Wednesday, November 16, 2005

The paradox of the unexpected fire drill

A message came round in mid-October, saying that we were going to have a surprise fire alarm practice some time in November. All we were told is that it would happen at 3pm one day, but we would not know which day it was going to be until the alarm went off.

So I got to thinking about what day was most likely. It was immediately obvious that it can't be scheduled for the 30th, because if it has not happened before we come into work on that day, we would know it has to happen that afternoon, and therefore it would not be a surprise. But on the morning of the 29th, there would only be two possible days left, and having already ruled out the 30th as impossible, it would have to be the 29th. So there is no surprise there either. Having ruled out the 30th and 29th, it is easy to show that the 28th is similarly impossible. And so on, right back to the 1st. I conclude that we can't have the surprise fire drill at all!

I know you will all be fascinated to hear how things turn out.

Update 16 Nov

Well, it happened today. Was it a surprise? No not really, as someone had already told me it was planned for this day. In fact I made up the story about it being a surprise. But it was only after glancing at the Wikipedia page on the paradox of the unexpected hanging that I found out that this is also known as the paradox of the unexpected fire drill. If you still don't know what I'm talking about then read the page and all will become clear.

8 comments:

CapitalistImperialistPig said...

My guess is that if they really wanted to surprise you, they wouldn't have told you in advance. Administrators don't like surprizes because they can make them look bad. Count on it being on the twenty-eighth.

Steve Bloom said...

Somehow I'm unable to shake the image of all those OL's racing outside in their fur-lined lingerie. Please don't ruin it for me by describing what it really looked like.

EliRabett said...

I understand this is how executions are carried out in Japan

crandles said...

I have added the following comments to the talk page which say:

Hidden assumption that prisoner knows the method the executioner decides
The [wikipedia] article refers to several possible weaknesses in the situation that perhaps could be a way of finding a flaw in the logic of the paradox. But there seems little evidence presented that these weaknesses are the actual flaw. I think I can see the actual flaw (or perhaps there are also other alternatives) and that it is that there is a hidden assumption that the prisoner knows the method that will be used to decide the day. Implicit in the judge's sentence is the assumption that there has to be a method of deciding the day (but this doesn't cause the flaw). If I am right about the flaw, then removing that assumption should make the paradox go away. So let's try that:

If the judge passes a sentence saying:

1. You will be hanged at noon one day next week, Monday through Friday.
2. The execution will be a surprise to you because the executioner will choose a method of deciding the day so that in no circumstance will the prisoner know the day of the hanging until the executioner knocks on your cell door at noon that day.
3. In choosing a method of deciding, the executioner is to assume that the prisoner is able to guess all details about the method.

I believe the paradox has disappeared here and the prisoners logic works correctly. The executioner cannot find a method so at least one of the statements has to be broken.

Let's try some examples: If the method was to choose a random number up to 3, then the prisoner might be suprised on day 1 or 2. However if the random number was 3, the prisoner would not be suprised; so this method is no good and cannot be used by the executioner. Whatever method is chosen, there has to be a last possible day and this causes that method to be rejected as impossible. Therefore there is no inconsistency between being suprised and there being no method because the lack of the method prevents the suprise.

The paradox has disappeared with this 'know the method' assumption made explicit and clearly if the prisoner does not know the method he can be suprised. Therefore the flaw in the paradox is the hidden assumption that the prisoner knows the method.

So what do you think, have I found the flaw more precisely than the wikipedia page's efforts?

James Annan said...

Chris,

I think you are overcomplicating things a bit. I don't think you have to go as far as wondering about methods etc.

Consider the simpler version: "The fire drill will be at 2pm tomorrow, and it will be a surprise". There is no decision to make. It seems to me that it is closely related to the simple "this statement is false" paradoxes - or maybe the "this statement is unprovable" of Godel's incompleteness theorem. In fact, if you swap "provable" for "believable" then it seems to be an near-exact analogue. Consider "this statement is not believable". It is true, but cannot be believed to be true. The statement "you will be hanged tomorrow and will not expect it" is also true but cannot be believed.

crandles said...

I do agree that it is closely related to Godel's incompleteness theorem.

However being closely related means it is not exactly the same.

I feel a paradox is not fully resolved until you can explain the exact circumstances when the flawed logic would work and how the paradox situation differs from that. Otherwise you run the risk that you are saying you know the paradox's logic is wrong but not why and therefore not resolving the paradox.

This is a high standard to set before saying that the paradox is fully resolved; but I think it is a high standard that one should aim to achieve.

Having said this, I accept that the article saying "note the difference between the truth of a statement and knowledge about this truth. The judge's statements might be true, but the prisoner can't know that they are true." has correctly found the area of the flaw.

Having set the standard at what I think is a high level, I am not sure that the article meets my high standard.

Claiming that the wiki article resolves the paradox but not fully resolves it sounds like I am creating a ficticious level of proof required. I wouldn't want to do that. I am merely trying to suggest a high standard of documentation (whether that is for a proof or a good explanation).

OTOH I could be going wrong for different or opposite reasons. Perhaps I have found one precise assumption that allows the flawed logic to work but there could be others. If these could all be grouped in a 'similar to Godel's incompleteness theorem' category, then my notes would still be woefully incomplete and best avoided on the wiki page unless/until someone completes them. In this case, it seems sensible to just leave my notes on the talk page as a suggestion to someone to complete the documentation.

However, I could be completely wrong and it may indeed be an unnecessary complication.

I think I can make a case that the unexpected hanging is a different paradox from your "The fire drill will be at 2pm tomorrow, and it will be a surprise" paradox because my method notes do not apply. Therefore my high documentation standard serves a purpose and this makes them worthwhile.

James Annan said...

Chris,

I'm not convinced. The paradox is unchanged even when only one day remains - the sentence is still true but not believable. I also note that even the somewhat simpler "liar paradox" ("this sentence is false") seems to have generated a fair amount of philosophical debate which I am not qualified (or sufficiently motivated!) to wade through. Given that the English language contains simple syntactically-valid sentences which can be neither true nor false, it seems to me that the existence of a sentence which is true but not believable is not really such a big deal.

truth machine said...

"The fire drill will be at 2pm tomorrow, and it will be a surprise" is indeed not believable, but it may be either true or false. The same goes for the judge's statement. The prisoner reasons that he can't be hung on Friday, else the judge's statement would be false. But that's ridiculous -- indeed the judge's statement *could* be false, so the prisoner *can* be hung on Friday. And since he can be hung on Friday, there's certainly no reason why he can't be hung on Thursday, Wednesday, etc. Certainly a prisoner who is consigned to his fate but uncertain of the day of hanging won't be *surprised* when the hangman shows up, regardless of when that is. Only the prisoner who has bogusly reasoned that he can't be hung will be surprised -- but the judge can't be certain that the prisoner has so reasoned. The judge is just making utterances, possibly true and possibly not, and such utterances have no bearing on what is possible and what is not.