### time and uncertainty

I am sure you know this one already, but ...

The director announces that next week will be a fire drill. In order to make it more realistic the day of the drill will be a surprise.

Here is the problem: The drill cannot be on Friday (the last day of the work week), because everybody would know on Friday morning that if the drill did not happen yet it will have to be this day, so it would not be a surprise.

But for the same reason it cannot be on Thursday, because everybody knows it cannot be on Friday and on Thursday morning, knowing that it did not happen yet one would have to conclude it will be this day, so it would not be a surprise. etc.
Therefore the fire drill cannot be on any day.

But on Tuesday the alarm bell rings and of course nobody knew it would be that day...

C.F. v. Weizsäcker discussed the puzzle in his book 'Aufbau der Physik', assuming that it tells us something about the nature of time.

According to Wikipedia no consensus on its correct resolution has yet been established despite significant academic interest (*).

Maybe we should try to assign Bayesian probabilities. Obviously, we have p(Fri) = 0 but then it follows that ...

(*) Notice the citation of the famous remark made by Defense Secretary Donald Rumsfeld!

RZ said...

I have always been troubled by this paradox. Consider the following case: Its Thursday evening and the director announces that there will be a surprise drill by the end of the week. We know this is impossible so we conclude that the fire drill cannot be on any day. Yet on Friday etc...

So basically, the director made an inconsistent statement (which shows that this is a realistic scenario) from which we cannot conclude anything ( disregarding conclusions on the nature of our director's intellect), so it is OK to be surprised by the drill.

I am sure that the above view is not original and exists somewhere in the references you gave but I haven't bothered to go through them.

wolfgang said...

I agree that it is instructive to consider extreme cases.

N=1 Only one day (like in your example) and the director's statement is obviously self-contradictory.

N=2 The surprise fire drill is either Thursday or Friday. But it clearly cannot be on Friday, so it has to be Thursday, ergo no surprise. etc.

N=large The surprise fire drill is on an unspecified day next year.
Clearly there is nothing self-contradictory and one day in March (or plug in another month) the alarm bell actually rings ...

RZ said...

I still think there are two different things going on here.

1) the director makes a self contradictory statement. This is true no matter how large N is.

Nobody would have considered this a paradox if the inspector would have declared a drill on the day in the year which is the first even prime after 2. Then we would directly reject the statement, and resign ourselves to being surprised eventually, even though the date is not even and/or not prime.

2) we are "surprised". But why is that? suppose the director would say "either we have a surprise drill this week, or we do not". It that case there is no contradiction ( it's an empty statement) but we are still going to surprised. So the fact that we are surprised is decoupled from the statement of the director.

What is troublesome is the psychological feeling that the director is somehow conveying information by his statement. alternatively, the director could say "I have pre-picked a day this week for the drill." now, we of course can know on Thursday evening that the drill MUST be on Friday, but on Wednesday we are still in the dark. I think that most people have the first impression that that is what is being said, and that is the source of the confusion.