Re: Updated unexpected hanging paradox bibliography
- From: Ittay Weiss <ettaybn@xxxxxxxxx>
- Date: Fri, 09 Sep 2005 16:49:52 EDT
> George Weinberg says...
>
> >But to make it a paradox, the stipulations are added
> that
> >the professor MUST ALWAYS tell the truth, and the
> students
> >know the professor must always tell the truth.
>
> I don't agree. There is no "MUST" to it. The paradox
> is
> that the professor *happens* to tell the truth, even
> though
> the students seem to have an airtight argument that
> he *isn't*
> telling the truth.
right. You can even say that the prof knowingly told the truth, that is when the prof. announced the surprise examination he/she realy intended it to be a surprise, and managaes to do that.
>
> Intuitively, if something is true, it should be
> possible to *know*
> that it's true, yet this is a counterexample. The
> professor says
> something true, but it is *impossible* for the
> students to know that
> it's true.
I totaly agree. This is the solution to the paradox. This is a beautiful example of a statement which is true yet cannont be proved, so its a beautiful example of an instance of Goedel's incompletenss theorem.
>
> >Back to the modification: the professor must always
> tell
> >the truth, and the students must know the professor
> always tells
> >the truth, for the situation to be of any interest.
>
> I think it is interesting that the professor manages
> to
> say something that is true, yet is impossible for the
> students
> to believe.
>
> >Otherwise, the professor could leave out the
> "surprise" part
> >and just say "there will be a test tomorrow", and
> the students
> >will still be surprised.
>
> No. If the professor says "There will be a test
> tomorrow",
> and the students *believe* him, then they *won't* be
> surprised.
> The interesting thing is the fact that the professor
> manages
> to tell the truth (that there will be a test
> tomorrow) without
> ruining the surprise.
>
> >>So if there is an exam, then the professor's
> statement is true
> >>if and only if the students do not believe that it
> is true. In
> >>that sense, it is like the statement
> >>
> >> Tim Chow will never believe this statement.
> >>
> >>which is true if and only if Tim Chow doesn't
> believe it.
> >
> >Very much so. Tim Chow is a really bright guy, he
> won't believe that
> >statement, since he's smart enough to know that if
> he did it will
> >lead to a contradiction. So we have is allegedly a
> sentence which is
> >true, which any reasonably bright person except Tim
> Chow will know
> >is true, but which Tim himself can't know is true,
> despite being
> >brighter than a lot of other people that can figure
> out that it's
> >true and having access to the same information,
> right?
> >
> >I don't think so. Self-referential statements like
> the above aren't
> >really "true", they don't have a meaningful truth
> value,
>
> I think that is incorrect. As Godel and Tarski
> showed, there is no
> problem with self-referential sentences. There is
> only a problem
> with self-referential sentences that refer to their
> own truth
> value. A sentence can refer to its own provability
> without any
> problem. A sentence can refer to its own length
> without any problem.
> Self-reference is never a problem unless we are
> talking about
> truth values. Belief and truth are not the same
> thing. Belief is
> more akin to "theorem" than it is to "truth".
> Something can be
> true without being believed, and something can be
> believed without
> being true.
>
> >and it's a mistake to conclude they're "true" or
> "false"
> >if you can make an internally consistent assignment
> and
> >"paradoxical" otherwise.
>
> It isn't a matter of finding an internally consistent
> truth assignment. Let's suppose that Tim Chow has a
> notebook
> of things that he believes. In this notebook, he
> writes down
> things such as "2+2=4", "George Washington was the
> first
> President of the United States", etc. Let's assume
> that he
> only writes something in this book if he is
> absolutely,
> positively sure of it. He doesn't include anything
> that's
> ambiguous, or context-dependent (such as "Its raining
> outside"),
> or meaningless.
>
> Now, I have my own notebook, where I just jot down
> random
> sentences, carefully numbered. Sentence number 12 is
> the
> following:
>
> 12. The twelfth sentence in Daryl's notebook will
> ill never be
> written into Tim Chow's notebook.
>
> I don't think that there is any ambiguity about what
> sentence
> 12 means. I think we can be sure that Tim Chow will
> never write
> it into his notebook. So sentence 12 is a true
> sentence that will
> never be written into Tim Chow's notebook. We can
> certainly say
> the following:
>
> If Tim Chow never writes falsehoods into his
> his notebook,
> then he will never write sentence 12 into his
> his notebook.
>
> There is nothing at all paradoxical about this. Now,
> let's suppose
> that Tim Chow has some kind of psychological quirk,
> that as soon
> as he becomes absolutely convinced of the truth of
> any statement,
> he writes it down in his notebook. Further, we assume
> that Tim
> knows about his own quirk. It seems to me that
> sentence 12 is
> true if and only if Tim Chow never becomes absolutely
> convinced
> that it is true. So sentence 12 has the same effect
> as
>
> Tim Chow will never believe this sentence.
>
> >Gardner's book on the unexpected hanging gave the
> best illustration
> >of this (I think somebody else sent it in to him,
> but I don't
> >recall who). He had two statements in a box:
> >
> >1) Both statements in this box are false.
> >2) The egg is in box #5
>
> My Tim Chow statement is different in that it does
> not rely
> on *truth* or falsity. As Tarski shows, there is no
> consistent
> way to deal with "is true" in a self-referential
> language, but
> there is no comparable argument about "is believed by
> Tim Chow".
>
> --
> Daryl McCullough
> Ithaca, NY
>
.
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