From newshub.ccs.yorku.ca!torn!cs.utexas.edu!uwm.edu!spool.mu.edu!uunet!secapl!Cookie!frank Tue Nov 24 10:51:06 EST 1992
Article 7562 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
Path: newshub.ccs.yorku.ca!torn!cs.utexas.edu!uwm.edu!spool.mu.edu!uunet!secapl!Cookie!frank
>From: frank@Cookie.secapl.com (Frank Adams)
Subject: Re: The Paradox of the Unexpected Hanging
Message-ID: <1992Nov10.030302.44285@Cookie.secapl.com>
Date: Tue, 10 Nov 1992 03:03:02 GMT
References: <1992Nov3.051001.21374@oracorp.com>
Organization: Security APL, Inc.
Lines: 39

In article <1992Nov3.051001.21374@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>In article <gm4FTB4w165w@CODEWKS.nacjack.gen.nz>,
>system@CODEWKS.nacjack.gen.nz (Wayne McDougall) writes:
>>> Now, as John Baez said, the induction aspect working backward from
>>> Friday to Sunday is irrelevant. The heart of the paradox can be found
>>> in a simplified version where the judge says to the prisoner: "You
>>> will be executed today, but you will not be able to figure out that
>>> you will be executed today."
>>
>>Excuse me, but it seems inappropriate to transform the judge's sentence 
>>into a self-contradictory statement.
>
>The statement "You will be executed today, but you will not be able to
>figure out that you will be executed today" is *not*
>self-contradictory. Suppose the prisoner thinks the judge might be
>lying, then he can't figure out anything from what the judge says.
>Therefore the judge's second statement "...you will not be able to
>figure out that you will be executed today" will turn out to be true.
>If in addition, the judge *does* hang the prisoner, then the first
>statement "You will be executed today" will turn out to be true. If
>everything the judge says turns out to be true, then it can't be
>self-contradictory.

There are two forms of the unexpected hanging problem: one where the judge
says "... you will be surprised ...", and the other where he says "... you
will not be able to prove ...".  Both really work the same in the stripped-
down version as when multiple days are involved.

In the proof-theoretic version, the judge's statements (together with the
fact that he makes them to the prisoner) *are* contradictory.  There is
potentially a problem with levels, since formal systems in general can't
deal with statements about what they are able to prove.  But in any context
where they can be accepted, they are contradictory; hence you can prove
anything from them; so in particular the second statement is false (since
you can prove anything from a contradiction, you *can* prove that it will
occur on whatever day it actually does occur).

The other version presents a psychological problem.  The prisoner *can* make
the second statement false here, simply by expecting the hanging every day.


