Re: Updated unexpected hanging paradox bibliography


tchow@xxxxxxxxxxxxx wrote:
> In 1998, I published an article in the American
> Mathematical Monthly on
> the surprise examination or unexpected hanging paradox.

That is a commendable achievement completely irrespective
of the content of the article, which I personally find
unsatisfying. The claim from the logical school
that the announcement needs to be self-referential I find
unconvincing, to put it charitably, but that school is
irrelevant anyhow; the problem is obviously epistemological.
The exposition from the epistemological school goes far too
many pages WITHOUT mentioning the possibility that the professor
may just be lying. This is especially galling given that
it suggests collapsing the paradox to 1 day ("There will be
an exam tomorrow, but you do not know that"). If, when the
exam is given, the students react (as they must, for this
collapse to be legitimate), "You were right, we didn't
know", then it follows that they had to be open to the
possibility that the professor might lie to them about
whether the exam would be given at all. The mere opening
of that possibility is a resolution of the paradox in all
schools and all cases, yet it had not yet been mentioned at
all, up to that point in the article. I would think that
all the formalizations in general need to be dichotomized
according to whether they do or don't allow for the possibility
of the exam's not being given at all.

One tack from the epistemological school suggests that
the students hold inconsistent beliefs about how knowledge
works. I would take the opposite tack: the students beliefs
about knowledge are fine. In light of them, though, it follows
that the professor has made inconsistent statements.
Normally, if the professor says something,the students can
take themelves as informed of it and knowing it.
The collapsed-to-1-day case thus becomes "I'm telling you
you'll have an exam tomorrow, so you know that, but I'm
also telling you that you don't know that, so you don't know
that." If someone tells you both that he will do P and
that he will do ~P, he has basically told you nothing
you can consistently believe. When he then does P, he
does NOT get to say, "I told you I would do P,
so now that I've done it, I am proved reliable/trustworthy"
or proved "right". He is in fact proved nothing but
inconsistent.

.

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