From newshub.ccs.yorku.ca!torn!cs.utexas.edu!rutgers!psinntp!psinntp!scylla!daryl Sat Oct 24 20:44:49 EDT 1992
Article 7370 of comp.ai.philosophy:
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>From: daryl@oracorp.com (Daryl McCullough)
Newsgroups: sci.logic,comp.ai.philosophy
Subject: The Paradox of the Unexpected Hanging
Message-ID: <1992Oct21.141527.28698@oracorp.com>
Date: 21 Oct 92 14:15:27 GMT
Organization: ORA Corporation
Lines: 48

Assuming that the Prisoner's Paradox is the same as the Paradox of the
Unexpected Hanging, then this subject has come up simultaneously in
Comp.ai.philosophy (brought up by David Chalmers) and in Sci.logic
(brought up by Marvin Minsky). I confess to not having read Quine's
analysis of the paradox, but I believe that I understand the paradox
fairly well, so here is my unsolicited analysis.

As I understand it, the paradox involves a judge telling a prisoner
(on Sunday): "You will be executed some time before next Saturday, but
on that fateful day, your execution will be unexpected." The prisoner
then deduces that he can't be executed on Friday, because then on
Friday morning he will expect to be executed (since Friday would be
the last possible day). Similarly, he can't be executed on Thursday,
because he would expect *that* (having ruled out Friday). Thus, the
prisoner can work his way backwards to Sunday, and he will conclude
that he cannot be executed at all! Then Monday morning, the prisoner
is executed, and it is a complete surprise (just as the judge
predicted).

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."

In this stripped-down form, it is clear that *if* the prisoner
believes the judge, then the prisoner's beliefs are inconsistent: From
the judge's first statement, the prisoner can conclude that he will be
executed today. So the judge's second statement is a lie; the prisoner
*does* figure out that he will be executed. Since the prisoner
believes what is clearly a lie, the prisoner's beliefs are
inconsistent.

On the other hand, if the prisoner does not believe the judge, then
the judge is telling the truth. The prisoner cannot deduce that he
will be hanged today, since he doesn't know whether the judge is
telling the truth about the hanging.

There is no real antinomy (in the sense of a logical contradiction).
There is just a strange relationship between the prisoner's beliefs
and the judge's veracity: the prisoner believes the judge if and only
if the judge is lying. Therefore, the prisoner's beliefs are either
inconsistent (he believes the lies of the judge), or they are
incomplete (he fails to believe the true statements of the judge).

Daryl McCullough
ORA Corp.
Ithaca, NY


