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Article 5983 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May29.224704.22798@CSD-NewsHost.Stanford.EDU>
Date: 29 May 92 22:47:04 GMT
References: <atten.706786286@groucho.phil.ruu.nl> <1992May29.125208.13752@sics.se>
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Organization: Computer Science Department, Stanford University
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In article <1992May29.125208.13752@sics.se>, torkel@sics.se (Torkel Franzen) writes:
|> In article <1992May29.112749.14160@CSD-NewsHost.Stanford.EDU> costello@
|> CS.Stanford.EDU (T Costello) writes:
|> 
|>    >Thus I claim that your reflection principle does not hold in general.
|> 
|>   Ok, I'll spell it out. Of the axioms used to derive the contradiction,
|> 
|>       B( B x \supset x)  
|> 
|> does not hold if we are speaking about provability in T (for the T at issue).
|> Indeed, by Loeb's theorem, if "if B(+A+) then A" is provable in T, so is A.
|> 
|>   Frankly, I'm a bit surprised at your persistence, since the
|> extension I described added to the theory only axioms that are
|> obviously true (given that T is sound).
|> 
|> 
 
I think I understand my confusion on this issue.  You claim your reflection 
principle holds only for sound theories, not any consistent theory.  That is,
I assume a theory that proves only true statements, where truth is defined by
some Kripke like partial predicate.  But as you have defined truth by what
amounts to reflection principles, your result follows.

I on the other hand was considering, not "true" theories but consistent ones.
It does not hold that adding your reflection principle to a consistent theory
results in a consistent theory.  I thought (and still think) that assumptions 
about truth of theories often muddy the water, as it is a purely semantic
issue, and not fully open to logical analysis.

Tom


