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From: hansw@cs.vu.nl (Hans Weigand)
Newsgroups: sci.math,sci.math.symbolic,sci.philosophy.tech
Subject: Re: Russell's set of sets which... paradox
Message-ID: <1551@botter.cs.vu.nl>
Date: Tue, 4-Aug-87 08:34:53 EDT
Article-I.D.: botter.1551
Posted: Tue Aug  4 08:34:53 1987
Date-Received: Thu, 6-Aug-87 01:11:13 EDT
References: <1404@cullvax.UUCP> <902@bsu-cs.UUCP> <4901@j.cc.purdue.edu>
Reply-To: hansw@cs.vu.nl (Hans Weigand)
Organization: VU Informatica, Amsterdam
Lines: 43
Keywords: logic, paradoxes
Summary: beware of self-referential logic
Xref: mnetor sci.math:1741 sci.math.symbolic:118 sci.philosophy.tech:348

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In article <4901@j.cc.purdue.edu> ags@j.cc.purdue.edu.UUCP (Dave Seaman) writes:
>(..)
>The stone paradox can be formulated as follows:  Assume x is an omnipotent
>being.  Let P(x) be the statement, "x can make a stone so big that x can't
>lift it."  Is P(x) true, or is it false?  A little thought shows that it is
>neither.
>
>What do we conclude from the contradiction?  There must be something wrong
>with the original assumption that there exists an omnipotent being.
>Therefore no such being can exist.  (..)

I agree with you that P(x) has no reasonable truth-value.
However, your conclusion is not warranted, since you confuse
logic and ontology. What is the false assumption of the Liar
Paradox? There is none, but its property of self-referenence
makes it hard to manage for logic. This does not prove that
there are no liars, etc, only that our logic falls short.

The intentional object "a stone that x can't lift" is logically
self-contradictory, given the assumption, just as "round squares"
are. Therefore, in the world as we construct it (through our
definitions and logic) such a thing cannot exist. Then it is
useless to make ontological predications about such a thing
(woruber man nicht sprechen kann, daruber soll man schweigen,
Tractatus Wittgenstein). However, you think you can say
something about it, because your omnipotent being is allowed to
violate the logical norms.

What your argument says, then, is that the concept of an
omnipotent being may lead to logical paradoxes if we
assume the omnipotent being may violate the logical norms.
This seems pretty much a tautology. The moral of this may be:
beware of logic when it refers (negatively) to its own principles.
However, I do not see what ontological conclusion can be drawn from this.

For a more extensive treatment of the power and limitations of logic,
I may draw your attention to the works of the great Medieval mathematician
and theologian Nicolaus Cusanus, in particular "De Docta Ignorantia"
(Learned Ignorance). Kant will probably do also.

-
Hans Weigand
Vrije Universiteit, Amsterdam