Logical Lo knew it was a mistake to holiday in Wild West parts,
where cowboys and sheriffs were laws unto themselves – and to
visitors.Here sits Lo,sentenced to death by hanging – a hanging
with a twist, a very strange twist, in its noose.The sheriff had
taken a dislike to her when she had rebuffed his advances,
denounced him as sexist and danced a jig on his ten-gallon hat.
’Tis now Sunday.The sheriff has just announced to Lo that
she is to be hanged, at noon, one day this coming week – either
Monday,Tuesday,Wednesday,Thursday or Friday. Saturday is
siesta day,when no one hangs around.The hanging will be a surprise:on
the morning of the hanging,Lo will have no good reason
to believe that the hanging will take place that day.Perhaps,
for example, she will undergo the noonday noose on Tuesday:
if so, on Tuesday morning (and before then) she will have no
good reason to believe that the hanging will be on that day. Lo
knows, she just knows, that the sheriff speaks the truth.
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Things look bleak. Lo is too young to hang. Distraught, she
settles in her cell, but then she starts to reason and reflect.
After all, she is Logical Lo.
If I get through to Friday morning, unhung and hence
unhanged, my neck still intact, then the hanging would have
to occur that day at noon, for there are no other days left. So,
for the hanging to be a surprise, it cannot occur on Friday.
Friday is ruled out.Mind you,there are still all the other days.
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Hold on – as Friday is ruled out for the hanging, then, if I get
through unhanged to Thursday morning, I’d know the hanging
would have to be on that day’s noon.That’s the only day
left, given Friday is no real possibility; hence, I can rule out
Thursday . . .
So Lo reasons, ruling out Friday, then Thursday, then
Wednesday and so on down the days, ruling out Tuesday and
Monday by similar reasoning. Smugly, she concludes that the
sentence announced by the sheriff cannot be implemented.
‘My neck is safe,’ smirks Logical Lo, in confident mode,
ravished by reason.
Then, one day that week, perhaps Wednesday or even
Friday, the hangman intrudes upon Lo’s confident complacency
and hangs her. She is both hanged and surprised or, more
accurately, surprised and hanged.
Surprise hangings,surprise parties,surprise school examinations,
can be given, yet if both the spectrum of possible dates
and the fact that the event will be a surprise are guaranteed,did
Lo not prove that, paradoxically, such surprises cannot occur?
Where does Lo’s argument, that there can be no
surprise hanging, go wrong?
A natural way of handling the paradox is to point out that when
someone is told of a surprise under such conditions – and
schoolchildren often are given surprise examinations in this
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way, though currently not hangings – what is meant is that the
event will be a surprise, unless it happens on the last possible
day.If you have reached the last possible day,then obviously you
will not be surprised, so the backward reasoning does not get
started and the puzzle evaporates. It is then no more puzzling
than being told that we shall be unable to work out which of
four face down ace cards (Clubs? Diamonds? Hearts? Spades?)
will be turned up last, until three have had their faces turned
up; after three are shown, we may easily work out which ace
remains unturned.The puzzle also is no puzzle if Lo knows that
the sheriff could be having her on; for then possibly no hanging
will take place. Once again, the backward reasoning cannot
start. It starts once Lo believes that Friday is a possible day for
the surprise hanging – for then, what can she conclude?
The puzzle bites if you have excellent reason to think the
sheriff is telling the truth. If he is – and if you suppose you get
through to Friday morning – then there is a conflict in your
reasons for what to believe. He tells you that there will
definitely be a hanging; hence, on Friday morning, you think
that it must occur today (it is the last possible day).Yet he also
tells you that it will be a surprise, so, now on Friday morning,
you think his overall announcement cannot be true: maybe the
hanging will not occur. But then you realize that you now are
not believing that there will be a hanging; yet, in that state of
mind,you can reflect that his overall announcement could well
be true after all: you do not believe there will be a hanging, so
now it can both occur and be a surprise.That reflection reels
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you round into again believing that a hanging will happen – but
in that state of mind it will not be a surprise to you;and so you go
through the reasoning again. Such reeling means that no stable
belief can be reached.The reasoning loops round and round.
***
We should question whether Lo can have good reason to
believe that the sheriff is right in predicting both that there will
be a hanging and that she will not believe that there will be one,
if she is left hanging around, unhanged, until Friday. How can
the sheriff predict her state of mind? Perhaps he could for
people who are unable to grasp the reasoning – but Logical Lo
is logical, her reasoning both seemingly impeccable and also
reeling her between thinking her neck will meet the noose and
thinking it will be noose free. Hence the sheriff cannot know
what she will believe just before a hanging on Friday and hence
he cannot be certain that she will be surprised.Logical Lo,who
will also have worked this out, should realize that she has no
good reason to believe that the sheriff’s announcement must
be true with regard to both the hanging and its being a surprise.
To be true, the sheriff’s announcement means that if she is
left hanging around, unhanged, until Friday morning (or if she
projects herself to Friday morning),she will think ‘My hanging
will take place at noon but I don’t believe my hanging will take
place at noon.’ What she thinks has the same structure and
absurdity as the thought in Moore’s Paradox (yet to be discussed,
with reference to Private Pike). Surprisingly, the
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puzzle neither results from the sheriff’s telling Lo of the surprise
hanging, nor from there being a surprise hanging, but
from Lo’s believing what the sheriff tells her.