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: Paradoxes, Undecidability
Message-ID: <17219@iuvax.cs.indiana.edu>
Date: 4 Feb 89 05:03:47 GMT
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In article <44270@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes:
>
>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."

Yep, I'm with you Barry.  This sentence is almost like Godel's sentence
"This sentence is formally unprovable", with the extra twist if recognizing
it's provability at "one level up" (the level of meta-logic, if you like).

One minor problem: the sentence need not now be paradoxical or cause any
difficulities - it's quite OK for THIS sentence simply to be false.

One major problem: it would be difficult to make such a statement cause any
problems a la Godel.  To do this it would have to be grounded in a given
system (to make it really meaningful, as opposed to word play), where it
talked about itself in a concrete way.  But if you grounded it in "Level 0"
(normal) logic, then any claims it makes about "Level 1" ('informal') logic
won't have the viciously circular property that makes it bite it's own 
tail.  If you ground it in Level 1, informal logic (as I think you'd 
prefer), then there are no longer any problems, the statement is simply
true, and provable within the system.  (Though its truth might vary with
the formulation.)

>And here we have the seeds of intuitionist logic.

Why?  I guess the intuitionists were happy that unprovable statements 
exist, because this was what they had always thought, but for different
reasons.  I think that an intuitionist would deny the validity of "informal
provability," though.

    Dave Chalmers