Solving Hangman's Paradox: Axioms, Deductions & Probabilities

[pinball1970]

TL;DR Summary

What did the prisoner do wrong?

All the information is in the link. The condemned man deduced he could not executed.

https://en.wikipedia.org/wiki/Unexpected_hanging_paradox

This started from a fairly innocuous post until fresh42 posted this link. Perok gave me guidance to work through it and I would like views on this.

I kind of get it now I think, breaking the statements down into assumptions and deductions, combining the two is not permitted because you are asserting what you are trying to prove.

I am trying to rephrase the information so I am not making the same mistake as the prisoner or Judge.

The wiki example seemed complicated.

“The prisoner will be hanged next week and the date (of the hanging) will not be deducible the night before from the assumption that the hanging will occur during the week (A).[1]”

So I tried this.

Would this work?

Axioms

1/Executed on one week day Mon- Fri.

2/Not be told by anyone which day.
So this is the same situation using slightly different words.

What can I deduce from these?

If I am alive on Monday at 12.01 then I will be killed on one of the four remaining days and not be informed which

If I am alive on Tue at 12.01 then I will be killed on one of the three remaining days but not informed which and so on…..till Thursday

If I am alive on Thursday at 12.01 then I will be killed the following day, axiom 2/ is still correct because no one has informed me

Re “surprise,” the judge cannot use that word to apply to Friday as the prisoner can deduce Friday will be the day he is executed if he is alive Thurs 12.01.

The prisoners mistake was to then use “surprise” as part of his deductions (as that was part of the original axiom from the judge which was false)
Is there a mathematical breakdown? Probabilities? Monday morning he has a 20% chance of execution? Tuesday at 12.01 it moves to 33% for Wednesday but he can only be 'surprised' till Thursday 12.01 when 50% changes to 100%?

I know there must be I am missing things because whole critiques have been written about this with “philosophical” implications.
I am not a mathematician but please take your gloves off with this.
Thanks.

 

[FactChecker]

I have not studied this paradox, but it seems to me that the prisoner's logic is flawed. He determines that Friday is impossible under the assumption that he has not been hung on Thursday (or prior). Then he decides that Thursday is impossible under the assumption that he was not hung on Wednesday or prior and that Friday is impossible. Aren't the assumptions of Thursday and Friday dependent on each other? This seems like circular logic.

 

[anuttarasammyak]

If we introduce continuous time instead of digital time, date, the story would not be a paradox. The execution at only one second or any tiny time before the dead line would surprise us.

 

[PeroK]

Here's my analysis of this paradox:

First, let’s clarify the problem somewhat. We can also get rid of the unnecessarily grisly aspect involving execution. We start with a similar but valid problem.

Basic Problem

1) You are told that there is a bag with some numbered balls in it. The balls are numbered 1-5, but you are not told how many of each there are. Note that some numbers might be missing.

2) You are told that a single ball will be drawn at random. I.e. that a single number will be chosen from 1-5.

3) If the ball is a 1, you will be told on Monday at 10 am; if it is a 2, you will be told on Tuesday etc.

It’s clear that if the number 5 ever gets selected, then on the Friday morning before 10 am you know it must be 5 and in this case you are certain of the number before being told.

It’s also clear that if the number 5 is not in the bag, then this case never arises and you can never be certain of what number was chosen. The best you can do is on Thursday know that either a 4 or 5 must have been selected.

That's the problem from your perspective as a "player". The organisers of the game, however, know which balls are in the bag. Let's assume now that there is no number 5 in the bag. They know that the game will never get to Friday and you (the player) will never be certain of what ball was chosen before being told.

Note: if you play the same game enough times, you might suspect there is no 5 in the bag, but that is a different question altogether.

Tricky Problem

Now, however, we make the problem tricky by telling the player that they will never be certain.

4) You are told that you will never be certain of the ball that was chosen before being told.

At first sight, this appears simply to rule out the number 5 being in the bag. But, that reduces the game to one where only the numbers 1-4 are involved. And, of course, once you have excluded 5, you can proceed by the same logic to exclude the number 4, then the number 3, then the numbers 2 and 1. And, by what appears to be sound logic you conclude that given rule 4), then rule 4) itself cannot be true/valid.

The simplest resolution to the paradox is, therefore, that rule 4) is analogous to “this statement is false”. And, this self-reference allows the possibility of self-contradiction because rule 4) itself tells you what you may deduce by using rule 4).

Analysing Rule 4)

We first take a closer look at the original game, assuming there is no number 5 in the bag. We first consider that there is a chief organiser who puts the balls in the bag. The other organisers know that there is no number 5 in the bag - because they know that the player should never be certain of the ball before being told. But, unknown to them, the chief organiser only puts balls 1-3 in the bag. Also, they do not see which ball the chief organiser selects. From their perspective, all they know is that it can't be number 5.

Now, if we replay the original game, we find:

The player is never certain because the game never gets to Friday.

The other organisers are never certain because the game never gets to Thursday.

Only the chief orgnaniser has all the information.

This gives the clue of how to fix rule 4).

Fixing Rule 4

To fix rule 4), we define a level 0 player as someone who is given only rules 1)-3). A level 1 player is given rule 4), which is now:

4) A level 0 player can never be certain of the number before being told.

This means that a level 1 player is not using rule 4) self-referentially. And, a level 1 player may validly conclude what the other organisers know - that there is no number 5 in the bag. The level 1 player, however, can no longer exclude number 4, because there is no rule about them (a level 1 player) never being certain. And, the game works well on a level 0 player with number 4 in the bag.

We can also see that the chief organiser has made themselves a level 2 player, who has rule 5):

5) A level 1 player can never be certain of the number before being told.

That allows the level 2 player, who knows rule 5), to exclude number 4.

Note: we can extend this to five levels of player if we want.

Finally, we can see more clearly now where the logic following rule 4) went wrong. The player correctly concluded that there was no 5 in the bag (that implicitly applied to the level 0 player). But, when they excluded number 4, they were extending the surprise/certainty logic to include themselves. That's where the self-reference created a contradiction.

Conclusion

The problem as originally set-up has a self-referential rule that is analagous to "this statement is false".

A careful analysis shows that surprise/certainty can only be applied generally using a system of players of different levels of knowledge.

In this particular case, it's when the player applies rule 4) to themselves (as a level 1 player) that the contradiction appears.

 

[pinball1970]

Here's my analysis of this paradox:

First, let’s clarify the problem somewhat. We can also get rid of the unnecessarily grisly aspect involving execution. We start with a similar but valid problem.

Basic Problem

1) You are told that there is a bag with some numbered balls in it. The balls are numbered 1-5, but you are not told how many of each there are. Note that some numbers might be missing.

2) You are told that a single ball will be drawn at random. I.e. that a single number will be chosen from 1-5.

3) If the ball is a 1, you will be told on Monday at 10 am; if it is a 2, you will be told on Tuesday etc.

It’s clear that if the number 5 ever gets selected, then on the Friday morning before 10 am you know it must be 5 and in this case you are certain of the number before being told.

It’s also clear that if the number 5 is not in the bag, then this case never arises and you can never be certain of what number was chosen. The best you can do is on Thursday know that either a 4 or 5 must have been selected.

That's the problem from your perspective as a "player". The organisers of the game, however, know which balls are in the bag. Let's assume now that there is no number 5 in the bag. They know that the game will never get to Friday and you (the player) will never be certain of what ball was chosen before being told.

Note: if you play the same game enough times, you might suspect there is no 5 in the bag, but that is a different question altogether.

Tricky Problem

Now, however, we make the problem tricky by telling the player that they will never be certain.

4) You are told that you will never be certain of the ball that was chosen before being told.

At first sight, this appears simply to rule out the number 5 being in the bag. But, that reduces the game to one where only the numbers 1-4 are involved. And, of course, once you have excluded 5, you can proceed by the same logic to exclude the number 4, then the number 3, then the numbers 2 and 1. And, by what appears to be sound logic you conclude that given rule 4), then rule 4) itself cannot be true/valid.

The simplest resolution to the paradox is, therefore, that rule 4) is analogous to “this statement is false”. And, this self-reference allows the possibility of self-contradiction because rule 4) itself tells you what you may deduce by using rule 4).

Analysing Rule 4)

We first take a closer look at the original game, assuming there is no number 5 in the bag. We first consider that there is a chief organiser who puts the balls in the bag. The other organisers know that there is no number 5 in the bag - because they know that the player should never be certain of the ball before being told. But, unknown to them, the chief organiser only puts balls 1-3 in the bag. Also, they do not see which ball the chief organiser selects. From their perspective, all they know is that it can't be number 5.

Now, if we replay the original game, we find:

The player is never certain because the game never gets to Friday.

The other organisers are never certain because the game never gets to Thursday.

Only the chief orgnaniser has all the information.

This gives the clue of how to fix rule 4).

Fixing Rule 4

To fix rule 4), we define a level 0 player as someone who is given only rules 1)-3). A level 1 player is given rule 4), which is now:

4) A level 0 player can never be certain of the number before being told.

This means that a level 1 player is not using rule 4) self-referentially. And, a level 1 player may validly conclude what the other organisers know - that there is no number 5 in the bag. The level 1 player, however, can no longer exclude number 4, because there is no rule about them (a level 1 player) never being certain. And, the game works well on a level 0 player with number 4 in the bag.

We can also see that the chief organiser has made themselves a level 2 player, who has rule 5):

5) A level 1 player can never be certain of the number before being told.

That allows the level 2 player, who knows rule 5), to exclude number 4.

Note: we can extend this to five levels of player if we want.

Finally, we can see more clearly now where the logic following rule 4) went wrong. The player correctly concluded that there was no 5 in the bag (that implicitly applied to the level 0 player). But, when they excluded number 4, they were extending the surprise/certainty logic to include themselves. That's where the self-reference created a contradiction.

Conclusion

The problem as originally set-up has a self-referential rule that is analagous to "this statement is false".

A careful analysis shows that suprise/certainty can only be applied generally using a system of players of different levels of knowledge.

In this particular case, it's when the player applies rule 4) to themselves (as a level 1 player) that the contradiction appears.

I am going to read your post a few times I think.

 

[pinball1970]

If we introduce continuous time instead of digital time, date, the story would not be a paradox. The execution at only one second or any tiny time before the dead line would surprise us.

Possibly but we could still get an issue if you told the prisoner his execution would be a surprise.

As the clock struck the penultimate second we would immediately know the knock at the door would come on the final second. It would not be a surprise. We can eliminate the final second. Using this reasoning can we therefore eliminate the penultimate second? The one before? If not why not?

 

[anuttarasammyak]

As the clock struck the penultimate second we would immediately know the knock at the door would come on the final second. It would not be a surprise. We can eliminate the final second. Using this reasoning can we therefore eliminate the penultimate second? The one before? If not why not?

I said continuous time. You still seem to have digital or discrete time of unit second in mind.
Say the condition is at T=0:00AM of the next day I should be dead so the execution should be done before that say time t=T-$\epsilon$ where $\epsilon>0$. At that time I would be surprised and angry saying, " Why now ? I still have time $\epsilon$ to live!" $\epsilon$ can be any tiny amount of time e.g. 1/1000..00 second.

 

[PeroK]

I said continuous time. You still seem to have digital or discrete time of unit second in mind.
Say the condition is at T=0:00AM of the next day I should be dead. The execution should be done before that say time t=T-$\epsilon$ where $\epsilon>0$. At that time I would be surprised and angry saying, " Why now ? I still have time $\epsilon$ to live!" $\epsilon$ can be any tiny amount of time e.g. 1/1000..00 second.

This is not adding anything to the debate. It's a question of logic.

 

[anuttarasammyak]

This is not adding anything to the debate. It's a question of logic.

I admit a different condition makes a different problem with the original one. The condition I introduced would make the prisoner surprise for execution of any time before the deadline so logic of the executioner is not challenged.

 

[pinball1970]

I admit a different condition makes a different problem with the original one. The condition I introduced would make the prisoner surprise for execution of any time before the deadline so logic of the executioner is not challenged.

This problem has enough in it I think.
It is looking at a specific issue using the assumption as part of the deduction.
You can remove that by changing things completely but then we are not addressing the problem.
This problem has had me thinking for several days now. A sentence can make perfect grammatical sense but be flawed in terms of logic.
Assumptions can be used but only in a certain way, I don't understand this as I have never come across it before.
This gets at it.
Peroks breakdown of WHY the axiom / deduction does not work is excellent but when other players are introduced I got a bit lost.
I need to absorb that part.
Can all statements be analysed this way? I suppose they can. I have read about incompleteness and Gödel, ABOUT it.
I am now very interested.

 

[jbergman]

Terry Tao analyzes this puzzle in his blog along with the blue-eyed islander puzzle which feels similar but doesn't have a paradox. https://terrytao.wordpress.com/2011...e-eyed-islander-puzzle-lower-bound/#more-4878