Unexpected hanging paradox

#1

Post by amelius » Sat, Jun 30, 2018, 12:35 AM UTC

Unexpected hanging paradox
en.wikipedia.org

Re: Unexpected hanging paradox

#2

Post by iforgotpassword » Sat, Jun 30, 2018, 8:02 AM UTC

Of all those paradoxical thought experiments, this is still my favorite, or really high up at least. Maybe because I've heard it at a young age and my little brain was all flabbergasted. The impression must have lasted.

Re: Unexpected hanging paradox

#3

Post by jancsika » Sat, Jun 30, 2018, 10:56 PM UTC

The judge has underspecified the process.

Judge says "on a weekday," and that it will be a surprise to the person being executed. But the judge has not specified which weekdays are actually in play.

Now, let's attempt to specify it:

"Judge will schedule the execution for one of the following days in the upcoming week: Monday, Tuesday, Wednesday, Thursday, or Friday. Judge will pick one of these days at random. This random process will determine the day of the execution, such that the person being executed will not be able to use a process of elimination to predict the day of the execution."

Now we can see that there is no paradox, only a self-contradicting specification.

Re: Unexpected hanging paradox

#4

Post by tristanj » Sat, Jun 30, 2018, 11:03 PM UTC

There is a simple resolution to this "paradox": it's only a paradox if you consider the judge's initial statement to be True. If the judge's initial statement is False (i.e. he's lying) then the whole situation is logically consistent. The truth table for modus ponens helps clarify, both F->T and F->F evaluate as T https://en.wikipedia.org/wiki/Modus_ponens#Justification_via...

The judge is saying he can predict how someone else will feel in the future (i.e. he can predict the future), which unless he is omnipotent is a false premise in my book.

Re: Unexpected hanging paradox

#5

Post by paulddraper » Sat, Jun 30, 2018, 11:07 PM UTC

I enjoy the math versions.

"Theorem" version:

Every integer is interesting. (Otherwise, one number would be the smallest uninteresting integer. And that would be of notable interest.)

"Paradox" version:

What is the largest number that cannot be described in fewer than 100 characters?

Or even the classic paradox by Russell https://en.wikipedia.org/wiki/Russell%27s_paradox

Does the set of all sets that do not include themselves include itself?

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The essence of all these is the malformed self-referential definition. The surprise/interestingness/definition/inclusion is "defined" recursively in such a way as to be actually not well defined.

Re: Unexpected hanging paradox

#6

Post by robocat » Sat, Jun 30, 2018, 11:11 PM UTC

The prisoner introduces "not being hanged" as an outcome.

However the prisoner doesn't include that outcome in their analysis.

Re: Unexpected hanging paradox

#7

Post by frv103 » Sat, Jun 30, 2018, 11:12 PM UTC

Can this concept be generalized to the point where NOTHING is a surprise? For example if a surprise party is planned for someone at some point in $TIME_PERIOD, it can't occur during the very last hour/minute/second of $TIME_PERIOD because it wouldn't be a surprise. The surprise party can't occur during $TIME_PERIOD - 1 because this would be known as well. Repeat this ad infinum just like the weekdays in the article.

Re: Unexpected hanging paradox

#8

Post by mcphage » Sat, Jun 30, 2018, 11:14 PM UTC

If the prisoner considers “I won’t be hung at all” from the start, his argument falls apart on its first day—on Friday he could still be surprised, since he doesn’t know if he’ll be hung at all. Thus getting hung would still be a surprise.

Re: Unexpected hanging paradox

#9

Post by s-shellfish » Sat, Jun 30, 2018, 11:17 PM UTC

jancsika wrote: post #3

The judge has underspecified the process. Judge says "on a weekday," and that it will be a surprise to the person being executed. But the judge has not specified which weekdays are actually in play. Now, let's attempt to specify it: "Judge will schedule the execution for one of the following days in the upcoming week: Monday, Tuesday, Wednesday, Thursday, or Friday. Judge will pick one of these days at random. This r…

The surprise is a surprise in relation to the week. If we use the relation of truth value of surprise to week day, as the prisoner uses to reason, there's a chance it won't be a surprise. That's because the prisoner is treating the statement value as though the judge stated each day, that the hanging will be a surprise. I agree it's not a paradox.

Re: Unexpected hanging paradox

#10

Post by pieguy » Sat, Jun 30, 2018, 11:19 PM UTC

Here's my take. The prisoner considers the statement "I will be hanged, and it will be a surprise", and after "proving" it false, takes its negation "Either I won't be hanged, or it won't be a surprise". But this is not correct. The prisoner actually disproves the statement "I will be hanged, and it will be a surprise no matter which day it happens". When you negate this statement, you obtain "Either I won't be hanged, or there is a day that it would not be a surprise", which is consistent with the outcome.