The Prisoner’s Paradox (different from The Prisoner’s Dilemma), a popular logical paradox, is a paradox based on not knowing when an event will occur. Here is the problem, my solutions to the paradox, and why some of these solutions are flawed. Which of these solutions are correct?
You’re in for a pretty long read, so get ready to think about something as fun as, “How big is infinity? And, is the universe infinite?” A quick summary of the answers is listed at the end, but I recommend you read through the description of each answer if you’d like to know the reasoning behind the solutions.
The Prisoner’s Paradox
A prisoner is sentenced to death by a judge. The judge says the prisoner will be executed at noon, on a day between next Monday and next Friday, but the day will be a surprise. When will the prisoner be executed?
(Note: See Solution #3 for ambiguities and interpretations of the word “surprise”)
The Prisoner’s Thought Process (SOLUTION #1)
“The judge said, the day of the execution will be a surprise.“
1. Let’s assume the execution is scheduled for Friday:
If it’s noon on Thursday, and the prisoner hasn’t been executed yet, he knows he will be executed the next day, because Friday will be last possible day that he will be executed. There will be no surprise that he will be executed on Friday.
If Friday comes along, the prisoner knows that he will be executed today, so it’s no surprise to him if the execution is on Friday.
Therefore, if the execution is scheduled for Friday, it will not be a surprise, so the execution can’t happen on Friday.
2. Let’s assume the execution is scheduled for Thursday:
Since Friday is ruled out because it wouldn’t be a surprise to the prisoner, what if the execution was planned to be on Thursday?
If it’s Wednesday at noon, and the prisoner still hasn’t been executed, the prisoner knows that the execution will take place on Thursday, because Friday has already been ruled out. Once Thursday comes around, then an execution on Thursday isn’t going to be surprising, because the prisoner already knows that the execution won’t be on Friday, because a Friday execution wouldn’t be a surprise, so the execution can’t happen on Thursday either.
3. Let’s assume the execution is scheduled for Wednesday, Tuesday, or Monday:
The prisoner can use the exact same logic as above to rule out an execution happening on Wednesday, and then Tuesday. So, the execution can’t happen on Wednesday or Tuesday.
This leaves the only possible day for the execution that would be a surprise to be Monday. But, if the only possible day for the execution to happen is Monday, because all of the other days have been ruled out because they wouldn’t be surprising, then Monday will also not be a surprise. Therefore, the execution can’t happen on Monday.
Following the prisoner’s logic, every day has been ruled out as a possibility for the day of the execution, because he will not be surprised to be executed on the day of the execution, so there will be no execution (SOLUTION #1).
Is that wrong? Did you catch the mistake? Let’s go through some other solutions, and let’s see if any of them are better solutions.
Problem with the prisoner’s logic: uncertainty of execution before noon (SOLUTION #2)
The fault in the prisoner’s logic above was when he ruled out the possibility of the execution happening on Friday (and thus Thursday, Wednesday, etc). This is because of a flawed chronologically backwards logic (depending on our interpretation of “surprised”). Let me explain:
- If it’s Friday (or, after 12:00 PM on Thursday) and the prisoner hasn’t been executed yet, then yes, being executed on Friday wouldn’t be a surprise.
- If it’s Thursday before noon, then it’s still unknown whether the execution will occur today or on Friday.
The uncertainty is only removed AFTER 12:00 PM on Thursday, when the prisoner is either executed or spared for another day.
To further illustrate this, let’s use another example of a prisoner that will be executed on one of two days. Let’s pretend there’s another prisoner, and the judge tells them, “You will be executed on either Saturday at noon, or Sunday at noon, but it’s a surprise.” When does the prisoner get executed? When it’s Saturday, 11:59 AM, it’s still a surprise whether the prisoner gets executed in one minute, or the next day.
The prisoner can only rule out the last day as the day of execution if the element of surprise must necessarily be present on the day of the execution.
Therefore, the original prisoner’s execution can take place on any day of the week (SOLUTION #2), unless the ambiguity of this “surprise” is specified to mean that the prisoner must be surprised on the day of the execution (and after noon of the previous day).
“Surprise” meaning any random day could be unexpected (SOLUTION #3)
When the judge says “the day will be a surprise,” do they mean that it can occur on any random day? Any day could be ruled out by the prisoner depending on their reasoning, so any day of the week could be a surprise to the prisoner.
This is an issue of equivocation: when the judge says “the day will be a surprise,” do they mean:
- On the day of the execution, the prisoner will be surprised and not expect it? or,
- The day of the execution will be a surprise unless it’s after 12:00 PM on Thursday, after which, certainty may be removed, but uncertainty of the day would still exist before this point, or
- The day will be random.
If (1) is true, see see Solution #1, 3, 4, 5, or 6.
If (2) is true, see Solution #2, 3, 4, 5, or 6.
If (3) is true, see Solution #2, 3, 4, 5, or 6.
Otherwise, see Solution #7, 8, 9, 10, or 11.
If (3) is true, Solution #3 suggests that execution could happen on any day of the week (SOLUTION #3). (note: This depends on whether we interpret “surprise” as in Solution #1 or Solution #2).
Certainty of Monday being the day of execution (SOLUTION #4)
Let’s again pretend that there’s a prisoner who will either be executed on Saturday or Sunday, and it will be a surprise for them. This prisoner can guarantee that the execution will happen on Saturday, because if they aren’t executed on Saturday, then the execution would be on Sunday, and they would expect it. (Note: But if the prisoner is certain that the execution will happen on Saturday, it wouldn’t be a surprise, so they can’t be executed on that day!)
Let’s say our prisoner in our original paradox follows this line of reasoning (up until the contradiction) and comes to the conclusion that they’re certain that they will be executed on Monday. Noon will roll around on Monday, and maybe they won’t be executed. They’ll now follow the same line of reasoning, and be certain that today will be the day of their execution. But again, since they’re certain, it’s not a surprise, so they’re not executed. On to Wednesday, Thursday, and Friday, and again, just like Solution #1, no execution can take place (SOLUTION #4), except this time, it’s because there was certainty of the execution happening on each day starting from Monday, as opposed to Solution #1 where there was certainty that the execution would not happen starting from Friday moving backwards.
Uncertainty that the execution should happen on any day (SOLUTION #5)
If the prisoner realizes that with their reasoning, they could rule out any day where the execution was to take place (as seen in Solution #1), then they would realize that they would be surprised to be executed every single day, so the execution could take place on ANY DAY (except possibly Friday). (SOLUTION #5).
The execution will be on Friday (SOLUTION #6)
If the prisoner follows their logic from Solution #5, then Friday may be the only day that a random, surprising execution couldn’t take place, because it wouldn’t be expected for the entire week. By excluding Friday and considering all other days possibilities for the execution, the prisoner can guarantee that the execution would be on Friday. (SOLUTION #6)… But this same circular reasoning can guarantee that any day either would or wouldn’t be the day of the execution.
The execution is on… Saturday?( SOLUTION #7-9)
What if the judge lied? What if the prisoner is killed right away, before Monday? What if the prisoner gets to Friday, being certain that he will be executed today, and his execution is postponed one day just to make sure that it’s a surprise day of execution? What if the prisoner is executed on Sunday, or the following Monday, or another later day for the same reason?
There is no prisoner (SOLUTION #10)
There is no prisoner. There is no judge. The hypothetical does not proceed to the random or preordained day of execution. The prisoner is saved, or found to be innocent. Or dies before his execution. Or a social revolution takes place.
The prisoner is the judge (SOLUTION #11)
Every day, the prisoner’s uneasiness increases as the clock approaches 12:00 PM. When noon passes and there is no executioner’s knock at his door, he breathes a heavy sigh of relief that he has another 24 hours to live, only to be met with heightened anxiety the following day. On Friday, the prisoner finally acknowledges that today is his day. The executioner knocks at his door. The prisoner is brought to be executed. The executioner’s mask slips off. The executioner is the judge. The judge swings his axe of justice at the prisoner. The axe does not harm the prisoner. The prisoner, having solved the paradox, is no longer mortal. The prisoner now sentences the judge to be executed next week, between Monday and Friday, on a day that will be a surprise. The prisoner becomes the judge, and the judge the prisoner. The cycle continues.
So, here’s a list of all the above solutions as to when/ if the execution will happen:
- No execution, because there will be no surprise starting from Friday moving backwards.
- Any day execution, because any day’s execution would be a surprise at some point in time.
- Any day execution, because “the day will be a surprise” could be an equivocation for “a random day.”
- No execution, because no days would surprise the prisoner.
- Any day execution (except Friday), because the prisoner cannot have 100% certainty on any day.
- Friday execution, because according to #5, not expecting Friday would make Friday a possibility (and the circular reasoning continues).
- Immediate execution, before Monday comes, to really surprise the prisoner.
- Saturday execution, because the prisoner would be certain of a Friday execution on Friday.
- Sunday execution, because the prisoner would be certain of a Saturday execution if the Friday execution didn’t take place (and so on, and so on).
- No execution. There is no prisoner. Or the prisoner is saved. Or the hypothetical ends.
- No execution. The prisoner reaches Friday, an execution fails as the prisoner becomes immortal, and then sentences the judge to same fate. The cycle continues.
Some of those last few might be a joke, but I’ll leave it to you to interpret when I stopped being serious. Maybe the joke started when I wrote 11 Solutions to “The Prisoner’s Paradox.” ¯\_(ツ)_/¯