logic at its foundation—i.e., if we have a rational explanation that threads the three cases we observed. I now wish to show the flip side: sometimes the very logic in favor of generalization actually undermines it and leads to error.

The law of small numbers

In Column 38 I brought several examples reflecting what Daniel Kahneman called “the law of small numbers.” We saw there that when working with a small number of examples one must beware of generalizations drawn from them. The reason is that in small samples the result can be very far from the average outcome—that is, from the general law.

For example, a survey once found that the schools with the best achievements were small schools. The obvious explanation suggested itself: in a small school one can give personal attention to each student, and clearly the student will be better nurtured. But later it was found that the schools with the worst achievements were also small schools. It turned out the explanation was much simpler than we thought. Suppose for argument’s sake the results are normally distributed (Gaussian)—a bell curve—with most schools near the mean and a few far out in the tails. By the law of large numbers, large schools will cluster around the mean (each contains many students, so the school’s average score will be roughly the national average), whereas small schools will appear at all parts of the curve. Therefore, when we look at the very best and the very worst achievements (the far tails of the Gaussian), we will find only small schools there.

A similar thing happened in surveys examining kidney cancer incidence: the places with the lowest incidence were small places. At first, people thought the explanation was that small places are quiet and rural and thus healthier. But on further examination they discovered that the places with the highest incidence were also small. The explanation is, of course, the same. At both extremes of the incidence curve you will always find small places.

Why did researchers jump hastily to the initial conclusions in both cases? Because they had a convincing explanation—a good story that persuaded them that, indeed, small places are healthier and small schools are more successful. The conclusion is that one must beware of biases stemming from our a priori reasoning. In Column 537 I gave another example of this fallacy. A professor from the Technion wrote to a newspaper’s letters section that Israel must increase investment in higher education, for global data show that investment in higher education correlates with higher GDP. His conclusion: if one invests in higher education, GDP rises. But he inexplicably ignored an alternative interpretation of the same data: countries with high GDP have the money to invest in higher education. What led him to infer the first conclusion and ignore the second? Perhaps personal interest (he works at the Technion), and likely also confirmation bias. He had a good story behind the conclusion, and that convinced him this was the correct interpretive direction.

Kahneman’s conclusion is that we must be very careful about generalizations based on an explanation that seems a priori convincing. That very sense of conviction biases us and leads to incorrect generalizations. Seemingly, this contradicts what we saw above—that there is room to generalize from a small sample only if it has a convincing explanation. So we must ask: does the existence of an a priori explanation help our generalizations or harm them?

Explanation and a return to Halakha

On further reflection there is no real contradiction. The conclusion is that generalizations must be made on a solid statistical basis, but one should not disparage our a priori assumptions either. Once we have data and a statistically significant conclusion (i.e., after examining both high and low incidence, both high- and low-achieving schools), it is proper to seek an explanation for the findings. We must be very wary of confirmation bias—our tendency to leap to conclusions that confirm what we already thought and to look at only those findings that suit us (see Column 663 on “cherry-picking”)—or to forgo serious statistical testing. That is Kahneman’s correct conclusion. But in the absence of statistical information, if we have several cases in which we noticed some phenomenon, it is reasonable to assume these are instances of a general phenomenon. This is the three-times presumption in Halakha. As we shall see immediately, the presumption merely assumes there is an explanation; it does not necessarily propose what it is.

How will we reach a conclusion as to what, precisely, the general phenomenon is that emerges from the sample? That is another question. For that we must employ our a priori reasoning, and from among the possibilities it raises we should choose one based on regressions and other statistical tests. Thus, in the schools case, it was reasonable to assume at the outset that school size is relevant to achievement, since reason suggests a connection. But it was important to test the statistical significance to see whether the data truly support that conclusion. If indeed the data had shown that small schools appear only on the right side of the Gaussian, it would have been correct to infer that small schools achieve better. But a serious examination shows that is not the case. Hence, there one should have abandoned the a priori assumption. So far, Kahneman’s warning.

Now consider what we should conclude if we discovered that all high-achieving schools have names beginning with a letter between Alef and Het. I presume we would all agree that here one should not generalize and infer conclusions about changing school names. Why not? Because, a priori, it is not plausible that a school’s name is relevant to its achievements. In such a case, we would prefer to assume this is coincidence—precisely as Mekor Chaim does. In other words, our a priori assumptions have an important role in statistical inference, even though of course one must be cautious with them. Respect them—but suspect them. I expanded on this phenomenon in Column 715, where we saw that intuition (our a priori reasoning) guides us toward the correct generalization, and without it there is no possibility of making a generalization. The notion that the process of generalization is objective and done solely on the basis of data is a misunderstanding.

Note that in the time of Hazal and the early authorities they did not, of course, have these statistical tools; therefore they required statistical significance—i.e., recurrence of at least two or three times—and when that occurred they sufficed with an a priori rational examination to conjecture what the explanation could be (what the correct generalization is). Note also, as I remarked above, that in all these cases it is a matter of establishing a presumption that we must be concerned for, not a positive conclusion. Hazal did not conclude that the fourth husband will also die, but rather that there is sufficient basis for concern that he will die to forbid him from marrying her. The attribution to the woman’s bodily fluids or to her mazal is not a positive determination but merely a possibility indicating that there is room for such concern. It is a possible explanation. Thus, in all these cases, it is not a scientific conclusion but the formation of a balanced concern in light of the data. This is certainly a logical and very rational path—at least in the absence of advanced statistical tools.

In the second part of Column 308 you will find an example of analyzing the logic underpinning a generalization, and of elimination as a rational method for winnowing among various possibilities.

Conclusions for our time

Today we have such statistical and scientific tools. Therefore, in my view, Halakha today should change and give greater weight to systematic statistical examination and to scientific knowledge (which itself is obtained through statistically significant methods), and not rely blindly on three-times presumptions. Note that this is not said only about the inability to determine the general law, for the three-times presumption does not determine a law but creates a concern. My claim is that even to create a concern it is insufficient.

For example, if three husbands of a given woman died in different ways, I would not prohibit a fourth husband from marrying her—at least as long as he himself wishes to do so. A priori, I do not believe in the influence of mazal; therefore I would not regard those events as significant. In effect I would say “her mazal caused it,” but not in the sense of the stars’ influence—rather in the contemporary sense of the word mazal, i.e., sheer randomness. There you have the influence of an a priori stance.

Some may accuse me of conservatism and mental rigidity, since the data stand against me and I cling to my a priori views—just like in Kahneman’s fallacy. In response I would ask: what would you say about a case in which successful schools are characterized by the first letter of their name? I assume that there you would not generalize on the basis of the results you found. Is that mental rigidity as well?

My claim is that it is not. In the next column I will offer a logical explanation for this.