[FOM] "This sentence is true"

[gerdes@invariant.org]

On Thu, Sep 05, 2002 at 07:42:54PM -0500, Matt Insall wrote:
> Richard Grandy writes:  
> 
> Sometimes it is hard to tell if a sentence is a Liar:
> 
> This sentence is false iff Goldbach's conjecture is true.
> 
> 
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
> 
> 
> I understand, and I agree.  
> This is what makes the question of what 
> to forbid a question about 
> what is true and what is false, 
> and which requires investigation, 
> and is not prescribable by a machine.  
> 

It seems from the phrasing of this reply that we are somehow allowed to 
choose which sentences we forbid.  In such a case it seems that we can
simply take the trivial solution and forbid those collections of speech
acts (as the liar paradox may of course be repeated as a collection of
several speech acts) which are paradoxical.

As whether or not a sentence is paradoxical may be made to depend on 
arbitrary mathematical statements, as demonstrated above, such a
solution makes the question of which statements to forbid undecidable
(in the formal sense of there not being a turing machine which upon
being given a sentence can determine if it is forbiden).  I fail to see
why this is a problem as we are not seeking a practical device to be
applied in normal speech.  In fact Prof. Friedman cites an alternative
definition of truth which seemingly does result in the question of
whether a sentence gets an assignment of true or false being decidable.

These solutions, and in fact any method which tries to solve the liar by
defining truth in such a way as to avoid the paradox seems to miss the
essential problem of the liar.  For instance a minor modification I have
seen in the literature produces the much more troubling sentence:

*) This sentence is either false or is neither true nor false.

Unless you are willing to claim that the above sentence is both true and
false the paradox remains in full force.   Certainly * is either true,
or false, or not (true or false).  As it is easy to eliminate the
possibilities * is true or star is false we are left with the only
possible conclusion that * is neither true nor false.  If * really does
assert that * is either false or is neither true nor false we run into
the obvious difficulties.

The only solution to this improved paradox seems to be the claim that
the sentence * fails to correspond to an assertion.  Just like 'blah
blah blah' * doesn't actually mean anything.  

In short a solution to liar's paradox type cases needs to explain why
our intuition that * corresponds to an assertion is flawed.  

If * does not correspond to an assertion the fact that 'this statement
is true' is non-paradoxical does not seem to be evidence that it
corresponds to a valid assertion.  In fact because of its similarity to
* we should expect our intuition about what this sentence asserts to be
quite suspect.

In fact it seems that we should expect any convincing explanation of why
* fails to correspond to an assertion to also convince us that other
seemingly sensical statements which are not themselves paradoxical to
fail to be assertions as well.

Peter