Path: utzoo!utgpu!watmath!iuvax!silver!chalmer
From: chalmer@silver.bacs.indiana.edu (david chalmers)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Keywords: Paradoxes, Undecidability
Message-ID: <3091@silver.bacs.indiana.edu>
Date: 1 Feb 89 18:44:52 GMT
References: <1883@buengc.BU.EDU> <2996@uhccux.uhcc.hawaii.edu> <905@ubu.warwick.UUCP> <479@aipna.ed.ac.uk> <1036@hudson.acc.virginia.edu> <3715@uklirb.UUCP> <44071@linus.UUCP>
Reply-To: dave@cogsci.indiana.edu (david chalmers)
Organization: Center for Research on Concepts and Cognition, Indiana University
Lines: 58

In article <44071@linus.UUCP> bwk@mbunix.mitre (Barry Kort) writes:
>Consider, if you will, the following pair of sentences:
>
>	"The following sentence is provable."
>	"The preceding sentence is unprovable."
>
>The paradox seems to have vanished.  The first statement
>can be both True and Unprovable.  The second sentence
>can be both True and Provable.  (But please don't ask
>me to supply the proof.  I didn't say they were provable
>by *me*!)
>
>The point is twofold:  Not all True sentences are provable
>and not all unprovable sentences are False.  Thus we need
>a third category: Undecidable.
>
>We can then resolve the paradox by chastising both sentences
>for overstating the case.  They could have gotten along
>very nicely if they had scaled back their dogmatic assertions
>along the lines of the second, more harmonious pair.

Sorry.  It's kind of obvious that if a sentence is provable, then
it's PROVABLE that it's provable.  Because if a proof exists, 
a PROOF that the proof exists just consists in displaying the 
original proof.  Got that?  (There may be weird logics in which
this is untrue - for instance where a proof could 'exist' but be
unable to be constructed - but this kind of thing isn't relevant
here.)

Here are your two statements again:
     1:  Sentence 2 is provable.
     2:  Sentence 1 is unprovable.

So: if Sentence 1 above is True, then (by reading what it says)
Sentence 2 is provable.  But as we've just said, this entails
that the statement "Sentence 2 is provable" is itself provable.
But this statement is just Sentence 1!  So now we see that 
Sentence 1 is provable, CONTRADICTING Sentence 2.  So, oops,
now Sentence 2 is FALSE, so it can't be provable.  

Continuing on the merry-go-round of paradox...this means that
Sentence 1 has to be false.  But if that's the case, then of
course it's unprovable, so Statement 2 is TRUE after all!

Aaargh!  Stop!  I want to get off.  But in fact this yields the
only resolution of this 'paradox.'  That is:  Sentence 1 is
false (and thus unprovable), and Sentence 2 is true but unprovable.
Check it out - it's consistent.  

(Now someone might say to me: haven't you just PROVED that
Sentence 1 is unprovable, so doesn't that make Sentence 2 provable
after all...but I'm going to quit while I'm ahead.)

Never underestimate the power of a paradox to bite its own tail.

                Dave Chalmers    (dave@cogsci.indiana.edu)
                Center for Research on Concepts and Cognition
                Indiana University