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From: bwk@mbunix.mitre.org (Barry W. Kort)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Summary: See if you can follow my inventive derivation.
Keywords: Paradoxes, Undecidability
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Date: 2 Feb 89 04:44:11 GMT
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Organization: International Malefactor and Fulminator, Roaring Rapids, ME
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In article <3091@silver.bacs.indiana.edu> dave@cogsci.indiana.edu
(David Chalmers) brilliantly analyzes the 2-sentence paradox:

	"The following sentence is provable."
	"The preceding sentence is unprovable."

David, a bit dizzy from his ride, steps off the merry-go-round
and concludes:

 > Sentence 1 is false (and thus unprovable),
 > and Sentence 2 is true but unprovable.

 > Check it out - it's consistent.  

Actually, I think we have to clarify the meaning of "provable".
We have a Goedel sentence here which is formally unprovable
(underivable) using deduction.  But we can nevertheless see that
it is true if we permit ourselves to transcend the rules of formal
derivation.

What would happen if I edited the sentences to read:

  "The following sentence is formally unprovable but informally provable."
  "The preceding sentence is formally unprovable but informally provable."

We can see that the two sentences have now become identical:

  "This sentence is formally unprovable but informally provable."

Or, if you prefer,

  "It is evidently the case that this sentence is not formally derivable."

And here we have the seeds of intuitionist logic.

--Barry Kort