The Anna Karenina Principle (Column 286)
From Tolstoy to Diamond: domestication, empires, and families
The essay opens with Tolstoy’s motto about happy and unhappy families, and shows how Jared Diamond gave it a systematic meaning: animals that can be domesticated, or societies that manage to survive, share a narrow set of necessary conditions, whereas failures split into many different forms. The successful group is therefore smaller and more uniform, while the failed group is larger and more varied.
Conjunction versus disjunction: the logic that generates the probability
The rabbi stresses that this is not primarily a probabilistic principle but a logical one: if success depends on a and b and c together, then failure can result from the absence of a or b or c. Only after understanding that structure can one say that, probabilistically too, in a random draw of situations it is more likely to get mismatch than fit; from here also comes the link to entropy and the second law of thermodynamics.
A Brisker tool: begin with the more specific position
From here the essay derives a Brisker analytical tool: in a Talmudic dispute one should begin with the position that assumes more premises. If one view rests on several cumulative conditions, then the opposing view can immediately be mapped by the condition or conditions it gives up, and thus the tree of possibilities in the sugya is built.
Why we instinctively treat happiness as the special case
The essay admits that the principle by itself does not determine that happiness, specifically, is the side that requires a conjunction of conditions; one could imagine the reverse as well. Still, the intuition is that success is a rare fit between a person and his circumstances, and so when circumstances are effectively drawn at random, misery is more likely. In the same spirit, when one finds a pair of opposed phenomena, one more uniform and the other more diverse, it is worth looking for a logical structure of this kind beneath them.
The principle governs lotteries, not necessarily designed systems
At this point comes an important qualification: the principle does not imply that in practice most societies collapse or that most spacecraft crash. When human beings design a society or a spacecraft, the event is not random; good design pushes the system precisely toward the narrow zone of success. So the principle predicts a multiplication of failures mainly when we are dealing with a lottery, or with something close to a lottery.
The theological proof in the form the world fits us seems natural — but fails
The essay shows why it is tempting to formulate the physico-theological proof from here: if a world suitable for human life is a special case, then the very fit seemingly points to a guiding hand. But here the anthropic objection enters: the human being is not a fixed datum against which circumstances are drawn, but a product of those very circumstances. In a different world, different creatures would have emerged, so there is no reason to marvel that our world suits us in particular.
The more precise formulation: not fit to us but complexity as such
The rabbi therefore suggests that the correct formulation of the proof, and apparently also the one intended by Hovot Halevavot, does not rest on the Anna Karenina principle but on the claim that a complex, composite thing does not make itself. Suitability for human life can serve as an indicator of complexity, but not as the proof itself. The conclusion is that the Anna Karenina principle remains entirely valid; what fails is only an application that ignores dependence, correlation, and the order of events.
With God’s help
A few days ago my son sent me an article that deals with the well-known two-envelope paradox in probability theory, entitled, in the Ishmaelite language, "Anna Karenina and the two envelope problem". The intriguing title of this article led me to look into what the Anna Karenina principle is, and then to think about paradoxes in general and the two-envelope paradox in particular. In this column I will present the Anna Karenina principle, and in the next column I will turn to the two-envelope paradox and the above-mentioned article.
The Anna Karenina Principle
The famous opening line of Leo Tolstoy’s Anna Karenina is quoted in many places:
All happy families are alike; each unhappy family is unhappy in its own way.
In the past (column 17) I wondered how true this is, or whether perhaps it is one of those sayings that sounds profound but, on a second look, turns out to be mere verbiage. My conclusion there was that "there is, after all, something to it" (Alterman, There, There).
And now it has become clear to me that there is a principle called the Anna Karenina principle, and it offers a probabilistic interpretation of this sentence. This principle was formulated by the historian and physiologist Jared Diamond, in his book Guns, Germs, and Steel, when he wrote in the context of animal domestication:
All animals suitable for domestication are alike; every animal that is hard to domesticate is hard in its own way.
The basic claim is that, for an animal to be domesticated, it must possess several very specific traits. The absence of even one of them excludes it from the class of animals fit for domestication. Therefore it is clear that the population of animals that can be domesticated will consist, more or less, of the same types and probably in small numbers, as opposed to the population of those that cannot be domesticated, which will be more varied and richer than that of the domesticated ones. Diamond repeated this logic in another of his books, Collapse, which deals with the success and failure of human societies (such as empires). You can readily imagine that a successful society requires several conditions, and the absence of any one of them leads to an unsuccessful society (that is, to a society’s collapse). See here for many additional implications and applications of this principle (spacecraft crashes and more).
I will now offer a few comments on this principle.
Note A: Probability and logic
The connection to the two-envelope paradox may suggest a probabilistic connotation for this principle. But this phenomenon is, at root, not probabilistic but logical. If a claim X is based on a conjunction ("and") of conditions: 
then its opposite is based on a disjunction ("or") of the negation of each of them:
To be sure, this logical phenomenon has a probabilistic implication. The diversity of failures as against the scarcity of successes is a logical consequence, but from this it follows that if we randomly generate hypotheses, or circumstances, chances are that we will get unsuccessful and unsuitable circumstances (that is, not special ones). This is the second law of thermodynamics, according to which in a closed system entropy, that is, the amount of order, does not increase. For a system to become more ordered, a guiding hand is needed.
Note B: An implication for Talmudic analysis
This principle has an implication for Talmudic analysis. I repeatedly tell my students that when analyzing a Talmudic dispute (between tannaim, amoraim, or commentators), one should begin the discussion with the more distinctive opinion, that is, the one that assumes the greatest number of premises. After we have found that one sage holds X, and the claim X is based on the fulfillment of all the assumptions a, b, and c, we can immediately say that there are at least three possibilities for explaining the position of the sage who disagrees with him: either he disputes assumption a, or b, or c.[1] In this way you will quickly and easily arrive at the Brisker tree of possibilities for the passage, and you will be able to characterize each of the positions of the medieval authorities (Rishonim) and place it on that tree. Note well: this is a simple but powerful tool of conceptual analysis.
Back to Tolstoy
For our purposes, the probabilistic interpretation that Diamond gave to Tolstoy’s motto is that happy families require the existence of a cluster of several parameters, and the absence of one or more of them causes a family to be unhappy. Therefore happy families resemble one another (they have the same characteristics), whereas unhappy ones can be of several different kinds. Mark this well. This is an excellent example of a principle that strikes us as intuitively correct, but it is hard to understand why that should be so; in logical or probabilistic analysis it receives a simple explanation (cf. evolution).
Note C: A non-probabilistic premise
But of course this is not the end of the story. One may ask why the more distinctive side (the one that requires the fulfillment of a cluster of conditions) is specifically happiness rather than unhappiness. Why not assume that specifically unhappiness requires a cluster of conditions, whereas happiness is produced by the absence of one or more of them. There is an intuition that the former is correct, because happiness is a kind of success, and that should be a special case, reached only rarely (and therefore usually deliberately); if things are left to luck, then in this lottery it is specifically unhappiness that will usually result. Put differently: for a person (or a family) to be happy, circumstances must obtain that suit him in one way or another. If we randomly draw circumstances, it is likely that most of them will not suit him, and therefore he will be unhappy in them. Let our cousins in Brazil, who advocate the manyana method, take note (which brings us back to column 17).
It may be worthwhile to use this principle in the opposite direction. When we see two opposing phenomena, one diverse and the other more uniform, it is worth looking for characterizations of the two groups such that they fit the Anna Karenina principle. By this I mean that one should try to extract from their overall characteristics a logical structure of the kind described above. The more uniform population is characterized by a cluster of features, whereas the more diverse population is composed of groups, each of which lacks one or more of the features of the other group.[2]
Note D: Probabilistic considerations in a non-random event
Now the question naturally arises: must the number of collapsing societies necessarily be greater than the number of successful ones? Not at all. The reason is that the success or failure of societies depends on their structure, and the structure of a society is not random. People plan societies according to some scheme that is supposed to take into account their ability to function and succeed. True, sometimes they fail, but that is only because the planning was deficient (or because of a higher power). If we were simply to randomize social structures, then this principle would dictate that most societies created in that way would indeed collapse. Likewise, most spacecraft do not crash, because the people who design spacecraft are experts. Spacecraft that were built by children would probably crash (because building by a non-expert is an almost random event).
This distinction between planned and random events underlies the physico-theological argument for God’s existence.
Note E: Implications for the physico-theological argument – the author of Duties of the Heart
At first glance, this analysis can be applied to the physico-theological argument. For example, in the formulation of the author of Duties of the Heart, "The Gate of Divine Unity," chapter 6:
But the way in which we make use of the premises we have mentioned in proving the existence of the exalted Creator is this: when we reflect on this world, we find it joined and compounded; no part of it exists without connection and order. For we perceive it with our senses and our intellect as like a built house, in which everything needed for it has been prepared: the heavens above like a ceiling, the earth spread out like a floor-covering, the stars arranged like lamps, and all physical things gathered within it like stored treasures, each thing for what is needed from it; and man is like the master of the house, making use of all that is in it. The various kinds of plants are prepared for his benefit, and the various kinds of animals serve his enjoyment, as David, peace be upon him, said: “You have given him dominion over the works of Your hands; You have placed everything beneath his feet: sheep and oxen, all of them, and also the beasts of the field, the birds of the heavens, and the fish of the sea, whatever passes through the paths of the seas.” The orderly rising and setting of the sun establish the times of day and night, and its ascent and descent produce cold and heat, summer and winter, the various seasons and their benefits, and their constant recurrence in a single fixed order without interruption, as it is written: “Who speaks to the sun and it does not rise, and seals up the stars,” and it is written: “You make darkness, and it is night,” etc. And the revolutions of the spheres, whose motions differ, and the stars and constellations, proceed under measured governance and precise proportion, not swerving and not changing. The purpose in every one of these things is a purpose of benefit and order for rational beings, as Solomon, peace be upon him, said: “He has made everything beautiful in its time; He has also placed the world in their heart,” etc.; and he said: “For everything there is a season, and a time for every purpose under heaven.” All this—its joining and composition—is visible both in the whole and in its parts. And when we examine plants and animals, we find that they are composed of the four elements—fire, air, water, and earth. These are divided and separate, and we have no power to combine them in their natural combination, because some of them are transformed into others and some oppose others. And if we do combine any of them, they quickly change and alter. But the composition joined by nature is a well-ordered and enduring composition, lasting until its appointed end. Some philosophers have thought that the spheres, the stars, and the higher beings are derived from the element of fire, and similar to this is what David, peace be upon him, said: “He makes His angels spirits, His ministers a flaming fire.” There is in this some support for that view, and not that there is a fifth element, as Aristotle thought. Since existing things come to be from the elements and are compounded from them, and we know that they did not mix by themselves and were not joined together by their own nature, because of the opposition among them, it arose in our minds and became clear to our souls that the One who joins them is other than they, who binds them without their own agency and composes them against their nature and despite themselves—He is their Creator, may He be exalted, who ordered their bond and established their composition.
At first glance, he seems to argue that we have a world that suits us and our lives in every respect, and therefore it is unlikely that it came into being on its own. There is a guiding hand that created it. In other words, if we randomly generate different worlds, there is a very small chance of getting a world suited to human life (because according to the Anna Karenina principle, the number of successful worlds is far smaller than the number of unsuccessful ones). This is the theological argument that was originally called the "anthropic principle".
However, this formulation of the argument suffers from a probabilistic fallacy. It is true that once a person of a certain constitution is given, and we now randomize the circumstances that will prevail around him, the chance of drawing suitable circumstances is small. But the person is himself a product of those very circumstances. Another world with different circumstances would produce different creatures that would exist within it. Therefore the fact that circumstances prevail around us that suit us proves nothing. This is the objection that is nowadays called the "anthropic principle" (in its atheistic formulation).
In the third and fourth notebooks (and in the third and fourth conversations in the book The First Existent) I argued that the more precise formulation of the argument is not based on the suitability of the world for human life, but on the complexity of the world as such. The claim is that a complex thing does not arise on its own. Incidentally, from a close reading of chapter 5 of Duties of the Heart there, it appears that he probably meant precisely this more accurate formulation:
But the premises from which it can be demonstrated that this world has a Creator who created it out of nothing are three. The first is that a thing does not make itself. The second is that a series of beginnings must have a finite number, and once their number is finite, they must have a first, before which there is no prior first. The third is that every composite is originated. Once these three premises are established, the conclusion follows for one who knows how to use and combine them: that the world has a Creator who created it out of nothing, as will become clear from what we are about to explain, with God’s help.
Here he speaks about complexity (the composite that does not compose itself). Incidentally, it seems that his words here combine the cosmological and the physico-theological arguments.
One may ask which of these two formulations fits the Anna Karenina principle. At first glance, it is specifically the first one, that is, the fallacious one. Since a person’s success requires quite a few conditions, the circumstances that make his existence possible are special, and their number is presumably far smaller than the number of circumstances that would not make it possible. But as noted, this is a fallacious argument. Of course, that does not mean that the Anna Karenina principle is fallacious. It is a correct principle: given a particular creature, the number of circumstances that suit its life is far smaller than the number of circumstances that do not suit it. This is the logical dimension of the Anna Karenina principle. Therefore, on the probabilistic plane, if we now randomize the circumstances within which it operates, we will probably arrive at unsuccessful circumstances (that is, miserable life, if any). And if the circumstances we have received are suited to human life, one may seemingly derive from this the conclusion that there is a guiding hand. From here one would seemingly infer that the creation of the world was not a lottery (a closed system in which order does not increase and uniqueness can only disappear), but something like a factory or a spacecraft—that is, a planned creation by an intelligent being (who is involved within the system, and therefore it is not closed).
What fails in the first formulation, which as noted uses the Anna Karenina principle, is not the principle itself—neither the logic nor the probability. As we have seen, the argument also uses the distinction between directed and random events. The problem lies in the analysis of reality that precedes the analysis. The randomization of the circumstances is not carried out independently of the creature that lives within them, and therefore it is wrong to marvel at the fit between them. The probabilistic consideration fails here because of the order of events and the correlations that are ignored. Mark this well.
The second formulation, by contrast, apparently does not use the Anna Karenina principle at all. We are not speaking about suitability to certain conditions, but about complexity pure and simple. Admittedly, suitability for human life is evidence of complexity. Moreover, the very existence of human beings is itself a kind of complexity that would not be expected in a system that came into being through a random process (a lottery). I think that the author of Duties of the Heart, who uses suitability for human life, does not mean the first formulation and the Anna Karenina principle, but rather the second formulation, that is, the correct one. This is a probabilistic consideration without the logical component, and it has nothing to do with the Anna Karenina principle.
[1] Why not two of them? Because of some version of Ockham’s razor, which in the language of the sages is formulated as follows: We do not proliferate disputes. ("we do not multiply disputes"). On this matter, see my article on the dichotomy of yeshiva-style inquiries.
[2] In this context it is worth looking at the methodological preliminaries and the analysis of poetry versus prose, in columns 107–113.
Discussion
In paragraph 2, line 2
… but he will not lose heart because he has not attained everything he wanted, …
Shatz, what you write merely moves the discussion inward, into the soul. But the logic remains intact. Such a successful soul is exceptional, and other souls will not feel that way.
Thank you very much.
Can you explain in more detail what the probabilistic fallacy is when people speak of the fit between man and the world, which disappears when people speak of the world fitting itself?
Do you mean that when one speaks of the fit between man and the world, theoretically there were many attempts to create life, only one of which succeeded, and therefore that is the only one we see? Could you elaborate?
I seem to recall there was a long polemic here about this.
Correction: in the third passage from the end (s.v. “One can ask”) — ‘as stated’ instead of ‘as donkicated’ (the small number of mistakes shows that there is a guiding hand here working to nullify the ‘Anna Karenina’ mechanism 🙂 ).
And unlike the domestication of animals, where the capacity to be domesticated depends on certain natural traits, the ability to attain inner happiness independent of external conditions does not depend on natural traits, but on a person’s decision and willingness to invest effort and persevere.
What may perhaps be correct is to define Anna Karenina’s principle through an inversion: one who emphasizes in his consciousness how he is similar to others, the ‘common denominator’ between himself and others, will find it easier to reach happiness, understanding that he is not alone in his difficulties. Everyone experiences one kind of difficulty or another. Such a person will be more patient and accommodating, and as a result happier as well. By contrast, one who is self-absorbed sees all his difficulties as mountains and is full of envy toward others—and then he truly is more miserable.
In short: the more a person connects to others, identifies with their difficulties, and tries to improve things for them, the happier he indeed will be!
With blessings, Shatz
The claim is that man, in his structure, came into being as a result of the conditions in the world, and therefore those circumstances and conditions cannot be treated as an independent variable relative to the structure of man. He was formed in adaptation to the conditions in the world (because everything that was not adapted did not survive).
Thanks for the correction.
Thank you very much.
Why can’t one say that the very complexity of the world expresses the principle?
After all, one could say that only a limited number of conditions would sustain the complexity of the world, and if any condition were not met, this complex world would not exist..
Why is this specifically a probabilistic matter?
I didn’t understand the question
Why does the principle not apply in the second formulation? You wrote that it is probabilistic without the logical part… It isn’t clear to me why this is a probabilistic consideration without the logic.
The logical consideration is what I described above here. Suitability for achieving a certain result requires many conditions, and therefore its negation is composed of a collection of many possibilities. That component does not exist when we are not speaking of suitability but of sheer complexity. Complexity raises a probabilistic consideration (assuming worlds are drawn at random, the chance of getting a complex world is small).
With God’s help, 27 Adar 5780
You don’t need a ‘cluster of parameters.’ To be happy, you need one parameter: ‘the ability to build happiness from within.’ Someone who thinks his happiness depends on one success or another will always find a reason for frustration. Something in their ‘conditions for happiness’ that they expected did not work out, and so they are resentful and disappointed. Sometimes this can even be a person whose life is good, but one small thing ruins him with the feeling that ‘all this is worth nothing to me.’
The moment a person adopts the approach that his happiness does not depend on acquiring this thing or that, but on his inner wholeness—he will manage and find something to rejoice in even with the little he has. He will focus on it and try to improve his situation, but he will not lose heart because he has not attained everything he wanted. In this parameter, ‘the ability to build happiness from within,’ indeed all happy people are alike; but the path to this insight passes for each person along his own unique road, strewn with quite a few crises and hesitations.
With blessings, Shatz