How to Corax your theory of mind

Don’t know why I decided to take a course in Greek Rhetoric this semester… But readers will be glad to learn that I really had fun! I came across a jewel of a mind-twister, an argument called the Corax. I think the Corax and its use might shed light on the structure of our folk psychology (a.k.a. Theory of Mind).

The cognitive arts (such as mnemotechnics, prayer, marketing, rhetoric…) are an underestimated mine of insight for a psychologist; the rhetoric art of advocacy is a case in point, since it must tap the jury’s psychological intuitions about the accused (did he have any rational motive for doing what he’s accused of? is he a good guy? etc.) and know quite a bit about the workings of the typical jury member’s mind in order to manipulate it. See for example the Corax, a rhetoric argument that taps into our naive theory of what rational actions are:

Suppose that you are a ruthless individual with many enemies and a reputation for violence (we’ll call you Vladimir P.); one of your greatest foes has just been killed (let us call her Anna P.). You’re in trouble. One possible line of defense for you is the Corax: it consists in arguing that the charges against you are so overwhelming that you can’t have murdered Anna P. How could you, since you knew that everyone would think of you as the prime suspect? Knowing all that, you could not reasonably plan a murder, whereas those who had less reasons to kill Anna P., took little risk in doing so, since they knew that all the fingers would point at you at the end of the day.

Greek sophists presented the Corax as a flawed argument, which it is not: it makes perfect sense to assume that one took the consequences of his crime into account before acting, one of the things to be weighed being his reputation and the likeliness of being accused. The sophists want to have us think that the Corax can turn plausibility into implausibility and vice-versa: the more I am suspect, the more I am innocent (this bold allegation earned them much spite from Socrates and Plato). This is only partially true: for the Corax to work, one must not only be suspect, but also be aware, at the time of the crime, that he would be held in suspicion for that crime. This is regular folk-psychological reasoning.

There is a twist, however: the Corax is reversible. Vladimir P. wants us to believe that he cannot have murdered Anna P. because he knew that he would be the prime suspect and could not want to take such a risk. But he could also have anticipated the fact that his being the prime suspect would allow him to defend himself by saying that he could not murder Anna P. because he knew that this would make him the prime suspect. In other words, Vladimir P. could have anticipated the fact that he could use a Corax to disculpate himself (by the way, I think that it’s what’s really happening with Mr. Putin). Such an argument is what I would call a second-degree Corax. You can destroy it in turn with a third degree Corax (Vladimir P. knew that he could not defend himself by saying that he knew he would be the prime suspect, since he knew that if he did so, the accusation would claim that he knew all along that he would be able to defend himself by saying that he knew all along that everyone would suspect him if he killed Anna P.; hence, Vladimir P; did not kill Anna P.), which is vulnerable to a fourth-degree Corax (OK, I’ll spare you with this one), etc.

Surely, Greek rhetoric must have known about second-degree Corax? The astounding answer is : no! In Antiphon’s First Tetralogy, the accused is defending himself with what came to be known as the canonical example of Corax. The accusation puts the whole rhetorical apparatus to use in order to make him guilty; some paralogisms are so far-fetched and so subtle that you have to read them four times to grasp the flaw. Most appeal in some way or other to psychological assumptions about the murderer. But second-degree Corax is never resorted to. The teacher, a respected hellenist who wrote a classical Histoire de la littérature grecque, assured me that she had never heard of a second-degree Corax in greek litterature, be it rhetoric or whatever.

There is a debate among game-theorists, philosophers and psychologists on the question of knowing the real depth of our folk psychology. Philosophers have argued that we can perfectly understand up to 7 or 8 embedded propositional attitudes (a propositional attitude is a sentence in the form: he thinks that the sun is bright; embedded propositional attitudes are sentences like: he thinks that she thinks that the sun is bright). However, in the real world, it seems that we very rarely predict or anticipate other people’s behavior (their beliefs, in psychology, or their moves, in game theory) up to the third order. Some think it very difficult for a human to compute even second-order anticipations. Our module for predicting beliefs and intentions might not work very smoothly very deep (which it would do if it were, for example, a general recursive mechanism).

The puzzling lack of second-degree Coraxes among greek sophists might point to a limit in our processing the mental states of others.

When I went to see the teacher after the class, she told me it was no wonder why no second-degree Corax was ever produced; a second-degree Corax, she explained, would be exposed to a 3rd-degree Corax, which in turn could be destroyed by a 4th-degree Corax, and so on. She argued that this potential regressio ad infinitum (which she called a sorite – erroneously I think, but this is pure pedantry – in the mood for which I most definitely am); ahem. She argued that this potential regressio ad infinitum amounted to a full-blown counter-argument.

“But” I replied, “This regress argument also applies to the (first-degree) Corax itself, doesn’t it?

– Yes it does; that is why the Corax is a paralogism.

– And people realize that when the Corax is 2d-degree or 3d-degree…

– Yes! Isn’t that obvious?

– Then how is it that a first-degree Corax works?”

She giggled and we left it at that.

That’s what’s baffling with the Humanities people: they catch a glimpse of amazing things about human nature, then giggle and leave it at that. Most often the psychological consequences of their work does not interest them in the slightest; think only of the progress they could bring to psychology, not by importing our jargon and dogmas into their field (for that is all we can offer them right now), but by mining their own insights into the human mind, taking their conclusions seriously and trying to prove them wrong.

Sigh… there are some things that they, and they alone, could find, that they will never be looking for – while we are looking for things which we may never find.

This post was originally published in 2006 on the Alphapsy blog.



1. On Wednesday 11 October 2006 by hugo

I’d like to comment on the case of regressio. I’m not sure it is “psychologically valid”. You can’t attribute to someone an infinity of embedding (or an infinity of anything actually). So even if the argument, in its abstract form, is clearly valid (and as much so for the first, the second or any degree of corax), I don’t think you can use it against someone who does the first (or any degree of) corax.

To draw an analogy from game theory, when you are engaged in a game and you have to predict your opponent’s move, you shouldn’t attribute to him an infinity of embedding. Likewise, in the case of the corax, you can’t attribute to the defendant this infinite regress. (by the way, if you did so, that you embark you on another infinite regress: after all, if you attribute this infinite regress to the defendant, then his case is not valid. so if he thought that you would attribute the infinite regress to him, then he would have known that you wouldn’t fall for the corax. so he wouldn’t have committed the crime since he would have had no way to defend himself. and so on and so forth. please Cantor, help me my head hurts (you can add an infinite number of dimensions of this sort…))

Moreover, if you really want to go psychological, then you have to admit that the plausibility of each step of the corax depends on a complex host of factors, most of them psychological (such as: does the jury think that the defendant would have been able/motivated to think of the possibility of a first degree corax? of a second degree? and so forth)

2. On Wednesday 11 October 2006 by onclepsycho

People in humanities are quite satisfied with Freud, Pavlov and Milgram. They don’t care about furthering their psychological insights, they merely want something to illustrate them.
Thanks for the Corax, never heard of that before. Could be useful to illustrate some points in our field…

3. On Wednesday 11 October 2006 by olivier

If you want to counter an argument on the ground that it leads to a possible regression, you need not assume that such a regression is actually taking place in a human mind; you can content yourself with showing that such a regression is possible by taking one first step in that direction. In the case we’re discussing, the 2d order Corax is one such step. I think that theoretically, it can destroy a first-order Corax.

Since you do not have to make the strong psychological assumption that someone, somewhere, is capable of an infinite series of reasonings, your argument from psychological implausibility is (I think) rebutted (but I had a hard time with it and am still not sure, so congratulations anyway).

There are mathematicians who consider that regressio ad infinitum is not a flaw in a demonstration; perhaps they are right in their own domain (you can define objects through recursion, etc.), but my attitude towards regression is the same as my mother’s attitude towards homosexuality: perhaps in some places they consider it normal; I won’t interfere with that; it might even make some people happy. But it is wrong, wrong, wrong.

4. On Wednesday 11 October 2006 by olivier

in answer to Oncle Psycho: thank you for your comment; may I ask what your field is?

5. On Wednesday 11 October 2006 by Andreas

You might be interested in Rosemarie Nagel’s paper “Unraveling in Guessing Games: An Experimental Study”, American Economc Review 85(5), pp. 1313-26. Nagel tries to elicit how many steps of higher order reasoning experimental subjects use in a game which economists call a Beauty Contest ( as you’ve been talking about Werner Güth’s Ultimatum Game yesterday, here’s yet another fancy name economists have come up with for a game. I believe the term Beauty Contest goes back to John Maynard Keynes. ) The rules of the game are as follows: A group of people are asked to individually submit a number between 0 and 100. The person that gets closest to two thirds of the group average wins a price. The game can be solved by elimination a (weakly) dominated strategies. The unique equilibrium should be for all participants to submit 0. ( Just to give you the flavour: First step: Suppose everybody submitted the highest possible number 100. 2/3 of 100 is 66,6…, so you should never submit a number higher than 66. Second step: Given that everybody understands this, nobody will submit a number higher than 66. Suppose everybody submits 66. Then you should go for 2/3*66 … and so on. The game unravels to an equilibrium bid of 0. ) This is however not what happens experimentally. It turns out that the distribution of numbers submitted can be meaningfully related to the number of steps of reasoning subjects use in trying to infer the behaviour of their competitors. More than 3 steps of reasoning are rare. There a mass points around 0, 1, and 2 steps of reasoning.

6. On Wednesday 11 October 2006 by hugo

thanks for the comment

I think that theoretically, it can destroy a first-order Corax

yes! (this is why the argument is not formally valid)
but not psychologically!
I maintain that if someone manages to make a convincing point that we knew all along that he would be a designed culprit, but also manages to convince you that he hasn’t thought about the second degree corax, then his defense is a good defense. it is a factual defense: what the guy actually though. if he didn’t actually though about the second degree corax, then you cannot debunk his first degree corax with a second degree corax. the mere possibility of him thinking about it is not enough: what matters is whether he actually though about it or not. the problem, obviously, is that it is a very hard case to make (how can you prove that you were thinking or not thinking about something). however, this kind of problem is very common: when juries have to determine whether there was premeditation or not, or when lawyers have to make a case that there was or not premeditation, they have to perform the same kind of analysis.

by the way, since you seem to have some expertise in these fields, do you think that the point about the infinite regress of infinite regresses is valid?

7. On Wednesday 11 October 2006 by hugo

[the preceding comment was for onclepsycho] to Andreas:
this game is quite interesting. have people tried to correlate it with other abilities? some (dubious maybe) claims might be made. if the ability to simulate these steps of reasoning for the other participants depends on theory of mind, then women might be better at it. at least, it should correlate with other measures of ToM. On the other hand, if it is a more ‘logical’ kind of reasoning, it should correlate with other stuff (IQ perhaps).
And in the end, it’s not so much a question of ability. it is possible that some people find the equilibrium (the reasoning is not that complex), but what should they do then? it all depends on whether they think that the others will also find it. and on whether the others think that the other others (ahah) will also find it. and then you have another regress.
does the result change if the participants are told that the other participants are very smart or very dumb?

8. On Wednesday 11 October 2006 by olivier

thank you very much Andreas; I knew about the Beauty Contest (which as you point out was invented by Keynes – some say adapted from a radio game), and I had this example in mind: another example making the case that people rarely go beyond 2d-order strategic reasoning. I haven’t read the paper you are alluding to, though (I will).

Hugo: Sorry to disappoint you, but I don’t know what you are talking about; could you explain?

9. On Wednesday 11 October 2006 by Andreas

the original paper is from 1995, I’m quite sure that there have been follow up studies trying to correlate the findings with education, sex, and so on. I’m however not aware of any such paper. (I could imagine that either Ernst Fehr at the University of Zurich or Colin Camerer at Caltech have done such research.)
I know of a study by University College, London psychologists, who play Ultimatum Games with autistic children and adults. They show that autistic children, when they play the proposer role, tend to play the unique subgame perfect equilibrium of offering 0 to the responder. As almost all of these offers are rejected they tend to do worse than the average person playing this game. The authors attribute this behaviour to the inability of the autistic children to produce a representation of other people’s behaviour in their mind. Adult autists don’t do any worse than the average person playing this game, even though they suffer from the same problem as the children. The authors conjecture that adult autists have learned to follow a social convention of playing fair, even though they are unable to anticipate their opponents behaviour. (I’m not a psychologist, my knowledge of autistic behaviour is zero, so I just reproduce what the authors claim.)
So this would point to the fact that logical capacities (here figuring out the unique subgame perfect equilibrium) don’t help much to do well in this sort of games.
By the way, my game theory professor used to play the Beauty Contest in class with his own money under the condition of being allowed to participate in the game himself. He would never lose his money, even though he didn’t submit the equilibrium bid of zero. I think his number was 37 which was 2/3 of the average of Nagel’s experimental data. Using logic to figure out this number should prove to be very difficult indeed.

10. On Wednesday 11 October 2006 by hugo

thanks for mentioning the study with autistic participants. the results are interesting indeed. i’m not sure i would agree with either of the two conclusions though (the one regarding the children and the one regarding the adults), but since i haven’t read the paper, i’m not going to argue over that now (but perhaps later? would you have the references of the paper?)
as for you teacher, he was using the best guide you could have in this case: his own experience and his knowledge of the results in the literature. it’s quite a good game to play, pedagogically speaking (for example if you want to illustrate the difference between the theoretical equilibrium and what people actually do)

11. On Wednesday 11 October 2006 by onclepsycho

In answer to comment n°4, my field is neuropsychology/cognitive neuroscience.
This discussion on regression ad infinitum could be reducible to the storage capacity of working memory, which is about 4 (pace Miller’s magical number 7). Keith Stanovich has a nice discussion of the human ability to make n-order judgments and preferences. Here’s an example from his book “The Robot’s Rebellion”, chap.8:
“Tessa prefers to celebrate Christmas more than she prefers her preference to prefer not to celebrate it”
which he models this was:
[XMAS pref NOXMAS] pref [(NOXMAS pref XMAS) pref (XMAS pref NOXMAS)] I’m lame in that kind of thing, but maybe some of you might offer some formula of that kind for the Corax thing, though it might get even more confusing as it involves several minds…
It occurs to me that the example of Vladimir P might be a good test with frontal or demented patients. I’ll try it next time, see if they can catch the 2nd order level (most ToM tests are boring or embarrassingly easy for patients, this at least could be funny).

12. On Wednesday 11 October 2006 by olivier

Oncle Psycho:

It occurs to me that the example of Vladimir P might be a good test with frontal or demented patients. I’ll try it next time, see if they can catch the 2nd order level (most ToM tests are boring or embarrassingly easy for patients, this at least could be funny).

That’s a very great idea! let us know the results.

13. On Thursday 12 October 2006 by BobW

Interesting stuff. Though only vaguely related, this put me in mind of a Martin Gardner column from long ago, that went something like this:

Three men are seated along a straight line all looking in the same direction such that the man in the back can see the other two, the man in the middle can only see the one in the front, and the man in the front can’t see either of the other two men. All three men are blindfolded. The men are told that each will get a hat from a bin which contains 5 hats total, 3 black and 2 white in color. Next, 3 hats are arbitrarily drawn out of the bin and placed on the men’s heads. The blindfolds are then removed and each man is
asked to figure out the color of his hat.

Some time passes, then the man in front (the one who can’t see any hats at all) says, “I have a black hat!” … and he’s right!

The man in back didn’t give an answer right away, which he could have done if he saw two white hats.

The second man reasoned from the first man’s delay that there were not two white hats, but that wasn’t enough information know what color *his* hat was. If he’d been looking at a white hat, though, he’d have known that his was black.

The front man noted that the lack of response from the first man, followed by the lack of response from the second, meant that he must be wearing a black hat!


14. On Thursday 12 October 2006 by Andreas

I’m still looking for the paper I’ve mentioned yesterday, but couldn’t find it so far. Here’s a very similar paper: Hill and Sally “Dilemmas and bargains: Autism, theory of mind, cooperation and fairness”, mimeo UCL and Yale.

It seems to go against some of the things I’ve said yesterday. The effect for autistic subjects is still there, but only in the first round of play. Also here they cannot find a strong difference between autistic children and adults. However looking at the distribution of offers of autistic subjects, it still seems to be the case that autists are more likely to follow learned social rules, i.e. if they share, they tend to predominantely go for the equal share rule. Also they are more likely to offer zero.

I’ll try to find you the reference for the research I’ve alluded to, however this paper here seems to be more carefully done.

15. On Thursday 12 October 2006 by hugo

thanks for the reference!

16. On Thursday 12 October 2006 by karim

People interested in the topic of recursive thinking might be interested in reading a sharp review by Andrew M. Colman of experimental work regarding recursive reasoning in games (Colman, A. M. (2003). Depth of strategic reasoning in games. Trends in Cognitive Sciences7(1), 2-4.) and the reply by the experimenters (Zhang, J., & Hedden, T. (2003). Two paradigms for depth of strategic reasoning in games: Response to Colman. Trends in Cognitive Sciences7(1), 4-5.) …

17. On Friday 13 October 2006 by olivier

To Bob W.:

Martin Gardner’s riddle is rephrasing a problem known among game theorists as the Bagdadi Husband. It became the story of the women of Sevitan in Gintis’handbook, Game Theory Evolving, that gives a complete mathematical account. The structure is roughly the same, but the story is a little more folkloric. There is a rumor in France that French OuLiPo writer Jacques Roubaud invented it but I never knew for sure.

The relationship with the Corax argument is not as shallow as you seem to think, since both imply recursive reasoning about common knowledge. What gives the bagdadi/hats/sevitan women story its spice is that the awareness of common knowledge is delayed. Thank you for this version, anyway!


18. On Friday 13 October 2006 by Andreas

Adam Branderburger of NYU has a short technical note on the modeling of common knowledge in game theory online. A more complete treatment of modeling higher order beliefs and knowledge can be found in John Geanakoplos’ review article “Common Knowledge” in the Handbook of Game Theory, Chapter 40, you will need access to science direct to follow this link.) Geanakoplos provides a detailed analysis of the hat-wearing puzzle.

19. On Tuesday 24 October 2006 by olivier

I would like everyone to receive my (belated) thanks for their very interesting comments. Second, to complete what Andreas said, I just found a very comprehensive paper on Common Knowledge in the Stanford Encyclopedia of Philosophy.

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