In a logical problem that Lacan examines in Écrits, there are three prisoners (to simplify the pronouns let’s say it’s a men’s prison). The warden makes them play a game and offers freedom to the winner. In this game each prisoner is made to wear a hat the color of which he doesn’t know and cannot observe. Each prisoner can only observe the color of the other two prisoners’ hats, and they all know that there is a total of two black and three white hats. The warden makes all three of them wear white hats and challenges them to guess the color of their own hats. The first prisoner to run for the door and give the correct answer with a valid logical explanation will gain his freedom.
How can a prisoner in this situation conclude that he is wearing a white hat, and how can he logically explain his reasoning? This problem is solved by constructing three situations that refer to each other. All three situations are presented within the present text below. The prisoner finds himself in Situation 3 and imagines Situation 2 and Situation 1 to articulate the virtual effect of unrealized possibilities that clarify his present situation. In this text, we examine these three situations in terms of Lacan’s four discourses. Let us first name the four terms involved: Master-Signifier (S1) names the prisoner’s decision about the answer, barred subject ($) names the prisoner’s indecision and hesitation about the answer, S2 names the prisoner’s knowledge about the situation, and objet a names the validation of the prisoner’s answer.
Situation 1: The prisoner observes two black hats. He immediately sees that his hat must be white and runs for the door.
Observing two black hats is a direct proof that the prisoner has a white hat. The prisoner’s initial indecision ($) is immediately covered by his decision (S1) because his knowledge (S2) directly produces validation (a). This simple decision of Situation 1 has the form of the Master’s discourse. The agency of the Master-Signifier acts on the instant of seeing. In this case the decision of S1 is a direct actuality and the indecision ($) never needs to appeal for validation (a), these two elements are disconnected.
Situation 2: The prisoner observes one white hat and one black hat. He thinks that if his hat was black, then the other prisoner with the white hat would be in Situation 1 and he would immediately see the answer and run for the door. Since this does not happen, he understands that his hat must be white and runs for the door.
Now the prisoner does not have direct proof (two black hats) but only indirect evidence (one black hat), so he must consider both possibilities (white or black). The decision of S1 is no longer a direct actuality but a mere possibility. Due to this new complication, the Master’s discourse in Situation 1 takes a quarter (90°) turn and two new gaps open up: Gap 1 is between indecision ($) and decision (S1): The prisoners may either hesitate or decide the answer. Gap 2 is between knowledge (S2) and validation (a): The knowledge may or may not validate the answer.
Initially, Gap 1 has the upper hand and the prisoner is hystericized: The possibility of validation (a) underlies the prisoner’s indecision ($) about the answer which depends on the possibility of the other prisoners’ decisions (S1) that are supposed to produce the knowledge (S2) he needs for validating the answer. This is the hysteric’s discourse. The knowledge (S2) is never able to validate (a) any answer because these two elements are disconnected.
But he sees that the other prisoners cannot decide either and they continue to hesitate. The simple reasoning of the Master’s discourse in Situation 1 is imagined but it does not come to pass. This leads him to an understanding: He notices that Gap 1 and Gap 2 are two sides of the same coin, these two gaps determine each other: The possibility of validation (a) directly relies on the possibility of decision (S1). The absence of the others’ decision gives the prisoner the validation that he needs. This is his time of understanding that puts an end to his hystericization.
Now the prisoner can give a half (180°) turn to the hysteric’s discourse and reach the University discourse, in which case Gap 2 gets the upper hand: The unrealized possibility of the other prisoners’ decisions (S1) underlies the prisoner’s acquired knowledge (S2) which validates (a) the answer that explains their indecision ($). In fact, the understanding of the University discourse is a trick that only works because it keeps the hysteric hesitation at bay by isolating the decision (S1) from the indecision ($), by disconnecting these two elements. It’s just an imposture that obsessively defends against indecision by inflicting it on to the others.
Situation 3: The prisoner observes two white hats. He thinks that if his hat was black, then each of the other prisoners would be in Situation 2 and they would think and understand the correct answer and run for the door. Since this does not happen, he concludes that his hat must be white and runs for the door.
The prisoner is again hystericized exactly as in Situation 2, but in this case he does not even have any evidence (no black hat) so he is unable to make the half (180°) turn into the University discourse: In Situation 3, the decision (S1) is no longer an unrealized possibility, since the decisive proof of two black hats has actually become impossible. Instead, the decision (S1) is now a virtuality.
Gap 1 and Gap 2 are thereby disconnected from each other. Knowledge (S2) can no longer cover decision (S1), and validation (a) can no longer cover indecision ($). The prisoner can no longer perform the understanding by himself. Instead, he is going to wait for the other prisoners to perform their own understanding and draw his own conclusions from their consequent behavior.
Thus the hysteric’s discourse makes another quarter (90°) turn and we get the analytic discourse: The prisoner lacks the knowledge (S2) upon which to base his validation (a) that will be able to turn his indecision ($) into a decision (S1). Instead of the positive knowledge of understanding in the University discourse, the analytic discourse contains a negative empty place that marks the lack of knowledge.
So the prisoner waits for the other prisoners to understand the correct answer. When they display their inability to understand it, the prisoner is able to conclude that his lack of knowledge is universal. The universality of this lack of knowledge turns the prisoner’s ignorance into a strength. Like the analyst, the prisoner can now “recognize in his own knowledge the symptom of his own ignorance” and use this “nonknowledge” to arrive at the correct conclusion (Lacan, Écrits, p. 297). This nonknowledge is the knowledge in the place of truth. It is a knowledge (S2) that is disconnected from the decision (S1).