[FOM] The liar "revenge"?

Nik Weaver nweaver at math.wustl.edu
Mon Jul 20 22:16:57 EDT 2015

Arnon Avron wrote:

>> So you would agree that the liar sentence is not true?
> My immediate answer is: No, I do not agree, because a bunch of words
> should be a meaningful sentence before it deserves the honor of
> asking whether it is true or not.

Okay.  But the problem with that kind of answer is that it generates
what is known in the literature as a "strengthened liar".  In this case,
that could be the sentence

"This sentence either is not true, or does not even deserve the honor
of asking whether it is true or not."

Is this sentence true?  Not true?  Does it not deserve the honor of
asking whether it is true or not?  Or something else?

Perhaps these questions sound flippant, and if so I apologize.  I
am making a serious point: there is a raft of proposed solutions to
the liar paradox which run into similar difficulties.  The general
phenomenon is known as the "revenge problem"; it occurs when the
original liar paradox is defeated with the aid of a new concept
which, if introduced into the object language, generates a new
paradox.  In your case, this would be the concept of not deserving
the honor of asking whether it is true or false.

If you read a little further in my book you will find that I have a
bit to say about the revenge problem.  However, based on your reaction
to the first page of the book I fear that there is little hope of your
getting much out of it.


Maybe the above makes my point well enough, but out of politeness I
will try to answer your question "Would you agree that the Eiffel
tower is not true?"  (Do I really have to answer all six questions?
I think my answer to this one will say enough.)

My answer is that the Eiffel tower is not the sort of thing that
can be true, and therefore it is not true.  But I would not insist
on this.  It depends on how we want to formalize a fragment of English
in which the question could be asked.  If you prefer to do this in such
a way that neither "The Eiffel tower is true" nor its negation is true,
that would be fine with me.  (The technical term here is "introducing
truth-value gaps".)  However, doing this would not rescue you from
liar-type paradoxes, for the reasons I explained above.


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