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| Author |
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md
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Paradoxes
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May 27 13:52 UTC 1994 |
In a book about paradoxes I once came across the following two.
In the book, they were presented as two of the great unsolved
mysteries of the human mind, but to me the solutions seemed
easy - too easy, in fact. So, I'm entering the paradoxes here
in the hope that someone can enlighten me.
1. THE UNEXPECTED HANGING
A judge says to a murderer: "You stand convicted of one of the
most brutal crimes I've ever seen. The death penalty is too
good for you. But unfortunately it's the worst sentence I can
pass, so I'm going to have to add a little twist to it. I'm
going to tell you that you will be hanged at 8:00 AM some day
next week, but I won't tell you which day. It will come as a
total surprise. You won't know what day you will be hanged
until the minute they come to take you away!"
The murderer's lawyer chuckles to himself, and the murderer
says, "I'm glad *you* find this funny. I think it's horrible!"
The lawyer says, "But don't you see? They *can't* hang you
now! The judge said you would be hanged next week. That means
the last day they can possibly hang you is next Saturday,
right? But if 8:00 AM Friday comes and goes and you're still
alive, then you'll *know* the hanging will take place on
Saturday. But the judge said it will come as a total surprise,
so the hanging can't take place on Saturday, and the last day
it *can* take place is Friday. But if Friday is the last day,
and if 8:00 AM Thursday comes and goes and you're still alive,
then you'll know the hanging will take place on Friday. But
the judge said it will come as a total surprise, so the hanging
can't take place on Friday. That means the last day it *can*
take place is Thursday..." In this way, the lawyer reasons all
the way back to Sunday, and concludes beyond any doubt that his
client can't possibly be hanged.
The murderer, now chuckling like his lawyer, goes to his cell
on death row. At 7:55 on Tuesday morning the guards come and
take him away to be hanged, and, sure enough, it comes as a
total surprise, just as the judge said it would.
2. NEWCOMBE'S PARADOX
A psychic whose predictions are known to be 90% accurate puts
two boxes in front of you, marked "A" and "B," and says:
"You can choose either the contents of both boxes, or the
contents of just box B. Last night, I predicted which you
would choose. If I predicted that you would choose both boxes,
I put $1,000 in box A and left box B empty. If I predicted
that you choose just box B, I put $1,000 in box A and
$1,000,000 in box B."
Which do you choose? Both boxes, or just box B?
Sounds simple at first. Obviously, you choose just box B
because you want that $1,000,000. But wait a minute...
Whatever the psychic predicted and however much money he put in
the boxes, that was last night, right? Either the $1,000,000
is in box B or it isn't. Nothing can change that now.
Certainly no choice you make now is going to have any effect on
the contents of the boxes. How could it? Choosing just box B
can't make the $1,000,000 appear if it wasn't in there to begin
with; and choosing both boxes can't make the $1,000,000
disappear from box B if it's already there. So, why *not*
choose both boxes? Why lose that $1,000 in box A?
So, which do you choose? Both boxes, or just box B?
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| 138 responses total. |
canis
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response 1 of 138:
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May 27 13:56 UTC 1994 |
Choose both. Take the sure thing.
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carl
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response 2 of 138:
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May 27 22:36 UTC 1994 |
Even as I read the lawyer's logic, I knew it wasn't right.
I don't see where the paradoxes are either.
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carson
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response 3 of 138:
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May 29 06:16 UTC 1994 |
(as far as Newcombe's Paradox goes, of COURSE you take both boxes!! If the
psychic is wrong, the psychic is REALLY wrong! What's so hard about that?
It's not like the psychic is 100% accurate anyway!)
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remmers
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response 4 of 138:
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May 30 12:44 UTC 1994 |
Are you so sure? It's given that the psychic is known to be 90%
accurate, and $10,000,000 is far more than ten times $1,000, so if
you believe in playing the odds (as the best gamblers do) ...
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carson
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response 5 of 138:
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May 30 16:35 UTC 1994 |
(yes I'm sure. No matter what the psychic predicts, the money in the two
boxes isn't going to change, because the psychic has already put it there.
If the psychic predicted wrong, oh well! It'll be the easiest cash I ever
made.)
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md
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response 6 of 138:
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May 31 17:58 UTC 1994 |
Remmers is right. If the psychic is 90% accurate, then when you
choose both boxes aren't you in effect betting $1,000,000 that
you got him on a 10% night? Does that sound very wise to you?
("90% accurate" is how the book worded it, and I've never
understood why. Maybe a 90%-accurate psychic supposed to be
easier to believe in than a 100%-accurate psychic?)
Well, let's suppose the psychic is known to be 100% accurate.
What then? If the psychic was known to be 100% accurate,
wouldn't you have to choose just box B if you wanted the
$1,000,000? If you choose both boxes, aren't you merely
fulfiling the prediction the psychic made the night before, and
aren't you guaranteed in that case to find box B empty?
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carson
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response 7 of 138:
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Jun 1 10:35 UTC 1994 |
(Remmers is *wrong*! Why would you *not* take *both* boxes? Is there some
sort of penalty that I'm missing as far as TAKING the second box as well?
I'm also trying to figure out what the psychic has to do with ANYTHING.
Am I to believe that with a wave of her/his hand, the psychic can change
the contents of the boxes? The psychic put whatever was going to be in the
boxes into the boxes THE DAY BEFORE. Not now, not a couple of minutes ago,
and most definitely NOT after you choose which boxes to take. This would
be very different if you had to choose between box A and B, but you DON'T.
You *can* take both! You *will* get at LEAST $1,000, no matter what.)
(I don't understand why this is so hard. There are two boxes. Do you take
one or both? That's it. That's the only question being asked, when you get
down to it.)
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jason242
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response 8 of 138:
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Jun 1 13:31 UTC 1994 |
Its not that easy, carson. If I choose to take one box there is a 90% chance
that there is a hulluvalota money there. If I choose both boxes there is
a 90% chance there is 1000 dollars there. So, I'd choose the one box.
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jason242
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response 9 of 138:
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Jun 1 13:35 UTC 1994 |
Dangit, got pulled, so I'll finish. The timing of the psychic is not relevant.
The psychic is correct 90% of the time. So whichever you choose, the psychic
has a 90% chance of correctly guessing which you will pick. If you could
somehow change your mind without the psychic knowing then the 90% would drop
accordingly. I would choose box B.
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carson
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response 10 of 138:
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Jun 1 16:12 UTC 1994 |
(It *is* that easy! It's either there or it isn't! Who cares what the
psychic thought yesterday?!? Whatever is going to be in those boxes is
there when you make the selection! It's *not* going to change because
you're trying to outthink some half-baked fortune teller! Sure, I guess
you *could* have spent most of the day before thinking, "gee, that psychic
sure is a nice person. I'll only take one box so that s/he doesn't lose
their life savings", but FOR CRYIN' OUT LOUD, TAKE BOTH BOXES!)
(I guess the point that the one-boxers are trying to make is that if you
want to take both boxes, it will somehow change the contents of the boxes.
While what goes on in your head might have affected *yesterday*, when the
psychic put the money in the boxes, it's not affecting *today*, when
you're actually making the choice. If there's going to be a "hulluvalota"
money in one of the boxes, why wouldn't you take BOTH just to make sure?)
(Please, if I'm REALLY missing something, PLEASE point it out.)
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md
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response 11 of 138:
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Jun 1 17:53 UTC 1994 |
Interesting. There does seem to be a paradox here after all, or at
least room for major disagreement.
It's not exactly as if there's a penalty for taking the second box,
carson, although it does seem that way. Your choice isn't what
determines the contents of the boxes. The psychic's prediction
last night is what determined that. If the psychic is 100%
accurate, and if you choose both boxes, then obviously that is what
he must have predicted you would do.
(Btw, *how* you will reach the decision to choose both boxes is
irrelevant as far as the psychic is concerned. Belief in psychic
ability, unbelief in psychic ability, belief in free will, belief
in fate, stubbornness, pure logic, pure greed, or whatever - in
short, your entire thought process - counts for nothing in the
psychic's mind. The one and only thing he did was to predict what
choice you would make. He is 100% accurate. Whatever choice you
make, no matter how you arrive at it, *by definition* has to be the
very choice he predicted you would make.)
That is not to say that the psychic is robbing you of your free
will, or that his prediction somehow forces you to make a certain
choice. In fact, you could almost say the opposite: that it's the
foreknowledge of your choice that forces *him* to make the
prediction he makes. So, if you choose just box B, then that has
to be what he predicted you'd do, and there will be $1,000,000 in
box B (and $1,000 in box A); and if you choose both boxes, then
*that* has to be what he predicted you'd do, and there will be
$1,000 in box A and nothing at all in box B.
So, I repeat, what's your choice?
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gerund
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response 12 of 138:
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Jun 1 18:20 UTC 1994 |
I hate to ask a dumb question, since I'm not really following the arguments
here, but even if the psychic is 100% to date what's to say he's going to
be right NOW? I know, it's probably irrelavant, but I just thought I'd
ask a stupid question. :-)
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md
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response 13 of 138:
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Jun 1 20:59 UTC 1994 |
That's a very good question. If a psychic who was 100% accurate predicted
that you'd wear a hat today, couldn't you deliberately avoid wearing
a hat just to confound his prediction? For that matter, if some
future supercomputer could analyse all of the data in a room you
were in, and caculated the exact position of your left arm 5 seconds
from now, couldn't you deliberately not hold your arm in that position?
Your question comes very close to the heart of this "paradox."
Hint: What I noticed about "The Unexpected Hanging" is that the
terms contradict each other. If the judge simply said that the
prisoner would be hanged on some unspecified date in the future,
it would indeed come as a complete surprise when it came, and
there's no paradox. But by telling the prisoner that he was
specifying a last possible day (next Saturday), the judge makes
it impossible for the hanging to be a surprise if it takes place
on that day. The judge is making a promise he can't keep in
every case: a date can't be both specified and unspecified at the
same time. If the prisoner guesses a day at random, he'll
initially have a 1/7 chance of being right; if he's still alive
Sunday night, he'll have a 1/6 chance of guessing right; on
Monday night, a 1/5 chance; Tuesday night, 1/4; Wednesday night,
1/3; Thursday night, 1/2; and Friday night (and here is where the
judge messes up), 1/1. When the lawyer concludes that the
hanging can't take place on Saturday, he's wrong. Of course it
can take place on Saturday; it just won't be a surprise.
Having figured that one out, I began to suspect that there had to
be a similar contradiction in the setup for Newcombe's Paradox.
I *think* I know what it is, but I'd like to hear what others
have to say first.
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carson
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response 14 of 138:
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Jun 2 02:37 UTC 1994 |
(actually, regarding "The Unexpected Hanging", the reason it was a
surprise was because he had "expected" it, and they killed him anyway. At
least, that's what I'd gathered.)
(I think what we're forgetting about Newcombe's Paradox is that our
decision on which boxes to choose *isn't* going to affect what's already
in the boxes. The $1,000 will either be in box A or B. The other box will
either have $1,000,000 or it will be empty. I'm really failing to see
what's so difficult about this.)
(BTW, the psychic is only *90%* accurate. I guess it's guys like me that
throw the accuracy out. ;) )
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jason242
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response 15 of 138:
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Jun 3 14:40 UTC 1994 |
Obviously you choose box B. Why? Because when you do, the psychic knew
you would, and put a cool million in it for ya. Well, at least there is a
90% chance that it it there. Since we gotta believe inb the power of psychics
for this, we gotta think like the psychic put the money there after we made
the choice. If you were able to alter that, then the 90% would go down
accordingly.
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carson
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response 16 of 138:
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Jun 3 19:12 UTC 1994 |
(OBVIOUSLY YOU CHOOSE *BOTH* BOXES!!!)
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remmers
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response 17 of 138:
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Jun 6 16:10 UTC 1994 |
Not obvious to me. If the psychic is 100% accurate, then if you
choose box B, then that means she predicted you'd choose box B,
and that means she put $1,000,000 in it. In other words, if
you choose box B, you are certain to win $1,000,000. Similarly,
if you chose both boxes, that means she predicted you'd choose
both boxes, and that means you win only $1000. So "box B" is
a thousand times preferable to "both boxes".
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vishnu
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response 18 of 138:
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Jun 6 21:17 UTC 1994 |
It is very not obvious. I'd say Box B, probably.
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carson
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response 19 of 138:
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Jun 8 16:22 UTC 1994 |
re #17: (She's only *90%* accurate.)
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md
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response 20 of 138:
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Jun 8 16:33 UTC 1994 |
[Somewhere up there I tried to simplify the problem by asking
everyone to assume the psychic is 100% accurate. If she's 90%
accurate, then you have to add "nine times out of ten" to the
statements in remmers' response #17. That doesn't make your
choice any more obvious, unless I'm missing something.]
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jason242
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response 21 of 138:
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Jun 8 17:17 UTC 1994 |
I agree. To figure out how much money you would get on average simply multiply
by .9. It doesn't really change anything.
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carson
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response 22 of 138:
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Jun 8 19:26 UTC 1994 |
(I really don't know how I can make this any clearer. You *can't* remove
the fact that the psychic is *90%* accurate if you're going to base your
guess on the psychic's prediction. My point couldn't care less *what* the
psychic predicted. The money is going to be in the boxes or it isn't.)
(I bet the reason that the psychic only claims *90%* accuracy is so that
when you people choose "only box B" [It still seems stupid to me, but I
digress], and there is only $1,000 in it, the psychic can say, "oops!
those pesky little fates...")
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jason242
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response 23 of 138:
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Jun 8 19:31 UTC 1994 |
Then you are changing the situation. This problem mandates acceptance of the
psychic. It is fundamental, and cannot be looked at as a skam. If you remove
the psychic then you must choose both boxes, because that is the only way
money will be there for sure. If you keep the psychic there is money in the
box you pick 90% of teh time. Not bad, eh?
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carson
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response 24 of 138:
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Jun 9 08:16 UTC 1994 |
(jason242, please type "only 0" after reading this and look at the
original paradox. I haven't changed the situation. There is no need to
accept the psychic's word as gold. There's no reason to even believe in
psychics for the sake of the problem. I'm simply trying to show why I
agree so strongly with the sentiments expressed in #0 in regards to why
anyone would pick both boxes. I don't think the psychic's prediction is at
all relevant at the time of the choosing of the boxes because the money is
already in the boxes and the amounts are not going to change because of
your choice of boxes.)
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