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From: stuart@gargoyle.UChicago.UUCP (Stuart Kurtz)
Newsgroups: net.philosophy,net.religion
Subject: Re: Logical paradoxes in the notion of omnipotence?
Message-ID: <204@gargoyle.UChicago.UUCP>
Date: Wed, 19-Sep-84 09:35:29 EDT
Article-I.D.: gargoyle.204
Posted: Wed Sep 19 09:35:29 1984
Date-Received: Tue, 25-Sep-84 09:03:22 EDT
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Organization: U. Chicago - Computer Science
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Don Steiny argues that paradoxes such as "This sentence is false" can be
resolved by disallowing them on the grounds of a confusion of levels.  I
have never been satisfied by such an analysis: self-reference is too useful
a technical tool in mathematical logic/theoretic computer science for me to
part with it easily.

The approach suggested is reminicient of Russell's types.

It is clear that this approach is viable, but it seems to prohibit analysis
an innocuous sentence such as "I am Stuart Kurtz", which is certain true
when stated by this writer.  Indeed, if you study the foundations of
mathematics, you'll find that the Russell-Whitehead type system (the formal
version of the levels Don S. speaks of) is not the logical system of choice
today.

Let us consider for a moment how the prevalent 1st order logics deal with
paradoxes such as "This sentence is false."

The key to the analysis is that the notions of "true" and "false" are
defined in the meta-theory.  Therefore there is no paradox unless the
sentence "This sentence is false" can be translated into the 1st order
language in question.  The Epimenidies paradox proves that it cannot be.
Now, it is possible to express the notion "This sentence is not provable"
within sufficiently powerful 1st order systems (this observation leads
immediately to the incompleteness theorem); however provability is the
merest shadow of truth.

Summarizing, the usual analysis of "This sentence is false" rejects the
sentence as ill-formed because the term "false" cannot be adequately
expressed within the system, not because the sentence refers to itself.

Now, back to the notion of omnipotence.  Perhaps it can indeed be shown to
be paradoxical, although I remain unconvinced of this.  At this point, all I
can agree with is that if the notion of omnipotence is consistent, it has
some unexpected consequences.

Stu

ihnp4!gargoyle!stuart