Puzzles, prisoners, and discs: how time structures human subjectivity

     If you look carefully enough, there’s a paradox at the heart of our experience. Consider the flash of a walk signal, inviting you to cross an intersection. It seems to happen immediately: an observation which seems obvious enough. The image presents itself to our consciousness in an all-or-nothing, singular moment of experience.
     But neuroscience, so far, has not corroborated this everyday claim.  Here’s what it says about the matter. Light reaches the photosensitive cells of your retina and a signal is sent to the LGN, a region of the walnut-sized thalamus which sits at the middle of the brain. From here, the signal is projected to the visual processing centers located in the occipital lobe, which are responsible for recognizing both overall contrast and movement of objects. Next, two ‘streams’ emerge — the dorsal and the ventral — which ultimately relay the information to your frontal and parietal cortices.
     This is the situation from a neural perspective; but something obvious is missing. When exactly do you become aware of the walk signal? This has been the object of considerable debate in cognitive science, and the discussion can be summarized simply: we have yet to determine a specific ‘spot’ that’s responsible for the subjective awareness. The paradox is now apparent: neuroscience suggests that the experience is ‘smeared out’ across time, but the snapshot of our conscious knowledge contradicts this.
     According to Jacques Lacan, our common conception of time is also made paradoxical by a certain kind of logic: the logic that pervades intersubjective relationships. He even went so far as to say that our typical conception of time — chronological time — is of a different quality than the time of intersubjective interaction: logical time. This idea, which developed early on in Lacan’s thought and came to pervade every aspect of his philosophy, was first illustrated by using a logical puzzle similar to the following.
     At a progressive but somewhat sadistic penitentiary, the warden selects 4 prisoners. A hat is placed on each of their heads and they’re made to assume the following positions:

prisoner puzzle

     Each one can see the hat color of the prisoners in front of him, with the exception of 4, who remains concealed behind a wall. The warden tells them that they wear either a white or black hat, and that he used 2 white and 2 black hats between them all. If a prisoner deduces his hat color and shouts out the answer, he will be freed.
     Let’s assume all the prisoners are well-educated in logical reasoning. The question is: what happens?
     The peculiarity of this logical puzzle lies in the fact that the passage of time plays a crucial role in its solution. Most logical problems need time in order to be solved (as takers of the LSAT are painfully aware), but once its solution is found, its underlying logic exists as an immediate, synchronous solution. By contrast, the prisoner problem involves some duration as a necessary aspect of its solution; time is an implicit factor in the logic of its solution. And this is because the concept of intersubjectivity is at play.
     If this hasn’t been enough to reveal the solution, here it is. Prisoner 1 sees a white and black hat in front of him. He’s therefore unable to deduce whether he wears a white or black hat, although he could have done so if he had seen either two whites or two blacks. Noting this hesitation, prisoner 2 can therefore deduce that whatever color hat he wears, it is not the same color as prisoner 3, since otherwise prisoner 1 would have shouted his answer. Seeing a black hat in front of him, prisoner 2 therefore deduces that he must wear a white hat.
     As this puzzle demonstrates, human intersubjectivity introduces the passage of time as a necessary component of our logical calculus. We are often required to gauge the response of others: to learn from their action or inaction, to intuit when a pause was just too long or an intervention was just too quick. Duration is an essential aspect of social interaction, and further still, the formation of our own identities.
     In Lacan’s paper Logical Time, he uses a slightly more complicated variant of this problem. 3 prisoners wear either a white or black disc, can see the disc of everyone else, and are granted freedom if they can logically deduce the color of their own disc. The warden has chosen from 3 white discs and 2 black discs, and in fact gives all 3 prisoners a white disc to wear. Can a prisoner figure out his color?
     Let’s call the initial subject who thinks A, the second subject B, and the third subject C, as indicated by the picture. A proceeds with the following line of thought: if I’m wearing a black disc, B would see a black disc and a white disc in front of him. From there, B can ask himself the question, ‘what would happen if I was wearing a black disc?’ and find that, if he were, C would know what disc he wears, since only 2 black discs in total were used. Now, since B does not move, this indicates to A that he was unable to make that deduction. This means A must not be wearing a black disc, and instead wears white. Now, because all 3 subjects can make the same deduction, they do so at the same time: each proceeds to the exit, having solved the puzzle.

logical time puzzle

     This explanation, as stated, is a kind of idealization: the strictly logical solution. Lacan is adamant on making a distinction here between the theoretical solution to the problem, which involves all 3 subjects coming to the exact same conclusion and exiting simultaneously, with the actual result of conducting this experiment. Practically speaking, there would be some difficulties involved with this problem:

“It will perhaps turn out to be of some scientific value to the psychologist, at least if we can trust what seemed to us to result from having tried it with various groups of appropriately chosen, qualified intellectuals: a very particular misunderstanding [méconnaissance] on the part of these subjects concerning the reality of others.” (Lacan, Logical Time, pg 6)

     This “particular misunderstanding”, which presents itself as a complication for the idealized solution of the problem, can be understood if we put ourselves in the shoes of one of the subjects. Suppose you are A and you just figured out that, in the absence of B’s motion, you must be wearing a white disc. The moment that you begin to move, however, you see B also start for the door. This observation must give you a moment of hesitation, since your entire logical edifice was founded on the fact that B would not move, owing to the fact that you (A) do not wear a black disc. A flash of doubt intrudes — could I actually be wearing a black disc all along?
     This first moment of hesitation, however, is soon supplemented by another observation. B himself has also stopped moving for a moment. Well then, if B is also stupefied by your motion, he must be in the exact same situation as you — thus, you do not in fact wear a black disc. If you had worn a black disc, the confidence of the other’s answer would not be jeopardized by the motion of anyone else. It is only after this realization that you can exit the room in full confidence. Your certainty — which was suspended for a moment by the motion of the other subjects — is now affirmed by the fact that you see the others giving a momentary pause themselves.
     So we see that the idealized solution of the problem, which involves all three subjects exiting the room simultaneously, is contrasted by the experimental scenario, where the recognition of the other’s motion causes some hiccups and interruptions of certainty. In fact, in order for the subject to be sure that he wears white, he must sense two distinct moments of hesitation on the part of the others. The first is the initial moment of hesitation, which signals that the others do not see one white and one black disc. The second is the moment of hesitation which interrupts their movement to the door, which ensures him in his own state of hesitation that they are in the same situation as he is. Given these two moments, we can divide the practical solution of the problem into 3 “evidential moments”:

1. the first phase, when all 3 subjects look at each other in mutual hesitation
2. the second phase, when all 3 subjects start towards the door then pause
3. the third phase, when all 3 subjects head towards the door with renewed certainty

     Lacan maps these phases onto a tripartite structure which he positions at the heart of human intersubjectivity: the instant of seeing, the time for understanding, and the moment of concluding. In the context of this problem, the instance of seeing involves the immediate recognition of the other two’s disc colors and the hypothesis “if I see 2 blacks, I know I’m a white”. Seeing 2 whites, we might think that this phase gives way to inutility, but armed as we our with a theory of mind — that ability to recognize subjectivity in others — the hypothesis takes on a more generalized form: “seeing 2 blacks, one knows that he is white”. Once this shift from the individual to the general is accomplished, the subject can recognize that were he black, the other two whites would immediately recognize that they are white. This point ushers in the time for understanding. Given that the other two whites have not moved, the subject interprets their immobility as evidence that he is not black. This is because, had he been black, they would have preceded him ‘by a beat’: their movement would have been quicker than his, simply because they didn’t waste time on the very hypothesis that the subject is now considering. Knowing this, the subject hastens to declare himself white because if he sees the others move before him, he can no longer be sure that they do not see him with a black disc. This quick move to assert himself comprises the moment of concluding.
     Lacan generalizes these phases and states that in intersubjective relationships, time operates in a parallel way. We make a logical hypothesis about ourselves, and this hypothesis can only be confirmed or denied by pure reciprocity: like the prisoner in the puzzle, we can only come to know about ourselves through the Other. Because this hypothesis is ‘cast outside of ourselves’ in this way, there is necessarily a temporal delay. As Lacan puts it, “the ‘I’, the subject of the conclusive assertion, is isolated from the other, that is, from the relation of reciprocity, by a logical beat”. This logical beat is the exact same beat that separates the subject during the time for understanding from the other prisoners, and it is a beat that compels him to quickly assert about himself so that he can gain the certainty of his conviction.
     In the context of Lacan’s intellectual development, Logical Time can be seen as an early reiteration of his concept of the mirror stage. The mirror stage comprises the revolutionary moment in which the young child begins to understand himself as a unified whole which is capable of being observed by others. This is dramatically illustrated by the first instance in which a baby recognizes itself in a mirror. Lacan posits that the development of the subject’s conclusion about himself in the prisoner puzzle is parallel to the child’s acquisition of the psychological ‘I’. Since birth, the subject recognizes himself through the activities and relations of others, that pure reciprocity towards which multiple hypothesis and potential interpretations are lobbed. What we come to assert about ourselves is characterized by a fundamental haste, what is called an ‘anticipating subjective assertion’.
     There’s still much to be said about Logical Time, specifically the metaphysical difference between intersubjective logic, which involves a certain irreducible haste, and pure logic, which gives us synchronous all-present solutions. But the lesson even from this brief outline is clear: the dimension of intersubjectivity introduces new complications in even our strongest intuitions about logic and time. In this way, psychoanalysis comes to knock at the door of philosophy.



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