Newsgroups: sci.logic,comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose's new book
Message-ID: <1994Oct26.172830.3987@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Wed, 26 Oct 1994 17:28:30 GMT
Lines: 39
Xref: glinda.oz.cs.cmu.edu sci.logic:8721 comp.ai.philosophy:21394

rar@birch.csl.sri.com (Bob Riemenschneider) writes:

>daryl@oracorp.com (Daryl McCullough) writes:
>
>> ... Basically, Penrose assumes that
>> human intuition can show us that human intuition is infallible. But
>> the belief that one is infallible, together with sufficient closure
>> properties of ones beliefs, implies that one is in fact inconsistent. ...
>
>Only if you formalize epistemic logic incorrectly :-)
>
>Since there are formalizations in which this problem doesn't arise -- see,
>e.g., L"ob's mid-50s _JSL_ paper on the consistency of ramified modal
>logics -- you at least have the option of blaming the inconsistency on a
>bug in some of the common formalizations rather than the belief in
>infalibillity.

Well, L"ob gave one of the most general proofs of the incompleteness
theorem for any theory with sufficient self-reference to have (1) A
"provability" (or belief) operator, (2) the existence of fixed-points
(diagonalization). So, if he then showed a system that did not
suffer from Godel's second incompleteness theorem, it must be
that he gave up some expressiveness.

I haven't read L"ob's paper, but...If you have a "ramified" hierarchy
of modal provability (or belief) operators, B0, B1, ... you can
certainly have B1(con(B0)), B2(con(B1)), etc. (where con(Bj) = not
B(false)).  However, such a ramified theory doesn't contain a single
statement asserting that the whole theory is consistent.

So, you can get around the incompleteness theorem by giving up
expressibility.

Daryl McCullough
ORA Corp.
Ithaca, NY



