Paradox and Anti-Paradox – Examples (Column 196)

With God's help

In the previous column I presented two types of statements: paradox (a logical loop that does not allow a truth value to be assigned to a statement) and anti-paradox (a statement that can receive two truth values simultaneously). I mentioned there that two interesting associations occurred to me in connection with this topic, and I will present them here: Alexander in Afriki and Buridan's Ass.

Alexander in Afriki

In Midrash Vayikra Rabbah, Emor, parashah 27, we find a story about Alexander of Macedon:

He went to another land called Afriki. They came out before him with golden carrots, golden pomegranates, and golden bread. He said, "What, is gold eaten in your land?" They said to him, "And is it not so in your land?" He said to them, "I did not come to see your business dealings, but to see your justice." While they were sitting, two men came before the king for judgment. One said, "My lord king, I bought a carob tree from this man, and when I dug around it I found a treasure in it. I said to him: Take your treasure, for I bought the carob tree, but I did not buy the treasure." And the other said, "Just as you fear the punishment for theft, so do I fear it; when I sold you the carob tree, everything in it I sold to you."

He came to the land of Afriki and wanted to see the local king conduct a legal proceeding. Two men came before him: one had sold a carob tree to the other, and the latter found a treasure in it. He wanted to return the treasure to the seller, arguing that he had bought a carob tree and not a treasure. But the seller did not want to accept it, because he too feared theft.

Is this connected to our discussion?

What is the law in such a case, really? At first glance, this seems similar to the liar paradox and to the dispute between Rav and Shmuel that we saw above. Each side claims that the other is right. But in fact it seems more similar to an anti-paradox, since there is no loop here. The seller is not saying that the buyer is right, but rather that the buyer ought to keep the treasure (which means that the seller himself is in fact mistaken, since he claims that the other party should keep it). We have here two rulings, both of them consistent.

But even that is not precise, because in fact this is not a matter of two halakhic opinions (as with Rav and Shmuel) but of two claims by litigants. On the legal plane, apparently only one of them is correct, and we are simply trying to determine the law. Therefore, at the halakhic level, what each litigant claims is not really important. What matters are the facts and the Jewish law that applies to them.

So what is the law?

An interesting thought crosses my mind every time I encounter this midrash (what else would you expect from a Litvak?!): why does Jewish law not discuss such a case? What really is the law in a similar situation? It seems to me that this is no accident. Halakhic jurisprudence deals with a situation in which one person complains that someone has violated his rights, and asks a religious court to protect him. In such a situation the court must decide the law. In our case, however, this is not a matter for legal dispute, since there is no party here claiming that his rights were violated. On the contrary, each claims that the other was wronged. But if the allegedly wronged party himself does not agree that his rights were violated, then there is nothing to discuss (this is an admission by a litigant, or a waiver).

But that is still not enough. Even if a religious court is not supposed to deal with such a case, the two men genuinely want to know what they should do. This is indeed not a legal question, because no one's rights were violated here, but it is certainly a halakhic question that either of them can ask. They should not go to a religious court, but rather to a halakhic decisor who will tell them what is correct and what is halakhically required of them. So what is the Jewish law in such a case? One could say that the treasure reverts to the presumptive ownership of the original owner (mara kama), though one would need to discuss the issue of presumption without a claim, and also whether in such a situation this really counts as his not making a claim. After all, he is not making a factual claim, but expressing his halakhic view that the treasure belongs to the other.

But beyond the formal law, there is room to seek a practical solution, even if it does not really resolve the halakhic problem. This is as we saw regarding stipulating away the law of overcharging: the issue can be decided on the basis of doubt considerations even without solving the paradox.

What happened in the end?

Surely you are wondering what happened in the end. The midrash goes on to say:

The king called one of them and said to him, "Do you have a son?" He said, "Yes." He called the other and said to him, "Do you have a daughter?" He said, "Yes." He said to them, "Let these marry each other, and let the two of them enjoy the treasure."

The king suggested matching their children with one another and giving them the treasure. Thus we have a practical solution that bypasses the need for a legal ruling.

But Alexander was quite a Litvak:

Alexander of Macedon began to marvel. The king said to him, "Why are you astonished? Did I not judge well?" He said, "Yes." He said, "If such a case had occurred in your land…"

He wrinkled his nose when he heard the solution found by the king of Afriki. Though presumably this was not exactly out of concern for Jewish law and love of truth. When he was asked what he would have done in such a case, he answered:

"What would you have done?" He said, "We would cut off this one's head and that one's head, and the treasure would go up to the king's palace."

He would have taken off both their heads and brought the treasure to the royal palace. It seems that in Macedon there was not much room for overly righteous people. This is apparently the Macedonian version of judicial discretion (shuda de-dayyanei).

And to this the king of Afriki replied:

He said, "Does the sun shine in your land?" He said, "Yes." "And does rain fall in your land?" He said, "Yes." He said, "Perhaps you have small cattle in your land?" He said, "Yes." He said, "May that man's spirit burst! It is by the merit of the small cattle that the sun shines upon you and the rain falls upon you; because of the cattle you survive. As it is written (Psalms 36), 'Man and beast You save, O Lord'—man is saved by the merit of beast."

The king of Afriki solemnly informed Alexander that the only reason life continues in Macedon (the sun rises and the rain falls) is on account of the animals.

Buridan's Ass[1]

Another situation that resembles an anti-paradox is Buridan's Ass. Jean Buridan was rector of the University of Paris during the fourteenth century. He used his famous ass in order to discuss, and in fact to challenge, the concept of rationality.

Imagine a donkey standing at equal distance from two identical mangers, one on either side. Everything around him is perfectly symmetrical in both directions, and the assumption is that his body too possesses similar symmetry (physicists always say that one should first solve the problem for a point donkey, for the sake of simplicity, and only afterward move on to a real donkey). Our point donkey, then, stands resolutely facing the two mangers while happily nibbling grass, his face pointed exactly forward. After the grass in his vicinity is exhausted, he rests, and once he feels hungry again he wonders to which of the two mangers at his sides he should approach. Because of the symmetry, he sees no way to prefer one over the other, and so he stands in place, lost in thought, unable to reach a decision. In the story's tragic ending, our donkey dies of hunger. His clinically severe inability to decide where to turn costs him his life.

A paraphrase on Buridan: an American cartoon from the beginning of the twentieth century, expressing Congress's hesitation over whether to dig a canal through Panama or through Nicaragua.

This situation is very reminiscent of an anti-paradox. There are two solutions here, each of which is consistent (it achieves the goal), but precisely for that reason it is impossible to justify either one. There is no 'correct' solution here, and therefore, according to the first definition of rationality, which we shall call strict rationality, adopting either one cannot be considered rational.

What is rationality?

As noted, the parable of the donkey was originally intended to criticize concepts of rationality. The claim is that if the donkey insists on strict rationality, he will inevitably die of hunger. Why should this really be so? At first glance, this is obviously irrational behavior. If he really is a rational point donkey, then let him not be such an ass! Let him choose one of the mangers as he wishes, and eat heartily.

The concept of strict rationality that Buridan attacks assumes that every rational action must have a determinate rationale. Buridan's assumption is that a rational creature chooses to perform action A rather than B only if it has a genuinely good reason to perform precisely that action. But here our donkey has no good reason to choose manger A, and of course none to choose B either, and therefore a strictly rational donkey will die of hunger. The same is true of a strictly rational person who finds himself in such a situation: he too will die of hunger.

Why not draw lots?

In practice, of course, this will not happen, and that is precisely why Buridan offered this example. He wanted to show that the concept of strict rationality he attacked is apparently mistaken. A truly rational person in such a situation would draw lots and choose one of the mangers. And if we asked him why he chooses A rather than B, he would answer: because I want to live. A fairly good reason on the rational plane, no? Buridan argued something that sounds sensible and plausible to many people: that this is, in fact, true rationality. It certainly seems much more rational than the conception he attacked.

It is important to understand that this route is not strictly rational. One can provide a determinate explanation for conducting the lottery, since it is the only way for him to survive. But we have no way to explain why, after the lottery has decided in favor of the right side, he actually walks right rather than left. That stage has no explanation in terms of strict rationality. Buridan therefore concludes that rationality does not mean activity in which every step has a determinate justification (why specifically X and not Y), but rather action that advances toward its goals in the best possible way (to live). According to this definition, conducting a lottery and obeying its result are also rational acts. This is Buridan's rationality.

From the analogy to our discussion, it would seem that when we have an anti-paradox we should not seek a determinate solution. It would be better to conduct a lottery that chooses arbitrarily (and not determinately) between the solutions. But this again mixes different planes of discussion. If in Buridan's case this sounds like a reasonable solution (it achieves the goal—to live), then with respect to an anti-paradox, as in the example of the dispute between Beit Shammai and Beit Hillel, there is no sense in it. There we are looking for a genuine solution, that is, we are asking what Jewish law actually determines in such a case. Practical rules of conduct that solve the problem we already have (we encountered such rules even with respect to paradox—as in the dispute between Rav and Shmuel regarding stipulating about overcharging), and that is not the issue here.

A strictly rational solution

In my last lecture in Petah Tikva, it suddenly occurred to me that one can propose a solution to the person in Buridan's situation even in the terms of the strict rationality that he attacked and rejected. Instead of speaking about the matter abstractly and explaining it from above, I will offer you a description of how a strictly rational person would behave in a situation in which he stands between two identical tables laden with every good thing, positioned on his two sides in complete symmetry. You will see that every stage in this process can be given a determinate rationale, and therefore, in my view, it meets the strict criterion that Buridan attacked.

1. The person creates a random generator that decides by lot between right and left. Justification: if he does not create such a generator, he will die of hunger. Therefore there is a determinate justification for creating it.
2. He builds a motor and connects it to the generator, so that the generator will move him in the chosen direction until he reaches the food table. Again, the justification is that without this he will die of hunger.
3. He straps himself to the device and waits with great anticipation for what follows. Notice that here too there is a determinate justification. Without this action he will die of hunger, because if he is not strapped down, he himself will have to decide whether to turn right, as determined by the generator (parallel to the lottery described in Buridan's conception of rationality), or left. Neither of those decisions has a determinate justification.
4. Now the motor carries him to the table on the right (in accordance with the generator's determination). At this stage he is obviously not deciding on his own to turn right, and therefore this step does not require a determinate justification.
5. He arrives beside the table and begins to eat. Here there is a decision, but it is very sensible, because now he chooses the nearer table, and that of course has a determinate justification.

If so, Buridan's argument against strict rationality is apparently invalid. A strictly rational person is not doomed to die of hunger. Even evolution would leave him alive.

A brief logical discussion

One may of course wonder why a rational person would do all this instead of simply choosing one side and beginning to eat on his own. What is rational about deciding on this complicated process rather than conducting a simple lottery? But that is not the question when we are discussing within the framework of determinate justification. Rationality in the strict sense looks for a justification for each and every step, and not necessarily for the whole.

Perhaps this can be explained differently. Part of rationality is to act rationally. We saw that conducting a lottery is not strictly rational, and therefore this person chooses the more roundabout route, because it enables him to be strictly rational.

My application: Buridan's man

In my book I used Buridan's example to argue a different claim. It served me in conducting a thought experiment on the question of determinism. In the mechanistic-physicalist picture of the human being, every action is performed because of a prior physical cause. Suppose that a person standing before the two mangers feels hunger. This means that a region of the brain comes to life and cries out, 'I am hungry!' The determinist explains to us that this cry is nothing but an epiphenomenon, that is, an accompanying phenomenon. What is really happening is neural activation, and this passes an instruction to the organs to perform some action (= walking to the manger and eating). At the same time, and independently of this, the feeling of hunger arises in our mental dimension, but this does not affect the neural mechanisms in any way, and therefore not our behavior either. It merely accompanies them passively. If so, the action we perform in such a situation is the result of physical processes alone. The human being is a physical creature, and all his actions are explained according to the laws of physics.

For a physical system to begin operating, some physical cause is required. If physicalist determinism sees the human being as a physical system, then every action of his requires a physical cause. Such a cause can be described through some physical law, formulated in mathematical language through equations together with boundary and initial conditions. In mathematics there is a very useful and important principle, the principle of symmetry, according to which in an environment possessing some symmetry (usually described in the equation and the boundary and initial conditions), the result (= the solution of the equation) must possess at least the same symmetry as the circumstances.[2] If the symmetry of the equation and the boundary and initial conditions is spherical, then the solution too must necessarily have spherical symmetry. If the symmetry is translational symmetry (that is, shifting by a certain distance does not change the forces and data of the equation and of the environment), then the solution too will have such symmetry. In our example (Buridan's man) there is symmetry between the two sides, both in space (the two mangers) and in the structure of the person himself. Therefore the solution (= the person's motion) must also have that symmetry (at least).

The symmetry of the problem makes the two sides equivalent, and therefore the symmetry of the solution must be similar as well. There are only two kinds of outcomes that have this symmetry: (a) to move in both directions simultaneously, which is of course impossible (again because of our accursed and stubborn laws of physics). (b) to remain in place (or move forward along the straight line between the mangers) and die of hunger. This is a mathematical proof that in the deterministic picture, under the conditions of our thought experiment, a (point) donkey found in a Buridanian situation cannot move toward one of the tables.

This leads to a very surprising conclusion about donkeys. A donkey in such a situation would indeed die of hunger. But not because it is strictly rational, as Buridan claimed, but because of the laws of physics. In a symmetric situation there is no force acting toward either side, and therefore the motion of a physical object toward one side cannot occur. The donkey will be bound to its place by the laws of physics and causality, regardless of the question of rationality.

And what about a human being? Anyone who sees the human being as a physical object and understands his action as the deterministic result of physical laws must conclude that a person's action too will possess the symmetry of the circumstances in which it is performed. Therefore a person too, with all his wisdom, will in such a situation die of hunger. Exactly like a donkey. This has nothing to do with strict rationality or any other kind, but with the laws of physics. As noted, the materialist determinist believes that the human being is a physical machine, and in that sense he is exactly like the donkey. He is admittedly more complex and more intelligent, but the question is not one of intelligence and not of rationality, but of the laws of physics. Whether we like it or not, these laws apply to human beings exactly as they do to donkeys.

Spontaneous symmetry breaking

In the physics of the late twentieth century, Buridan-like situations of this kind are used to illustrate what is called 'spontaneous symmetry breaking.' For example, think of a small ball perched on the tip of a round mountain peak. There too the situation is symmetric, and we have no ability to predict in advance where it will fall—just as we have no way of knowing from which manger the person, or the donkey, will decide to eat. In principle it ought to remain on that tip and not move. But in the practical world there will always eventually be some minor event of one kind or another, originating in the ball's environment (or that of the person/donkey), that will break the symmetry, and thereby create a tiny force in one direction. For example, a light puff of wind may come, or the density of the air may change on one side, and this will move our little ball from its place to one of the sides.

In a real experiment, then, there will always be something in the environment that causes symmetry breaking. True, because this is a spontaneous event (not initiated by us), we cannot predict from which manger the donkey, or the person, will eat, but it is clear to us that in the end it will come to eat from one of the mangers. It certainly will not die of hunger. But our concern here is not with practical solutions (of the 'tiny nudges' variety, or the principle that doubt does not override certainty), for we are dealing here with a thought experiment. The experiment is theoretical, and within it we ask what will happen when the environment is completely neutral and completely symmetric. In this hypothetical situation, our little ball will of course remain forever at the top of the mountain, and by the same token our donkey will of course die of hunger. And what shall we say about a person standing in such a thought experiment? In the materialist-mechanistic picture, the person too in such a situation will die of hunger, since there is no physical cause that can make him go to one side.

Invoking symmetry breaking is a practical solution of the 'tiny nudges' type, not a real solution. The real question is the theoretical one that deals with an ideal environment that is perfectly symmetric. The fact that in practice the problem does not arise (that is, we have a practical solution) is irrelevant to the discussion.

Can the strict solution be applied here?

Is the solution I proposed above to Buridan's problem relevant to this question as well? At first glance, in a deterministic world one could create a random generator and bind oneself to it, and that would solve the problem of determinism as well. The determinist will say that in such a situation a person would not die of hunger if he adopted this method.

But this is nonsense, of course. First of all because intuition says that he will simply go to one side and eat, and will not begin constructing generators that he knows nothing about and has no idea how to build. Can a solution that rescues the intuition that he will not die be so counterintuitive?! Put differently: does the determinist really think that in such a situation that person will begin carrying out this whole crazy procedure? After all, the question here is what will happen in practice, not what he can do. But beyond that, there is also a difficulty in the matter itself. In a physicalist world there is no random generator. Randomness too is forbidden, not only free-willed choice. Buridan dealt with rationality, so there was no obstacle there to using a random generator. But here we are dealing with determinism, and within its conceptual framework there simply is no random generator (at these scales).

On donkeys and human beings

To the best of my judgment, a point donkey found in such a hypothetical situation really would die of hunger. The reason is that I assume (perhaps mistakenly), as Descartes assumed, that a donkey truly acts only causally, and therefore the practical outcome in its case must be symmetric, as I proved above. But what about a person found in such a situation? Would he too die of hunger? My claim is that he would not. A human being is not entirely subject to the laws of physics, and here we return from the question of physical causality to the question of rationality. A person has the ability to deviate from the laws of physics if he rationally decides to do so. The desire to live will lead him either to draw lots or, alternatively, to bind himself to the generator described above.

This thought experiment is intended to hold a mirror up to the materialist. He must ask himself whether, in his opinion, under this hypothetical circumstance a point human being really would die of hunger. If so—then he truly is a materialist. But if he suddenly realizes that in his opinion this is not what would happen, he has thereby discovered that he is in fact a covert dualist or libertarian (see columns 191 and 194 on covert worldviews).

If a symmetric person in such a situation can make a decision, on the basis of his own judgment, to turn toward one of the tables, that would be a decision against the laws of physics.[3] The meaning of such a decision is that the person's action is not determined by the physical circumstances that prevailed before it (for the circumstances were symmetric and the decision is asymmetric). That is, there was something beyond physics that participated in determining the physical state at the next moment. But do not worry: if and when the Buridanian person is sued in the deterministic court for violating the laws of physics, he will of course be able to defend himself by arguing that it was a case of saving a life.

[1] See chapter 16 of my book The Science of Freedom.

[2] The mathematician Emmy Noether published in 1918 the famous theorems that bear her name, which establish that from every such symmetry there follows the existence of a conserved quantity, and thus explained all the conservation laws in physics (such as conservation of momentum and energy). All these theorems are a result of the phenomenon mentioned here. On symmetry one may read in Mario Livio's book, The Language of Symmetry – The Equation That Couldn't Be Solved, Arie Nir, 2006; and also in Marcus du Sautoy's book, Symmetry: A Journey into the Patterns of Nature, translated by Uriel Givon, Sifrei Aliyat HaGag, Yedioth Books, 2010.

[3] In that book I point out that in the materialist picture even conducting a lottery is impossible. At the scale relevant to our discussion, the relevant physics is classical rather than quantum.