The Unexpected Hanging Paradox

Factual Questions
21

I don’t see where the problem is. The basic jist of it is that the prisoner cannot be hung if he’s expecting it.

Let’s look at it like this

The prisoner is told this on Friday the 1st.

The time limit starts on Saturday(2nd), Sunday(3rd), Monday(4th), Tuesday(5th), Wednesday(6th), Thursday(7th) and Friday(8th).

That’s 7 options. Only right away you know you can’t be hung on Friday so you’d be expecting it.

The day before the countdown begins: Friday the first, there is no chance of being hung since the countdown starts on Saturday

On Saturday morning the odds are 1:6 he’ll be hung.

At 12:01pm on Satuday he can’t be hung anymore on that date 'cause the execution must take place at noon, and it’s past that

So at 12:01pm the odds are now 1:5 he’ll be hung on Sunday. He could be hung on Sunday, Monday, Tuesday, Wednesday or Thursday.

At 12:01pm on Sunday no hanging takes place, so he’s safe for Sunday.

So the odds are now 1:4. He could be hung on Monday, Tuesday, Wednesday or Thursday.

At 12:01pm on Monday no hanging takes place. So the odds fall to 1:3, he could be hung on Tuesday, Wednesday or Thursday.

Now you can figure out the rest.

There is no paradox because it’s a poorly worded puzzles.

Odds are based on UNEXPECTED outcomes.

The Prisoner was taking the outcome and working backward, while the hangman is taking the situation and working forward.

Statistics and Odds don’t tell you what WILL happen, they tell you what is likely to happen.

Jane has a baby, it’s a boy and see this as so.

Now I go up to Jane and say “what are the odds that the baby you have is a girl.” She’d say “0”. Why? Because she knows the end result.

Now I go to a stranger and say “My friend Jane just had a baby, what are the odds that she has a girl.” He’d say “50:50”

Same question, same facts but the odds are different? Are either correct? Yes they are BOTH correct.

Because Jim and Jane have different frames of reference both can state correct odds.

Odds are based on the TOTAL amount of information your given.

This “paradox” is an example of how the same facts can yield different results depending on which way you look at them (forward in the hangman’s point, and backwards from the crim’s point) and both wind up with correct results.

Another example.

I say to Tom, “There are three doors which one has the tiger behind it.” He picks door one and I open it. Tom sees no tiger."

Then Harry walks into the room. Now I ask the same question to Harry. “Which one has the tiger behind it.” Now Harry is not allowed to ask Tom which one he picked.

At that point in time Harry’s odds are 1:3 of picking the tiger. But Tom’s odds at the same point and time are 1:2 because he has MORE information than Harry.

This is actually a logic puzzle. This is actually a logic puzzle to teach you how to think rather than a puzzle to arrive at a definate answer.

It’s kind of like saying:

“I’ll get you when you least expect it.”

“Well then I’ll always being expecting it. So you can’t get me.”

“But if you’re always expecting it, you’d assume I wouldn’t get you, so in reality that would mean you actually DON’T expect it.”

I think Dick Van Dyke said it best:

Laura) Rob things like that don’t happen to people like us

Rob) That’s what people like us say until something happens then they cease to be people like us

:slight_smile:

22

I studied this paradox as part of my philosophy degree. I have also read Martin Gardner’s excellent commentary on the problem, which is probably the most lucid and succinct.

The judge’s assertions constitute a contradiction, but the wording disguises this fact so it is not immediately obvious.

Thr simplest way to see this is to consider a case where there is only one possible day for the hanging, and that day is tomorrow. The judge’s assertions would sound like this, ‘You will be hanged tomorrow, but you will not know in advance that you will be hanged tomorrow’. The first assertion informs the prisoner that something is going to happen, but the second assertion informs the prisoner that he will not know it is going to happen. If the judge expects the first assertion to be believed (considered true), then he cannot consistently expect the second part to be believed - and vice-versa. It has to be one or the other: either he will be hanged tomorrow or he will not know it is going to happen. One cannot consistently assert both.

The situtation is analogous to me pointing to a box I’ve placed in front of you and saying, ‘When you open the box and find a penny inside, you will be surprised’. At most, one of these assertions can be true. But not both.

The customary wording of the problem involves a range of possible days for the hanging (= all of the supposedly possible days involved in ‘next week’). This tends to mask or disguise the self-contradictory nature of the judge’s set of assertions; nonetheless the self-contradiction remains, which is why the prisoner’s reasoning leads to an apparent contradiction.

23

I am not seeing it as contradictory - the prisoner is, after all, unexpectedly hanged.

Then again, I don’t see the paradox. The prisoner is correct that he cannot be hanged if he doesn’t receive the knock by Thursday afternoon, but every moment before that he doesn’t know when he will be hanged. So they can unexpectedly hang him anytime before Thursday afternoon.

I am not real good at these sorts of things - I didn’t get the paradox in the bellboy and the missing dollar either.

Regards,
Shodan

24

Right, so I think “There is no way that box contains a penny because it wouldn’t be a surprise”. Then I open the box and am surprised to see a penny.

Both of your assertions came true.

25

The simplest resolution is that the prisoner’s reasoning must be invalid because it leads to a logical paradox – ergo, no date for the hanging can be ruled out because the ruling-out is based on flawed reasoning.

26

No the jist is that the prisoner can be hanged on any day, including Friday, because he cannot logically expect it.

The point is, you can’t assume the statement to be true, even though it is. By assuming it to be true, you would make it self-contradictory. This is different from normal true statements that you are unsure about, because normally there is no contradiction in assuming them to be true. Your neighbour says “I once met Buzz Aldrin.” Maybe he did, maybe not, but if you assume it be true there is no contradiction. It could still be true.

So the paradox is of a true statement that becomes self-contradictory if you believe it.

27

If you were not in such a rush to dismiss what I wrote, perhaps you would have read it accurately.

I said ‘When you open the box…’, not ‘Before you open it…’.

I’ll try one more time. Suppose that I tell you something is going to happen: ‘When you open the box, you’ll find a penny inside’. Okay, you open it, find the penny, and it’s a big ‘So what?’.

However, suppose I tell you something is going to happen and I also assert that it will be a surprise or that the outcome will be unexpected: ‘When you open the box, you’ll find a penny inside, and this will be a surprise’. The problem is, can you legitimately claim to be surprised if events unfold exactly as I said they would? If one takes the view that no, one cannot, then my original assertion embodies a contradiction, and the same may be said of the judge’s assertion in the original form of the ‘hanging’ paradox.

As I said before, it helps to consider the one day scenario. The judge says, ‘You will be hanged tomorrow, but this will be unexpected’. If the latter part is true, then the first part cannot be; but if the first part is void then the latter part has no meaning. This is why the prisoner’s reasoning doesn’t resolve satisfactorily to one conclusion. If he believes the latter part (that it will be unexpected) then he cannot also believe the first part (you will be hanged).

In the traditional presentation of the paradox, the range of possible days and the ‘chain’ of reasoning attributed to the prisoner just serve to disguise the fundamental contradiction a little, and to give the paradox some narrative flavour. But in essence it’s the same as the one day version I’ve just given.

28

Friday can actually be ruled out because you simply would not be surprised on that day assuming you believe the judge that you will be hanged that week.

29

Well, the paradox is that since he knows he can’t be hanged on a Friday (because a lack of a knock would indicate he knows he’s being hanged on Friday), then if Wednesday rolls around without hearing a knock, he knows he’s being hanged on Thursday, because if he weren’t being hanged on Thursday, then being hanged on Friday is not a surprise. And so on.

30

You are right and this is the resolution of the paradox. I said the same thing back in post #15. As Martin Gardner put it in his early 1960’s SciAm article on this paradox: “The judge speaks truly and the condemned man reasons falsely. The very first step in his chain of reasoning – that he cannot be hanged on the last day – is faulty.” The article, with addendum, was republished in this book. But why is the judge’s statement true when it turns out to be contradictory from the prisoner’s standpoint? Here’s why (quoting Gardner who paraphrased T. H. O’Beirne): “… the key to resolving the paradox lies in recognizing that a statement about a future event can be known to be be true to one person but not known to be true by another until after the event. … The judge knows that his prediction is sound. But the prediction cannot be used to support a chain of arguments that results eventually in discrediting the prediction itself. It is this roundabout self-reference that, like the sentence on the face of Jourdain’s card, tosses the monkey wrench into all attempts to prove the prediction unsound.”

31

But he can only definitely rule out Friday - ahead of time. That’s because on Thursday afternoon, there is only one remaining day. He doesn’t know he won’t be hanged on Thursday until Thursday afternoon.

At the beginning of the week, there are four possible days when he could be hanged - MTWR. On Monday afternoon, TWR. Etc. It’s only when he gets to Thursday afternoon that the number of possible hanging days is zero.

Or something like that.

Regards,
Shodan

32

But even on Monday he knows it can’t possibly be Thursday, because if it’s set for Thursday once he makes it to Wednesday afternoon he’ll know tommorrow has to be the day (since Friday was already ruled out), and then it can’t possibly be a surprise.

33

Nope, Shodan and pravnik, dear friends, you still don’t see it.

Consider the situation: it’s practically the end of Thursday and the prisoner still hasn’t been hanged. What does the prisoner know? The judge has told him that he will be hanged, but it will be unexpected.

According to the first part of what the judge said, he will be hanged on Friday, the only possible remaining day. So the prisoner expects to be hanged on Friday.

But if he expects it, then the second part of what the judge said will be false. Ergo, he will not be hanged. So now the prisoner does not expect to be hanged. But if he does not expect to be hanged, the first part of his [the prisoner’s] reasoning was incorrect.

Hence it is impossible to say the judge’s statement is either true or not true, because the conditions he lays down are self-contradictory.

And that is the resolution of the paradiox: although it is not immediately obvious, the judge’s assertion [taken as a whole] cannot be ascribed a truth value of either 1 (true) or 0 (false). Any attempt to do so leads to contradiction.

34

ianzin, I’m not getting it either but then I’m usually behind the 8-ball on these things.

I understand the academic nature of your reasoning but the practical side in me sees that as over-thinking the problem. As I stated up thread, the prisoner reasons that he won’t be hanged at all. The judge shows up on Monday, knocks on his cell - surprise! Why can’t the real world resolution be that the prisoners reasoning was wrong? Clearly in a real application (assuming the prisoner’s reasoning) it is.

35

This is almost as bad as the paradox about the two bags that might contain money. The perfect predicter says that he knows what you will decide ahead of time, and if you decide to take just one bag, it and the one you leave behind will both contain $1000. But if you get greedy and try to take both bags they will be filled with blank paper. Are there any logical similiarities between the paradoxes?

36

Calm down there, cowboy. As far as I can tell, Jtgain was posting in good faith.

In any case, “When you open the box…” is not a sentence and can’t be true or false. So it’s not clear to me what contradiction you’re talking about when you say there is one in “When you open the box and find a penny, you’ll be surprised.”

Here are two sentences:

  1. There is a penny inside the box.
  2. You will be surprised when you see the penny inside the box.

There is no contradiction between these two sentences–in fact, both are true in the scenario described. There is a penny inside the box, and the person opening the box is surprised to find it there.

I suspect the contradiction you’re talking about arises when you add a third implicit assumption, something like:

  1. There is a penny inside the box.
    1A. I hereby, by this action of telling you about the penny, make it such that you can not be surprised when you see the penny inside the box.
  2. You will be surprised when you see the penny inside the box.

(In other words, I think the contradiction you’re talking about arises not from a logical analysis of what the judge says, but rather, from an analysis of what sentences like those he uttered are typically intended to accomplish. For example, when I say “I hereby christen thee ‘Jacob’”, a logical analysis of what I said reveals that I made the claim that I christened thee Jacob. But an analysis of what a sentence like that is typically intended to accomplish reveals that I intended to make it the case that someone was named Jacob. Sentences don’t just mean things, they do things as well.)

In fact, there’s no need for that first sentence now. Here’s the contradiction:

A. Here by this action of telling you there is a penny in the box, I make it such that you can not be surprised when you see the penny inside the box.
B. You will see a penny in the box and be surprised.

I think those two sentences are indeed contradictory. The first claims you can’t be surprised, the second claims you will be surprised.

And since they are in contradiction, we can’t draw any decent conclusions from them. This means we should draw no conclusions at all about whether there’s a penny in the box.

The question is whether A and B are fair representations of what is happening in the scenario. (They’re not fair representations of what the puzzlemaster says, but they may be fair representations of what the puzzlemaster does. I.e., he doesn’t say “I make it such that you can’t be surprised etc…” but what he does amounts to an act normally intended to make it such that you can’t be surprised etc…

37

No, I see it fine, thanks. I was just attempting to explain the prisoner’s reasoning to Shodan, who didn’t see any paradox.

38

You are correct.

But - but -

Both the judge’s statements are true. The prisoner was hanged. And he didn’t expect it.

Unless you mean something different than what I think you do.

Regards,
Shodan

39

[ Ianzin throws in the towel ]

“Well, I tried, didn’t I, goddamit? At least I did that.”

40

I’m with you. I still don’t get it either for the reason that you said. The statements are not self-contradictory because they both ended up being true.