Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!iuvax!cogsci!dave
From: dave@cogsci.indiana.edu (David Chalmers)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Keywords: Types, Paradoxes
Message-ID: <17220@iuvax.cs.indiana.edu>
Date: 4 Feb 89 05:18:11 GMT
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Reply-To: dave@duckie.cogsci.indiana.edu (David Chalmers)
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In article <44268@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes:
>In article <3781@uklirb.UUCP> kerber@uklirb.UUCP (Manfred Kerber) writes:
>
> > One gets S1 <==> NOT S1.  It is a real paradox.
>
>The paradox goes away if you admit the possibility that S1
>is unprovable (or undecidable).  Then we merely have
>that S1 is unprovable if and only if NOT S1 is unprovable.
>
Sure, you've shown that S1 is unprovable.  But the paradox is still there.
It still seems that S1 can be neither true nor false.  According to the 
views of most mathematicians, meaningful mathematical statements must have 
some truth-value, irrespective of whether or not they are provable.  Even
Godel's result never argued with this (his famous unprovable sentence G,
say, was in fact true, although the proof had to be outside its 
particular system.)  To abandon the notion that a meaningful mathematical
statement must be either true or false leads one straight into the arms of
the intuitionists.  (Well, I guess there are a few of them about.)

All the problems with S1 stem from the fact that it is not meaningful - it
talks about nothing apart from it's own truth-value, so it is not grounded
in reality.  So any 'paradox' shouldn't worry us.

    Dave Chalmers

P.S.  Will people please shut up about the French royal family.  This is
really semantic games with absolutely no interesting conceptual problems
behind it all.