Return to my Logic pages
Go to my home page
© Copyright 1997, Jim Loy
The Unexpected Hanging is a famous logical paradox. Let me state it for you:
A prisoner had been condemned to be hanged. The warden knew that the prisoner enjoyed logic puzzles. So, he informed the prisoner of the execution date in this way:
"1. You will be hanged on one of the days of next week (Sunday through Saturday). And, 2. you will not be able to logically deduce which day you are to be hanged."
The prisoner realized that this is a puzzle which lends itself to working backward. He reasoned, "If today were Saturday (the last day), and I had not been executed yet, then I would know that today I would be hanged. In other words, if today were Saturday, then I could logically deduce that I would hang today. But, the warden stated that I would not be able to logically deduce which day that I would be hanged. Therefore, I will not be hanged on Saturday."
He reasoned further, "If today were Friday (the next to the last day), and I had not been executed yet, then I would know that today I would be hanged, since I won't be hanged on Saturday. In other words, if today were Friday, then I could logically deduce that I would hang today. But, the warden stated that I would not be able to logically deduce which day that I would be hanged. Therefore, I will not be hanged on Friday."
He similarly eliminated Thursday, since he had previously eliminated Friday and Saturday, and Thursday would then be no surprise. And, he similarly eliminated Wednesday, Tuesday, Monday, and even Sunday. He had eliminated Monday through Saturday. So, he could deduce that Sunday was the execution day, which eliminated Sunday, because of the Warden's original statements. So the prisoner had deduced that he could not be hanged, and still fulfill the Warden's original statements.
When the warden came to visit, on Sunday, the prisoner told him of his reasoning. He added, "You seem to have made a mistake. One of your two statements must be false. I don't see how you can hang me this week and still make it impossible for me to deduce which day it will be."
The warden replied, "Wrong! String him up, boys." And both of the warden's original statements were true.
Now, the prisoner's reasoning seems fairly good. Where is the flaw in the reasoning? Think about it.
The prisoner's reasoning for Sunday is seriously flawed. He has not logically eliminated the other days (Monday through Saturday). We will get to the reason for that, in a moment. Therefore, there is no way to reason, one way or the other, concerning Sunday. If the warden chooses to hang him on Sunday, the prisoner couldn't possibly have deduced that, and the warden's original statements will be true. The same can be said for all of the days except Saturday. The warden can have him hanged, on any day from Sunday to Friday, with no contradiction of his two original statements.
So, it seems that he cannot be hanged on Saturday. So, what is wrong with working backward, to eliminate each of the preceding days? The problem with working backward from Saturday, is that he can be hanged on Saturday. Let's look at Saturday:
It is Saturday, and the prisoner has not been hanged yet. He reasons, "I have not been executed yet. So I know that today I will be hanged. In other words, I can logically deduce that I will be hanged today. But, the warden stated that I would not be able to logically deduce which day that I would be hanged. Therefore, I will not be hanged today."
So the warden comes in, says "Hang him, boys." And his two statements were true after all. We don't need seven days of backward reasoning. We don't need any backward reasoning. The prisoner's initial deductions about Saturday are flawed.
Let's look at Saturday, in more detail. First, the prisoner assumes that the warden's first statement ("You will be hanged on one of the days of next week.") is true:
The prisoner has not been hanged. The prisoner CAN logically deduce that he will now be hanged. The warden's second statement was, "2. you will not be able to logically deduce which day you are to be hanged." This was false, whether the prisoner actually deduced the execution day or not. If the prisoner continues to reason, "So, they won't hang me today," then he is now making an error. That is an erroneous deduction, it does not even follow. The warden's second statement is false, and the prisoner just went off on a tangent of incorrect reasoning, if he actually failed to predict the execution day. The prisoner is hanged on Saturday.
Obviously, one of the warden's statements is false. I think that, for Saturday, the warden's second statement contradicts his first statement. And any deduction that either assumes that both are true, or proves that both are true must be viewed with skepticism.
Well, what about the other days? Can we eliminate Friday, since we have eliminated Saturday. No, we never proved that the prisoner could not be hanged on Saturday. We proved that the warden's statements are inconsistent on Saturday, and the prisoner will be hanged anyway, on Saturday. Assuming that they are consistent on Saturday leads to a contradiction, but does not prove that the prisoner will not be hanged.
Assume that today is Friday, and that the warden's first statement is true. Well, the warden's second statement is true, now, on Friday. The prisoner cannot predict which day he will be hanged. The warden's second statement may become false tomorrow. But, that does not affect its truth value today (Friday).
Are you convinced?
The warden's first statement is a simple statement of fact, it is true or false. The second statement is a circular reference. It is a statement about both of the warden's statements. And, like "This statement is false," it may be self-contradictory and have no truth value at all.
Every day (Sunday through Saturday), the prisoner despairs of making sense of this circular reference. And that is what makes the warden's second statement true.
The big event does not have to be an execution. It can be a test or quiz (much like an execution, as I recall). It can even be a wedding, which may be more like an execution than an execution is. Never had one, so I don't know.