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From: bwk@mbunix.mitre.org (Barry W. Kort)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Summary: These sentences have just gone too far!
Keywords: Paradoxes, Undecidability
Message-ID: <44071@linus.UUCP>
Date: 28 Jan 89 23:39:49 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>
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Reply-To: bwk@mbunix.mitre (Barry Kort)
Organization: The Taolight Zone, Ste. Elsewhen
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In an article long, long ago, someone asked how to resolve
the following paradox:

 >        "The following sentence is true."
 >        "The preceeding sentence is false."    

If one adopts Aristotle's Law of the Excluded Middle, then
one has that the above pair of sentences is mutually inconsistent.
But remember our discussion about Intuitionist Logic, where
we threw away the Law of the Excluded Middle, and invented
a few middle possibilities besides True and False.

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.

--Barry Kort